Hindawi Publishing CorporationInternational Journal of CombinatoricsVolume 2013 Article ID 595210 34 pageshttpdxdoiorg1011552013595210

Review ArticleOn Bondage Numbers of Graphs A Survey withSome Comments

Jun-Ming Xu

School of Mathematical Sciences University of Science and Technology of China Wentsun Wu Key Laboratory of CAS HefeiAnhui 230026 China

Correspondence should be addressed to Jun-Ming Xu xujmustceducn

Received 15 December 2012 Revised 20 February 2013 Accepted 11 March 2013

Academic Editor Chınh T Hoang

Copyright copy 2013 Jun-Ming XuThis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The domination number of a graph G is the smallest number of vertices which dominate all remaining vertices by edges of G Thebondage number of a nonempty graphG is the smallest number of edges whose removal fromG results in a graph with dominationnumber greater than the domination number of G The concept of the bondage number was formally introduced by Fink et al in1990 Since then this topic has received considerable research attention and made some progress variations and generalizationsThis paper gives a survey on the bondage number including known results conjectures problems and some comments alsoselectively summarizes other types of bondage numbers

1 Introduction

For terminology andnotation on graph theory not given herethe reader is referred to Xu [1] Let 119866 = (119881 119864) be a finiteundirected and simple graph We call |119881| and |119864| the orderand size of 119866 and denote them by 120592 = 120592(119866) and 120576 = 120576(119866)respectively unless otherwise specified Through this paperthe notations 119875

119899 119862

119899 and 119870

119899always denote a path a cycle

and a complete graph of order 119899 respectively the notation11987011989911198992119899119905

denotes a complete 119905-partite graph with 1198991⩽ 119899

2⩽

sdot sdot sdot ⩽ 119899119905and 119899

119905gt 1 119870

119905(119898) = 119870

11989911198992119899119905

with 1198991= sdot sdot sdot = 119899

119905=

119898 and1198701119899minus1

is a star For two vertices 119909 and 119910 in a connectedgraph 119866 we use 119889

119866(119909 119910) to denote the distance between 119909

and 119910 in 119866For a vertex 119909 in 119866 let119873

119866(119909) be the open set of neighbors

of 119909 and 119873119866[119909] = 119873[119909] = 119873

119866(119909) cup 119909 the closed

set of neighbors of 119909 For a subset 119883 sub 119881(119866) 119873119866(119883) =

(cup119909isin119883119873

119866(119909)) cap 119883 and 119873

119866[119883] = 119873

119866(119883) cup 119883 where 119883 =

119881(119866) 119883 Let 119864119909be the set of edges incident with 119909 in 119866

that is 119864119909= 119909119910 isin 119864(119866) 119910 isin 119873

119866(119909) We denote the

degree of119909 by119889119866(119909) = |119864

119909|Themaximumand theminimum

degrees of 119866 are denoted by Δ(119866) and 120575(119866) respectively Avertex of degree zero is called an isolated vertex An edgeincident with a vertex of degree one is called a pendantedge

The bondage number is an important parameter of graphswhich is based upon the well-known domination number

A subset 119878 sube 119881(119866) is called a dominating set of 119866 if119873

119866[119878] = 119881(119866) that is every vertex 119909 in 119878 has at least one

neighbor in 119878The domination number of119866 denoted by 120574(119866)is the minimum cardinality among all dominating sets thatis

120574 (119866) = min |119878| 119878 sube 119881 (119866) 119873119866[119878] = 119881 (119866) (1)

A dominating set 119878 is called a 120574-set of 119866 if |119878| = 120574(119866)The domination is such an important and classic concep-

tion that it has become one of the most widely studied topicsin graph theory and also is frequently used to study propertyof networks The domination with many variations andgeneralizations is now well studied in graph and networkstheory The early vast literature on domination includes thebibliography compiled by Hedetniemi and Laskar [2] anda thorough study of domination appears in the books byHaynes et al [3 4] However the problem determining thedomination number for general graphs was early proved tobe NP-complete (see GT2 in Appendix in Garey and Johnson[5] 1979)

Among various problems related with the dominationnumber some focus on graph alterations and their effects

2 International Journal of Combinatorics

on the domination number Here we are concerned witha particular graph alternation the removal of edges from agraph

Graphs with domination numbers changed upon theremoval of an edge were first investigated by Walikar andAcharya [6] in 1979 A graph is called edge-domination-critical graph if 120574(119866 minus 119890) gt 120574(119866) for every edge 119890 in 119866 Theedge-domination-critical graph was characterized by Baueret al [7] in 1983 that is a graph is edge-domination-criticalif and only if it is the union of stars

However for lots of graphs the domination number is outof the range of one-edge removal It is immediate that 120574(119867) ⩾120574(119866) for any spanning subgraph 119867 of 119866 Every graph 119866 hasa spanning forest 119879 with 120574(119866) = 120574(119879) and so in general agraph has a nonempty subset 119865 sube 119864(119866) for which 120574(119866 minus 119865) =120574(119866)

Then it is natural for the alternation to be generalized tothe removal of several edges which is just enough to enlargethe domination number That is the idea of the bondagenumber

Ameasure of the efficiency of a domination in graphs wasfirst given by Bauer et al [7] in 1983 who called this measureas domination line-stability defined as theminimumnumberof lines (ie edges) which when removed from119866 increases 120574

In 1990 Fink et al [8] formally introduced the bondagenumber as a parameter for measuring the vulnerability of theinterconnection network under link failure The minimumdominating set of sites plays an important role in the networkfor it dominates the whole network with the minimum costSowemust consider whether its function remains goodwhenthe network is attacked Suppose that someone such as asaboteur does not know which sites in the network take partin the dominating role but does know that the set of thesespecial sites corresponds to aminimumdominating set in therelated graph Then how many links does he has to attack sothat the cost cannot remains the same in order to dominatethewhole networkThatminimumnumber of links is just thebondage number

The bondage number 119887(119866) of a nonempty undirectedgraph 119866 is the minimum number of edges whose removalfrom119866 results in a graphwith larger domination numberTheprecise definition of the bondage number is as follows

119887 (119866) = min |119861| 119861 sube 119864 (119866) 120574 (119866 minus 119861) gt 120574 (119866) (2)

Since the domination number of every spanning subgraph ofa nonempty graph 119866 is at least as great as 120574(119866) the bondagenumber of a nonempty graph is well defined

We call such an edge-set 119861 with 120574(119866 minus 119861) gt 120574(119866)

the bondage set and the minimum one the minimumbondage set In fact if 119861 is a minimum bondage set then120574(119866 minus 119861) = 120574(119866) + 1 because the removal of one single edgecannot increase the domination number by more than oneIf 119887(119866) does not exist for example empty graphs we define119887(119866) = infin

It is quite difficult to compute the exact value of thebondage number for general graphs since it strongly dependson the domination number of the graphsMuchwork focusedon the bounds on the bondage number aswell as the restraints

on some particular classes of graphs or networks Thepurpose of this paper is to give a survey of results and researchmethods related to these topics for graphs and digraphsFor some results and research methods we will make somecomments to develop our further study

The rest of the paper is organized as follows Section 2gives some preliminary results and complexity Sections3 and 4 survey some upper bounds and lower boundsrespectively The results for some special classes of graphsand planar graphs are stated in Sections 5 and 6 respectivelyIn Section 7 we introduce some results on crossing numberrestraints In Sections 8 and 9 we are concerned about otherand generalized types of bondage numbers respectively InSection 10 we introduce some results for digraphs In thelast section we introduce some results for vertex-transitivegraphs by applying efficient dominating sets

2 Simplicity and Complexity

As we have known from Introduction the bondage numberis an important parameter for measuring the stability orthe vulnerability of a domination in a graph or a networkOur aim is to compute the bondage number for any givengraphs or networks One has determined the exact value ofthe bondage number for some graphs with simple structureFor arbitrarily given graph however it has been proved thatdetermining its bondage number is NP-hard

21 Exact Values for Ordinary Graphs We begin our inves-tigation of the bondage number by computing its value forseveral well-known classes of graphs with simple structureIn 1990 Fink et al [8] proposed the concept of the bondagenumber and completely determined the exact values ofbondage numbers of some ordinary graphs such as completegraphs paths cycles and complete multipartite graphs

Theorem 1 (Fink et al [8] 1990) The exact values of bondagenumbers of the following class of graphs are completely deter-mined

119887 (119870119899) = lceil

119899

2rceil

119887 (119875119899) =

2 119894119891 119899 equiv 1 (mod 3) 1 119900119905ℎ119890119903119908119894119904119890

119887 (119862119899) =

3 119894119891 119899 equiv 1 (mod 3) 2 119900119905ℎ119890119903119908119894119904119890

119887 (11987011989911198992119899119905

)

=

lceil119895

2rceil 119894119891 119899

119895= 1 119886119899119889 119899

119895+1⩾ 2 119891119900119903 119904119900119898119890 119895

1 ⩽ 119895 lt 119905

2119905 minus 1 119894119891 1198991= 119899

2= sdot sdot sdot = 119899

119905= 2

119905minus1

sum

119894=1

119899119894

119900119905ℎ119890119903119908119894119904119890

(3)

The complete graph119870119899is the unique (119899minus1)-regular graph

of order 119899 ⩾ 2 119887(119870119899) = lceil1198992rceil by Theorem 1 The 119905-partite

International Journal of Combinatorics 3

graph 119870119905(2) with 119905 = 1198992 is an (119899 minus 2)-regular graph 119866 of

order 119899 ⩾ 2 and not unique 119887(119866) = 119899 minus 1 by Theorem 1for an even integer 119899 ⩾ 4 For an (119899 minus 3)-regular graph119866 of order 119899 ⩾ 4 Hu and Xu [9] obtained the followingresult

Theorem 2 (Hu and Xu [9]) 119887(119866) = 119899 minus 3 for any (119899 minus 3)-regular graph 119866 of order 119899 ⩾ 4

Up to the present no results have been known for 119896-regular graphs with 3 ⩽ 119896 ⩽ 119899 minus 4 For general graphs thereare the following two results

Theorem 3 (Teschner [10] 1997) If 119866 is a nonempty graphwith a unique minimum dominating set then 119887(119866) = 1

Theorem4 (Bauer et al [7] 1983) If any vertex of a graph119866 isadjacent with two ormore vertices of degree one then 119887(119866) = 1

Bauer et al [7] observed that the star is the uniquegraph with the property that the bondage number is 1 andthe deletion of any edge results in the domination numberincreasing Motivated by this fact Hartnell and Rall [11]proposed the concept of uniformly bonded graphs A graph iscalled to be uniformly bonded if it has bondage number 119887 andthe deletion of any 119887 edges results in a graph with increaseddomination number Unfortunately there are a few uniformlybonded graphs

Theorem 5 (Hartnell and Rall [11] 1999) The only uniformlybonded graphs with bondage number 2 are 119862

3and 119875

4 The

unique uniformly bonded graph with bondage number 3 is 1198624

There are no such graphs for bondage number greater than 3

As we mentioned to compute the exact value of bondagenumber for a graph strongly depends upon its dominationnumber In this sense studying the bondage number cangreatly inspire onersquos research concerned with dominationsHowever determining the exact value of domination numberfor a given graph is quite difficult In fact even if theexact value of the domination number for some graph isdetermined it is still very difficulty to compute the valueof the bondage number for that graph For example for thehypercube 119876

119899 we have 120574(119876

119899) = 2

119899minus1 but we have not yetdetermined 119887(119876

119899) for any 119899 ⩾ 2

Perhaps Theorems 3 and 4 provide an approach tocompute the exact value of bondage number for some graphsby establishing some sufficient conditions for 119887(119866) = 119887In fact we will see later that Theorem 3 plays an importantrole in determining the exact values of the bondage numbersfor some graphs Thus to study the bondage number it isimportant to present various characterizations of graphs witha unique minimum dominating set

22 Characterizations of Trees For trees Bauer et al [7] in1983 from the point of view of the domination line-stabilityindependently and Fink et al [8] in 1990 from the pointof view of the domination edge-vulnerability obtained thefollowing result

119908

Figure 1 Tree 119865119905

Theorem 6 For any nontrivial tree 119879 119887(119879) ⩽ 2

By Theorem 6 it is natural to classify all trees accordingto their bondage numbers Fink et al [8] proved that aforbidden subgraph characterization to classify trees withdifferent bondage numbers is impossible since they provedthat if 119865 is a forest then 119865 is an induced subgraph of a tree 119879with 119887(119879) = 1 and a tree 1198791015840 with 119887(1198791015840) = 2 However theypointed out that the complexity of calculating the bondagenumber of a tree of order 119899 is at most 119874(1198992) by methodicallyremoving each pair of edges

Even so some characterizations whether a tree hasbondage number 1 or 2 have been found by several authorssee for example [10 12 13]

First we describe the method due to Hartnell and Rall[12] by which all trees with bondage number 2 can be con-structed inductively An important tree119865

119905in the construction

is shown in Figure 1 To characterize this construction weneed some terminologies

(1) attach a path 119875119899to a vertex 119909 of a tree which means

to link 119909 and one end-vertex of the 119875119899by an edge

(2) attach 119865119905to a vertex 119909 which means to link 119909 and a

vertex 119910 of 119865119905by an edge

The following are four operations on a tree 119879

Type 1 attach a 1198752to 119909 isin 119881(119879) where 120574(119879 minus 119909) = 120574(119879) and 119909

belongs to at least one 120574-set of 119879 (such a vertex existssay one end-vertex of 119875

5)

Type 2 attach a 1198753to 119909 isin 119881(119879) where 120574(119879 minus 119909) lt 120574(119879)

Type 3 attach 1198651to 119909 isin 119881(119879) where 119909 belongs to at least one

120574-set of 119879Type 4 attach 119865

119905 119905 ⩾ 2 to 119909 isin 119881(119879) where 119909 can be any

vertex of 119879

LetC = 119879 119879 is a tree and 119879 = 1198701 119879 = 119875

4 and 119879 = 119865

119905

for some 119905 ⩾ 2 or 119879 can be obtained from 1198754or 119865

119905(119905 ⩾ 2) by

a finite sequence of operations of Types 1 2 3 4

Theorem 7 (Hartnell and Rall [12] 1992) A tree has bondagenumber 2 if and only if it belongs toC

Looking at different minimum dominating sets of a treeTeschner [10] presented a totally different characterizationof the set of trees having bondage number 1 They defineda vertex to be universal if it belongs to each minimumdominating set and to be idle if it does not belong to anyminimum dominating set

4 International Journal of Combinatorics

Theorem 8 (Teschner [10] 1997) A tree 119879 has bondagenumber 1 if and only if 119879 has a universal vertex or an edge119909119910 satisfying

(1) 119909 and 119910 are neither universal nor idle and

(2) all neighbors of 119909 and 119910 (except for 119909 and 119910) are idle

For a positive integer 119896 a subset 119868 sube 119881(119866) is called a 119896-independent set (also called a 119896-packing) if 119889

119866(119909 119910) gt 119896 for

any two distinct vertices 119909 and 119910 in 119868 When 119896 = 1 1-set is thenormal independent setThemaximumcardinality among all119896-independent sets is called the 119896-independence number (or119896-packing number) of 119866 denoted by 120572

119896(119866) A 119896-independent

set 119868 is called an 120572119896-set if |119868| = 120572

119896(119866) A graph 119866 is said to be

120572119896-stable if 120572

119896(119866) = 120572

119896(119866 minus 119890) for every edge 119890 of 119866 There are

two important results on 119896-independent sets

Proposition 9 (Topp and Vestergaard [13] 2000) A tree 119879 is120572119896-stable if and only if 119879 has a unique 120572

119896-set

Proposition 10 (Meir and Moon [14] 1975) 1205722(119866) ⩽ 120574(119866)

for any connected graph 119866 with equality for any tree

Hartnell et al [15] independently and Topp and Vester-gaard [13] also gave a constructive characterization of treeswith bondage number 2 and applying Proposition 10 pre-sented another characterization of those trees

Theorem 11 (Hartnell et al [15] 1998 Top and Vestergaard[13] 2000) 119887(119879) = 2 for a tree 119879 if and only if 119879 has a unique1205722-set

According to this characterization Hartnell et al [15]presented a linear algorithm for determining the bondagenumber of a tree

In this subsection we introduce three characterizationsfor trees with bondage number 1 or 2 The characterizationin Theorem 7 is constructive constructing all trees withbondage number 2 a natural and straightforward methodby a series of graph-operations The characterization inTheorem 8 is a little advisable by describing the inherentproperty of trees with bondage number 1 The characteriza-tion in Theorem 11 is wonderful by using a strong graph-theoretic concept 120572

119896-set In fact this characterization is a

byproduct of some results related to 120572119896-sets for trees It is

that this characterization closed the relation between twoconcepts the bondage number and the 119896-independent setand hence is of research value and important significance

23 Complexity for General Graphs Asmentioned above thebondage number of a tree can be determined by a linear timealgorithm According to this algorithm we can determinewithin polynomial time the domination number of any treeby removing each edge and verifyingwhether the dominationnumber is enlarged according to the known linear timealgorithm for domination numbers of trees

However it is impossible to find a polynomial timealgorithm for bondage numbers of general graphs If suchan algorithm 119860 exists then the domination number of any

nonempty undirected graph 119866 can be determined withinpolynomial time by repeatedly using 119860 Let 119866

0= 119866 and

119866119894+1= 119866

119894minus119861

119894 where 119861

119894is the minimum edge set of 119866

119894found

by 119860 such that 120574(119866119894minus 119861

119894) = 120574(119866

119894minus1) + 1 for each 119894 = 0 1

we can always find the minimum 119861119894whose removal from 119866

119894

enlarges the domination number until 119866119896= 119866

119896minus1minus 119861

119896minus1

is empty for some 119896 ⩾ 1 though 119861119896minus1

is not empty Then120574(119866) = 120574(119866

119896) minus 119896 = 120592(119866) minus 119896 As known to all if NP = 119875

the minimum dominating set problem is NP-complete andso polynomial time algorithms for the bondage number donot exist unless NP = 119875

In fact Hu and Xu [16] have recently shown that theproblem determining the bondage number of general graphsis NP-hard

Problem 1 Consider the decision problemBondage ProblemInstance a graph 119866 and a positive integer 119887 (⩽ 120576(119866))Question is 119887(119866) ⩽ 119887

Theorem 12 (Hu and Xu [16] 2012) The bondage problem isNP-hard

Thebasicway of the proof is to followGarey and Johnsonrsquostechniques for proving NP-hardness [5] by describing apolynomial transformation from the known NP-completeproblem 3-satisfiability problem

Theorem 12 shows that we are unable to find a polynomialtime algorithm to determine bondage numbers of generalgraphs unless NP = 119875 At the same time this result alsoshows that the following study on the bondage number is ofimportant significance

(i) Find approximation polynomial algorithms with per-formance ratio as small as possible

(ii) Find the lower and upper bounds with difference assmall as possible

(iii) Determine exact values for some graphs speciallywell-known networks

Unfortunately we cannot provewhether or not determin-ing the bondage number is NP-problem since for any subset119861 sub 119864(119866) it is not clear that there is a polynomial algorithmto verify 120574(119866 minus 119861) gt 120574(119866) Since the problem of determiningthe domination number is NP-complete we conjecture thatit is not in NPThis is a worthwhile task to be studied further

Motivated by the linear time algorithm of Hartnell et alto compute the bondage number of a tree we can makean attempt to consider whether there is a polynomial timealgorithm to compute the bondage number for some specialclasses of graphs such as planar graphs Cayley graphs orgraphs with some restrictions of graph-theoretical parame-ters such as degree diameter connectivity and dominationnumber

3 Upper Bounds

ByTheorem 12 since we cannot find a polynomial time algo-rithm for determining the exact values of bondage numbers

International Journal of Combinatorics 5

of general graphs it is weightily significative to establishsome sharp bounds on the bondage number of a graph Inthis section we survey several known upper bounds on thebondage number in terms of some other graph-theoreticalparameters

31 Most Basic Upper Bounds We start this subsection witha simple observation

Observation 1 (Teschner [10] 1997) Let 119867 be a spanningsubgraph obtained from a graph119866 by removing 119896 edgesThen119887(119866) ⩽ 119887(119867) + 119896

If we select a spanning subgraph 119867 such that 119887(119867) = 1thenObservation 1 yields some upper bounds on the bondagenumber of a graph 119866 For example take119867 = 119866 minus 119864

119909+ 119890 and

119896 = 119889119866(119909) minus 1 where 119890 isin 119864

119909 the following result can be

obtained

Theorem 13 (Bauer et al [7] 1983) If there exists at least onevertex 119909 in a graph 119866 such that 120574(119866 minus 119909) ⩾ 120574(119866) then 119887(119866) ⩽119889119866(119909) ⩽ Δ(119866)

The following early result obtained can be derived fromObservation 1 by taking119867 = 119866minus119864

119909119910and 119896 = 119889(119909)+119889(119910)minus2

Theorem 14 (Bauer et al [7] 1983 Fink et al [8] 1990)119887(119866) ⩽ 119889(119909) + 119889(119910) minus 1 for any two adjacent vertices 119909 and119910 in a graph 119866 that is

119887 (119866) ⩽ min119909119910isin119864(119866)

119889119866(119909) + 119889

119866(119910) minus 1 (4)

This theorem gives a natural corollary obtained by severalauthors

Corollary 15 (Bauer et al [7] 1983 Fink et al [8] 1990) If 119866is a graphwithout isolated vertices then 119887(119866) ⩽ Δ(119866)+120575(119866)minus1

In 1999 Hartnell and Rall [11] extended Theorem 14 tothe following more general case which can be also derivedfrom Observation 1 by taking119867 = 119866 minus 119864

119909minus 119864

119910+ (119909 119911 119910) if

119889119866(119909 119910) = 2 where (119909 119911 119910) is a path of length 2 in 119866

Theorem 16 (Hartnell and Rall [11] 1999) 119887(119866) ⩽ 119889(119909) +119889(119910)minus1 for any distinct two vertices 119909 and 119910 in a graph119866with119889119866(119909 119910) ⩽ 2 that is

119887 (119866) ⩽ min119889119866(119909119910)⩽2

119889119866(119909) + 119889

119866(119910) minus 1 (5)

Corollary 17 (Fink et al [8] 1990) If a vertex of a graph 119866 isadjacent with two ormore vertices of degree one then 119887(119866) = 1

We remark that the bounds stated in Corollary 15 andTheorem 16 are sharp As indicated byTheorem 1 one class ofgraphs in which the bondage number achieves these boundsis the class of cycles whose orders are congruent to 1 modulo3

On the other hand Hartnell and Rall [17] sharpened theupper bound in Theorem 14 as follows which can be alsoderived from Observation 1

1198792(119909 119910)

1198794(119909 119910)

1198793(119909 119910)

1198791(119909 119910)

119909

119910

Figure 2 Illustration of 119879119894(119909 119910) for 119894 = 1 2 3 4

Theorem 18 (Hartnell and Rall [17] 1994) 119887(119866) ⩽ 119889119866(119909) +

119889119866(119910) minus 1 minus |119873

119866(119909) cap 119873

119866(119910)| for any two adjacent vertices 119909

and 119910 in a graph 119866 that is

119887 (119866) ⩽ min119909119910isin119864(119866)

119889119866(119909) + 119889

119866(119910) minus 1 minus

1003816100381610038161003816119873119866(119909) cap 119873

119866(119910)1003816100381610038161003816

(6)

These results give simple but important upper bounds onthe bondage number of a graph and is also the foundationof almost all results on bondage numbers upper boundsobtained till now

By careful consideration of the nature of the edges fromthe neighbors of119909 and119910Wang [18] further refined the boundin Theorem 18 For any edge 119909119910 isin 119864(119866) 119873

119866(119910) contains the

following four subsets

(1) 1198791(119909 119910) = 119873

119866[119909] cap 119873

119866(119910)

(2) 1198792(119909 119910) = 119908 isin 119873

119866(119910) 119873

119866(119908) sube 119873

119866(119910) minus 119909

(3) 1198793(119909 119910) = 119908 isin 119873

119866(119910) 119873

119866(119908) sube 119873

119866(119911)minus119909 for some

119911 isin 119873119866(119909) cap 119873

119866(119910)

(4) 1198794(119909 119910) = 119873

119866(119910) (119879

1(119909 119910) cup 119879

2(119909 119910) cup 119879

3(119909 119910))

The illustrations of 1198791(119909 119910) 119879

2(119909 119910) 119879

3(119909 119910) and

1198794(119909 119910) are shown in Figure 2 (corresponding vertices

pointed by dashed arrows)

Theorem 19 (Wang [18] 1996) For any nonempty graph 119866

119887 (119866) ⩽ min119909119910isin119864(119866)

119889119866(119909) +

10038161003816100381610038161198794 (119909 119910)1003816100381610038161003816 (7)

The graph in Figure 2 shows that the upper bound givenin Theorem 19 is better than that in Theorems 16 and 18for the upper bounds obtained from these two theorems are119889119866(119909)+119889

119866(119910)minus1 = 11 and 119889

119866(119909)+119889

119866(119910)minus|119873

119866(119909)cap119873

119866(119910)| =

9 respectively while the upper bound given byTheorem 19 is119889119866(119909) + |119879

4(119909 119910)| = 6

The following result is also an improvement ofTheorem 14 in which 119905 = 2

6 International Journal of Combinatorics

Theorem 20 (Teschner [10] 1997) If 119866 contains a completesubgraph 119870

119905with 119905 ⩾ 2 then 119887(119866) ⩽ min

119909119910isin119864(119870119905)119889

119866(119909) +

119889119866(119910) minus 119905 + 1

Following Fricke et al [19] a vertex 119909 of a graph 119866 is 120574-good if 119909 belongs to some 120574-set of119866 and 120574-bad if 119909 belongs tono 120574-set of 119866 Let 119860(119866) be the set of 120574-good vertices and let119861(119866) be the set of 120574-bad vertices in 119866 Clearly 119860(119866) 119861(119866)is a partition of119881(119866) Note there exists some 119909 isin 119860 such that120574(119866 minus 119909) = 120574(119866) say one end-vertex of 119875

5 Samodivkin [20]

presented some sharp upper bounds for 119887(119866) in terms of 120574-good and 120574-bad vertices of 119866

Theorem 21 (Samodivkin [20] 2008) Let 119866 be a graph

(a) Let 119862(119866) = 119909 isin 119881(119866) 120574(119866minus119909) ⩾ 120574(119866) If 119862(119866) = 0then

119887 (119866) ⩽ min 119889119866(119909) + 120574 (119866) minus 120574 (119866 minus 119909) 119909 isin 119862 (119866)

(8)

(b) If 119861 = 0 then

119887 (119866) ⩽ min 1003816100381610038161003816119873119866(119909) cap 119860

1003816100381610038161003816 119909 isin 119861 (119866) (9)

Theorem 22 (Samodivkin [20] 2008) Let 119866 be a graph If119860+

(119866) = 119909 isin 119860(119866) 119889119866(119909) ⩾ 1 and 120574(119866 minus 119909) lt 120574(119866) = 0

then

119887 (119866) ⩽ min119909isin119860+(119866)119910isin119861(119866minus119909)

119889119866(119909) +

1003816100381610038161003816119873119866(119910) cap 119860 (119866 minus 119909)

1003816100381610038161003816

(10)

Proposition 23 (Samodivkin [20] 2008) Under the notationof Theorem 19 if 119909 isin 119860+

(119866) then (1198791(119909 119910) minus 119909) cup 119879

2(119909 119910) cup

1198793(119909 119910) sube 119873

119866(119910) 119861(119866 minus 119909)

By Proposition 23 if 119909 isin 119860+

(119866) then

119889119866(119909) + min

119910isin119873119866(119909)

10038161003816100381610038161198794 (119909 119910)

1003816100381610038161003816

⩾ 119889119866(119909) + min

119910isin119873119866(119909)

1003816100381610038161003816119873119866

(119910) cap 119860 (119866 minus 119909)1003816100381610038161003816

⩾ 119889119866(119909) + min

119910isin119861(119866minus119909)

1003816100381610038161003816119873119866

(119910) cap 119860 (119866 minus 119909)1003816100381610038161003816

(11)

Hence Theorem 19 can be seen to follow from Theorem 22Any graph 119866 with 119887(119866) achieving the upper bound inTheorem 19 can be used to show that the bound inTheorem 22 is sharp

Let 119905 ⩾ 2 be an integer Samodivkin [20] con-structed a very interesting graph 119866

119905to show that the upper

bound in Theorem 22 is better than the known bounds Let119867

0 119867

1 119867

2 119867

119905+1be mutually vertex-disjoint graphs such

that 1198670cong 119870

2 119867

119905+1cong 119870

119905+3and 119867

119894cong 119870

119905+3minus 119890 for each

119894 = 1 2 119905 Let 119881(1198670) = 119909 119910 119909

119905+1isin 119881(119867

119905+1) and

1198671

1198672

1198673

11990911

11990912

11990921 11990922

1199093

119909

119910

Figure 3 The graph 1198662

1199091198941 119909

1198942isin 119881(119867

119894) 119909

11989411199091198942notin 119864(119867

119894) for each 119894 = 1 2 119905 The

graph 119866119905is defined as follows

119881 (119866119905) =

119905+1

⋃

119896=0

119881 (119867119896)

119864 (119866119905)=(

119905+1

⋃

119896=0

119864 (119867119896))cup(

119905

⋃

119894=1

1199091199091198941 119909119909

1198942)cup119909119909

119905+1

(12)

Such a constructed graph119866119905is shown in Figure 3 when 119905 = 2

Observe that 120574(119866119905) = 119905 + 2 119860(119866

119905) = 119881(119866

119905) 119889

119866119905

(119909) =

2119905 + 2 119889119866119905

(119909119905+1) = 119905 + 3 119889

119866119905

(119910) = 1 and 119889119866119905

(119911) = 119905 + 2

for each 119911 isin 119881(119866119905minus 119909 119910 119909

119905+1) Moreover 120574(119866 minus 119910) lt 120574(119866)

and 120574(119866119905minus119911) = 120574(119866

119905) for any 119911 isin 119881(119866

119905) minus 119910 Hence each of

the bounds stated inTheorems 13ndash20 is greater than or equals119905 + 2

Consider the graph 119866119905minus 119909119910 Clearly 120574(119866

119905minus 119909119910) = 120574(119866

119905)

and

119861 (119866119905minus 119909119910) = 119861 (119866

119905minus 119910)

= 119909 cup 119881 (119867119905+1minus 119909

119905+1) cup (

119905

⋃

119896=1

1199091198961

1199091198962

)

(13)

Therefore 119873119866119905

(119909) cap 119866(119866119905minus 119910) = 119909

119905+1 which implies that

the upper bound stated in Theorem 22 is equal to 119889119866119905

(119910) +

|119909119905+1| = 2 Clearly 119887(119866

119905) = 2 and hence this bound is sharp

for 119866119905

From the graph 119866119905 we obtain the following statement

immediately

Proposition 24 For every integer 119905 ⩾ 2 there is a graph 119866such that the difference between any upper bound stated inTheorems 13ndash20 and the upper bound in Theorem 22 is equalto 119905

Although Theorem 22 supplies us with the upper boundthat is closer to 119887(119866) for some graph 119866 than what any one of

International Journal of Combinatorics 7

Theorems 13ndash20 provides it is not easy to determine the sets119860+

(119866) and 119861(119866) mentioned in Theorem 22 for an arbitrarygraph 119866 Thus the upper bound given in Theorem 22 is oftheoretical importance but not applied since until now wehave not found a new class of graphswhose bondage numbersare determined byTheorem 22

The above-mentioned upper bounds on the bondagenumber are involved in only degrees of two vertices Hartnelland Rall [11] established an upper bound on 119887(119866) in terms ofthe numbers of vertices and edges of 119866 with order 119899 For anyconnected graph 119866 let 120575(119866) represent the average degree ofvertices in 119866 that is 120575(119866) = (1119899)sum

119909isin119881119889119866(119909) Hartnell and

Rall first discovered the following proposition

Proposition 25 For any connected graph 119866 there exist twovertices 119909 and 119910 with distance at most two and with theproperty that 119889

119866(119909) + 119889

119866(119910) ⩽ 2120575(119866)

Using Proposition 25 and Theorem 16 Hartnell and Rallgave the following bound

Theorem 26 (Hartnell and Rall [11] 1999) 119887(119866) ⩽ 2120575(119866) minus 1for any connected graph 119866

Note that 2120576 = 119899120575(119866) for any graph 119866 with order 119899 andsize 120576 Theorem 26 implies the following bound in terms oforder 119899 and size 120576

Corollary 27 119887(119866) ⩽ (4120576119899) minus 1 for any connected graph 119866with order 119899 and size 120576

A lower bound on 120576 in terms of order 119899 and the bondagenumber is obtained from Corollary 28 immediately

Corollary 28 120576 ⩾ (1198994)(119887(119866) + 1) for any connected graph 119866with order 119899 and size 120576

Hartnell and Rall [11] gave some graphs 119866 with order 119899to show that for each value of 119887(119866) the lower bound on 120576(119866)given in the Corollary 28 is sharp for some values of 119899

32 Bounds Implied by Connectivity Use 120581(119866) and 120582(119866) todenote the vertex-connectivity and the edge-connectivity ofa connected graph 119866 respectively which are the minimumnumbers of vertices and edges whose removal result in 119866disconnected The famous Whitney inequality is stated as120581(119866) ⩽ 120582(119866) ⩽ 120575(119866) for any graph or digraph 119866 Corollary 15is improved by several authors as follows

Theorem 29 (Hartnell and Rall [17] 1994 Teschner [10]1997) If 119866 is a connected graph then 119887(119866) ⩽ 120582(119866)+Δ(119866)minus 1

The upper bound given in Theorem 29 can be attainedFor example a cycle 119862

3119896+1with 119896 ⩾ 1 119887(119862

3119896+1) = 3 by

Theorem 1 Since 120581(1198623119896+1) = 120582(119862

3119896+1) = 2 we have 120581(119862

3119896+1)+

120582(1198623119896+1) minus 1 = 2 + 2 minus 1 = 3 = 119887(119862

3119896+1)

Motivated byCorollary 15Theorems 29 and theWhitneyinequality Dunbar et al [21] naturally proposed the followingconjecture

Figure 4 A graph119867 with 120574(119867) = 3 and 119887(119867) = 5

Conjecture 30 If119866 is a connected graph then 119887(119866) ⩽ Δ(119866)+120581(119866) minus 1

However Liu and Sun [22] presented a counterexampleto this conjecture They first constructed a graph 119867 showedin Figure 4 with 120574(119867) = 3 and 119887(119867) = 5 Then let 119866 be thedisjoint union of two copies of119867 by identifying two verticesof degree two They proved that 119887(119866) ⩾ 5 Clearly 119866 is a 4-regular graph with 120581(119866) = 1 and 120582(119866) = 2 and so 119887(119866) ⩽ 5byTheorem 29 Thus 119887(119866) = 5 gt 4 = Δ(119866) + 120581(119866) minus 1

With a suspicion of the relationship between the bondagenumber and the vertex-connectivity of a graph the followingconjecture is proposed

Conjecture 31 (Liu and Sun [22] 2003) For any positiveinteger 119903 there exists a connected graph 119866 such that 119887(119866) ⩾Δ(119866) + 120581(119866) + 119903

To the knowledge of the author until now no results havebeen known about this conjecture

We conclude this subsection with the following remarksFromTheorem 29 if Conjecture 31 holds for some connectedgraph119866 then 120582(119866) gt 120581(119866)+119903 which implies that119866 is of largeedge-connectivity and small vertex-connectivity Use 120585(119866) todenote the minimum edge-degree of 119866 that is

120585 (119866) = min119909119910isin119864(119866)

119889119866(119909) + 119889

119866(119910) minus 2 (14)

Theorem 14 implies the following result

Proposition 32 119887(119866) ⩽ 120585(119866) + 1 for any graph 119866

Use 1205821015840(119866) to denote the restricted edge-connectivity ofa connected graph 119866 which is the minimum number ofedgeswhose removal result in119866 disconnected and no isolatedvertices

Proposition 33 (Esfahanian and Hakimi [23] 1988) If 119866 isneither 119870

1119899nor 119870

3 then 1205821015840(119866) ⩽ 120585(119866)

Combining Propositions 32 and 33 we propose a conjec-ture

Conjecture 34 If a connected 119866 is neither 1198701119899

nor 1198703 then

119887(119866) ⩽ 120575(119866) + 1205821015840

(119866) minus 1

8 International Journal of Combinatorics

A cycle 1198623119896+1

also satisfies 119887(1198623119896+1) = 3 = 120575(119862

3119896+1) +

1205821015840

(1198623119896+1) minus 1 since 1205821015840(119862

3119896+1) = 120575(119862

3119896+1) = 2 for any integer

119896 ⩾ 1For the graph119867 shown in Figure 41205821015840(119867) = 4 and 120575(119867) =

2 and so 119887(119867) = 5 = 120575(119867) + 1205821015840(119867) minus 1For the 4-regular graph119866 constructed by Liu and Sun [22]

obtained from the disjoint union of two copies of the graph119867 showed in Figure 4 by identifying two vertices of degreetwo we have 119887(119866) ⩾ 5 Clearly 1205821015840(119866) = 2 Thus 119887(119866) = 5 =120575(119866) + 120582

1015840

(119866) minus 1For the 4-regular graph 119866

119905constructed by Samodivkin

[20] see Figure 3 for 119905 = 2 119887(119866119905) = 2 Clearly 120575(119866

119905) = 1

and 1205821015840(119866) = 2 Thus 119887(119866) = 2 = 120575(119866) + 1205821015840(119866) minus 1These examples show that if Conjecture 34 is true then the

given upper bound is tight

33 Bounds Implied by Degree Sequence Now let us return toTheorem 16 from which Teschner [10] obtained some otherbounds in terms of the degree sequencesThe degree sequence120587(119866) of a graph 119866 with vertex-set 119881 = 119909

1 119909

2 119909

119899 is the

sequence 120587 = (1198891 119889

2 119889

119899) with 119889

1⩽ 119889

2⩽ sdot sdot sdot ⩽ 119889

119899 where

119889119894= 119889

119866(119909

119894) for each 119894 = 1 2 119899 The following result is

essentially a corollary of Theorem 16

Theorem35 (Teschner [10] 1997) Let119866 be a nonempty graphwith degree sequence120587(119866) If120572

2(119866) = 119905 then 119887(119866) ⩽ 119889

119905+119889

119905+1minus

1

CombiningTheorem 35with Proposition 10 (ie 1205722(119866) ⩽

120574(119866)) we have the following corollary

Corollary 36 (Teschner [10] 1997) Let 119866 be a nonemptygraph with the degree sequence 120587(119866) If 120574(119866) = 120574 then 119887(119866) ⩽119889120574+ 119889

120574+1minus 1

In [10] Teschner showed that these two bounds are sharpfor arbitrarily many graphs Let119867 = 119862

3119896+1+ 119909

11199094 119909

11199093119896minus1

where 1198623119896+1

is a cycle (1199091 119909

2 119909

3119896+1 119909

1) for any integer

119896 ⩾ 2 Then 120574(119867) = 119896 + 1 and so 119887(119867) ⩽ 2 + 2 minus 1 = 3by Corollary 36 Since119862

3119896+1is a spanning subgraph of119867 and

120574(119866) = 120574(119867) Observation 1 yields that 119887(119867) ⩾ 119887(1198623119896+1) = 3

and so 119887(119867) = 3Although various upper bounds have been established

as the above we find that the appearance of these boundsis essentially based upon the local structures of a graphprecisely speaking the structures of the neighborhoods oftwo vertices within distance 2 Even if these bounds can beachieved by some special graphs it is more often not the caseThe reason lies essentially in the definition of the bondagenumber which is theminimumvalue among all bondage setsan integral property of a graph While it easy to find upperbounds just by choosing some bondage set the gap betweenthe exact value of the bondage number and such a boundobtained only from local structures of a graph is often largeFor example a star 119870

1Δ however large Δ is 119887(119870

1Δ) = 1

Therefore one has been longing for better bounds upon someintegral parameters However as what we will see below it isdifficult to establish such upper bounds

34 Bounds in 120574-Critical Graphs A graph119866 is called a vertexdomination-critical graph (vc-graph or 120574-critical for short) if120574(119866 minus 119909) lt 120574(119866) for any 119909 isin 119881(119866) proposed by Brighamet al [24] in 1988 Several families of graphs are known to be120574-critical From definition it is clear that if 119866 is a 120574-criticalgraph then 120574(119866) ⩾ 2The class of 120574-critical graphs with 120574 = 2is characterized as follows

Proposition 37 (Brigham et al [24] 1988) A graph 119866 with120574(119866) = 2 is a 120574-critical graph if and only if 119866 is a completegraph 119870

2119905(119905 ⩾ 2) with a perfect matching removed

A more interesting family is composed of the 119899-criticalgraphs 119866

119898119899defined for 119898 119899 ⩾ 2 by the circulant undirected

graph 119866(119873 plusmn119878) where 119873 = (119898 + 1)(119899 minus 1) + 1 and119878 = 1 2 lfloor1198982rfloor in which 119881(119866(119873 plusmn119878)) = 119873 and119864(119866(119873 plusmn119878)) = 119894119895 |119895 minus 119894| equiv 119904 (mod 119873) 119904 isin 119878

The reason why 120574-critical graphs are of special interest inthis context is easy to see that they play an important role instudy of the bondage number For instance it immediatelyfollows from Theorem 13 that if 119887(119866) gt Δ(119866) then 119866 is a 120574-critical graphThe 120574-critical graphs are defined exactly in thisway In order to find graphs 119866 with a high bondage number(ie higher than Δ(119866)) and beyond its general upper boundsfor the bondage number we therefore have to look at 120574-critical graphs

The bondage numbers of some 120574-critical graphs havebeen examined by several authors see for example [1020 25 26] From Theorem 13 we know that the bondagenumber of a graph 119866 is bounded from above by Δ(119866)if 119866 is not a 120574-critical graph For 120574-critical graphs it ismore difficult to find an upper bound We will see that thebondage numbers of 120574-critical graphs in general are noteven bounded from above by Δ + 119888 for any fixed naturalnumber 119888

In this subsection we introduce some upper bounds forthe bondage number of a 120574-critical graph By Proposition 37we easily obtain the following result

Theorem 38 If 119866 is a 120574-critical graph with 120574(119866) = 2 then119887(119866) ⩽ Δ + 1

In Section 4 by Theorem 59 we will see that the equalityin Theorem 38 holds that is 119887(119866) = Δ + 1 if 119866 is a 120574-criticalgraph with 120574(119866) = 2

Theorem 39 (Teschner [10] 1997) Let119866 be a 120574-critical graphwith degree sequence 120587(119866) Then 119887(119866) ⩽ maxΔ(119866) + 1 119889

1+

1198892+ sdot sdot sdot + 119889

120574minus1 where 120574 = 120574(119866)

As we mentioned above if 119866 is a 120574-critical graph with120574(119866) = 2 then 119887(119866) = Δ + 1 which shows that thebound given in Theorem 39 can be attained for 120574 = 2However we have not known whether this bound is tightfor general 120574 ⩾ 3 Theorem 39 gives the following corollaryimmediately

Corollary 40 (Teschner [10] 1997) Let 119866 be a 120574-criticalgraph with degree sequence 120587(119866) If 120574(119866) = 3 then 119887(119866) ⩽maxΔ(119866) + 1 120575(119866) + 119889

2

International Journal of Combinatorics 9

From Theorem 38 and Corollary 40 we have 119887(119866) ⩽2Δ(119866) if 119866 is a 120574-critical graph with 120574(119866) ⩽ 3 The followingresult shows that this bound is not tight

Theorem 41 (Teschner [26] 1995) Let119866 be a 120574-critical graphgraph with 120574(119866) ⩽ 3 Then 119887(119866) ⩽ (32)Δ(119866)

Until now we have not known whether the bound givenin Theorem 41 is tight or not We state two conjectures on120574-critical graphs proposed by Samodivkin [20] The first ofthem is motivated byTheorems 22 and 19

Conjecture 42 (Samodivkin [20] 2008) For every connectednontrivial 120574-critical graph 119866

min119909isin119860+(119866)119910isin119861(119866minus119909)

119889119866(119909) +

1003816100381610038161003816119873119866(119910) cap 119860 (119866 minus 119909)

1003816100381610038161003816 ⩽3

2Δ (119866)

(15)

To state the second conjecture we need the followingresult on 120574-critical graphs

Proposition 43 If119866 is a 120574-critical graph then 120592(119866) ⩽ (Δ(119866)+1)(120574(119866)minus1)+1 moreover if the equality holds then119866 is regular

The upper bound in Proposition 43 is sharp in thesense that equality holds for the infinite class of 120574-criticalgraphs 119866

119898119899defined in the beginning of this subsection In

Proposition 43 the first result is due to Brigham et al [24] in1988 the second is due to Fulman et al [27] in 1995

For a 120574-critical graph 119866 with 120574 = 3 by Proposition 43120592(119866) ⩽ 2Δ(119866) + 3

Theorem 44 (Teschner [26] 1995) If 119866 is a 120574-critical graphwith 120574 = 3 and 120592(119866) = 2Δ(119866) + 3 then 119887(119866) ⩽ Δ + 1

We have not known yet whether the equality inTheorem 44holds or notHowever Samodivkin proposed thefollowing conjecture

Conjecture 45 (Samodivkin [20] 2008) If 119866 is a 120574-criticalgraph with (Δ(119866)+1)(120574minus1)+1 vertices then 119887(119866) = Δ(119866)+1

In general based on Theorem 41 Teschner proposed thefollowing conjecture

Conjecture 46 (Teschner [26] 1995) If119866 is a 120574-critical graphthen 119887(119866) ⩽ (32)Δ(119866) for 120574 ⩾ 4

We conclude this subsection with some remarks Graphswhich are minimal or critical with respect to a given prop-erty or parameter frequently play an important role in theinvestigation of that property or parameter Not only are suchgraphs of considerable interest in their own right but also aknowledge of their structure often aids in the developmentof the general theory In particular when investigating anyfinite structure a great number of results are proven byinduction Consequently it is desirable to learn as much aspossible about those graphs that are critical with respect toa given property or parameter so as to aid and abet suchinvestigations

In this subsection we survey some results on the bondagenumber for 120574-critical graphs Although these results are notvery perfect they provides a feasible method to approach thebondage number from different viewpoints In particular themethods given in Teschner [26] worth further explorationand development

The following proposition is maybe useful for us tofurther investigate the bondage number of a 120574-critical graph

Proposition 47 (Brigham et al [24] 1988) If 119866 has anonisolated vertex 119909 such that the subgraph induced by119873

119866(119909)

is a complete graph then 119866 is not 120574-critical

This simple fact shows that any 120574-critical graph containsno vertices of degree one

35 Bounds Implied by Domination In the preceding sub-section we introduced some upper bounds on the bondagenumbers for 120574-critical graphs by consideration of domina-tions In this subsection we introduce related results for ageneral graph with given domination number

Theorem 48 (Fink et al [8] 1990) For any connected graph119866 of order 119899 (⩾2)

(a) 119887(119866) ⩽ 119899 minus 1

(b) 119887(119866) ⩽ Δ(119866) + 1 if 120574(119866) = 2

(c) 119887(119866) ⩽ 119899 minus 120574(119866) + 1

While the upper bound of 119899minus1 on 119887(119866) is not particularlygood for many classes of graphs (eg trees and cycles) it isan attainable bound For example if 119866 is a complete 119905-partitegraph 119870

119905(2) for 119905 = 1198992 and 119899 ⩽ 4 an even integer then the

three bounds on 119887(119866) in Theorem 48 are sharpTeschner [26] consider a graphwith 120574(119866) = 1 or 120574(119866) = 2

The next result is almost trivial but useful

Lemma 49 Let 119866 be a graph with order 119899 and 120574(119866) = 1 119905 thenumber of vertices of degree 119899 minus 1 Then 119887(119866) = lceil1199052rceil

Since 119905 ⩽ Δ(119866) clearly Lemma 49 yields the followingresult immediately

Theorem 50 (Teschner [26] 1995) 119887(119866) ⩽ (12)Δ+1 for anygraph 119866 with 120574(119866) = 1

For a complete graph1198702119896+1

119887(1198702119896+1) = 119896+1 = (12)Δ+1

which shows that the upper bound given in Theorem 50 canbe attained For a graph119866with 120574(119866) = 2 byTheorem 59 laterthe upper bound given inTheorem 48 (b) can be also attainedby a 2-critical graph (see Proposition 37)

36 Two Conjectures In 1990 when Fink et al [8] introducedthe concept of the bondage number they proposed thefollowing conjecture

Conjecture 51 119887(119866) ⩽ Δ(119866) + 1 for any nonempty graph 119866

10 International Journal of Combinatorics

Although some results partially support Conjecture 51Teschner [28] in 1993 found a counterexample to this con-jecture the cartesian product 119870

3times 119870

3 which shows that

119887(119866) = Δ(119866) + 2If a graph 119866 is a counterexample to Conjecture 51 it

must be a 120574-critical graph by Theorem 13 That is why thevertex domination-critical graphs are of special interest in theliterature

Now we return to Conjecture 51 Hartnell and Rall [17]and Teschner [29] independently proved that 119887(119866) can bemuch greater than Δ(119866) by showing the following result

Theorem 52 (Hartnell and Rall [17] 1994 Teschner [29]1996) For an integer 119899 ⩾ 3 let 119866

119899be the cartesian product

119870119899times 119870

119899 Then 119887(119866

119899) = 3(119899 minus 1) = (32)Δ(119866

119899)

This theorem shows that there exist no upper boundsof the form 119887(119866) ⩽ Δ(119866) + 119888 for any integer 119888 Teschner[26] proved that 119887(119866) ⩽ (32)Δ(119866) for any graph 119866 with120574(119866) ⩽ 2 (see Theorems 50 and 48) and for 120574-critical graphswith 120574(119866) = 3 and proposed the following conjecture

Conjecture 53 (Teschner [26] 1995) 119887(119866) ⩽ (32)Δ(119866) forany graph 119866

We believe that this conjecture is true but so far no muchwork about it has been known

4 Lower Bounds

Since the bondage number is defined as the smallest numberof edges whose removal results in an increase of dominationnumber each constructive method that creates a concretebondage set leads to an upper bound on the bondage numberFor that reason it is hard to find lower bounds Neverthelessthere are still a few lower bounds

41 Bounds Implied by Subgraphs The first lower boundon the bondage number is gotten in terms of its spanningsubgraph

Theorem 54 (Teschner [10] 1997) Let 119867 be a spanningsubgraph of a nonempty graph119866 If 120574(119867) = 120574(119866) then 119887(119867) ⩽119887(119866)

By Theorem 1 119887(119862119899) = 3 if 119899 equiv 1 (mod 3) and 119887(119862

119899) =

2 otherwise 119887(119875119899) = 2 if 119899 equiv 1 (mod 3) and 119887(119875

119899) = 1

otherwise From these results and Theorem 54 we get thefollowing two corollaries

Corollary 55 If119866 is hamiltonianwith order 119899 ⩾ 3 and 120574(119866) =lceil1198993rceil then 119887(119866) ⩾ 2 and in addition 119887(119866) ⩾ 3 if 119899 equiv 1 (mod3)

Corollary 56 If 119866 with order 119899 ⩾ 2 has a hamiltonian pathand 120574(119866) = lceil1198993rceil then 119887(119866) ⩾ 2 if 119899 equiv 1 (mod 3)

42 Other Bounds The vertex covering number 120573(119866) of 119866 isthe minimum number of vertices that are incident with all

Figure 5 A 120574-critical graph with 120573 = 120574 = 4 120575 = 2 and 119887 = 3

edges in 119866 If 119866 has no isolated vertices then 120574(119866) ⩽ 120573(119866)clearly In [30] Volkmann gave a lot of graphs with 120573 = 120574

Theorem 57 (Teschner [10] 1997) Let119866 be a graph If 120574(119866) =120573(119866) then

(a) 119887(119866) ⩾ 120575(119866)

(b) 119887(119866) ⩾ 120575(119866) + 1 if 119866 is a 120574-critical graph

The graph shown in Figure 5 shows that the bound givenin Theorem 57(b) is sharp For the graph 119866 it is a 120574-criticalgraph and 120574(119866) = 120573(119866) = 4 By Theorem 57 we have 119887(119866) ⩾3 On the other hand 119887(119866) ⩽ 3 by Theorem 16 Thus 119887(119866) =3

Proposition 58 (Sanchis [31] 1991) Let 119866 be a graph 119866 oforder 119899 (⩾ 6) and size 120576 If 119866 has no isolated vertices and3 ⩽ 120574(119866) ⩽ 1198992 then 120576 ⩽ ( 119899minus120574(119866)+1

2)

Using the idea in the proof of Theorem 57 every upperbound for 120574(119866) can lead to a lower bound for 119887(119866) Inthis way Teschner [10] obtained another lower bound fromProposition 58

Theorem59 (Teschner [10] 1997) Let119866 be a graph119866 of order119899 (⩾6) and size 120576 If 2 ⩽ 120574(119866) ⩽ 1198992 minus 1 then

(a) 119887(119866) ⩾ min120575(119866) 120576 minus ( 119899minus120574(119866)2)

(b) 119887(119866) ⩾ min120575(119866) + 1 120576 minus ( 119899minus120574(119866)2) if 119866 is a 120574-critical

graph

The lower bound in Theorem 59(b) is sharp for a classof 120574-critical graphs with domination number 2 Let 119866 be thegraph obtained from complete graph119870

2119905(119905 ⩾ 2) by removing

a perfect matching By Proposition 37 119866 is a 120574-critical graphwith 120574(119866) = 2 Then 119887(119866) ⩾ 120575(119866) + 1 = Δ(119866) + 1 byTheorem 59 and 119887(119866) ⩽ Δ(119866) + 1 by Theorem 38 and so119887(119866) = Δ(119866) + 1 = 2119905 minus 1

As far as the author knows there are no more lowerbounds in the present literature In view of applications ofthe bondage number a network is vulnerable if its bondagenumber is small while it is stable if its bondage number islargeTherefore better lower bounds let us learn better aboutthe stability of networks from this point of view Thus it isof great significance to seek more lower bounds for variousclasses of graphs

International Journal of Combinatorics 11

5 Results on Graph Operations

Generally speaking it is quite difficult to determine the exactvalue of the bondage number for a given graph since itstrongly depends on the dominating number of the graphThus determining bondage numbers for some special graphsis interesting if the dominating numbers of those graphs areknown or can be easily determined In this section we willintroduce some known results on bondage numbers for somespecial classes of graphs obtained by some graph operations

51 Cartesian Product Graphs Let 1198661= (119881

1 119864

1) and 119866

2=

(1198812 119864

2) be two graphs The Cartesian product of 119866

1and 119866

2

is an undirected graph denoted by 1198661times 119866

2 where 119881(119866

1times

1198662) = 119881

1times 119881

2 two distinct vertices 119909

11199092and 119910

11199102 where

1199091 119910

1isin 119881(119866

1) and 119909

2 119910

2isin 119881(119866

2) are linked by an edge in

1198661times 119866

2if and only if either 119909

1= 119910

1and 119909

21199102isin 119864(119866

2) or

1199092= 119910

2and 119909

11199101isin 119864(119866

1) The Cartesian product is a very

effective method for constructing a larger graph from severalspecified small graphs

Theorem 60 (Dunbar et al [21] 1998) For 119899 ⩾ 3

119887 (119862119899times 119870

2) =

2 119894119891 119899 equiv 0 119900119903 1 (mod 4) 3 119894119891 119899 equiv 3 (mod 4) 4 119894119891 119899 equiv 2 (mod 4)

(16)

For the Cartesian product 119862119899times 119862

119898of two cycles 119862

119899and

119862119898 where 119899 ⩾ 4 and 119898 ⩾ 3 Klavzar and Seifter [32]

determined 120574(119862119899times 119862

119898) for some 119899 and 119898 For example

120574(119862119899times119862

4) = 119899 for 119899 ⩾ 3 and 120574(119862

119899times119862

3) = 119899minuslfloor1198994rfloor for 119899 ⩾ 4

Kim [33] determined 119887(1198623times 119862

3) = 6 and 119887(119862

4times 119862

4) = 4 For

a general 119899 ⩾ 4 the exact values of the bondage numbers of119862119899times 119862

3and 119862

119899times 119862

4are determined as follows

Theorem 61 (Sohn et al [34] 2007) For 119899 ⩾ 4

119887 (119862119899times 119862

3) =

2 119894119891 119899 equiv 0 (mod 4) 4 119894119891 119899 equiv 1 119900119903 2 (mod 4) 5 119894119891 119899 equiv 3 (mod 4)

(17)

Theorem62 (Kang et al [35] 2005) 119887(119862119899times119862

4) = 4 for 119899 ⩾ 4

For larger 119898 and 119899 Huang and Xu [36] obtained thefollowing result see Theorem 197 for more details

Theorem 63 (Huang and Xu [36] 2008) 119887(1198625119894times119862

5119895) = 3 for

any positive integers 119894 and 119895

Xiang et al [37] determined that for 119899 ⩾ 5

119887 (119862119899times 119862

5)

= 3 if 119899 equiv 0 or 1 (mod 5) = 4 if 119899 equiv 2 or 4 (mod 5) ⩽ 7 if 119899 equiv 3 (mod 5)

(18)

For the Cartesian product 119875119899times119875

119898of two paths 119875

119899and 119875

119898

Jacobson and Kinch [38] determined 120574(119875119899times119875

2) = lceil(119899+1)2rceil

120574(119875119899times 119875

3) = 119899 minus lfloor(119899 minus 1)4rfloor and 120574(119875

119899times 119875

4) = 119899 + 1 if 119899 =

1 2 3 5 6 9 and 119899 otherwiseThe bondage number of119875119899times119875

119898

for 119899 ⩾ 4 and 2 ⩽ 119898 ⩽ 4 is determined as follows (Li [39] alsodetermined 119887(119875

119899times 119875

2))

Theorem 64 (Hu and Xu [40] 2012) For 119899 ⩾ 4

119887 (119875119899times 119875

2) =

1 119894119891 119899 equiv 1 (mod 2)2 119894119891 119899 equiv 0 (mod 2)

119887 (119875119899times 119875

3) =

1 119894119891 119899 equiv 1 119900119903 2 (mod 4)2 119894119891 119899 equiv 0 119900119903 3 (mod 4)

119887 (119875119899times 119875

4) = 1 119891119900119903 119899 ⩾ 14

(19)

From the proof of Theorem 64 we find that if 119875119899=

0 1 119899 minus 1 and 119875119898= 0 1 119898 minus 1 then the removal

of the vertex (0 0) in 119875119899times119875

119898does not change the domination

number If119898 increases the effect of (0 0) for the dominationnumber will be smaller and smaller from the probabilityTherefore we expect it is possible that 120574(119875

119899times 119875

119898minus (0 0)) =

120574(119875119899times 119875

119898) for119898 ⩾ 5 and give the following conjecture

Conjecture 65 119887(119875119899times 119875

119898) ⩽ 2 for119898 ⩾ 5 and 119899 ⩾ 4

52 Block Graphs and Cactus Graphs In this subsection weintroduce some results for corona graphs block graphs andcactus graphs

The corona1198661∘ 119866

2 proposed by Frucht and Harary [41]

is the graph formed from a copy of 1198661and 120592(119866

1) copies of

1198662by joining the 119894th vertex of 119866

1to the 119894th copy of 119866

2 In

particular we are concernedwith the corona119867∘1198701 the graph

formed by adding a new vertex V119894 and a new edge 119906

119894V119894for

every vertex 119906119894in119867

Theorem 66 (Carlson and Develin [42] 2006) 119887(119867 ∘ 1198701) =

120575(119867) + 1

This is a very important result which is often used toconstruct a graph with required bondage number In otherwords this result implies that for any given positive integer 119887there is a graph 119866 such that 119887(119866) = 119887

A block graph is a graph whose blocks are completegraphs Each block in a cactus graph is either a cycle or a119870

2

If each block of a graph is either a complete graph or a cyclethen we call this graph a block-cactus graph

Theorem67 (Teschner [43] 1997) 119887(119866) le Δ(119866) for any blockgraph 119866

Teschner [43] characterized all block graphs with 119887(119866) =Δ(119866) In the same paper Teschner found that 120574-criticalgraphs are instrumental in determining bounds for thebondage number of cactus and block graphs and obtained thefollowing result

Theorem 68 (Teschner [43] 1997) 119887(119866) ⩽ 3 for anynontrivial cactus graph 119866

This bound can be achieved by 119867 ∘ 1198701where 119867 is

a nontrivial cactus graph without vertices of degree one

12 International Journal of Combinatorics

by Theorem 66 In 1998 Dunbar et al [21] proposed thefollowing problem

Problem 2 Characterize all cactus graphs with bondagenumber 3

Some upper bounds for block-cactus graphs were alsoobtained

Theorem 69 (Dunbar et al [21] 1998) Let 119866 be a connectedblock-cactus graph with at least two blocksThen 119887(119866) ⩽ Δ(119866)

Theorem 70 (Dunbar et al [21] 1998) Let 119866 be a connectedblock-cactus graph which is neither a cactus graph nor a blockgraph Then 119887(119866) ⩽ Δ(119866) minus 1

53 Edge-Deletion and Edge-Contraction In order to obtainfurther results of the bondage number we may consider howthe bondage number changes under some operations of agraph 119866 which remain the domination number unchangedor preserve some property of 119866 such as planarity Twosimple operations satisfying these requirements are the edge-deletion and the edge-contraction

Theorem 71 (Huang and Xu [44] 2012) Let 119866 be any graphand 119890 isin 119864(119866) Then 119887(119866 minus 119890) ⩾ 119887(119866) minus 1 Moreover 119887(119866 minus 119890) ⩽119887(119866) if 120574(119866 minus 119890) = 120574(119866)

FromTheorem 1 119887(119862119899minus 119890) = 119887(119875

119899) = 119887(119862

119899) minus 1 for any 119890

in119862119899 which shows that the lower bound on 119887(119866minus119890) is sharp

The upper bound can reach for any graph 119866 with 119887(119866) ⩾ 2Next we consider the effect of the edge-contraction on

the bondage number Given a graph 119866 the contraction of 119866by the edge 119890 = 119909119910 denoted by 119866119909119910 is the graph obtainedfrom 119866 by contracting two vertices 119909 and 119910 to a new vertexand then deleting all multiedges It is easy to observe that forany graph 119866 120574(119866) minus 1 ⩽ 120574(119866119909119910) ⩽ 120574(119866) for any edge 119909119910 of119866

Theorem 72 (Huang and Xu [44] 2012) Let 119866 be any graphand 119909119910 be any edge in119866 If119873

119866(119909)cap119873

119866(119910) = 0 and 120574(119866119909119910) =

120574(119866) then 119887(119866119909119910) ⩾ 119887(119866)

We can use examples 119862119899and 119870

119899to show that the

conditions ofTheorem 72 are necessary and the lower boundinTheorem 72 is tight Clearly 120574(119862

119899) = lceil1198993rceil and 120574(119870

119899) = 1

for any edge 119909119910 119862119899119909119910 = 119862

119899minus1and 119870

119899119909119910 = 119870

119899minus1 By

Theorem 1 if 119899 equiv 1 (mod 3) then 120574(119862119899119909119910) lt 120574(119866) and

119887(119862119899119909119910) = 2 lt 3 = 119887(119862

119899) Thus the result in Theorem 72

is generally invalid without the hypothesis 120574(119866119909119910) = 120574(119866)Furthermore the condition 119873

119866(119909) cap 119873

119866(119910) = 0 cannot be

omitted even if 120574(119866119909119910) = 120574(119866) since for odd 119899 119887(119870119899119909119910) =

lceil(119899 minus 1)2rceil lt lceil1198992rceil = 119887(119870119899) by Theorem 1

On the other hand 119887(119862119899119909119910) = 119887(119862

119899) = 2 if 119899 equiv 0 (mod

3) and 119887(119870119899119909119910) = 119887(119870

119899) if 119899 is even which shows that the

equality in 119887(119866119909119910) ⩾ 119887(119866) may hold Thus the bound inTheorem 72 is tight However provided all the conditions119887(119866119909119910) can be arbitrarily larger than 119887(119866) Given a graph119867let119866 be the graph formed from119867∘119870

1by adding a new vertex

119909 and joining it to an vertex 119910 of degree one in119867 ∘ 1198701 Then

119866119909119910 = 119867 ∘ 1198701 120574(119866) = 120574(119866119909119910) and 119873

119866(119909) cap 119873

119866(119910) = 0

But 119887(119866) = 1 since 120574(119866 minus 119909119910) = 120574(119866119909119910) + 1 and 119887(119866119909119910) =120575(119867) + 1 byTheorem 66The gap between 119887(119866) and 119887(119866119909119910)is 120575(119867)

6 Results on Planar Graphs

From Section 2 we have seen that the bondage numberfor a tree has been completely solved Moreover a lineartime algorithm to compute the bondage number of a tree isdesigned by Hartnell et al [15] It is quite natural to considerthe bondage number for a planar graph In this section westate some results and problems on the bondage number fora planar graph

61 Conjecture on Bondage Number As mentioned inSection 3 the bondage number can be much larger than themaximum degree But for a planar graph 119866 the bondagenumber 119887(119866) cannot exceed Δ(119866) too much It is clear that119887(119866) ⩽ Δ(119866) + 4 by Corollary 15 since 120575(119866) ⩽ 5 for anyplanar graph119866 In 1998Dunbar et al [21] posed the followingconjecture

Conjecture 73 If 119866 is a planar graph then 119887(119866) ⩽ Δ(119866) + 1

Because of its attraction it immediately became the focusof attention when this conjecture is proposed In fact themain aim concerning the research of the bondage number fora planar graph is focused on this conjecture

It has been mentioned in Theorems 1 and 60 that119887(119862

3119896+1) = 3 = Δ + 1 and 119887(119862

4119896+2times 119870

2) = 4 = Δ + 1 It

is known that 119887(1198706minus 119872) = 5 = Δ + 1 where119872 is a perfect

matching of the complete graph1198706These examples show that

if Conjecture 73 is true then the upper bound is sharp for2 ⩽ Δ ⩽ 4

Here we note that it is sufficient to prove this conjecturefor connected planar graphs since the bondage number of adisconnected graph is simply the minimum of the bondagenumbers of its components

The first paper attacking this conjecture is due to Kangand Yuan [45] which confirmed the conjecture for everyconnected planar graph 119866 with Δ ⩾ 7 The proofs are mainlybased onTheorems 16 and 18

Theorem 74 (Kang and Yuan [45] 2000) If 119866 is a connectedplanar graph then 119887(119866) ⩽ min8 Δ(119866) + 2

Obviously in view of Theorem 74 Conjecture 73 is truefor any connected planar graph with Δ ⩾ 7

62 Bounds Implied by Degree Conditions As we have seenfrom Theorem 74 to attack Conjecture 73 we only need toconsider connected planar graphs with maximum Δ ⩽ 6Thus studying the bondage number of planar graphs bydegree conditions is of significanceThe first result on boundsimplied by degree conditions is obtained by Kang and Yuan[45]

International Journal of Combinatorics 13

Theorem 75 (Kang and Yuan [45] 2000) If 119866 is a connectedplanar graph without vertices of degree 5 then 119887(119866) ⩽ 7

Fischermann et al [46] generalized Theorem 75 as fol-lows

Theorem 76 (Fischermann et al [46] 2003) Let 119866 be aconnected planar graph and 119883 the set of vertices of degree 5which have distance at least 3 to vertices of degrees 1 2 and3 If all vertices in 119883 not adjacent with vertices of degree 4are independent and not adjacent to vertices of degree 6 then119887(119866) ⩽ 7

Clearly if119866 has no vertices of degree 5 then119883 = 0 ThusTheorem 76 yields Theorem 75

Use 119899119894to denote the number of vertices of degree 119894 in 119866

for each 119894 = 1 2 Δ(119866) Using Theorem 18 Fischermannet al obtained the following two theorems

Theorem 77 (Fischermann et al [46] 2003) For any con-nected planar graph 119866

(1) 119887(119866) ⩽ 7 if 1198995lt 2119899

2+ 3119899

3+ 2119899

4+ 12

(2) 119887(119866) ⩽ 6 if 119866 contains no vertices of degrees 4 and 5

Theorem 78 (Fischermann et al [46] 2003) For any con-nected planar graph 119866 119887(119866) ⩽ Δ(119866) + 1 if

(1) Δ(119866) = 6 and every edge 119890 = 119909119910 with 119889119866(119909) = 5 and

119889119866(119910) = 6 is contained in at most one triangle

(2) Δ(119866) = 5 and no triangle contains an edge 119890 = 119909119910 with119889119866(119909) = 5 and 4 ⩽ 119889

119866(119910) ⩽ 5

63 Bounds Implied by Girth Conditions The girth 119892(119866) of agraph 119866 is the length of the shortest cycle in 119866 If 119866 has nocycles we define 119892(119866) = infin

Combining Theorem 74 with Theorem 78 we find thatif a planar graph contains no triangles and has maximumdegree Δ ⩾ 5 then Conjecture 73 holds This fact motivatedFischermann et alrsquos [46] attempt to attack Conjecture 73 bygirth restraints

Theorem 79 (Fischermann et al [46] 2003) For any con-nected planar graph 119866

119887 (119866) ⩽

6 119894119891 119892 (119866) ⩾ 4

5 119894119891 119892 (119866) ⩾ 5

4 119894119891 119892 (119866) ⩾ 6

3 119894119891 119892 (119866) ⩾ 8

(20)

The first result in Theorem 79 shows that Conjecture 73is valid for all connected planar graphs with 119892(119866) ⩾ 4 andΔ(119866) ⩾ 5 It is easy to verify that the second result inTheorem 79 implies that Conjecture 73 is valid for all not3-regular graphs of girth 119892(119866) ⩾ 5 which is stated in thefollowing corollary

Corollary 80 For any connected planar graph 119866 if 119866 is not3-regular and 119892(119866) ⩾ 5 then 119887(119866) ⩽ Δ(119866) + 1

The first result in Theorem 79 also implies that if 119866 isa connected planar graph with no cycles of length 3 then119887(119866) ⩽ 6 Hou et al [47] improved this result as follows

Theorem 81 (Hou et al [47] 2011) For any connected planargraph 119866 if 119866 contains no cycles of length 119894 (4 ⩽ 119894 ⩽ 6) then119887(119866) ⩽ 6

Since 119887(1198623119896+1) = 3 for 119896 ⩾ 3 (see Theorem 1) the

last bound in Theorem 79 is tight Whether other bounds inTheorem 79 are tight remains open In 2003 Fischermann etal [46] posed the following conjecture

Conjecture 82 For any connected planar graph 119866 119887(119866) ⩽ 7and

119887 (119866) ⩽ 5 119894119891 119892 (119866) ⩾ 4

4 119894119891 119892 (119866) ⩾ 5(21)

We conclude this subsection with a question on bondagenumbers of planar graphs

Question 1 (Fischermann et al [46] 2003) Is there a planargraph 119866 with 6 ⩽ 119887(119866) ⩽ 8

In 2006 Carlson andDevelin [42] showed that the corona119866 = 119867 ∘ 119870

1for a planar graph 119867 with 120575(119867) = 5 has the

bondage number 119887(119866) = 120575(119867) + 1 = 6 (see Theorem 66)Since the minimum degree of planar graphs is at most 5then 119887(119866) can attach 6 If we take 119867 as the graph of theicosahedron then 119866 = 119867 ∘ 119870

1is such an example The

question for the existence of planar graphs with bondagenumber 7 or 8 remains open

64 Comments on the Conjectures Conjecture 73 is true forall connected planar graphs with minimum degree 120575 ⩽ 2 byTheorem 16 or maximum degree Δ ⩾ 7 by Theorem 74 ornot 120574-critical planar graphs by Theorem 13 Thus to attackConjecture 73 we only need to consider connected criticalplanar graphs with degree-restriction 3 ⩽ 120575 ⩽ Δ ⩽ 6

Recalling and anatomizing the proofs of all results men-tioned in the preceding subsections on the bondage numberfor connected planar graphs we find that the proofs of theseresults strongly depend upon Theorem 16 or Theorem 18 Inother words a basic way used in the proofs is to find twovertices 119909 and 119910 with distance at most two in a consideredplanar graph 119866 such that

119889119866(119909) + 119889

119866(119910) or 119889

119866(119909) + 119889

119866(119910) minus

1003816100381610038161003816119873119866(119909) cap 119873

119866(119910)1003816100381610038161003816

(22)

which bounds 119887(119866) is as small as possible Very recentlyHuang and Xu [44] have considered the following parameter

119861 (119866) = min119909119910isin119881(119866)

119889119909119910 1 ⩽ 119889

119866(119909 119910) ⩽ 2

cup 119889119909119910minus 119899

119909119910 119889 (119909 119910) = 1

(23)

where 119889119909119910= 119889

119866(119909) + 119889

119866(119910) minus 1 and 119899

119909119910= |119873

119866(119909) cap 119873

119866(119910)|

14 International Journal of Combinatorics

It follows fromTheorems 16 and 18 that

119887 (119866) ⩽ 119861 (119866) (24)

The proofs given in Theorems 74 and 79 indeed imply thefollowing stronger results

Theorem 83 If 119866 is a connected planar graph then

119861 (119866) ⩽

min 8 Δ (119866) + 26 119894119891 119892 (119866) ⩾ 4

5 119894119891 119892 (119866) ⩾ 5

4 119894119891 119892 (119866) ⩾ 6

3 119894119891 119892 (119866) ⩾ 8

(25)

Thus using Theorem 16 or Theorem 18 if we can proveConjectures 73 and 82 then we can prove the followingstatement

Statement If 119866 is a connected planar graph then

119861 (119866) ⩽

Δ (119866) + 1

7

5 if 119892 (119866) ⩾ 44 if 119892 (119866) ⩾ 5

(26)

It follows from (24) that Statement implies Conjectures 73and 82 However Huang and Xu [44] gave examples to showthat none of conclusions in Statement is true As a result theystated the following conclusion

Theorem 84 (Huang and Xu [44] 2012) It is not possible toprove Conjectures 73 and 82 if they are right using Theorems16 and 18

Therefore a new method is needed to prove or disprovethese conjectures At the present time one cannot prove theseconjectures and may consider to disprove them If one ofthese conjectures is invalid then there exists a minimumcounterexample 119866 with respect to 120592(119866) + 120576(119866) In order toobtain a minimum counterexample one may consider twosimple operations satisfying these requirements the edge-deletion and the edge-contractionwhich decrease 120592(119866)+120576(119866)and preserve planarity However applying Theorems 71 and72 Huang and Xu [44] presented the following results

Theorem 85 (Huang and Xu [44] 2012) It is not possible toconstruct minimum counterexamples to Conjectures 73 and 82by the operation of an edge-deletion or an edge-contraction

7 Results on Crossing Number Restraints

It is quite natural to generalize the known results on thebondage number for planar graphs to formore general graphsin terms of graph-theoretical parameters In this section weconsider graphs with crossing number restraints

The crossing number cr(119866) of 119866 is the smallest numberof pairwise intersections of its edges when 119866 is drawn in theplane If cr(119866) = 0 then 119866 is a planar graph

71 General Methods Use 119899119894to denote the number of vertices

of degree 119894 in119866 for 119894 = 1 2 Δ(119866) Huang and Xu obtainedthe following results

Theorem 86 (Huang and Xu [48] 2007) For any connectedgraph 119866

119887 (119866) ⩽

6 119894119891 119892 (119866) ⩾ 4 119886119899119889 2cr (119866) lt 121198861

5 119894119891 119892 (119866) ⩾ 5 119886119899119889 6cr (119866) lt 161198862

4 119894119891 119892 (119866) ⩾ 6 119886119899119889 4cr (119866) lt 141198863

3 119894119891 119892 (119866) ⩾ 8 119886119899119889 6cr (119866) lt 161198864

(27)

where 1198861= 119899

1+ 2119899

2+ 2119899

3+sum

Δ

119894=8(119894 minus 7)119899

119894+ 8 119886

2= 3119899

1+ 6119899

2+

51198993+sum

Δ

119894=7(3119894minus18)119899

119894+20 119886

3= 119899

1+2119899

2+sum

Δ

119894=6(2119894minus10)119899

119894+12

and 1198864= sum

Δ

119894=5(3119894 minus 12)119899

119894+ 16

Simplifying the conditions in Theorem 86 we obtain thefollowing corollaries immediately

Corollary 87 (Huang and Xu [48] 2007) For any connectedgraph 119866

119887 (119866) ⩽

6 119894119891 119892 (119866) ⩾ 4 119886119899119889 cr (119866) ⩽ 3

5 119894119891 119892 (119866) ⩾ 5 119886119899119889 cr (119866) ⩽ 4

4 119894119891 119892 (119866) ⩾ 6 119886119899119889 cr (119866) ⩽ 2

3 119894119891 119892 (119866) ⩾ 8 119886119899119889 cr (119866) ⩽ 2

(28)

Corollary 88 (Huang and Xu [48] 2007) For any connectedgraph 119866

(a) 119887(119866) ⩽ 6 if 119866 is not 4-regular cr(119866) = 4 and 119892(119866) ⩾ 4

(b) 119887(119866) ⩽ Δ(119866) + 1 if 119866 is not 3-regular cr(119866) ⩽ 4 and119892(119866) ⩾ 5

(c) 119887(119866) ⩽ 4 if 119866 is not 3-regular cr(119866) = 3 and 119892(119866) ⩾ 6

(d) 119887(119866) ⩽ 3 if cr(119866) = 3 119892(119866) ⩾ 8 and Δ(119866) ⩾ 5

These corollaries generalize some known results for pla-nar graphs For example Corollary 87 contains Theorem 79Corollary 88 (b) contains Corollary 80

Theorem 89 (Huang and Xu [48] 2007) For any connectedgraph 119866 if cr(119866) ⩽ 119899

3(119866) + 119899

4(119866) + 3 then 119887(119866) ⩽ 8

Corollary 90 (Huang and Xu [48] 2007) For any connectedgraph 119866 if cr(119866) ⩽ 3 then 119887(119866) ⩽ 8

Perhaps being unaware of this result in 2010 Ma et al[49] proved that 119887(119866) ⩽ 12 for any graph 119866 with cr(119866) = 1

Theorem91 (Huang and Xu [48] 2007) Let119866 be a connectedgraph and 119868 = 119909 isin 119881(119866) 119889

119866(119909) = 5 119889

119866(119909 119910) ⩾ 3 if

International Journal of Combinatorics 15

119889119866(119910) ⩽ 3 and 119889

119866(119910) = 4 for every 119910 isin 119873

119866(119909) If 119868 is

independent no vertices adjacent to vertices of degree 6 and

cr (119866) lt max5119899

3+ |119868| minus 2119899

4+ 28

117119899

3+ 40

16 (29)

then 119887(119866) ⩽ 7

Corollary 92 (Huang and Xu [48] 2007) Let 119866 be aconnected graph with cr(119866) ⩽ 2 If 119868 = 119909 isin 119881(119866) 119889

119866(119909) =

5 119889119866(119910 119909) ⩾ 3 if 119889

119866(119910) ⩽ 3 and 119889

119866(119910) = 4 for every 119910 isin

119873119866(119909) is independent and no vertices adjacent to vertices of

degree 6 then 119887(119866) ⩽ 7

Theorem93 (Huang andXu [48] 2007) Let119866 be a connectedgraph If 119866 satisfies

(a) 5cr(119866) + 1198995lt 2119899

2+ 3119899

3+ 2119899

4+ 12 or

(b) 7cr(119866) + 21198995lt 3119899

2+ 4119899

4+ 24

then 119887(119866) ⩽ 7

Proposition 94 (Huang and Xu [48] 2007) Let 119866 be aconnected graph with no vertices of degrees four and five Ifcr(119866) ⩽ 2 then 119887(119866) ⩽ 6

Theorem 95 (Huang and Xu [48] 2007) If 119866 is a connectedgraph with cr(119866) ⩽ 4 and not 4-regular when cr(119866) = 4 then119887(119866) ⩽ Δ(119866) + 2

The above results generalize some known results forplanar graphs For example Corollary 90 and Theorem 95contain Theorem 74 the first condition in Theorem 93 con-tains the second condition inTheorem 77

From Corollary 88 and Theorem 95 we suggest the fol-lowing questions

Question 2 Is there a

(a) 4-regular graph 119866 with cr(119866) = 4 such that 119887(119866) ⩾ 7

(b) 3-regular graph 119866 with cr ⩽ 4 and 119892(119866) ⩾ 5 such that119887(119866) = 5

(c) 3-regular graph 119866 with cr = 3 and 119892(119866) = 6 or 7 suchthat 119887(119866) = 5

72 Carlson-Develin Methods In this subsection we intro-duce an elegant method presented by Carlson and Develin[42] to obtain some upper bounds for the bondage numberof a graph

Suppose that 119866 is a connected graph We say that 119866 hasgenus 120588 if 119866 can be embedded in a surface 119878 with 120588 handlessuch that edges are pairwise disjoint except possibly for end-vertices Let119866 be an embedding of119866 in surface 119878 and let120601(119866)denote the number of regions in 119866 The boundary of everyregion contains at least three edges and every edge is on theboundary of atmost two regions (the two regions are identicalwhen 119890 is a cut-edge) For any edge 119890 of 119866 let 1199031

119866(119890) and 1199032

119866(119890)

be the numbers of edges comprising the regions in 119866 whichthe edge 119890 borders It is clear that every 119890 = 119909119910 isin 119864(119866)

1199032

119866(119890) ⩾ 119903

1

119866(119890) ⩾ 4 if 1003816100381610038161003816119873119866

(119909) cap 119873119866(119910)1003816100381610038161003816 = 0

1199032

119866(119890) ⩾ 4 119903

1

119866(119890) ⩾ 3 if 1003816100381610038161003816119873119866

(119909) cap 119873119866(119910)1003816100381610038161003816 = 1

1199032

119866(119890) ⩾ 119903

1

119866(119890) ⩾ 3 if 1003816100381610038161003816119873119866

(119909) cap 119873119866(119910)1003816100381610038161003816 ⩾ 2

(30)

Following Carlson and Develin [42] for any edge 119890 = 119909119910 of119866 we define

119863119866(119890) =

1

119889119866(119909)+

1

119889119866(119910)

+1

1199031

119866(119890)+

1

1199032

119866(119890)minus 1 (31)

By the well-known Eulerrsquos formula

120592 (119866) minus 120576 (119866) + 120601 (119866) = 2 minus 2120588 (119866) (32)

it is easy to see that

sum

119890isin119864(119866)

119863119866(119890) = 120592 (119866) minus 120576 (119866) + 120601 (119866) = 2 minus 2120588 (119866) (33)

If 119866 is a planar graph that is 120588(119866) = 0 then

sum

119890isin119864(119866)

119863119866(119890) = 120592 (119866) minus 120576 (119866) + 120601 (119866) = 2 (34)

Combining these formulas withTheorem 18 Carlson andDevelin [42] gave a simple and intuitive proof ofTheorem 74and obtained the following result

Theorem 96 (Carlson and Develin [42] 2006) Let 119866 be aconnected graph which can be embedded on a torus Then119887(119866) ⩽ Δ(119866) + 3

Recently Hou and Liu [50] improved Theorem 96 asfollows

Theorem 97 (Hou and Liu [50] 2012) Let 119866 be a connectedgraph which can be embedded on a torus Then 119887(119866) ⩽ 9Moreover if Δ(119866) = 6 then 119887(119866) ⩽ 8

By the way Cao et al [51] generalized the result inTheorem 79 to a connected graph 119866 that can be embeddedon a torus that is

119887 (119866) le

6 if 119892 (119866) ⩾ 4 and 119866 is not 4-regular5 if 119892 (119866) ⩾ 54 if 119892 (119866) ⩾ 6 and 119866 is not 3-regular3 if 119892 (119866) ⩾ 8

(35)

Several authors used this method to obtain some resultson the bondage number For example Fischermann et al[46] used this method to prove the second conclusion inTheorem 78 Recently Cao et al [52] have used thismethod todeal with more general graphs with small crossing numbersFirst they found the following property

Lemma 98 120575(119866) ⩽ 5 for any graph 119866 with cr(119866) ⩽ 5

16 International Journal of Combinatorics

This result implies that 119887(119866) ⩽ Δ(119866) + 4 by Corollary 15for any graph 119866 with cr(119866) ⩽ 5 Cao et al established thefollowing relation in terms of maximum planar subgraphs

Lemma 99 (Cao et al [52]) Let 119866 be a connected graph withcrossing number cr(119866) For any maximum planar subgraph119867of 119866

sum

119890isin119864(119867)

119863119866(119890) ⩾ 2 minus

2cr (119866)120575 (119866)

(36)

Using Carlson-Develin method and combiningLemma 98 with Lemma 99 Cao et al proved the followingresults

Theorem 100 (Cao et al [52]) For any connected graph 119866119887(119866) ⩽ Δ(119866) + 2 if 119866 satisfies one of the following conditions

(a) cr(119866) ⩽ 3(b) cr(119866) = 4 and 119866 is not 4-regular(c) cr(119866) = 5 and 119866 contains no vertices of degree 4

Theorem101 (Cao et al [52]) Let119866 be a connected graphwithΔ(119866) = 5 and cr(119866) ⩽ 4 If no triangles contain two vertices ofdegree 5 then 119887(119866) ⩽ 6 = Δ(119866) + 1

Theorem 102 (Cao et al [52]) Let 119866 be a connected graphwith Δ(119866) ⩾ 6 and cr(119866) ⩽ 3 If Δ(119866) ⩾ 7 or if Δ(119866) = 6120575(119866) = 3 and every edge 119890 = 119909119910 with 119889

119866(119909) = 5 and 119889

119866(119910) = 6

is contained in atmost one triangle then 119887(119866) ⩽ min8 Δ(119866)+1

Using Carlson-Develin method it can be proved that119887(119866) ⩽ Δ(119866) + 2 if cr(119866) ⩽ 3 (see the first conclusion inTheorem 100) and not yet proved that 119887(119866) ⩽ 8 if cr(119866) ⩽3 although it has been proved by using other method (seeCorollary 90)

By using Carlson-Develin method Samodivkin [25]obtained some results on the bondage number for graphswithsome given properties

Kim [53] showed that 119887(119866) ⩽ Δ(119866) + 2 for a connectedgraph 119866 with genus 1 Δ(119866) ⩽ 5 and having a toroidalembedding of which at least one region is not 4-sidedRecently Gagarin and Zverovich [54] further have extendedCarlson-Develin ideas to establish a nice upper bound forarbitrary graphs that can be embedded on orientable ornonorientable topological surface

Theorem 103 (Gagarin and Zverovich [54] 2012) Let 119866 bea graph embeddable on an orientable surface of genus ℎ and anonorientable surface of genus 119896 Then 119887(119866) ⩽ minΔ(119866)+ℎ+2 Δ(119866) + 119896 + 1

This result generalizes the corresponding upper boundsin Theorems 74 and 96 for any orientable or nonori-entable topological surface By investigating the proof ofTheorem 103 Huang [55] found that the issue of the ori-entability can be avoided by using the Euler characteristic120594(= 120592(119866) minus 120576(119866) + 120601(119866)) instead of the genera ℎ and 119896 the

relations are 120594 = 2minus2ℎ and 120594 = 2minus119896 To obtain the best resultfromTheorem 103 onewants ℎ and 119896 as small as possible thisis equivalent to having 120594 as large as possible

According toTheorem 103 if119866 is planar (ℎ = 0 120594 = 2) orcan be embedded on the real projective plane (119896 = 1 120594 = 1)then 119887(119866) ⩽ Δ(119866) + 2 In all other cases Huang [55] had thefollowing improvement for Theorem 103 the proof is basedon the technique developed byCarlson-Develin andGagarin-Zverovich and includes some elementary calculus as a newingredient mainly the intermediate-value theorem and themean-value theorem

Theorem 104 (Huang [55] 2012) Let 119866 be a graph embed-dable on a surface whose Euler characteristic 120594 is as large aspossible If 120594 ⩽ 0 then 119887(119866) ⩽ Δ(119866)+lfloor119903rfloor where 119903 is the largestreal root of the following cubic equation in 119911

1199113

+ 21199112

+ (6120594 minus 7) 119911 + 18120594 minus 24 = 0 (37)

In addition if 120594 decreases then 119903 increases

The following result is asymptotically equivalent toTheorem 104

Theorem 105 (Huang [55] 2012) Let 119866 be a graph embed-dable on a surface whose Euler characteristic 120594 is as large aspossible If 120594 ⩽ 0 then 119887(119866) lt Δ(119866) + radic12 minus 6120594 + 12 orequivalently 119887(119866) ⩽ Δ(119866) + lceilradic12 minus 6120594 minus 12rceil

Also Gagarin andZverovich [54] indicated that the upperbound in Theorem 103 can be improved for larger values ofthe genera ℎ and 119896 by adjusting the proofs and stated thefollowing general conjecture

Conjecture 106 (Gagarin and Zverovich [54] 2012) For aconnected graph G of orientable genus ℎ and nonorientablegenus 119896

119887 (119866) ⩽ min 119888ℎ 119888

1015840

119896 Δ (119866) + 119900 (ℎ) Δ (119866) + 119900 (119896) (38)

where 119888ℎand 1198881015840

119896are constants depending respectively on the

orientable and nonorientable genera of 119866

In the recent paper Gagarin and Zverovich [56] providedconstant upper bounds for the bondage number of graphs ontopological surfaces which can be used as the first estimationfor the constants 119888

ℎand 1198881015840

119896of Conjecture 106

Theorem 107 (Gagarin and Zverovich [56] 2013) Let 119866 bea connected graph of order 119899 If 119866 can be embedded on thesurface of its orientable or nonorientable genus of the Eulercharacteristic 120594 then

(a) 120594 ⩾ 1 implies 119887(119866) ⩽ 10(b) 120594 ⩽ 0 and 119899 gt minus12120594 imply 119887(119866) ⩽ 11(c) 120594 ⩽ minus1 and 119899 ⩽ minus12120594 imply 119887(119866) ⩽ 11 +

3120594(radic17 minus 8120594 minus 3)(120594 minus 1) = 11 + 119874(radicminus120594)

Theorem 79 shows that a connected planar triangle-freegraph 119866 has 119887(119866) ⩽ 6 The following result generalizes to itall the other topological surfaces

International Journal of Combinatorics 17

Theorem 108 (Gagarin and Zverovich [56] 2013) Let 119866 be aconnected triangle-free graph of order 119899 If 119866 can be embeddedon the surface of its orientable or nonorientable genus of theEuler characteristic 120594 then

(a) 120594 ⩾ 1 implies 119887(119866) ⩽ 6(b) 120594 ⩽ 0 and 119899 gt minus8120594 imply 119887(119866) ⩽ 7(c) 120594 ⩽ minus1 and 119899 ⩽ minus8120594 imply 119887(119866) ⩽ 7minus4120594(1+radic2 minus 120594)

In [56] the authors also explicitly improved upperbounds in Theorem 103 and gave tight lower bounds forthe number of vertices of graphs 2-cell embeddable ontopological surfaces of a given genusThe interested reader isreferred to the original paper for details The following resultis interesting although its proof is easy

Theorem 109 (Samodivkin [57]) Let119866 be a graph with order119899 and girth 119892 If 119866 is embeddable on a surface whose Eulercharacteristic 120594 is as large as possible then

119887 (119866) ⩽ 3 +8

119892 minus 2minus

4120594119892

119899 (119892 minus 2) (39)

If 119866 is a planar graph with girth 119892 ⩾ 4 + 119894 for any 119894 isin0 1 2 then Theorem 109 leads to 119887(119866) ⩽ 6 minus 119894 which isthe result inTheorem 79 obtained by Fischermann et al [46]Also inmany cases the bound stated inTheorem 109 is betterthan those given byTheorems 103 and 105

8 Conditional Bondage Numbers

Since the concept of the bondage number is based upon thedomination all sorts of dominations which are variationsof the normal domination by adding restricted conditionsto the normal dominating set can develop a new ldquobondagenumberrdquo as long as a variation of domination number isgiven In this section we survey results on the bondagenumber under some restricted conditions

81 Total Bondage Numbers A dominating set 119878 of a graph119866 without isolated vertices is called total if the subgraphinduced by 119878 contains no isolated vertices The minimumcardinality of a total dominating set is called the totaldomination number of 119866 and denoted by 120574

119879(119866) It is clear

that 120574(119866) ⩽ 120574119879(119866) ⩽ 2120574(119866) for any graph 119866 without isolated

verticesThe total domination in graphs was introduced by Cock-

ayne et al [58] in 1980 Pfaff et al [59 60] in 1983 showedthat the problem determining total domination number forgeneral graphs is NP-complete even for bipartite graphs andchordal graphs Even now total domination in graphs hasbeen extensively studied in the literature In 2009 Henning[61] gave a survey of selected recent results on total domina-tion in graphs

The total bondage number of 119866 denoted by 119887119879(119866) is the

smallest cardinality of a subset 119861 sube 119864(119866) with the propertythat119866minus119861 contains no isolated vertices and 120574

119879(119866minus119861) gt 120574

119879(119866)

From definition 119887119879(119866)may not exist for some graphs for

example119866 = 1198701119899 We put 119887

119879(119866) = infin if 119887

119879(119866) does not exist

It is easy to see that 119887119879(119866) is finite for any connected graph 119866

other than 1198701 119870

2 119870

3 and 119870

1119899 In fact for such a graph 119866

since there is a path of length 3 in 119866 we can find 1198611sube 119864(119866)

such that 119866 minus 1198611is a spanning tree 119879 of 119866 containing a path

of length 3 So 120574119879(119879) = 120574

119879(119866minus119861

1) ⩾ 120574

119879(119866) For the tree119879 we

find 1198612sube 119864(119879) such that 120574

119879(119879 minus 119861

2) gt 120574

119879(119879) ⩾ 120574

119879(119866) Thus

we have 120574119879(119866minus119861

2minus119861

1) gt 120574

119879(119866) and so 119887

119879(119866) ⩽ |119861

1|+|119861

2| In

the following discussion we always assume that 119887119879(119866) exists

when a graph 119866 is mentionedIn 1991 Kulli and Patwari [62] first studied the total

bondage number of a graph and calculated the exact valuesof 119887

119879(119866) for some standard graphs

Theorem 110 (Kulli and Patwari [62] 1991) For 119899 ⩾ 4

119887119879(119862

119899) =

3 119894119891 119899 equiv 2 (mod 4) 2 119900119905ℎ119890119903119908119894119904119890

119887119879(119875

119899) =

2 119894119891 119899 equiv 2 (mod 4) 1 119900119905ℎ119890119903119908119894119904119890

119887119879(119870

119898119899) = 119898 119908119894119905ℎ 2 ⩽ 119898 ⩽ 119899

119887119879(119870

119899) =

4 119891119900119903 119899 = 4

2119899 minus 5 119891119900119903 119899 ⩾ 5

(40)

Recently Hu et al [63] have obtained some results on thetotal bondage number of the Cartesian product119875

119898times119875

119899of two

paths 119875119898and 119875

119899

Theorem 111 (Hu et al [63] 2012) For the Cartesian product119875119898times 119875

119899of two paths 119875

119898and 119875

119899

119887119879(119875

2times 119875

119899) =

1 119894119891 119899 equiv 0 (mod 3) 2 119894119891 119899 equiv 2 (mod 3) 3 119894119891 119899 equiv 1 (mod 3)

119887119879(119875

3times 119875

119899) = 1

119887119879(119875

4times 119875

119899)

= 1 119894119891 119899 equiv 1 (mod 5) = 2 119894119891 119899 equiv 4 (mod 5) ⩽ 3 119894119891 119899 equiv 2 (mod 5) ⩽ 4 119894119891 119899 equiv 0 3 (mod 5)

(41)

Generalized Petersen graphs are an important class ofcommonly used interconnection networks and have beenstudied recently By constructing a family of minimum totaldominating sets Cao et al [64] determined the total bondagenumber of the generalized Petersen graphs

FromTheorem 6 we know that 119887(119879) ⩽ 2 for a nontrivialtree 119879 But given any positive integer 119896 Sridharan et al [65]constructed a tree 119879 for which 119887

119879(119879) = 119896 Let119867

119896be the tree

obtained from a star1198701119896+1

by subdividing 119896 edges twice Thetree 119867

7is shown in Figure 6 It can be easily verified that

119887119879(119867

119896) = 119896 This fact can be stated as the following theorem

Theorem 112 (Sridharan et al [65] 2007) For any positiveinteger 119896 there exists a tree 119879 with 119887

119879(119879) = 119896

18 International Journal of Combinatorics

Figure 61198677

Combining Theorem 1 with Theorem 112 we have whatfollows

Corollary 113 (Sridharan et al [65] 2007) The bondagenumber and the total bondage number are unrelated even fortrees

However Sridharan et al [65] gave an upper bound onthe total bondage number of a tree in terms of its maximumdegree For any tree 119879 of order 119899 if 119879 =119870

1119899minus1 then 119887

119879(119879) =

minΔ(119879) (13)(119899 minus 1) Rad and Raczek [66] improved thisupper bound and gave a constructive characterization of acertain class of trees attaching the upper bound

Theorem 114 (Rad and Raczek [66] 2011) 119887119879(119879) ⩽ Δ(119879) minus 1

for any tree 119879 with maximum degree at least three

In general the decision problem for 119887119879(119866) is NP-complete

for any graph 119866 We state the decision problem for the totalbondage as follows

Problem 3 Consider the decision problemTotal Bondage ProblemInstance a graph 119866 and a positive integer 119887Question is 119887

119879(119866) ⩽ 119887

Theorem 115 (Hu and Xu [16] 2012) The total bondageproblem is NP-complete

Consequently in view of its computational hardnessit is significative to establish some sharp bounds on thetotal bondage number of a graph in terms of other graphicparameters In 1991 Kulli and Patwari [62] showed that119887119879(119866) ⩽ 2119899 minus 5 for any graph 119866 with order 119899 ⩾ 5 Sridharan

et al [65] improved this result as follows

Theorem 116 (Sridharan et al [65] 2007) For any graph 119866with order 119899 if 119899 ⩾ 5 then

119887119879(119866) ⩽

1

3(119899 minus 1) 119894119891 119866 119888119900119899119905119886119894119899119904 119899119900 119888119910119888119897119890119904

119899 minus 1 119894119891 119892 (119866) ⩾ 5

119899 minus 2 119894119891 119892 (119866) = 4

(42)

Rad and Raczek [66] also established some upperbounds on 119887

119879(119866) for a general graph 119866 In particular they

gave an upper bound on 119887119879(119866) in terms of the girth of

119866

Theorem 117 (Rad and Raczek [66] 2011) 119887119879(119866)⩽4Δ(119866) minus 5

for any graph 119866 with 119892(119866) ⩾ 4

82 Paired Bondage Numbers A dominating set 119878 of 119866 iscalled to be paired if the subgraph induced by 119878 containsa perfect matching The paired domination number of 119866denoted by 120574

119875(119866) is the minimum cardinality of a paired

dominating set of 119866 Clearly 120574119879(119866) ⩽ 120574

119875(119866) for every

connected graph 119866 with order at least two where 120574119879(119866) is

the total domination number of 119866 and 120574119875(119866) ⩽ 2120574(119866) for

any graph119866without isolated vertices Paired domination wasintroduced by Haynes and Slater [67 68] and further studiedin [69ndash72]

The paired bondage number of 119866 with 120575(119866) ⩾ 1 denotedby 119887P(119866) is the minimum cardinality among all sets of edges119861 sube 119864 such that 120575(119866 minus 119861) ⩾ 1 and 120574

119875(119866 minus 119861) gt 120574

119875(119866)

The concept of the paired bondage number was firstproposed by Raczek [73] in 2008The following observationsfollow immediately from the definition of the paired bondagenumber

Observation 2 Let 119866 be a graph with 120575(119866) ⩾ 1

(a) If 119867 is a subgraph of 119866 such that 120575(119867) ⩾ 1 then119887119875(119867) ⩽ 119887

119875(119866)

(b) If 119867 is a subgraph of 119866 such that 119887119875(119867) = 1 and 119896

is the number of edges removed to form119867 then 1 ⩽119887119875(119866) ⩽ 119896 + 1

(c) If 119909119910 isin 119864(119866) such that 119889119866(119909) 119889

119866(119910) gt 1 and 119909119910

belongs to each perfect matching of each minimumpaired dominating set of 119866 then 119887

119875(119866) = 1

Based on these observations 119887119875(119875

119899) ⩽ 2 In fact the

paired bondage number of a path has been determined

Theorem 118 (Raczek [73] 2008) For any positive integer 119896if 119899 ⩾ 2 then

119887119875(119875

119899) =

0 119894119891 119899 = 2 3 119900119903 5

1 119894119891 119899 = 4119896 4119896 + 3 119900119903 4119896 + 6

2 119900119905ℎ119890119903119908119894119904119890

(43)

For a cycle 119862119899of order 119899 ⩾ 3 since 120574

119875(119862

119899) = 120574

119875(119875

119899) from

Theorem 118 the following result holds immediately

Corollary 119 (Raczek [73] 2008) For any positive integer 119896if 119899 ⩾ 3 then

119887119875(119862

119899) =

0 119894119891 119899 = 3 119900119903 5

2 119894119891 119899 = 4119896 4119896 + 3 119900119903 4119896 + 6

3 119900119905ℎ119890119903119908119894119904119890

(44)

A wheel 119882119899 where 119899 ⩾ 4 is a graph with 119899 vertices

formed by connecting a single vertex to all vertices of a cycle119862119899minus1

Of course 120574119875(119882

119899) = 2

International Journal of Combinatorics 19

119896

Figure 7 A tree 119879 with 119887119875(119879) = 119896

Theorem 120 (Raczek [73] 2008) For positive integers 119899 and119898

119887119875(119870

119898119899) = 119898 119891119900119903 1 lt 119898 ⩽ 119899

119887119875(119882

119899) =

4 119894119891 119899 = 4

3 119894119891 119899 = 5

2 119900119905ℎ119890119903119908119894119904119890

(45)

Theorem 16 shows that if 119879 is a nontrivial tree then119887(119879) ⩽ 2 However no similar result exists for pairedbondage For any nonnegative integer 119896 let 119879

119896be a tree

obtained by subdividing all but one edge of a star 1198701119896+1

(asshown in Figure 7) It is easy to see that 119887

119875(119879

119896) = 119896 We

state this fact as the following theorem which is similar toTheorem 112

Theorem121 (Raczek [73] 2008) For any nonnegative integer119896 there exists a tree 119879 with 119887

119875(119879) = 119896

Consider the tree defined in Figure 5 for 119896 = 3 Then119887(119879

3) ⩽ 2 and 119887

119875(119879

3) = 3 On the other hand 119887(119875

4) = 2

and 119887119875(119875

4) = 1 Thus we have what follows which is similar

to Corollary 113

Corollary 122 The bondage number and the paired bondagenumber are unrelated even for trees

A constructive characterization of trees with 119887(119879) = 2is given by Hartnell and Rall in [12] Raczek [73] provideda constructive characterization of trees with 119887

119875(119879) = 0 In

order to state the characterization we define a labeling andthree simple operations on a tree119879 Let 119910 isin 119881(119879) and let ℓ(119910)be the label assigned to 119910

Operation T1 If ℓ(119910) = 119861 add a vertex 119909 and the

edge 119910119909 and let ℓ(119909) = 119860OperationT

2 If ℓ(119910) = 119862 add a path (119909

1 119909

2) and the

edge 1199101199091 and let ℓ(119909

1) = 119861 ℓ(119909

2) = 119860

Operation T3 If ℓ(119910) = 119861 add a path (119909

1 119909

2 119909

3)

and the edge 1199101199091 and let ℓ(119909

1) = 119862 ℓ(119909

2) = 119861 and

ℓ(1199093) = 119860

Let 1198752= (119906 V) with ℓ(119906) = 119860 and ℓ(V) = 119861 Let T be

the class of all trees obtained from the labeled 1198752by a finite

sequence of operationsT1 T

2 T

3

A tree 119879 in Figure 8 belongs to the familyTRaczek [73] obtained the following characterization of all

trees 119879 with 119887119875(119879) = 0

119860

119860

119860

119860

119860

119861 119862

119861 119862 119861

119861119860

Figure 8 A tree 119879 belong to the familyT

Theorem 123 (Raczek [73] 2008) For a tree 119879 119887119875(119879) = 0 if

and only if 119879 is inT

We state the decision problem for the paired bondage asfollows

Problem 4 Consider the decision problem

Paired Bondage ProblemInstance a graph 119866 and a positive integer 119887Question is 119887

119875(119866) ⩽ 119887

Conjecture 124 The paired bondage problem is NP-complete

83 Restrained Bondage Numbers A dominating set 119878 of 119866 iscalled to be restrained if the subgraph induced by 119878 containsno isolated vertices where 119878 = 119881(119866) 119878 The restraineddomination number of 119866 denoted by 120574

119877(119866) is the smallest

cardinality of a restrained dominating set of 119866 It is clear that120574119877(119866) exists and 120574(119866) ⩽ 120574

119877(119866) for any nonempty graph 119866

The concept of restrained domination was introducedby Telle and Proskurowski [74] in 1997 albeit indirectly asa vertex partitioning problem Concerning the study of therestrained domination numbers the reader is referred to [74ndash79]

In 2008 Hattingh and Plummer [80] defined therestrained bondage number 119887R(119866) of a nonempty graph 119866 tobe the minimum cardinality among all subsets 119861 sube 119864(119866) forwhich 120574

119877(119866 minus 119861) gt 120574

119877(119866)

For some simple graphs their restrained dominationnumbers can be easily determined and so restrained bondagenumbers have been also determined For example it is clearthat 120574

119877(119870

119899) = 1 Domke et al [77] showed that 120574

119877(119862

119899) =

119899 minus 2lfloor1198993rfloor for 119899 ⩾ 3 and 120574119877(119875

119899) = 119899 minus 2lfloor(119899 minus 1)3rfloor for 119899 ⩾ 1

Using these results Hattingh and Plummer [80] obtained therestricted bondage numbers for119870

119899 119862

119899 119875

119899 and119866 = 119870

11989911198992119899119905

as follows

Theorem 125 (Hattingh and Plummer [80] 2008) For 119899 ⩾ 3

119887R (119870119899) =

1 119894119891 119899 = 3

lceil119899

2rceil 119900119905ℎ119890119903119908119894119904119890

119887R (119862119899) =

1 119894119891 119899 equiv 0 (mod 3) 2 119900119905ℎ119890119903119908119894119904119890

119887R (119875119899) = 1 119899 ⩾ 4

20 International Journal of Combinatorics

119887R (119866)

=

lceil119898

2rceil 119894119891 119899

119898= 1 119886119899119889 119899

119898+1⩾ 2 (1 ⩽ 119898 lt 119905)

2119905 minus 2 119894119891 1198991= 119899

2= sdot sdot sdot = 119899

119905= 2 (119905 ⩾ 2)

2 119894119891 1198991= 2 119886119899119889 119899

2⩾ 3 (119905 = 2)

119905minus1

sum

119894=1

119899119894minus 1 119900119905ℎ119890119903119908119894119904119890

(46)

Theorem 126 (Hattingh and Plummer [80] 2008) For a tree119879 of order 119899 (⩾ 4) 119879 ≇ 119870

1119899minus1if and only if 119887R(119879) = 1

Theorem 126 shows that the restrained bondage numberof a tree can be computed in constant time However thedecision problem for 119887R(119866) is NP-complete even for bipartitegraphs

Problem 5 Consider the decision problem

Restrained Bondage Problem

Instance a graph 119866 and a positive integer 119887

Question is 119887R(119866) ⩽ 119887

Theorem 127 (Hattingh and Plummer [80] 2008) Therestrained bondage problem is NP-complete even for bipartitegraphs

Consequently in view of its computational hardnessit is significative to establish some sharp bounds on therestrained bondage number of a graph in terms of othergraphic parameters

Theorem 128 (Hattingh and Plummer [80] 2008) For anygraph 119866 with 120575(119866) ⩾ 2

119887R (119866) ⩽ min119909119910isin119864(119866)

119889119866(119909) + 119889

119866(119910) minus 2 (47)

Corollary 129 119887R(119866) ⩽ Δ(119866)+120575(119866)minus2 for any graph119866 with120575(119866) ⩾ 2

Notice that the bounds stated in Theorem 128 andCorollary 129 are sharp Indeed the class of cycles whoseorders are congruent to 1 2 (mod 3) have a restrained bondagenumber achieving these bounds (see Theorem 125)

Theorem 128 is an analogue of Theorem 14 A quitenatural problem is whether or not for restricted bondagenumber there are analogues of Theorem 16 Theorem 18Theorem 29 and so on Indeed we should consider all resultsfor the bondage number whether or not there are analoguesfor restricted bondage number

Theorem 130 (Hattingh and Plummer [80] 2008) For anygraph 119866 with 120574

119877(119866) = 2

119887R (119866) ⩽ Δ (119866) + 1 (48)

We now consider the relation between 119887119877(119866) and 119887(119866)

Theorem 131 (Hattingh and Plummer [80] 2008) If 120574119877(119866) =

120574(119866) for some graph 119866 then 119887R(119866) ⩽ 119887(119866)

Corollary 129 and Theorem 131 provide a possibility toattack Conjecture 73 by characterizing planar graphs with120574119877= 120574 and 119887R = 119887 when 3 ⩽ Δ ⩽ 6However we do not have 119887R(119866) = 119887(119866) for any graph

119866 even if 120574119877(119866) = 120574(119866) Observe that 120574

119877(119870

3) = 120574(119870

3) yet

119887119877(119870

3) = 1 and 119887(119870

3) = 2 We still may not claim that

119887119877(119866) = 119887(119866) even in the case that every 120574-set is a 120574

119877-set

The example 1198703again demonstrates this

Theorem 132 (Hattingh and Plummer [80] 2008) There isan infinite class of graphs inwhich each graph119866 satisfies 119887(119866) lt119887R(119866)

In fact such an infinite class of graphs can be obtained asfollows Let119867 be a connected graph BC(119867) a graph obtainedfrom 119867 by attaching ℓ (⩾ 2) vertices of degree 1 to eachvertex in 119867 and B = 119866 119866 = BC(119867) for some graph 119867such that 120575(119867) ⩾ 2 By Theorem 3 we can easily check that119887(119866) = 1 lt 2 ⩽ min120575(119867) ℓ = 119887R(119866) for any 119866 isin BMoreover there exists a graph 119866 isin B such that 119887R(119866) canbe much larger than 119887(119866) which is stated as the followingtheorem

Theorem 133 (Hattingh and Plummer [80] 2008) For eachpositive integer 119896 there is a graph119866 such that 119896 = 119887R(119866)minus119887(119866)

Combining Theorem 133 with Theorem 132 we have thefollowing corollary

Corollary 134 The bondage number and the restrainedbondage number are unrelated

Very recently Jafari Rad et al [81] have consideredthe total restrained bondage in graphs based on both totaldominating sets and restrained dominating sets and obtainedseveral properties exact values and bounds for the totalrestrained bondage number of a graph

A restrained dominating set 119878 of a graph 119866 without iso-lated vertices is called to be a total restrained dominating setif 119878 is a total dominating set The total restrained dominationnumber of119866 denoted by 120574TR (119866) is theminimum cardinalityof a total restrained dominating set of 119866 For referenceson this domination in graphs see [3 61 82ndash85] The totalrestrained bondage number 119887TR (119866) of a graph 119866 with noisolated vertex is the cardinality of a smallest set of edges 119861 sube119864(119866) for which119866minus119861 has no isolated vertex and 120574TR (119866minus119861) gt120574TR (119866) In the case that there is no such subset 119861 we define119887TR (119866) = infin

Ma et al [84] determined that 120574TR (119875119899) = 119899minus2lfloor(119899minus2)4rfloorfor 119899 ⩾ 3 120574TR (119862119899

) = 119899 minus 2lfloor1198994rfloor for 119899 ⩾ 3 120574TR (1198703) =

3 and 120574TR (119870119899) = 2 for 119899 = 3 and 120574TR (1198701119899

) = 1 + 119899

and 120574TR (119870119898119899) = 2 for 2 ⩽ 119898 ⩽ 119899 According to these

results Jafari Rad et al [81] determined the exact valuesof total restrained bondage numbers for the correspondinggraphs

International Journal of Combinatorics 21

Theorem 135 (Jafari Rad et al [81]) For an integer 119899

119887TR (119875119899) = infin 119894119891 2 ⩽ 119899 ⩽ 5

1 119894119891 119899 ⩾ 5

119887TR (119862119899) =

infin 119894119891 119899 = 3

1 119894119891 3 lt 119899 119899 equiv 0 1 (mod 4)2 119894119891 3 lt 119899 119899 equiv 2 3 (mod 4)

119887TR (119882119899) = 2 119891119900119903 119899 ⩾ 6

119887TR (119870119899) = 119899 minus 1 119891119900119903 119899 ⩾ 4

119887TR (119870119898119899) = 119898 minus 1 119891119900119903 2 ⩽ 119898 ⩽ 119899

(49)

119887TR (119870119899) = 119899minus1 shows the fact that for any integer 119899 ⩾ 3 there

exists a connected graph namely119870119899+1

such that 119887TR (119870119899+1) =

119899 that is the total restrained bondage number is unboundedfrom the above

A vertex is said to be a support vertex if it is adjacent toa vertex of degree one Let A be the family of all connectedgraphs such that119866 belongs toA if and only if every edge of119866is incident to a support vertex or 119866 is a cycle of length three

Proposition 136 (Cyman and Raczek [82] 2006) For aconnected graph119866 of order 119899 120574TR (119866) = 119899 if and only if119866 isin A

Hence if 119866 isin A then the removal of any subset of edgesof119866 cannot increase the total restrained domination numberof 119866 and thus 119887TR (119866) = infin When 119866 notin A a result similar toTheorem 6 is obtained

Theorem 137 (Jafari Rad et al [81]) For any tree 119879 notin A119887TR (119879) ⩽ 2 This bound is sharp

In the samepaper the authors provided a characterizationfor trees 119879 notin A with 119887TR (119879) = 1 or 2 The following result issimilar to Observation 1

Observation 3 (Jafari Rad et al [81]) If 119896 edges can beremoved from a graph 119866 to obtain a graph 119867 without anisolated vertex and with 119887TR (119867) = 119905 then 119887TR (119866) ⩽ 119896 + 119905

ApplyingTheorem 137 andObservation 3 Jafari Rad et alcharacterized all connected graphs 119866 with 119887TR (119866) = infin

Theorem 138 (Jafari Rad et al [81]) For a connected graph119866119887TR (119866) = infin if and only if 119866 isin A

84 Other Conditional Bondage Numbers A subset 119868 sube 119881(119866)is called an independent set if no two vertices in 119868 are adjacentin 119866 The maximum cardinality among all independent setsis called the independence number of 119866 denoted by 120572(119866)

A dominating set 119878 of a graph 119866 is called to be inde-pendent if 119878 is an independent set of 119866 The minimumcardinality among all independent dominating set is calledthe independence domination number of 119866 and denoted by120574119868(119866)

Since an independent dominating set is not only adominating set but also an independent set 120574(119866) ⩽ 120574

119868(119866) ⩽

120572(119866) for any graph 119866It is clear that a maximal independent set is certainly a

dominating set Thus an independent set is maximal if andonly if it is an independent dominating set and so 120574

119868(119866) is the

minimum cardinality among all maximal independent setsof 119866 This graph-theoretical invariant has been well studiedin the literature see for example Haynes et al [3] for earlyresults Goddard and Henning [86] for recent results onindependent domination in graphs

In 2003 Zhang et al [87] defined the independencebondage number 119887

119868(119866) of a nonempty graph 119866 to be the

minimum cardinality among all subsets 119861 sube 119864(119866) for which120574119868(119866 minus 119861) gt 120574

119868(119866) For some ordinary graphs their indepen-

dence domination numbers can be easily determined and soindependence bondage numbers have been also determinedClearly 119887

119868(119870

119899) = 1 if 119899 ⩾ 2

Theorem 139 (Zhang et al [87] 2003) For any integer 119899 ⩾ 4

119887119868(119862

119899) =

1 119894119891 119899 equiv 1 (mod 2) 2 119894119891 119899 equiv 0 (mod 2)

119887119868(119875

119899) =

1 119894119891 119899 equiv 0 (mod 2) 2 119894119891 119899 equiv 1 (mod 2)

119887119868(119870

119898119899) = 119899 119891119900119903 119898 ⩽ 119899

(50)

Apart from the above-mentioned results as far as we findno other results on the independence bondage number in thepresent literature we never knew much of any result on thisparameter for a tree

A subset 119878 of 119881(119866) is called an equitable dominatingset if for every 119910 isin 119881(119866 minus 119878) there exists a vertex 119909 isin 119878such that 119909119910 isin 119864(119866) and |119889

119866(119909) minus 119889

119866(119910)| ⩽ 1 proposed

by Dharmalingam [88] The minimum cardinality of such adominating set is called the equitable domination number andis denoted by 120574

119864(119866) Deepak et al [89] defined the equitable

bondage number 119887119864(119866) of a graph 119866 to be the cardinality of a

smallest set 119861 sube 119864(119866) of edges for which 120574119864(119866 minus 119861) gt 120574

119864(119866)

and determined 119887119864(119866) for some special graphs

Theorem 140 (Deepak et al [89] 2011) For any integer 119899 ⩾ 2

119887119864(119870

119899) = lceil

119899

2rceil

119887119864(119870

119898119899) = 119898 119891119900119903 0 ⩽ 119899 minus 119898 ⩽ 1

119887119864(119862

119899) =

3 119894119891 119899 equiv 1 (mod 3)2 119900119905ℎ119890119903119908119894119904119890

119887119864(119875

119899) =

2 119894119891 119899 equiv 1 (mod 3)1 119900119905ℎ119890119903119908119894119904119890

119887119864(119879) ⩽ 2 119891119900119903 119886119899119910 119899119900119899119905119903119894119907119894119886119897 119905119903119890119890 119879

119887119864(119866) ⩽ 119899 minus 1 119891119900119903 119886119899119910 119888119900119899119899119890119888119905119890119889 119892119903119886119901ℎ 119866 119900119891 119900119903119889119890119903 119899

(51)

22 International Journal of Combinatorics

As far as we know there are no more results on theequitable bondage number of graphs in the present literature

There are other variations of the normal domination byadding restricted conditions to the normal dominating setFor example the connected domination set of a graph119866 pro-posed by Sampathkumar and Walikar [90] is a dominatingset 119878 such that the subgraph induced by 119878 is connected Theproblem of finding a minimum connected domination set ina graph is equivalent to the problemof finding a spanning treewithmaximumnumber of vertices of degree one [91] and hassome important applications in wireless networks [92] thusit has received much research attention However for suchan important domination there is no result on its bondagenumber We believe that the topic is of significance for futureresearch

9 Generalized Bondage Numbers

There are various generalizations of the classical dominationsuch as 119901-domination distance domination fractional dom-ination Roman domination and rainbow domination Everysuch a generalization can lead to a corresponding bondageIn this section we introduce some of them

91 119901-Bondage Numbers In 1985 Fink and Jacobson [93]introduced the concept of 119901-domination Let 119901 be a positiveinteger A subset 119878 of 119881(119866) is a 119901-dominating set of 119866 if|119878 cap 119873

119866(119910)| ⩾ 119901 for every 119910 isin 119878 The 119901-domination number

120574119901(119866) is the minimum cardinality among all 119901-dominating

sets of 119866 Any 119901-dominating set of 119866 with cardinality 120574119901(119866)

is called a 120574119901-set of 119866 Note that a 120574

1-set is a classic minimum

dominating set Notice that every graph has a 119901-dominatingset since the vertex-set119881(119866) is such a setWe also note that a 1-dominating set is a dominating set and so 120574(119866) = 120574

1(119866) The

119901-domination number has received much research attentionsee a state-of-the-art survey article by Chellali et al [94]

It is clear from definition that every 119901-dominating setof a graph certainly contains all vertices of degree at most119901 minus 1 By this simple observation to avoid the trivial caseoccurrence we always assume that Δ(119866) ⩾ 119901 For 119901 ⩾ 2 Luet al [95] gave a constructive characterization of trees withunique minimum 119901-dominating sets

Recently Lu and Xu [96] have introduced the conceptto the 119901-bondage number of 119866 denoted by 119887

119901(119866) as the

minimum cardinality among all sets of edges 119861 sube 119864(119866) suchthat 120574

119901(119866 minus 119861) gt 120574

119901(119866) Clearly 119887

1(119866) = 119887(119866)

Lu and Xu [96] established a lower bound and an upperbound on 119887

119901(119879) for any integer 119901 ⩾ 2 and any tree 119879 with

Δ(119879) ⩾ 119901 and characterized all trees achieving the lowerbound and the upper bound respectively

Theorem 141 (Lu and Xu [96] 2011) For any integer 119901 ⩾ 2and any tree 119879 with Δ(119879) ⩾ 119901

1 ⩽ 119887119901(119879) ⩽ Δ (119879) minus 119901 + 1 (52)

Let 119878 be a given subset of 119881(119866) and for any 119909 isin 119878 let

119873119901(119909 119878 119866) = 119910 isin 119878 cap 119873

119866(119909)

1003816100381610038161003816119873119866(119910) cap 119878

1003816100381610038161003816 = 119901

(53)

Then all trees achieving the lower bound 1 can be character-ized as follows

Theorem 142 (Lu and Xu [96] 2011) Let 119879 be a tree withΔ(119879) ⩾ 119901 ⩾ 2 Then 119887

119901(119879) = 1 if and only if for any 120574

119901-set

119878 of 119879 there exists an edge 119909119910 isin (119878 119878) such that 119910 isin 119873119901(119909 119878 119879)

The notation 119878(119886 119887) denotes a double star obtained byadding an edge between the central vertices of two stars119870

1119886minus1

and1198701119887minus1

And the vertex with degree 119886 (resp 119887) in 119878(119886 119887) iscalled the 119871-central vertex (resp 119877-central vertex) of 119878(119886 119887)

To characterize all trees attaining the upper bound givenin Theorem 141 we define three types of operations on a tree119879 with Δ(119879) = Δ ⩾ 119901 + 1

Type 1 attach a pendant edge to a vertex 119910with 119889119879(119910) ⩽ 119901minus

2 in 119879

Type 2 attach a star 1198701Δminus1

to a vertex 119910 of 119879 by joining itscentral vertex to 119910 where 119910 in a 120574

119901-set of 119879 and

119889119879(119910) ⩽ Δ minus 1

Type 3 attach a double star 119878(119901 Δ minus 1) to a pendant vertex 119910of 119879 by coinciding its 119877-central vertex with 119910 wherethe unique neighbor of 119910 is in a 120574

119901-set of 119879

Let B = 119879 a tree obtained from 1198701Δ

or 119878(119901 Δ) by afinite sequence of operations of Types 1 2 and 3

Theorem 143 (Lu and Xu [96] 2011) A tree with the maxi-mum Δ ⩾ 119901 + 1 has 119901-bondage number Δ minus 119901 + 1 if and onlyif it belongs toB

Theorem 144 (Lu and Xu [97] 2012) Let 119866119898119899= 119875

119898times 119875

119899

Then

1198872(119866

2119899) = 1 119891119900119903 119899 ⩾ 2

1198872(119866

3119899) =

2 119894119891 119899 equiv 1 (mod 3) 1 119900119905ℎ119890119903119908119894119904119890

for 119899 ⩾ 2

1198872(119866

4119899) =

1 119894119891 119899 equiv 3 (mod 4) 2 119900119905ℎ119890119903119908119894119904119890

for 119899 ⩾ 7

(54)

Recently Krzywkowski [98] has proposed the conceptof the nonisolating 2-bondage of a graph The nonisolating2-bondage number of a graph 119866 denoted by 1198871015840

2(119866) is the

minimum cardinality among all sets of edges 119861 sube 119864(119866) suchthat 119866 minus 119861 has no isolated vertices and 120574

2(119866 minus 119861) gt 120574

2(119866) If

for every 119861 sube 119864(119866) either 1205742(119866 minus 119861) = 120574

2(119866) or 119866 minus 119861 has

isolated vertices then define 11988710158402(119866) = 0 and say that 119866 is a 2-

nonisolatedly strongly stable graph Krzywkowski presentedsome basic properties of nonisolating 2-bondage in graphsshowed that for every nonnegative integer 119896 there exists atree 119879 such 1198871015840

2(119879) = 119896 characterized all 2-non-isolatedly

strongly stable trees and determined 11988710158402(119866) for several special

graphs

International Journal of Combinatorics 23

Theorem 145 (Krzywkowski [98] 2012) For any positiveintegers 119899 and119898 with119898 ⩽ 119899

1198871015840

2(119870

119899) =

0 119894119891 119899 = 1 2 3

lfloor2119899

3rfloor 119900119905ℎ119890119903119908119894119904119890

1198871015840

2(119875

119899) =

0 119894119891 119899 = 1 2 3

1 119900119905ℎ119890119903119908119894119904119890

1198871015840

2(119862

119899) =

0 119894119891 119899 = 3

1 119894119891 119899 119894119904 119890V119890119899

2 119894119891 119899 119894119904 119900119889119889

1198871015840

2(119870

119898119899) =

3 119894119891 119898 = 119899 = 3

1 119894119891 119898 = 119899 = 4

2 119900119905ℎ119890119903119908119894119904119890

(55)

92 Distance Bondage Numbers A subset 119878 of vertices of agraph119866 is said to be a distance 119896-dominating set for119866 if everyvertex in 119866 not in 119878 is at distance at most 119896 from some vertexof 119878 The minimum cardinality of all distance 119896-dominatingsets is called the distance 119896-domination number of 119866 anddenoted by 120574

119896(119866) (do not confuse with the above-mentioned

120574119901(119866)) When 119896 = 1 a distance 1-dominating set is a normal

dominating set and so 1205741(119866) = 120574(119866) for any graph 119866 Thus

the distance 119896-domination is a generalization of the classicaldomination

The relation between 120574119896and 120572

119896(the 119896-independence

number defined in Section 22) for a tree obtained by Meirand Moon [14] who proved that 120574

119896(119879) = 120572

2119896(119879) for any tree

119879 (a special case for 119896 = 1 is stated in Proposition 10) Thefurther research results can be found in Henning et al [99]Tian and Xu [100ndash103] and Liu et al [104]

In 1998 Hartnell et al [15] defined the distance 119896-bondagenumber of 119866 denoted by 119887

119896(119866) to be the cardinality of a

smallest subset 119861 of edges of 119866 with the property that 120574119896(119866 minus

119861) gt 120574119896(119866) FromTheorem 6 it is clear that if119879 is a nontrivial

tree then 1 ⩽ 1198871(119879) ⩽ 2 Hartnell et al [15] generalized this

result to any integer 119896 ⩾ 1

Theorem 146 (Hartnell et al [15] 1998) For every nontrivialtree 119879 and positive integer 119896 1 ⩽ 119887

119896(119879) ⩽ 2

Hartnell et al [15] and Topp and Vestergaard [13] alsocharacterized the trees having distance 119896-bondage number 2In particular the class of trees for which 119887

1(119879) = 2 are just

those which have a unique maximum 2-independent set (seeTheorem 11)

Since when 119896 = 1 the distance 1-bondage number 1198871(119866)

is the classical bondage number 119887(119866) Theorem 12 gives theNP-hardness of deciding the distance 119896-bondage number ofgeneral graphs

Theorem 147 Given a nonempty undirected graph 119866 andpositive integers 119896 and 119887 with 119887 ⩽ 120576(119866) determining wetheror not 119887

119896(119866) ⩽ 119887 is NP-hard

For a vertex 119909 in119866 the open 119896-neighborhood119873119896(119909) of 119909 is

defined as119873119896(119909) = 119910 isin 119881(119866) 1 ⩽ 119889

119866(119909 119910) le 119896The closed

119896-neighborhood119873119896[119909] of 119909 in 119866 is defined as119873

119896(119909) cup 119909 Let

Δ119896(119866) = max 1003816100381610038161003816119873119896

(119909)1003816100381610038161003816 for any 119909 isin 119881 (119866) (56)

Clearly Δ1(119866) = Δ(119866) The 119896th power of a graph 119866 is the

graph119866119896 with vertex-set119881(119866119896

) = 119881(119866) and edge-set119864(119866119896

) =

119909119910 1 ⩽ 119889119866(119909 119910) ⩽ 119896 The following lemma holds directly

from the definition of 119866119896

Proposition 148 (Tian and Xu [101]) Δ(119866119896

) = Δ119896(119866) and

120574(119866119896

) = 120574119896(119866) for any graph 119866 and each 119896 ⩾ 1

A graph 119866 is 119896-distance domination-critical or 120574119896-critical

for short if 120574119896(119866 minus 119909) lt 120574

119896(119866) for every vertex 119909 in 119866

proposed by Henning et al [105]

Proposition 149 (Tian and Xu [101]) For each 119896 ⩾ 1 a graph119866 is 120574

119896-critical if and only if 119866119896 is 120574

119896-critical

By the above facts we suggest the following worthwhileproblem

Problem 6 Can we generalize the results on the bondage for119866 to 119866119896 In particular do the following propositions hold

(a) 119887(119866119896

) = 119887119896(119866) for any graph 119866 and each 119896 ⩾ 1

(b) 119887(119866119896

) ⩽ Δ119896(119866) if 119866 is not 120574

119896-critical

Let 119896 and 119901 be positive integers A subset 119878 of 119881(119866) isdefined to be a (119896 119901)-dominating set of 119866 if for any vertex119909 isin 119878 |119873

119896(119909) cap 119878| ⩾ 119901 The (119896 119901)-domination number of

119866 denoted by 120574119896119901(119866) is the minimum cardinality among

all (119896 119901)-dominating sets of 119866 Clearly for a graph 119866 a(1 1)-dominating set is a classical dominating set a (119896 1)-dominating set is a distance 119896-dominating set and a (1 119901)-dominating set is the above-mentioned 119901-dominating setThat is 120574

11(119866) = 120574(119866) 120574

1198961(119866) = 120574

119896(119866) and 120574

1119901(119866) = 120574

119901(119866)

The concept of (119896 119901)-domination in a graph 119866 is a gen-eralized domination which combines distance 119896-dominationand 119901-domination in 119866 So the investigation of (119896 119901)-domination of 119866 is more interesting and has received theattention of many researchers see for example [106ndash109]

More general Li and Xu [110] proposed the concept of the(ℓ 119908)-domination number of a119908-connected graph A subset119878 of 119881(119866) is defined to be an (ℓ 119908)-dominating set of 119866 iffor any vertex 119909 isin 119878 there are 119908 internally disjoint pathsof length at most ℓ from 119909 to some vertex in 119878 The (ℓ 119908)-domination number of119866 denoted by 120574

ℓ119908(119866) is theminimum

cardinality among all (ℓ 119908)-dominating sets of 119866 Clearly1205741198961(119866) = 120574

119896(119866) and 120574

1119901(119866) = 120574

119901(119866)

It is quite natural to propose the concept of bondage num-ber for (119896 119901)-domination or (ℓ 119908)-domination However asfar as we know none has proposed this concept until todayThis is a worthwhile topic for further research

24 International Journal of Combinatorics

93 Fractional Bondage Numbers If 120590 is a function mappingthe vertex-set 119881 into some set of real numbers then for anysubset 119878 sube 119881 let 120590(119878) = sum

119909isin119878120590(119909) Also let |120590| = 120590(119881)

A real-value function 120590 119881 rarr [0 1] is a dominatingfunction of a graph 119866 if for every 119909 isin 119881(119866) 120590(119873

119866[119909]) ⩾ 1

Thus if 119878 is a dominating set of 119866 120590 is a function where

120590 (119909) = 1 if 119909 isin 1198780 otherwise

(57)

then 120590 is a dominating function of119866 The fractional domina-tion number of 119866 denoted by 120574

119865(119866) is defined as follows

120574119865(119866) = min |120590| 120590 is a domination function of 119866

(58)

The 120574119865-bondage number of 119866 denoted by 119887

119865(119866) is

defined as the minimum cardinality of a subset 119861 sube 119864 whoseremoval results in 120574

119865(119866 minus 119861) gt 120574

119865(119866)

Hedetniemi et al [111] are the first to study fractionaldomination although Farber [112] introduced the idea indi-rectly The concept of the fractional domination numberwas proposed by Domke and Laskar [113] in 1997 Thefractional domination numbers for some ordinary graphs aredetermined

Proposition 150 (Domke et al [114] 1998 Domke and Laskar[113] 1997) (a) If 119866 is a 119896-regular graph with order 119899 then120574119865(119866) = 119899119896 + 1(b) 120574

119865(119862

119899) = 1198993 120574

119865(119875

119899) = lceil1198993rceil 120574

119865(119870

119899) = 1

(c) 120574119865(119866) = 1 if and only if Δ(119866) = 119899 minus 1

(d) 120574119865(119879) = 120574(119879) for any tree 119879

(e) 120574119865(119870

119899119898) = (119899(119898 + 1) + 119898(119899 + 1))(119899119898 minus 1) where

max119899119898 gt 1

The assertions (a)ndash(d) are due to Domke et al [114] andthe assertion (e) is due to Domke and Laskar [113] Accordingto these results Domke and Laskar [113] determined the 120574

119865-

bondage numbers for these graphs

Theorem 151 (Domke and Laskar [113] 1997) 119887119865(119870

119899) =

lceil1198992rceil 119887119865(119870

119899119898) = 1 wheremax119899119898 gt 1

119887119865(119862

119899) =

2 119894119891 119899 equiv 0 (mod 3) 1 119900119905ℎ119890119903119908119894119904119890

119887119865(119875

119899) =

2 119894119891 119899 equiv 1 (mod 3) 1 119900119905ℎ119890119903119908119894119904119890

(59)

It is easy to see that for any tree119879 119887119865(119879) ⩽ 2 In fact since

120574119865(119879) = 120574(119879) for any tree 119879 120574

119865(119879

1015840

) = 120574(1198791015840

) for any subgraph1198791015840 of 119879 It follows that

119887119865(119879) = min 119861 sube 119864 (119879) 120574

119865(119879 minus 119861) gt 120574

119865(119879)

= min 119861 sube 119864 (119879) 120574 (119879 minus 119861) gt 120574 (119879)

= 119887 (119879) ⩽ 2

(60)

Except for these no results on 119887119865(119866) have been known as

yet

94 Roman Bondage Numbers A Roman dominating func-tion on a graph 119866 is a labeling 119891 119881 rarr 0 1 2 such thatevery vertex with label 0 has at least one neighbor with label2 The weight of a Roman dominating function 119891 on 119866 is thevalue

119891 (119866) = sum

119906isin119881(119866)

119891 (119906) (61)

The minimum weight of a Roman dominating function ona graph 119866 is called the Roman domination number of 119866denoted by 120574RM (119866)

A Roman dominating function 119891 119881 rarr 0 1 2

can be represented by the ordered partition (1198810 119881

1 119881

2) (or

(119881119891

0 119881

119891

1 119881

119891

2) to refer to119891) of119881 where119881

119894= V isin 119881 | 119891(V) = 119894

In this representation its weight 119891(119866) = |1198811| + 2|119881

2| It is

clear that119881119891

1cup119881

119891

2is a dominating set of 119866 called the Roman

dominating set denoted by 119863119891

R = (1198811 1198812) Since 119881119891

1cup 119881

119891

2is

a dominating set when 119891 is a Roman dominating functionand since placing weight 2 at the vertices of a dominating setyields a Roman dominating function in [115] it is observedthat

120574 (119866) ⩽ 120574RM (119866) ⩽ 2120574 (119866) (62)

A graph 119866 is called to be Roman if 120574RM (119866) = 2120574(119866)The definition of the Roman domination function is

proposed implicitly by Stewart [116] and ReVelle and Rosing[117] Roman dominating numbers have been deeply studiedIn particular Bahremandpour et al [118] showed that theproblem determining the Roman domination number is NP-complete even for bipartite graphs

Let 119866 be a graph with maximum degree at least twoThe Roman bondage number 119887RM (119866) of 119866 is the minimumcardinality of all sets 1198641015840 sube 119864 for which 120574RM (119866 minus 119864

1015840

) gt

120574RM (119866) Since in study of Roman bondage number theassumption Δ(119866) ⩾ 2 is necessary we always assume thatwhen we discuss 119887RM (119866) all graphs involved satisfy Δ(119866) ⩾2 The Roman bondage number 119887RM (119866) is introduced byJafari Rad and Volkmann in [119]

Recently Bahremandpour et al [118] have shown that theproblem determining the Roman bondage number is NP-hard even for bipartite graphs

Problem 7 Consider the decision problem

Roman Bondage Problem

Instance a nonempty bipartite graph119866 and a positiveinteger 119896

Question is 119887RM (119866) ⩽ 119896

Theorem 152 (Bahremandpour et al [118] 2013) TheRomanbondage problem is NP-hard even for bipartite graphs

The exact value of 119887RM(119866) is known only for a few familyof graphs including the complete graphs cycles and paths

International Journal of Combinatorics 25

Theorem 153 (Jafari Rad and Volkmann [119] 2011) For anypositive integers 119899 and119898 with119898 ⩽ 119899

119887RM (119875119899) = 2 119894119891 119899 equiv 2 (mod 3) 1 119900119905ℎ119890119903119908119894119904119890

119887RM (119862119899) =

3 119894119891 119899 = 2 (mod 3) 2 119900119905ℎ119890119903119908119894119904119890

119887RM (119870119899) = lceil

119899

2rceil 119891119900119903 119899 ⩾ 3

119887RM (119870119898119899) =

1 119894119891 119898 = 1 119886119899119889 119899 = 1

4 119894119891 119898 = 119899 = 3

119898 119900119905ℎ119890119903119908119894119904119890

(63)

Lemma 154 (Cockayne et al [115] 2004) If 119866 is a graph oforder 119899 and contains vertices of degree 119899 minus 1 then 120574RM(119866) = 2

Using Lemma 154 the third conclusion in Theorem 153can be generalized to more general case which is similar toLemma 49

Theorem 155 (Jafari Rad and Volkmann [119] 2011) Let119866 bea graph with order 119899 ge 3 and 119905 the number of vertices of degree119899 minus 1 in 119866 If 119905 ⩾ 1 then 119887RM(119866) = lceil1199052rceil

Ebadi and PushpaLatha [120] conjectured that 119887RM(119866) ⩽119899 minus 1 for any graph 119866 of order 119899 ⩾ 3 Dehgardi et al[121] Akbari andQajar [122] independently showed that thisconjecture is true

Theorem 156 119887RM(119866) ⩽ 119899 minus 1 for any connected graph 119866 oforder 119899 ⩾ 3

For a complete 119905-partite graph we have the followingresult

Theorem 157 (Hu and Xu [123] 2011) Let 119866 = 11987011989811198982119898119905

bea complete 119905-partite graph with 119898

1= sdot sdot sdot = 119898

119894lt 119898

119894+1le sdot sdot sdot ⩽

119898119905 119905 ⩾ 2 and 119899 = sum119905

119895=1119898

119895 Then

119887RM (119866) =

lceil119894

2rceil 119894119891 119898

119894= 1 119886119899119889 119899 ⩾ 3

2 119894119891 119898119894= 2 119886119899119889 119894 = 1

119894 119894119891 119898119894= 2 119886119899119889 119894 ⩾ 2

4 119894119891 119898119894= 3 119886119899119889 119894 = 119905 = 2

119899 minus 1 119894119891 119898119894= 3 119886119899119889 119894 = 119905 ⩾ 3

119899 minus 119898119905

119894119891 119898119894⩾ 3 119886119899119889 119898

119905⩾ 4

(64)

Consider a complete 119905-partite graph 119870119905(3) for 119905 ⩾ 3

119887RM(119870119905(3)) = 119899 minus 1 by Theorem 157 where 119899 = 3119905

which shows that this upper bound of 119899 minus 1 on 119887RM(119866) inTheorem 156 is sharp This fact and 120574

119877(119870

119905(3)) = 4 give

a negative answer to the following two problems posed byDehgardi et al [121]Prove or Disprove for any connected graph 119866 of order 119899 ge 3(i) 119887RM(119866) = 119899 minus 1 if and only if 119866 cong 119870

3 (ii) 119887RM(119866) ⩽ 119899 minus

120574119877(119866) + 1Note that a complete 119905-partite graph 119870

119905(3) is (119899 minus 3)-

regular and 119887RM(119870119905(3)) = 119899 minus 1 for 119905 ⩾ 3 where 119899 = 3119905

Hu and Xu further determined that 119887119877(119866) = 119899 minus 2 for any

(119899 minus 3)-regular graph 119866 of order 119899 ⩽ 5 except 119870119905(3)

Theorem 158 (Hu and Xu [123] 2011) For any (119899minus3)-regulargraph 119866 of order 119899 other than119870

119905(3) 119887RM(119866) = 119899 minus 2 for 119899 ⩾ 5

For a tree 119879 with order 119899 ⩾ 3 Ebadi and PushpaLatha[120] and Jafari Rad and Volkmann [119] independentlyobtained an upper bound on 119887RM(119879)

Theorem 159 119887RM(119879) ⩽ 3 for any tree 119879 with order 119899 ⩾ 3

Theorem 160 (Bahremandpour et al [118] 2013) 119887RM(1198752 times119875119899) = 2 for 119899 ⩾ 2

Theorem 161 (Jafari Rad and Volkmann [119] 2011) Let119866 bea graph with order at least three

(a) If (119909 119910 119911) is a path of length 2 in 119866 then 119887RM(119866) ⩽119889119866(119909) + 119889

119866(119910) + 119889

119866(119911) minus 3 minus |119873

119866(119909) cap 119873

119866(119910)|

(b) If 119866 is connected then 119887RM(119866) ⩽ 120582(119866) + 2Δ(119866) minus 3

Theorem 161 (b) implies 119887RM(119866) ⩽ 120575(119866) + 2Δ(119866) minus 3 for aconnected graph 119866 Note that for a planar graph 119866 120575(119866) ⩽ 5moreover 120575(119866) ⩽ 3 if the girth at least 4 and 120575(119866) ⩽ 2 if thegirth at least 6 These two facts show that 119887RM(119866) ⩽ 2Δ(119866) +2 for a connected planar graph 119866 Jafari Rad and Volkmann[124] improved this bound

Theorem 162 (Jafari Rad and Volkmann [124] 2011) For anyconnected planar graph 119866 of order 119899 ⩾ 3 with girth 119892(119866)

119887RM (119866)

⩽

2Δ (119866)

Δ (119866) + 6

Δ (119866) + 5 119894119891 119866 119888119900119899119905119886119894119899119904 119899119900 V119890119903119905119894119888119890119904 119900119891 119889119890119892119903119890119890 119891119894V119890

Δ (119866) + 4 119894119891 119892 (119866) ⩾ 4

Δ (119866) + 3 119894119891 119892 (119866) ⩾ 5

Δ (119866) + 2 119894119891 119892 (119866) ⩾ 6

Δ (119866) + 1 119894119891 119892 (119866) ⩾ 8

(65)

According toTheorem 157 119887RM(119862119899) = 3 = Δ(119862

119899)+1 for a

cycle 119862119899of length 119899 ⩾ 8 with 119899 equiv 2 (mod 3) and therefore

the last upper bound inTheorem 162 is sharp at least for Δ =2

Combining the fact that every planar graph 119866 withminimum degree 5 contains an edge 119909119910 with 119889

119866(119909) = 5

and 119889119866(119910) isin 5 6 with Theorem 161 (a) Akbari et al [125]

obtained the following result

26 International Journal of Combinatorics

middot middot middot

middot middot middot

middot middot middot

middot middot middot

119866

Figure 9 The graph 119866 is constructed from 119866

Theorem 163 (Akbari et al [125] 2012) 119887RM(119866) ⩽ 15 forevery planar graph 119866

It remains open to show whether the bound inTheorem 163 is sharp or not Though finding a planargraph 119866 with 119887RM(119866) = 15 seems to be difficult Akbari etal [125] constructed an infinite family of planar graphs withRoman bondage number equal to 7 by proving the followingresult

Theorem 164 (Akbari et al [125] 2012) Let 119866 be a graph oforder 119899 and 119866 is the graph of order 5119899 obtained from 119866 byattaching the central vertex of a copy of a path 119875

5to each vertex

of119866 (see Figure 9) Then 120574RM(119866) = 4119899 and 119887RM(119866) = 120575(119866)+2

ByTheorem 164 infinitemany planar graphswith Romanbondage number 7 can be obtained by considering any planargraph 119866 with 120575(119866) = 5 (eg the icosahedron graph)

Conjecture 165 (Akbari et al [125] 2012) The Romanbondage number of every planar graph is at most 7

For general bounds the following observation is directlyobtained which is similar to Observation 1

Observation 4 Let 119866 be a graph of order 119899 with maximumdegree at least two Assume that 119867 is a spanning subgraphobtained from 119866 by removing 119896 edges If 120574RM(119867) = 120574RM(119866)then 119887

119877(119867) ⩽ 119887RM(119866) ⩽ 119887RM(119867) + 119896

Theorem 166 (Bahremandpour et al [118] 2013) For anyconnected graph 119866 of order 119899 if 119899 ⩾ 3 and 120574RM(119866) = 120574(119866) + 1then

119887RM (119866) ⩽ min 119887 (119866) 119899Δ (66)

where 119899Δis the number of vertices with maximum degree Δ in

119866

Observation 5 For any graph 119866 if 120574(119866) = 120574RM(119866) then119887RM(119866) ⩽ 119887(119866)

Theorem 167 (Bahremandpour et al [118] 2013) For everyRoman graph 119866

119887RM (119866) ⩾ 119887 (119866) (67)

The bound is sharp for cycles on 119899 vertices where 119899 equiv 0 (mod3)

The strict inequality in Theorem 167 can hold for exam-ple 119887(119862

3119896+2) = 2 lt 3 = 119887RM(1198623119896+2

) byTheorem 157A graph 119866 is called to be vertex Roman domination-

critical if 120574RM(119866 minus 119909) lt 120574RM(119866) for every vertex 119909 in 119866 If119866 has no isolated vertices then 120574RM(119866) ⩽ 2120574(119866) ⩽ 2120573(119866)If 120574RM(119866) = 2120573(119866) then 120574RM(119866) = 2120574(119866) and hence 119866 is aRoman graph In [30] Volkmann gave a lot of graphs with120574(119866) = 120573(119866) The following result is similar to Theorem 57

Theorem 168 (Bahremandpour et al [118] 2013) For anygraph 119866 if 120574RM(119866) = 2120573(119866) then

(a) 119887RM(119866) ⩾ 120575(119866)

(b) 119887RM(119866) ⩾ 120575(119866)+1 if119866 is a vertex Roman domination-critical graph

If 120574RM(119866) = 2 then obviously 119887RM(119866) ⩽ 120575(119866) So weassume 120574RM(119866) ⩾ 3 Dehgardi et al [121] proved 119887RM(119866) ⩽(120574RM(119866) minus 2)Δ(119866) + 1 and posed the following problem if 119866is a connected graph of order 119899 ⩾ 4 with 120574RM(119866) ⩾ 3 then

119887RM (119866) ⩽ (120574RM (119866) minus 2) Δ (119866) (68)

Theorem 161 (a) shows that the inequality (68) holds if120574RM(119866) ⩾ 5 Thus the bound in (68) is of interest only when120574RM(119866) is 3 or 4

Lemma 169 (Bahremandpour et al [118] 2013) For anynonempty graph 119866 of order 119899 ⩾ 3 120574RM(119866) = 3 if and onlyif Δ(119866) = 119899 minus 2

The following result shows that (68) holds for all graphs119866 of order 119899 ⩾ 4 with 120574RM(119866) = 3 or 4 which improvesTheorem 161 (b)

Theorem 170 (Bahremandpour et al [118] 2013) For anyconnected graph 119866 of order 119899 ⩾ 4

119887RM (119866) le Δ (119866) = 119899 minus 2 119894119891 120574RM (119866) = 3

Δ (119866) + 120575 (119866) minus 1 119894119891 120574RM (119866) = 4(69)

with the first equality if and only if 119866 cong 1198624

Recently Akbari and Qajar [122] have proved the follow-ing result

Theorem 171 (Akbari and Qajar [122]) For any connectedgraph 119866 of order 119899 ⩾ 3

119887RM (119866) ⩽ 119899 minus 120574RM (119866) + 5 (70)

International Journal of Combinatorics 27

We conclude this section with the following problems

Problem 8 Characterize all connected graphs 119866 of order 119899 ⩾3 for which 119887RM(119866) = 119899 minus 1

Problem 9 Prove or disprove if 119866 is a connected graph oforder 119899 ⩾ 3 then

119887RM (119866) ⩽ 119899 minus 120574RM (119866) + 3 (71)

Note that 120574119877(119870

119905(3)) = 4 and 119887RM(119870119905

(3)) = 119899minus1 where 119899 =3119905 for 119905 ⩾ 3 The upper bound on 119887RM(119866) given in Problem 9is sharp if Problem 9 has positive answer

95 Rainbow Bondage Numbers For a positive integer 119896 a119896-rainbow dominating function of a graph 119866 is a function 119891from the vertex-set 119881(119866) to the set of all subsets of the set1 2 119896 such that for any vertex 119909 in 119866 with 119891(119909) = 0 thecondition

⋃

119910isin119873119866(119909)

119891 (119910) = 1 2 119896 (72)

is fulfilledThe weight of a 119896-rainbow dominating function 119891on 119866 is the value

119891 (119866) = sum

119909isin119881(119866)

1003816100381610038161003816119891 (119909)1003816100381610038161003816 (73)

The 119896-rainbow domination number of a graph 119866 denoted by120574119903119896(119866) is the minimum weight of a 119896-rainbow dominating

function 119891 of 119866Note that 120574

1199031(119866) is the classical domination number 120574(119866)

The 119896-rainbow domination number is introduced by Bresaret al [126] Rainbow domination of a graph 119866 coincides withordinary domination of the Cartesian product of 119866 with thecomplete graph in particular 120574

119903119896(119866) = 120574(119866 times 119870

119896) for any

positive integer 119896 and any graph 119866 [126] Hartnell and Rall[127] proved that 120574(119879) lt 120574

1199032(119879) for any tree 119879 no graph has

120574 = 120574119903119896for 119896 ⩾ 3 for any 119896 ⩾ 2 and any graph 119866 of order 119899

min 119899 120574 (119866) + 119896 minus 2 ⩽ 120574119903119896(119866) ⩽ 119896120574 (119866) (74)

The introduction of rainbow domination is motivatedby the study of paired domination in Cartesian productsof graphs where certain upper bounds can be expressed interms of rainbow domination that is for any graph 119866 andany graph 119867 if 119881(119867) can be partitioned into 119896120574

119875-sets then

120574119875(119866 times 119867) ⩽ (1119896)120592(119867)120574

119903119896(119866) proved by Bresar et al [126]

Dehgardi et al [128] introduced the concept of 119896-rainbowbondage number of graphs Let 119866 be a graph with maximumdegree at least two The 119896-rainbow bondage number 119887

119903119896(119866)

of 119866 is the minimum cardinality of all sets 119861 sube 119864(119866) forwhich 120574

119903119896(119866 minus 119861) gt 120574

119903119896(119866) Since in the study of 119896-rainbow

bondage number the assumption Δ(119866) ⩾ 2 is necessarywe always assume that when we discuss 119887

119903119896(119866) all graphs

involved satisfy Δ(119866) ⩾ 2The 2-rainbow bondage number of some special families

of graphs has been determined

Theorem 172 (Dehgardi et al [128]) For any integer 119899 ⩾ 3

1198871199032(119875

119899) = 1 119891119900119903 119899 ⩾ 2

1198871199032(119862

119899) =

1 119894119891 119899 equiv 0 (mod 4) 2 119900119905ℎ119890119903119908119894119904119890

1198871199032(119870

119899119899) =

1 119894119891 119899 = 2

3 119894119891 119899 = 3

6 119894119891 119899 = 4

119898 119894119891 119899 ⩾ 5

(75)

Theorem 173 (Dehgardi et al [128]) For any connected graph119866 of order 119899 ⩾ 3 119887

1199032(119866) ⩽ 120582(119866) + 2Δ(119866) minus 3

Theorem 174 (Dehgardi et al [128]) For any tree 119879 of order119899 ⩾ 3 119887

1199032(119879) ⩽ 2

Theorem 175 (Dehgardi et al [128]) If 119866 is a planar graphwithmaximumdegree at least two then 119887

1199032(119866) ⩽ 15Moreover

if the girth 119892(119866) ⩾ 4 then 1198871199032(119866) ⩽ 11 and if 119892(119866) ⩾ 6 then

1198871199032(119866) ⩽ 9

Theorem 176 (Dehgardi et al [128]) For a complete bipartitegraph 119870

119901119902 if 2119896 + 1 ⩽ 119901 ⩽ 119902 then 119887

119903119896(119870

119901119902) = 119901

Theorem177 (Dehgardi et al [128]) For a complete graph119870119899

if 119899 ⩾ 119896 + 1 then 119887119903119896(119870

119899) = lceil119896119899(119896 + 1)rceil

The special case 119896 = 1 of Theorem 177 can be found inTheorem 1 The following is a simple observation similar toObservation 1

Observation 6 (Dehgardi et al [128]) Let 119866 be a graphwith maximum degree at least two 119867 a spanning subgraphobtained from 119866 by removing 119896 edges If 120574

119903119896(119867) = 120574

119903119896(119866)

then 119887119903119896(119867) ⩽ 119887

119903119896(119866) ⩽ 119887

119903119896(119867) + 119896

Theorem 178 (Dehgardi et al [128]) For every graph 119866 with120574119903119896(119866) = 119896120574(119866) 119887

119903119896(119866) ⩾ 119887(119866)

A graph 119866 is said to be vertex 119896-rainbow domination-critical if 120574

119903119896(119866 minus 119909) lt 120574

119903119896(119866) for every vertex 119909 in 119866 It is

clear that if 119866 has no isolated vertices then 120574119903119896(119866) ⩽ 119896120574(119866) ⩽

119896120573(119866) where 120573(119866) is the vertex-covering number of119866 Thusif 120574

119903119896(119866) = 119896120573(119866) then 120574

119903119896(119866) = 119896120574(119866)

Theorem 179 (Dehgardi et al [128]) For any graph 119866 if120574119903119896(119866) = 119896120573(119866) then

(a) 119887119903119896(119866) ⩾ 120575(119866)

(b) 119887119903119896(119866) ⩾ 120575(119866)+1 if119866 is vertex 119896-rainbow domination-

critical

As far as we know there are no more results on the 119896-rainbow bondage number of graphs in the literature

28 International Journal of Combinatorics

10 Results on Digraphs

Although domination has been extensively studied in undi-rected graphs it is natural to think of a dominating setas a one-way relationship between vertices of the graphIndeed among the earliest literatures on this subject vonNeumann and Morgenstern [129] used what is now calleddomination in digraphs to find solution (or kernels whichare independent dominating sets) for cooperative 119899-persongames Most likely the first formulation of domination byBerge [130] in 1958 was given in the context of digraphs andonly some years latter by Ore [131] in 1962 for undirectedgraphs Despite this history examination of domination andits variants in digraphs has been essentially overlooked (see[4] for an overview of the domination literature) Thus thereare few if any such results on domination for digraphs in theliterature

The bondage number and its related topics for undirectedgraph have become one of major areas both in theoreticaland applied researches However until recently Carlsonand Develin [42] Shan and Kang [132] and Huang andXu [133 134] studied the bondage number for digraphsindependently In this section we introduce their resultsfor general digraphs Results for some special digraphs suchas vertex-transitive digraphs are introduced in the nextsection

101 Upper Bounds for General Digraphs Let 119866 = (119881 119864)

be a digraph without loops and parallel edges A digraph 119866is called to be asymmetric if whenever (119909 119910) isin 119864(119866) then(119910 119909) notin 119864(119866) and to be symmetric if (119909 119910) isin 119864(119866) implies(119910 119909) isin 119864(119866) For a vertex 119909 of 119881(119866) the sets of out-neighbors and in-neighbors of 119909 are respectively defined as119873

+

119866(119909) = 119910 isin 119881(119866) (119909 119910) isin 119864(119866) and 119873minus

119866(119909) = 119909 isin

119881(119866) (119909 119910) isin 119864(119866) the out-degree and the in-degree of119909 are respectively defined as 119889+

119866(119909) = |119873

+

119866(119909)| and 119889minus

119866(119909) =

|119873+

119866(119909)| Denote themaximum and theminimum out-degree

(resp in-degree) of 119866 by Δ+(119866) and 120575+(119866) (resp Δminus(119866) and120575minus

(119866))The degree 119889119866(119909) of 119909 is defined as 119889+

119866(119909)+119889

minus

119866(119909) and

the maximum and the minimum degrees of119866 are denoted byΔ(119866) and 120575(119866) respectively that is Δ(119866) = max119889

119866(119909) 119909 isin

119881(119866) and 120575(119866) = min119889119866(119909) 119909 isin 119881(119866) Note that the

definitions here are different from the ones in the textbookon digraphs see for example Xu [1]

A subset 119878 of119881(119866) is called a dominating set if119881(119866) = 119878cup119873

+

119866(119878) where119873+

119866(119878) is the set of out-neighbors of 119878Then just

as for undirected graphs 120574(119866) is the minimum cardinalityof a dominating set and the bondage number 119887(119866) is thesmallest cardinality of a set 119861 of edges such that 120574(119866 minus 119861) gt120574(119866) if such a subset 119861 sube 119864(119866) exists Otherwise we put119887(119866) = infin

Some basic results for undirected graphs stated inSection 3 can be generalized to digraphs For exampleTheorem 18 is generalized by Carlson and Develin [42]Huang andXu [133] and Shan andKang [132] independentlyas follows

Theorem 180 Let 119866 be a digraph and (119909 119910) isin 119864(119866) Then119887(119866) ⩽ 119889

119866(119910) + 119889

minus

119866(119909) minus |119873

minus

119866(119909) cap 119873

minus

119866(119910)|

Corollary 181 For a digraph 119866 119887(119866) ⩽ 120575minus(119866) + Δ(119866)

Since 120575minus

(119866) ⩽ (12)Δ(119866) from Corollary 181Conjecture 53 is valid for digraphs

Corollary 182 For a digraph 119866 119887(119866) ⩽ (32)Δ(119866)

In the case of undirected graphs the bondage number of119866119899= 119870

119899times 119870

119899achieves this bound However it was shown

by Carlson and Develin in [42] that if we take the symmetricdigraph119866

119899 we haveΔ(119866

119899) = 4(119899minus1) 120574(119866

119899) = 119899 and 119887(119866

119899) ⩽

4(119899 minus 1) + 1 = Δ(119866119899) + 1 So this family of digraphs cannot

show that the bound in Corollary 182 is tightCorresponding to Conjecture 51 which is discredited

for undirected graphs and is valid for digraphs the sameconjecture for digraphs can be proposed as follows

Conjecture 183 (Carlson and Develin [42] 2006) 119887(119866) ⩽Δ(119866) + 1 for any digraph 119866

If Conjecture 183 is true then the following results showsthat this upper bound is tight since 119887(119870

119899times119870

119899) = Δ(119870

119899times119870

119899)+1

for a complete digraph 119870119899

In 2007 Shan and Kang [132] gave some tight upperbounds on the bondage numbers for some asymmetricdigraphs For example 119887(119879) ⩽ Δ(119879) for any asymmetricdirected tree 119879 119887(119866) ⩽ Δ(119866) for any asymmetric digraph 119866with order at least 4 and 120574(119866) ⩽ 2 For planar digraphs theyobtained the following results

Theorem 184 (Shan and Kang [132] 2007) Let 119866 be aasymmetric planar digraphThen 119887(119866) ⩽ Δ(119866)+2 and 119887(119866) ⩽Δ(119866) + 1 if Δ(119866) ⩾ 5 and 119889minus

119866(119909) ⩾ 3 for every vertex 119909 with

119889119866(119909) ⩾ 4

102 Results for Some Special Digraphs The exact values andbounds on 119887(119866) for some standard digraphs are determined

Theorem185 (Huang andXu [133] 2006) For a directed cycle119862119899and a directed path 119875

119899

119887 (119862119899) =

3 119894119891 119899 119894119904 119900119889119889

2 119894119891 119899 119894119904 119890V119890119899

119887 (119875119899) =

2 119894119891 119899 119894119904 119900119889119889

1 119894119891 119899 119894119904 119890V119890119899

(76)

For the de Bruijn digraph 119861(119889 119899) and the Kautz digraph119870(119889 119899)

119887 (119861 (119889 119899)) = 119889 119894119891 119899 119894119904 119900119889119889

119889 ⩽ 119887 (119861 (119889 119899)) ⩽ 2119889 119894119891 119899 119894119904 119890V119890119899

119887 (119870 (119889 119899)) = 119889 + 1

(77)

Like undirected graphs we can define the total domi-nation number and the total bondage number On the totalbondage numbers for some special digraphs the knownresults are as follows

International Journal of Combinatorics 29

Theorem 186 (Huang and Xu [134] 2007) For a directedcycle 119862

119899and a directed path 119875

119899 119887

119879(119875

119899) and 119887

119879(119862

119899) all do not

exist For a complete digraph 119870119899

119887119879(119870

119899) = infin 119894119891 119899 = 1 2

119887119879(119870

119899) = 3 119894119891 119899 = 3

119899 ⩽ 119887119879(119870

119899) ⩽ 2119899 minus 3 119894119891 119899 ⩾ 4

(78)

The extended de Bruijn digraph EB(119889 119899 1199021 119902

119901) and

the extended Kautz digraph EK(119889 119899 1199021 119902

119901) are intro-

duced by Shibata and Gonda [135] If 119901 = 1 then theyare the de Bruijn digraph B(119889 119899) and the Kautz digraphK(119889 119899) respectively Huang and Xu [134] determined theirtotal domination numbers In particular their total bondagenumbers for general cases are determined as follows

Theorem 187 (Huang andXu [134] 2007) If 119889 ⩾ 2 and 119902119894⩾ 2

for each 119894 = 1 2 119901 then

119887119879(EB(119889 119899 119902

1 119902

119901)) = 119889

119901

minus 1

119887119879(EK(119889 119899 119902

1 119902

119901)) = 119889

119901

(79)

In particular for the de Bruijn digraph B(119889 119899) and the Kautzdigraph K(119889 119899)

119887119879(B (119889 119899)) = 119889 minus 1 119887

119879(K (119889 119899)) = 119889 (80)

Zhang et al [136] determined the bondage number incomplete 119905-partite digraphs

Theorem 188 (Zhang et al [136] 2009) For a complete 119905-partite digraph 119870

11989911198992119899119905

where 1198991⩽ 119899

2⩽ sdot sdot sdot ⩽ 119899

119905

119887 (11987011989911198992119899119905

)

=

119898 119894119891 119899119898= 1 119899

119898+1⩾ 2 119891119900119903 119904119900119898119890

119898 (1 ⩽ 119898 lt 119905)

4119905 minus 3 119894119891 1198991= 119899

2= sdot sdot sdot = 119899

119905= 2

119905minus1

sum

119894=1

119899119894+ 2 (119905 minus 1) 119900119905ℎ119890119903119908119894119904119890

(81)

Since an undirected graph can be thought of as a sym-metric digraph any result for digraphs has an analogy forundirected graphs in general In view of this point studyingthe bondage number for digraphs is more significant thanfor undirected graphs Thus we should further study thebondage number of digraphs and try to generalize knownresults on the bondage number and related variants for undi-rected graphs to digraphs prove or disprove Conjecture 183In particular determine the exact values of 119887(B(119889 119899)) for aneven 119899 and 119887

119879(119870

119899) for 119899 ⩾ 4

11 Efficient Dominating Sets

A dominating set 119878 of a graph 119866 is called to be efficient if forevery vertex 119909 in119866 |119873

119866[119909]cap119878| = 1 if119866 is a undirected graph

or |119873minus

119866[119909] cap 119878| = 1 if 119866 is a directed graph The concept of an

efficient set was proposed by Bange et al [137] in 1988 In theliterature an efficient set is also called a perfect dominatingset (see Livingston and Stout [138] for an explanation)

From definition if 119878 is an efficient dominating set of agraph 119866 then 119878 is certainly an independent set and everyvertex not in 119878 is adjacent to exactly one vertex in 119878

It is also clear from definition that a dominating set 119878 isefficient if and only if N

119866[119878] = 119873

119866[119909] 119909 isin 119878 for the

undirected graph 119866 or N+

119866[119878] = 119873

+

119866[119909] 119909 isin 119878 for the

digraph119866 is a partition of119881(119866) where the induced subgraphby119873

119866[119909] or119873+

119866[119909] is a star or an out-star with the root 119909

The efficient domination has important applications inmany areas such as error-correcting codes and receivesmuch attention in the late years

The concept of efficient dominating sets is a measureof the efficiency of domination in graphs Unfortunately asshown in [137] not every graph has an efficient dominatingset andmoreover it is anNP-complete problem to determinewhether a given graph has an efficient dominating set Inaddition it was shown by Clark [139] in 1993 that for a widerange of 119901 almost every random undirected graph 119866 isin

G(120592 119901) has no efficient dominating sets This means thatundirected graphs possessing an efficient dominating set arerare However it is easy to show that every undirected graphhas an orientationwith an efficient dominating set (see Bangeet al [140])

In 1993 Barkauskas and Host [141] showed that deter-mining whether an arbitrary oriented graph has an efficientdominating set is NP-complete Even so the existence ofefficient dominating sets for some special classes of graphs hasbeen examined see for example Dejter and Serra [142] andLee [143] for Cayley graph Gu et al [144] formeshes and toriObradovic et al [145] Huang and Xu [36] Reji Kumar andMacGillivray [146] for circulant graphs Harary graphs andtori and Van Wieren et al [147] for cube-connected cycles

In this section we introduce results of the bondagenumber for some graphs with efficient dominating sets dueto Huang and Xu [36]

111 Results for General Graphs In this subsection we intro-duce some results on bondage numbers obtained by applyingefficient dominating sets We first state the two followinglemmas

Lemma 189 For any 119896-regular graph or digraph 119866 of order 119899120574(119866) ⩾ 119899(119896 + 1) with equality if and only if 119866 has an efficientdominating set In addition if 119866 has an efficient dominatingset then every efficient dominating set is certainly a 120574-set andvice versa

Let 119890 be an edge and 119878 a dominating set in 119866 We say that119890 supports 119878 if 119890 isin (119878 119878) where (119878 119878) = (119909 119910) isin 119864(119866) 119909 isin 119878 119910 isin 119878 Denote by 119904(119866) the minimum number of edgeswhich support all 120574-sets in 119866

Lemma 190 For any graph or digraph 119866 119887(119866) ⩾ 119904(119866) withequality if 119866 is regular and has an efficient dominating set

30 International Journal of Combinatorics

A graph 119866 is called to be vertex-transitive if its automor-phism group Aut(119866) acts transitively on its vertex-set 119881(119866)A vertex-transitive graph is regular Applying Lemmas 189and 190 Huang and Xu obtained some results on bondagenumbers for vertex-transitive graphs or digraphs

Theorem 191 (Huang and Xu [36] 2008) For any vertex-transitive graph or digraph 119866 of order 119899

119887 (119866) ⩾

lceil119899

2120574 (119866)rceil 119894119891 119866 119894119904 119906119899119889119894119903119890119888119905119890119889

lceil119899

120574 (119866)rceil 119894119891 119866 119894119904 119889119894119903119890119888119905119890119889

(82)

Theorem192 (Huang andXu [36] 2008) Let119866 be a 119896-regulargraph of order 119899 Then

119887(119866) ⩽ 119896 if119866 is undirected and 119899 ⩾ 120574(119866)(119896+1)minus119896+1119887(119866) ⩽ 119896+1+ℓ if119866 is directed and 119899 ⩾ 120574(119866)(119896+1)minusℓwith 0 ⩽ ℓ ⩽ 119896 minus 1

Next we introduce a better upper bound on 119887(119866) To thisaim we need the following concept which is a generalizationof the concept of the edge-covering of a graph 119866 For 1198811015840

sube

119881(119866) and 1198641015840 sube 119864(119866) we say that 1198641015840covers 1198811015840 and call 1198641015840an edge-covering for 1198811015840 if there exists an edge (119909 119910) isin 1198641015840 forany vertex 119909 isin 1198811015840 For 119910 isin 119881(119866) let 1205731015840[119910] be the minimumcardinality over all edge-coverings for119873minus

119866[119910]

Theorem 193 (Huang and Xu [36] 2008) If 119866 is a 119896-regulargraph with order 120574(119866)(119896 + 1) then 119887(119866) ⩽ 1205731015840[119910] for any 119910 isin119881(119866)

Theupper bound on 119887(119866) given inTheorem 193 is tight inview of 119887(119862

119899) for a cycle or a directed cycle 119862

119899(seeTheorems

1 and 185 resp)It is easy to see that for a 119896-regular graph 119866 lceil(119896 + 1)2rceil ⩽

1205731015840

[119910] ⩽ 119896 when 119866 is undirected and 1205731015840[119910] = 119896 + 1 when 119866 isdirected By this fact and Lemma 189 the following theoremis merely a simple combination of Theorems 191 and 193

Theorem 194 (Huang and Xu [36] 2008) Let 119866 be a vertex-transitive graph of degree 119896 If 119866 has an efficient dominatingset then

lceil119896 + 1

2rceil ⩽ 119887 (119866) ⩽ 119896 119894119891 119866 119894119904 119906119899119889119894119903119890119888119905119890119889

119887 (119866) = 119896 + 1 119894119891 119866 119894119904 119889119894119903119890119888119905119890119889

(83)

Theorem 195 (Huang and Xu [36] 2008) If 119866 is an undi-rected vertex-transitive cubic graph with order 4120574(119866) and girth119892(119866) ⩽ 5 then 119887(119866) = 2

The proof of Theorem 195 leads to a byproduct If 119892 =5 let (119906

1 119906

2 119906

3 119906

4 119906

5) be a cycle and let D

119894be the family

of efficient dominating sets containing 119906119894for each 119894 isin

1 2 3 4 5 ThenD119894capD

119895= 0 for 119894 = 1 2 5 Then 119866 has

at least 5119904 efficient dominating sets where 119904 = 119904(119866) But thereare only 4s (=ns120574) distinct efficient dominating sets in119866This

contradiction implies that an undirected vertex-transitivecubic graph with girth five has no efficient dominating setsBut a similar argument for 119892(119866) = 3 4 or 119892(119866) ⩾ 6 couldnot give any contradiction This is consistent with the resultthat CCC(119899) a vertex-transitive cubic graph with girth 119899 if3 ⩽ 119899 ⩽ 8 or girth 8 if 119899 ⩾ 9 has efficient dominating setsfor all 119899 ⩾ 3 except 119899 = 5 (see Theorem 199 in the followingsubsection)

112 Results for Cayley Graphs In this subsection we useTheorem 194 to determine the exact values or approximativevalues of bondage numbers for some special vertex-transitivegraphs by characterizing the existence of efficient dominatingsets in these graphs

Let Γ be a nontrivial finite group 119878 a nonempty subset ofΓ without the identity element of Γ A digraph 119866 is defined asfollows

119881 (119866) = Γ (119909 119910) isin 119864 (119866) lArrrArr 119909minus1

119910 isin 119878

for any 119909 119910 isin Γ(84)

is called a Cayley digraph of the group Γ with respect to119878 denoted by 119862

Γ(119878) If 119878minus1 = 119904minus1 119904 isin 119878 = 119878 then

119862Γ(119878) is symmetric and is called a Cayley undirected graph

a Cayley graph for short Cayley graphs or digraphs arecertainly vertex-transitive (see Xu [1])

A circulant graph 119866(119899 119878) of order 119899 is a Cayley graph119862119885119899

(119878) where 119885119899= 0 119899 minus 1 is the addition group of

order 119899 and 119878 is a nonempty subset of119885119899without the identity

element and hence is a vertex-transitive digraph of degree|119878| If 119878minus1 = 119878 then119866(119899 119878) is an undirected graph If 119878 = 1 119896where 2 ⩽ 119896 ⩽ 119899 minus 2 we write 119866(119899 1 119896) for 119866(119899 1 119896) or119866(119899 plusmn1 plusmn119896) and call it a double-loop circulant graph

Huang and Xu [36] showed that for directed 119866 =

119866(119899 1 119896) lceil1198993rceil ⩽ 120574(119866) ⩽ lceil1198992rceil and 119866 has an efficientdominating set if and only if 3 | 119899 and 119896 equiv 2 (mod 3) fordirected 119866 = 119866(119899 1 119896) and 119896 = 1198992 lceil1198995rceil ⩽ 120574(119866) ⩽ lceil1198993rceiland 119866 has an efficient dominating set if and only if 5 | 119899 and119896 equiv plusmn2 (mod 5) By Theorem 194 we obtain the bondagenumber of a double loop circulant graph if it has an efficientdominating set

Theorem 196 (Huang and Xu [36] 2008) Let 119866 be a double-loop circulant graph 119866(119899 1 119896) If 119866 is directed with 3 | 119899and 119896 equiv 2 (mod 3) or 119866 is undirected with 5 | 119899 and119896 equiv plusmn2 (mod 5) then 119887(119866) = 3

The 119898 times 119899 torus is the Cartesian product 119862119898times 119862

119899of

two cycles and is a Cayley graph 119862119885119898times119885119899

(119878) where 119878 =

(0 1) (1 0) for directed cycles and 119878 = (0 plusmn1) (plusmn1 0) forundirected cycles and hence is vertex-transitive Gu et al[144] showed that the undirected torus119862

119898times119862

119899has an efficient

dominating set if and only if both 119898 and 119899 are multiplesof 5 Huang and Xu [36] showed that the directed torus119862119898times 119862

119899has an efficient dominating set if and only if both

119898 and 119899 are multiples of 3 Moreover they found a necessarycondition for a dominating set containing the vertex (0 0)in 119862

119898times 119862

119899to be efficient and obtained the following

result

International Journal of Combinatorics 31

Theorem 197 (Huang and Xu [36] 2008) Let 119866 = 119862119898times 119862

119899

If 119866 is undirected and both119898 and 119899 are multiples of 5 or if 119866 isdirected and both119898 and 119899 are multiples of 3 then 119887(119866) = 3

The hypercube 119876119899

is the Cayley graph 119862Γ(119878)

where Γ = 1198852times sdot sdot sdot times 119885

2= (119885

2)119899 and 119878 =

100 sdot sdot sdot 0 010 sdot sdot sdot 0 00 sdot sdot sdot 01 Lee [143] showed that119876119899has an efficient dominating set if and only if 119899 = 2119898 minus 1

for a positive integer119898 ByTheorem 194 the following resultis obtained immediately

Theorem 198 (Huang and Xu [36] 2008) If 119899 = 2119898 minus 1 for apositive integer119898 then 2119898minus1

⩽ 119887(119876119899) ⩽ 2

119898

minus 1

Problem 10 Determine the exact value of 119887(119876119899) for 119899 = 2119898minus1

The 119899-dimensional cube-connected cycle denoted byCCC(119899) is constructed from the 119899-dimensional hypercube119876119899by replacing each vertex 119909 in 119876

119899with an undirected cycle

119862119899of length 119899 and linking the 119894th vertex of the 119862

119899to the 119894th

neighbor of 119909 It has been proved that CCC(119899) is a Cayleygraph and hence is a vertex-transitive graph with degree 3Van Wieren et al [147] proved that CCC(119899) has an efficientdominating set if and only if 119899 = 5 From Theorems 194 and195 the following result can be derived

Theorem 199 (Huang and Xu [36] 2008) Let 119866 = 119862119862119862(119899)be the 119899-dimensional cube-connected cycles with 119899 ⩾ 3 and119899 = 5 Then 120574(119866) = 1198992119899minus2 and 2 ⩽ 119887(119866) ⩽ 3 In addition119887(119862119862119862(3)) = 119887(119862119862119862(4)) = 2

Problem 11 Determine the exact value of 119887(CCC(119899)) for 119899 ⩾5

Acknowledgments

This work was supported by NNSF of China (Grants nos11071233 61272008) The author would like to express hisgratitude to the anonymous referees for their kind sugges-tions and comments on the original paper to ProfessorSheikholeslami and Professor Jafari Rad for sending thereferences [128] and [81] which resulted in this version

References

[1] J-M Xu Theory and Application of Graphs vol 10 of NetworkTheory and Applications Kluwer Academic Dordrecht TheNetherlands 2003

[2] S THedetniemi andR C Laskar ldquoBibliography on dominationin graphs and some basic definitions of domination parame-tersrdquo Discrete Mathematics vol 86 no 1ndash3 pp 257ndash277 1990

[3] TW Haynes S T Hedetniemi and P J Slater Fundamentals ofDomination in Graphs vol 208 ofMonographs and Textbooks inPure and Applied Mathematics Marcel Dekker New York NYUSA 1998

[4] TWHaynes S THedetniemi and P J Slater EdsDominationin Graphs vol 209 of Monographs and Textbooks in Pure andAppliedMathematicsMarcelDekker NewYorkNYUSA 1998

[5] M R Garey andD S JohnsonComputers and Intractability WH Freeman and Company San Francisco Calif USA 1979

[6] H B Walikar and B D Acharya ldquoDomination critical graphsrdquoNational Academy Science Letters vol 2 pp 70ndash72 1979

[7] D Bauer F Harary J Nieminen and C L Suffel ldquoDominationalteration sets in graphsrdquo Discrete Mathematics vol 47 no 2-3pp 153ndash161 1983

[8] J F Fink M S Jacobson L F Kinch and J Roberts ldquoThebondage number of a graphrdquo Discrete Mathematics vol 86 no1ndash3 pp 47ndash57 1990

[9] F-T Hu and J-M Xu ldquoThe bondage number of (119899 minus 3)-regulargraph G of order nrdquo Ars Combinatoria to appear

[10] U Teschner ldquoNew results about the bondage number of agraphrdquoDiscreteMathematics vol 171 no 1ndash3 pp 249ndash259 1997

[11] B L Hartnell and D F Rall ldquoA bound on the size of a graphwith given order and bondage numberrdquo Discrete Mathematicsvol 197-198 pp 409ndash413 1999 16th British CombinatorialConference (London 1997)

[12] B L Hartnell and D F Rall ldquoA characterization of trees inwhich no edge is essential to the domination numberrdquo ArsCombinatoria vol 33 pp 65ndash76 1992

[13] J Topp and P D Vestergaard ldquo120572119896- and 120574

119896-stable graphsrdquo

Discrete Mathematics vol 212 no 1-2 pp 149ndash160 2000[14] A Meir and J W Moon ldquoRelations between packing and

covering numbers of a treerdquo Pacific Journal of Mathematics vol61 no 1 pp 225ndash233 1975

[15] B L Hartnell L K Joslashrgensen P D Vestergaard and CA Whitehead ldquoEdge stability of the k-domination numberof treesrdquo Bulletin of the Institute of Combinatorics and ItsApplications vol 22 pp 31ndash40 1998

[16] F-T Hu and J-M Xu ldquoOn the complexity of the bondage andreinforcement problemsrdquo Journal of Complexity vol 28 no 2pp 192ndash201 2012

[17] B L Hartnell and D F Rall ldquoBounds on the bondage numberof a graphrdquo Discrete Mathematics vol 128 no 1ndash3 pp 173ndash1771994

[18] Y-L Wang ldquoOn the bondage number of a graphrdquo DiscreteMathematics vol 159 no 1ndash3 pp 291ndash294 1996

[19] G H Fricke TW Haynes S M Hedetniemi S T Hedetniemiand R C Laskar ldquoExcellent treesrdquo Bulletin of the Institute ofCombinatorics and Its Applications vol 34 pp 27ndash38 2002

[20] V Samodivkin ldquoThe bondage number of graphs good and badverticesrdquoDiscussiones Mathematicae GraphTheory vol 28 no3 pp 453ndash462 2008

[21] J E Dunbar T W Haynes U Teschner and L VolkmannldquoBondage insensitivity and reinforcementrdquo in Domination inGraphs vol 209 of Monographs and Textbooks in Pure andAppliedMathematics pp 471ndash489 Dekker NewYork NY USA1998

[22] H Liu and L Sun ldquoThe bondage and connectivity of a graphrdquoDiscrete Mathematics vol 263 no 1ndash3 pp 289ndash293 2003

[23] A-H Esfahanian and S L Hakimi ldquoOn computing a con-ditional edge-connectivity of a graphrdquo Information ProcessingLetters vol 27 no 4 pp 195ndash199 1988

[24] R C Brigham P Z Chinn and R D Dutton ldquoVertexdomination-critical graphsrdquoNetworks vol 18 no 3 pp 173ndash1791988

[25] V Samodivkin ldquoDomination with respect to nondegenerateproperties bondage numberrdquoThe Australasian Journal of Com-binatorics vol 45 pp 217ndash226 2009

[26] U Teschner ldquoA new upper bound for the bondage numberof graphs with small domination numberrdquo The AustralasianJournal of Combinatorics vol 12 pp 27ndash35 1995

32 International Journal of Combinatorics

[27] J Fulman D Hanson and G MacGillivray ldquoVertex dom-ination-critical graphsrdquoNetworks vol 25 no 2 pp 41ndash43 1995

[28] U Teschner ldquoA counterexample to a conjecture on the bondagenumber of a graphrdquo Discrete Mathematics vol 122 no 1ndash3 pp393ndash395 1993

[29] U Teschner ldquoThe bondage number of a graph G can be muchgreater than Δ(G)rdquo Ars Combinatoria vol 43 pp 81ndash87 1996

[30] L Volkmann ldquoOn graphs with equal domination and coveringnumbersrdquoDiscrete AppliedMathematics vol 51 no 1-2 pp 211ndash217 1994

[31] L A Sanchis ldquoMaximum number of edges in connected graphswith a given domination numberrdquoDiscreteMathematics vol 87no 1 pp 65ndash72 1991

[32] S Klavzar and N Seifter ldquoDominating Cartesian products ofcyclesrdquoDiscrete AppliedMathematics vol 59 no 2 pp 129ndash1361995

[33] H K Kim ldquoA note on Vizingrsquos conjecture about the bondagenumberrdquo Korean Journal of Mathematics vol 10 no 2 pp 39ndash42 2003

[34] M Y Sohn X Yuan and H S Jeong ldquoThe bondage number of1198623times119862

119899rdquo Journal of the KoreanMathematical Society vol 44 no

6 pp 1213ndash1231 2007[35] L Kang M Y Sohn and H K Kim ldquoBondage number of the

discrete torus 119862119899times 119862

4rdquo Discrete Mathematics vol 303 no 1ndash3

pp 80ndash86 2005[36] J Huang and J-M Xu ldquoThe bondage numbers and efficient

dominations of vertex-transitive graphsrdquoDiscrete Mathematicsvol 308 no 4 pp 571ndash582 2008

[37] C J Xiang X Yuan andM Y Sohn ldquoDomination and bondagenumber of119862

3times119862

119899rdquoArs Combinatoria vol 97 pp 299ndash310 2010

[38] M S Jacobson and L F Kinch ldquoOn the domination numberof products of graphs Irdquo Ars Combinatoria vol 18 pp 33ndash441984

[39] Y-Q Li ldquoA note on the bondage number of a graphrdquo ChineseQuarterly Journal of Mathematics vol 9 no 4 pp 1ndash3 1994

[40] F-T Hu and J-M Xu ldquoBondage number of mesh networksrdquoFrontiers ofMathematics inChina vol 7 no 5 pp 813ndash826 2012

[41] R Frucht and F Harary ldquoOn the corona of two graphsrdquoAequationes Mathematicae vol 4 pp 322ndash325 1970

[42] K Carlson and M Develin ldquoOn the bondage number of planarand directed graphsrdquoDiscreteMathematics vol 306 no 8-9 pp820ndash826 2006

[43] U Teschner ldquoOn the bondage number of block graphsrdquo ArsCombinatoria vol 46 pp 25ndash32 1997

[44] J Huang and J-M Xu ldquoNote on conjectures of bondagenumbers of planar graphsrdquo Applied Mathematical Sciences vol6 no 65ndash68 pp 3277ndash3287 2012

[45] L Kang and J Yuan ldquoBondage number of planar graphsrdquoDiscrete Mathematics vol 222 no 1ndash3 pp 191ndash198 2000

[46] M Fischermann D Rautenbach and L Volkmann ldquoRemarkson the bondage number of planar graphsrdquo Discrete Mathemat-ics vol 260 no 1ndash3 pp 57ndash67 2003

[47] J Hou G Liu and J Wu ldquoBondage number of planar graphswithout small cyclesrdquoUtilitasMathematica vol 84 pp 189ndash1992011

[48] J Huang and J-M Xu ldquoThe bondage numbers of graphs withsmall crossing numbersrdquo Discrete Mathematics vol 307 no 15pp 1881ndash1897 2007

[49] Q Ma S Zhang and J Wang ldquoBondage number of 1-planargraphrdquo Applied Mathematics vol 1 no 2 pp 101ndash103 2010

[50] J Hou and G Liu ldquoA bound on the bondage number of toroidalgraphsrdquoDiscreteMathematics Algorithms and Applications vol4 no 3 Article ID 1250046 5 pages 2012

[51] Y-C Cao J-M Xu and X-R Xu ldquoOn bondage number oftoroidal graphsrdquo Journal of University of Science and Technologyof China vol 39 no 3 pp 225ndash228 2009

[52] Y-C Cao J Huang and J-M Xu ldquoThe bondage number ofgraphs with crossing number less than fourrdquo Ars Combinatoriato appear

[53] H K Kim ldquoOn the bondage numbers of some graphsrdquo KoreanJournal of Mathematics vol 11 no 2 pp 11ndash15 2004

[54] A Gagarin and V Zverovich ldquoUpper bounds for the bondagenumber of graphs on topological surfacesrdquo Discrete Mathemat-ics 2012

[55] J Huang ldquoAn improved upper bound for the bondage numberof graphs on surfacesrdquoDiscrete Mathematics vol 312 no 18 pp2776ndash2781 2012

[56] A Gagarin and V Zverovich ldquoThe bondage number of graphson topological surfaces and Teschnerrsquos conjecturerdquo DiscreteMathematics vol 313 no 6 pp 796ndash808 2013

[57] V Samodivkin ldquoNote on the bondage number of graphs ontopological surfacesrdquo httparxivorgabs12086203

[58] E J Cockayne R M Dawes and S T Hedetniemi ldquoTotaldomination in graphsrdquoNetworks vol 10 no 3 pp 211ndash219 1980

[59] R Laskar J Pfaff S M Hedetniemi and S T HedetniemildquoOn the algorithmic complexity of total dominationrdquo Society forIndustrial and Applied Mathematics vol 5 no 3 pp 420ndash4251984

[60] J Pfaff R C Laskar and S T Hedetniemi ldquoNP-completeness oftotal and connected domination and irredundance for bipartitegraphsrdquo Tech Rep 428 Department of Mathematical SciencesClemson University 1983

[61] M A Henning ldquoA survey of selected recent results on totaldomination in graphsrdquoDiscrete Mathematics vol 309 no 1 pp32ndash63 2009

[62] V R Kulli and D K Patwari ldquoThe total bondage number ofa graphrdquo in Advances in Graph Theory pp 227ndash235 VishwaGulbarga India 1991

[63] F-T Hu Y Lu and J-M Xu ldquoThe total bondage number ofgrid graphsrdquo Discrete Applied Mathematics vol 160 no 16-17pp 2408ndash2418 2012

[64] J Cao M Shi and B Wu ldquoResearch on the total bondagenumber of a spectial networkrdquo in Proceedings of the 3rdInternational Joint Conference on Computational Sciences andOptimization (CSO rsquo10) pp 456ndash458 May 2010

[65] N Sridharan MD Elias and V S A Subramanian ldquoTotalbondage number of a graphrdquo AKCE International Journal ofGraphs and Combinatorics vol 4 no 2 pp 203ndash209 2007

[66] N Jafari Rad and J Raczek ldquoSome progress on total bondage ingraphsrdquo Graphs and Combinatorics 2011

[67] T W Haynes and P J Slater ldquoPaired-domination and thepaired-domatic numberrdquoCongressusNumerantium vol 109 pp65ndash72 1995

[68] T W Haynes and P J Slater ldquoPaired-domination in graphsrdquoNetworks vol 32 no 3 pp 199ndash206 1998

[69] X Hou ldquoA characterization of 2120574 120574119875-treesrdquoDiscrete Mathemat-

ics vol 308 no 15 pp 3420ndash3426 2008[70] K E Proffitt T W Haynes and P J Slater ldquoPaired-domination

in grid graphsrdquo Congressus Numerantium vol 150 pp 161ndash1722001

International Journal of Combinatorics 33

[71] H Qiao L KangM Cardei andD-Z Du ldquoPaired-dominationof treesrdquo Journal of Global Optimization vol 25 no 1 pp 43ndash542003

[72] E Shan L Kang and M A Henning ldquoA characterizationof trees with equal total domination and paired-dominationnumbersrdquo The Australasian Journal of Combinatorics vol 30pp 31ndash39 2004

[73] J Raczek ldquoPaired bondage in treesrdquo Discrete Mathematics vol308 no 23 pp 5570ndash5575 2008

[74] J A Telle and A Proskurowski ldquoAlgorithms for vertex parti-tioning problems on partial k-treesrdquo SIAM Journal on DiscreteMathematics vol 10 no 4 pp 529ndash550 1997

[75] G S Domke J H Hattingh M A Henning and L R MarkusldquoRestrained domination in treesrdquoDiscreteMathematics vol 211no 1ndash3 pp 1ndash9 2000

[76] G S Domke J H Hattingh M A Henning and L R MarkusldquoRestrained domination in graphs with minimum degree twordquoJournal of Combinatorial Mathematics and Combinatorial Com-puting vol 35 pp 239ndash254 2000

[77] G S Domke J H Hattingh S T Hedetniemi R C Laskarand L R Markus ldquoRestrained domination in graphsrdquo DiscreteMathematics vol 203 no 1ndash3 pp 61ndash69 1999

[78] J H Hattingh and E J Joubert ldquoAn upper bound for therestrained domination number of a graph with minimumdegree at least two in terms of order and minimum degreerdquoDiscrete Applied Mathematics vol 157 no 13 pp 2846ndash28582009

[79] M A Henning ldquoGraphs with large restrained dominationnumberrdquo Discrete Mathematics vol 197-198 pp 415ndash429 199916th British Combinatorial Conference (London 1997)

[80] J H Hattingh and A R Plummer ldquoRestrained bondage ingraphsrdquo Discrete Mathematics vol 308 no 23 pp 5446ndash54532008

[81] N Jafari Rad R Hasni J Raczek and L Volkmann ldquoTotalrestrained bondage in graphsrdquoActaMathematica Sinica vol 29no 6 pp 1033ndash1042 2013

[82] J Cyman and J Raczek ldquoOn the total restrained dominationnumber of a graphrdquoThe Australasian Journal of Combinatoricsvol 36 pp 91ndash100 2006

[83] M A Henning ldquoGraphs with large total domination numberrdquoJournal of Graph Theory vol 35 no 1 pp 21ndash45 2000

[84] D-X Ma X-G Chen and L Sun ldquoOn total restraineddomination in graphsrdquoCzechoslovakMathematical Journal vol55 no 1 pp 165ndash173 2005

[85] B Zelinka ldquoRemarks on restrained domination and totalrestrained domination in graphsrdquo Czechoslovak MathematicalJournal vol 55 no 2 pp 393ndash396 2005

[86] W Goddard and M A Henning ldquoIndependent domination ingraphs a survey and recent resultsrdquo Discrete Mathematics vol313 no 7 pp 839ndash854 2013

[87] J H Zhang H L Liu and L Sun ldquoIndependence bondagenumber and reinforcement number of some graphsrdquo Transac-tions of Beijing Institute of Technology vol 23 no 2 pp 140ndash1422003

[88] K D Dharmalingam Studies in Graph Theorymdashequitabledomination and bottleneck domination [PhD thesis] MaduraiKamaraj University Madurai India 2006

[89] GDeepakND Soner andAAlwardi ldquoThe equitable bondagenumber of a graphrdquo Research Journal of Pure Algebra vol 1 no9 pp 209ndash212 2011

[90] E Sampathkumar and H B Walikar ldquoThe connected domina-tion number of a graphrdquo Journal of Mathematical and PhysicalSciences vol 13 no 6 pp 607ndash613 1979

[91] K White M Farber and W Pulleyblank ldquoSteiner trees con-nected domination and strongly chordal graphsrdquoNetworks vol15 no 1 pp 109ndash124 1985

[92] M Cadei M X Cheng X Cheng and D-Z Du ldquoConnecteddomination in Ad Hoc wireless networksrdquo in Proceedings ofthe 6th International Conference on Computer Science andInformatics (CSampI rsquo02) 2002

[93] J F Fink and M S Jacobson ldquon-domination in graphsrdquo inGraph Theory with Applications to Algorithms and ComputerScience Wiley-Interscience Publications pp 283ndash300 WileyNew York NY USA 1985

[94] M Chellali O Favaron A Hansberg and L Volkmann ldquok-domination and k-independence in graphs a surveyrdquo Graphsand Combinatorics vol 28 no 1 pp 1ndash55 2012

[95] Y Lu X Hou J-M Xu andN Li ldquoTrees with uniqueminimump-dominating setsrdquo Utilitas Mathematica vol 86 pp 193ndash2052011

[96] Y Lu and J-M Xu ldquoThe p-bondage number of treesrdquo Graphsand Combinatorics vol 27 no 1 pp 129ndash141 2011

[97] Y Lu and J-M Xu ldquoThe 2-domination and 2-bondage numbersof grid graphs A manuscriptrdquo httparxivorgabs12044514

[98] M Krzywkowski ldquoNon-isolating 2-bondage in graphsrdquo TheJournal of the Mathematical Society of Japan vol 64 no 4 2012

[99] M A Henning O R Oellermann and H C Swart ldquoBoundson distance domination parametersrdquo Journal of CombinatoricsInformation amp System Sciences vol 16 no 1 pp 11ndash18 1991

[100] F Tian and J-M Xu ldquoBounds for distance domination numbersof graphsrdquo Journal of University of Science and Technology ofChina vol 34 no 5 pp 529ndash534 2004

[101] F Tian and J-M Xu ldquoDistance domination-critical graphsrdquoApplied Mathematics Letters vol 21 no 4 pp 416ndash420 2008

[102] F Tian and J-M Xu ldquoA note on distance domination numbersof graphsrdquo The Australasian Journal of Combinatorics vol 43pp 181ndash190 2009

[103] F Tian and J-M Xu ldquoAverage distances and distance domina-tion numbersrdquo Discrete Applied Mathematics vol 157 no 5 pp1113ndash1127 2009

[104] Z-L Liu F Tian and J-M Xu ldquoProbabilistic analysis of upperbounds for 2-connected distance k-dominating sets in graphsrdquoTheoretical Computer Science vol 410 no 38ndash40 pp 3804ndash3813 2009

[105] M A Henning O R Oellermann and H C Swart ldquoDistancedomination critical graphsrdquo Journal of Combinatorial Mathe-matics and Combinatorial Computing vol 44 pp 33ndash45 2003

[106] T J Bean M A Henning and H C Swart ldquoOn the integrityof distance domination in graphsrdquo The Australasian Journal ofCombinatorics vol 10 pp 29ndash43 1994

[107] M Fischermann and L Volkmann ldquoA remark on a conjecturefor the (kp)-domination numberrdquoUtilitasMathematica vol 67pp 223ndash227 2005

[108] T Korneffel D Meierling and L Volkmann ldquoA remark on the(22)-domination numberrdquo Discussiones Mathematicae GraphTheory vol 28 no 2 pp 361ndash366 2008

[109] Y Lu X Hou and J-M Xu ldquoOn the (22)-domination numberof treesrdquo Discussiones Mathematicae Graph Theory vol 30 no2 pp 185ndash199 2010

34 International Journal of Combinatorics

[110] H Li and J-M Xu ldquo(dm)-domination numbers of m-connected graphsrdquo Rapports de Recherche 1130 LRI UniversiteParis-Sud 1997

[111] S M Hedetniemi S T Hedetniemi and T V Wimer ldquoLineartime resource allocation algorithms for treesrdquo Tech Rep URI-014 Department of Mathematical Sciences Clemson Univer-sity 1987

[112] M Farber ldquoDomination independent domination and dualityin strongly chordal graphsrdquo Discrete Applied Mathematics vol7 no 2 pp 115ndash130 1984

[113] G S Domke and R C Laskar ldquoThe bondage and reinforcementnumbers of 120574

119891for some graphsrdquoDiscrete Mathematics vol 167-

168 pp 249ndash259 1997[114] G S Domke S T Hedetniemi and R C Laskar ldquoFractional

packings coverings and irredundance in graphsrdquo vol 66 pp227ndash238 1988

[115] E J Cockayne P A Dreyer Jr S M Hedetniemi andS T Hedetniemi ldquoRoman domination in graphsrdquo DiscreteMathematics vol 278 no 1ndash3 pp 11ndash22 2004

[116] I Stewart ldquoDefend the roman empirerdquo Scientific American vol281 pp 136ndash139 1999

[117] C S ReVelle and K E Rosing ldquoDefendens imperiumromanum a classical problem in military strategyrdquoThe Ameri-can Mathematical Monthly vol 107 no 7 pp 585ndash594 2000

[118] A Bahremandpour F-T Hu S M Sheikholeslami and J-MXu ldquoOn the Roman bondage number of a graphrdquo DiscreteMathematics Algorithms and Applications vol 5 no 1 ArticleID 1350001 15 pages 2013

[119] N Jafari Rad and L Volkmann ldquoRoman bondage in graphsrdquoDiscussiones Mathematicae Graph Theory vol 31 no 4 pp763ndash773 2011

[120] K Ebadi and L PushpaLatha ldquoRoman bondage and Romanreinforcement numbers of a graphrdquo International Journal ofContemporary Mathematical Sciences vol 5 pp 1487ndash14972010

[121] N Dehgardi S M Sheikholeslami and L Volkman ldquoOn theRoman k-bondage number of a graphrdquo AKCE InternationalJournal of Graphs and Combinatorics vol 8 pp 169ndash180 2011

[122] S Akbari and S Qajar ldquoA note on Roman bondage number ofgraphsrdquo Ars Combinatoria to Appear

[123] F-T Hu and J-M Xu ldquoRoman bondage numbers of somegraphsrdquo httparxivorgabs11093933

[124] N Jafari Rad and L Volkmann ldquoOn the Roman bondagenumber of planar graphsrdquo Graphs and Combinatorics vol 27no 4 pp 531ndash538 2011

[125] S Akbari M Khatirinejad and S Qajar ldquoA note on the romanbondage number of planar graphsrdquo Graphs and Combinatorics2012

[126] B Bresar M A Henning and D F Rall ldquoRainbow dominationin graphsrdquo Taiwanese Journal of Mathematics vol 12 no 1 pp213ndash225 2008

[127] B L Hartnell and D F Rall ldquoOn dominating the Cartesianproduct of a graph and 120572krdquo Discussiones Mathematicae GraphTheory vol 24 no 3 pp 389ndash402 2004

[128] N Dehgardi S M Sheikholeslami and L Volkman ldquoThe k-rainbow bondage number of a graphrdquo to appear in DiscreteApplied Mathematics

[129] J von Neumann and O Morgenstern Theory of Games andEconomic Behavior Princeton University Press Princeton NJUSA 1944

[130] C BergeTheorıe des Graphes et ses Applications 1958[131] O Ore Theory of Graphs vol 38 American Mathematical

Society Colloquium Publications 1962[132] E Shan and L Kang ldquoBondage number in oriented graphsrdquoArs

Combinatoria vol 84 pp 319ndash331 2007[133] J Huang and J-M Xu ldquoThe bondage numbers of extended

de Bruijn and Kautz digraphsrdquo Computers amp Mathematics withApplications vol 51 no 6-7 pp 1137ndash1147 2006

[134] J Huang and J-M Xu ldquoThe total domination and total bondagenumbers of extended de Bruijn and Kautz digraphsrdquoComputersamp Mathematics with Applications vol 53 no 8 pp 1206ndash12132007

[135] Y Shibata and Y Gonda ldquoExtension of de Bruijn graph andKautz graphrdquo Computers amp Mathematics with Applications vol30 no 9 pp 51ndash61 1995

[136] X Zhang J Liu and JMeng ldquoThebondage number in completet-partite digraphsrdquo Information Processing Letters vol 109 no17 pp 997ndash1000 2009

[137] D W Bange A E Barkauskas and P J Slater ldquoEfficient dom-inating sets in graphsrdquo in Applications of Discrete Mathematicspp 189ndash199 SIAM Philadelphia Pa USA 1988

[138] M Livingston and Q F Stout ldquoPerfect dominating setsrdquoCongressus Numerantium vol 79 pp 187ndash203 1990

[139] L Clark ldquoPerfect domination in random graphsrdquo Journal ofCombinatorial Mathematics and Combinatorial Computing vol14 pp 173ndash182 1993

[140] D W Bange A E Barkauskas L H Host and L H ClarkldquoEfficient domination of the orientations of a graphrdquo DiscreteMathematics vol 178 no 1ndash3 pp 1ndash14 1998

[141] A E Barkauskas and L H Host ldquoFinding efficient dominatingsets in oriented graphsrdquo Congressus Numerantium vol 98 pp27ndash32 1993

[142] I J Dejter and O Serra ldquoEfficient dominating sets in CayleygraphsrdquoDiscrete AppliedMathematics vol 129 no 2-3 pp 319ndash328 2003

[143] J Lee ldquoIndependent perfect domination sets in Cayley graphsrdquoJournal of Graph Theory vol 37 no 4 pp 213ndash219 2001

[144] WGu X Jia and J Shen ldquoIndependent perfect domination setsin meshes tori and treesrdquo Unpublished

[145] N Obradovic J Peters and G Ruzic ldquoEfficient domination incirculant graphswith two chord lengthsrdquo Information ProcessingLetters vol 102 no 6 pp 253ndash258 2007

[146] K Reji Kumar and G MacGillivray ldquoEfficient domination incirculant graphsrdquo Discrete Mathematics vol 313 no 6 pp 767ndash771 2013

[147] D Van Wieren M Livingston and Q F Stout ldquoPerfectdominating sets on cube-connected cyclesrdquo Congressus Numer-antium vol 97 pp 51ndash70 1993

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 International Journal of Combinatorics

on the domination number Here we are concerned witha particular graph alternation the removal of edges from agraph

Graphs with domination numbers changed upon theremoval of an edge were first investigated by Walikar andAcharya [6] in 1979 A graph is called edge-domination-critical graph if 120574(119866 minus 119890) gt 120574(119866) for every edge 119890 in 119866 Theedge-domination-critical graph was characterized by Baueret al [7] in 1983 that is a graph is edge-domination-criticalif and only if it is the union of stars

However for lots of graphs the domination number is outof the range of one-edge removal It is immediate that 120574(119867) ⩾120574(119866) for any spanning subgraph 119867 of 119866 Every graph 119866 hasa spanning forest 119879 with 120574(119866) = 120574(119879) and so in general agraph has a nonempty subset 119865 sube 119864(119866) for which 120574(119866 minus 119865) =120574(119866)

Then it is natural for the alternation to be generalized tothe removal of several edges which is just enough to enlargethe domination number That is the idea of the bondagenumber

Ameasure of the efficiency of a domination in graphs wasfirst given by Bauer et al [7] in 1983 who called this measureas domination line-stability defined as theminimumnumberof lines (ie edges) which when removed from119866 increases 120574

In 1990 Fink et al [8] formally introduced the bondagenumber as a parameter for measuring the vulnerability of theinterconnection network under link failure The minimumdominating set of sites plays an important role in the networkfor it dominates the whole network with the minimum costSowemust consider whether its function remains goodwhenthe network is attacked Suppose that someone such as asaboteur does not know which sites in the network take partin the dominating role but does know that the set of thesespecial sites corresponds to aminimumdominating set in therelated graph Then how many links does he has to attack sothat the cost cannot remains the same in order to dominatethewhole networkThatminimumnumber of links is just thebondage number

The bondage number 119887(119866) of a nonempty undirectedgraph 119866 is the minimum number of edges whose removalfrom119866 results in a graphwith larger domination numberTheprecise definition of the bondage number is as follows

119887 (119866) = min |119861| 119861 sube 119864 (119866) 120574 (119866 minus 119861) gt 120574 (119866) (2)

Since the domination number of every spanning subgraph ofa nonempty graph 119866 is at least as great as 120574(119866) the bondagenumber of a nonempty graph is well defined

We call such an edge-set 119861 with 120574(119866 minus 119861) gt 120574(119866)

the bondage set and the minimum one the minimumbondage set In fact if 119861 is a minimum bondage set then120574(119866 minus 119861) = 120574(119866) + 1 because the removal of one single edgecannot increase the domination number by more than oneIf 119887(119866) does not exist for example empty graphs we define119887(119866) = infin

It is quite difficult to compute the exact value of thebondage number for general graphs since it strongly dependson the domination number of the graphsMuchwork focusedon the bounds on the bondage number aswell as the restraints

on some particular classes of graphs or networks Thepurpose of this paper is to give a survey of results and researchmethods related to these topics for graphs and digraphsFor some results and research methods we will make somecomments to develop our further study

The rest of the paper is organized as follows Section 2gives some preliminary results and complexity Sections3 and 4 survey some upper bounds and lower boundsrespectively The results for some special classes of graphsand planar graphs are stated in Sections 5 and 6 respectivelyIn Section 7 we introduce some results on crossing numberrestraints In Sections 8 and 9 we are concerned about otherand generalized types of bondage numbers respectively InSection 10 we introduce some results for digraphs In thelast section we introduce some results for vertex-transitivegraphs by applying efficient dominating sets

2 Simplicity and Complexity

As we have known from Introduction the bondage numberis an important parameter for measuring the stability orthe vulnerability of a domination in a graph or a networkOur aim is to compute the bondage number for any givengraphs or networks One has determined the exact value ofthe bondage number for some graphs with simple structureFor arbitrarily given graph however it has been proved thatdetermining its bondage number is NP-hard

21 Exact Values for Ordinary Graphs We begin our inves-tigation of the bondage number by computing its value forseveral well-known classes of graphs with simple structureIn 1990 Fink et al [8] proposed the concept of the bondagenumber and completely determined the exact values ofbondage numbers of some ordinary graphs such as completegraphs paths cycles and complete multipartite graphs

Theorem 1 (Fink et al [8] 1990) The exact values of bondagenumbers of the following class of graphs are completely deter-mined

119887 (119870119899) = lceil

119899

2rceil

119887 (119875119899) =

2 119894119891 119899 equiv 1 (mod 3) 1 119900119905ℎ119890119903119908119894119904119890

119887 (119862119899) =

3 119894119891 119899 equiv 1 (mod 3) 2 119900119905ℎ119890119903119908119894119904119890

119887 (11987011989911198992119899119905

)

=

lceil119895

2rceil 119894119891 119899

119895= 1 119886119899119889 119899

119895+1⩾ 2 119891119900119903 119904119900119898119890 119895

1 ⩽ 119895 lt 119905

2119905 minus 1 119894119891 1198991= 119899

2= sdot sdot sdot = 119899

119905= 2

119905minus1

sum

119894=1

119899119894

119900119905ℎ119890119903119908119894119904119890

(3)

The complete graph119870119899is the unique (119899minus1)-regular graph

of order 119899 ⩾ 2 119887(119870119899) = lceil1198992rceil by Theorem 1 The 119905-partite

International Journal of Combinatorics 3

graph 119870119905(2) with 119905 = 1198992 is an (119899 minus 2)-regular graph 119866 of

order 119899 ⩾ 2 and not unique 119887(119866) = 119899 minus 1 by Theorem 1for an even integer 119899 ⩾ 4 For an (119899 minus 3)-regular graph119866 of order 119899 ⩾ 4 Hu and Xu [9] obtained the followingresult

Theorem 2 (Hu and Xu [9]) 119887(119866) = 119899 minus 3 for any (119899 minus 3)-regular graph 119866 of order 119899 ⩾ 4

Up to the present no results have been known for 119896-regular graphs with 3 ⩽ 119896 ⩽ 119899 minus 4 For general graphs thereare the following two results

Theorem 3 (Teschner [10] 1997) If 119866 is a nonempty graphwith a unique minimum dominating set then 119887(119866) = 1

Theorem4 (Bauer et al [7] 1983) If any vertex of a graph119866 isadjacent with two ormore vertices of degree one then 119887(119866) = 1

Bauer et al [7] observed that the star is the uniquegraph with the property that the bondage number is 1 andthe deletion of any edge results in the domination numberincreasing Motivated by this fact Hartnell and Rall [11]proposed the concept of uniformly bonded graphs A graph iscalled to be uniformly bonded if it has bondage number 119887 andthe deletion of any 119887 edges results in a graph with increaseddomination number Unfortunately there are a few uniformlybonded graphs

Theorem 5 (Hartnell and Rall [11] 1999) The only uniformlybonded graphs with bondage number 2 are 119862

3and 119875

4 The

unique uniformly bonded graph with bondage number 3 is 1198624

There are no such graphs for bondage number greater than 3

As we mentioned to compute the exact value of bondagenumber for a graph strongly depends upon its dominationnumber In this sense studying the bondage number cangreatly inspire onersquos research concerned with dominationsHowever determining the exact value of domination numberfor a given graph is quite difficult In fact even if theexact value of the domination number for some graph isdetermined it is still very difficulty to compute the valueof the bondage number for that graph For example for thehypercube 119876

119899 we have 120574(119876

119899) = 2

119899minus1 but we have not yetdetermined 119887(119876

119899) for any 119899 ⩾ 2

Perhaps Theorems 3 and 4 provide an approach tocompute the exact value of bondage number for some graphsby establishing some sufficient conditions for 119887(119866) = 119887In fact we will see later that Theorem 3 plays an importantrole in determining the exact values of the bondage numbersfor some graphs Thus to study the bondage number it isimportant to present various characterizations of graphs witha unique minimum dominating set

22 Characterizations of Trees For trees Bauer et al [7] in1983 from the point of view of the domination line-stabilityindependently and Fink et al [8] in 1990 from the pointof view of the domination edge-vulnerability obtained thefollowing result

119908

Figure 1 Tree 119865119905

Theorem 6 For any nontrivial tree 119879 119887(119879) ⩽ 2

By Theorem 6 it is natural to classify all trees accordingto their bondage numbers Fink et al [8] proved that aforbidden subgraph characterization to classify trees withdifferent bondage numbers is impossible since they provedthat if 119865 is a forest then 119865 is an induced subgraph of a tree 119879with 119887(119879) = 1 and a tree 1198791015840 with 119887(1198791015840) = 2 However theypointed out that the complexity of calculating the bondagenumber of a tree of order 119899 is at most 119874(1198992) by methodicallyremoving each pair of edges

Even so some characterizations whether a tree hasbondage number 1 or 2 have been found by several authorssee for example [10 12 13]

First we describe the method due to Hartnell and Rall[12] by which all trees with bondage number 2 can be con-structed inductively An important tree119865

119905in the construction

is shown in Figure 1 To characterize this construction weneed some terminologies

(1) attach a path 119875119899to a vertex 119909 of a tree which means

to link 119909 and one end-vertex of the 119875119899by an edge

(2) attach 119865119905to a vertex 119909 which means to link 119909 and a

vertex 119910 of 119865119905by an edge

The following are four operations on a tree 119879

Type 1 attach a 1198752to 119909 isin 119881(119879) where 120574(119879 minus 119909) = 120574(119879) and 119909

belongs to at least one 120574-set of 119879 (such a vertex existssay one end-vertex of 119875

5)

Type 2 attach a 1198753to 119909 isin 119881(119879) where 120574(119879 minus 119909) lt 120574(119879)

Type 3 attach 1198651to 119909 isin 119881(119879) where 119909 belongs to at least one

120574-set of 119879Type 4 attach 119865

119905 119905 ⩾ 2 to 119909 isin 119881(119879) where 119909 can be any

vertex of 119879

LetC = 119879 119879 is a tree and 119879 = 1198701 119879 = 119875

4 and 119879 = 119865

119905

for some 119905 ⩾ 2 or 119879 can be obtained from 1198754or 119865

119905(119905 ⩾ 2) by

a finite sequence of operations of Types 1 2 3 4

Theorem 7 (Hartnell and Rall [12] 1992) A tree has bondagenumber 2 if and only if it belongs toC

Looking at different minimum dominating sets of a treeTeschner [10] presented a totally different characterizationof the set of trees having bondage number 1 They defineda vertex to be universal if it belongs to each minimumdominating set and to be idle if it does not belong to anyminimum dominating set

4 International Journal of Combinatorics

Theorem 8 (Teschner [10] 1997) A tree 119879 has bondagenumber 1 if and only if 119879 has a universal vertex or an edge119909119910 satisfying

(1) 119909 and 119910 are neither universal nor idle and

(2) all neighbors of 119909 and 119910 (except for 119909 and 119910) are idle

For a positive integer 119896 a subset 119868 sube 119881(119866) is called a 119896-independent set (also called a 119896-packing) if 119889

119866(119909 119910) gt 119896 for

any two distinct vertices 119909 and 119910 in 119868 When 119896 = 1 1-set is thenormal independent setThemaximumcardinality among all119896-independent sets is called the 119896-independence number (or119896-packing number) of 119866 denoted by 120572

119896(119866) A 119896-independent

set 119868 is called an 120572119896-set if |119868| = 120572

119896(119866) A graph 119866 is said to be

120572119896-stable if 120572

119896(119866) = 120572

119896(119866 minus 119890) for every edge 119890 of 119866 There are

two important results on 119896-independent sets

Proposition 9 (Topp and Vestergaard [13] 2000) A tree 119879 is120572119896-stable if and only if 119879 has a unique 120572

119896-set

Proposition 10 (Meir and Moon [14] 1975) 1205722(119866) ⩽ 120574(119866)

for any connected graph 119866 with equality for any tree

Hartnell et al [15] independently and Topp and Vester-gaard [13] also gave a constructive characterization of treeswith bondage number 2 and applying Proposition 10 pre-sented another characterization of those trees

Theorem 11 (Hartnell et al [15] 1998 Top and Vestergaard[13] 2000) 119887(119879) = 2 for a tree 119879 if and only if 119879 has a unique1205722-set

According to this characterization Hartnell et al [15]presented a linear algorithm for determining the bondagenumber of a tree

In this subsection we introduce three characterizationsfor trees with bondage number 1 or 2 The characterizationin Theorem 7 is constructive constructing all trees withbondage number 2 a natural and straightforward methodby a series of graph-operations The characterization inTheorem 8 is a little advisable by describing the inherentproperty of trees with bondage number 1 The characteriza-tion in Theorem 11 is wonderful by using a strong graph-theoretic concept 120572

119896-set In fact this characterization is a

byproduct of some results related to 120572119896-sets for trees It is

that this characterization closed the relation between twoconcepts the bondage number and the 119896-independent setand hence is of research value and important significance

23 Complexity for General Graphs Asmentioned above thebondage number of a tree can be determined by a linear timealgorithm According to this algorithm we can determinewithin polynomial time the domination number of any treeby removing each edge and verifyingwhether the dominationnumber is enlarged according to the known linear timealgorithm for domination numbers of trees

However it is impossible to find a polynomial timealgorithm for bondage numbers of general graphs If suchan algorithm 119860 exists then the domination number of any

nonempty undirected graph 119866 can be determined withinpolynomial time by repeatedly using 119860 Let 119866

0= 119866 and

119866119894+1= 119866

119894minus119861

119894 where 119861

119894is the minimum edge set of 119866

119894found

by 119860 such that 120574(119866119894minus 119861

119894) = 120574(119866

119894minus1) + 1 for each 119894 = 0 1

we can always find the minimum 119861119894whose removal from 119866

119894

enlarges the domination number until 119866119896= 119866

119896minus1minus 119861

119896minus1

is empty for some 119896 ⩾ 1 though 119861119896minus1

is not empty Then120574(119866) = 120574(119866

119896) minus 119896 = 120592(119866) minus 119896 As known to all if NP = 119875

the minimum dominating set problem is NP-complete andso polynomial time algorithms for the bondage number donot exist unless NP = 119875

In fact Hu and Xu [16] have recently shown that theproblem determining the bondage number of general graphsis NP-hard

Problem 1 Consider the decision problemBondage ProblemInstance a graph 119866 and a positive integer 119887 (⩽ 120576(119866))Question is 119887(119866) ⩽ 119887

Theorem 12 (Hu and Xu [16] 2012) The bondage problem isNP-hard

Thebasicway of the proof is to followGarey and Johnsonrsquostechniques for proving NP-hardness [5] by describing apolynomial transformation from the known NP-completeproblem 3-satisfiability problem

Theorem 12 shows that we are unable to find a polynomialtime algorithm to determine bondage numbers of generalgraphs unless NP = 119875 At the same time this result alsoshows that the following study on the bondage number is ofimportant significance

(i) Find approximation polynomial algorithms with per-formance ratio as small as possible

(ii) Find the lower and upper bounds with difference assmall as possible

(iii) Determine exact values for some graphs speciallywell-known networks

Unfortunately we cannot provewhether or not determin-ing the bondage number is NP-problem since for any subset119861 sub 119864(119866) it is not clear that there is a polynomial algorithmto verify 120574(119866 minus 119861) gt 120574(119866) Since the problem of determiningthe domination number is NP-complete we conjecture thatit is not in NPThis is a worthwhile task to be studied further

Motivated by the linear time algorithm of Hartnell et alto compute the bondage number of a tree we can makean attempt to consider whether there is a polynomial timealgorithm to compute the bondage number for some specialclasses of graphs such as planar graphs Cayley graphs orgraphs with some restrictions of graph-theoretical parame-ters such as degree diameter connectivity and dominationnumber

3 Upper Bounds

ByTheorem 12 since we cannot find a polynomial time algo-rithm for determining the exact values of bondage numbers

International Journal of Combinatorics 5

of general graphs it is weightily significative to establishsome sharp bounds on the bondage number of a graph Inthis section we survey several known upper bounds on thebondage number in terms of some other graph-theoreticalparameters

31 Most Basic Upper Bounds We start this subsection witha simple observation

Observation 1 (Teschner [10] 1997) Let 119867 be a spanningsubgraph obtained from a graph119866 by removing 119896 edgesThen119887(119866) ⩽ 119887(119867) + 119896

If we select a spanning subgraph 119867 such that 119887(119867) = 1thenObservation 1 yields some upper bounds on the bondagenumber of a graph 119866 For example take119867 = 119866 minus 119864

119909+ 119890 and

119896 = 119889119866(119909) minus 1 where 119890 isin 119864

119909 the following result can be

obtained

Theorem 13 (Bauer et al [7] 1983) If there exists at least onevertex 119909 in a graph 119866 such that 120574(119866 minus 119909) ⩾ 120574(119866) then 119887(119866) ⩽119889119866(119909) ⩽ Δ(119866)

The following early result obtained can be derived fromObservation 1 by taking119867 = 119866minus119864

119909119910and 119896 = 119889(119909)+119889(119910)minus2

Theorem 14 (Bauer et al [7] 1983 Fink et al [8] 1990)119887(119866) ⩽ 119889(119909) + 119889(119910) minus 1 for any two adjacent vertices 119909 and119910 in a graph 119866 that is

119887 (119866) ⩽ min119909119910isin119864(119866)

119889119866(119909) + 119889

119866(119910) minus 1 (4)

This theorem gives a natural corollary obtained by severalauthors

Corollary 15 (Bauer et al [7] 1983 Fink et al [8] 1990) If 119866is a graphwithout isolated vertices then 119887(119866) ⩽ Δ(119866)+120575(119866)minus1

In 1999 Hartnell and Rall [11] extended Theorem 14 tothe following more general case which can be also derivedfrom Observation 1 by taking119867 = 119866 minus 119864

119909minus 119864

119910+ (119909 119911 119910) if

119889119866(119909 119910) = 2 where (119909 119911 119910) is a path of length 2 in 119866

Theorem 16 (Hartnell and Rall [11] 1999) 119887(119866) ⩽ 119889(119909) +119889(119910)minus1 for any distinct two vertices 119909 and 119910 in a graph119866with119889119866(119909 119910) ⩽ 2 that is

119887 (119866) ⩽ min119889119866(119909119910)⩽2

119889119866(119909) + 119889

119866(119910) minus 1 (5)

Corollary 17 (Fink et al [8] 1990) If a vertex of a graph 119866 isadjacent with two ormore vertices of degree one then 119887(119866) = 1

We remark that the bounds stated in Corollary 15 andTheorem 16 are sharp As indicated byTheorem 1 one class ofgraphs in which the bondage number achieves these boundsis the class of cycles whose orders are congruent to 1 modulo3

On the other hand Hartnell and Rall [17] sharpened theupper bound in Theorem 14 as follows which can be alsoderived from Observation 1

1198792(119909 119910)

1198794(119909 119910)

1198793(119909 119910)

1198791(119909 119910)

119909

119910

Figure 2 Illustration of 119879119894(119909 119910) for 119894 = 1 2 3 4

Theorem 18 (Hartnell and Rall [17] 1994) 119887(119866) ⩽ 119889119866(119909) +

119889119866(119910) minus 1 minus |119873

119866(119909) cap 119873

119866(119910)| for any two adjacent vertices 119909

and 119910 in a graph 119866 that is

119887 (119866) ⩽ min119909119910isin119864(119866)

119889119866(119909) + 119889

119866(119910) minus 1 minus

1003816100381610038161003816119873119866(119909) cap 119873

119866(119910)1003816100381610038161003816

(6)

These results give simple but important upper bounds onthe bondage number of a graph and is also the foundationof almost all results on bondage numbers upper boundsobtained till now

By careful consideration of the nature of the edges fromthe neighbors of119909 and119910Wang [18] further refined the boundin Theorem 18 For any edge 119909119910 isin 119864(119866) 119873

119866(119910) contains the

following four subsets

(1) 1198791(119909 119910) = 119873

119866[119909] cap 119873

119866(119910)

(2) 1198792(119909 119910) = 119908 isin 119873

119866(119910) 119873

119866(119908) sube 119873

119866(119910) minus 119909

(3) 1198793(119909 119910) = 119908 isin 119873

119866(119910) 119873

119866(119908) sube 119873

119866(119911)minus119909 for some

119911 isin 119873119866(119909) cap 119873

119866(119910)

(4) 1198794(119909 119910) = 119873

119866(119910) (119879

1(119909 119910) cup 119879

2(119909 119910) cup 119879

3(119909 119910))

The illustrations of 1198791(119909 119910) 119879

2(119909 119910) 119879

3(119909 119910) and

1198794(119909 119910) are shown in Figure 2 (corresponding vertices

pointed by dashed arrows)

Theorem 19 (Wang [18] 1996) For any nonempty graph 119866

119887 (119866) ⩽ min119909119910isin119864(119866)

119889119866(119909) +

10038161003816100381610038161198794 (119909 119910)1003816100381610038161003816 (7)

The graph in Figure 2 shows that the upper bound givenin Theorem 19 is better than that in Theorems 16 and 18for the upper bounds obtained from these two theorems are119889119866(119909)+119889

119866(119910)minus1 = 11 and 119889

119866(119909)+119889

119866(119910)minus|119873

119866(119909)cap119873

119866(119910)| =

9 respectively while the upper bound given byTheorem 19 is119889119866(119909) + |119879

4(119909 119910)| = 6

The following result is also an improvement ofTheorem 14 in which 119905 = 2

6 International Journal of Combinatorics

Theorem 20 (Teschner [10] 1997) If 119866 contains a completesubgraph 119870

119905with 119905 ⩾ 2 then 119887(119866) ⩽ min

119909119910isin119864(119870119905)119889

119866(119909) +

119889119866(119910) minus 119905 + 1

Following Fricke et al [19] a vertex 119909 of a graph 119866 is 120574-good if 119909 belongs to some 120574-set of119866 and 120574-bad if 119909 belongs tono 120574-set of 119866 Let 119860(119866) be the set of 120574-good vertices and let119861(119866) be the set of 120574-bad vertices in 119866 Clearly 119860(119866) 119861(119866)is a partition of119881(119866) Note there exists some 119909 isin 119860 such that120574(119866 minus 119909) = 120574(119866) say one end-vertex of 119875

5 Samodivkin [20]

presented some sharp upper bounds for 119887(119866) in terms of 120574-good and 120574-bad vertices of 119866

Theorem 21 (Samodivkin [20] 2008) Let 119866 be a graph

(a) Let 119862(119866) = 119909 isin 119881(119866) 120574(119866minus119909) ⩾ 120574(119866) If 119862(119866) = 0then

119887 (119866) ⩽ min 119889119866(119909) + 120574 (119866) minus 120574 (119866 minus 119909) 119909 isin 119862 (119866)

(8)

(b) If 119861 = 0 then

119887 (119866) ⩽ min 1003816100381610038161003816119873119866(119909) cap 119860

1003816100381610038161003816 119909 isin 119861 (119866) (9)

Theorem 22 (Samodivkin [20] 2008) Let 119866 be a graph If119860+

(119866) = 119909 isin 119860(119866) 119889119866(119909) ⩾ 1 and 120574(119866 minus 119909) lt 120574(119866) = 0

then

119887 (119866) ⩽ min119909isin119860+(119866)119910isin119861(119866minus119909)

119889119866(119909) +

1003816100381610038161003816119873119866(119910) cap 119860 (119866 minus 119909)

1003816100381610038161003816

(10)

Proposition 23 (Samodivkin [20] 2008) Under the notationof Theorem 19 if 119909 isin 119860+

(119866) then (1198791(119909 119910) minus 119909) cup 119879

2(119909 119910) cup

1198793(119909 119910) sube 119873

119866(119910) 119861(119866 minus 119909)

By Proposition 23 if 119909 isin 119860+

(119866) then

119889119866(119909) + min

119910isin119873119866(119909)

10038161003816100381610038161198794 (119909 119910)

1003816100381610038161003816

⩾ 119889119866(119909) + min

119910isin119873119866(119909)

1003816100381610038161003816119873119866

(119910) cap 119860 (119866 minus 119909)1003816100381610038161003816

⩾ 119889119866(119909) + min

119910isin119861(119866minus119909)

1003816100381610038161003816119873119866

(119910) cap 119860 (119866 minus 119909)1003816100381610038161003816

(11)

Hence Theorem 19 can be seen to follow from Theorem 22Any graph 119866 with 119887(119866) achieving the upper bound inTheorem 19 can be used to show that the bound inTheorem 22 is sharp

Let 119905 ⩾ 2 be an integer Samodivkin [20] con-structed a very interesting graph 119866

119905to show that the upper

bound in Theorem 22 is better than the known bounds Let119867

0 119867

1 119867

2 119867

119905+1be mutually vertex-disjoint graphs such

that 1198670cong 119870

2 119867

119905+1cong 119870

119905+3and 119867

119894cong 119870

119905+3minus 119890 for each

119894 = 1 2 119905 Let 119881(1198670) = 119909 119910 119909

119905+1isin 119881(119867

119905+1) and

1198671

1198672

1198673

11990911

11990912

11990921 11990922

1199093

119909

119910

Figure 3 The graph 1198662

1199091198941 119909

1198942isin 119881(119867

119894) 119909

11989411199091198942notin 119864(119867

119894) for each 119894 = 1 2 119905 The

graph 119866119905is defined as follows

119881 (119866119905) =

119905+1

⋃

119896=0

119881 (119867119896)

119864 (119866119905)=(

119905+1

⋃

119896=0

119864 (119867119896))cup(

119905

⋃

119894=1

1199091199091198941 119909119909

1198942)cup119909119909

119905+1

(12)

Such a constructed graph119866119905is shown in Figure 3 when 119905 = 2

Observe that 120574(119866119905) = 119905 + 2 119860(119866

119905) = 119881(119866

119905) 119889

119866119905

(119909) =

2119905 + 2 119889119866119905

(119909119905+1) = 119905 + 3 119889

119866119905

(119910) = 1 and 119889119866119905

(119911) = 119905 + 2

for each 119911 isin 119881(119866119905minus 119909 119910 119909

119905+1) Moreover 120574(119866 minus 119910) lt 120574(119866)

and 120574(119866119905minus119911) = 120574(119866

119905) for any 119911 isin 119881(119866

119905) minus 119910 Hence each of

the bounds stated inTheorems 13ndash20 is greater than or equals119905 + 2

Consider the graph 119866119905minus 119909119910 Clearly 120574(119866

119905minus 119909119910) = 120574(119866

119905)

and

119861 (119866119905minus 119909119910) = 119861 (119866

119905minus 119910)

= 119909 cup 119881 (119867119905+1minus 119909

119905+1) cup (

119905

⋃

119896=1

1199091198961

1199091198962

)

(13)

Therefore 119873119866119905

(119909) cap 119866(119866119905minus 119910) = 119909

119905+1 which implies that

the upper bound stated in Theorem 22 is equal to 119889119866119905

(119910) +

|119909119905+1| = 2 Clearly 119887(119866

119905) = 2 and hence this bound is sharp

for 119866119905

From the graph 119866119905 we obtain the following statement

immediately

Proposition 24 For every integer 119905 ⩾ 2 there is a graph 119866such that the difference between any upper bound stated inTheorems 13ndash20 and the upper bound in Theorem 22 is equalto 119905

Although Theorem 22 supplies us with the upper boundthat is closer to 119887(119866) for some graph 119866 than what any one of

International Journal of Combinatorics 7

Theorems 13ndash20 provides it is not easy to determine the sets119860+

(119866) and 119861(119866) mentioned in Theorem 22 for an arbitrarygraph 119866 Thus the upper bound given in Theorem 22 is oftheoretical importance but not applied since until now wehave not found a new class of graphswhose bondage numbersare determined byTheorem 22

The above-mentioned upper bounds on the bondagenumber are involved in only degrees of two vertices Hartnelland Rall [11] established an upper bound on 119887(119866) in terms ofthe numbers of vertices and edges of 119866 with order 119899 For anyconnected graph 119866 let 120575(119866) represent the average degree ofvertices in 119866 that is 120575(119866) = (1119899)sum

119909isin119881119889119866(119909) Hartnell and

Rall first discovered the following proposition

Proposition 25 For any connected graph 119866 there exist twovertices 119909 and 119910 with distance at most two and with theproperty that 119889

119866(119909) + 119889

119866(119910) ⩽ 2120575(119866)

Using Proposition 25 and Theorem 16 Hartnell and Rallgave the following bound

Theorem 26 (Hartnell and Rall [11] 1999) 119887(119866) ⩽ 2120575(119866) minus 1for any connected graph 119866

Note that 2120576 = 119899120575(119866) for any graph 119866 with order 119899 andsize 120576 Theorem 26 implies the following bound in terms oforder 119899 and size 120576

Corollary 27 119887(119866) ⩽ (4120576119899) minus 1 for any connected graph 119866with order 119899 and size 120576

A lower bound on 120576 in terms of order 119899 and the bondagenumber is obtained from Corollary 28 immediately

Corollary 28 120576 ⩾ (1198994)(119887(119866) + 1) for any connected graph 119866with order 119899 and size 120576

Hartnell and Rall [11] gave some graphs 119866 with order 119899to show that for each value of 119887(119866) the lower bound on 120576(119866)given in the Corollary 28 is sharp for some values of 119899

32 Bounds Implied by Connectivity Use 120581(119866) and 120582(119866) todenote the vertex-connectivity and the edge-connectivity ofa connected graph 119866 respectively which are the minimumnumbers of vertices and edges whose removal result in 119866disconnected The famous Whitney inequality is stated as120581(119866) ⩽ 120582(119866) ⩽ 120575(119866) for any graph or digraph 119866 Corollary 15is improved by several authors as follows

Theorem 29 (Hartnell and Rall [17] 1994 Teschner [10]1997) If 119866 is a connected graph then 119887(119866) ⩽ 120582(119866)+Δ(119866)minus 1

The upper bound given in Theorem 29 can be attainedFor example a cycle 119862

3119896+1with 119896 ⩾ 1 119887(119862

3119896+1) = 3 by

Theorem 1 Since 120581(1198623119896+1) = 120582(119862

3119896+1) = 2 we have 120581(119862

3119896+1)+

120582(1198623119896+1) minus 1 = 2 + 2 minus 1 = 3 = 119887(119862

3119896+1)

Motivated byCorollary 15Theorems 29 and theWhitneyinequality Dunbar et al [21] naturally proposed the followingconjecture

Figure 4 A graph119867 with 120574(119867) = 3 and 119887(119867) = 5

Conjecture 30 If119866 is a connected graph then 119887(119866) ⩽ Δ(119866)+120581(119866) minus 1

However Liu and Sun [22] presented a counterexampleto this conjecture They first constructed a graph 119867 showedin Figure 4 with 120574(119867) = 3 and 119887(119867) = 5 Then let 119866 be thedisjoint union of two copies of119867 by identifying two verticesof degree two They proved that 119887(119866) ⩾ 5 Clearly 119866 is a 4-regular graph with 120581(119866) = 1 and 120582(119866) = 2 and so 119887(119866) ⩽ 5byTheorem 29 Thus 119887(119866) = 5 gt 4 = Δ(119866) + 120581(119866) minus 1

With a suspicion of the relationship between the bondagenumber and the vertex-connectivity of a graph the followingconjecture is proposed

Conjecture 31 (Liu and Sun [22] 2003) For any positiveinteger 119903 there exists a connected graph 119866 such that 119887(119866) ⩾Δ(119866) + 120581(119866) + 119903

To the knowledge of the author until now no results havebeen known about this conjecture

We conclude this subsection with the following remarksFromTheorem 29 if Conjecture 31 holds for some connectedgraph119866 then 120582(119866) gt 120581(119866)+119903 which implies that119866 is of largeedge-connectivity and small vertex-connectivity Use 120585(119866) todenote the minimum edge-degree of 119866 that is

120585 (119866) = min119909119910isin119864(119866)

119889119866(119909) + 119889

119866(119910) minus 2 (14)

Theorem 14 implies the following result

Proposition 32 119887(119866) ⩽ 120585(119866) + 1 for any graph 119866

Use 1205821015840(119866) to denote the restricted edge-connectivity ofa connected graph 119866 which is the minimum number ofedgeswhose removal result in119866 disconnected and no isolatedvertices

Proposition 33 (Esfahanian and Hakimi [23] 1988) If 119866 isneither 119870

1119899nor 119870

3 then 1205821015840(119866) ⩽ 120585(119866)

Combining Propositions 32 and 33 we propose a conjec-ture

Conjecture 34 If a connected 119866 is neither 1198701119899

nor 1198703 then

119887(119866) ⩽ 120575(119866) + 1205821015840

(119866) minus 1

8 International Journal of Combinatorics

A cycle 1198623119896+1

also satisfies 119887(1198623119896+1) = 3 = 120575(119862

3119896+1) +

1205821015840

(1198623119896+1) minus 1 since 1205821015840(119862

3119896+1) = 120575(119862

3119896+1) = 2 for any integer

119896 ⩾ 1For the graph119867 shown in Figure 41205821015840(119867) = 4 and 120575(119867) =

2 and so 119887(119867) = 5 = 120575(119867) + 1205821015840(119867) minus 1For the 4-regular graph119866 constructed by Liu and Sun [22]

obtained from the disjoint union of two copies of the graph119867 showed in Figure 4 by identifying two vertices of degreetwo we have 119887(119866) ⩾ 5 Clearly 1205821015840(119866) = 2 Thus 119887(119866) = 5 =120575(119866) + 120582

1015840

(119866) minus 1For the 4-regular graph 119866

119905constructed by Samodivkin

[20] see Figure 3 for 119905 = 2 119887(119866119905) = 2 Clearly 120575(119866

119905) = 1

and 1205821015840(119866) = 2 Thus 119887(119866) = 2 = 120575(119866) + 1205821015840(119866) minus 1These examples show that if Conjecture 34 is true then the

given upper bound is tight

33 Bounds Implied by Degree Sequence Now let us return toTheorem 16 from which Teschner [10] obtained some otherbounds in terms of the degree sequencesThe degree sequence120587(119866) of a graph 119866 with vertex-set 119881 = 119909

1 119909

2 119909

119899 is the

sequence 120587 = (1198891 119889

2 119889

119899) with 119889

1⩽ 119889

2⩽ sdot sdot sdot ⩽ 119889

119899 where

119889119894= 119889

119866(119909

119894) for each 119894 = 1 2 119899 The following result is

essentially a corollary of Theorem 16

Theorem35 (Teschner [10] 1997) Let119866 be a nonempty graphwith degree sequence120587(119866) If120572

2(119866) = 119905 then 119887(119866) ⩽ 119889

119905+119889

119905+1minus

1

CombiningTheorem 35with Proposition 10 (ie 1205722(119866) ⩽

120574(119866)) we have the following corollary

Corollary 36 (Teschner [10] 1997) Let 119866 be a nonemptygraph with the degree sequence 120587(119866) If 120574(119866) = 120574 then 119887(119866) ⩽119889120574+ 119889

120574+1minus 1

In [10] Teschner showed that these two bounds are sharpfor arbitrarily many graphs Let119867 = 119862

3119896+1+ 119909

11199094 119909

11199093119896minus1

where 1198623119896+1

is a cycle (1199091 119909

2 119909

3119896+1 119909

1) for any integer

119896 ⩾ 2 Then 120574(119867) = 119896 + 1 and so 119887(119867) ⩽ 2 + 2 minus 1 = 3by Corollary 36 Since119862

3119896+1is a spanning subgraph of119867 and

120574(119866) = 120574(119867) Observation 1 yields that 119887(119867) ⩾ 119887(1198623119896+1) = 3

and so 119887(119867) = 3Although various upper bounds have been established

as the above we find that the appearance of these boundsis essentially based upon the local structures of a graphprecisely speaking the structures of the neighborhoods oftwo vertices within distance 2 Even if these bounds can beachieved by some special graphs it is more often not the caseThe reason lies essentially in the definition of the bondagenumber which is theminimumvalue among all bondage setsan integral property of a graph While it easy to find upperbounds just by choosing some bondage set the gap betweenthe exact value of the bondage number and such a boundobtained only from local structures of a graph is often largeFor example a star 119870

1Δ however large Δ is 119887(119870

1Δ) = 1

Therefore one has been longing for better bounds upon someintegral parameters However as what we will see below it isdifficult to establish such upper bounds

34 Bounds in 120574-Critical Graphs A graph119866 is called a vertexdomination-critical graph (vc-graph or 120574-critical for short) if120574(119866 minus 119909) lt 120574(119866) for any 119909 isin 119881(119866) proposed by Brighamet al [24] in 1988 Several families of graphs are known to be120574-critical From definition it is clear that if 119866 is a 120574-criticalgraph then 120574(119866) ⩾ 2The class of 120574-critical graphs with 120574 = 2is characterized as follows

Proposition 37 (Brigham et al [24] 1988) A graph 119866 with120574(119866) = 2 is a 120574-critical graph if and only if 119866 is a completegraph 119870

2119905(119905 ⩾ 2) with a perfect matching removed

A more interesting family is composed of the 119899-criticalgraphs 119866

119898119899defined for 119898 119899 ⩾ 2 by the circulant undirected

graph 119866(119873 plusmn119878) where 119873 = (119898 + 1)(119899 minus 1) + 1 and119878 = 1 2 lfloor1198982rfloor in which 119881(119866(119873 plusmn119878)) = 119873 and119864(119866(119873 plusmn119878)) = 119894119895 |119895 minus 119894| equiv 119904 (mod 119873) 119904 isin 119878

The reason why 120574-critical graphs are of special interest inthis context is easy to see that they play an important role instudy of the bondage number For instance it immediatelyfollows from Theorem 13 that if 119887(119866) gt Δ(119866) then 119866 is a 120574-critical graphThe 120574-critical graphs are defined exactly in thisway In order to find graphs 119866 with a high bondage number(ie higher than Δ(119866)) and beyond its general upper boundsfor the bondage number we therefore have to look at 120574-critical graphs

The bondage numbers of some 120574-critical graphs havebeen examined by several authors see for example [1020 25 26] From Theorem 13 we know that the bondagenumber of a graph 119866 is bounded from above by Δ(119866)if 119866 is not a 120574-critical graph For 120574-critical graphs it ismore difficult to find an upper bound We will see that thebondage numbers of 120574-critical graphs in general are noteven bounded from above by Δ + 119888 for any fixed naturalnumber 119888

In this subsection we introduce some upper bounds forthe bondage number of a 120574-critical graph By Proposition 37we easily obtain the following result

Theorem 38 If 119866 is a 120574-critical graph with 120574(119866) = 2 then119887(119866) ⩽ Δ + 1

In Section 4 by Theorem 59 we will see that the equalityin Theorem 38 holds that is 119887(119866) = Δ + 1 if 119866 is a 120574-criticalgraph with 120574(119866) = 2

Theorem 39 (Teschner [10] 1997) Let119866 be a 120574-critical graphwith degree sequence 120587(119866) Then 119887(119866) ⩽ maxΔ(119866) + 1 119889

1+

1198892+ sdot sdot sdot + 119889

120574minus1 where 120574 = 120574(119866)

As we mentioned above if 119866 is a 120574-critical graph with120574(119866) = 2 then 119887(119866) = Δ + 1 which shows that thebound given in Theorem 39 can be attained for 120574 = 2However we have not known whether this bound is tightfor general 120574 ⩾ 3 Theorem 39 gives the following corollaryimmediately

Corollary 40 (Teschner [10] 1997) Let 119866 be a 120574-criticalgraph with degree sequence 120587(119866) If 120574(119866) = 3 then 119887(119866) ⩽maxΔ(119866) + 1 120575(119866) + 119889

2

International Journal of Combinatorics 9

From Theorem 38 and Corollary 40 we have 119887(119866) ⩽2Δ(119866) if 119866 is a 120574-critical graph with 120574(119866) ⩽ 3 The followingresult shows that this bound is not tight

Theorem 41 (Teschner [26] 1995) Let119866 be a 120574-critical graphgraph with 120574(119866) ⩽ 3 Then 119887(119866) ⩽ (32)Δ(119866)

Until now we have not known whether the bound givenin Theorem 41 is tight or not We state two conjectures on120574-critical graphs proposed by Samodivkin [20] The first ofthem is motivated byTheorems 22 and 19

Conjecture 42 (Samodivkin [20] 2008) For every connectednontrivial 120574-critical graph 119866

min119909isin119860+(119866)119910isin119861(119866minus119909)

119889119866(119909) +

1003816100381610038161003816119873119866(119910) cap 119860 (119866 minus 119909)

1003816100381610038161003816 ⩽3

2Δ (119866)

(15)

To state the second conjecture we need the followingresult on 120574-critical graphs

Proposition 43 If119866 is a 120574-critical graph then 120592(119866) ⩽ (Δ(119866)+1)(120574(119866)minus1)+1 moreover if the equality holds then119866 is regular

The upper bound in Proposition 43 is sharp in thesense that equality holds for the infinite class of 120574-criticalgraphs 119866

119898119899defined in the beginning of this subsection In

Proposition 43 the first result is due to Brigham et al [24] in1988 the second is due to Fulman et al [27] in 1995

For a 120574-critical graph 119866 with 120574 = 3 by Proposition 43120592(119866) ⩽ 2Δ(119866) + 3

Theorem 44 (Teschner [26] 1995) If 119866 is a 120574-critical graphwith 120574 = 3 and 120592(119866) = 2Δ(119866) + 3 then 119887(119866) ⩽ Δ + 1

We have not known yet whether the equality inTheorem 44holds or notHowever Samodivkin proposed thefollowing conjecture

Conjecture 45 (Samodivkin [20] 2008) If 119866 is a 120574-criticalgraph with (Δ(119866)+1)(120574minus1)+1 vertices then 119887(119866) = Δ(119866)+1

In general based on Theorem 41 Teschner proposed thefollowing conjecture

Conjecture 46 (Teschner [26] 1995) If119866 is a 120574-critical graphthen 119887(119866) ⩽ (32)Δ(119866) for 120574 ⩾ 4

We conclude this subsection with some remarks Graphswhich are minimal or critical with respect to a given prop-erty or parameter frequently play an important role in theinvestigation of that property or parameter Not only are suchgraphs of considerable interest in their own right but also aknowledge of their structure often aids in the developmentof the general theory In particular when investigating anyfinite structure a great number of results are proven byinduction Consequently it is desirable to learn as much aspossible about those graphs that are critical with respect toa given property or parameter so as to aid and abet suchinvestigations

In this subsection we survey some results on the bondagenumber for 120574-critical graphs Although these results are notvery perfect they provides a feasible method to approach thebondage number from different viewpoints In particular themethods given in Teschner [26] worth further explorationand development

The following proposition is maybe useful for us tofurther investigate the bondage number of a 120574-critical graph

Proposition 47 (Brigham et al [24] 1988) If 119866 has anonisolated vertex 119909 such that the subgraph induced by119873

119866(119909)

is a complete graph then 119866 is not 120574-critical

This simple fact shows that any 120574-critical graph containsno vertices of degree one

35 Bounds Implied by Domination In the preceding sub-section we introduced some upper bounds on the bondagenumbers for 120574-critical graphs by consideration of domina-tions In this subsection we introduce related results for ageneral graph with given domination number

Theorem 48 (Fink et al [8] 1990) For any connected graph119866 of order 119899 (⩾2)

(a) 119887(119866) ⩽ 119899 minus 1

(b) 119887(119866) ⩽ Δ(119866) + 1 if 120574(119866) = 2

(c) 119887(119866) ⩽ 119899 minus 120574(119866) + 1

While the upper bound of 119899minus1 on 119887(119866) is not particularlygood for many classes of graphs (eg trees and cycles) it isan attainable bound For example if 119866 is a complete 119905-partitegraph 119870

119905(2) for 119905 = 1198992 and 119899 ⩽ 4 an even integer then the

three bounds on 119887(119866) in Theorem 48 are sharpTeschner [26] consider a graphwith 120574(119866) = 1 or 120574(119866) = 2

The next result is almost trivial but useful

Lemma 49 Let 119866 be a graph with order 119899 and 120574(119866) = 1 119905 thenumber of vertices of degree 119899 minus 1 Then 119887(119866) = lceil1199052rceil

Since 119905 ⩽ Δ(119866) clearly Lemma 49 yields the followingresult immediately

Theorem 50 (Teschner [26] 1995) 119887(119866) ⩽ (12)Δ+1 for anygraph 119866 with 120574(119866) = 1

For a complete graph1198702119896+1

119887(1198702119896+1) = 119896+1 = (12)Δ+1

which shows that the upper bound given in Theorem 50 canbe attained For a graph119866with 120574(119866) = 2 byTheorem 59 laterthe upper bound given inTheorem 48 (b) can be also attainedby a 2-critical graph (see Proposition 37)

36 Two Conjectures In 1990 when Fink et al [8] introducedthe concept of the bondage number they proposed thefollowing conjecture

Conjecture 51 119887(119866) ⩽ Δ(119866) + 1 for any nonempty graph 119866

10 International Journal of Combinatorics

Although some results partially support Conjecture 51Teschner [28] in 1993 found a counterexample to this con-jecture the cartesian product 119870

3times 119870

3 which shows that

119887(119866) = Δ(119866) + 2If a graph 119866 is a counterexample to Conjecture 51 it

must be a 120574-critical graph by Theorem 13 That is why thevertex domination-critical graphs are of special interest in theliterature

Now we return to Conjecture 51 Hartnell and Rall [17]and Teschner [29] independently proved that 119887(119866) can bemuch greater than Δ(119866) by showing the following result

Theorem 52 (Hartnell and Rall [17] 1994 Teschner [29]1996) For an integer 119899 ⩾ 3 let 119866

119899be the cartesian product

119870119899times 119870

119899 Then 119887(119866

119899) = 3(119899 minus 1) = (32)Δ(119866

119899)

This theorem shows that there exist no upper boundsof the form 119887(119866) ⩽ Δ(119866) + 119888 for any integer 119888 Teschner[26] proved that 119887(119866) ⩽ (32)Δ(119866) for any graph 119866 with120574(119866) ⩽ 2 (see Theorems 50 and 48) and for 120574-critical graphswith 120574(119866) = 3 and proposed the following conjecture

Conjecture 53 (Teschner [26] 1995) 119887(119866) ⩽ (32)Δ(119866) forany graph 119866

We believe that this conjecture is true but so far no muchwork about it has been known

4 Lower Bounds

Since the bondage number is defined as the smallest numberof edges whose removal results in an increase of dominationnumber each constructive method that creates a concretebondage set leads to an upper bound on the bondage numberFor that reason it is hard to find lower bounds Neverthelessthere are still a few lower bounds

41 Bounds Implied by Subgraphs The first lower boundon the bondage number is gotten in terms of its spanningsubgraph

Theorem 54 (Teschner [10] 1997) Let 119867 be a spanningsubgraph of a nonempty graph119866 If 120574(119867) = 120574(119866) then 119887(119867) ⩽119887(119866)

By Theorem 1 119887(119862119899) = 3 if 119899 equiv 1 (mod 3) and 119887(119862

119899) =

2 otherwise 119887(119875119899) = 2 if 119899 equiv 1 (mod 3) and 119887(119875

119899) = 1

otherwise From these results and Theorem 54 we get thefollowing two corollaries

Corollary 55 If119866 is hamiltonianwith order 119899 ⩾ 3 and 120574(119866) =lceil1198993rceil then 119887(119866) ⩾ 2 and in addition 119887(119866) ⩾ 3 if 119899 equiv 1 (mod3)

Corollary 56 If 119866 with order 119899 ⩾ 2 has a hamiltonian pathand 120574(119866) = lceil1198993rceil then 119887(119866) ⩾ 2 if 119899 equiv 1 (mod 3)

42 Other Bounds The vertex covering number 120573(119866) of 119866 isthe minimum number of vertices that are incident with all

Figure 5 A 120574-critical graph with 120573 = 120574 = 4 120575 = 2 and 119887 = 3

edges in 119866 If 119866 has no isolated vertices then 120574(119866) ⩽ 120573(119866)clearly In [30] Volkmann gave a lot of graphs with 120573 = 120574

Theorem 57 (Teschner [10] 1997) Let119866 be a graph If 120574(119866) =120573(119866) then

(a) 119887(119866) ⩾ 120575(119866)

(b) 119887(119866) ⩾ 120575(119866) + 1 if 119866 is a 120574-critical graph

The graph shown in Figure 5 shows that the bound givenin Theorem 57(b) is sharp For the graph 119866 it is a 120574-criticalgraph and 120574(119866) = 120573(119866) = 4 By Theorem 57 we have 119887(119866) ⩾3 On the other hand 119887(119866) ⩽ 3 by Theorem 16 Thus 119887(119866) =3

Proposition 58 (Sanchis [31] 1991) Let 119866 be a graph 119866 oforder 119899 (⩾ 6) and size 120576 If 119866 has no isolated vertices and3 ⩽ 120574(119866) ⩽ 1198992 then 120576 ⩽ ( 119899minus120574(119866)+1

2)

Using the idea in the proof of Theorem 57 every upperbound for 120574(119866) can lead to a lower bound for 119887(119866) Inthis way Teschner [10] obtained another lower bound fromProposition 58

Theorem59 (Teschner [10] 1997) Let119866 be a graph119866 of order119899 (⩾6) and size 120576 If 2 ⩽ 120574(119866) ⩽ 1198992 minus 1 then

(a) 119887(119866) ⩾ min120575(119866) 120576 minus ( 119899minus120574(119866)2)

(b) 119887(119866) ⩾ min120575(119866) + 1 120576 minus ( 119899minus120574(119866)2) if 119866 is a 120574-critical

graph

The lower bound in Theorem 59(b) is sharp for a classof 120574-critical graphs with domination number 2 Let 119866 be thegraph obtained from complete graph119870

2119905(119905 ⩾ 2) by removing

a perfect matching By Proposition 37 119866 is a 120574-critical graphwith 120574(119866) = 2 Then 119887(119866) ⩾ 120575(119866) + 1 = Δ(119866) + 1 byTheorem 59 and 119887(119866) ⩽ Δ(119866) + 1 by Theorem 38 and so119887(119866) = Δ(119866) + 1 = 2119905 minus 1

As far as the author knows there are no more lowerbounds in the present literature In view of applications ofthe bondage number a network is vulnerable if its bondagenumber is small while it is stable if its bondage number islargeTherefore better lower bounds let us learn better aboutthe stability of networks from this point of view Thus it isof great significance to seek more lower bounds for variousclasses of graphs

International Journal of Combinatorics 11

5 Results on Graph Operations

Generally speaking it is quite difficult to determine the exactvalue of the bondage number for a given graph since itstrongly depends on the dominating number of the graphThus determining bondage numbers for some special graphsis interesting if the dominating numbers of those graphs areknown or can be easily determined In this section we willintroduce some known results on bondage numbers for somespecial classes of graphs obtained by some graph operations

51 Cartesian Product Graphs Let 1198661= (119881

1 119864

1) and 119866

2=

(1198812 119864

2) be two graphs The Cartesian product of 119866

1and 119866

2

is an undirected graph denoted by 1198661times 119866

2 where 119881(119866

1times

1198662) = 119881

1times 119881

2 two distinct vertices 119909

11199092and 119910

11199102 where

1199091 119910

1isin 119881(119866

1) and 119909

2 119910

2isin 119881(119866

2) are linked by an edge in

1198661times 119866

2if and only if either 119909

1= 119910

1and 119909

21199102isin 119864(119866

2) or

1199092= 119910

2and 119909

11199101isin 119864(119866

1) The Cartesian product is a very

effective method for constructing a larger graph from severalspecified small graphs

Theorem 60 (Dunbar et al [21] 1998) For 119899 ⩾ 3

119887 (119862119899times 119870

2) =

2 119894119891 119899 equiv 0 119900119903 1 (mod 4) 3 119894119891 119899 equiv 3 (mod 4) 4 119894119891 119899 equiv 2 (mod 4)

(16)

For the Cartesian product 119862119899times 119862

119898of two cycles 119862

119899and

119862119898 where 119899 ⩾ 4 and 119898 ⩾ 3 Klavzar and Seifter [32]

determined 120574(119862119899times 119862

119898) for some 119899 and 119898 For example

120574(119862119899times119862

4) = 119899 for 119899 ⩾ 3 and 120574(119862

119899times119862

3) = 119899minuslfloor1198994rfloor for 119899 ⩾ 4

Kim [33] determined 119887(1198623times 119862

3) = 6 and 119887(119862

4times 119862

4) = 4 For

a general 119899 ⩾ 4 the exact values of the bondage numbers of119862119899times 119862

3and 119862

119899times 119862

4are determined as follows

Theorem 61 (Sohn et al [34] 2007) For 119899 ⩾ 4

119887 (119862119899times 119862

3) =

2 119894119891 119899 equiv 0 (mod 4) 4 119894119891 119899 equiv 1 119900119903 2 (mod 4) 5 119894119891 119899 equiv 3 (mod 4)

(17)

Theorem62 (Kang et al [35] 2005) 119887(119862119899times119862

4) = 4 for 119899 ⩾ 4

For larger 119898 and 119899 Huang and Xu [36] obtained thefollowing result see Theorem 197 for more details

Theorem 63 (Huang and Xu [36] 2008) 119887(1198625119894times119862

5119895) = 3 for

any positive integers 119894 and 119895

Xiang et al [37] determined that for 119899 ⩾ 5

119887 (119862119899times 119862

5)

= 3 if 119899 equiv 0 or 1 (mod 5) = 4 if 119899 equiv 2 or 4 (mod 5) ⩽ 7 if 119899 equiv 3 (mod 5)

(18)

For the Cartesian product 119875119899times119875

119898of two paths 119875

119899and 119875

119898

Jacobson and Kinch [38] determined 120574(119875119899times119875

2) = lceil(119899+1)2rceil

120574(119875119899times 119875

3) = 119899 minus lfloor(119899 minus 1)4rfloor and 120574(119875

119899times 119875

4) = 119899 + 1 if 119899 =

1 2 3 5 6 9 and 119899 otherwiseThe bondage number of119875119899times119875

119898

for 119899 ⩾ 4 and 2 ⩽ 119898 ⩽ 4 is determined as follows (Li [39] alsodetermined 119887(119875

119899times 119875

2))

Theorem 64 (Hu and Xu [40] 2012) For 119899 ⩾ 4

119887 (119875119899times 119875

2) =

1 119894119891 119899 equiv 1 (mod 2)2 119894119891 119899 equiv 0 (mod 2)

119887 (119875119899times 119875

3) =

1 119894119891 119899 equiv 1 119900119903 2 (mod 4)2 119894119891 119899 equiv 0 119900119903 3 (mod 4)

119887 (119875119899times 119875

4) = 1 119891119900119903 119899 ⩾ 14

(19)

From the proof of Theorem 64 we find that if 119875119899=

0 1 119899 minus 1 and 119875119898= 0 1 119898 minus 1 then the removal

of the vertex (0 0) in 119875119899times119875

119898does not change the domination

number If119898 increases the effect of (0 0) for the dominationnumber will be smaller and smaller from the probabilityTherefore we expect it is possible that 120574(119875

119899times 119875

119898minus (0 0)) =

120574(119875119899times 119875

119898) for119898 ⩾ 5 and give the following conjecture

Conjecture 65 119887(119875119899times 119875

119898) ⩽ 2 for119898 ⩾ 5 and 119899 ⩾ 4

52 Block Graphs and Cactus Graphs In this subsection weintroduce some results for corona graphs block graphs andcactus graphs

The corona1198661∘ 119866

2 proposed by Frucht and Harary [41]

is the graph formed from a copy of 1198661and 120592(119866

1) copies of

1198662by joining the 119894th vertex of 119866

1to the 119894th copy of 119866

2 In

particular we are concernedwith the corona119867∘1198701 the graph

formed by adding a new vertex V119894 and a new edge 119906

119894V119894for

every vertex 119906119894in119867

Theorem 66 (Carlson and Develin [42] 2006) 119887(119867 ∘ 1198701) =

120575(119867) + 1

This is a very important result which is often used toconstruct a graph with required bondage number In otherwords this result implies that for any given positive integer 119887there is a graph 119866 such that 119887(119866) = 119887

A block graph is a graph whose blocks are completegraphs Each block in a cactus graph is either a cycle or a119870

2

If each block of a graph is either a complete graph or a cyclethen we call this graph a block-cactus graph

Theorem67 (Teschner [43] 1997) 119887(119866) le Δ(119866) for any blockgraph 119866

Teschner [43] characterized all block graphs with 119887(119866) =Δ(119866) In the same paper Teschner found that 120574-criticalgraphs are instrumental in determining bounds for thebondage number of cactus and block graphs and obtained thefollowing result

Theorem 68 (Teschner [43] 1997) 119887(119866) ⩽ 3 for anynontrivial cactus graph 119866

This bound can be achieved by 119867 ∘ 1198701where 119867 is

a nontrivial cactus graph without vertices of degree one

12 International Journal of Combinatorics

by Theorem 66 In 1998 Dunbar et al [21] proposed thefollowing problem

Problem 2 Characterize all cactus graphs with bondagenumber 3

Some upper bounds for block-cactus graphs were alsoobtained

Theorem 69 (Dunbar et al [21] 1998) Let 119866 be a connectedblock-cactus graph with at least two blocksThen 119887(119866) ⩽ Δ(119866)

Theorem 70 (Dunbar et al [21] 1998) Let 119866 be a connectedblock-cactus graph which is neither a cactus graph nor a blockgraph Then 119887(119866) ⩽ Δ(119866) minus 1

53 Edge-Deletion and Edge-Contraction In order to obtainfurther results of the bondage number we may consider howthe bondage number changes under some operations of agraph 119866 which remain the domination number unchangedor preserve some property of 119866 such as planarity Twosimple operations satisfying these requirements are the edge-deletion and the edge-contraction

Theorem 71 (Huang and Xu [44] 2012) Let 119866 be any graphand 119890 isin 119864(119866) Then 119887(119866 minus 119890) ⩾ 119887(119866) minus 1 Moreover 119887(119866 minus 119890) ⩽119887(119866) if 120574(119866 minus 119890) = 120574(119866)

FromTheorem 1 119887(119862119899minus 119890) = 119887(119875

119899) = 119887(119862

119899) minus 1 for any 119890

in119862119899 which shows that the lower bound on 119887(119866minus119890) is sharp

The upper bound can reach for any graph 119866 with 119887(119866) ⩾ 2Next we consider the effect of the edge-contraction on

the bondage number Given a graph 119866 the contraction of 119866by the edge 119890 = 119909119910 denoted by 119866119909119910 is the graph obtainedfrom 119866 by contracting two vertices 119909 and 119910 to a new vertexand then deleting all multiedges It is easy to observe that forany graph 119866 120574(119866) minus 1 ⩽ 120574(119866119909119910) ⩽ 120574(119866) for any edge 119909119910 of119866

Theorem 72 (Huang and Xu [44] 2012) Let 119866 be any graphand 119909119910 be any edge in119866 If119873

119866(119909)cap119873

119866(119910) = 0 and 120574(119866119909119910) =

120574(119866) then 119887(119866119909119910) ⩾ 119887(119866)

We can use examples 119862119899and 119870

119899to show that the

conditions ofTheorem 72 are necessary and the lower boundinTheorem 72 is tight Clearly 120574(119862

119899) = lceil1198993rceil and 120574(119870

119899) = 1

for any edge 119909119910 119862119899119909119910 = 119862

119899minus1and 119870

119899119909119910 = 119870

119899minus1 By

Theorem 1 if 119899 equiv 1 (mod 3) then 120574(119862119899119909119910) lt 120574(119866) and

119887(119862119899119909119910) = 2 lt 3 = 119887(119862

119899) Thus the result in Theorem 72

is generally invalid without the hypothesis 120574(119866119909119910) = 120574(119866)Furthermore the condition 119873

119866(119909) cap 119873

119866(119910) = 0 cannot be

omitted even if 120574(119866119909119910) = 120574(119866) since for odd 119899 119887(119870119899119909119910) =

lceil(119899 minus 1)2rceil lt lceil1198992rceil = 119887(119870119899) by Theorem 1

On the other hand 119887(119862119899119909119910) = 119887(119862

119899) = 2 if 119899 equiv 0 (mod

3) and 119887(119870119899119909119910) = 119887(119870

119899) if 119899 is even which shows that the

equality in 119887(119866119909119910) ⩾ 119887(119866) may hold Thus the bound inTheorem 72 is tight However provided all the conditions119887(119866119909119910) can be arbitrarily larger than 119887(119866) Given a graph119867let119866 be the graph formed from119867∘119870

1by adding a new vertex

119909 and joining it to an vertex 119910 of degree one in119867 ∘ 1198701 Then

119866119909119910 = 119867 ∘ 1198701 120574(119866) = 120574(119866119909119910) and 119873

119866(119909) cap 119873

119866(119910) = 0

But 119887(119866) = 1 since 120574(119866 minus 119909119910) = 120574(119866119909119910) + 1 and 119887(119866119909119910) =120575(119867) + 1 byTheorem 66The gap between 119887(119866) and 119887(119866119909119910)is 120575(119867)

6 Results on Planar Graphs

From Section 2 we have seen that the bondage numberfor a tree has been completely solved Moreover a lineartime algorithm to compute the bondage number of a tree isdesigned by Hartnell et al [15] It is quite natural to considerthe bondage number for a planar graph In this section westate some results and problems on the bondage number fora planar graph

61 Conjecture on Bondage Number As mentioned inSection 3 the bondage number can be much larger than themaximum degree But for a planar graph 119866 the bondagenumber 119887(119866) cannot exceed Δ(119866) too much It is clear that119887(119866) ⩽ Δ(119866) + 4 by Corollary 15 since 120575(119866) ⩽ 5 for anyplanar graph119866 In 1998Dunbar et al [21] posed the followingconjecture

Conjecture 73 If 119866 is a planar graph then 119887(119866) ⩽ Δ(119866) + 1

Because of its attraction it immediately became the focusof attention when this conjecture is proposed In fact themain aim concerning the research of the bondage number fora planar graph is focused on this conjecture

It has been mentioned in Theorems 1 and 60 that119887(119862

3119896+1) = 3 = Δ + 1 and 119887(119862

4119896+2times 119870

2) = 4 = Δ + 1 It

is known that 119887(1198706minus 119872) = 5 = Δ + 1 where119872 is a perfect

matching of the complete graph1198706These examples show that

if Conjecture 73 is true then the upper bound is sharp for2 ⩽ Δ ⩽ 4

Here we note that it is sufficient to prove this conjecturefor connected planar graphs since the bondage number of adisconnected graph is simply the minimum of the bondagenumbers of its components

The first paper attacking this conjecture is due to Kangand Yuan [45] which confirmed the conjecture for everyconnected planar graph 119866 with Δ ⩾ 7 The proofs are mainlybased onTheorems 16 and 18

Theorem 74 (Kang and Yuan [45] 2000) If 119866 is a connectedplanar graph then 119887(119866) ⩽ min8 Δ(119866) + 2

Obviously in view of Theorem 74 Conjecture 73 is truefor any connected planar graph with Δ ⩾ 7

62 Bounds Implied by Degree Conditions As we have seenfrom Theorem 74 to attack Conjecture 73 we only need toconsider connected planar graphs with maximum Δ ⩽ 6Thus studying the bondage number of planar graphs bydegree conditions is of significanceThe first result on boundsimplied by degree conditions is obtained by Kang and Yuan[45]

International Journal of Combinatorics 13

Theorem 75 (Kang and Yuan [45] 2000) If 119866 is a connectedplanar graph without vertices of degree 5 then 119887(119866) ⩽ 7

Fischermann et al [46] generalized Theorem 75 as fol-lows

Theorem 76 (Fischermann et al [46] 2003) Let 119866 be aconnected planar graph and 119883 the set of vertices of degree 5which have distance at least 3 to vertices of degrees 1 2 and3 If all vertices in 119883 not adjacent with vertices of degree 4are independent and not adjacent to vertices of degree 6 then119887(119866) ⩽ 7

Clearly if119866 has no vertices of degree 5 then119883 = 0 ThusTheorem 76 yields Theorem 75

Use 119899119894to denote the number of vertices of degree 119894 in 119866

for each 119894 = 1 2 Δ(119866) Using Theorem 18 Fischermannet al obtained the following two theorems

Theorem 77 (Fischermann et al [46] 2003) For any con-nected planar graph 119866

(1) 119887(119866) ⩽ 7 if 1198995lt 2119899

2+ 3119899

3+ 2119899

4+ 12

(2) 119887(119866) ⩽ 6 if 119866 contains no vertices of degrees 4 and 5

Theorem 78 (Fischermann et al [46] 2003) For any con-nected planar graph 119866 119887(119866) ⩽ Δ(119866) + 1 if

(1) Δ(119866) = 6 and every edge 119890 = 119909119910 with 119889119866(119909) = 5 and

119889119866(119910) = 6 is contained in at most one triangle

(2) Δ(119866) = 5 and no triangle contains an edge 119890 = 119909119910 with119889119866(119909) = 5 and 4 ⩽ 119889

119866(119910) ⩽ 5

63 Bounds Implied by Girth Conditions The girth 119892(119866) of agraph 119866 is the length of the shortest cycle in 119866 If 119866 has nocycles we define 119892(119866) = infin

Combining Theorem 74 with Theorem 78 we find thatif a planar graph contains no triangles and has maximumdegree Δ ⩾ 5 then Conjecture 73 holds This fact motivatedFischermann et alrsquos [46] attempt to attack Conjecture 73 bygirth restraints

Theorem 79 (Fischermann et al [46] 2003) For any con-nected planar graph 119866

119887 (119866) ⩽

6 119894119891 119892 (119866) ⩾ 4

5 119894119891 119892 (119866) ⩾ 5

4 119894119891 119892 (119866) ⩾ 6

3 119894119891 119892 (119866) ⩾ 8

(20)

The first result in Theorem 79 shows that Conjecture 73is valid for all connected planar graphs with 119892(119866) ⩾ 4 andΔ(119866) ⩾ 5 It is easy to verify that the second result inTheorem 79 implies that Conjecture 73 is valid for all not3-regular graphs of girth 119892(119866) ⩾ 5 which is stated in thefollowing corollary

Corollary 80 For any connected planar graph 119866 if 119866 is not3-regular and 119892(119866) ⩾ 5 then 119887(119866) ⩽ Δ(119866) + 1

The first result in Theorem 79 also implies that if 119866 isa connected planar graph with no cycles of length 3 then119887(119866) ⩽ 6 Hou et al [47] improved this result as follows

Theorem 81 (Hou et al [47] 2011) For any connected planargraph 119866 if 119866 contains no cycles of length 119894 (4 ⩽ 119894 ⩽ 6) then119887(119866) ⩽ 6

Since 119887(1198623119896+1) = 3 for 119896 ⩾ 3 (see Theorem 1) the

last bound in Theorem 79 is tight Whether other bounds inTheorem 79 are tight remains open In 2003 Fischermann etal [46] posed the following conjecture

Conjecture 82 For any connected planar graph 119866 119887(119866) ⩽ 7and

119887 (119866) ⩽ 5 119894119891 119892 (119866) ⩾ 4

4 119894119891 119892 (119866) ⩾ 5(21)

We conclude this subsection with a question on bondagenumbers of planar graphs

Question 1 (Fischermann et al [46] 2003) Is there a planargraph 119866 with 6 ⩽ 119887(119866) ⩽ 8

In 2006 Carlson andDevelin [42] showed that the corona119866 = 119867 ∘ 119870

1for a planar graph 119867 with 120575(119867) = 5 has the

bondage number 119887(119866) = 120575(119867) + 1 = 6 (see Theorem 66)Since the minimum degree of planar graphs is at most 5then 119887(119866) can attach 6 If we take 119867 as the graph of theicosahedron then 119866 = 119867 ∘ 119870

1is such an example The

question for the existence of planar graphs with bondagenumber 7 or 8 remains open

64 Comments on the Conjectures Conjecture 73 is true forall connected planar graphs with minimum degree 120575 ⩽ 2 byTheorem 16 or maximum degree Δ ⩾ 7 by Theorem 74 ornot 120574-critical planar graphs by Theorem 13 Thus to attackConjecture 73 we only need to consider connected criticalplanar graphs with degree-restriction 3 ⩽ 120575 ⩽ Δ ⩽ 6

Recalling and anatomizing the proofs of all results men-tioned in the preceding subsections on the bondage numberfor connected planar graphs we find that the proofs of theseresults strongly depend upon Theorem 16 or Theorem 18 Inother words a basic way used in the proofs is to find twovertices 119909 and 119910 with distance at most two in a consideredplanar graph 119866 such that

119889119866(119909) + 119889

119866(119910) or 119889

119866(119909) + 119889

119866(119910) minus

1003816100381610038161003816119873119866(119909) cap 119873

119866(119910)1003816100381610038161003816

(22)

which bounds 119887(119866) is as small as possible Very recentlyHuang and Xu [44] have considered the following parameter

119861 (119866) = min119909119910isin119881(119866)

119889119909119910 1 ⩽ 119889

119866(119909 119910) ⩽ 2

cup 119889119909119910minus 119899

119909119910 119889 (119909 119910) = 1

(23)

where 119889119909119910= 119889

119866(119909) + 119889

119866(119910) minus 1 and 119899

119909119910= |119873

119866(119909) cap 119873

119866(119910)|

14 International Journal of Combinatorics

It follows fromTheorems 16 and 18 that

119887 (119866) ⩽ 119861 (119866) (24)

The proofs given in Theorems 74 and 79 indeed imply thefollowing stronger results

Theorem 83 If 119866 is a connected planar graph then

119861 (119866) ⩽

min 8 Δ (119866) + 26 119894119891 119892 (119866) ⩾ 4

5 119894119891 119892 (119866) ⩾ 5

4 119894119891 119892 (119866) ⩾ 6

3 119894119891 119892 (119866) ⩾ 8

(25)

Thus using Theorem 16 or Theorem 18 if we can proveConjectures 73 and 82 then we can prove the followingstatement

Statement If 119866 is a connected planar graph then

119861 (119866) ⩽

Δ (119866) + 1

7

5 if 119892 (119866) ⩾ 44 if 119892 (119866) ⩾ 5

(26)

It follows from (24) that Statement implies Conjectures 73and 82 However Huang and Xu [44] gave examples to showthat none of conclusions in Statement is true As a result theystated the following conclusion

Theorem 84 (Huang and Xu [44] 2012) It is not possible toprove Conjectures 73 and 82 if they are right using Theorems16 and 18

Therefore a new method is needed to prove or disprovethese conjectures At the present time one cannot prove theseconjectures and may consider to disprove them If one ofthese conjectures is invalid then there exists a minimumcounterexample 119866 with respect to 120592(119866) + 120576(119866) In order toobtain a minimum counterexample one may consider twosimple operations satisfying these requirements the edge-deletion and the edge-contractionwhich decrease 120592(119866)+120576(119866)and preserve planarity However applying Theorems 71 and72 Huang and Xu [44] presented the following results

Theorem 85 (Huang and Xu [44] 2012) It is not possible toconstruct minimum counterexamples to Conjectures 73 and 82by the operation of an edge-deletion or an edge-contraction

7 Results on Crossing Number Restraints

It is quite natural to generalize the known results on thebondage number for planar graphs to formore general graphsin terms of graph-theoretical parameters In this section weconsider graphs with crossing number restraints

The crossing number cr(119866) of 119866 is the smallest numberof pairwise intersections of its edges when 119866 is drawn in theplane If cr(119866) = 0 then 119866 is a planar graph

71 General Methods Use 119899119894to denote the number of vertices

of degree 119894 in119866 for 119894 = 1 2 Δ(119866) Huang and Xu obtainedthe following results

Theorem 86 (Huang and Xu [48] 2007) For any connectedgraph 119866

119887 (119866) ⩽

6 119894119891 119892 (119866) ⩾ 4 119886119899119889 2cr (119866) lt 121198861

5 119894119891 119892 (119866) ⩾ 5 119886119899119889 6cr (119866) lt 161198862

4 119894119891 119892 (119866) ⩾ 6 119886119899119889 4cr (119866) lt 141198863

3 119894119891 119892 (119866) ⩾ 8 119886119899119889 6cr (119866) lt 161198864

(27)

where 1198861= 119899

1+ 2119899

2+ 2119899

3+sum

Δ

119894=8(119894 minus 7)119899

119894+ 8 119886

2= 3119899

1+ 6119899

2+

51198993+sum

Δ

119894=7(3119894minus18)119899

119894+20 119886

3= 119899

1+2119899

2+sum

Δ

119894=6(2119894minus10)119899

119894+12

and 1198864= sum

Δ

119894=5(3119894 minus 12)119899

119894+ 16

Simplifying the conditions in Theorem 86 we obtain thefollowing corollaries immediately

Corollary 87 (Huang and Xu [48] 2007) For any connectedgraph 119866

119887 (119866) ⩽

6 119894119891 119892 (119866) ⩾ 4 119886119899119889 cr (119866) ⩽ 3

5 119894119891 119892 (119866) ⩾ 5 119886119899119889 cr (119866) ⩽ 4

4 119894119891 119892 (119866) ⩾ 6 119886119899119889 cr (119866) ⩽ 2

3 119894119891 119892 (119866) ⩾ 8 119886119899119889 cr (119866) ⩽ 2

(28)

Corollary 88 (Huang and Xu [48] 2007) For any connectedgraph 119866

(a) 119887(119866) ⩽ 6 if 119866 is not 4-regular cr(119866) = 4 and 119892(119866) ⩾ 4

(b) 119887(119866) ⩽ Δ(119866) + 1 if 119866 is not 3-regular cr(119866) ⩽ 4 and119892(119866) ⩾ 5

(c) 119887(119866) ⩽ 4 if 119866 is not 3-regular cr(119866) = 3 and 119892(119866) ⩾ 6

(d) 119887(119866) ⩽ 3 if cr(119866) = 3 119892(119866) ⩾ 8 and Δ(119866) ⩾ 5

These corollaries generalize some known results for pla-nar graphs For example Corollary 87 contains Theorem 79Corollary 88 (b) contains Corollary 80

Theorem 89 (Huang and Xu [48] 2007) For any connectedgraph 119866 if cr(119866) ⩽ 119899

3(119866) + 119899

4(119866) + 3 then 119887(119866) ⩽ 8

Corollary 90 (Huang and Xu [48] 2007) For any connectedgraph 119866 if cr(119866) ⩽ 3 then 119887(119866) ⩽ 8

Perhaps being unaware of this result in 2010 Ma et al[49] proved that 119887(119866) ⩽ 12 for any graph 119866 with cr(119866) = 1

Theorem91 (Huang and Xu [48] 2007) Let119866 be a connectedgraph and 119868 = 119909 isin 119881(119866) 119889

119866(119909) = 5 119889

119866(119909 119910) ⩾ 3 if

International Journal of Combinatorics 15

119889119866(119910) ⩽ 3 and 119889

119866(119910) = 4 for every 119910 isin 119873

119866(119909) If 119868 is

independent no vertices adjacent to vertices of degree 6 and

cr (119866) lt max5119899

3+ |119868| minus 2119899

4+ 28

117119899

3+ 40

16 (29)

then 119887(119866) ⩽ 7

Corollary 92 (Huang and Xu [48] 2007) Let 119866 be aconnected graph with cr(119866) ⩽ 2 If 119868 = 119909 isin 119881(119866) 119889

119866(119909) =

5 119889119866(119910 119909) ⩾ 3 if 119889

119866(119910) ⩽ 3 and 119889

119866(119910) = 4 for every 119910 isin

119873119866(119909) is independent and no vertices adjacent to vertices of

degree 6 then 119887(119866) ⩽ 7

Theorem93 (Huang andXu [48] 2007) Let119866 be a connectedgraph If 119866 satisfies

(a) 5cr(119866) + 1198995lt 2119899

2+ 3119899

3+ 2119899

4+ 12 or

(b) 7cr(119866) + 21198995lt 3119899

2+ 4119899

4+ 24

then 119887(119866) ⩽ 7

Proposition 94 (Huang and Xu [48] 2007) Let 119866 be aconnected graph with no vertices of degrees four and five Ifcr(119866) ⩽ 2 then 119887(119866) ⩽ 6

Theorem 95 (Huang and Xu [48] 2007) If 119866 is a connectedgraph with cr(119866) ⩽ 4 and not 4-regular when cr(119866) = 4 then119887(119866) ⩽ Δ(119866) + 2

The above results generalize some known results forplanar graphs For example Corollary 90 and Theorem 95contain Theorem 74 the first condition in Theorem 93 con-tains the second condition inTheorem 77

From Corollary 88 and Theorem 95 we suggest the fol-lowing questions

Question 2 Is there a

(a) 4-regular graph 119866 with cr(119866) = 4 such that 119887(119866) ⩾ 7

(b) 3-regular graph 119866 with cr ⩽ 4 and 119892(119866) ⩾ 5 such that119887(119866) = 5

(c) 3-regular graph 119866 with cr = 3 and 119892(119866) = 6 or 7 suchthat 119887(119866) = 5

72 Carlson-Develin Methods In this subsection we intro-duce an elegant method presented by Carlson and Develin[42] to obtain some upper bounds for the bondage numberof a graph

Suppose that 119866 is a connected graph We say that 119866 hasgenus 120588 if 119866 can be embedded in a surface 119878 with 120588 handlessuch that edges are pairwise disjoint except possibly for end-vertices Let119866 be an embedding of119866 in surface 119878 and let120601(119866)denote the number of regions in 119866 The boundary of everyregion contains at least three edges and every edge is on theboundary of atmost two regions (the two regions are identicalwhen 119890 is a cut-edge) For any edge 119890 of 119866 let 1199031

119866(119890) and 1199032

119866(119890)

be the numbers of edges comprising the regions in 119866 whichthe edge 119890 borders It is clear that every 119890 = 119909119910 isin 119864(119866)

1199032

119866(119890) ⩾ 119903

1

119866(119890) ⩾ 4 if 1003816100381610038161003816119873119866

(119909) cap 119873119866(119910)1003816100381610038161003816 = 0

1199032

119866(119890) ⩾ 4 119903

1

119866(119890) ⩾ 3 if 1003816100381610038161003816119873119866

(119909) cap 119873119866(119910)1003816100381610038161003816 = 1

1199032

119866(119890) ⩾ 119903

1

119866(119890) ⩾ 3 if 1003816100381610038161003816119873119866

(119909) cap 119873119866(119910)1003816100381610038161003816 ⩾ 2

(30)

Following Carlson and Develin [42] for any edge 119890 = 119909119910 of119866 we define

119863119866(119890) =

1

119889119866(119909)+

1

119889119866(119910)

+1

1199031

119866(119890)+

1

1199032

119866(119890)minus 1 (31)

By the well-known Eulerrsquos formula

120592 (119866) minus 120576 (119866) + 120601 (119866) = 2 minus 2120588 (119866) (32)

it is easy to see that

sum

119890isin119864(119866)

119863119866(119890) = 120592 (119866) minus 120576 (119866) + 120601 (119866) = 2 minus 2120588 (119866) (33)

If 119866 is a planar graph that is 120588(119866) = 0 then

sum

119890isin119864(119866)

119863119866(119890) = 120592 (119866) minus 120576 (119866) + 120601 (119866) = 2 (34)

Combining these formulas withTheorem 18 Carlson andDevelin [42] gave a simple and intuitive proof ofTheorem 74and obtained the following result

Theorem 96 (Carlson and Develin [42] 2006) Let 119866 be aconnected graph which can be embedded on a torus Then119887(119866) ⩽ Δ(119866) + 3

Recently Hou and Liu [50] improved Theorem 96 asfollows

Theorem 97 (Hou and Liu [50] 2012) Let 119866 be a connectedgraph which can be embedded on a torus Then 119887(119866) ⩽ 9Moreover if Δ(119866) = 6 then 119887(119866) ⩽ 8

By the way Cao et al [51] generalized the result inTheorem 79 to a connected graph 119866 that can be embeddedon a torus that is

119887 (119866) le

6 if 119892 (119866) ⩾ 4 and 119866 is not 4-regular5 if 119892 (119866) ⩾ 54 if 119892 (119866) ⩾ 6 and 119866 is not 3-regular3 if 119892 (119866) ⩾ 8

(35)

Several authors used this method to obtain some resultson the bondage number For example Fischermann et al[46] used this method to prove the second conclusion inTheorem 78 Recently Cao et al [52] have used thismethod todeal with more general graphs with small crossing numbersFirst they found the following property

Lemma 98 120575(119866) ⩽ 5 for any graph 119866 with cr(119866) ⩽ 5

16 International Journal of Combinatorics

This result implies that 119887(119866) ⩽ Δ(119866) + 4 by Corollary 15for any graph 119866 with cr(119866) ⩽ 5 Cao et al established thefollowing relation in terms of maximum planar subgraphs

Lemma 99 (Cao et al [52]) Let 119866 be a connected graph withcrossing number cr(119866) For any maximum planar subgraph119867of 119866

sum

119890isin119864(119867)

119863119866(119890) ⩾ 2 minus

2cr (119866)120575 (119866)

(36)

Using Carlson-Develin method and combiningLemma 98 with Lemma 99 Cao et al proved the followingresults

Theorem 100 (Cao et al [52]) For any connected graph 119866119887(119866) ⩽ Δ(119866) + 2 if 119866 satisfies one of the following conditions

(a) cr(119866) ⩽ 3(b) cr(119866) = 4 and 119866 is not 4-regular(c) cr(119866) = 5 and 119866 contains no vertices of degree 4

Theorem101 (Cao et al [52]) Let119866 be a connected graphwithΔ(119866) = 5 and cr(119866) ⩽ 4 If no triangles contain two vertices ofdegree 5 then 119887(119866) ⩽ 6 = Δ(119866) + 1

Theorem 102 (Cao et al [52]) Let 119866 be a connected graphwith Δ(119866) ⩾ 6 and cr(119866) ⩽ 3 If Δ(119866) ⩾ 7 or if Δ(119866) = 6120575(119866) = 3 and every edge 119890 = 119909119910 with 119889

119866(119909) = 5 and 119889

119866(119910) = 6

is contained in atmost one triangle then 119887(119866) ⩽ min8 Δ(119866)+1

Using Carlson-Develin method it can be proved that119887(119866) ⩽ Δ(119866) + 2 if cr(119866) ⩽ 3 (see the first conclusion inTheorem 100) and not yet proved that 119887(119866) ⩽ 8 if cr(119866) ⩽3 although it has been proved by using other method (seeCorollary 90)

By using Carlson-Develin method Samodivkin [25]obtained some results on the bondage number for graphswithsome given properties

Kim [53] showed that 119887(119866) ⩽ Δ(119866) + 2 for a connectedgraph 119866 with genus 1 Δ(119866) ⩽ 5 and having a toroidalembedding of which at least one region is not 4-sidedRecently Gagarin and Zverovich [54] further have extendedCarlson-Develin ideas to establish a nice upper bound forarbitrary graphs that can be embedded on orientable ornonorientable topological surface

Theorem 103 (Gagarin and Zverovich [54] 2012) Let 119866 bea graph embeddable on an orientable surface of genus ℎ and anonorientable surface of genus 119896 Then 119887(119866) ⩽ minΔ(119866)+ℎ+2 Δ(119866) + 119896 + 1

This result generalizes the corresponding upper boundsin Theorems 74 and 96 for any orientable or nonori-entable topological surface By investigating the proof ofTheorem 103 Huang [55] found that the issue of the ori-entability can be avoided by using the Euler characteristic120594(= 120592(119866) minus 120576(119866) + 120601(119866)) instead of the genera ℎ and 119896 the

relations are 120594 = 2minus2ℎ and 120594 = 2minus119896 To obtain the best resultfromTheorem 103 onewants ℎ and 119896 as small as possible thisis equivalent to having 120594 as large as possible

According toTheorem 103 if119866 is planar (ℎ = 0 120594 = 2) orcan be embedded on the real projective plane (119896 = 1 120594 = 1)then 119887(119866) ⩽ Δ(119866) + 2 In all other cases Huang [55] had thefollowing improvement for Theorem 103 the proof is basedon the technique developed byCarlson-Develin andGagarin-Zverovich and includes some elementary calculus as a newingredient mainly the intermediate-value theorem and themean-value theorem

Theorem 104 (Huang [55] 2012) Let 119866 be a graph embed-dable on a surface whose Euler characteristic 120594 is as large aspossible If 120594 ⩽ 0 then 119887(119866) ⩽ Δ(119866)+lfloor119903rfloor where 119903 is the largestreal root of the following cubic equation in 119911

1199113

+ 21199112

+ (6120594 minus 7) 119911 + 18120594 minus 24 = 0 (37)

In addition if 120594 decreases then 119903 increases

The following result is asymptotically equivalent toTheorem 104

Theorem 105 (Huang [55] 2012) Let 119866 be a graph embed-dable on a surface whose Euler characteristic 120594 is as large aspossible If 120594 ⩽ 0 then 119887(119866) lt Δ(119866) + radic12 minus 6120594 + 12 orequivalently 119887(119866) ⩽ Δ(119866) + lceilradic12 minus 6120594 minus 12rceil

Also Gagarin andZverovich [54] indicated that the upperbound in Theorem 103 can be improved for larger values ofthe genera ℎ and 119896 by adjusting the proofs and stated thefollowing general conjecture

Conjecture 106 (Gagarin and Zverovich [54] 2012) For aconnected graph G of orientable genus ℎ and nonorientablegenus 119896

119887 (119866) ⩽ min 119888ℎ 119888

1015840

119896 Δ (119866) + 119900 (ℎ) Δ (119866) + 119900 (119896) (38)

where 119888ℎand 1198881015840

119896are constants depending respectively on the

orientable and nonorientable genera of 119866

In the recent paper Gagarin and Zverovich [56] providedconstant upper bounds for the bondage number of graphs ontopological surfaces which can be used as the first estimationfor the constants 119888

ℎand 1198881015840

119896of Conjecture 106

Theorem 107 (Gagarin and Zverovich [56] 2013) Let 119866 bea connected graph of order 119899 If 119866 can be embedded on thesurface of its orientable or nonorientable genus of the Eulercharacteristic 120594 then

(a) 120594 ⩾ 1 implies 119887(119866) ⩽ 10(b) 120594 ⩽ 0 and 119899 gt minus12120594 imply 119887(119866) ⩽ 11(c) 120594 ⩽ minus1 and 119899 ⩽ minus12120594 imply 119887(119866) ⩽ 11 +

3120594(radic17 minus 8120594 minus 3)(120594 minus 1) = 11 + 119874(radicminus120594)

Theorem 79 shows that a connected planar triangle-freegraph 119866 has 119887(119866) ⩽ 6 The following result generalizes to itall the other topological surfaces

International Journal of Combinatorics 17

Theorem 108 (Gagarin and Zverovich [56] 2013) Let 119866 be aconnected triangle-free graph of order 119899 If 119866 can be embeddedon the surface of its orientable or nonorientable genus of theEuler characteristic 120594 then

(a) 120594 ⩾ 1 implies 119887(119866) ⩽ 6(b) 120594 ⩽ 0 and 119899 gt minus8120594 imply 119887(119866) ⩽ 7(c) 120594 ⩽ minus1 and 119899 ⩽ minus8120594 imply 119887(119866) ⩽ 7minus4120594(1+radic2 minus 120594)

In [56] the authors also explicitly improved upperbounds in Theorem 103 and gave tight lower bounds forthe number of vertices of graphs 2-cell embeddable ontopological surfaces of a given genusThe interested reader isreferred to the original paper for details The following resultis interesting although its proof is easy

Theorem 109 (Samodivkin [57]) Let119866 be a graph with order119899 and girth 119892 If 119866 is embeddable on a surface whose Eulercharacteristic 120594 is as large as possible then

119887 (119866) ⩽ 3 +8

119892 minus 2minus

4120594119892

119899 (119892 minus 2) (39)

If 119866 is a planar graph with girth 119892 ⩾ 4 + 119894 for any 119894 isin0 1 2 then Theorem 109 leads to 119887(119866) ⩽ 6 minus 119894 which isthe result inTheorem 79 obtained by Fischermann et al [46]Also inmany cases the bound stated inTheorem 109 is betterthan those given byTheorems 103 and 105

8 Conditional Bondage Numbers

Since the concept of the bondage number is based upon thedomination all sorts of dominations which are variationsof the normal domination by adding restricted conditionsto the normal dominating set can develop a new ldquobondagenumberrdquo as long as a variation of domination number isgiven In this section we survey results on the bondagenumber under some restricted conditions

81 Total Bondage Numbers A dominating set 119878 of a graph119866 without isolated vertices is called total if the subgraphinduced by 119878 contains no isolated vertices The minimumcardinality of a total dominating set is called the totaldomination number of 119866 and denoted by 120574

119879(119866) It is clear

that 120574(119866) ⩽ 120574119879(119866) ⩽ 2120574(119866) for any graph 119866 without isolated

verticesThe total domination in graphs was introduced by Cock-

ayne et al [58] in 1980 Pfaff et al [59 60] in 1983 showedthat the problem determining total domination number forgeneral graphs is NP-complete even for bipartite graphs andchordal graphs Even now total domination in graphs hasbeen extensively studied in the literature In 2009 Henning[61] gave a survey of selected recent results on total domina-tion in graphs

The total bondage number of 119866 denoted by 119887119879(119866) is the

smallest cardinality of a subset 119861 sube 119864(119866) with the propertythat119866minus119861 contains no isolated vertices and 120574

119879(119866minus119861) gt 120574

119879(119866)

From definition 119887119879(119866)may not exist for some graphs for

example119866 = 1198701119899 We put 119887

119879(119866) = infin if 119887

119879(119866) does not exist

It is easy to see that 119887119879(119866) is finite for any connected graph 119866

other than 1198701 119870

2 119870

3 and 119870

1119899 In fact for such a graph 119866

since there is a path of length 3 in 119866 we can find 1198611sube 119864(119866)

such that 119866 minus 1198611is a spanning tree 119879 of 119866 containing a path

of length 3 So 120574119879(119879) = 120574

119879(119866minus119861

1) ⩾ 120574

119879(119866) For the tree119879 we

find 1198612sube 119864(119879) such that 120574

119879(119879 minus 119861

2) gt 120574

119879(119879) ⩾ 120574

119879(119866) Thus

we have 120574119879(119866minus119861

2minus119861

1) gt 120574

119879(119866) and so 119887

119879(119866) ⩽ |119861

1|+|119861

2| In

the following discussion we always assume that 119887119879(119866) exists

when a graph 119866 is mentionedIn 1991 Kulli and Patwari [62] first studied the total

bondage number of a graph and calculated the exact valuesof 119887

119879(119866) for some standard graphs

Theorem 110 (Kulli and Patwari [62] 1991) For 119899 ⩾ 4

119887119879(119862

119899) =

3 119894119891 119899 equiv 2 (mod 4) 2 119900119905ℎ119890119903119908119894119904119890

119887119879(119875

119899) =

2 119894119891 119899 equiv 2 (mod 4) 1 119900119905ℎ119890119903119908119894119904119890

119887119879(119870

119898119899) = 119898 119908119894119905ℎ 2 ⩽ 119898 ⩽ 119899

119887119879(119870

119899) =

4 119891119900119903 119899 = 4

2119899 minus 5 119891119900119903 119899 ⩾ 5

(40)

Recently Hu et al [63] have obtained some results on thetotal bondage number of the Cartesian product119875

119898times119875

119899of two

paths 119875119898and 119875

119899

Theorem 111 (Hu et al [63] 2012) For the Cartesian product119875119898times 119875

119899of two paths 119875

119898and 119875

119899

119887119879(119875

2times 119875

119899) =

1 119894119891 119899 equiv 0 (mod 3) 2 119894119891 119899 equiv 2 (mod 3) 3 119894119891 119899 equiv 1 (mod 3)

119887119879(119875

3times 119875

119899) = 1

119887119879(119875

4times 119875

119899)

= 1 119894119891 119899 equiv 1 (mod 5) = 2 119894119891 119899 equiv 4 (mod 5) ⩽ 3 119894119891 119899 equiv 2 (mod 5) ⩽ 4 119894119891 119899 equiv 0 3 (mod 5)

(41)

Generalized Petersen graphs are an important class ofcommonly used interconnection networks and have beenstudied recently By constructing a family of minimum totaldominating sets Cao et al [64] determined the total bondagenumber of the generalized Petersen graphs

FromTheorem 6 we know that 119887(119879) ⩽ 2 for a nontrivialtree 119879 But given any positive integer 119896 Sridharan et al [65]constructed a tree 119879 for which 119887

119879(119879) = 119896 Let119867

119896be the tree

obtained from a star1198701119896+1

by subdividing 119896 edges twice Thetree 119867

7is shown in Figure 6 It can be easily verified that

119887119879(119867

119896) = 119896 This fact can be stated as the following theorem

Theorem 112 (Sridharan et al [65] 2007) For any positiveinteger 119896 there exists a tree 119879 with 119887

119879(119879) = 119896

18 International Journal of Combinatorics

Figure 61198677

Combining Theorem 1 with Theorem 112 we have whatfollows

Corollary 113 (Sridharan et al [65] 2007) The bondagenumber and the total bondage number are unrelated even fortrees

However Sridharan et al [65] gave an upper bound onthe total bondage number of a tree in terms of its maximumdegree For any tree 119879 of order 119899 if 119879 =119870

1119899minus1 then 119887

119879(119879) =

minΔ(119879) (13)(119899 minus 1) Rad and Raczek [66] improved thisupper bound and gave a constructive characterization of acertain class of trees attaching the upper bound

Theorem 114 (Rad and Raczek [66] 2011) 119887119879(119879) ⩽ Δ(119879) minus 1

for any tree 119879 with maximum degree at least three

In general the decision problem for 119887119879(119866) is NP-complete

for any graph 119866 We state the decision problem for the totalbondage as follows

Problem 3 Consider the decision problemTotal Bondage ProblemInstance a graph 119866 and a positive integer 119887Question is 119887

119879(119866) ⩽ 119887

Theorem 115 (Hu and Xu [16] 2012) The total bondageproblem is NP-complete

Consequently in view of its computational hardnessit is significative to establish some sharp bounds on thetotal bondage number of a graph in terms of other graphicparameters In 1991 Kulli and Patwari [62] showed that119887119879(119866) ⩽ 2119899 minus 5 for any graph 119866 with order 119899 ⩾ 5 Sridharan

et al [65] improved this result as follows

Theorem 116 (Sridharan et al [65] 2007) For any graph 119866with order 119899 if 119899 ⩾ 5 then

119887119879(119866) ⩽

1

3(119899 minus 1) 119894119891 119866 119888119900119899119905119886119894119899119904 119899119900 119888119910119888119897119890119904

119899 minus 1 119894119891 119892 (119866) ⩾ 5

119899 minus 2 119894119891 119892 (119866) = 4

(42)

Rad and Raczek [66] also established some upperbounds on 119887

119879(119866) for a general graph 119866 In particular they

gave an upper bound on 119887119879(119866) in terms of the girth of

119866

Theorem 117 (Rad and Raczek [66] 2011) 119887119879(119866)⩽4Δ(119866) minus 5

for any graph 119866 with 119892(119866) ⩾ 4

82 Paired Bondage Numbers A dominating set 119878 of 119866 iscalled to be paired if the subgraph induced by 119878 containsa perfect matching The paired domination number of 119866denoted by 120574

119875(119866) is the minimum cardinality of a paired

dominating set of 119866 Clearly 120574119879(119866) ⩽ 120574

119875(119866) for every

connected graph 119866 with order at least two where 120574119879(119866) is

the total domination number of 119866 and 120574119875(119866) ⩽ 2120574(119866) for

any graph119866without isolated vertices Paired domination wasintroduced by Haynes and Slater [67 68] and further studiedin [69ndash72]

The paired bondage number of 119866 with 120575(119866) ⩾ 1 denotedby 119887P(119866) is the minimum cardinality among all sets of edges119861 sube 119864 such that 120575(119866 minus 119861) ⩾ 1 and 120574

119875(119866 minus 119861) gt 120574

119875(119866)

The concept of the paired bondage number was firstproposed by Raczek [73] in 2008The following observationsfollow immediately from the definition of the paired bondagenumber

Observation 2 Let 119866 be a graph with 120575(119866) ⩾ 1

(a) If 119867 is a subgraph of 119866 such that 120575(119867) ⩾ 1 then119887119875(119867) ⩽ 119887

119875(119866)

(b) If 119867 is a subgraph of 119866 such that 119887119875(119867) = 1 and 119896

is the number of edges removed to form119867 then 1 ⩽119887119875(119866) ⩽ 119896 + 1

(c) If 119909119910 isin 119864(119866) such that 119889119866(119909) 119889

119866(119910) gt 1 and 119909119910

belongs to each perfect matching of each minimumpaired dominating set of 119866 then 119887

119875(119866) = 1

Based on these observations 119887119875(119875

119899) ⩽ 2 In fact the

paired bondage number of a path has been determined

Theorem 118 (Raczek [73] 2008) For any positive integer 119896if 119899 ⩾ 2 then

119887119875(119875

119899) =

0 119894119891 119899 = 2 3 119900119903 5

1 119894119891 119899 = 4119896 4119896 + 3 119900119903 4119896 + 6

2 119900119905ℎ119890119903119908119894119904119890

(43)

For a cycle 119862119899of order 119899 ⩾ 3 since 120574

119875(119862

119899) = 120574

119875(119875

119899) from

Theorem 118 the following result holds immediately

Corollary 119 (Raczek [73] 2008) For any positive integer 119896if 119899 ⩾ 3 then

119887119875(119862

119899) =

0 119894119891 119899 = 3 119900119903 5

2 119894119891 119899 = 4119896 4119896 + 3 119900119903 4119896 + 6

3 119900119905ℎ119890119903119908119894119904119890

(44)

A wheel 119882119899 where 119899 ⩾ 4 is a graph with 119899 vertices

formed by connecting a single vertex to all vertices of a cycle119862119899minus1

Of course 120574119875(119882

119899) = 2

International Journal of Combinatorics 19

119896

Figure 7 A tree 119879 with 119887119875(119879) = 119896

Theorem 120 (Raczek [73] 2008) For positive integers 119899 and119898

119887119875(119870

119898119899) = 119898 119891119900119903 1 lt 119898 ⩽ 119899

119887119875(119882

119899) =

4 119894119891 119899 = 4

3 119894119891 119899 = 5

2 119900119905ℎ119890119903119908119894119904119890

(45)

Theorem 16 shows that if 119879 is a nontrivial tree then119887(119879) ⩽ 2 However no similar result exists for pairedbondage For any nonnegative integer 119896 let 119879

119896be a tree

obtained by subdividing all but one edge of a star 1198701119896+1

(asshown in Figure 7) It is easy to see that 119887

119875(119879

119896) = 119896 We

state this fact as the following theorem which is similar toTheorem 112

Theorem121 (Raczek [73] 2008) For any nonnegative integer119896 there exists a tree 119879 with 119887

119875(119879) = 119896

Consider the tree defined in Figure 5 for 119896 = 3 Then119887(119879

3) ⩽ 2 and 119887

119875(119879

3) = 3 On the other hand 119887(119875

4) = 2

and 119887119875(119875

4) = 1 Thus we have what follows which is similar

to Corollary 113

Corollary 122 The bondage number and the paired bondagenumber are unrelated even for trees

A constructive characterization of trees with 119887(119879) = 2is given by Hartnell and Rall in [12] Raczek [73] provideda constructive characterization of trees with 119887

119875(119879) = 0 In

order to state the characterization we define a labeling andthree simple operations on a tree119879 Let 119910 isin 119881(119879) and let ℓ(119910)be the label assigned to 119910

Operation T1 If ℓ(119910) = 119861 add a vertex 119909 and the

edge 119910119909 and let ℓ(119909) = 119860OperationT

2 If ℓ(119910) = 119862 add a path (119909

1 119909

2) and the

edge 1199101199091 and let ℓ(119909

1) = 119861 ℓ(119909

2) = 119860

Operation T3 If ℓ(119910) = 119861 add a path (119909

1 119909

2 119909

3)

and the edge 1199101199091 and let ℓ(119909

1) = 119862 ℓ(119909

2) = 119861 and

ℓ(1199093) = 119860

Let 1198752= (119906 V) with ℓ(119906) = 119860 and ℓ(V) = 119861 Let T be

the class of all trees obtained from the labeled 1198752by a finite

sequence of operationsT1 T

2 T

3

A tree 119879 in Figure 8 belongs to the familyTRaczek [73] obtained the following characterization of all

trees 119879 with 119887119875(119879) = 0

119860

119860

119860

119860

119860

119861 119862

119861 119862 119861

119861119860

Figure 8 A tree 119879 belong to the familyT

Theorem 123 (Raczek [73] 2008) For a tree 119879 119887119875(119879) = 0 if

and only if 119879 is inT

We state the decision problem for the paired bondage asfollows

Problem 4 Consider the decision problem

Paired Bondage ProblemInstance a graph 119866 and a positive integer 119887Question is 119887

119875(119866) ⩽ 119887

Conjecture 124 The paired bondage problem is NP-complete

83 Restrained Bondage Numbers A dominating set 119878 of 119866 iscalled to be restrained if the subgraph induced by 119878 containsno isolated vertices where 119878 = 119881(119866) 119878 The restraineddomination number of 119866 denoted by 120574

119877(119866) is the smallest

cardinality of a restrained dominating set of 119866 It is clear that120574119877(119866) exists and 120574(119866) ⩽ 120574

119877(119866) for any nonempty graph 119866

The concept of restrained domination was introducedby Telle and Proskurowski [74] in 1997 albeit indirectly asa vertex partitioning problem Concerning the study of therestrained domination numbers the reader is referred to [74ndash79]

In 2008 Hattingh and Plummer [80] defined therestrained bondage number 119887R(119866) of a nonempty graph 119866 tobe the minimum cardinality among all subsets 119861 sube 119864(119866) forwhich 120574

119877(119866 minus 119861) gt 120574

119877(119866)

For some simple graphs their restrained dominationnumbers can be easily determined and so restrained bondagenumbers have been also determined For example it is clearthat 120574

119877(119870

119899) = 1 Domke et al [77] showed that 120574

119877(119862

119899) =

119899 minus 2lfloor1198993rfloor for 119899 ⩾ 3 and 120574119877(119875

119899) = 119899 minus 2lfloor(119899 minus 1)3rfloor for 119899 ⩾ 1

Using these results Hattingh and Plummer [80] obtained therestricted bondage numbers for119870

119899 119862

119899 119875

119899 and119866 = 119870

11989911198992119899119905

as follows

Theorem 125 (Hattingh and Plummer [80] 2008) For 119899 ⩾ 3

119887R (119870119899) =

1 119894119891 119899 = 3

lceil119899

2rceil 119900119905ℎ119890119903119908119894119904119890

119887R (119862119899) =

1 119894119891 119899 equiv 0 (mod 3) 2 119900119905ℎ119890119903119908119894119904119890

119887R (119875119899) = 1 119899 ⩾ 4

20 International Journal of Combinatorics

119887R (119866)

=

lceil119898

2rceil 119894119891 119899

119898= 1 119886119899119889 119899

119898+1⩾ 2 (1 ⩽ 119898 lt 119905)

2119905 minus 2 119894119891 1198991= 119899

2= sdot sdot sdot = 119899

119905= 2 (119905 ⩾ 2)

2 119894119891 1198991= 2 119886119899119889 119899

2⩾ 3 (119905 = 2)

119905minus1

sum

119894=1

119899119894minus 1 119900119905ℎ119890119903119908119894119904119890

(46)

Theorem 126 (Hattingh and Plummer [80] 2008) For a tree119879 of order 119899 (⩾ 4) 119879 ≇ 119870

1119899minus1if and only if 119887R(119879) = 1

Theorem 126 shows that the restrained bondage numberof a tree can be computed in constant time However thedecision problem for 119887R(119866) is NP-complete even for bipartitegraphs

Problem 5 Consider the decision problem

Restrained Bondage Problem

Instance a graph 119866 and a positive integer 119887

Question is 119887R(119866) ⩽ 119887

Theorem 127 (Hattingh and Plummer [80] 2008) Therestrained bondage problem is NP-complete even for bipartitegraphs

Consequently in view of its computational hardnessit is significative to establish some sharp bounds on therestrained bondage number of a graph in terms of othergraphic parameters

Theorem 128 (Hattingh and Plummer [80] 2008) For anygraph 119866 with 120575(119866) ⩾ 2

119887R (119866) ⩽ min119909119910isin119864(119866)

119889119866(119909) + 119889

119866(119910) minus 2 (47)

Corollary 129 119887R(119866) ⩽ Δ(119866)+120575(119866)minus2 for any graph119866 with120575(119866) ⩾ 2

Notice that the bounds stated in Theorem 128 andCorollary 129 are sharp Indeed the class of cycles whoseorders are congruent to 1 2 (mod 3) have a restrained bondagenumber achieving these bounds (see Theorem 125)

Theorem 128 is an analogue of Theorem 14 A quitenatural problem is whether or not for restricted bondagenumber there are analogues of Theorem 16 Theorem 18Theorem 29 and so on Indeed we should consider all resultsfor the bondage number whether or not there are analoguesfor restricted bondage number

Theorem 130 (Hattingh and Plummer [80] 2008) For anygraph 119866 with 120574

119877(119866) = 2

119887R (119866) ⩽ Δ (119866) + 1 (48)

We now consider the relation between 119887119877(119866) and 119887(119866)

Theorem 131 (Hattingh and Plummer [80] 2008) If 120574119877(119866) =

120574(119866) for some graph 119866 then 119887R(119866) ⩽ 119887(119866)

Corollary 129 and Theorem 131 provide a possibility toattack Conjecture 73 by characterizing planar graphs with120574119877= 120574 and 119887R = 119887 when 3 ⩽ Δ ⩽ 6However we do not have 119887R(119866) = 119887(119866) for any graph

119866 even if 120574119877(119866) = 120574(119866) Observe that 120574

119877(119870

3) = 120574(119870

3) yet

119887119877(119870

3) = 1 and 119887(119870

3) = 2 We still may not claim that

119887119877(119866) = 119887(119866) even in the case that every 120574-set is a 120574

119877-set

The example 1198703again demonstrates this

Theorem 132 (Hattingh and Plummer [80] 2008) There isan infinite class of graphs inwhich each graph119866 satisfies 119887(119866) lt119887R(119866)

In fact such an infinite class of graphs can be obtained asfollows Let119867 be a connected graph BC(119867) a graph obtainedfrom 119867 by attaching ℓ (⩾ 2) vertices of degree 1 to eachvertex in 119867 and B = 119866 119866 = BC(119867) for some graph 119867such that 120575(119867) ⩾ 2 By Theorem 3 we can easily check that119887(119866) = 1 lt 2 ⩽ min120575(119867) ℓ = 119887R(119866) for any 119866 isin BMoreover there exists a graph 119866 isin B such that 119887R(119866) canbe much larger than 119887(119866) which is stated as the followingtheorem

Theorem 133 (Hattingh and Plummer [80] 2008) For eachpositive integer 119896 there is a graph119866 such that 119896 = 119887R(119866)minus119887(119866)

Combining Theorem 133 with Theorem 132 we have thefollowing corollary

Corollary 134 The bondage number and the restrainedbondage number are unrelated

Very recently Jafari Rad et al [81] have consideredthe total restrained bondage in graphs based on both totaldominating sets and restrained dominating sets and obtainedseveral properties exact values and bounds for the totalrestrained bondage number of a graph

A restrained dominating set 119878 of a graph 119866 without iso-lated vertices is called to be a total restrained dominating setif 119878 is a total dominating set The total restrained dominationnumber of119866 denoted by 120574TR (119866) is theminimum cardinalityof a total restrained dominating set of 119866 For referenceson this domination in graphs see [3 61 82ndash85] The totalrestrained bondage number 119887TR (119866) of a graph 119866 with noisolated vertex is the cardinality of a smallest set of edges 119861 sube119864(119866) for which119866minus119861 has no isolated vertex and 120574TR (119866minus119861) gt120574TR (119866) In the case that there is no such subset 119861 we define119887TR (119866) = infin

Ma et al [84] determined that 120574TR (119875119899) = 119899minus2lfloor(119899minus2)4rfloorfor 119899 ⩾ 3 120574TR (119862119899

) = 119899 minus 2lfloor1198994rfloor for 119899 ⩾ 3 120574TR (1198703) =

3 and 120574TR (119870119899) = 2 for 119899 = 3 and 120574TR (1198701119899

) = 1 + 119899

and 120574TR (119870119898119899) = 2 for 2 ⩽ 119898 ⩽ 119899 According to these

results Jafari Rad et al [81] determined the exact valuesof total restrained bondage numbers for the correspondinggraphs

International Journal of Combinatorics 21

Theorem 135 (Jafari Rad et al [81]) For an integer 119899

119887TR (119875119899) = infin 119894119891 2 ⩽ 119899 ⩽ 5

1 119894119891 119899 ⩾ 5

119887TR (119862119899) =

infin 119894119891 119899 = 3

1 119894119891 3 lt 119899 119899 equiv 0 1 (mod 4)2 119894119891 3 lt 119899 119899 equiv 2 3 (mod 4)

119887TR (119882119899) = 2 119891119900119903 119899 ⩾ 6

119887TR (119870119899) = 119899 minus 1 119891119900119903 119899 ⩾ 4

119887TR (119870119898119899) = 119898 minus 1 119891119900119903 2 ⩽ 119898 ⩽ 119899

(49)

119887TR (119870119899) = 119899minus1 shows the fact that for any integer 119899 ⩾ 3 there

exists a connected graph namely119870119899+1

such that 119887TR (119870119899+1) =

119899 that is the total restrained bondage number is unboundedfrom the above

A vertex is said to be a support vertex if it is adjacent toa vertex of degree one Let A be the family of all connectedgraphs such that119866 belongs toA if and only if every edge of119866is incident to a support vertex or 119866 is a cycle of length three

Proposition 136 (Cyman and Raczek [82] 2006) For aconnected graph119866 of order 119899 120574TR (119866) = 119899 if and only if119866 isin A

Hence if 119866 isin A then the removal of any subset of edgesof119866 cannot increase the total restrained domination numberof 119866 and thus 119887TR (119866) = infin When 119866 notin A a result similar toTheorem 6 is obtained

Theorem 137 (Jafari Rad et al [81]) For any tree 119879 notin A119887TR (119879) ⩽ 2 This bound is sharp

In the samepaper the authors provided a characterizationfor trees 119879 notin A with 119887TR (119879) = 1 or 2 The following result issimilar to Observation 1

Observation 3 (Jafari Rad et al [81]) If 119896 edges can beremoved from a graph 119866 to obtain a graph 119867 without anisolated vertex and with 119887TR (119867) = 119905 then 119887TR (119866) ⩽ 119896 + 119905

ApplyingTheorem 137 andObservation 3 Jafari Rad et alcharacterized all connected graphs 119866 with 119887TR (119866) = infin

Theorem 138 (Jafari Rad et al [81]) For a connected graph119866119887TR (119866) = infin if and only if 119866 isin A

84 Other Conditional Bondage Numbers A subset 119868 sube 119881(119866)is called an independent set if no two vertices in 119868 are adjacentin 119866 The maximum cardinality among all independent setsis called the independence number of 119866 denoted by 120572(119866)

A dominating set 119878 of a graph 119866 is called to be inde-pendent if 119878 is an independent set of 119866 The minimumcardinality among all independent dominating set is calledthe independence domination number of 119866 and denoted by120574119868(119866)

Since an independent dominating set is not only adominating set but also an independent set 120574(119866) ⩽ 120574

119868(119866) ⩽

120572(119866) for any graph 119866It is clear that a maximal independent set is certainly a

dominating set Thus an independent set is maximal if andonly if it is an independent dominating set and so 120574

119868(119866) is the

minimum cardinality among all maximal independent setsof 119866 This graph-theoretical invariant has been well studiedin the literature see for example Haynes et al [3] for earlyresults Goddard and Henning [86] for recent results onindependent domination in graphs

In 2003 Zhang et al [87] defined the independencebondage number 119887

119868(119866) of a nonempty graph 119866 to be the

minimum cardinality among all subsets 119861 sube 119864(119866) for which120574119868(119866 minus 119861) gt 120574

119868(119866) For some ordinary graphs their indepen-

dence domination numbers can be easily determined and soindependence bondage numbers have been also determinedClearly 119887

119868(119870

119899) = 1 if 119899 ⩾ 2

Theorem 139 (Zhang et al [87] 2003) For any integer 119899 ⩾ 4

119887119868(119862

119899) =

1 119894119891 119899 equiv 1 (mod 2) 2 119894119891 119899 equiv 0 (mod 2)

119887119868(119875

119899) =

1 119894119891 119899 equiv 0 (mod 2) 2 119894119891 119899 equiv 1 (mod 2)

119887119868(119870

119898119899) = 119899 119891119900119903 119898 ⩽ 119899

(50)

Apart from the above-mentioned results as far as we findno other results on the independence bondage number in thepresent literature we never knew much of any result on thisparameter for a tree

A subset 119878 of 119881(119866) is called an equitable dominatingset if for every 119910 isin 119881(119866 minus 119878) there exists a vertex 119909 isin 119878such that 119909119910 isin 119864(119866) and |119889

119866(119909) minus 119889

119866(119910)| ⩽ 1 proposed

by Dharmalingam [88] The minimum cardinality of such adominating set is called the equitable domination number andis denoted by 120574

119864(119866) Deepak et al [89] defined the equitable

bondage number 119887119864(119866) of a graph 119866 to be the cardinality of a

smallest set 119861 sube 119864(119866) of edges for which 120574119864(119866 minus 119861) gt 120574

119864(119866)

and determined 119887119864(119866) for some special graphs

Theorem 140 (Deepak et al [89] 2011) For any integer 119899 ⩾ 2

119887119864(119870

119899) = lceil

119899

2rceil

119887119864(119870

119898119899) = 119898 119891119900119903 0 ⩽ 119899 minus 119898 ⩽ 1

119887119864(119862

119899) =

3 119894119891 119899 equiv 1 (mod 3)2 119900119905ℎ119890119903119908119894119904119890

119887119864(119875

119899) =

2 119894119891 119899 equiv 1 (mod 3)1 119900119905ℎ119890119903119908119894119904119890

119887119864(119879) ⩽ 2 119891119900119903 119886119899119910 119899119900119899119905119903119894119907119894119886119897 119905119903119890119890 119879

119887119864(119866) ⩽ 119899 minus 1 119891119900119903 119886119899119910 119888119900119899119899119890119888119905119890119889 119892119903119886119901ℎ 119866 119900119891 119900119903119889119890119903 119899

(51)

22 International Journal of Combinatorics

As far as we know there are no more results on theequitable bondage number of graphs in the present literature

There are other variations of the normal domination byadding restricted conditions to the normal dominating setFor example the connected domination set of a graph119866 pro-posed by Sampathkumar and Walikar [90] is a dominatingset 119878 such that the subgraph induced by 119878 is connected Theproblem of finding a minimum connected domination set ina graph is equivalent to the problemof finding a spanning treewithmaximumnumber of vertices of degree one [91] and hassome important applications in wireless networks [92] thusit has received much research attention However for suchan important domination there is no result on its bondagenumber We believe that the topic is of significance for futureresearch

9 Generalized Bondage Numbers

There are various generalizations of the classical dominationsuch as 119901-domination distance domination fractional dom-ination Roman domination and rainbow domination Everysuch a generalization can lead to a corresponding bondageIn this section we introduce some of them

91 119901-Bondage Numbers In 1985 Fink and Jacobson [93]introduced the concept of 119901-domination Let 119901 be a positiveinteger A subset 119878 of 119881(119866) is a 119901-dominating set of 119866 if|119878 cap 119873

119866(119910)| ⩾ 119901 for every 119910 isin 119878 The 119901-domination number

120574119901(119866) is the minimum cardinality among all 119901-dominating

sets of 119866 Any 119901-dominating set of 119866 with cardinality 120574119901(119866)

is called a 120574119901-set of 119866 Note that a 120574

1-set is a classic minimum

dominating set Notice that every graph has a 119901-dominatingset since the vertex-set119881(119866) is such a setWe also note that a 1-dominating set is a dominating set and so 120574(119866) = 120574

1(119866) The

119901-domination number has received much research attentionsee a state-of-the-art survey article by Chellali et al [94]

It is clear from definition that every 119901-dominating setof a graph certainly contains all vertices of degree at most119901 minus 1 By this simple observation to avoid the trivial caseoccurrence we always assume that Δ(119866) ⩾ 119901 For 119901 ⩾ 2 Luet al [95] gave a constructive characterization of trees withunique minimum 119901-dominating sets

Recently Lu and Xu [96] have introduced the conceptto the 119901-bondage number of 119866 denoted by 119887

119901(119866) as the

minimum cardinality among all sets of edges 119861 sube 119864(119866) suchthat 120574

119901(119866 minus 119861) gt 120574

119901(119866) Clearly 119887

1(119866) = 119887(119866)

Lu and Xu [96] established a lower bound and an upperbound on 119887

119901(119879) for any integer 119901 ⩾ 2 and any tree 119879 with

Δ(119879) ⩾ 119901 and characterized all trees achieving the lowerbound and the upper bound respectively

Theorem 141 (Lu and Xu [96] 2011) For any integer 119901 ⩾ 2and any tree 119879 with Δ(119879) ⩾ 119901

1 ⩽ 119887119901(119879) ⩽ Δ (119879) minus 119901 + 1 (52)

Let 119878 be a given subset of 119881(119866) and for any 119909 isin 119878 let

119873119901(119909 119878 119866) = 119910 isin 119878 cap 119873

119866(119909)

1003816100381610038161003816119873119866(119910) cap 119878

1003816100381610038161003816 = 119901

(53)

Then all trees achieving the lower bound 1 can be character-ized as follows

Theorem 142 (Lu and Xu [96] 2011) Let 119879 be a tree withΔ(119879) ⩾ 119901 ⩾ 2 Then 119887

119901(119879) = 1 if and only if for any 120574

119901-set

119878 of 119879 there exists an edge 119909119910 isin (119878 119878) such that 119910 isin 119873119901(119909 119878 119879)

The notation 119878(119886 119887) denotes a double star obtained byadding an edge between the central vertices of two stars119870

1119886minus1

and1198701119887minus1

And the vertex with degree 119886 (resp 119887) in 119878(119886 119887) iscalled the 119871-central vertex (resp 119877-central vertex) of 119878(119886 119887)

To characterize all trees attaining the upper bound givenin Theorem 141 we define three types of operations on a tree119879 with Δ(119879) = Δ ⩾ 119901 + 1

Type 1 attach a pendant edge to a vertex 119910with 119889119879(119910) ⩽ 119901minus

2 in 119879

Type 2 attach a star 1198701Δminus1

to a vertex 119910 of 119879 by joining itscentral vertex to 119910 where 119910 in a 120574

119901-set of 119879 and

119889119879(119910) ⩽ Δ minus 1

Type 3 attach a double star 119878(119901 Δ minus 1) to a pendant vertex 119910of 119879 by coinciding its 119877-central vertex with 119910 wherethe unique neighbor of 119910 is in a 120574

119901-set of 119879

Let B = 119879 a tree obtained from 1198701Δ

or 119878(119901 Δ) by afinite sequence of operations of Types 1 2 and 3

Theorem 143 (Lu and Xu [96] 2011) A tree with the maxi-mum Δ ⩾ 119901 + 1 has 119901-bondage number Δ minus 119901 + 1 if and onlyif it belongs toB

Theorem 144 (Lu and Xu [97] 2012) Let 119866119898119899= 119875

119898times 119875

119899

Then

1198872(119866

2119899) = 1 119891119900119903 119899 ⩾ 2

1198872(119866

3119899) =

2 119894119891 119899 equiv 1 (mod 3) 1 119900119905ℎ119890119903119908119894119904119890

for 119899 ⩾ 2

1198872(119866

4119899) =

1 119894119891 119899 equiv 3 (mod 4) 2 119900119905ℎ119890119903119908119894119904119890

for 119899 ⩾ 7

(54)

Recently Krzywkowski [98] has proposed the conceptof the nonisolating 2-bondage of a graph The nonisolating2-bondage number of a graph 119866 denoted by 1198871015840

2(119866) is the

minimum cardinality among all sets of edges 119861 sube 119864(119866) suchthat 119866 minus 119861 has no isolated vertices and 120574

2(119866 minus 119861) gt 120574

2(119866) If

for every 119861 sube 119864(119866) either 1205742(119866 minus 119861) = 120574

2(119866) or 119866 minus 119861 has

isolated vertices then define 11988710158402(119866) = 0 and say that 119866 is a 2-

nonisolatedly strongly stable graph Krzywkowski presentedsome basic properties of nonisolating 2-bondage in graphsshowed that for every nonnegative integer 119896 there exists atree 119879 such 1198871015840

2(119879) = 119896 characterized all 2-non-isolatedly

strongly stable trees and determined 11988710158402(119866) for several special

graphs

International Journal of Combinatorics 23

Theorem 145 (Krzywkowski [98] 2012) For any positiveintegers 119899 and119898 with119898 ⩽ 119899

1198871015840

2(119870

119899) =

0 119894119891 119899 = 1 2 3

lfloor2119899

3rfloor 119900119905ℎ119890119903119908119894119904119890

1198871015840

2(119875

119899) =

0 119894119891 119899 = 1 2 3

1 119900119905ℎ119890119903119908119894119904119890

1198871015840

2(119862

119899) =

0 119894119891 119899 = 3

1 119894119891 119899 119894119904 119890V119890119899

2 119894119891 119899 119894119904 119900119889119889

1198871015840

2(119870

119898119899) =

3 119894119891 119898 = 119899 = 3

1 119894119891 119898 = 119899 = 4

2 119900119905ℎ119890119903119908119894119904119890

(55)

92 Distance Bondage Numbers A subset 119878 of vertices of agraph119866 is said to be a distance 119896-dominating set for119866 if everyvertex in 119866 not in 119878 is at distance at most 119896 from some vertexof 119878 The minimum cardinality of all distance 119896-dominatingsets is called the distance 119896-domination number of 119866 anddenoted by 120574

119896(119866) (do not confuse with the above-mentioned

120574119901(119866)) When 119896 = 1 a distance 1-dominating set is a normal

dominating set and so 1205741(119866) = 120574(119866) for any graph 119866 Thus

the distance 119896-domination is a generalization of the classicaldomination

The relation between 120574119896and 120572

119896(the 119896-independence

number defined in Section 22) for a tree obtained by Meirand Moon [14] who proved that 120574

119896(119879) = 120572

2119896(119879) for any tree

119879 (a special case for 119896 = 1 is stated in Proposition 10) Thefurther research results can be found in Henning et al [99]Tian and Xu [100ndash103] and Liu et al [104]

In 1998 Hartnell et al [15] defined the distance 119896-bondagenumber of 119866 denoted by 119887

119896(119866) to be the cardinality of a

smallest subset 119861 of edges of 119866 with the property that 120574119896(119866 minus

119861) gt 120574119896(119866) FromTheorem 6 it is clear that if119879 is a nontrivial

tree then 1 ⩽ 1198871(119879) ⩽ 2 Hartnell et al [15] generalized this

result to any integer 119896 ⩾ 1

Theorem 146 (Hartnell et al [15] 1998) For every nontrivialtree 119879 and positive integer 119896 1 ⩽ 119887

119896(119879) ⩽ 2

Hartnell et al [15] and Topp and Vestergaard [13] alsocharacterized the trees having distance 119896-bondage number 2In particular the class of trees for which 119887

1(119879) = 2 are just

those which have a unique maximum 2-independent set (seeTheorem 11)

Since when 119896 = 1 the distance 1-bondage number 1198871(119866)

is the classical bondage number 119887(119866) Theorem 12 gives theNP-hardness of deciding the distance 119896-bondage number ofgeneral graphs

Theorem 147 Given a nonempty undirected graph 119866 andpositive integers 119896 and 119887 with 119887 ⩽ 120576(119866) determining wetheror not 119887

119896(119866) ⩽ 119887 is NP-hard

For a vertex 119909 in119866 the open 119896-neighborhood119873119896(119909) of 119909 is

defined as119873119896(119909) = 119910 isin 119881(119866) 1 ⩽ 119889

119866(119909 119910) le 119896The closed

119896-neighborhood119873119896[119909] of 119909 in 119866 is defined as119873

119896(119909) cup 119909 Let

Δ119896(119866) = max 1003816100381610038161003816119873119896

(119909)1003816100381610038161003816 for any 119909 isin 119881 (119866) (56)

Clearly Δ1(119866) = Δ(119866) The 119896th power of a graph 119866 is the

graph119866119896 with vertex-set119881(119866119896

) = 119881(119866) and edge-set119864(119866119896

) =

119909119910 1 ⩽ 119889119866(119909 119910) ⩽ 119896 The following lemma holds directly

from the definition of 119866119896

Proposition 148 (Tian and Xu [101]) Δ(119866119896

) = Δ119896(119866) and

120574(119866119896

) = 120574119896(119866) for any graph 119866 and each 119896 ⩾ 1

A graph 119866 is 119896-distance domination-critical or 120574119896-critical

for short if 120574119896(119866 minus 119909) lt 120574

119896(119866) for every vertex 119909 in 119866

proposed by Henning et al [105]

Proposition 149 (Tian and Xu [101]) For each 119896 ⩾ 1 a graph119866 is 120574

119896-critical if and only if 119866119896 is 120574

119896-critical

By the above facts we suggest the following worthwhileproblem

Problem 6 Can we generalize the results on the bondage for119866 to 119866119896 In particular do the following propositions hold

(a) 119887(119866119896

) = 119887119896(119866) for any graph 119866 and each 119896 ⩾ 1

(b) 119887(119866119896

) ⩽ Δ119896(119866) if 119866 is not 120574

119896-critical

Let 119896 and 119901 be positive integers A subset 119878 of 119881(119866) isdefined to be a (119896 119901)-dominating set of 119866 if for any vertex119909 isin 119878 |119873

119896(119909) cap 119878| ⩾ 119901 The (119896 119901)-domination number of

119866 denoted by 120574119896119901(119866) is the minimum cardinality among

all (119896 119901)-dominating sets of 119866 Clearly for a graph 119866 a(1 1)-dominating set is a classical dominating set a (119896 1)-dominating set is a distance 119896-dominating set and a (1 119901)-dominating set is the above-mentioned 119901-dominating setThat is 120574

11(119866) = 120574(119866) 120574

1198961(119866) = 120574

119896(119866) and 120574

1119901(119866) = 120574

119901(119866)

The concept of (119896 119901)-domination in a graph 119866 is a gen-eralized domination which combines distance 119896-dominationand 119901-domination in 119866 So the investigation of (119896 119901)-domination of 119866 is more interesting and has received theattention of many researchers see for example [106ndash109]

More general Li and Xu [110] proposed the concept of the(ℓ 119908)-domination number of a119908-connected graph A subset119878 of 119881(119866) is defined to be an (ℓ 119908)-dominating set of 119866 iffor any vertex 119909 isin 119878 there are 119908 internally disjoint pathsof length at most ℓ from 119909 to some vertex in 119878 The (ℓ 119908)-domination number of119866 denoted by 120574

ℓ119908(119866) is theminimum

cardinality among all (ℓ 119908)-dominating sets of 119866 Clearly1205741198961(119866) = 120574

119896(119866) and 120574

1119901(119866) = 120574

119901(119866)

It is quite natural to propose the concept of bondage num-ber for (119896 119901)-domination or (ℓ 119908)-domination However asfar as we know none has proposed this concept until todayThis is a worthwhile topic for further research

24 International Journal of Combinatorics

93 Fractional Bondage Numbers If 120590 is a function mappingthe vertex-set 119881 into some set of real numbers then for anysubset 119878 sube 119881 let 120590(119878) = sum

119909isin119878120590(119909) Also let |120590| = 120590(119881)

A real-value function 120590 119881 rarr [0 1] is a dominatingfunction of a graph 119866 if for every 119909 isin 119881(119866) 120590(119873

119866[119909]) ⩾ 1

Thus if 119878 is a dominating set of 119866 120590 is a function where

120590 (119909) = 1 if 119909 isin 1198780 otherwise

(57)

then 120590 is a dominating function of119866 The fractional domina-tion number of 119866 denoted by 120574

119865(119866) is defined as follows

120574119865(119866) = min |120590| 120590 is a domination function of 119866

(58)

The 120574119865-bondage number of 119866 denoted by 119887

119865(119866) is

defined as the minimum cardinality of a subset 119861 sube 119864 whoseremoval results in 120574

119865(119866 minus 119861) gt 120574

119865(119866)

Hedetniemi et al [111] are the first to study fractionaldomination although Farber [112] introduced the idea indi-rectly The concept of the fractional domination numberwas proposed by Domke and Laskar [113] in 1997 Thefractional domination numbers for some ordinary graphs aredetermined

Proposition 150 (Domke et al [114] 1998 Domke and Laskar[113] 1997) (a) If 119866 is a 119896-regular graph with order 119899 then120574119865(119866) = 119899119896 + 1(b) 120574

119865(119862

119899) = 1198993 120574

119865(119875

119899) = lceil1198993rceil 120574

119865(119870

119899) = 1

(c) 120574119865(119866) = 1 if and only if Δ(119866) = 119899 minus 1

(d) 120574119865(119879) = 120574(119879) for any tree 119879

(e) 120574119865(119870

119899119898) = (119899(119898 + 1) + 119898(119899 + 1))(119899119898 minus 1) where

max119899119898 gt 1

The assertions (a)ndash(d) are due to Domke et al [114] andthe assertion (e) is due to Domke and Laskar [113] Accordingto these results Domke and Laskar [113] determined the 120574

119865-

bondage numbers for these graphs

Theorem 151 (Domke and Laskar [113] 1997) 119887119865(119870

119899) =

lceil1198992rceil 119887119865(119870

119899119898) = 1 wheremax119899119898 gt 1

119887119865(119862

119899) =

2 119894119891 119899 equiv 0 (mod 3) 1 119900119905ℎ119890119903119908119894119904119890

119887119865(119875

119899) =

2 119894119891 119899 equiv 1 (mod 3) 1 119900119905ℎ119890119903119908119894119904119890

(59)

It is easy to see that for any tree119879 119887119865(119879) ⩽ 2 In fact since

120574119865(119879) = 120574(119879) for any tree 119879 120574

119865(119879

1015840

) = 120574(1198791015840

) for any subgraph1198791015840 of 119879 It follows that

119887119865(119879) = min 119861 sube 119864 (119879) 120574

119865(119879 minus 119861) gt 120574

119865(119879)

= min 119861 sube 119864 (119879) 120574 (119879 minus 119861) gt 120574 (119879)

= 119887 (119879) ⩽ 2

(60)

Except for these no results on 119887119865(119866) have been known as

yet

94 Roman Bondage Numbers A Roman dominating func-tion on a graph 119866 is a labeling 119891 119881 rarr 0 1 2 such thatevery vertex with label 0 has at least one neighbor with label2 The weight of a Roman dominating function 119891 on 119866 is thevalue

119891 (119866) = sum

119906isin119881(119866)

119891 (119906) (61)

The minimum weight of a Roman dominating function ona graph 119866 is called the Roman domination number of 119866denoted by 120574RM (119866)

A Roman dominating function 119891 119881 rarr 0 1 2

can be represented by the ordered partition (1198810 119881

1 119881

2) (or

(119881119891

0 119881

119891

1 119881

119891

2) to refer to119891) of119881 where119881

119894= V isin 119881 | 119891(V) = 119894

In this representation its weight 119891(119866) = |1198811| + 2|119881

2| It is

clear that119881119891

1cup119881

119891

2is a dominating set of 119866 called the Roman

dominating set denoted by 119863119891

R = (1198811 1198812) Since 119881119891

1cup 119881

119891

2is

a dominating set when 119891 is a Roman dominating functionand since placing weight 2 at the vertices of a dominating setyields a Roman dominating function in [115] it is observedthat

120574 (119866) ⩽ 120574RM (119866) ⩽ 2120574 (119866) (62)

A graph 119866 is called to be Roman if 120574RM (119866) = 2120574(119866)The definition of the Roman domination function is

proposed implicitly by Stewart [116] and ReVelle and Rosing[117] Roman dominating numbers have been deeply studiedIn particular Bahremandpour et al [118] showed that theproblem determining the Roman domination number is NP-complete even for bipartite graphs

Let 119866 be a graph with maximum degree at least twoThe Roman bondage number 119887RM (119866) of 119866 is the minimumcardinality of all sets 1198641015840 sube 119864 for which 120574RM (119866 minus 119864

1015840

) gt

120574RM (119866) Since in study of Roman bondage number theassumption Δ(119866) ⩾ 2 is necessary we always assume thatwhen we discuss 119887RM (119866) all graphs involved satisfy Δ(119866) ⩾2 The Roman bondage number 119887RM (119866) is introduced byJafari Rad and Volkmann in [119]

Recently Bahremandpour et al [118] have shown that theproblem determining the Roman bondage number is NP-hard even for bipartite graphs

Problem 7 Consider the decision problem

Roman Bondage Problem

Instance a nonempty bipartite graph119866 and a positiveinteger 119896

Question is 119887RM (119866) ⩽ 119896

Theorem 152 (Bahremandpour et al [118] 2013) TheRomanbondage problem is NP-hard even for bipartite graphs

The exact value of 119887RM(119866) is known only for a few familyof graphs including the complete graphs cycles and paths

International Journal of Combinatorics 25

Theorem 153 (Jafari Rad and Volkmann [119] 2011) For anypositive integers 119899 and119898 with119898 ⩽ 119899

119887RM (119875119899) = 2 119894119891 119899 equiv 2 (mod 3) 1 119900119905ℎ119890119903119908119894119904119890

119887RM (119862119899) =

3 119894119891 119899 = 2 (mod 3) 2 119900119905ℎ119890119903119908119894119904119890

119887RM (119870119899) = lceil

119899

2rceil 119891119900119903 119899 ⩾ 3

119887RM (119870119898119899) =

1 119894119891 119898 = 1 119886119899119889 119899 = 1

4 119894119891 119898 = 119899 = 3

119898 119900119905ℎ119890119903119908119894119904119890

(63)

Lemma 154 (Cockayne et al [115] 2004) If 119866 is a graph oforder 119899 and contains vertices of degree 119899 minus 1 then 120574RM(119866) = 2

Using Lemma 154 the third conclusion in Theorem 153can be generalized to more general case which is similar toLemma 49

Theorem 155 (Jafari Rad and Volkmann [119] 2011) Let119866 bea graph with order 119899 ge 3 and 119905 the number of vertices of degree119899 minus 1 in 119866 If 119905 ⩾ 1 then 119887RM(119866) = lceil1199052rceil

Ebadi and PushpaLatha [120] conjectured that 119887RM(119866) ⩽119899 minus 1 for any graph 119866 of order 119899 ⩾ 3 Dehgardi et al[121] Akbari andQajar [122] independently showed that thisconjecture is true

Theorem 156 119887RM(119866) ⩽ 119899 minus 1 for any connected graph 119866 oforder 119899 ⩾ 3

For a complete 119905-partite graph we have the followingresult

Theorem 157 (Hu and Xu [123] 2011) Let 119866 = 11987011989811198982119898119905

bea complete 119905-partite graph with 119898

1= sdot sdot sdot = 119898

119894lt 119898

119894+1le sdot sdot sdot ⩽

119898119905 119905 ⩾ 2 and 119899 = sum119905

119895=1119898

119895 Then

119887RM (119866) =

lceil119894

2rceil 119894119891 119898

119894= 1 119886119899119889 119899 ⩾ 3

2 119894119891 119898119894= 2 119886119899119889 119894 = 1

119894 119894119891 119898119894= 2 119886119899119889 119894 ⩾ 2

4 119894119891 119898119894= 3 119886119899119889 119894 = 119905 = 2

119899 minus 1 119894119891 119898119894= 3 119886119899119889 119894 = 119905 ⩾ 3

119899 minus 119898119905

119894119891 119898119894⩾ 3 119886119899119889 119898

119905⩾ 4

(64)

Consider a complete 119905-partite graph 119870119905(3) for 119905 ⩾ 3

119887RM(119870119905(3)) = 119899 minus 1 by Theorem 157 where 119899 = 3119905

which shows that this upper bound of 119899 minus 1 on 119887RM(119866) inTheorem 156 is sharp This fact and 120574

119877(119870

119905(3)) = 4 give

a negative answer to the following two problems posed byDehgardi et al [121]Prove or Disprove for any connected graph 119866 of order 119899 ge 3(i) 119887RM(119866) = 119899 minus 1 if and only if 119866 cong 119870

3 (ii) 119887RM(119866) ⩽ 119899 minus

120574119877(119866) + 1Note that a complete 119905-partite graph 119870

119905(3) is (119899 minus 3)-

regular and 119887RM(119870119905(3)) = 119899 minus 1 for 119905 ⩾ 3 where 119899 = 3119905

Hu and Xu further determined that 119887119877(119866) = 119899 minus 2 for any

(119899 minus 3)-regular graph 119866 of order 119899 ⩽ 5 except 119870119905(3)

Theorem 158 (Hu and Xu [123] 2011) For any (119899minus3)-regulargraph 119866 of order 119899 other than119870

119905(3) 119887RM(119866) = 119899 minus 2 for 119899 ⩾ 5

For a tree 119879 with order 119899 ⩾ 3 Ebadi and PushpaLatha[120] and Jafari Rad and Volkmann [119] independentlyobtained an upper bound on 119887RM(119879)

Theorem 159 119887RM(119879) ⩽ 3 for any tree 119879 with order 119899 ⩾ 3

Theorem 160 (Bahremandpour et al [118] 2013) 119887RM(1198752 times119875119899) = 2 for 119899 ⩾ 2

Theorem 161 (Jafari Rad and Volkmann [119] 2011) Let119866 bea graph with order at least three

(a) If (119909 119910 119911) is a path of length 2 in 119866 then 119887RM(119866) ⩽119889119866(119909) + 119889

119866(119910) + 119889

119866(119911) minus 3 minus |119873

119866(119909) cap 119873

119866(119910)|

(b) If 119866 is connected then 119887RM(119866) ⩽ 120582(119866) + 2Δ(119866) minus 3

Theorem 161 (b) implies 119887RM(119866) ⩽ 120575(119866) + 2Δ(119866) minus 3 for aconnected graph 119866 Note that for a planar graph 119866 120575(119866) ⩽ 5moreover 120575(119866) ⩽ 3 if the girth at least 4 and 120575(119866) ⩽ 2 if thegirth at least 6 These two facts show that 119887RM(119866) ⩽ 2Δ(119866) +2 for a connected planar graph 119866 Jafari Rad and Volkmann[124] improved this bound

Theorem 162 (Jafari Rad and Volkmann [124] 2011) For anyconnected planar graph 119866 of order 119899 ⩾ 3 with girth 119892(119866)

119887RM (119866)

⩽

2Δ (119866)

Δ (119866) + 6

Δ (119866) + 5 119894119891 119866 119888119900119899119905119886119894119899119904 119899119900 V119890119903119905119894119888119890119904 119900119891 119889119890119892119903119890119890 119891119894V119890

Δ (119866) + 4 119894119891 119892 (119866) ⩾ 4

Δ (119866) + 3 119894119891 119892 (119866) ⩾ 5

Δ (119866) + 2 119894119891 119892 (119866) ⩾ 6

Δ (119866) + 1 119894119891 119892 (119866) ⩾ 8

(65)

According toTheorem 157 119887RM(119862119899) = 3 = Δ(119862

119899)+1 for a

cycle 119862119899of length 119899 ⩾ 8 with 119899 equiv 2 (mod 3) and therefore

the last upper bound inTheorem 162 is sharp at least for Δ =2

Combining the fact that every planar graph 119866 withminimum degree 5 contains an edge 119909119910 with 119889

119866(119909) = 5

and 119889119866(119910) isin 5 6 with Theorem 161 (a) Akbari et al [125]

obtained the following result

26 International Journal of Combinatorics

middot middot middot

middot middot middot

middot middot middot

middot middot middot

119866

Figure 9 The graph 119866 is constructed from 119866

Theorem 163 (Akbari et al [125] 2012) 119887RM(119866) ⩽ 15 forevery planar graph 119866

It remains open to show whether the bound inTheorem 163 is sharp or not Though finding a planargraph 119866 with 119887RM(119866) = 15 seems to be difficult Akbari etal [125] constructed an infinite family of planar graphs withRoman bondage number equal to 7 by proving the followingresult

Theorem 164 (Akbari et al [125] 2012) Let 119866 be a graph oforder 119899 and 119866 is the graph of order 5119899 obtained from 119866 byattaching the central vertex of a copy of a path 119875

5to each vertex

of119866 (see Figure 9) Then 120574RM(119866) = 4119899 and 119887RM(119866) = 120575(119866)+2

ByTheorem 164 infinitemany planar graphswith Romanbondage number 7 can be obtained by considering any planargraph 119866 with 120575(119866) = 5 (eg the icosahedron graph)

Conjecture 165 (Akbari et al [125] 2012) The Romanbondage number of every planar graph is at most 7

For general bounds the following observation is directlyobtained which is similar to Observation 1

Observation 4 Let 119866 be a graph of order 119899 with maximumdegree at least two Assume that 119867 is a spanning subgraphobtained from 119866 by removing 119896 edges If 120574RM(119867) = 120574RM(119866)then 119887

119877(119867) ⩽ 119887RM(119866) ⩽ 119887RM(119867) + 119896

Theorem 166 (Bahremandpour et al [118] 2013) For anyconnected graph 119866 of order 119899 if 119899 ⩾ 3 and 120574RM(119866) = 120574(119866) + 1then

119887RM (119866) ⩽ min 119887 (119866) 119899Δ (66)

where 119899Δis the number of vertices with maximum degree Δ in

119866

Observation 5 For any graph 119866 if 120574(119866) = 120574RM(119866) then119887RM(119866) ⩽ 119887(119866)

Theorem 167 (Bahremandpour et al [118] 2013) For everyRoman graph 119866

119887RM (119866) ⩾ 119887 (119866) (67)

The bound is sharp for cycles on 119899 vertices where 119899 equiv 0 (mod3)

The strict inequality in Theorem 167 can hold for exam-ple 119887(119862

3119896+2) = 2 lt 3 = 119887RM(1198623119896+2

) byTheorem 157A graph 119866 is called to be vertex Roman domination-

critical if 120574RM(119866 minus 119909) lt 120574RM(119866) for every vertex 119909 in 119866 If119866 has no isolated vertices then 120574RM(119866) ⩽ 2120574(119866) ⩽ 2120573(119866)If 120574RM(119866) = 2120573(119866) then 120574RM(119866) = 2120574(119866) and hence 119866 is aRoman graph In [30] Volkmann gave a lot of graphs with120574(119866) = 120573(119866) The following result is similar to Theorem 57

Theorem 168 (Bahremandpour et al [118] 2013) For anygraph 119866 if 120574RM(119866) = 2120573(119866) then

(a) 119887RM(119866) ⩾ 120575(119866)

(b) 119887RM(119866) ⩾ 120575(119866)+1 if119866 is a vertex Roman domination-critical graph

If 120574RM(119866) = 2 then obviously 119887RM(119866) ⩽ 120575(119866) So weassume 120574RM(119866) ⩾ 3 Dehgardi et al [121] proved 119887RM(119866) ⩽(120574RM(119866) minus 2)Δ(119866) + 1 and posed the following problem if 119866is a connected graph of order 119899 ⩾ 4 with 120574RM(119866) ⩾ 3 then

119887RM (119866) ⩽ (120574RM (119866) minus 2) Δ (119866) (68)

Theorem 161 (a) shows that the inequality (68) holds if120574RM(119866) ⩾ 5 Thus the bound in (68) is of interest only when120574RM(119866) is 3 or 4

Lemma 169 (Bahremandpour et al [118] 2013) For anynonempty graph 119866 of order 119899 ⩾ 3 120574RM(119866) = 3 if and onlyif Δ(119866) = 119899 minus 2

The following result shows that (68) holds for all graphs119866 of order 119899 ⩾ 4 with 120574RM(119866) = 3 or 4 which improvesTheorem 161 (b)

Theorem 170 (Bahremandpour et al [118] 2013) For anyconnected graph 119866 of order 119899 ⩾ 4

119887RM (119866) le Δ (119866) = 119899 minus 2 119894119891 120574RM (119866) = 3

Δ (119866) + 120575 (119866) minus 1 119894119891 120574RM (119866) = 4(69)

with the first equality if and only if 119866 cong 1198624

Recently Akbari and Qajar [122] have proved the follow-ing result

Theorem 171 (Akbari and Qajar [122]) For any connectedgraph 119866 of order 119899 ⩾ 3

119887RM (119866) ⩽ 119899 minus 120574RM (119866) + 5 (70)

International Journal of Combinatorics 27

We conclude this section with the following problems

Problem 8 Characterize all connected graphs 119866 of order 119899 ⩾3 for which 119887RM(119866) = 119899 minus 1

Problem 9 Prove or disprove if 119866 is a connected graph oforder 119899 ⩾ 3 then

119887RM (119866) ⩽ 119899 minus 120574RM (119866) + 3 (71)

Note that 120574119877(119870

119905(3)) = 4 and 119887RM(119870119905

(3)) = 119899minus1 where 119899 =3119905 for 119905 ⩾ 3 The upper bound on 119887RM(119866) given in Problem 9is sharp if Problem 9 has positive answer

95 Rainbow Bondage Numbers For a positive integer 119896 a119896-rainbow dominating function of a graph 119866 is a function 119891from the vertex-set 119881(119866) to the set of all subsets of the set1 2 119896 such that for any vertex 119909 in 119866 with 119891(119909) = 0 thecondition

⋃

119910isin119873119866(119909)

119891 (119910) = 1 2 119896 (72)

is fulfilledThe weight of a 119896-rainbow dominating function 119891on 119866 is the value

119891 (119866) = sum

119909isin119881(119866)

1003816100381610038161003816119891 (119909)1003816100381610038161003816 (73)

The 119896-rainbow domination number of a graph 119866 denoted by120574119903119896(119866) is the minimum weight of a 119896-rainbow dominating

function 119891 of 119866Note that 120574

1199031(119866) is the classical domination number 120574(119866)

The 119896-rainbow domination number is introduced by Bresaret al [126] Rainbow domination of a graph 119866 coincides withordinary domination of the Cartesian product of 119866 with thecomplete graph in particular 120574

119903119896(119866) = 120574(119866 times 119870

119896) for any

positive integer 119896 and any graph 119866 [126] Hartnell and Rall[127] proved that 120574(119879) lt 120574

1199032(119879) for any tree 119879 no graph has

120574 = 120574119903119896for 119896 ⩾ 3 for any 119896 ⩾ 2 and any graph 119866 of order 119899

min 119899 120574 (119866) + 119896 minus 2 ⩽ 120574119903119896(119866) ⩽ 119896120574 (119866) (74)

The introduction of rainbow domination is motivatedby the study of paired domination in Cartesian productsof graphs where certain upper bounds can be expressed interms of rainbow domination that is for any graph 119866 andany graph 119867 if 119881(119867) can be partitioned into 119896120574

119875-sets then

120574119875(119866 times 119867) ⩽ (1119896)120592(119867)120574

119903119896(119866) proved by Bresar et al [126]

Dehgardi et al [128] introduced the concept of 119896-rainbowbondage number of graphs Let 119866 be a graph with maximumdegree at least two The 119896-rainbow bondage number 119887

119903119896(119866)

of 119866 is the minimum cardinality of all sets 119861 sube 119864(119866) forwhich 120574

119903119896(119866 minus 119861) gt 120574

119903119896(119866) Since in the study of 119896-rainbow

bondage number the assumption Δ(119866) ⩾ 2 is necessarywe always assume that when we discuss 119887

119903119896(119866) all graphs

involved satisfy Δ(119866) ⩾ 2The 2-rainbow bondage number of some special families

of graphs has been determined

Theorem 172 (Dehgardi et al [128]) For any integer 119899 ⩾ 3

1198871199032(119875

119899) = 1 119891119900119903 119899 ⩾ 2

1198871199032(119862

119899) =

1 119894119891 119899 equiv 0 (mod 4) 2 119900119905ℎ119890119903119908119894119904119890

1198871199032(119870

119899119899) =

1 119894119891 119899 = 2

3 119894119891 119899 = 3

6 119894119891 119899 = 4

119898 119894119891 119899 ⩾ 5

(75)

Theorem 173 (Dehgardi et al [128]) For any connected graph119866 of order 119899 ⩾ 3 119887

1199032(119866) ⩽ 120582(119866) + 2Δ(119866) minus 3

Theorem 174 (Dehgardi et al [128]) For any tree 119879 of order119899 ⩾ 3 119887

1199032(119879) ⩽ 2

Theorem 175 (Dehgardi et al [128]) If 119866 is a planar graphwithmaximumdegree at least two then 119887

1199032(119866) ⩽ 15Moreover

if the girth 119892(119866) ⩾ 4 then 1198871199032(119866) ⩽ 11 and if 119892(119866) ⩾ 6 then

1198871199032(119866) ⩽ 9

Theorem 176 (Dehgardi et al [128]) For a complete bipartitegraph 119870

119901119902 if 2119896 + 1 ⩽ 119901 ⩽ 119902 then 119887

119903119896(119870

119901119902) = 119901

Theorem177 (Dehgardi et al [128]) For a complete graph119870119899

if 119899 ⩾ 119896 + 1 then 119887119903119896(119870

119899) = lceil119896119899(119896 + 1)rceil

The special case 119896 = 1 of Theorem 177 can be found inTheorem 1 The following is a simple observation similar toObservation 1

Observation 6 (Dehgardi et al [128]) Let 119866 be a graphwith maximum degree at least two 119867 a spanning subgraphobtained from 119866 by removing 119896 edges If 120574

119903119896(119867) = 120574

119903119896(119866)

then 119887119903119896(119867) ⩽ 119887

119903119896(119866) ⩽ 119887

119903119896(119867) + 119896

Theorem 178 (Dehgardi et al [128]) For every graph 119866 with120574119903119896(119866) = 119896120574(119866) 119887

119903119896(119866) ⩾ 119887(119866)

A graph 119866 is said to be vertex 119896-rainbow domination-critical if 120574

119903119896(119866 minus 119909) lt 120574

119903119896(119866) for every vertex 119909 in 119866 It is

clear that if 119866 has no isolated vertices then 120574119903119896(119866) ⩽ 119896120574(119866) ⩽

119896120573(119866) where 120573(119866) is the vertex-covering number of119866 Thusif 120574

119903119896(119866) = 119896120573(119866) then 120574

119903119896(119866) = 119896120574(119866)

Theorem 179 (Dehgardi et al [128]) For any graph 119866 if120574119903119896(119866) = 119896120573(119866) then

(a) 119887119903119896(119866) ⩾ 120575(119866)

(b) 119887119903119896(119866) ⩾ 120575(119866)+1 if119866 is vertex 119896-rainbow domination-

critical

As far as we know there are no more results on the 119896-rainbow bondage number of graphs in the literature

28 International Journal of Combinatorics

10 Results on Digraphs

Although domination has been extensively studied in undi-rected graphs it is natural to think of a dominating setas a one-way relationship between vertices of the graphIndeed among the earliest literatures on this subject vonNeumann and Morgenstern [129] used what is now calleddomination in digraphs to find solution (or kernels whichare independent dominating sets) for cooperative 119899-persongames Most likely the first formulation of domination byBerge [130] in 1958 was given in the context of digraphs andonly some years latter by Ore [131] in 1962 for undirectedgraphs Despite this history examination of domination andits variants in digraphs has been essentially overlooked (see[4] for an overview of the domination literature) Thus thereare few if any such results on domination for digraphs in theliterature

The bondage number and its related topics for undirectedgraph have become one of major areas both in theoreticaland applied researches However until recently Carlsonand Develin [42] Shan and Kang [132] and Huang andXu [133 134] studied the bondage number for digraphsindependently In this section we introduce their resultsfor general digraphs Results for some special digraphs suchas vertex-transitive digraphs are introduced in the nextsection

101 Upper Bounds for General Digraphs Let 119866 = (119881 119864)

be a digraph without loops and parallel edges A digraph 119866is called to be asymmetric if whenever (119909 119910) isin 119864(119866) then(119910 119909) notin 119864(119866) and to be symmetric if (119909 119910) isin 119864(119866) implies(119910 119909) isin 119864(119866) For a vertex 119909 of 119881(119866) the sets of out-neighbors and in-neighbors of 119909 are respectively defined as119873

+

119866(119909) = 119910 isin 119881(119866) (119909 119910) isin 119864(119866) and 119873minus

119866(119909) = 119909 isin

119881(119866) (119909 119910) isin 119864(119866) the out-degree and the in-degree of119909 are respectively defined as 119889+

119866(119909) = |119873

+

119866(119909)| and 119889minus

119866(119909) =

|119873+

119866(119909)| Denote themaximum and theminimum out-degree

(resp in-degree) of 119866 by Δ+(119866) and 120575+(119866) (resp Δminus(119866) and120575minus

(119866))The degree 119889119866(119909) of 119909 is defined as 119889+

119866(119909)+119889

minus

119866(119909) and

the maximum and the minimum degrees of119866 are denoted byΔ(119866) and 120575(119866) respectively that is Δ(119866) = max119889

119866(119909) 119909 isin

119881(119866) and 120575(119866) = min119889119866(119909) 119909 isin 119881(119866) Note that the

definitions here are different from the ones in the textbookon digraphs see for example Xu [1]

A subset 119878 of119881(119866) is called a dominating set if119881(119866) = 119878cup119873

+

119866(119878) where119873+

119866(119878) is the set of out-neighbors of 119878Then just

as for undirected graphs 120574(119866) is the minimum cardinalityof a dominating set and the bondage number 119887(119866) is thesmallest cardinality of a set 119861 of edges such that 120574(119866 minus 119861) gt120574(119866) if such a subset 119861 sube 119864(119866) exists Otherwise we put119887(119866) = infin

Some basic results for undirected graphs stated inSection 3 can be generalized to digraphs For exampleTheorem 18 is generalized by Carlson and Develin [42]Huang andXu [133] and Shan andKang [132] independentlyas follows

Theorem 180 Let 119866 be a digraph and (119909 119910) isin 119864(119866) Then119887(119866) ⩽ 119889

119866(119910) + 119889

minus

119866(119909) minus |119873

minus

119866(119909) cap 119873

minus

119866(119910)|

Corollary 181 For a digraph 119866 119887(119866) ⩽ 120575minus(119866) + Δ(119866)

Since 120575minus

(119866) ⩽ (12)Δ(119866) from Corollary 181Conjecture 53 is valid for digraphs

Corollary 182 For a digraph 119866 119887(119866) ⩽ (32)Δ(119866)

In the case of undirected graphs the bondage number of119866119899= 119870

119899times 119870

119899achieves this bound However it was shown

by Carlson and Develin in [42] that if we take the symmetricdigraph119866

119899 we haveΔ(119866

119899) = 4(119899minus1) 120574(119866

119899) = 119899 and 119887(119866

119899) ⩽

4(119899 minus 1) + 1 = Δ(119866119899) + 1 So this family of digraphs cannot

show that the bound in Corollary 182 is tightCorresponding to Conjecture 51 which is discredited

for undirected graphs and is valid for digraphs the sameconjecture for digraphs can be proposed as follows

Conjecture 183 (Carlson and Develin [42] 2006) 119887(119866) ⩽Δ(119866) + 1 for any digraph 119866

If Conjecture 183 is true then the following results showsthat this upper bound is tight since 119887(119870

119899times119870

119899) = Δ(119870

119899times119870

119899)+1

for a complete digraph 119870119899

In 2007 Shan and Kang [132] gave some tight upperbounds on the bondage numbers for some asymmetricdigraphs For example 119887(119879) ⩽ Δ(119879) for any asymmetricdirected tree 119879 119887(119866) ⩽ Δ(119866) for any asymmetric digraph 119866with order at least 4 and 120574(119866) ⩽ 2 For planar digraphs theyobtained the following results

Theorem 184 (Shan and Kang [132] 2007) Let 119866 be aasymmetric planar digraphThen 119887(119866) ⩽ Δ(119866)+2 and 119887(119866) ⩽Δ(119866) + 1 if Δ(119866) ⩾ 5 and 119889minus

119866(119909) ⩾ 3 for every vertex 119909 with

119889119866(119909) ⩾ 4

102 Results for Some Special Digraphs The exact values andbounds on 119887(119866) for some standard digraphs are determined

Theorem185 (Huang andXu [133] 2006) For a directed cycle119862119899and a directed path 119875

119899

119887 (119862119899) =

3 119894119891 119899 119894119904 119900119889119889

2 119894119891 119899 119894119904 119890V119890119899

119887 (119875119899) =

2 119894119891 119899 119894119904 119900119889119889

1 119894119891 119899 119894119904 119890V119890119899

(76)

For the de Bruijn digraph 119861(119889 119899) and the Kautz digraph119870(119889 119899)

119887 (119861 (119889 119899)) = 119889 119894119891 119899 119894119904 119900119889119889

119889 ⩽ 119887 (119861 (119889 119899)) ⩽ 2119889 119894119891 119899 119894119904 119890V119890119899

119887 (119870 (119889 119899)) = 119889 + 1

(77)

Like undirected graphs we can define the total domi-nation number and the total bondage number On the totalbondage numbers for some special digraphs the knownresults are as follows

International Journal of Combinatorics 29

Theorem 186 (Huang and Xu [134] 2007) For a directedcycle 119862

119899and a directed path 119875

119899 119887

119879(119875

119899) and 119887

119879(119862

119899) all do not

exist For a complete digraph 119870119899

119887119879(119870

119899) = infin 119894119891 119899 = 1 2

119887119879(119870

119899) = 3 119894119891 119899 = 3

119899 ⩽ 119887119879(119870

119899) ⩽ 2119899 minus 3 119894119891 119899 ⩾ 4

(78)

The extended de Bruijn digraph EB(119889 119899 1199021 119902

119901) and

the extended Kautz digraph EK(119889 119899 1199021 119902

119901) are intro-

duced by Shibata and Gonda [135] If 119901 = 1 then theyare the de Bruijn digraph B(119889 119899) and the Kautz digraphK(119889 119899) respectively Huang and Xu [134] determined theirtotal domination numbers In particular their total bondagenumbers for general cases are determined as follows

Theorem 187 (Huang andXu [134] 2007) If 119889 ⩾ 2 and 119902119894⩾ 2

for each 119894 = 1 2 119901 then

119887119879(EB(119889 119899 119902

1 119902

119901)) = 119889

119901

minus 1

119887119879(EK(119889 119899 119902

1 119902

119901)) = 119889

119901

(79)

In particular for the de Bruijn digraph B(119889 119899) and the Kautzdigraph K(119889 119899)

119887119879(B (119889 119899)) = 119889 minus 1 119887

119879(K (119889 119899)) = 119889 (80)

Zhang et al [136] determined the bondage number incomplete 119905-partite digraphs

Theorem 188 (Zhang et al [136] 2009) For a complete 119905-partite digraph 119870

11989911198992119899119905

where 1198991⩽ 119899

2⩽ sdot sdot sdot ⩽ 119899

119905

119887 (11987011989911198992119899119905

)

=

119898 119894119891 119899119898= 1 119899

119898+1⩾ 2 119891119900119903 119904119900119898119890

119898 (1 ⩽ 119898 lt 119905)

4119905 minus 3 119894119891 1198991= 119899

2= sdot sdot sdot = 119899

119905= 2

119905minus1

sum

119894=1

119899119894+ 2 (119905 minus 1) 119900119905ℎ119890119903119908119894119904119890

(81)

Since an undirected graph can be thought of as a sym-metric digraph any result for digraphs has an analogy forundirected graphs in general In view of this point studyingthe bondage number for digraphs is more significant thanfor undirected graphs Thus we should further study thebondage number of digraphs and try to generalize knownresults on the bondage number and related variants for undi-rected graphs to digraphs prove or disprove Conjecture 183In particular determine the exact values of 119887(B(119889 119899)) for aneven 119899 and 119887

119879(119870

119899) for 119899 ⩾ 4

11 Efficient Dominating Sets

A dominating set 119878 of a graph 119866 is called to be efficient if forevery vertex 119909 in119866 |119873

119866[119909]cap119878| = 1 if119866 is a undirected graph

or |119873minus

119866[119909] cap 119878| = 1 if 119866 is a directed graph The concept of an

efficient set was proposed by Bange et al [137] in 1988 In theliterature an efficient set is also called a perfect dominatingset (see Livingston and Stout [138] for an explanation)

From definition if 119878 is an efficient dominating set of agraph 119866 then 119878 is certainly an independent set and everyvertex not in 119878 is adjacent to exactly one vertex in 119878

It is also clear from definition that a dominating set 119878 isefficient if and only if N

119866[119878] = 119873

119866[119909] 119909 isin 119878 for the

undirected graph 119866 or N+

119866[119878] = 119873

+

119866[119909] 119909 isin 119878 for the

digraph119866 is a partition of119881(119866) where the induced subgraphby119873

119866[119909] or119873+

119866[119909] is a star or an out-star with the root 119909

The efficient domination has important applications inmany areas such as error-correcting codes and receivesmuch attention in the late years

The concept of efficient dominating sets is a measureof the efficiency of domination in graphs Unfortunately asshown in [137] not every graph has an efficient dominatingset andmoreover it is anNP-complete problem to determinewhether a given graph has an efficient dominating set Inaddition it was shown by Clark [139] in 1993 that for a widerange of 119901 almost every random undirected graph 119866 isin

G(120592 119901) has no efficient dominating sets This means thatundirected graphs possessing an efficient dominating set arerare However it is easy to show that every undirected graphhas an orientationwith an efficient dominating set (see Bangeet al [140])

In 1993 Barkauskas and Host [141] showed that deter-mining whether an arbitrary oriented graph has an efficientdominating set is NP-complete Even so the existence ofefficient dominating sets for some special classes of graphs hasbeen examined see for example Dejter and Serra [142] andLee [143] for Cayley graph Gu et al [144] formeshes and toriObradovic et al [145] Huang and Xu [36] Reji Kumar andMacGillivray [146] for circulant graphs Harary graphs andtori and Van Wieren et al [147] for cube-connected cycles

In this section we introduce results of the bondagenumber for some graphs with efficient dominating sets dueto Huang and Xu [36]

111 Results for General Graphs In this subsection we intro-duce some results on bondage numbers obtained by applyingefficient dominating sets We first state the two followinglemmas

Lemma 189 For any 119896-regular graph or digraph 119866 of order 119899120574(119866) ⩾ 119899(119896 + 1) with equality if and only if 119866 has an efficientdominating set In addition if 119866 has an efficient dominatingset then every efficient dominating set is certainly a 120574-set andvice versa

Let 119890 be an edge and 119878 a dominating set in 119866 We say that119890 supports 119878 if 119890 isin (119878 119878) where (119878 119878) = (119909 119910) isin 119864(119866) 119909 isin 119878 119910 isin 119878 Denote by 119904(119866) the minimum number of edgeswhich support all 120574-sets in 119866

Lemma 190 For any graph or digraph 119866 119887(119866) ⩾ 119904(119866) withequality if 119866 is regular and has an efficient dominating set

30 International Journal of Combinatorics

A graph 119866 is called to be vertex-transitive if its automor-phism group Aut(119866) acts transitively on its vertex-set 119881(119866)A vertex-transitive graph is regular Applying Lemmas 189and 190 Huang and Xu obtained some results on bondagenumbers for vertex-transitive graphs or digraphs

Theorem 191 (Huang and Xu [36] 2008) For any vertex-transitive graph or digraph 119866 of order 119899

119887 (119866) ⩾

lceil119899

2120574 (119866)rceil 119894119891 119866 119894119904 119906119899119889119894119903119890119888119905119890119889

lceil119899

120574 (119866)rceil 119894119891 119866 119894119904 119889119894119903119890119888119905119890119889

(82)

Theorem192 (Huang andXu [36] 2008) Let119866 be a 119896-regulargraph of order 119899 Then

119887(119866) ⩽ 119896 if119866 is undirected and 119899 ⩾ 120574(119866)(119896+1)minus119896+1119887(119866) ⩽ 119896+1+ℓ if119866 is directed and 119899 ⩾ 120574(119866)(119896+1)minusℓwith 0 ⩽ ℓ ⩽ 119896 minus 1

Next we introduce a better upper bound on 119887(119866) To thisaim we need the following concept which is a generalizationof the concept of the edge-covering of a graph 119866 For 1198811015840

sube

119881(119866) and 1198641015840 sube 119864(119866) we say that 1198641015840covers 1198811015840 and call 1198641015840an edge-covering for 1198811015840 if there exists an edge (119909 119910) isin 1198641015840 forany vertex 119909 isin 1198811015840 For 119910 isin 119881(119866) let 1205731015840[119910] be the minimumcardinality over all edge-coverings for119873minus

119866[119910]

Theorem 193 (Huang and Xu [36] 2008) If 119866 is a 119896-regulargraph with order 120574(119866)(119896 + 1) then 119887(119866) ⩽ 1205731015840[119910] for any 119910 isin119881(119866)

Theupper bound on 119887(119866) given inTheorem 193 is tight inview of 119887(119862

119899) for a cycle or a directed cycle 119862

119899(seeTheorems

1 and 185 resp)It is easy to see that for a 119896-regular graph 119866 lceil(119896 + 1)2rceil ⩽

1205731015840

[119910] ⩽ 119896 when 119866 is undirected and 1205731015840[119910] = 119896 + 1 when 119866 isdirected By this fact and Lemma 189 the following theoremis merely a simple combination of Theorems 191 and 193

Theorem 194 (Huang and Xu [36] 2008) Let 119866 be a vertex-transitive graph of degree 119896 If 119866 has an efficient dominatingset then

lceil119896 + 1

2rceil ⩽ 119887 (119866) ⩽ 119896 119894119891 119866 119894119904 119906119899119889119894119903119890119888119905119890119889

119887 (119866) = 119896 + 1 119894119891 119866 119894119904 119889119894119903119890119888119905119890119889

(83)

Theorem 195 (Huang and Xu [36] 2008) If 119866 is an undi-rected vertex-transitive cubic graph with order 4120574(119866) and girth119892(119866) ⩽ 5 then 119887(119866) = 2

The proof of Theorem 195 leads to a byproduct If 119892 =5 let (119906

1 119906

2 119906

3 119906

4 119906

5) be a cycle and let D

119894be the family

of efficient dominating sets containing 119906119894for each 119894 isin

1 2 3 4 5 ThenD119894capD

119895= 0 for 119894 = 1 2 5 Then 119866 has

at least 5119904 efficient dominating sets where 119904 = 119904(119866) But thereare only 4s (=ns120574) distinct efficient dominating sets in119866This

contradiction implies that an undirected vertex-transitivecubic graph with girth five has no efficient dominating setsBut a similar argument for 119892(119866) = 3 4 or 119892(119866) ⩾ 6 couldnot give any contradiction This is consistent with the resultthat CCC(119899) a vertex-transitive cubic graph with girth 119899 if3 ⩽ 119899 ⩽ 8 or girth 8 if 119899 ⩾ 9 has efficient dominating setsfor all 119899 ⩾ 3 except 119899 = 5 (see Theorem 199 in the followingsubsection)

112 Results for Cayley Graphs In this subsection we useTheorem 194 to determine the exact values or approximativevalues of bondage numbers for some special vertex-transitivegraphs by characterizing the existence of efficient dominatingsets in these graphs

Let Γ be a nontrivial finite group 119878 a nonempty subset ofΓ without the identity element of Γ A digraph 119866 is defined asfollows

119881 (119866) = Γ (119909 119910) isin 119864 (119866) lArrrArr 119909minus1

119910 isin 119878

for any 119909 119910 isin Γ(84)

is called a Cayley digraph of the group Γ with respect to119878 denoted by 119862

Γ(119878) If 119878minus1 = 119904minus1 119904 isin 119878 = 119878 then

119862Γ(119878) is symmetric and is called a Cayley undirected graph

a Cayley graph for short Cayley graphs or digraphs arecertainly vertex-transitive (see Xu [1])

A circulant graph 119866(119899 119878) of order 119899 is a Cayley graph119862119885119899

(119878) where 119885119899= 0 119899 minus 1 is the addition group of

order 119899 and 119878 is a nonempty subset of119885119899without the identity

element and hence is a vertex-transitive digraph of degree|119878| If 119878minus1 = 119878 then119866(119899 119878) is an undirected graph If 119878 = 1 119896where 2 ⩽ 119896 ⩽ 119899 minus 2 we write 119866(119899 1 119896) for 119866(119899 1 119896) or119866(119899 plusmn1 plusmn119896) and call it a double-loop circulant graph

Huang and Xu [36] showed that for directed 119866 =

119866(119899 1 119896) lceil1198993rceil ⩽ 120574(119866) ⩽ lceil1198992rceil and 119866 has an efficientdominating set if and only if 3 | 119899 and 119896 equiv 2 (mod 3) fordirected 119866 = 119866(119899 1 119896) and 119896 = 1198992 lceil1198995rceil ⩽ 120574(119866) ⩽ lceil1198993rceiland 119866 has an efficient dominating set if and only if 5 | 119899 and119896 equiv plusmn2 (mod 5) By Theorem 194 we obtain the bondagenumber of a double loop circulant graph if it has an efficientdominating set

Theorem 196 (Huang and Xu [36] 2008) Let 119866 be a double-loop circulant graph 119866(119899 1 119896) If 119866 is directed with 3 | 119899and 119896 equiv 2 (mod 3) or 119866 is undirected with 5 | 119899 and119896 equiv plusmn2 (mod 5) then 119887(119866) = 3

The 119898 times 119899 torus is the Cartesian product 119862119898times 119862

119899of

two cycles and is a Cayley graph 119862119885119898times119885119899

(119878) where 119878 =

(0 1) (1 0) for directed cycles and 119878 = (0 plusmn1) (plusmn1 0) forundirected cycles and hence is vertex-transitive Gu et al[144] showed that the undirected torus119862

119898times119862

119899has an efficient

dominating set if and only if both 119898 and 119899 are multiplesof 5 Huang and Xu [36] showed that the directed torus119862119898times 119862

119899has an efficient dominating set if and only if both

119898 and 119899 are multiples of 3 Moreover they found a necessarycondition for a dominating set containing the vertex (0 0)in 119862

119898times 119862

119899to be efficient and obtained the following

result

International Journal of Combinatorics 31

Theorem 197 (Huang and Xu [36] 2008) Let 119866 = 119862119898times 119862

119899

If 119866 is undirected and both119898 and 119899 are multiples of 5 or if 119866 isdirected and both119898 and 119899 are multiples of 3 then 119887(119866) = 3

The hypercube 119876119899

is the Cayley graph 119862Γ(119878)

where Γ = 1198852times sdot sdot sdot times 119885

2= (119885

2)119899 and 119878 =

100 sdot sdot sdot 0 010 sdot sdot sdot 0 00 sdot sdot sdot 01 Lee [143] showed that119876119899has an efficient dominating set if and only if 119899 = 2119898 minus 1

for a positive integer119898 ByTheorem 194 the following resultis obtained immediately

Theorem 198 (Huang and Xu [36] 2008) If 119899 = 2119898 minus 1 for apositive integer119898 then 2119898minus1

⩽ 119887(119876119899) ⩽ 2

119898

minus 1

Problem 10 Determine the exact value of 119887(119876119899) for 119899 = 2119898minus1

The 119899-dimensional cube-connected cycle denoted byCCC(119899) is constructed from the 119899-dimensional hypercube119876119899by replacing each vertex 119909 in 119876

119899with an undirected cycle

119862119899of length 119899 and linking the 119894th vertex of the 119862

119899to the 119894th

neighbor of 119909 It has been proved that CCC(119899) is a Cayleygraph and hence is a vertex-transitive graph with degree 3Van Wieren et al [147] proved that CCC(119899) has an efficientdominating set if and only if 119899 = 5 From Theorems 194 and195 the following result can be derived

Theorem 199 (Huang and Xu [36] 2008) Let 119866 = 119862119862119862(119899)be the 119899-dimensional cube-connected cycles with 119899 ⩾ 3 and119899 = 5 Then 120574(119866) = 1198992119899minus2 and 2 ⩽ 119887(119866) ⩽ 3 In addition119887(119862119862119862(3)) = 119887(119862119862119862(4)) = 2

Problem 11 Determine the exact value of 119887(CCC(119899)) for 119899 ⩾5

Acknowledgments

This work was supported by NNSF of China (Grants nos11071233 61272008) The author would like to express hisgratitude to the anonymous referees for their kind sugges-tions and comments on the original paper to ProfessorSheikholeslami and Professor Jafari Rad for sending thereferences [128] and [81] which resulted in this version

References

[1] J-M Xu Theory and Application of Graphs vol 10 of NetworkTheory and Applications Kluwer Academic Dordrecht TheNetherlands 2003

[2] S THedetniemi andR C Laskar ldquoBibliography on dominationin graphs and some basic definitions of domination parame-tersrdquo Discrete Mathematics vol 86 no 1ndash3 pp 257ndash277 1990

[3] TW Haynes S T Hedetniemi and P J Slater Fundamentals ofDomination in Graphs vol 208 ofMonographs and Textbooks inPure and Applied Mathematics Marcel Dekker New York NYUSA 1998

[4] TWHaynes S THedetniemi and P J Slater EdsDominationin Graphs vol 209 of Monographs and Textbooks in Pure andAppliedMathematicsMarcelDekker NewYorkNYUSA 1998

[5] M R Garey andD S JohnsonComputers and Intractability WH Freeman and Company San Francisco Calif USA 1979

[6] H B Walikar and B D Acharya ldquoDomination critical graphsrdquoNational Academy Science Letters vol 2 pp 70ndash72 1979

[7] D Bauer F Harary J Nieminen and C L Suffel ldquoDominationalteration sets in graphsrdquo Discrete Mathematics vol 47 no 2-3pp 153ndash161 1983

[8] J F Fink M S Jacobson L F Kinch and J Roberts ldquoThebondage number of a graphrdquo Discrete Mathematics vol 86 no1ndash3 pp 47ndash57 1990

[9] F-T Hu and J-M Xu ldquoThe bondage number of (119899 minus 3)-regulargraph G of order nrdquo Ars Combinatoria to appear

[10] U Teschner ldquoNew results about the bondage number of agraphrdquoDiscreteMathematics vol 171 no 1ndash3 pp 249ndash259 1997

[11] B L Hartnell and D F Rall ldquoA bound on the size of a graphwith given order and bondage numberrdquo Discrete Mathematicsvol 197-198 pp 409ndash413 1999 16th British CombinatorialConference (London 1997)

[12] B L Hartnell and D F Rall ldquoA characterization of trees inwhich no edge is essential to the domination numberrdquo ArsCombinatoria vol 33 pp 65ndash76 1992

[13] J Topp and P D Vestergaard ldquo120572119896- and 120574

119896-stable graphsrdquo

Discrete Mathematics vol 212 no 1-2 pp 149ndash160 2000[14] A Meir and J W Moon ldquoRelations between packing and

covering numbers of a treerdquo Pacific Journal of Mathematics vol61 no 1 pp 225ndash233 1975

[15] B L Hartnell L K Joslashrgensen P D Vestergaard and CA Whitehead ldquoEdge stability of the k-domination numberof treesrdquo Bulletin of the Institute of Combinatorics and ItsApplications vol 22 pp 31ndash40 1998

[16] F-T Hu and J-M Xu ldquoOn the complexity of the bondage andreinforcement problemsrdquo Journal of Complexity vol 28 no 2pp 192ndash201 2012

[17] B L Hartnell and D F Rall ldquoBounds on the bondage numberof a graphrdquo Discrete Mathematics vol 128 no 1ndash3 pp 173ndash1771994

[18] Y-L Wang ldquoOn the bondage number of a graphrdquo DiscreteMathematics vol 159 no 1ndash3 pp 291ndash294 1996

[19] G H Fricke TW Haynes S M Hedetniemi S T Hedetniemiand R C Laskar ldquoExcellent treesrdquo Bulletin of the Institute ofCombinatorics and Its Applications vol 34 pp 27ndash38 2002

[20] V Samodivkin ldquoThe bondage number of graphs good and badverticesrdquoDiscussiones Mathematicae GraphTheory vol 28 no3 pp 453ndash462 2008

[21] J E Dunbar T W Haynes U Teschner and L VolkmannldquoBondage insensitivity and reinforcementrdquo in Domination inGraphs vol 209 of Monographs and Textbooks in Pure andAppliedMathematics pp 471ndash489 Dekker NewYork NY USA1998

[22] H Liu and L Sun ldquoThe bondage and connectivity of a graphrdquoDiscrete Mathematics vol 263 no 1ndash3 pp 289ndash293 2003

[23] A-H Esfahanian and S L Hakimi ldquoOn computing a con-ditional edge-connectivity of a graphrdquo Information ProcessingLetters vol 27 no 4 pp 195ndash199 1988

[24] R C Brigham P Z Chinn and R D Dutton ldquoVertexdomination-critical graphsrdquoNetworks vol 18 no 3 pp 173ndash1791988

[25] V Samodivkin ldquoDomination with respect to nondegenerateproperties bondage numberrdquoThe Australasian Journal of Com-binatorics vol 45 pp 217ndash226 2009

[26] U Teschner ldquoA new upper bound for the bondage numberof graphs with small domination numberrdquo The AustralasianJournal of Combinatorics vol 12 pp 27ndash35 1995

32 International Journal of Combinatorics

[27] J Fulman D Hanson and G MacGillivray ldquoVertex dom-ination-critical graphsrdquoNetworks vol 25 no 2 pp 41ndash43 1995

[28] U Teschner ldquoA counterexample to a conjecture on the bondagenumber of a graphrdquo Discrete Mathematics vol 122 no 1ndash3 pp393ndash395 1993

[29] U Teschner ldquoThe bondage number of a graph G can be muchgreater than Δ(G)rdquo Ars Combinatoria vol 43 pp 81ndash87 1996

[30] L Volkmann ldquoOn graphs with equal domination and coveringnumbersrdquoDiscrete AppliedMathematics vol 51 no 1-2 pp 211ndash217 1994

[31] L A Sanchis ldquoMaximum number of edges in connected graphswith a given domination numberrdquoDiscreteMathematics vol 87no 1 pp 65ndash72 1991

[32] S Klavzar and N Seifter ldquoDominating Cartesian products ofcyclesrdquoDiscrete AppliedMathematics vol 59 no 2 pp 129ndash1361995

[33] H K Kim ldquoA note on Vizingrsquos conjecture about the bondagenumberrdquo Korean Journal of Mathematics vol 10 no 2 pp 39ndash42 2003

[34] M Y Sohn X Yuan and H S Jeong ldquoThe bondage number of1198623times119862

119899rdquo Journal of the KoreanMathematical Society vol 44 no

6 pp 1213ndash1231 2007[35] L Kang M Y Sohn and H K Kim ldquoBondage number of the

discrete torus 119862119899times 119862

4rdquo Discrete Mathematics vol 303 no 1ndash3

pp 80ndash86 2005[36] J Huang and J-M Xu ldquoThe bondage numbers and efficient

dominations of vertex-transitive graphsrdquoDiscrete Mathematicsvol 308 no 4 pp 571ndash582 2008

[37] C J Xiang X Yuan andM Y Sohn ldquoDomination and bondagenumber of119862

3times119862

119899rdquoArs Combinatoria vol 97 pp 299ndash310 2010

[38] M S Jacobson and L F Kinch ldquoOn the domination numberof products of graphs Irdquo Ars Combinatoria vol 18 pp 33ndash441984

[39] Y-Q Li ldquoA note on the bondage number of a graphrdquo ChineseQuarterly Journal of Mathematics vol 9 no 4 pp 1ndash3 1994

[40] F-T Hu and J-M Xu ldquoBondage number of mesh networksrdquoFrontiers ofMathematics inChina vol 7 no 5 pp 813ndash826 2012

[41] R Frucht and F Harary ldquoOn the corona of two graphsrdquoAequationes Mathematicae vol 4 pp 322ndash325 1970

[42] K Carlson and M Develin ldquoOn the bondage number of planarand directed graphsrdquoDiscreteMathematics vol 306 no 8-9 pp820ndash826 2006

[43] U Teschner ldquoOn the bondage number of block graphsrdquo ArsCombinatoria vol 46 pp 25ndash32 1997

[44] J Huang and J-M Xu ldquoNote on conjectures of bondagenumbers of planar graphsrdquo Applied Mathematical Sciences vol6 no 65ndash68 pp 3277ndash3287 2012

[45] L Kang and J Yuan ldquoBondage number of planar graphsrdquoDiscrete Mathematics vol 222 no 1ndash3 pp 191ndash198 2000

[46] M Fischermann D Rautenbach and L Volkmann ldquoRemarkson the bondage number of planar graphsrdquo Discrete Mathemat-ics vol 260 no 1ndash3 pp 57ndash67 2003

[47] J Hou G Liu and J Wu ldquoBondage number of planar graphswithout small cyclesrdquoUtilitasMathematica vol 84 pp 189ndash1992011

[48] J Huang and J-M Xu ldquoThe bondage numbers of graphs withsmall crossing numbersrdquo Discrete Mathematics vol 307 no 15pp 1881ndash1897 2007

[49] Q Ma S Zhang and J Wang ldquoBondage number of 1-planargraphrdquo Applied Mathematics vol 1 no 2 pp 101ndash103 2010

[50] J Hou and G Liu ldquoA bound on the bondage number of toroidalgraphsrdquoDiscreteMathematics Algorithms and Applications vol4 no 3 Article ID 1250046 5 pages 2012

[51] Y-C Cao J-M Xu and X-R Xu ldquoOn bondage number oftoroidal graphsrdquo Journal of University of Science and Technologyof China vol 39 no 3 pp 225ndash228 2009

[52] Y-C Cao J Huang and J-M Xu ldquoThe bondage number ofgraphs with crossing number less than fourrdquo Ars Combinatoriato appear

[53] H K Kim ldquoOn the bondage numbers of some graphsrdquo KoreanJournal of Mathematics vol 11 no 2 pp 11ndash15 2004

[54] A Gagarin and V Zverovich ldquoUpper bounds for the bondagenumber of graphs on topological surfacesrdquo Discrete Mathemat-ics 2012

[55] J Huang ldquoAn improved upper bound for the bondage numberof graphs on surfacesrdquoDiscrete Mathematics vol 312 no 18 pp2776ndash2781 2012

[56] A Gagarin and V Zverovich ldquoThe bondage number of graphson topological surfaces and Teschnerrsquos conjecturerdquo DiscreteMathematics vol 313 no 6 pp 796ndash808 2013

[57] V Samodivkin ldquoNote on the bondage number of graphs ontopological surfacesrdquo httparxivorgabs12086203

[58] E J Cockayne R M Dawes and S T Hedetniemi ldquoTotaldomination in graphsrdquoNetworks vol 10 no 3 pp 211ndash219 1980

[59] R Laskar J Pfaff S M Hedetniemi and S T HedetniemildquoOn the algorithmic complexity of total dominationrdquo Society forIndustrial and Applied Mathematics vol 5 no 3 pp 420ndash4251984

[60] J Pfaff R C Laskar and S T Hedetniemi ldquoNP-completeness oftotal and connected domination and irredundance for bipartitegraphsrdquo Tech Rep 428 Department of Mathematical SciencesClemson University 1983

[61] M A Henning ldquoA survey of selected recent results on totaldomination in graphsrdquoDiscrete Mathematics vol 309 no 1 pp32ndash63 2009

[62] V R Kulli and D K Patwari ldquoThe total bondage number ofa graphrdquo in Advances in Graph Theory pp 227ndash235 VishwaGulbarga India 1991

[63] F-T Hu Y Lu and J-M Xu ldquoThe total bondage number ofgrid graphsrdquo Discrete Applied Mathematics vol 160 no 16-17pp 2408ndash2418 2012

[64] J Cao M Shi and B Wu ldquoResearch on the total bondagenumber of a spectial networkrdquo in Proceedings of the 3rdInternational Joint Conference on Computational Sciences andOptimization (CSO rsquo10) pp 456ndash458 May 2010

[65] N Sridharan MD Elias and V S A Subramanian ldquoTotalbondage number of a graphrdquo AKCE International Journal ofGraphs and Combinatorics vol 4 no 2 pp 203ndash209 2007

[66] N Jafari Rad and J Raczek ldquoSome progress on total bondage ingraphsrdquo Graphs and Combinatorics 2011

[67] T W Haynes and P J Slater ldquoPaired-domination and thepaired-domatic numberrdquoCongressusNumerantium vol 109 pp65ndash72 1995

[68] T W Haynes and P J Slater ldquoPaired-domination in graphsrdquoNetworks vol 32 no 3 pp 199ndash206 1998

[69] X Hou ldquoA characterization of 2120574 120574119875-treesrdquoDiscrete Mathemat-

ics vol 308 no 15 pp 3420ndash3426 2008[70] K E Proffitt T W Haynes and P J Slater ldquoPaired-domination

in grid graphsrdquo Congressus Numerantium vol 150 pp 161ndash1722001

International Journal of Combinatorics 33

[71] H Qiao L KangM Cardei andD-Z Du ldquoPaired-dominationof treesrdquo Journal of Global Optimization vol 25 no 1 pp 43ndash542003

[72] E Shan L Kang and M A Henning ldquoA characterizationof trees with equal total domination and paired-dominationnumbersrdquo The Australasian Journal of Combinatorics vol 30pp 31ndash39 2004

[73] J Raczek ldquoPaired bondage in treesrdquo Discrete Mathematics vol308 no 23 pp 5570ndash5575 2008

[74] J A Telle and A Proskurowski ldquoAlgorithms for vertex parti-tioning problems on partial k-treesrdquo SIAM Journal on DiscreteMathematics vol 10 no 4 pp 529ndash550 1997

[75] G S Domke J H Hattingh M A Henning and L R MarkusldquoRestrained domination in treesrdquoDiscreteMathematics vol 211no 1ndash3 pp 1ndash9 2000

[76] G S Domke J H Hattingh M A Henning and L R MarkusldquoRestrained domination in graphs with minimum degree twordquoJournal of Combinatorial Mathematics and Combinatorial Com-puting vol 35 pp 239ndash254 2000

[77] G S Domke J H Hattingh S T Hedetniemi R C Laskarand L R Markus ldquoRestrained domination in graphsrdquo DiscreteMathematics vol 203 no 1ndash3 pp 61ndash69 1999

[78] J H Hattingh and E J Joubert ldquoAn upper bound for therestrained domination number of a graph with minimumdegree at least two in terms of order and minimum degreerdquoDiscrete Applied Mathematics vol 157 no 13 pp 2846ndash28582009

[79] M A Henning ldquoGraphs with large restrained dominationnumberrdquo Discrete Mathematics vol 197-198 pp 415ndash429 199916th British Combinatorial Conference (London 1997)

[80] J H Hattingh and A R Plummer ldquoRestrained bondage ingraphsrdquo Discrete Mathematics vol 308 no 23 pp 5446ndash54532008

[81] N Jafari Rad R Hasni J Raczek and L Volkmann ldquoTotalrestrained bondage in graphsrdquoActaMathematica Sinica vol 29no 6 pp 1033ndash1042 2013

[82] J Cyman and J Raczek ldquoOn the total restrained dominationnumber of a graphrdquoThe Australasian Journal of Combinatoricsvol 36 pp 91ndash100 2006

[83] M A Henning ldquoGraphs with large total domination numberrdquoJournal of Graph Theory vol 35 no 1 pp 21ndash45 2000

[84] D-X Ma X-G Chen and L Sun ldquoOn total restraineddomination in graphsrdquoCzechoslovakMathematical Journal vol55 no 1 pp 165ndash173 2005

[85] B Zelinka ldquoRemarks on restrained domination and totalrestrained domination in graphsrdquo Czechoslovak MathematicalJournal vol 55 no 2 pp 393ndash396 2005

[86] W Goddard and M A Henning ldquoIndependent domination ingraphs a survey and recent resultsrdquo Discrete Mathematics vol313 no 7 pp 839ndash854 2013

[87] J H Zhang H L Liu and L Sun ldquoIndependence bondagenumber and reinforcement number of some graphsrdquo Transac-tions of Beijing Institute of Technology vol 23 no 2 pp 140ndash1422003

[88] K D Dharmalingam Studies in Graph Theorymdashequitabledomination and bottleneck domination [PhD thesis] MaduraiKamaraj University Madurai India 2006

[89] GDeepakND Soner andAAlwardi ldquoThe equitable bondagenumber of a graphrdquo Research Journal of Pure Algebra vol 1 no9 pp 209ndash212 2011

[90] E Sampathkumar and H B Walikar ldquoThe connected domina-tion number of a graphrdquo Journal of Mathematical and PhysicalSciences vol 13 no 6 pp 607ndash613 1979

[91] K White M Farber and W Pulleyblank ldquoSteiner trees con-nected domination and strongly chordal graphsrdquoNetworks vol15 no 1 pp 109ndash124 1985

[92] M Cadei M X Cheng X Cheng and D-Z Du ldquoConnecteddomination in Ad Hoc wireless networksrdquo in Proceedings ofthe 6th International Conference on Computer Science andInformatics (CSampI rsquo02) 2002

[93] J F Fink and M S Jacobson ldquon-domination in graphsrdquo inGraph Theory with Applications to Algorithms and ComputerScience Wiley-Interscience Publications pp 283ndash300 WileyNew York NY USA 1985

[94] M Chellali O Favaron A Hansberg and L Volkmann ldquok-domination and k-independence in graphs a surveyrdquo Graphsand Combinatorics vol 28 no 1 pp 1ndash55 2012

[95] Y Lu X Hou J-M Xu andN Li ldquoTrees with uniqueminimump-dominating setsrdquo Utilitas Mathematica vol 86 pp 193ndash2052011

[96] Y Lu and J-M Xu ldquoThe p-bondage number of treesrdquo Graphsand Combinatorics vol 27 no 1 pp 129ndash141 2011

[97] Y Lu and J-M Xu ldquoThe 2-domination and 2-bondage numbersof grid graphs A manuscriptrdquo httparxivorgabs12044514

[98] M Krzywkowski ldquoNon-isolating 2-bondage in graphsrdquo TheJournal of the Mathematical Society of Japan vol 64 no 4 2012

[99] M A Henning O R Oellermann and H C Swart ldquoBoundson distance domination parametersrdquo Journal of CombinatoricsInformation amp System Sciences vol 16 no 1 pp 11ndash18 1991

[100] F Tian and J-M Xu ldquoBounds for distance domination numbersof graphsrdquo Journal of University of Science and Technology ofChina vol 34 no 5 pp 529ndash534 2004

[101] F Tian and J-M Xu ldquoDistance domination-critical graphsrdquoApplied Mathematics Letters vol 21 no 4 pp 416ndash420 2008

[102] F Tian and J-M Xu ldquoA note on distance domination numbersof graphsrdquo The Australasian Journal of Combinatorics vol 43pp 181ndash190 2009

[103] F Tian and J-M Xu ldquoAverage distances and distance domina-tion numbersrdquo Discrete Applied Mathematics vol 157 no 5 pp1113ndash1127 2009

[104] Z-L Liu F Tian and J-M Xu ldquoProbabilistic analysis of upperbounds for 2-connected distance k-dominating sets in graphsrdquoTheoretical Computer Science vol 410 no 38ndash40 pp 3804ndash3813 2009

[105] M A Henning O R Oellermann and H C Swart ldquoDistancedomination critical graphsrdquo Journal of Combinatorial Mathe-matics and Combinatorial Computing vol 44 pp 33ndash45 2003

[106] T J Bean M A Henning and H C Swart ldquoOn the integrityof distance domination in graphsrdquo The Australasian Journal ofCombinatorics vol 10 pp 29ndash43 1994

[107] M Fischermann and L Volkmann ldquoA remark on a conjecturefor the (kp)-domination numberrdquoUtilitasMathematica vol 67pp 223ndash227 2005

[108] T Korneffel D Meierling and L Volkmann ldquoA remark on the(22)-domination numberrdquo Discussiones Mathematicae GraphTheory vol 28 no 2 pp 361ndash366 2008

[109] Y Lu X Hou and J-M Xu ldquoOn the (22)-domination numberof treesrdquo Discussiones Mathematicae Graph Theory vol 30 no2 pp 185ndash199 2010

34 International Journal of Combinatorics

[110] H Li and J-M Xu ldquo(dm)-domination numbers of m-connected graphsrdquo Rapports de Recherche 1130 LRI UniversiteParis-Sud 1997

[111] S M Hedetniemi S T Hedetniemi and T V Wimer ldquoLineartime resource allocation algorithms for treesrdquo Tech Rep URI-014 Department of Mathematical Sciences Clemson Univer-sity 1987

[112] M Farber ldquoDomination independent domination and dualityin strongly chordal graphsrdquo Discrete Applied Mathematics vol7 no 2 pp 115ndash130 1984

[113] G S Domke and R C Laskar ldquoThe bondage and reinforcementnumbers of 120574

119891for some graphsrdquoDiscrete Mathematics vol 167-

168 pp 249ndash259 1997[114] G S Domke S T Hedetniemi and R C Laskar ldquoFractional

packings coverings and irredundance in graphsrdquo vol 66 pp227ndash238 1988

[115] E J Cockayne P A Dreyer Jr S M Hedetniemi andS T Hedetniemi ldquoRoman domination in graphsrdquo DiscreteMathematics vol 278 no 1ndash3 pp 11ndash22 2004

[116] I Stewart ldquoDefend the roman empirerdquo Scientific American vol281 pp 136ndash139 1999

[117] C S ReVelle and K E Rosing ldquoDefendens imperiumromanum a classical problem in military strategyrdquoThe Ameri-can Mathematical Monthly vol 107 no 7 pp 585ndash594 2000

[118] A Bahremandpour F-T Hu S M Sheikholeslami and J-MXu ldquoOn the Roman bondage number of a graphrdquo DiscreteMathematics Algorithms and Applications vol 5 no 1 ArticleID 1350001 15 pages 2013

[119] N Jafari Rad and L Volkmann ldquoRoman bondage in graphsrdquoDiscussiones Mathematicae Graph Theory vol 31 no 4 pp763ndash773 2011

[120] K Ebadi and L PushpaLatha ldquoRoman bondage and Romanreinforcement numbers of a graphrdquo International Journal ofContemporary Mathematical Sciences vol 5 pp 1487ndash14972010

[121] N Dehgardi S M Sheikholeslami and L Volkman ldquoOn theRoman k-bondage number of a graphrdquo AKCE InternationalJournal of Graphs and Combinatorics vol 8 pp 169ndash180 2011

[122] S Akbari and S Qajar ldquoA note on Roman bondage number ofgraphsrdquo Ars Combinatoria to Appear

[123] F-T Hu and J-M Xu ldquoRoman bondage numbers of somegraphsrdquo httparxivorgabs11093933

[124] N Jafari Rad and L Volkmann ldquoOn the Roman bondagenumber of planar graphsrdquo Graphs and Combinatorics vol 27no 4 pp 531ndash538 2011

[125] S Akbari M Khatirinejad and S Qajar ldquoA note on the romanbondage number of planar graphsrdquo Graphs and Combinatorics2012

[126] B Bresar M A Henning and D F Rall ldquoRainbow dominationin graphsrdquo Taiwanese Journal of Mathematics vol 12 no 1 pp213ndash225 2008

[127] B L Hartnell and D F Rall ldquoOn dominating the Cartesianproduct of a graph and 120572krdquo Discussiones Mathematicae GraphTheory vol 24 no 3 pp 389ndash402 2004

[128] N Dehgardi S M Sheikholeslami and L Volkman ldquoThe k-rainbow bondage number of a graphrdquo to appear in DiscreteApplied Mathematics

[129] J von Neumann and O Morgenstern Theory of Games andEconomic Behavior Princeton University Press Princeton NJUSA 1944

[130] C BergeTheorıe des Graphes et ses Applications 1958[131] O Ore Theory of Graphs vol 38 American Mathematical

Society Colloquium Publications 1962[132] E Shan and L Kang ldquoBondage number in oriented graphsrdquoArs

Combinatoria vol 84 pp 319ndash331 2007[133] J Huang and J-M Xu ldquoThe bondage numbers of extended

de Bruijn and Kautz digraphsrdquo Computers amp Mathematics withApplications vol 51 no 6-7 pp 1137ndash1147 2006

[134] J Huang and J-M Xu ldquoThe total domination and total bondagenumbers of extended de Bruijn and Kautz digraphsrdquoComputersamp Mathematics with Applications vol 53 no 8 pp 1206ndash12132007

[135] Y Shibata and Y Gonda ldquoExtension of de Bruijn graph andKautz graphrdquo Computers amp Mathematics with Applications vol30 no 9 pp 51ndash61 1995

[136] X Zhang J Liu and JMeng ldquoThebondage number in completet-partite digraphsrdquo Information Processing Letters vol 109 no17 pp 997ndash1000 2009

[137] D W Bange A E Barkauskas and P J Slater ldquoEfficient dom-inating sets in graphsrdquo in Applications of Discrete Mathematicspp 189ndash199 SIAM Philadelphia Pa USA 1988

[138] M Livingston and Q F Stout ldquoPerfect dominating setsrdquoCongressus Numerantium vol 79 pp 187ndash203 1990

[139] L Clark ldquoPerfect domination in random graphsrdquo Journal ofCombinatorial Mathematics and Combinatorial Computing vol14 pp 173ndash182 1993

[140] D W Bange A E Barkauskas L H Host and L H ClarkldquoEfficient domination of the orientations of a graphrdquo DiscreteMathematics vol 178 no 1ndash3 pp 1ndash14 1998

[141] A E Barkauskas and L H Host ldquoFinding efficient dominatingsets in oriented graphsrdquo Congressus Numerantium vol 98 pp27ndash32 1993

[142] I J Dejter and O Serra ldquoEfficient dominating sets in CayleygraphsrdquoDiscrete AppliedMathematics vol 129 no 2-3 pp 319ndash328 2003

[143] J Lee ldquoIndependent perfect domination sets in Cayley graphsrdquoJournal of Graph Theory vol 37 no 4 pp 213ndash219 2001

[144] WGu X Jia and J Shen ldquoIndependent perfect domination setsin meshes tori and treesrdquo Unpublished

[145] N Obradovic J Peters and G Ruzic ldquoEfficient domination incirculant graphswith two chord lengthsrdquo Information ProcessingLetters vol 102 no 6 pp 253ndash258 2007

[146] K Reji Kumar and G MacGillivray ldquoEfficient domination incirculant graphsrdquo Discrete Mathematics vol 313 no 6 pp 767ndash771 2013

[147] D Van Wieren M Livingston and Q F Stout ldquoPerfectdominating sets on cube-connected cyclesrdquo Congressus Numer-antium vol 97 pp 51ndash70 1993

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Stochastic AnalysisInternational Journal of

International Journal of Combinatorics 3

graph 119870119905(2) with 119905 = 1198992 is an (119899 minus 2)-regular graph 119866 of

order 119899 ⩾ 2 and not unique 119887(119866) = 119899 minus 1 by Theorem 1for an even integer 119899 ⩾ 4 For an (119899 minus 3)-regular graph119866 of order 119899 ⩾ 4 Hu and Xu [9] obtained the followingresult

Theorem 2 (Hu and Xu [9]) 119887(119866) = 119899 minus 3 for any (119899 minus 3)-regular graph 119866 of order 119899 ⩾ 4

Up to the present no results have been known for 119896-regular graphs with 3 ⩽ 119896 ⩽ 119899 minus 4 For general graphs thereare the following two results

Theorem 3 (Teschner [10] 1997) If 119866 is a nonempty graphwith a unique minimum dominating set then 119887(119866) = 1

Theorem4 (Bauer et al [7] 1983) If any vertex of a graph119866 isadjacent with two ormore vertices of degree one then 119887(119866) = 1

Bauer et al [7] observed that the star is the uniquegraph with the property that the bondage number is 1 andthe deletion of any edge results in the domination numberincreasing Motivated by this fact Hartnell and Rall [11]proposed the concept of uniformly bonded graphs A graph iscalled to be uniformly bonded if it has bondage number 119887 andthe deletion of any 119887 edges results in a graph with increaseddomination number Unfortunately there are a few uniformlybonded graphs

Theorem 5 (Hartnell and Rall [11] 1999) The only uniformlybonded graphs with bondage number 2 are 119862

3and 119875

4 The

unique uniformly bonded graph with bondage number 3 is 1198624

There are no such graphs for bondage number greater than 3

As we mentioned to compute the exact value of bondagenumber for a graph strongly depends upon its dominationnumber In this sense studying the bondage number cangreatly inspire onersquos research concerned with dominationsHowever determining the exact value of domination numberfor a given graph is quite difficult In fact even if theexact value of the domination number for some graph isdetermined it is still very difficulty to compute the valueof the bondage number for that graph For example for thehypercube 119876

119899 we have 120574(119876

119899) = 2

119899minus1 but we have not yetdetermined 119887(119876

119899) for any 119899 ⩾ 2

Perhaps Theorems 3 and 4 provide an approach tocompute the exact value of bondage number for some graphsby establishing some sufficient conditions for 119887(119866) = 119887In fact we will see later that Theorem 3 plays an importantrole in determining the exact values of the bondage numbersfor some graphs Thus to study the bondage number it isimportant to present various characterizations of graphs witha unique minimum dominating set

22 Characterizations of Trees For trees Bauer et al [7] in1983 from the point of view of the domination line-stabilityindependently and Fink et al [8] in 1990 from the pointof view of the domination edge-vulnerability obtained thefollowing result

119908

Figure 1 Tree 119865119905

Theorem 6 For any nontrivial tree 119879 119887(119879) ⩽ 2

By Theorem 6 it is natural to classify all trees accordingto their bondage numbers Fink et al [8] proved that aforbidden subgraph characterization to classify trees withdifferent bondage numbers is impossible since they provedthat if 119865 is a forest then 119865 is an induced subgraph of a tree 119879with 119887(119879) = 1 and a tree 1198791015840 with 119887(1198791015840) = 2 However theypointed out that the complexity of calculating the bondagenumber of a tree of order 119899 is at most 119874(1198992) by methodicallyremoving each pair of edges

Even so some characterizations whether a tree hasbondage number 1 or 2 have been found by several authorssee for example [10 12 13]

First we describe the method due to Hartnell and Rall[12] by which all trees with bondage number 2 can be con-structed inductively An important tree119865

119905in the construction

is shown in Figure 1 To characterize this construction weneed some terminologies

(1) attach a path 119875119899to a vertex 119909 of a tree which means

to link 119909 and one end-vertex of the 119875119899by an edge

(2) attach 119865119905to a vertex 119909 which means to link 119909 and a

vertex 119910 of 119865119905by an edge

The following are four operations on a tree 119879

Type 1 attach a 1198752to 119909 isin 119881(119879) where 120574(119879 minus 119909) = 120574(119879) and 119909

belongs to at least one 120574-set of 119879 (such a vertex existssay one end-vertex of 119875

5)

Type 2 attach a 1198753to 119909 isin 119881(119879) where 120574(119879 minus 119909) lt 120574(119879)

Type 3 attach 1198651to 119909 isin 119881(119879) where 119909 belongs to at least one

120574-set of 119879Type 4 attach 119865

119905 119905 ⩾ 2 to 119909 isin 119881(119879) where 119909 can be any

vertex of 119879

LetC = 119879 119879 is a tree and 119879 = 1198701 119879 = 119875

4 and 119879 = 119865

119905

for some 119905 ⩾ 2 or 119879 can be obtained from 1198754or 119865

119905(119905 ⩾ 2) by

a finite sequence of operations of Types 1 2 3 4

Theorem 7 (Hartnell and Rall [12] 1992) A tree has bondagenumber 2 if and only if it belongs toC

Looking at different minimum dominating sets of a treeTeschner [10] presented a totally different characterizationof the set of trees having bondage number 1 They defineda vertex to be universal if it belongs to each minimumdominating set and to be idle if it does not belong to anyminimum dominating set

4 International Journal of Combinatorics

Theorem 8 (Teschner [10] 1997) A tree 119879 has bondagenumber 1 if and only if 119879 has a universal vertex or an edge119909119910 satisfying

(1) 119909 and 119910 are neither universal nor idle and

(2) all neighbors of 119909 and 119910 (except for 119909 and 119910) are idle

For a positive integer 119896 a subset 119868 sube 119881(119866) is called a 119896-independent set (also called a 119896-packing) if 119889

119866(119909 119910) gt 119896 for

any two distinct vertices 119909 and 119910 in 119868 When 119896 = 1 1-set is thenormal independent setThemaximumcardinality among all119896-independent sets is called the 119896-independence number (or119896-packing number) of 119866 denoted by 120572

119896(119866) A 119896-independent

set 119868 is called an 120572119896-set if |119868| = 120572

119896(119866) A graph 119866 is said to be

120572119896-stable if 120572

119896(119866) = 120572

119896(119866 minus 119890) for every edge 119890 of 119866 There are

two important results on 119896-independent sets

Proposition 9 (Topp and Vestergaard [13] 2000) A tree 119879 is120572119896-stable if and only if 119879 has a unique 120572

119896-set

Proposition 10 (Meir and Moon [14] 1975) 1205722(119866) ⩽ 120574(119866)

for any connected graph 119866 with equality for any tree

Hartnell et al [15] independently and Topp and Vester-gaard [13] also gave a constructive characterization of treeswith bondage number 2 and applying Proposition 10 pre-sented another characterization of those trees

Theorem 11 (Hartnell et al [15] 1998 Top and Vestergaard[13] 2000) 119887(119879) = 2 for a tree 119879 if and only if 119879 has a unique1205722-set

According to this characterization Hartnell et al [15]presented a linear algorithm for determining the bondagenumber of a tree

In this subsection we introduce three characterizationsfor trees with bondage number 1 or 2 The characterizationin Theorem 7 is constructive constructing all trees withbondage number 2 a natural and straightforward methodby a series of graph-operations The characterization inTheorem 8 is a little advisable by describing the inherentproperty of trees with bondage number 1 The characteriza-tion in Theorem 11 is wonderful by using a strong graph-theoretic concept 120572

119896-set In fact this characterization is a

byproduct of some results related to 120572119896-sets for trees It is

that this characterization closed the relation between twoconcepts the bondage number and the 119896-independent setand hence is of research value and important significance

23 Complexity for General Graphs Asmentioned above thebondage number of a tree can be determined by a linear timealgorithm According to this algorithm we can determinewithin polynomial time the domination number of any treeby removing each edge and verifyingwhether the dominationnumber is enlarged according to the known linear timealgorithm for domination numbers of trees

However it is impossible to find a polynomial timealgorithm for bondage numbers of general graphs If suchan algorithm 119860 exists then the domination number of any

nonempty undirected graph 119866 can be determined withinpolynomial time by repeatedly using 119860 Let 119866

0= 119866 and

119866119894+1= 119866

119894minus119861

119894 where 119861

119894is the minimum edge set of 119866

119894found

by 119860 such that 120574(119866119894minus 119861

119894) = 120574(119866

119894minus1) + 1 for each 119894 = 0 1

we can always find the minimum 119861119894whose removal from 119866

119894

enlarges the domination number until 119866119896= 119866

119896minus1minus 119861

119896minus1

is empty for some 119896 ⩾ 1 though 119861119896minus1

is not empty Then120574(119866) = 120574(119866

119896) minus 119896 = 120592(119866) minus 119896 As known to all if NP = 119875

the minimum dominating set problem is NP-complete andso polynomial time algorithms for the bondage number donot exist unless NP = 119875

In fact Hu and Xu [16] have recently shown that theproblem determining the bondage number of general graphsis NP-hard

Problem 1 Consider the decision problemBondage ProblemInstance a graph 119866 and a positive integer 119887 (⩽ 120576(119866))Question is 119887(119866) ⩽ 119887

Theorem 12 (Hu and Xu [16] 2012) The bondage problem isNP-hard

Thebasicway of the proof is to followGarey and Johnsonrsquostechniques for proving NP-hardness [5] by describing apolynomial transformation from the known NP-completeproblem 3-satisfiability problem

Theorem 12 shows that we are unable to find a polynomialtime algorithm to determine bondage numbers of generalgraphs unless NP = 119875 At the same time this result alsoshows that the following study on the bondage number is ofimportant significance

(i) Find approximation polynomial algorithms with per-formance ratio as small as possible

(ii) Find the lower and upper bounds with difference assmall as possible

(iii) Determine exact values for some graphs speciallywell-known networks

Unfortunately we cannot provewhether or not determin-ing the bondage number is NP-problem since for any subset119861 sub 119864(119866) it is not clear that there is a polynomial algorithmto verify 120574(119866 minus 119861) gt 120574(119866) Since the problem of determiningthe domination number is NP-complete we conjecture thatit is not in NPThis is a worthwhile task to be studied further

Motivated by the linear time algorithm of Hartnell et alto compute the bondage number of a tree we can makean attempt to consider whether there is a polynomial timealgorithm to compute the bondage number for some specialclasses of graphs such as planar graphs Cayley graphs orgraphs with some restrictions of graph-theoretical parame-ters such as degree diameter connectivity and dominationnumber

3 Upper Bounds

ByTheorem 12 since we cannot find a polynomial time algo-rithm for determining the exact values of bondage numbers

International Journal of Combinatorics 5

of general graphs it is weightily significative to establishsome sharp bounds on the bondage number of a graph Inthis section we survey several known upper bounds on thebondage number in terms of some other graph-theoreticalparameters

31 Most Basic Upper Bounds We start this subsection witha simple observation

Observation 1 (Teschner [10] 1997) Let 119867 be a spanningsubgraph obtained from a graph119866 by removing 119896 edgesThen119887(119866) ⩽ 119887(119867) + 119896

If we select a spanning subgraph 119867 such that 119887(119867) = 1thenObservation 1 yields some upper bounds on the bondagenumber of a graph 119866 For example take119867 = 119866 minus 119864

119909+ 119890 and

119896 = 119889119866(119909) minus 1 where 119890 isin 119864

119909 the following result can be

obtained

Theorem 13 (Bauer et al [7] 1983) If there exists at least onevertex 119909 in a graph 119866 such that 120574(119866 minus 119909) ⩾ 120574(119866) then 119887(119866) ⩽119889119866(119909) ⩽ Δ(119866)

The following early result obtained can be derived fromObservation 1 by taking119867 = 119866minus119864

119909119910and 119896 = 119889(119909)+119889(119910)minus2

Theorem 14 (Bauer et al [7] 1983 Fink et al [8] 1990)119887(119866) ⩽ 119889(119909) + 119889(119910) minus 1 for any two adjacent vertices 119909 and119910 in a graph 119866 that is

119887 (119866) ⩽ min119909119910isin119864(119866)

119889119866(119909) + 119889

119866(119910) minus 1 (4)

This theorem gives a natural corollary obtained by severalauthors

Corollary 15 (Bauer et al [7] 1983 Fink et al [8] 1990) If 119866is a graphwithout isolated vertices then 119887(119866) ⩽ Δ(119866)+120575(119866)minus1

In 1999 Hartnell and Rall [11] extended Theorem 14 tothe following more general case which can be also derivedfrom Observation 1 by taking119867 = 119866 minus 119864

119909minus 119864

119910+ (119909 119911 119910) if

119889119866(119909 119910) = 2 where (119909 119911 119910) is a path of length 2 in 119866

Theorem 16 (Hartnell and Rall [11] 1999) 119887(119866) ⩽ 119889(119909) +119889(119910)minus1 for any distinct two vertices 119909 and 119910 in a graph119866with119889119866(119909 119910) ⩽ 2 that is

119887 (119866) ⩽ min119889119866(119909119910)⩽2

119889119866(119909) + 119889

119866(119910) minus 1 (5)

Corollary 17 (Fink et al [8] 1990) If a vertex of a graph 119866 isadjacent with two ormore vertices of degree one then 119887(119866) = 1

We remark that the bounds stated in Corollary 15 andTheorem 16 are sharp As indicated byTheorem 1 one class ofgraphs in which the bondage number achieves these boundsis the class of cycles whose orders are congruent to 1 modulo3

On the other hand Hartnell and Rall [17] sharpened theupper bound in Theorem 14 as follows which can be alsoderived from Observation 1

1198792(119909 119910)

1198794(119909 119910)

1198793(119909 119910)

1198791(119909 119910)

119909

119910

Figure 2 Illustration of 119879119894(119909 119910) for 119894 = 1 2 3 4

Theorem 18 (Hartnell and Rall [17] 1994) 119887(119866) ⩽ 119889119866(119909) +

119889119866(119910) minus 1 minus |119873

119866(119909) cap 119873

119866(119910)| for any two adjacent vertices 119909

and 119910 in a graph 119866 that is

119887 (119866) ⩽ min119909119910isin119864(119866)

119889119866(119909) + 119889

119866(119910) minus 1 minus

1003816100381610038161003816119873119866(119909) cap 119873

119866(119910)1003816100381610038161003816

(6)

These results give simple but important upper bounds onthe bondage number of a graph and is also the foundationof almost all results on bondage numbers upper boundsobtained till now

By careful consideration of the nature of the edges fromthe neighbors of119909 and119910Wang [18] further refined the boundin Theorem 18 For any edge 119909119910 isin 119864(119866) 119873

119866(119910) contains the

following four subsets

(1) 1198791(119909 119910) = 119873

119866[119909] cap 119873

119866(119910)

(2) 1198792(119909 119910) = 119908 isin 119873

119866(119910) 119873

119866(119908) sube 119873

119866(119910) minus 119909

(3) 1198793(119909 119910) = 119908 isin 119873

119866(119910) 119873

119866(119908) sube 119873

119866(119911)minus119909 for some

119911 isin 119873119866(119909) cap 119873

119866(119910)

(4) 1198794(119909 119910) = 119873

119866(119910) (119879

1(119909 119910) cup 119879

2(119909 119910) cup 119879

3(119909 119910))

The illustrations of 1198791(119909 119910) 119879

2(119909 119910) 119879

3(119909 119910) and

1198794(119909 119910) are shown in Figure 2 (corresponding vertices

pointed by dashed arrows)

Theorem 19 (Wang [18] 1996) For any nonempty graph 119866

119887 (119866) ⩽ min119909119910isin119864(119866)

119889119866(119909) +

10038161003816100381610038161198794 (119909 119910)1003816100381610038161003816 (7)

The graph in Figure 2 shows that the upper bound givenin Theorem 19 is better than that in Theorems 16 and 18for the upper bounds obtained from these two theorems are119889119866(119909)+119889

119866(119910)minus1 = 11 and 119889

119866(119909)+119889

119866(119910)minus|119873

119866(119909)cap119873

119866(119910)| =

9 respectively while the upper bound given byTheorem 19 is119889119866(119909) + |119879

4(119909 119910)| = 6

The following result is also an improvement ofTheorem 14 in which 119905 = 2

6 International Journal of Combinatorics

Theorem 20 (Teschner [10] 1997) If 119866 contains a completesubgraph 119870

119905with 119905 ⩾ 2 then 119887(119866) ⩽ min

119909119910isin119864(119870119905)119889

119866(119909) +

119889119866(119910) minus 119905 + 1

Following Fricke et al [19] a vertex 119909 of a graph 119866 is 120574-good if 119909 belongs to some 120574-set of119866 and 120574-bad if 119909 belongs tono 120574-set of 119866 Let 119860(119866) be the set of 120574-good vertices and let119861(119866) be the set of 120574-bad vertices in 119866 Clearly 119860(119866) 119861(119866)is a partition of119881(119866) Note there exists some 119909 isin 119860 such that120574(119866 minus 119909) = 120574(119866) say one end-vertex of 119875

5 Samodivkin [20]

presented some sharp upper bounds for 119887(119866) in terms of 120574-good and 120574-bad vertices of 119866

Theorem 21 (Samodivkin [20] 2008) Let 119866 be a graph

(a) Let 119862(119866) = 119909 isin 119881(119866) 120574(119866minus119909) ⩾ 120574(119866) If 119862(119866) = 0then

119887 (119866) ⩽ min 119889119866(119909) + 120574 (119866) minus 120574 (119866 minus 119909) 119909 isin 119862 (119866)

(8)

(b) If 119861 = 0 then

119887 (119866) ⩽ min 1003816100381610038161003816119873119866(119909) cap 119860

1003816100381610038161003816 119909 isin 119861 (119866) (9)

Theorem 22 (Samodivkin [20] 2008) Let 119866 be a graph If119860+

(119866) = 119909 isin 119860(119866) 119889119866(119909) ⩾ 1 and 120574(119866 minus 119909) lt 120574(119866) = 0

then

119887 (119866) ⩽ min119909isin119860+(119866)119910isin119861(119866minus119909)

119889119866(119909) +

1003816100381610038161003816119873119866(119910) cap 119860 (119866 minus 119909)

1003816100381610038161003816

(10)

Proposition 23 (Samodivkin [20] 2008) Under the notationof Theorem 19 if 119909 isin 119860+

(119866) then (1198791(119909 119910) minus 119909) cup 119879

2(119909 119910) cup

1198793(119909 119910) sube 119873

119866(119910) 119861(119866 minus 119909)

By Proposition 23 if 119909 isin 119860+

(119866) then

119889119866(119909) + min

119910isin119873119866(119909)

10038161003816100381610038161198794 (119909 119910)

1003816100381610038161003816

⩾ 119889119866(119909) + min

119910isin119873119866(119909)

1003816100381610038161003816119873119866

(119910) cap 119860 (119866 minus 119909)1003816100381610038161003816

⩾ 119889119866(119909) + min

119910isin119861(119866minus119909)

1003816100381610038161003816119873119866

(119910) cap 119860 (119866 minus 119909)1003816100381610038161003816

(11)

Hence Theorem 19 can be seen to follow from Theorem 22Any graph 119866 with 119887(119866) achieving the upper bound inTheorem 19 can be used to show that the bound inTheorem 22 is sharp

Let 119905 ⩾ 2 be an integer Samodivkin [20] con-structed a very interesting graph 119866

119905to show that the upper

bound in Theorem 22 is better than the known bounds Let119867

0 119867

1 119867

2 119867

119905+1be mutually vertex-disjoint graphs such

that 1198670cong 119870

2 119867

119905+1cong 119870

119905+3and 119867

119894cong 119870

119905+3minus 119890 for each

119894 = 1 2 119905 Let 119881(1198670) = 119909 119910 119909

119905+1isin 119881(119867

119905+1) and

1198671

1198672

1198673

11990911

11990912

11990921 11990922

1199093

119909

119910

Figure 3 The graph 1198662

1199091198941 119909

1198942isin 119881(119867

119894) 119909

11989411199091198942notin 119864(119867

119894) for each 119894 = 1 2 119905 The

graph 119866119905is defined as follows

119881 (119866119905) =

119905+1

⋃

119896=0

119881 (119867119896)

119864 (119866119905)=(

119905+1

⋃

119896=0

119864 (119867119896))cup(

119905

⋃

119894=1

1199091199091198941 119909119909

1198942)cup119909119909

119905+1

(12)

Such a constructed graph119866119905is shown in Figure 3 when 119905 = 2

Observe that 120574(119866119905) = 119905 + 2 119860(119866

119905) = 119881(119866

119905) 119889

119866119905

(119909) =

2119905 + 2 119889119866119905

(119909119905+1) = 119905 + 3 119889

119866119905

(119910) = 1 and 119889119866119905

(119911) = 119905 + 2

for each 119911 isin 119881(119866119905minus 119909 119910 119909

119905+1) Moreover 120574(119866 minus 119910) lt 120574(119866)

and 120574(119866119905minus119911) = 120574(119866

119905) for any 119911 isin 119881(119866

119905) minus 119910 Hence each of

the bounds stated inTheorems 13ndash20 is greater than or equals119905 + 2

Consider the graph 119866119905minus 119909119910 Clearly 120574(119866

119905minus 119909119910) = 120574(119866

119905)

and

119861 (119866119905minus 119909119910) = 119861 (119866

119905minus 119910)

= 119909 cup 119881 (119867119905+1minus 119909

119905+1) cup (

119905

⋃

119896=1

1199091198961

1199091198962

)

(13)

Therefore 119873119866119905

(119909) cap 119866(119866119905minus 119910) = 119909

119905+1 which implies that

the upper bound stated in Theorem 22 is equal to 119889119866119905

(119910) +

|119909119905+1| = 2 Clearly 119887(119866

119905) = 2 and hence this bound is sharp

for 119866119905

From the graph 119866119905 we obtain the following statement

immediately

Proposition 24 For every integer 119905 ⩾ 2 there is a graph 119866such that the difference between any upper bound stated inTheorems 13ndash20 and the upper bound in Theorem 22 is equalto 119905

Although Theorem 22 supplies us with the upper boundthat is closer to 119887(119866) for some graph 119866 than what any one of

International Journal of Combinatorics 7

Theorems 13ndash20 provides it is not easy to determine the sets119860+

(119866) and 119861(119866) mentioned in Theorem 22 for an arbitrarygraph 119866 Thus the upper bound given in Theorem 22 is oftheoretical importance but not applied since until now wehave not found a new class of graphswhose bondage numbersare determined byTheorem 22

The above-mentioned upper bounds on the bondagenumber are involved in only degrees of two vertices Hartnelland Rall [11] established an upper bound on 119887(119866) in terms ofthe numbers of vertices and edges of 119866 with order 119899 For anyconnected graph 119866 let 120575(119866) represent the average degree ofvertices in 119866 that is 120575(119866) = (1119899)sum

119909isin119881119889119866(119909) Hartnell and

Rall first discovered the following proposition

Proposition 25 For any connected graph 119866 there exist twovertices 119909 and 119910 with distance at most two and with theproperty that 119889

119866(119909) + 119889

119866(119910) ⩽ 2120575(119866)

Using Proposition 25 and Theorem 16 Hartnell and Rallgave the following bound

Theorem 26 (Hartnell and Rall [11] 1999) 119887(119866) ⩽ 2120575(119866) minus 1for any connected graph 119866

Note that 2120576 = 119899120575(119866) for any graph 119866 with order 119899 andsize 120576 Theorem 26 implies the following bound in terms oforder 119899 and size 120576

Corollary 27 119887(119866) ⩽ (4120576119899) minus 1 for any connected graph 119866with order 119899 and size 120576

A lower bound on 120576 in terms of order 119899 and the bondagenumber is obtained from Corollary 28 immediately

Corollary 28 120576 ⩾ (1198994)(119887(119866) + 1) for any connected graph 119866with order 119899 and size 120576

Hartnell and Rall [11] gave some graphs 119866 with order 119899to show that for each value of 119887(119866) the lower bound on 120576(119866)given in the Corollary 28 is sharp for some values of 119899

32 Bounds Implied by Connectivity Use 120581(119866) and 120582(119866) todenote the vertex-connectivity and the edge-connectivity ofa connected graph 119866 respectively which are the minimumnumbers of vertices and edges whose removal result in 119866disconnected The famous Whitney inequality is stated as120581(119866) ⩽ 120582(119866) ⩽ 120575(119866) for any graph or digraph 119866 Corollary 15is improved by several authors as follows

Theorem 29 (Hartnell and Rall [17] 1994 Teschner [10]1997) If 119866 is a connected graph then 119887(119866) ⩽ 120582(119866)+Δ(119866)minus 1

The upper bound given in Theorem 29 can be attainedFor example a cycle 119862

3119896+1with 119896 ⩾ 1 119887(119862

3119896+1) = 3 by

Theorem 1 Since 120581(1198623119896+1) = 120582(119862

3119896+1) = 2 we have 120581(119862

3119896+1)+

120582(1198623119896+1) minus 1 = 2 + 2 minus 1 = 3 = 119887(119862

3119896+1)

Motivated byCorollary 15Theorems 29 and theWhitneyinequality Dunbar et al [21] naturally proposed the followingconjecture

Figure 4 A graph119867 with 120574(119867) = 3 and 119887(119867) = 5

Conjecture 30 If119866 is a connected graph then 119887(119866) ⩽ Δ(119866)+120581(119866) minus 1

However Liu and Sun [22] presented a counterexampleto this conjecture They first constructed a graph 119867 showedin Figure 4 with 120574(119867) = 3 and 119887(119867) = 5 Then let 119866 be thedisjoint union of two copies of119867 by identifying two verticesof degree two They proved that 119887(119866) ⩾ 5 Clearly 119866 is a 4-regular graph with 120581(119866) = 1 and 120582(119866) = 2 and so 119887(119866) ⩽ 5byTheorem 29 Thus 119887(119866) = 5 gt 4 = Δ(119866) + 120581(119866) minus 1

With a suspicion of the relationship between the bondagenumber and the vertex-connectivity of a graph the followingconjecture is proposed

Conjecture 31 (Liu and Sun [22] 2003) For any positiveinteger 119903 there exists a connected graph 119866 such that 119887(119866) ⩾Δ(119866) + 120581(119866) + 119903

To the knowledge of the author until now no results havebeen known about this conjecture

We conclude this subsection with the following remarksFromTheorem 29 if Conjecture 31 holds for some connectedgraph119866 then 120582(119866) gt 120581(119866)+119903 which implies that119866 is of largeedge-connectivity and small vertex-connectivity Use 120585(119866) todenote the minimum edge-degree of 119866 that is

120585 (119866) = min119909119910isin119864(119866)

119889119866(119909) + 119889

119866(119910) minus 2 (14)

Theorem 14 implies the following result

Proposition 32 119887(119866) ⩽ 120585(119866) + 1 for any graph 119866

Use 1205821015840(119866) to denote the restricted edge-connectivity ofa connected graph 119866 which is the minimum number ofedgeswhose removal result in119866 disconnected and no isolatedvertices

Proposition 33 (Esfahanian and Hakimi [23] 1988) If 119866 isneither 119870

1119899nor 119870

3 then 1205821015840(119866) ⩽ 120585(119866)

Combining Propositions 32 and 33 we propose a conjec-ture

Conjecture 34 If a connected 119866 is neither 1198701119899

nor 1198703 then

119887(119866) ⩽ 120575(119866) + 1205821015840

(119866) minus 1

8 International Journal of Combinatorics

A cycle 1198623119896+1

also satisfies 119887(1198623119896+1) = 3 = 120575(119862

3119896+1) +

1205821015840

(1198623119896+1) minus 1 since 1205821015840(119862

3119896+1) = 120575(119862

3119896+1) = 2 for any integer

119896 ⩾ 1For the graph119867 shown in Figure 41205821015840(119867) = 4 and 120575(119867) =

2 and so 119887(119867) = 5 = 120575(119867) + 1205821015840(119867) minus 1For the 4-regular graph119866 constructed by Liu and Sun [22]

obtained from the disjoint union of two copies of the graph119867 showed in Figure 4 by identifying two vertices of degreetwo we have 119887(119866) ⩾ 5 Clearly 1205821015840(119866) = 2 Thus 119887(119866) = 5 =120575(119866) + 120582

1015840

(119866) minus 1For the 4-regular graph 119866

119905constructed by Samodivkin

[20] see Figure 3 for 119905 = 2 119887(119866119905) = 2 Clearly 120575(119866

119905) = 1

and 1205821015840(119866) = 2 Thus 119887(119866) = 2 = 120575(119866) + 1205821015840(119866) minus 1These examples show that if Conjecture 34 is true then the

given upper bound is tight

33 Bounds Implied by Degree Sequence Now let us return toTheorem 16 from which Teschner [10] obtained some otherbounds in terms of the degree sequencesThe degree sequence120587(119866) of a graph 119866 with vertex-set 119881 = 119909

1 119909

2 119909

119899 is the

sequence 120587 = (1198891 119889

2 119889

119899) with 119889

1⩽ 119889

2⩽ sdot sdot sdot ⩽ 119889

119899 where

119889119894= 119889

119866(119909

119894) for each 119894 = 1 2 119899 The following result is

essentially a corollary of Theorem 16

Theorem35 (Teschner [10] 1997) Let119866 be a nonempty graphwith degree sequence120587(119866) If120572

2(119866) = 119905 then 119887(119866) ⩽ 119889

119905+119889

119905+1minus

1

CombiningTheorem 35with Proposition 10 (ie 1205722(119866) ⩽

120574(119866)) we have the following corollary

Corollary 36 (Teschner [10] 1997) Let 119866 be a nonemptygraph with the degree sequence 120587(119866) If 120574(119866) = 120574 then 119887(119866) ⩽119889120574+ 119889

120574+1minus 1

In [10] Teschner showed that these two bounds are sharpfor arbitrarily many graphs Let119867 = 119862

3119896+1+ 119909

11199094 119909

11199093119896minus1

where 1198623119896+1

is a cycle (1199091 119909

2 119909

3119896+1 119909

1) for any integer

119896 ⩾ 2 Then 120574(119867) = 119896 + 1 and so 119887(119867) ⩽ 2 + 2 minus 1 = 3by Corollary 36 Since119862

3119896+1is a spanning subgraph of119867 and

120574(119866) = 120574(119867) Observation 1 yields that 119887(119867) ⩾ 119887(1198623119896+1) = 3

and so 119887(119867) = 3Although various upper bounds have been established

as the above we find that the appearance of these boundsis essentially based upon the local structures of a graphprecisely speaking the structures of the neighborhoods oftwo vertices within distance 2 Even if these bounds can beachieved by some special graphs it is more often not the caseThe reason lies essentially in the definition of the bondagenumber which is theminimumvalue among all bondage setsan integral property of a graph While it easy to find upperbounds just by choosing some bondage set the gap betweenthe exact value of the bondage number and such a boundobtained only from local structures of a graph is often largeFor example a star 119870

1Δ however large Δ is 119887(119870

1Δ) = 1

Therefore one has been longing for better bounds upon someintegral parameters However as what we will see below it isdifficult to establish such upper bounds

34 Bounds in 120574-Critical Graphs A graph119866 is called a vertexdomination-critical graph (vc-graph or 120574-critical for short) if120574(119866 minus 119909) lt 120574(119866) for any 119909 isin 119881(119866) proposed by Brighamet al [24] in 1988 Several families of graphs are known to be120574-critical From definition it is clear that if 119866 is a 120574-criticalgraph then 120574(119866) ⩾ 2The class of 120574-critical graphs with 120574 = 2is characterized as follows

Proposition 37 (Brigham et al [24] 1988) A graph 119866 with120574(119866) = 2 is a 120574-critical graph if and only if 119866 is a completegraph 119870

2119905(119905 ⩾ 2) with a perfect matching removed

A more interesting family is composed of the 119899-criticalgraphs 119866

119898119899defined for 119898 119899 ⩾ 2 by the circulant undirected

graph 119866(119873 plusmn119878) where 119873 = (119898 + 1)(119899 minus 1) + 1 and119878 = 1 2 lfloor1198982rfloor in which 119881(119866(119873 plusmn119878)) = 119873 and119864(119866(119873 plusmn119878)) = 119894119895 |119895 minus 119894| equiv 119904 (mod 119873) 119904 isin 119878

The reason why 120574-critical graphs are of special interest inthis context is easy to see that they play an important role instudy of the bondage number For instance it immediatelyfollows from Theorem 13 that if 119887(119866) gt Δ(119866) then 119866 is a 120574-critical graphThe 120574-critical graphs are defined exactly in thisway In order to find graphs 119866 with a high bondage number(ie higher than Δ(119866)) and beyond its general upper boundsfor the bondage number we therefore have to look at 120574-critical graphs

The bondage numbers of some 120574-critical graphs havebeen examined by several authors see for example [1020 25 26] From Theorem 13 we know that the bondagenumber of a graph 119866 is bounded from above by Δ(119866)if 119866 is not a 120574-critical graph For 120574-critical graphs it ismore difficult to find an upper bound We will see that thebondage numbers of 120574-critical graphs in general are noteven bounded from above by Δ + 119888 for any fixed naturalnumber 119888

In this subsection we introduce some upper bounds forthe bondage number of a 120574-critical graph By Proposition 37we easily obtain the following result

Theorem 38 If 119866 is a 120574-critical graph with 120574(119866) = 2 then119887(119866) ⩽ Δ + 1

In Section 4 by Theorem 59 we will see that the equalityin Theorem 38 holds that is 119887(119866) = Δ + 1 if 119866 is a 120574-criticalgraph with 120574(119866) = 2

Theorem 39 (Teschner [10] 1997) Let119866 be a 120574-critical graphwith degree sequence 120587(119866) Then 119887(119866) ⩽ maxΔ(119866) + 1 119889

1+

1198892+ sdot sdot sdot + 119889

120574minus1 where 120574 = 120574(119866)

As we mentioned above if 119866 is a 120574-critical graph with120574(119866) = 2 then 119887(119866) = Δ + 1 which shows that thebound given in Theorem 39 can be attained for 120574 = 2However we have not known whether this bound is tightfor general 120574 ⩾ 3 Theorem 39 gives the following corollaryimmediately

Corollary 40 (Teschner [10] 1997) Let 119866 be a 120574-criticalgraph with degree sequence 120587(119866) If 120574(119866) = 3 then 119887(119866) ⩽maxΔ(119866) + 1 120575(119866) + 119889

2

International Journal of Combinatorics 9

From Theorem 38 and Corollary 40 we have 119887(119866) ⩽2Δ(119866) if 119866 is a 120574-critical graph with 120574(119866) ⩽ 3 The followingresult shows that this bound is not tight

Theorem 41 (Teschner [26] 1995) Let119866 be a 120574-critical graphgraph with 120574(119866) ⩽ 3 Then 119887(119866) ⩽ (32)Δ(119866)

Until now we have not known whether the bound givenin Theorem 41 is tight or not We state two conjectures on120574-critical graphs proposed by Samodivkin [20] The first ofthem is motivated byTheorems 22 and 19

Conjecture 42 (Samodivkin [20] 2008) For every connectednontrivial 120574-critical graph 119866

min119909isin119860+(119866)119910isin119861(119866minus119909)

119889119866(119909) +

1003816100381610038161003816119873119866(119910) cap 119860 (119866 minus 119909)

1003816100381610038161003816 ⩽3

2Δ (119866)

(15)

To state the second conjecture we need the followingresult on 120574-critical graphs

Proposition 43 If119866 is a 120574-critical graph then 120592(119866) ⩽ (Δ(119866)+1)(120574(119866)minus1)+1 moreover if the equality holds then119866 is regular

The upper bound in Proposition 43 is sharp in thesense that equality holds for the infinite class of 120574-criticalgraphs 119866

119898119899defined in the beginning of this subsection In

Proposition 43 the first result is due to Brigham et al [24] in1988 the second is due to Fulman et al [27] in 1995

For a 120574-critical graph 119866 with 120574 = 3 by Proposition 43120592(119866) ⩽ 2Δ(119866) + 3

Theorem 44 (Teschner [26] 1995) If 119866 is a 120574-critical graphwith 120574 = 3 and 120592(119866) = 2Δ(119866) + 3 then 119887(119866) ⩽ Δ + 1

We have not known yet whether the equality inTheorem 44holds or notHowever Samodivkin proposed thefollowing conjecture

Conjecture 45 (Samodivkin [20] 2008) If 119866 is a 120574-criticalgraph with (Δ(119866)+1)(120574minus1)+1 vertices then 119887(119866) = Δ(119866)+1

In general based on Theorem 41 Teschner proposed thefollowing conjecture

Conjecture 46 (Teschner [26] 1995) If119866 is a 120574-critical graphthen 119887(119866) ⩽ (32)Δ(119866) for 120574 ⩾ 4

We conclude this subsection with some remarks Graphswhich are minimal or critical with respect to a given prop-erty or parameter frequently play an important role in theinvestigation of that property or parameter Not only are suchgraphs of considerable interest in their own right but also aknowledge of their structure often aids in the developmentof the general theory In particular when investigating anyfinite structure a great number of results are proven byinduction Consequently it is desirable to learn as much aspossible about those graphs that are critical with respect toa given property or parameter so as to aid and abet suchinvestigations

In this subsection we survey some results on the bondagenumber for 120574-critical graphs Although these results are notvery perfect they provides a feasible method to approach thebondage number from different viewpoints In particular themethods given in Teschner [26] worth further explorationand development

The following proposition is maybe useful for us tofurther investigate the bondage number of a 120574-critical graph

Proposition 47 (Brigham et al [24] 1988) If 119866 has anonisolated vertex 119909 such that the subgraph induced by119873

119866(119909)

is a complete graph then 119866 is not 120574-critical

This simple fact shows that any 120574-critical graph containsno vertices of degree one

35 Bounds Implied by Domination In the preceding sub-section we introduced some upper bounds on the bondagenumbers for 120574-critical graphs by consideration of domina-tions In this subsection we introduce related results for ageneral graph with given domination number

Theorem 48 (Fink et al [8] 1990) For any connected graph119866 of order 119899 (⩾2)

(a) 119887(119866) ⩽ 119899 minus 1

(b) 119887(119866) ⩽ Δ(119866) + 1 if 120574(119866) = 2

(c) 119887(119866) ⩽ 119899 minus 120574(119866) + 1

While the upper bound of 119899minus1 on 119887(119866) is not particularlygood for many classes of graphs (eg trees and cycles) it isan attainable bound For example if 119866 is a complete 119905-partitegraph 119870

119905(2) for 119905 = 1198992 and 119899 ⩽ 4 an even integer then the

three bounds on 119887(119866) in Theorem 48 are sharpTeschner [26] consider a graphwith 120574(119866) = 1 or 120574(119866) = 2

The next result is almost trivial but useful

Lemma 49 Let 119866 be a graph with order 119899 and 120574(119866) = 1 119905 thenumber of vertices of degree 119899 minus 1 Then 119887(119866) = lceil1199052rceil

Since 119905 ⩽ Δ(119866) clearly Lemma 49 yields the followingresult immediately

Theorem 50 (Teschner [26] 1995) 119887(119866) ⩽ (12)Δ+1 for anygraph 119866 with 120574(119866) = 1

For a complete graph1198702119896+1

119887(1198702119896+1) = 119896+1 = (12)Δ+1

which shows that the upper bound given in Theorem 50 canbe attained For a graph119866with 120574(119866) = 2 byTheorem 59 laterthe upper bound given inTheorem 48 (b) can be also attainedby a 2-critical graph (see Proposition 37)

36 Two Conjectures In 1990 when Fink et al [8] introducedthe concept of the bondage number they proposed thefollowing conjecture

Conjecture 51 119887(119866) ⩽ Δ(119866) + 1 for any nonempty graph 119866

10 International Journal of Combinatorics

Although some results partially support Conjecture 51Teschner [28] in 1993 found a counterexample to this con-jecture the cartesian product 119870

3times 119870

3 which shows that

119887(119866) = Δ(119866) + 2If a graph 119866 is a counterexample to Conjecture 51 it

must be a 120574-critical graph by Theorem 13 That is why thevertex domination-critical graphs are of special interest in theliterature

Now we return to Conjecture 51 Hartnell and Rall [17]and Teschner [29] independently proved that 119887(119866) can bemuch greater than Δ(119866) by showing the following result

Theorem 52 (Hartnell and Rall [17] 1994 Teschner [29]1996) For an integer 119899 ⩾ 3 let 119866

119899be the cartesian product

119870119899times 119870

119899 Then 119887(119866

119899) = 3(119899 minus 1) = (32)Δ(119866

119899)

This theorem shows that there exist no upper boundsof the form 119887(119866) ⩽ Δ(119866) + 119888 for any integer 119888 Teschner[26] proved that 119887(119866) ⩽ (32)Δ(119866) for any graph 119866 with120574(119866) ⩽ 2 (see Theorems 50 and 48) and for 120574-critical graphswith 120574(119866) = 3 and proposed the following conjecture

Conjecture 53 (Teschner [26] 1995) 119887(119866) ⩽ (32)Δ(119866) forany graph 119866

We believe that this conjecture is true but so far no muchwork about it has been known

4 Lower Bounds

Since the bondage number is defined as the smallest numberof edges whose removal results in an increase of dominationnumber each constructive method that creates a concretebondage set leads to an upper bound on the bondage numberFor that reason it is hard to find lower bounds Neverthelessthere are still a few lower bounds

41 Bounds Implied by Subgraphs The first lower boundon the bondage number is gotten in terms of its spanningsubgraph

Theorem 54 (Teschner [10] 1997) Let 119867 be a spanningsubgraph of a nonempty graph119866 If 120574(119867) = 120574(119866) then 119887(119867) ⩽119887(119866)

By Theorem 1 119887(119862119899) = 3 if 119899 equiv 1 (mod 3) and 119887(119862

119899) =

2 otherwise 119887(119875119899) = 2 if 119899 equiv 1 (mod 3) and 119887(119875

119899) = 1

otherwise From these results and Theorem 54 we get thefollowing two corollaries

Corollary 55 If119866 is hamiltonianwith order 119899 ⩾ 3 and 120574(119866) =lceil1198993rceil then 119887(119866) ⩾ 2 and in addition 119887(119866) ⩾ 3 if 119899 equiv 1 (mod3)

Corollary 56 If 119866 with order 119899 ⩾ 2 has a hamiltonian pathand 120574(119866) = lceil1198993rceil then 119887(119866) ⩾ 2 if 119899 equiv 1 (mod 3)

42 Other Bounds The vertex covering number 120573(119866) of 119866 isthe minimum number of vertices that are incident with all

Figure 5 A 120574-critical graph with 120573 = 120574 = 4 120575 = 2 and 119887 = 3

edges in 119866 If 119866 has no isolated vertices then 120574(119866) ⩽ 120573(119866)clearly In [30] Volkmann gave a lot of graphs with 120573 = 120574

Theorem 57 (Teschner [10] 1997) Let119866 be a graph If 120574(119866) =120573(119866) then

(a) 119887(119866) ⩾ 120575(119866)

(b) 119887(119866) ⩾ 120575(119866) + 1 if 119866 is a 120574-critical graph

The graph shown in Figure 5 shows that the bound givenin Theorem 57(b) is sharp For the graph 119866 it is a 120574-criticalgraph and 120574(119866) = 120573(119866) = 4 By Theorem 57 we have 119887(119866) ⩾3 On the other hand 119887(119866) ⩽ 3 by Theorem 16 Thus 119887(119866) =3

Proposition 58 (Sanchis [31] 1991) Let 119866 be a graph 119866 oforder 119899 (⩾ 6) and size 120576 If 119866 has no isolated vertices and3 ⩽ 120574(119866) ⩽ 1198992 then 120576 ⩽ ( 119899minus120574(119866)+1

2)

Using the idea in the proof of Theorem 57 every upperbound for 120574(119866) can lead to a lower bound for 119887(119866) Inthis way Teschner [10] obtained another lower bound fromProposition 58

Theorem59 (Teschner [10] 1997) Let119866 be a graph119866 of order119899 (⩾6) and size 120576 If 2 ⩽ 120574(119866) ⩽ 1198992 minus 1 then

(a) 119887(119866) ⩾ min120575(119866) 120576 minus ( 119899minus120574(119866)2)

(b) 119887(119866) ⩾ min120575(119866) + 1 120576 minus ( 119899minus120574(119866)2) if 119866 is a 120574-critical

graph

The lower bound in Theorem 59(b) is sharp for a classof 120574-critical graphs with domination number 2 Let 119866 be thegraph obtained from complete graph119870

2119905(119905 ⩾ 2) by removing

a perfect matching By Proposition 37 119866 is a 120574-critical graphwith 120574(119866) = 2 Then 119887(119866) ⩾ 120575(119866) + 1 = Δ(119866) + 1 byTheorem 59 and 119887(119866) ⩽ Δ(119866) + 1 by Theorem 38 and so119887(119866) = Δ(119866) + 1 = 2119905 minus 1

As far as the author knows there are no more lowerbounds in the present literature In view of applications ofthe bondage number a network is vulnerable if its bondagenumber is small while it is stable if its bondage number islargeTherefore better lower bounds let us learn better aboutthe stability of networks from this point of view Thus it isof great significance to seek more lower bounds for variousclasses of graphs

International Journal of Combinatorics 11

5 Results on Graph Operations

Generally speaking it is quite difficult to determine the exactvalue of the bondage number for a given graph since itstrongly depends on the dominating number of the graphThus determining bondage numbers for some special graphsis interesting if the dominating numbers of those graphs areknown or can be easily determined In this section we willintroduce some known results on bondage numbers for somespecial classes of graphs obtained by some graph operations

51 Cartesian Product Graphs Let 1198661= (119881

1 119864

1) and 119866

2=

(1198812 119864

2) be two graphs The Cartesian product of 119866

1and 119866

2

is an undirected graph denoted by 1198661times 119866

2 where 119881(119866

1times

1198662) = 119881

1times 119881

2 two distinct vertices 119909

11199092and 119910

11199102 where

1199091 119910

1isin 119881(119866

1) and 119909

2 119910

2isin 119881(119866

2) are linked by an edge in

1198661times 119866

2if and only if either 119909

1= 119910

1and 119909

21199102isin 119864(119866

2) or

1199092= 119910

2and 119909

11199101isin 119864(119866

1) The Cartesian product is a very

effective method for constructing a larger graph from severalspecified small graphs

Theorem 60 (Dunbar et al [21] 1998) For 119899 ⩾ 3

119887 (119862119899times 119870

2) =

2 119894119891 119899 equiv 0 119900119903 1 (mod 4) 3 119894119891 119899 equiv 3 (mod 4) 4 119894119891 119899 equiv 2 (mod 4)

(16)

For the Cartesian product 119862119899times 119862

119898of two cycles 119862

119899and

119862119898 where 119899 ⩾ 4 and 119898 ⩾ 3 Klavzar and Seifter [32]

determined 120574(119862119899times 119862

119898) for some 119899 and 119898 For example

120574(119862119899times119862

4) = 119899 for 119899 ⩾ 3 and 120574(119862

119899times119862

3) = 119899minuslfloor1198994rfloor for 119899 ⩾ 4

Kim [33] determined 119887(1198623times 119862

3) = 6 and 119887(119862

4times 119862

4) = 4 For

a general 119899 ⩾ 4 the exact values of the bondage numbers of119862119899times 119862

3and 119862

119899times 119862

4are determined as follows

Theorem 61 (Sohn et al [34] 2007) For 119899 ⩾ 4

119887 (119862119899times 119862

3) =

2 119894119891 119899 equiv 0 (mod 4) 4 119894119891 119899 equiv 1 119900119903 2 (mod 4) 5 119894119891 119899 equiv 3 (mod 4)

(17)

Theorem62 (Kang et al [35] 2005) 119887(119862119899times119862

4) = 4 for 119899 ⩾ 4

For larger 119898 and 119899 Huang and Xu [36] obtained thefollowing result see Theorem 197 for more details

Theorem 63 (Huang and Xu [36] 2008) 119887(1198625119894times119862

5119895) = 3 for

any positive integers 119894 and 119895

Xiang et al [37] determined that for 119899 ⩾ 5

119887 (119862119899times 119862

5)

= 3 if 119899 equiv 0 or 1 (mod 5) = 4 if 119899 equiv 2 or 4 (mod 5) ⩽ 7 if 119899 equiv 3 (mod 5)

(18)

For the Cartesian product 119875119899times119875

119898of two paths 119875

119899and 119875

119898

Jacobson and Kinch [38] determined 120574(119875119899times119875

2) = lceil(119899+1)2rceil

120574(119875119899times 119875

3) = 119899 minus lfloor(119899 minus 1)4rfloor and 120574(119875

119899times 119875

4) = 119899 + 1 if 119899 =

1 2 3 5 6 9 and 119899 otherwiseThe bondage number of119875119899times119875

119898

for 119899 ⩾ 4 and 2 ⩽ 119898 ⩽ 4 is determined as follows (Li [39] alsodetermined 119887(119875

119899times 119875

2))

Theorem 64 (Hu and Xu [40] 2012) For 119899 ⩾ 4

119887 (119875119899times 119875

2) =

1 119894119891 119899 equiv 1 (mod 2)2 119894119891 119899 equiv 0 (mod 2)

119887 (119875119899times 119875

3) =

1 119894119891 119899 equiv 1 119900119903 2 (mod 4)2 119894119891 119899 equiv 0 119900119903 3 (mod 4)

119887 (119875119899times 119875

4) = 1 119891119900119903 119899 ⩾ 14

(19)

From the proof of Theorem 64 we find that if 119875119899=

0 1 119899 minus 1 and 119875119898= 0 1 119898 minus 1 then the removal

of the vertex (0 0) in 119875119899times119875

119898does not change the domination

number If119898 increases the effect of (0 0) for the dominationnumber will be smaller and smaller from the probabilityTherefore we expect it is possible that 120574(119875

119899times 119875

119898minus (0 0)) =

120574(119875119899times 119875

119898) for119898 ⩾ 5 and give the following conjecture

Conjecture 65 119887(119875119899times 119875

119898) ⩽ 2 for119898 ⩾ 5 and 119899 ⩾ 4

52 Block Graphs and Cactus Graphs In this subsection weintroduce some results for corona graphs block graphs andcactus graphs

The corona1198661∘ 119866

2 proposed by Frucht and Harary [41]

is the graph formed from a copy of 1198661and 120592(119866

1) copies of

1198662by joining the 119894th vertex of 119866

1to the 119894th copy of 119866

2 In

particular we are concernedwith the corona119867∘1198701 the graph

formed by adding a new vertex V119894 and a new edge 119906

119894V119894for

every vertex 119906119894in119867

Theorem 66 (Carlson and Develin [42] 2006) 119887(119867 ∘ 1198701) =

120575(119867) + 1

This is a very important result which is often used toconstruct a graph with required bondage number In otherwords this result implies that for any given positive integer 119887there is a graph 119866 such that 119887(119866) = 119887

A block graph is a graph whose blocks are completegraphs Each block in a cactus graph is either a cycle or a119870

2

If each block of a graph is either a complete graph or a cyclethen we call this graph a block-cactus graph

Theorem67 (Teschner [43] 1997) 119887(119866) le Δ(119866) for any blockgraph 119866

Teschner [43] characterized all block graphs with 119887(119866) =Δ(119866) In the same paper Teschner found that 120574-criticalgraphs are instrumental in determining bounds for thebondage number of cactus and block graphs and obtained thefollowing result

Theorem 68 (Teschner [43] 1997) 119887(119866) ⩽ 3 for anynontrivial cactus graph 119866

This bound can be achieved by 119867 ∘ 1198701where 119867 is

a nontrivial cactus graph without vertices of degree one

12 International Journal of Combinatorics

by Theorem 66 In 1998 Dunbar et al [21] proposed thefollowing problem

Problem 2 Characterize all cactus graphs with bondagenumber 3

Some upper bounds for block-cactus graphs were alsoobtained

Theorem 69 (Dunbar et al [21] 1998) Let 119866 be a connectedblock-cactus graph with at least two blocksThen 119887(119866) ⩽ Δ(119866)

Theorem 70 (Dunbar et al [21] 1998) Let 119866 be a connectedblock-cactus graph which is neither a cactus graph nor a blockgraph Then 119887(119866) ⩽ Δ(119866) minus 1

53 Edge-Deletion and Edge-Contraction In order to obtainfurther results of the bondage number we may consider howthe bondage number changes under some operations of agraph 119866 which remain the domination number unchangedor preserve some property of 119866 such as planarity Twosimple operations satisfying these requirements are the edge-deletion and the edge-contraction

Theorem 71 (Huang and Xu [44] 2012) Let 119866 be any graphand 119890 isin 119864(119866) Then 119887(119866 minus 119890) ⩾ 119887(119866) minus 1 Moreover 119887(119866 minus 119890) ⩽119887(119866) if 120574(119866 minus 119890) = 120574(119866)

FromTheorem 1 119887(119862119899minus 119890) = 119887(119875

119899) = 119887(119862

119899) minus 1 for any 119890

in119862119899 which shows that the lower bound on 119887(119866minus119890) is sharp

The upper bound can reach for any graph 119866 with 119887(119866) ⩾ 2Next we consider the effect of the edge-contraction on

the bondage number Given a graph 119866 the contraction of 119866by the edge 119890 = 119909119910 denoted by 119866119909119910 is the graph obtainedfrom 119866 by contracting two vertices 119909 and 119910 to a new vertexand then deleting all multiedges It is easy to observe that forany graph 119866 120574(119866) minus 1 ⩽ 120574(119866119909119910) ⩽ 120574(119866) for any edge 119909119910 of119866

Theorem 72 (Huang and Xu [44] 2012) Let 119866 be any graphand 119909119910 be any edge in119866 If119873

119866(119909)cap119873

119866(119910) = 0 and 120574(119866119909119910) =

120574(119866) then 119887(119866119909119910) ⩾ 119887(119866)

We can use examples 119862119899and 119870

119899to show that the

conditions ofTheorem 72 are necessary and the lower boundinTheorem 72 is tight Clearly 120574(119862

119899) = lceil1198993rceil and 120574(119870

119899) = 1

for any edge 119909119910 119862119899119909119910 = 119862

119899minus1and 119870

119899119909119910 = 119870

119899minus1 By

Theorem 1 if 119899 equiv 1 (mod 3) then 120574(119862119899119909119910) lt 120574(119866) and

119887(119862119899119909119910) = 2 lt 3 = 119887(119862

119899) Thus the result in Theorem 72

is generally invalid without the hypothesis 120574(119866119909119910) = 120574(119866)Furthermore the condition 119873

119866(119909) cap 119873

119866(119910) = 0 cannot be

omitted even if 120574(119866119909119910) = 120574(119866) since for odd 119899 119887(119870119899119909119910) =

lceil(119899 minus 1)2rceil lt lceil1198992rceil = 119887(119870119899) by Theorem 1

On the other hand 119887(119862119899119909119910) = 119887(119862

119899) = 2 if 119899 equiv 0 (mod

3) and 119887(119870119899119909119910) = 119887(119870

119899) if 119899 is even which shows that the

equality in 119887(119866119909119910) ⩾ 119887(119866) may hold Thus the bound inTheorem 72 is tight However provided all the conditions119887(119866119909119910) can be arbitrarily larger than 119887(119866) Given a graph119867let119866 be the graph formed from119867∘119870

1by adding a new vertex

119909 and joining it to an vertex 119910 of degree one in119867 ∘ 1198701 Then

119866119909119910 = 119867 ∘ 1198701 120574(119866) = 120574(119866119909119910) and 119873

119866(119909) cap 119873

119866(119910) = 0

But 119887(119866) = 1 since 120574(119866 minus 119909119910) = 120574(119866119909119910) + 1 and 119887(119866119909119910) =120575(119867) + 1 byTheorem 66The gap between 119887(119866) and 119887(119866119909119910)is 120575(119867)

6 Results on Planar Graphs

From Section 2 we have seen that the bondage numberfor a tree has been completely solved Moreover a lineartime algorithm to compute the bondage number of a tree isdesigned by Hartnell et al [15] It is quite natural to considerthe bondage number for a planar graph In this section westate some results and problems on the bondage number fora planar graph

61 Conjecture on Bondage Number As mentioned inSection 3 the bondage number can be much larger than themaximum degree But for a planar graph 119866 the bondagenumber 119887(119866) cannot exceed Δ(119866) too much It is clear that119887(119866) ⩽ Δ(119866) + 4 by Corollary 15 since 120575(119866) ⩽ 5 for anyplanar graph119866 In 1998Dunbar et al [21] posed the followingconjecture

Conjecture 73 If 119866 is a planar graph then 119887(119866) ⩽ Δ(119866) + 1

Because of its attraction it immediately became the focusof attention when this conjecture is proposed In fact themain aim concerning the research of the bondage number fora planar graph is focused on this conjecture

It has been mentioned in Theorems 1 and 60 that119887(119862

3119896+1) = 3 = Δ + 1 and 119887(119862

4119896+2times 119870

2) = 4 = Δ + 1 It

is known that 119887(1198706minus 119872) = 5 = Δ + 1 where119872 is a perfect

matching of the complete graph1198706These examples show that

if Conjecture 73 is true then the upper bound is sharp for2 ⩽ Δ ⩽ 4

Here we note that it is sufficient to prove this conjecturefor connected planar graphs since the bondage number of adisconnected graph is simply the minimum of the bondagenumbers of its components

The first paper attacking this conjecture is due to Kangand Yuan [45] which confirmed the conjecture for everyconnected planar graph 119866 with Δ ⩾ 7 The proofs are mainlybased onTheorems 16 and 18

Theorem 74 (Kang and Yuan [45] 2000) If 119866 is a connectedplanar graph then 119887(119866) ⩽ min8 Δ(119866) + 2

Obviously in view of Theorem 74 Conjecture 73 is truefor any connected planar graph with Δ ⩾ 7

62 Bounds Implied by Degree Conditions As we have seenfrom Theorem 74 to attack Conjecture 73 we only need toconsider connected planar graphs with maximum Δ ⩽ 6Thus studying the bondage number of planar graphs bydegree conditions is of significanceThe first result on boundsimplied by degree conditions is obtained by Kang and Yuan[45]

International Journal of Combinatorics 13

Theorem 75 (Kang and Yuan [45] 2000) If 119866 is a connectedplanar graph without vertices of degree 5 then 119887(119866) ⩽ 7

Fischermann et al [46] generalized Theorem 75 as fol-lows

Theorem 76 (Fischermann et al [46] 2003) Let 119866 be aconnected planar graph and 119883 the set of vertices of degree 5which have distance at least 3 to vertices of degrees 1 2 and3 If all vertices in 119883 not adjacent with vertices of degree 4are independent and not adjacent to vertices of degree 6 then119887(119866) ⩽ 7

Clearly if119866 has no vertices of degree 5 then119883 = 0 ThusTheorem 76 yields Theorem 75

Use 119899119894to denote the number of vertices of degree 119894 in 119866

for each 119894 = 1 2 Δ(119866) Using Theorem 18 Fischermannet al obtained the following two theorems

Theorem 77 (Fischermann et al [46] 2003) For any con-nected planar graph 119866

(1) 119887(119866) ⩽ 7 if 1198995lt 2119899

2+ 3119899

3+ 2119899

4+ 12

(2) 119887(119866) ⩽ 6 if 119866 contains no vertices of degrees 4 and 5

Theorem 78 (Fischermann et al [46] 2003) For any con-nected planar graph 119866 119887(119866) ⩽ Δ(119866) + 1 if

(1) Δ(119866) = 6 and every edge 119890 = 119909119910 with 119889119866(119909) = 5 and

119889119866(119910) = 6 is contained in at most one triangle

(2) Δ(119866) = 5 and no triangle contains an edge 119890 = 119909119910 with119889119866(119909) = 5 and 4 ⩽ 119889

119866(119910) ⩽ 5

63 Bounds Implied by Girth Conditions The girth 119892(119866) of agraph 119866 is the length of the shortest cycle in 119866 If 119866 has nocycles we define 119892(119866) = infin

Combining Theorem 74 with Theorem 78 we find thatif a planar graph contains no triangles and has maximumdegree Δ ⩾ 5 then Conjecture 73 holds This fact motivatedFischermann et alrsquos [46] attempt to attack Conjecture 73 bygirth restraints

Theorem 79 (Fischermann et al [46] 2003) For any con-nected planar graph 119866

119887 (119866) ⩽

6 119894119891 119892 (119866) ⩾ 4

5 119894119891 119892 (119866) ⩾ 5

4 119894119891 119892 (119866) ⩾ 6

3 119894119891 119892 (119866) ⩾ 8

(20)

The first result in Theorem 79 shows that Conjecture 73is valid for all connected planar graphs with 119892(119866) ⩾ 4 andΔ(119866) ⩾ 5 It is easy to verify that the second result inTheorem 79 implies that Conjecture 73 is valid for all not3-regular graphs of girth 119892(119866) ⩾ 5 which is stated in thefollowing corollary

Corollary 80 For any connected planar graph 119866 if 119866 is not3-regular and 119892(119866) ⩾ 5 then 119887(119866) ⩽ Δ(119866) + 1

The first result in Theorem 79 also implies that if 119866 isa connected planar graph with no cycles of length 3 then119887(119866) ⩽ 6 Hou et al [47] improved this result as follows

Theorem 81 (Hou et al [47] 2011) For any connected planargraph 119866 if 119866 contains no cycles of length 119894 (4 ⩽ 119894 ⩽ 6) then119887(119866) ⩽ 6

Since 119887(1198623119896+1) = 3 for 119896 ⩾ 3 (see Theorem 1) the

last bound in Theorem 79 is tight Whether other bounds inTheorem 79 are tight remains open In 2003 Fischermann etal [46] posed the following conjecture

Conjecture 82 For any connected planar graph 119866 119887(119866) ⩽ 7and

119887 (119866) ⩽ 5 119894119891 119892 (119866) ⩾ 4

4 119894119891 119892 (119866) ⩾ 5(21)

We conclude this subsection with a question on bondagenumbers of planar graphs

Question 1 (Fischermann et al [46] 2003) Is there a planargraph 119866 with 6 ⩽ 119887(119866) ⩽ 8

In 2006 Carlson andDevelin [42] showed that the corona119866 = 119867 ∘ 119870

1for a planar graph 119867 with 120575(119867) = 5 has the

bondage number 119887(119866) = 120575(119867) + 1 = 6 (see Theorem 66)Since the minimum degree of planar graphs is at most 5then 119887(119866) can attach 6 If we take 119867 as the graph of theicosahedron then 119866 = 119867 ∘ 119870

1is such an example The

question for the existence of planar graphs with bondagenumber 7 or 8 remains open

64 Comments on the Conjectures Conjecture 73 is true forall connected planar graphs with minimum degree 120575 ⩽ 2 byTheorem 16 or maximum degree Δ ⩾ 7 by Theorem 74 ornot 120574-critical planar graphs by Theorem 13 Thus to attackConjecture 73 we only need to consider connected criticalplanar graphs with degree-restriction 3 ⩽ 120575 ⩽ Δ ⩽ 6

Recalling and anatomizing the proofs of all results men-tioned in the preceding subsections on the bondage numberfor connected planar graphs we find that the proofs of theseresults strongly depend upon Theorem 16 or Theorem 18 Inother words a basic way used in the proofs is to find twovertices 119909 and 119910 with distance at most two in a consideredplanar graph 119866 such that

119889119866(119909) + 119889

119866(119910) or 119889

119866(119909) + 119889

119866(119910) minus

1003816100381610038161003816119873119866(119909) cap 119873

119866(119910)1003816100381610038161003816

(22)

which bounds 119887(119866) is as small as possible Very recentlyHuang and Xu [44] have considered the following parameter

119861 (119866) = min119909119910isin119881(119866)

119889119909119910 1 ⩽ 119889

119866(119909 119910) ⩽ 2

cup 119889119909119910minus 119899

119909119910 119889 (119909 119910) = 1

(23)

where 119889119909119910= 119889

119866(119909) + 119889

119866(119910) minus 1 and 119899

119909119910= |119873

119866(119909) cap 119873

119866(119910)|

14 International Journal of Combinatorics

It follows fromTheorems 16 and 18 that

119887 (119866) ⩽ 119861 (119866) (24)

The proofs given in Theorems 74 and 79 indeed imply thefollowing stronger results

Theorem 83 If 119866 is a connected planar graph then

119861 (119866) ⩽

min 8 Δ (119866) + 26 119894119891 119892 (119866) ⩾ 4

5 119894119891 119892 (119866) ⩾ 5

4 119894119891 119892 (119866) ⩾ 6

3 119894119891 119892 (119866) ⩾ 8

(25)

Thus using Theorem 16 or Theorem 18 if we can proveConjectures 73 and 82 then we can prove the followingstatement

Statement If 119866 is a connected planar graph then

119861 (119866) ⩽

Δ (119866) + 1

7

5 if 119892 (119866) ⩾ 44 if 119892 (119866) ⩾ 5

(26)

It follows from (24) that Statement implies Conjectures 73and 82 However Huang and Xu [44] gave examples to showthat none of conclusions in Statement is true As a result theystated the following conclusion

Theorem 84 (Huang and Xu [44] 2012) It is not possible toprove Conjectures 73 and 82 if they are right using Theorems16 and 18

Therefore a new method is needed to prove or disprovethese conjectures At the present time one cannot prove theseconjectures and may consider to disprove them If one ofthese conjectures is invalid then there exists a minimumcounterexample 119866 with respect to 120592(119866) + 120576(119866) In order toobtain a minimum counterexample one may consider twosimple operations satisfying these requirements the edge-deletion and the edge-contractionwhich decrease 120592(119866)+120576(119866)and preserve planarity However applying Theorems 71 and72 Huang and Xu [44] presented the following results

Theorem 85 (Huang and Xu [44] 2012) It is not possible toconstruct minimum counterexamples to Conjectures 73 and 82by the operation of an edge-deletion or an edge-contraction

7 Results on Crossing Number Restraints

It is quite natural to generalize the known results on thebondage number for planar graphs to formore general graphsin terms of graph-theoretical parameters In this section weconsider graphs with crossing number restraints

The crossing number cr(119866) of 119866 is the smallest numberof pairwise intersections of its edges when 119866 is drawn in theplane If cr(119866) = 0 then 119866 is a planar graph

71 General Methods Use 119899119894to denote the number of vertices

of degree 119894 in119866 for 119894 = 1 2 Δ(119866) Huang and Xu obtainedthe following results

Theorem 86 (Huang and Xu [48] 2007) For any connectedgraph 119866

119887 (119866) ⩽

6 119894119891 119892 (119866) ⩾ 4 119886119899119889 2cr (119866) lt 121198861

5 119894119891 119892 (119866) ⩾ 5 119886119899119889 6cr (119866) lt 161198862

4 119894119891 119892 (119866) ⩾ 6 119886119899119889 4cr (119866) lt 141198863

3 119894119891 119892 (119866) ⩾ 8 119886119899119889 6cr (119866) lt 161198864

(27)

where 1198861= 119899

1+ 2119899

2+ 2119899

3+sum

Δ

119894=8(119894 minus 7)119899

119894+ 8 119886

2= 3119899

1+ 6119899

2+

51198993+sum

Δ

119894=7(3119894minus18)119899

119894+20 119886

3= 119899

1+2119899

2+sum

Δ

119894=6(2119894minus10)119899

119894+12

and 1198864= sum

Δ

119894=5(3119894 minus 12)119899

119894+ 16

Simplifying the conditions in Theorem 86 we obtain thefollowing corollaries immediately

Corollary 87 (Huang and Xu [48] 2007) For any connectedgraph 119866

119887 (119866) ⩽

6 119894119891 119892 (119866) ⩾ 4 119886119899119889 cr (119866) ⩽ 3

5 119894119891 119892 (119866) ⩾ 5 119886119899119889 cr (119866) ⩽ 4

4 119894119891 119892 (119866) ⩾ 6 119886119899119889 cr (119866) ⩽ 2

3 119894119891 119892 (119866) ⩾ 8 119886119899119889 cr (119866) ⩽ 2

(28)

Corollary 88 (Huang and Xu [48] 2007) For any connectedgraph 119866

(a) 119887(119866) ⩽ 6 if 119866 is not 4-regular cr(119866) = 4 and 119892(119866) ⩾ 4

(b) 119887(119866) ⩽ Δ(119866) + 1 if 119866 is not 3-regular cr(119866) ⩽ 4 and119892(119866) ⩾ 5

(c) 119887(119866) ⩽ 4 if 119866 is not 3-regular cr(119866) = 3 and 119892(119866) ⩾ 6

(d) 119887(119866) ⩽ 3 if cr(119866) = 3 119892(119866) ⩾ 8 and Δ(119866) ⩾ 5

These corollaries generalize some known results for pla-nar graphs For example Corollary 87 contains Theorem 79Corollary 88 (b) contains Corollary 80

Theorem 89 (Huang and Xu [48] 2007) For any connectedgraph 119866 if cr(119866) ⩽ 119899

3(119866) + 119899

4(119866) + 3 then 119887(119866) ⩽ 8

Corollary 90 (Huang and Xu [48] 2007) For any connectedgraph 119866 if cr(119866) ⩽ 3 then 119887(119866) ⩽ 8

Perhaps being unaware of this result in 2010 Ma et al[49] proved that 119887(119866) ⩽ 12 for any graph 119866 with cr(119866) = 1

Theorem91 (Huang and Xu [48] 2007) Let119866 be a connectedgraph and 119868 = 119909 isin 119881(119866) 119889

119866(119909) = 5 119889

119866(119909 119910) ⩾ 3 if

International Journal of Combinatorics 15

119889119866(119910) ⩽ 3 and 119889

119866(119910) = 4 for every 119910 isin 119873

119866(119909) If 119868 is

independent no vertices adjacent to vertices of degree 6 and

cr (119866) lt max5119899

3+ |119868| minus 2119899

4+ 28

117119899

3+ 40

16 (29)

then 119887(119866) ⩽ 7

Corollary 92 (Huang and Xu [48] 2007) Let 119866 be aconnected graph with cr(119866) ⩽ 2 If 119868 = 119909 isin 119881(119866) 119889

119866(119909) =

5 119889119866(119910 119909) ⩾ 3 if 119889

119866(119910) ⩽ 3 and 119889

119866(119910) = 4 for every 119910 isin

119873119866(119909) is independent and no vertices adjacent to vertices of

degree 6 then 119887(119866) ⩽ 7

Theorem93 (Huang andXu [48] 2007) Let119866 be a connectedgraph If 119866 satisfies

(a) 5cr(119866) + 1198995lt 2119899

2+ 3119899

3+ 2119899

4+ 12 or

(b) 7cr(119866) + 21198995lt 3119899

2+ 4119899

4+ 24

then 119887(119866) ⩽ 7

Proposition 94 (Huang and Xu [48] 2007) Let 119866 be aconnected graph with no vertices of degrees four and five Ifcr(119866) ⩽ 2 then 119887(119866) ⩽ 6

Theorem 95 (Huang and Xu [48] 2007) If 119866 is a connectedgraph with cr(119866) ⩽ 4 and not 4-regular when cr(119866) = 4 then119887(119866) ⩽ Δ(119866) + 2

The above results generalize some known results forplanar graphs For example Corollary 90 and Theorem 95contain Theorem 74 the first condition in Theorem 93 con-tains the second condition inTheorem 77

From Corollary 88 and Theorem 95 we suggest the fol-lowing questions

Question 2 Is there a

(a) 4-regular graph 119866 with cr(119866) = 4 such that 119887(119866) ⩾ 7

(b) 3-regular graph 119866 with cr ⩽ 4 and 119892(119866) ⩾ 5 such that119887(119866) = 5

(c) 3-regular graph 119866 with cr = 3 and 119892(119866) = 6 or 7 suchthat 119887(119866) = 5

72 Carlson-Develin Methods In this subsection we intro-duce an elegant method presented by Carlson and Develin[42] to obtain some upper bounds for the bondage numberof a graph

Suppose that 119866 is a connected graph We say that 119866 hasgenus 120588 if 119866 can be embedded in a surface 119878 with 120588 handlessuch that edges are pairwise disjoint except possibly for end-vertices Let119866 be an embedding of119866 in surface 119878 and let120601(119866)denote the number of regions in 119866 The boundary of everyregion contains at least three edges and every edge is on theboundary of atmost two regions (the two regions are identicalwhen 119890 is a cut-edge) For any edge 119890 of 119866 let 1199031

119866(119890) and 1199032

119866(119890)

be the numbers of edges comprising the regions in 119866 whichthe edge 119890 borders It is clear that every 119890 = 119909119910 isin 119864(119866)

1199032

119866(119890) ⩾ 119903

1

119866(119890) ⩾ 4 if 1003816100381610038161003816119873119866

(119909) cap 119873119866(119910)1003816100381610038161003816 = 0

1199032

119866(119890) ⩾ 4 119903

1

119866(119890) ⩾ 3 if 1003816100381610038161003816119873119866

(119909) cap 119873119866(119910)1003816100381610038161003816 = 1

1199032

119866(119890) ⩾ 119903

1

119866(119890) ⩾ 3 if 1003816100381610038161003816119873119866

(119909) cap 119873119866(119910)1003816100381610038161003816 ⩾ 2

(30)

Following Carlson and Develin [42] for any edge 119890 = 119909119910 of119866 we define

119863119866(119890) =

1

119889119866(119909)+

1

119889119866(119910)

+1

1199031

119866(119890)+

1

1199032

119866(119890)minus 1 (31)

By the well-known Eulerrsquos formula

120592 (119866) minus 120576 (119866) + 120601 (119866) = 2 minus 2120588 (119866) (32)

it is easy to see that

sum

119890isin119864(119866)

119863119866(119890) = 120592 (119866) minus 120576 (119866) + 120601 (119866) = 2 minus 2120588 (119866) (33)

If 119866 is a planar graph that is 120588(119866) = 0 then

sum

119890isin119864(119866)

119863119866(119890) = 120592 (119866) minus 120576 (119866) + 120601 (119866) = 2 (34)

Combining these formulas withTheorem 18 Carlson andDevelin [42] gave a simple and intuitive proof ofTheorem 74and obtained the following result

Theorem 96 (Carlson and Develin [42] 2006) Let 119866 be aconnected graph which can be embedded on a torus Then119887(119866) ⩽ Δ(119866) + 3

Recently Hou and Liu [50] improved Theorem 96 asfollows

Theorem 97 (Hou and Liu [50] 2012) Let 119866 be a connectedgraph which can be embedded on a torus Then 119887(119866) ⩽ 9Moreover if Δ(119866) = 6 then 119887(119866) ⩽ 8

By the way Cao et al [51] generalized the result inTheorem 79 to a connected graph 119866 that can be embeddedon a torus that is

119887 (119866) le

6 if 119892 (119866) ⩾ 4 and 119866 is not 4-regular5 if 119892 (119866) ⩾ 54 if 119892 (119866) ⩾ 6 and 119866 is not 3-regular3 if 119892 (119866) ⩾ 8

(35)

Several authors used this method to obtain some resultson the bondage number For example Fischermann et al[46] used this method to prove the second conclusion inTheorem 78 Recently Cao et al [52] have used thismethod todeal with more general graphs with small crossing numbersFirst they found the following property

Lemma 98 120575(119866) ⩽ 5 for any graph 119866 with cr(119866) ⩽ 5

16 International Journal of Combinatorics

This result implies that 119887(119866) ⩽ Δ(119866) + 4 by Corollary 15for any graph 119866 with cr(119866) ⩽ 5 Cao et al established thefollowing relation in terms of maximum planar subgraphs

Lemma 99 (Cao et al [52]) Let 119866 be a connected graph withcrossing number cr(119866) For any maximum planar subgraph119867of 119866

sum

119890isin119864(119867)

119863119866(119890) ⩾ 2 minus

2cr (119866)120575 (119866)

(36)

Using Carlson-Develin method and combiningLemma 98 with Lemma 99 Cao et al proved the followingresults

Theorem 100 (Cao et al [52]) For any connected graph 119866119887(119866) ⩽ Δ(119866) + 2 if 119866 satisfies one of the following conditions

(a) cr(119866) ⩽ 3(b) cr(119866) = 4 and 119866 is not 4-regular(c) cr(119866) = 5 and 119866 contains no vertices of degree 4

Theorem101 (Cao et al [52]) Let119866 be a connected graphwithΔ(119866) = 5 and cr(119866) ⩽ 4 If no triangles contain two vertices ofdegree 5 then 119887(119866) ⩽ 6 = Δ(119866) + 1

Theorem 102 (Cao et al [52]) Let 119866 be a connected graphwith Δ(119866) ⩾ 6 and cr(119866) ⩽ 3 If Δ(119866) ⩾ 7 or if Δ(119866) = 6120575(119866) = 3 and every edge 119890 = 119909119910 with 119889

119866(119909) = 5 and 119889

119866(119910) = 6

is contained in atmost one triangle then 119887(119866) ⩽ min8 Δ(119866)+1

Using Carlson-Develin method it can be proved that119887(119866) ⩽ Δ(119866) + 2 if cr(119866) ⩽ 3 (see the first conclusion inTheorem 100) and not yet proved that 119887(119866) ⩽ 8 if cr(119866) ⩽3 although it has been proved by using other method (seeCorollary 90)

By using Carlson-Develin method Samodivkin [25]obtained some results on the bondage number for graphswithsome given properties

Kim [53] showed that 119887(119866) ⩽ Δ(119866) + 2 for a connectedgraph 119866 with genus 1 Δ(119866) ⩽ 5 and having a toroidalembedding of which at least one region is not 4-sidedRecently Gagarin and Zverovich [54] further have extendedCarlson-Develin ideas to establish a nice upper bound forarbitrary graphs that can be embedded on orientable ornonorientable topological surface

Theorem 103 (Gagarin and Zverovich [54] 2012) Let 119866 bea graph embeddable on an orientable surface of genus ℎ and anonorientable surface of genus 119896 Then 119887(119866) ⩽ minΔ(119866)+ℎ+2 Δ(119866) + 119896 + 1

This result generalizes the corresponding upper boundsin Theorems 74 and 96 for any orientable or nonori-entable topological surface By investigating the proof ofTheorem 103 Huang [55] found that the issue of the ori-entability can be avoided by using the Euler characteristic120594(= 120592(119866) minus 120576(119866) + 120601(119866)) instead of the genera ℎ and 119896 the

relations are 120594 = 2minus2ℎ and 120594 = 2minus119896 To obtain the best resultfromTheorem 103 onewants ℎ and 119896 as small as possible thisis equivalent to having 120594 as large as possible

According toTheorem 103 if119866 is planar (ℎ = 0 120594 = 2) orcan be embedded on the real projective plane (119896 = 1 120594 = 1)then 119887(119866) ⩽ Δ(119866) + 2 In all other cases Huang [55] had thefollowing improvement for Theorem 103 the proof is basedon the technique developed byCarlson-Develin andGagarin-Zverovich and includes some elementary calculus as a newingredient mainly the intermediate-value theorem and themean-value theorem

Theorem 104 (Huang [55] 2012) Let 119866 be a graph embed-dable on a surface whose Euler characteristic 120594 is as large aspossible If 120594 ⩽ 0 then 119887(119866) ⩽ Δ(119866)+lfloor119903rfloor where 119903 is the largestreal root of the following cubic equation in 119911

1199113

+ 21199112

+ (6120594 minus 7) 119911 + 18120594 minus 24 = 0 (37)

In addition if 120594 decreases then 119903 increases

The following result is asymptotically equivalent toTheorem 104

Theorem 105 (Huang [55] 2012) Let 119866 be a graph embed-dable on a surface whose Euler characteristic 120594 is as large aspossible If 120594 ⩽ 0 then 119887(119866) lt Δ(119866) + radic12 minus 6120594 + 12 orequivalently 119887(119866) ⩽ Δ(119866) + lceilradic12 minus 6120594 minus 12rceil

Also Gagarin andZverovich [54] indicated that the upperbound in Theorem 103 can be improved for larger values ofthe genera ℎ and 119896 by adjusting the proofs and stated thefollowing general conjecture

Conjecture 106 (Gagarin and Zverovich [54] 2012) For aconnected graph G of orientable genus ℎ and nonorientablegenus 119896

119887 (119866) ⩽ min 119888ℎ 119888

1015840

119896 Δ (119866) + 119900 (ℎ) Δ (119866) + 119900 (119896) (38)

where 119888ℎand 1198881015840

119896are constants depending respectively on the

orientable and nonorientable genera of 119866

In the recent paper Gagarin and Zverovich [56] providedconstant upper bounds for the bondage number of graphs ontopological surfaces which can be used as the first estimationfor the constants 119888

ℎand 1198881015840

119896of Conjecture 106

Theorem 107 (Gagarin and Zverovich [56] 2013) Let 119866 bea connected graph of order 119899 If 119866 can be embedded on thesurface of its orientable or nonorientable genus of the Eulercharacteristic 120594 then

(a) 120594 ⩾ 1 implies 119887(119866) ⩽ 10(b) 120594 ⩽ 0 and 119899 gt minus12120594 imply 119887(119866) ⩽ 11(c) 120594 ⩽ minus1 and 119899 ⩽ minus12120594 imply 119887(119866) ⩽ 11 +

3120594(radic17 minus 8120594 minus 3)(120594 minus 1) = 11 + 119874(radicminus120594)

Theorem 79 shows that a connected planar triangle-freegraph 119866 has 119887(119866) ⩽ 6 The following result generalizes to itall the other topological surfaces

International Journal of Combinatorics 17

Theorem 108 (Gagarin and Zverovich [56] 2013) Let 119866 be aconnected triangle-free graph of order 119899 If 119866 can be embeddedon the surface of its orientable or nonorientable genus of theEuler characteristic 120594 then

(a) 120594 ⩾ 1 implies 119887(119866) ⩽ 6(b) 120594 ⩽ 0 and 119899 gt minus8120594 imply 119887(119866) ⩽ 7(c) 120594 ⩽ minus1 and 119899 ⩽ minus8120594 imply 119887(119866) ⩽ 7minus4120594(1+radic2 minus 120594)

In [56] the authors also explicitly improved upperbounds in Theorem 103 and gave tight lower bounds forthe number of vertices of graphs 2-cell embeddable ontopological surfaces of a given genusThe interested reader isreferred to the original paper for details The following resultis interesting although its proof is easy

Theorem 109 (Samodivkin [57]) Let119866 be a graph with order119899 and girth 119892 If 119866 is embeddable on a surface whose Eulercharacteristic 120594 is as large as possible then

119887 (119866) ⩽ 3 +8

119892 minus 2minus

4120594119892

119899 (119892 minus 2) (39)

If 119866 is a planar graph with girth 119892 ⩾ 4 + 119894 for any 119894 isin0 1 2 then Theorem 109 leads to 119887(119866) ⩽ 6 minus 119894 which isthe result inTheorem 79 obtained by Fischermann et al [46]Also inmany cases the bound stated inTheorem 109 is betterthan those given byTheorems 103 and 105

8 Conditional Bondage Numbers

Since the concept of the bondage number is based upon thedomination all sorts of dominations which are variationsof the normal domination by adding restricted conditionsto the normal dominating set can develop a new ldquobondagenumberrdquo as long as a variation of domination number isgiven In this section we survey results on the bondagenumber under some restricted conditions

81 Total Bondage Numbers A dominating set 119878 of a graph119866 without isolated vertices is called total if the subgraphinduced by 119878 contains no isolated vertices The minimumcardinality of a total dominating set is called the totaldomination number of 119866 and denoted by 120574

119879(119866) It is clear

that 120574(119866) ⩽ 120574119879(119866) ⩽ 2120574(119866) for any graph 119866 without isolated

verticesThe total domination in graphs was introduced by Cock-

ayne et al [58] in 1980 Pfaff et al [59 60] in 1983 showedthat the problem determining total domination number forgeneral graphs is NP-complete even for bipartite graphs andchordal graphs Even now total domination in graphs hasbeen extensively studied in the literature In 2009 Henning[61] gave a survey of selected recent results on total domina-tion in graphs

The total bondage number of 119866 denoted by 119887119879(119866) is the

smallest cardinality of a subset 119861 sube 119864(119866) with the propertythat119866minus119861 contains no isolated vertices and 120574

119879(119866minus119861) gt 120574

119879(119866)

From definition 119887119879(119866)may not exist for some graphs for

example119866 = 1198701119899 We put 119887

119879(119866) = infin if 119887

119879(119866) does not exist

It is easy to see that 119887119879(119866) is finite for any connected graph 119866

other than 1198701 119870

2 119870

3 and 119870

1119899 In fact for such a graph 119866

since there is a path of length 3 in 119866 we can find 1198611sube 119864(119866)

such that 119866 minus 1198611is a spanning tree 119879 of 119866 containing a path

of length 3 So 120574119879(119879) = 120574

119879(119866minus119861

1) ⩾ 120574

119879(119866) For the tree119879 we

find 1198612sube 119864(119879) such that 120574

119879(119879 minus 119861

2) gt 120574

119879(119879) ⩾ 120574

119879(119866) Thus

we have 120574119879(119866minus119861

2minus119861

1) gt 120574

119879(119866) and so 119887

119879(119866) ⩽ |119861

1|+|119861

2| In

the following discussion we always assume that 119887119879(119866) exists

when a graph 119866 is mentionedIn 1991 Kulli and Patwari [62] first studied the total

bondage number of a graph and calculated the exact valuesof 119887

119879(119866) for some standard graphs

Theorem 110 (Kulli and Patwari [62] 1991) For 119899 ⩾ 4

119887119879(119862

119899) =

3 119894119891 119899 equiv 2 (mod 4) 2 119900119905ℎ119890119903119908119894119904119890

119887119879(119875

119899) =

2 119894119891 119899 equiv 2 (mod 4) 1 119900119905ℎ119890119903119908119894119904119890

119887119879(119870

119898119899) = 119898 119908119894119905ℎ 2 ⩽ 119898 ⩽ 119899

119887119879(119870

119899) =

4 119891119900119903 119899 = 4

2119899 minus 5 119891119900119903 119899 ⩾ 5

(40)

Recently Hu et al [63] have obtained some results on thetotal bondage number of the Cartesian product119875

119898times119875

119899of two

paths 119875119898and 119875

119899

Theorem 111 (Hu et al [63] 2012) For the Cartesian product119875119898times 119875

119899of two paths 119875

119898and 119875

119899

119887119879(119875

2times 119875

119899) =

1 119894119891 119899 equiv 0 (mod 3) 2 119894119891 119899 equiv 2 (mod 3) 3 119894119891 119899 equiv 1 (mod 3)

119887119879(119875

3times 119875

119899) = 1

119887119879(119875

4times 119875

119899)

= 1 119894119891 119899 equiv 1 (mod 5) = 2 119894119891 119899 equiv 4 (mod 5) ⩽ 3 119894119891 119899 equiv 2 (mod 5) ⩽ 4 119894119891 119899 equiv 0 3 (mod 5)

(41)

Generalized Petersen graphs are an important class ofcommonly used interconnection networks and have beenstudied recently By constructing a family of minimum totaldominating sets Cao et al [64] determined the total bondagenumber of the generalized Petersen graphs

FromTheorem 6 we know that 119887(119879) ⩽ 2 for a nontrivialtree 119879 But given any positive integer 119896 Sridharan et al [65]constructed a tree 119879 for which 119887

119879(119879) = 119896 Let119867

119896be the tree

obtained from a star1198701119896+1

by subdividing 119896 edges twice Thetree 119867

7is shown in Figure 6 It can be easily verified that

119887119879(119867

119896) = 119896 This fact can be stated as the following theorem

Theorem 112 (Sridharan et al [65] 2007) For any positiveinteger 119896 there exists a tree 119879 with 119887

119879(119879) = 119896

18 International Journal of Combinatorics

Figure 61198677

Combining Theorem 1 with Theorem 112 we have whatfollows

Corollary 113 (Sridharan et al [65] 2007) The bondagenumber and the total bondage number are unrelated even fortrees

However Sridharan et al [65] gave an upper bound onthe total bondage number of a tree in terms of its maximumdegree For any tree 119879 of order 119899 if 119879 =119870

1119899minus1 then 119887

119879(119879) =

minΔ(119879) (13)(119899 minus 1) Rad and Raczek [66] improved thisupper bound and gave a constructive characterization of acertain class of trees attaching the upper bound

Theorem 114 (Rad and Raczek [66] 2011) 119887119879(119879) ⩽ Δ(119879) minus 1

for any tree 119879 with maximum degree at least three

In general the decision problem for 119887119879(119866) is NP-complete

for any graph 119866 We state the decision problem for the totalbondage as follows

Problem 3 Consider the decision problemTotal Bondage ProblemInstance a graph 119866 and a positive integer 119887Question is 119887

119879(119866) ⩽ 119887

Theorem 115 (Hu and Xu [16] 2012) The total bondageproblem is NP-complete

Consequently in view of its computational hardnessit is significative to establish some sharp bounds on thetotal bondage number of a graph in terms of other graphicparameters In 1991 Kulli and Patwari [62] showed that119887119879(119866) ⩽ 2119899 minus 5 for any graph 119866 with order 119899 ⩾ 5 Sridharan

et al [65] improved this result as follows

Theorem 116 (Sridharan et al [65] 2007) For any graph 119866with order 119899 if 119899 ⩾ 5 then

119887119879(119866) ⩽

1

3(119899 minus 1) 119894119891 119866 119888119900119899119905119886119894119899119904 119899119900 119888119910119888119897119890119904

119899 minus 1 119894119891 119892 (119866) ⩾ 5

119899 minus 2 119894119891 119892 (119866) = 4

(42)

Rad and Raczek [66] also established some upperbounds on 119887

119879(119866) for a general graph 119866 In particular they

gave an upper bound on 119887119879(119866) in terms of the girth of

119866

Theorem 117 (Rad and Raczek [66] 2011) 119887119879(119866)⩽4Δ(119866) minus 5

for any graph 119866 with 119892(119866) ⩾ 4

82 Paired Bondage Numbers A dominating set 119878 of 119866 iscalled to be paired if the subgraph induced by 119878 containsa perfect matching The paired domination number of 119866denoted by 120574

119875(119866) is the minimum cardinality of a paired

dominating set of 119866 Clearly 120574119879(119866) ⩽ 120574

119875(119866) for every

connected graph 119866 with order at least two where 120574119879(119866) is

the total domination number of 119866 and 120574119875(119866) ⩽ 2120574(119866) for

any graph119866without isolated vertices Paired domination wasintroduced by Haynes and Slater [67 68] and further studiedin [69ndash72]

The paired bondage number of 119866 with 120575(119866) ⩾ 1 denotedby 119887P(119866) is the minimum cardinality among all sets of edges119861 sube 119864 such that 120575(119866 minus 119861) ⩾ 1 and 120574

119875(119866 minus 119861) gt 120574

119875(119866)

The concept of the paired bondage number was firstproposed by Raczek [73] in 2008The following observationsfollow immediately from the definition of the paired bondagenumber

Observation 2 Let 119866 be a graph with 120575(119866) ⩾ 1

(a) If 119867 is a subgraph of 119866 such that 120575(119867) ⩾ 1 then119887119875(119867) ⩽ 119887

119875(119866)

(b) If 119867 is a subgraph of 119866 such that 119887119875(119867) = 1 and 119896

is the number of edges removed to form119867 then 1 ⩽119887119875(119866) ⩽ 119896 + 1

(c) If 119909119910 isin 119864(119866) such that 119889119866(119909) 119889

119866(119910) gt 1 and 119909119910

belongs to each perfect matching of each minimumpaired dominating set of 119866 then 119887

119875(119866) = 1

Based on these observations 119887119875(119875

119899) ⩽ 2 In fact the

paired bondage number of a path has been determined

Theorem 118 (Raczek [73] 2008) For any positive integer 119896if 119899 ⩾ 2 then

119887119875(119875

119899) =

0 119894119891 119899 = 2 3 119900119903 5

1 119894119891 119899 = 4119896 4119896 + 3 119900119903 4119896 + 6

2 119900119905ℎ119890119903119908119894119904119890

(43)

For a cycle 119862119899of order 119899 ⩾ 3 since 120574

119875(119862

119899) = 120574

119875(119875

119899) from

Theorem 118 the following result holds immediately

Corollary 119 (Raczek [73] 2008) For any positive integer 119896if 119899 ⩾ 3 then

119887119875(119862

119899) =

0 119894119891 119899 = 3 119900119903 5

2 119894119891 119899 = 4119896 4119896 + 3 119900119903 4119896 + 6

3 119900119905ℎ119890119903119908119894119904119890

(44)

A wheel 119882119899 where 119899 ⩾ 4 is a graph with 119899 vertices

formed by connecting a single vertex to all vertices of a cycle119862119899minus1

Of course 120574119875(119882

119899) = 2

International Journal of Combinatorics 19

119896

Figure 7 A tree 119879 with 119887119875(119879) = 119896

Theorem 120 (Raczek [73] 2008) For positive integers 119899 and119898

119887119875(119870

119898119899) = 119898 119891119900119903 1 lt 119898 ⩽ 119899

119887119875(119882

119899) =

4 119894119891 119899 = 4

3 119894119891 119899 = 5

2 119900119905ℎ119890119903119908119894119904119890

(45)

Theorem 16 shows that if 119879 is a nontrivial tree then119887(119879) ⩽ 2 However no similar result exists for pairedbondage For any nonnegative integer 119896 let 119879

119896be a tree

obtained by subdividing all but one edge of a star 1198701119896+1

(asshown in Figure 7) It is easy to see that 119887

119875(119879

119896) = 119896 We

state this fact as the following theorem which is similar toTheorem 112

Theorem121 (Raczek [73] 2008) For any nonnegative integer119896 there exists a tree 119879 with 119887

119875(119879) = 119896

Consider the tree defined in Figure 5 for 119896 = 3 Then119887(119879

3) ⩽ 2 and 119887

119875(119879

3) = 3 On the other hand 119887(119875

4) = 2

and 119887119875(119875

4) = 1 Thus we have what follows which is similar

to Corollary 113

Corollary 122 The bondage number and the paired bondagenumber are unrelated even for trees

A constructive characterization of trees with 119887(119879) = 2is given by Hartnell and Rall in [12] Raczek [73] provideda constructive characterization of trees with 119887

119875(119879) = 0 In

order to state the characterization we define a labeling andthree simple operations on a tree119879 Let 119910 isin 119881(119879) and let ℓ(119910)be the label assigned to 119910

Operation T1 If ℓ(119910) = 119861 add a vertex 119909 and the

edge 119910119909 and let ℓ(119909) = 119860OperationT

2 If ℓ(119910) = 119862 add a path (119909

1 119909

2) and the

edge 1199101199091 and let ℓ(119909

1) = 119861 ℓ(119909

2) = 119860

Operation T3 If ℓ(119910) = 119861 add a path (119909

1 119909

2 119909

3)

and the edge 1199101199091 and let ℓ(119909

1) = 119862 ℓ(119909

2) = 119861 and

ℓ(1199093) = 119860

Let 1198752= (119906 V) with ℓ(119906) = 119860 and ℓ(V) = 119861 Let T be

the class of all trees obtained from the labeled 1198752by a finite

sequence of operationsT1 T

2 T

3

A tree 119879 in Figure 8 belongs to the familyTRaczek [73] obtained the following characterization of all

trees 119879 with 119887119875(119879) = 0

119860

119860

119860

119860

119860

119861 119862

119861 119862 119861

119861119860

Figure 8 A tree 119879 belong to the familyT

Theorem 123 (Raczek [73] 2008) For a tree 119879 119887119875(119879) = 0 if

and only if 119879 is inT

We state the decision problem for the paired bondage asfollows

Problem 4 Consider the decision problem

Paired Bondage ProblemInstance a graph 119866 and a positive integer 119887Question is 119887

119875(119866) ⩽ 119887

Conjecture 124 The paired bondage problem is NP-complete

83 Restrained Bondage Numbers A dominating set 119878 of 119866 iscalled to be restrained if the subgraph induced by 119878 containsno isolated vertices where 119878 = 119881(119866) 119878 The restraineddomination number of 119866 denoted by 120574

119877(119866) is the smallest

cardinality of a restrained dominating set of 119866 It is clear that120574119877(119866) exists and 120574(119866) ⩽ 120574

119877(119866) for any nonempty graph 119866

The concept of restrained domination was introducedby Telle and Proskurowski [74] in 1997 albeit indirectly asa vertex partitioning problem Concerning the study of therestrained domination numbers the reader is referred to [74ndash79]

In 2008 Hattingh and Plummer [80] defined therestrained bondage number 119887R(119866) of a nonempty graph 119866 tobe the minimum cardinality among all subsets 119861 sube 119864(119866) forwhich 120574

119877(119866 minus 119861) gt 120574

119877(119866)

For some simple graphs their restrained dominationnumbers can be easily determined and so restrained bondagenumbers have been also determined For example it is clearthat 120574

119877(119870

119899) = 1 Domke et al [77] showed that 120574

119877(119862

119899) =

119899 minus 2lfloor1198993rfloor for 119899 ⩾ 3 and 120574119877(119875

119899) = 119899 minus 2lfloor(119899 minus 1)3rfloor for 119899 ⩾ 1

Using these results Hattingh and Plummer [80] obtained therestricted bondage numbers for119870

119899 119862

119899 119875

119899 and119866 = 119870

11989911198992119899119905

as follows

Theorem 125 (Hattingh and Plummer [80] 2008) For 119899 ⩾ 3

119887R (119870119899) =

1 119894119891 119899 = 3

lceil119899

2rceil 119900119905ℎ119890119903119908119894119904119890

119887R (119862119899) =

1 119894119891 119899 equiv 0 (mod 3) 2 119900119905ℎ119890119903119908119894119904119890

119887R (119875119899) = 1 119899 ⩾ 4

20 International Journal of Combinatorics

119887R (119866)

=

lceil119898

2rceil 119894119891 119899

119898= 1 119886119899119889 119899

119898+1⩾ 2 (1 ⩽ 119898 lt 119905)

2119905 minus 2 119894119891 1198991= 119899

2= sdot sdot sdot = 119899

119905= 2 (119905 ⩾ 2)

2 119894119891 1198991= 2 119886119899119889 119899

2⩾ 3 (119905 = 2)

119905minus1

sum

119894=1

119899119894minus 1 119900119905ℎ119890119903119908119894119904119890

(46)

Theorem 126 (Hattingh and Plummer [80] 2008) For a tree119879 of order 119899 (⩾ 4) 119879 ≇ 119870

1119899minus1if and only if 119887R(119879) = 1

Theorem 126 shows that the restrained bondage numberof a tree can be computed in constant time However thedecision problem for 119887R(119866) is NP-complete even for bipartitegraphs

Problem 5 Consider the decision problem

Restrained Bondage Problem

Instance a graph 119866 and a positive integer 119887

Question is 119887R(119866) ⩽ 119887

Theorem 127 (Hattingh and Plummer [80] 2008) Therestrained bondage problem is NP-complete even for bipartitegraphs

Consequently in view of its computational hardnessit is significative to establish some sharp bounds on therestrained bondage number of a graph in terms of othergraphic parameters

Theorem 128 (Hattingh and Plummer [80] 2008) For anygraph 119866 with 120575(119866) ⩾ 2

119887R (119866) ⩽ min119909119910isin119864(119866)

119889119866(119909) + 119889

119866(119910) minus 2 (47)

Corollary 129 119887R(119866) ⩽ Δ(119866)+120575(119866)minus2 for any graph119866 with120575(119866) ⩾ 2

Notice that the bounds stated in Theorem 128 andCorollary 129 are sharp Indeed the class of cycles whoseorders are congruent to 1 2 (mod 3) have a restrained bondagenumber achieving these bounds (see Theorem 125)

Theorem 128 is an analogue of Theorem 14 A quitenatural problem is whether or not for restricted bondagenumber there are analogues of Theorem 16 Theorem 18Theorem 29 and so on Indeed we should consider all resultsfor the bondage number whether or not there are analoguesfor restricted bondage number

Theorem 130 (Hattingh and Plummer [80] 2008) For anygraph 119866 with 120574

119877(119866) = 2

119887R (119866) ⩽ Δ (119866) + 1 (48)

We now consider the relation between 119887119877(119866) and 119887(119866)

Theorem 131 (Hattingh and Plummer [80] 2008) If 120574119877(119866) =

120574(119866) for some graph 119866 then 119887R(119866) ⩽ 119887(119866)

Corollary 129 and Theorem 131 provide a possibility toattack Conjecture 73 by characterizing planar graphs with120574119877= 120574 and 119887R = 119887 when 3 ⩽ Δ ⩽ 6However we do not have 119887R(119866) = 119887(119866) for any graph

119866 even if 120574119877(119866) = 120574(119866) Observe that 120574

119877(119870

3) = 120574(119870

3) yet

119887119877(119870

3) = 1 and 119887(119870

3) = 2 We still may not claim that

119887119877(119866) = 119887(119866) even in the case that every 120574-set is a 120574

119877-set

The example 1198703again demonstrates this

Theorem 132 (Hattingh and Plummer [80] 2008) There isan infinite class of graphs inwhich each graph119866 satisfies 119887(119866) lt119887R(119866)

In fact such an infinite class of graphs can be obtained asfollows Let119867 be a connected graph BC(119867) a graph obtainedfrom 119867 by attaching ℓ (⩾ 2) vertices of degree 1 to eachvertex in 119867 and B = 119866 119866 = BC(119867) for some graph 119867such that 120575(119867) ⩾ 2 By Theorem 3 we can easily check that119887(119866) = 1 lt 2 ⩽ min120575(119867) ℓ = 119887R(119866) for any 119866 isin BMoreover there exists a graph 119866 isin B such that 119887R(119866) canbe much larger than 119887(119866) which is stated as the followingtheorem

Theorem 133 (Hattingh and Plummer [80] 2008) For eachpositive integer 119896 there is a graph119866 such that 119896 = 119887R(119866)minus119887(119866)

Combining Theorem 133 with Theorem 132 we have thefollowing corollary

Corollary 134 The bondage number and the restrainedbondage number are unrelated

Very recently Jafari Rad et al [81] have consideredthe total restrained bondage in graphs based on both totaldominating sets and restrained dominating sets and obtainedseveral properties exact values and bounds for the totalrestrained bondage number of a graph

A restrained dominating set 119878 of a graph 119866 without iso-lated vertices is called to be a total restrained dominating setif 119878 is a total dominating set The total restrained dominationnumber of119866 denoted by 120574TR (119866) is theminimum cardinalityof a total restrained dominating set of 119866 For referenceson this domination in graphs see [3 61 82ndash85] The totalrestrained bondage number 119887TR (119866) of a graph 119866 with noisolated vertex is the cardinality of a smallest set of edges 119861 sube119864(119866) for which119866minus119861 has no isolated vertex and 120574TR (119866minus119861) gt120574TR (119866) In the case that there is no such subset 119861 we define119887TR (119866) = infin

Ma et al [84] determined that 120574TR (119875119899) = 119899minus2lfloor(119899minus2)4rfloorfor 119899 ⩾ 3 120574TR (119862119899

) = 119899 minus 2lfloor1198994rfloor for 119899 ⩾ 3 120574TR (1198703) =

3 and 120574TR (119870119899) = 2 for 119899 = 3 and 120574TR (1198701119899

) = 1 + 119899

and 120574TR (119870119898119899) = 2 for 2 ⩽ 119898 ⩽ 119899 According to these

results Jafari Rad et al [81] determined the exact valuesof total restrained bondage numbers for the correspondinggraphs

International Journal of Combinatorics 21

Theorem 135 (Jafari Rad et al [81]) For an integer 119899

119887TR (119875119899) = infin 119894119891 2 ⩽ 119899 ⩽ 5

1 119894119891 119899 ⩾ 5

119887TR (119862119899) =

infin 119894119891 119899 = 3

1 119894119891 3 lt 119899 119899 equiv 0 1 (mod 4)2 119894119891 3 lt 119899 119899 equiv 2 3 (mod 4)

119887TR (119882119899) = 2 119891119900119903 119899 ⩾ 6

119887TR (119870119899) = 119899 minus 1 119891119900119903 119899 ⩾ 4

119887TR (119870119898119899) = 119898 minus 1 119891119900119903 2 ⩽ 119898 ⩽ 119899

(49)

119887TR (119870119899) = 119899minus1 shows the fact that for any integer 119899 ⩾ 3 there

exists a connected graph namely119870119899+1

such that 119887TR (119870119899+1) =

119899 that is the total restrained bondage number is unboundedfrom the above

A vertex is said to be a support vertex if it is adjacent toa vertex of degree one Let A be the family of all connectedgraphs such that119866 belongs toA if and only if every edge of119866is incident to a support vertex or 119866 is a cycle of length three

Proposition 136 (Cyman and Raczek [82] 2006) For aconnected graph119866 of order 119899 120574TR (119866) = 119899 if and only if119866 isin A

Hence if 119866 isin A then the removal of any subset of edgesof119866 cannot increase the total restrained domination numberof 119866 and thus 119887TR (119866) = infin When 119866 notin A a result similar toTheorem 6 is obtained

Theorem 137 (Jafari Rad et al [81]) For any tree 119879 notin A119887TR (119879) ⩽ 2 This bound is sharp

In the samepaper the authors provided a characterizationfor trees 119879 notin A with 119887TR (119879) = 1 or 2 The following result issimilar to Observation 1

Observation 3 (Jafari Rad et al [81]) If 119896 edges can beremoved from a graph 119866 to obtain a graph 119867 without anisolated vertex and with 119887TR (119867) = 119905 then 119887TR (119866) ⩽ 119896 + 119905

ApplyingTheorem 137 andObservation 3 Jafari Rad et alcharacterized all connected graphs 119866 with 119887TR (119866) = infin

Theorem 138 (Jafari Rad et al [81]) For a connected graph119866119887TR (119866) = infin if and only if 119866 isin A

84 Other Conditional Bondage Numbers A subset 119868 sube 119881(119866)is called an independent set if no two vertices in 119868 are adjacentin 119866 The maximum cardinality among all independent setsis called the independence number of 119866 denoted by 120572(119866)

A dominating set 119878 of a graph 119866 is called to be inde-pendent if 119878 is an independent set of 119866 The minimumcardinality among all independent dominating set is calledthe independence domination number of 119866 and denoted by120574119868(119866)

Since an independent dominating set is not only adominating set but also an independent set 120574(119866) ⩽ 120574

119868(119866) ⩽

120572(119866) for any graph 119866It is clear that a maximal independent set is certainly a

dominating set Thus an independent set is maximal if andonly if it is an independent dominating set and so 120574

119868(119866) is the

minimum cardinality among all maximal independent setsof 119866 This graph-theoretical invariant has been well studiedin the literature see for example Haynes et al [3] for earlyresults Goddard and Henning [86] for recent results onindependent domination in graphs

In 2003 Zhang et al [87] defined the independencebondage number 119887

119868(119866) of a nonempty graph 119866 to be the

minimum cardinality among all subsets 119861 sube 119864(119866) for which120574119868(119866 minus 119861) gt 120574

119868(119866) For some ordinary graphs their indepen-

dence domination numbers can be easily determined and soindependence bondage numbers have been also determinedClearly 119887

119868(119870

119899) = 1 if 119899 ⩾ 2

Theorem 139 (Zhang et al [87] 2003) For any integer 119899 ⩾ 4

119887119868(119862

119899) =

1 119894119891 119899 equiv 1 (mod 2) 2 119894119891 119899 equiv 0 (mod 2)

119887119868(119875

119899) =

1 119894119891 119899 equiv 0 (mod 2) 2 119894119891 119899 equiv 1 (mod 2)

119887119868(119870

119898119899) = 119899 119891119900119903 119898 ⩽ 119899

(50)

Apart from the above-mentioned results as far as we findno other results on the independence bondage number in thepresent literature we never knew much of any result on thisparameter for a tree

A subset 119878 of 119881(119866) is called an equitable dominatingset if for every 119910 isin 119881(119866 minus 119878) there exists a vertex 119909 isin 119878such that 119909119910 isin 119864(119866) and |119889

119866(119909) minus 119889

119866(119910)| ⩽ 1 proposed

by Dharmalingam [88] The minimum cardinality of such adominating set is called the equitable domination number andis denoted by 120574

119864(119866) Deepak et al [89] defined the equitable

bondage number 119887119864(119866) of a graph 119866 to be the cardinality of a

smallest set 119861 sube 119864(119866) of edges for which 120574119864(119866 minus 119861) gt 120574

119864(119866)

and determined 119887119864(119866) for some special graphs

Theorem 140 (Deepak et al [89] 2011) For any integer 119899 ⩾ 2

119887119864(119870

119899) = lceil

119899

2rceil

119887119864(119870

119898119899) = 119898 119891119900119903 0 ⩽ 119899 minus 119898 ⩽ 1

119887119864(119862

119899) =

3 119894119891 119899 equiv 1 (mod 3)2 119900119905ℎ119890119903119908119894119904119890

119887119864(119875

119899) =

2 119894119891 119899 equiv 1 (mod 3)1 119900119905ℎ119890119903119908119894119904119890

119887119864(119879) ⩽ 2 119891119900119903 119886119899119910 119899119900119899119905119903119894119907119894119886119897 119905119903119890119890 119879

119887119864(119866) ⩽ 119899 minus 1 119891119900119903 119886119899119910 119888119900119899119899119890119888119905119890119889 119892119903119886119901ℎ 119866 119900119891 119900119903119889119890119903 119899

(51)

22 International Journal of Combinatorics

As far as we know there are no more results on theequitable bondage number of graphs in the present literature

There are other variations of the normal domination byadding restricted conditions to the normal dominating setFor example the connected domination set of a graph119866 pro-posed by Sampathkumar and Walikar [90] is a dominatingset 119878 such that the subgraph induced by 119878 is connected Theproblem of finding a minimum connected domination set ina graph is equivalent to the problemof finding a spanning treewithmaximumnumber of vertices of degree one [91] and hassome important applications in wireless networks [92] thusit has received much research attention However for suchan important domination there is no result on its bondagenumber We believe that the topic is of significance for futureresearch

9 Generalized Bondage Numbers

There are various generalizations of the classical dominationsuch as 119901-domination distance domination fractional dom-ination Roman domination and rainbow domination Everysuch a generalization can lead to a corresponding bondageIn this section we introduce some of them

91 119901-Bondage Numbers In 1985 Fink and Jacobson [93]introduced the concept of 119901-domination Let 119901 be a positiveinteger A subset 119878 of 119881(119866) is a 119901-dominating set of 119866 if|119878 cap 119873

119866(119910)| ⩾ 119901 for every 119910 isin 119878 The 119901-domination number

120574119901(119866) is the minimum cardinality among all 119901-dominating

sets of 119866 Any 119901-dominating set of 119866 with cardinality 120574119901(119866)

is called a 120574119901-set of 119866 Note that a 120574

1-set is a classic minimum

dominating set Notice that every graph has a 119901-dominatingset since the vertex-set119881(119866) is such a setWe also note that a 1-dominating set is a dominating set and so 120574(119866) = 120574

1(119866) The

119901-domination number has received much research attentionsee a state-of-the-art survey article by Chellali et al [94]

It is clear from definition that every 119901-dominating setof a graph certainly contains all vertices of degree at most119901 minus 1 By this simple observation to avoid the trivial caseoccurrence we always assume that Δ(119866) ⩾ 119901 For 119901 ⩾ 2 Luet al [95] gave a constructive characterization of trees withunique minimum 119901-dominating sets

Recently Lu and Xu [96] have introduced the conceptto the 119901-bondage number of 119866 denoted by 119887

119901(119866) as the

minimum cardinality among all sets of edges 119861 sube 119864(119866) suchthat 120574

119901(119866 minus 119861) gt 120574

119901(119866) Clearly 119887

1(119866) = 119887(119866)

Lu and Xu [96] established a lower bound and an upperbound on 119887

119901(119879) for any integer 119901 ⩾ 2 and any tree 119879 with

Δ(119879) ⩾ 119901 and characterized all trees achieving the lowerbound and the upper bound respectively

Theorem 141 (Lu and Xu [96] 2011) For any integer 119901 ⩾ 2and any tree 119879 with Δ(119879) ⩾ 119901

1 ⩽ 119887119901(119879) ⩽ Δ (119879) minus 119901 + 1 (52)

Let 119878 be a given subset of 119881(119866) and for any 119909 isin 119878 let

119873119901(119909 119878 119866) = 119910 isin 119878 cap 119873

119866(119909)

1003816100381610038161003816119873119866(119910) cap 119878

1003816100381610038161003816 = 119901

(53)

Then all trees achieving the lower bound 1 can be character-ized as follows

Theorem 142 (Lu and Xu [96] 2011) Let 119879 be a tree withΔ(119879) ⩾ 119901 ⩾ 2 Then 119887

119901(119879) = 1 if and only if for any 120574

119901-set

119878 of 119879 there exists an edge 119909119910 isin (119878 119878) such that 119910 isin 119873119901(119909 119878 119879)

The notation 119878(119886 119887) denotes a double star obtained byadding an edge between the central vertices of two stars119870

1119886minus1

and1198701119887minus1

And the vertex with degree 119886 (resp 119887) in 119878(119886 119887) iscalled the 119871-central vertex (resp 119877-central vertex) of 119878(119886 119887)

To characterize all trees attaining the upper bound givenin Theorem 141 we define three types of operations on a tree119879 with Δ(119879) = Δ ⩾ 119901 + 1

Type 1 attach a pendant edge to a vertex 119910with 119889119879(119910) ⩽ 119901minus

2 in 119879

Type 2 attach a star 1198701Δminus1

to a vertex 119910 of 119879 by joining itscentral vertex to 119910 where 119910 in a 120574

119901-set of 119879 and

119889119879(119910) ⩽ Δ minus 1

Type 3 attach a double star 119878(119901 Δ minus 1) to a pendant vertex 119910of 119879 by coinciding its 119877-central vertex with 119910 wherethe unique neighbor of 119910 is in a 120574

119901-set of 119879

Let B = 119879 a tree obtained from 1198701Δ

or 119878(119901 Δ) by afinite sequence of operations of Types 1 2 and 3

Theorem 143 (Lu and Xu [96] 2011) A tree with the maxi-mum Δ ⩾ 119901 + 1 has 119901-bondage number Δ minus 119901 + 1 if and onlyif it belongs toB

Theorem 144 (Lu and Xu [97] 2012) Let 119866119898119899= 119875

119898times 119875

119899

Then

1198872(119866

2119899) = 1 119891119900119903 119899 ⩾ 2

1198872(119866

3119899) =

2 119894119891 119899 equiv 1 (mod 3) 1 119900119905ℎ119890119903119908119894119904119890

for 119899 ⩾ 2

1198872(119866

4119899) =

1 119894119891 119899 equiv 3 (mod 4) 2 119900119905ℎ119890119903119908119894119904119890

for 119899 ⩾ 7

(54)

Recently Krzywkowski [98] has proposed the conceptof the nonisolating 2-bondage of a graph The nonisolating2-bondage number of a graph 119866 denoted by 1198871015840

2(119866) is the

minimum cardinality among all sets of edges 119861 sube 119864(119866) suchthat 119866 minus 119861 has no isolated vertices and 120574

2(119866 minus 119861) gt 120574

2(119866) If

for every 119861 sube 119864(119866) either 1205742(119866 minus 119861) = 120574

2(119866) or 119866 minus 119861 has

isolated vertices then define 11988710158402(119866) = 0 and say that 119866 is a 2-

nonisolatedly strongly stable graph Krzywkowski presentedsome basic properties of nonisolating 2-bondage in graphsshowed that for every nonnegative integer 119896 there exists atree 119879 such 1198871015840

2(119879) = 119896 characterized all 2-non-isolatedly

strongly stable trees and determined 11988710158402(119866) for several special

graphs

International Journal of Combinatorics 23

Theorem 145 (Krzywkowski [98] 2012) For any positiveintegers 119899 and119898 with119898 ⩽ 119899

1198871015840

2(119870

119899) =

0 119894119891 119899 = 1 2 3

lfloor2119899

3rfloor 119900119905ℎ119890119903119908119894119904119890

1198871015840

2(119875

119899) =

0 119894119891 119899 = 1 2 3

1 119900119905ℎ119890119903119908119894119904119890

1198871015840

2(119862

119899) =

0 119894119891 119899 = 3

1 119894119891 119899 119894119904 119890V119890119899

2 119894119891 119899 119894119904 119900119889119889

1198871015840

2(119870

119898119899) =

3 119894119891 119898 = 119899 = 3

1 119894119891 119898 = 119899 = 4

2 119900119905ℎ119890119903119908119894119904119890

(55)

92 Distance Bondage Numbers A subset 119878 of vertices of agraph119866 is said to be a distance 119896-dominating set for119866 if everyvertex in 119866 not in 119878 is at distance at most 119896 from some vertexof 119878 The minimum cardinality of all distance 119896-dominatingsets is called the distance 119896-domination number of 119866 anddenoted by 120574

119896(119866) (do not confuse with the above-mentioned

120574119901(119866)) When 119896 = 1 a distance 1-dominating set is a normal

dominating set and so 1205741(119866) = 120574(119866) for any graph 119866 Thus

the distance 119896-domination is a generalization of the classicaldomination

The relation between 120574119896and 120572

119896(the 119896-independence

number defined in Section 22) for a tree obtained by Meirand Moon [14] who proved that 120574

119896(119879) = 120572

2119896(119879) for any tree

119879 (a special case for 119896 = 1 is stated in Proposition 10) Thefurther research results can be found in Henning et al [99]Tian and Xu [100ndash103] and Liu et al [104]

In 1998 Hartnell et al [15] defined the distance 119896-bondagenumber of 119866 denoted by 119887

119896(119866) to be the cardinality of a

smallest subset 119861 of edges of 119866 with the property that 120574119896(119866 minus

119861) gt 120574119896(119866) FromTheorem 6 it is clear that if119879 is a nontrivial

tree then 1 ⩽ 1198871(119879) ⩽ 2 Hartnell et al [15] generalized this

result to any integer 119896 ⩾ 1

Theorem 146 (Hartnell et al [15] 1998) For every nontrivialtree 119879 and positive integer 119896 1 ⩽ 119887

119896(119879) ⩽ 2

Hartnell et al [15] and Topp and Vestergaard [13] alsocharacterized the trees having distance 119896-bondage number 2In particular the class of trees for which 119887

1(119879) = 2 are just

those which have a unique maximum 2-independent set (seeTheorem 11)

Since when 119896 = 1 the distance 1-bondage number 1198871(119866)

is the classical bondage number 119887(119866) Theorem 12 gives theNP-hardness of deciding the distance 119896-bondage number ofgeneral graphs

Theorem 147 Given a nonempty undirected graph 119866 andpositive integers 119896 and 119887 with 119887 ⩽ 120576(119866) determining wetheror not 119887

119896(119866) ⩽ 119887 is NP-hard

For a vertex 119909 in119866 the open 119896-neighborhood119873119896(119909) of 119909 is

defined as119873119896(119909) = 119910 isin 119881(119866) 1 ⩽ 119889

119866(119909 119910) le 119896The closed

119896-neighborhood119873119896[119909] of 119909 in 119866 is defined as119873

119896(119909) cup 119909 Let

Δ119896(119866) = max 1003816100381610038161003816119873119896

(119909)1003816100381610038161003816 for any 119909 isin 119881 (119866) (56)

Clearly Δ1(119866) = Δ(119866) The 119896th power of a graph 119866 is the

graph119866119896 with vertex-set119881(119866119896

) = 119881(119866) and edge-set119864(119866119896

) =

119909119910 1 ⩽ 119889119866(119909 119910) ⩽ 119896 The following lemma holds directly

from the definition of 119866119896

Proposition 148 (Tian and Xu [101]) Δ(119866119896

) = Δ119896(119866) and

120574(119866119896

) = 120574119896(119866) for any graph 119866 and each 119896 ⩾ 1

A graph 119866 is 119896-distance domination-critical or 120574119896-critical

for short if 120574119896(119866 minus 119909) lt 120574

119896(119866) for every vertex 119909 in 119866

proposed by Henning et al [105]

Proposition 149 (Tian and Xu [101]) For each 119896 ⩾ 1 a graph119866 is 120574

119896-critical if and only if 119866119896 is 120574

119896-critical

By the above facts we suggest the following worthwhileproblem

Problem 6 Can we generalize the results on the bondage for119866 to 119866119896 In particular do the following propositions hold

(a) 119887(119866119896

) = 119887119896(119866) for any graph 119866 and each 119896 ⩾ 1

(b) 119887(119866119896

) ⩽ Δ119896(119866) if 119866 is not 120574

119896-critical

Let 119896 and 119901 be positive integers A subset 119878 of 119881(119866) isdefined to be a (119896 119901)-dominating set of 119866 if for any vertex119909 isin 119878 |119873

119896(119909) cap 119878| ⩾ 119901 The (119896 119901)-domination number of

119866 denoted by 120574119896119901(119866) is the minimum cardinality among

all (119896 119901)-dominating sets of 119866 Clearly for a graph 119866 a(1 1)-dominating set is a classical dominating set a (119896 1)-dominating set is a distance 119896-dominating set and a (1 119901)-dominating set is the above-mentioned 119901-dominating setThat is 120574

11(119866) = 120574(119866) 120574

1198961(119866) = 120574

119896(119866) and 120574

1119901(119866) = 120574

119901(119866)

The concept of (119896 119901)-domination in a graph 119866 is a gen-eralized domination which combines distance 119896-dominationand 119901-domination in 119866 So the investigation of (119896 119901)-domination of 119866 is more interesting and has received theattention of many researchers see for example [106ndash109]

More general Li and Xu [110] proposed the concept of the(ℓ 119908)-domination number of a119908-connected graph A subset119878 of 119881(119866) is defined to be an (ℓ 119908)-dominating set of 119866 iffor any vertex 119909 isin 119878 there are 119908 internally disjoint pathsof length at most ℓ from 119909 to some vertex in 119878 The (ℓ 119908)-domination number of119866 denoted by 120574

ℓ119908(119866) is theminimum

cardinality among all (ℓ 119908)-dominating sets of 119866 Clearly1205741198961(119866) = 120574

119896(119866) and 120574

1119901(119866) = 120574

119901(119866)

It is quite natural to propose the concept of bondage num-ber for (119896 119901)-domination or (ℓ 119908)-domination However asfar as we know none has proposed this concept until todayThis is a worthwhile topic for further research

24 International Journal of Combinatorics

93 Fractional Bondage Numbers If 120590 is a function mappingthe vertex-set 119881 into some set of real numbers then for anysubset 119878 sube 119881 let 120590(119878) = sum

119909isin119878120590(119909) Also let |120590| = 120590(119881)

A real-value function 120590 119881 rarr [0 1] is a dominatingfunction of a graph 119866 if for every 119909 isin 119881(119866) 120590(119873

119866[119909]) ⩾ 1

Thus if 119878 is a dominating set of 119866 120590 is a function where

120590 (119909) = 1 if 119909 isin 1198780 otherwise

(57)

then 120590 is a dominating function of119866 The fractional domina-tion number of 119866 denoted by 120574

119865(119866) is defined as follows

120574119865(119866) = min |120590| 120590 is a domination function of 119866

(58)

The 120574119865-bondage number of 119866 denoted by 119887

119865(119866) is

defined as the minimum cardinality of a subset 119861 sube 119864 whoseremoval results in 120574

119865(119866 minus 119861) gt 120574

119865(119866)

Hedetniemi et al [111] are the first to study fractionaldomination although Farber [112] introduced the idea indi-rectly The concept of the fractional domination numberwas proposed by Domke and Laskar [113] in 1997 Thefractional domination numbers for some ordinary graphs aredetermined

Proposition 150 (Domke et al [114] 1998 Domke and Laskar[113] 1997) (a) If 119866 is a 119896-regular graph with order 119899 then120574119865(119866) = 119899119896 + 1(b) 120574

119865(119862

119899) = 1198993 120574

119865(119875

119899) = lceil1198993rceil 120574

119865(119870

119899) = 1

(c) 120574119865(119866) = 1 if and only if Δ(119866) = 119899 minus 1

(d) 120574119865(119879) = 120574(119879) for any tree 119879

(e) 120574119865(119870

119899119898) = (119899(119898 + 1) + 119898(119899 + 1))(119899119898 minus 1) where

max119899119898 gt 1

The assertions (a)ndash(d) are due to Domke et al [114] andthe assertion (e) is due to Domke and Laskar [113] Accordingto these results Domke and Laskar [113] determined the 120574

119865-

bondage numbers for these graphs

Theorem 151 (Domke and Laskar [113] 1997) 119887119865(119870

119899) =

lceil1198992rceil 119887119865(119870

119899119898) = 1 wheremax119899119898 gt 1

119887119865(119862

119899) =

2 119894119891 119899 equiv 0 (mod 3) 1 119900119905ℎ119890119903119908119894119904119890

119887119865(119875

119899) =

2 119894119891 119899 equiv 1 (mod 3) 1 119900119905ℎ119890119903119908119894119904119890

(59)

It is easy to see that for any tree119879 119887119865(119879) ⩽ 2 In fact since

120574119865(119879) = 120574(119879) for any tree 119879 120574

119865(119879

1015840

) = 120574(1198791015840

) for any subgraph1198791015840 of 119879 It follows that

119887119865(119879) = min 119861 sube 119864 (119879) 120574

119865(119879 minus 119861) gt 120574

119865(119879)

= min 119861 sube 119864 (119879) 120574 (119879 minus 119861) gt 120574 (119879)

= 119887 (119879) ⩽ 2

(60)

Except for these no results on 119887119865(119866) have been known as

yet

94 Roman Bondage Numbers A Roman dominating func-tion on a graph 119866 is a labeling 119891 119881 rarr 0 1 2 such thatevery vertex with label 0 has at least one neighbor with label2 The weight of a Roman dominating function 119891 on 119866 is thevalue

119891 (119866) = sum

119906isin119881(119866)

119891 (119906) (61)

The minimum weight of a Roman dominating function ona graph 119866 is called the Roman domination number of 119866denoted by 120574RM (119866)

A Roman dominating function 119891 119881 rarr 0 1 2

can be represented by the ordered partition (1198810 119881

1 119881

2) (or

(119881119891

0 119881

119891

1 119881

119891

2) to refer to119891) of119881 where119881

119894= V isin 119881 | 119891(V) = 119894

In this representation its weight 119891(119866) = |1198811| + 2|119881

2| It is

clear that119881119891

1cup119881

119891

2is a dominating set of 119866 called the Roman

dominating set denoted by 119863119891

R = (1198811 1198812) Since 119881119891

1cup 119881

119891

2is

a dominating set when 119891 is a Roman dominating functionand since placing weight 2 at the vertices of a dominating setyields a Roman dominating function in [115] it is observedthat

120574 (119866) ⩽ 120574RM (119866) ⩽ 2120574 (119866) (62)

A graph 119866 is called to be Roman if 120574RM (119866) = 2120574(119866)The definition of the Roman domination function is

proposed implicitly by Stewart [116] and ReVelle and Rosing[117] Roman dominating numbers have been deeply studiedIn particular Bahremandpour et al [118] showed that theproblem determining the Roman domination number is NP-complete even for bipartite graphs

Let 119866 be a graph with maximum degree at least twoThe Roman bondage number 119887RM (119866) of 119866 is the minimumcardinality of all sets 1198641015840 sube 119864 for which 120574RM (119866 minus 119864

1015840

) gt

120574RM (119866) Since in study of Roman bondage number theassumption Δ(119866) ⩾ 2 is necessary we always assume thatwhen we discuss 119887RM (119866) all graphs involved satisfy Δ(119866) ⩾2 The Roman bondage number 119887RM (119866) is introduced byJafari Rad and Volkmann in [119]

Recently Bahremandpour et al [118] have shown that theproblem determining the Roman bondage number is NP-hard even for bipartite graphs

Problem 7 Consider the decision problem

Roman Bondage Problem

Instance a nonempty bipartite graph119866 and a positiveinteger 119896

Question is 119887RM (119866) ⩽ 119896

Theorem 152 (Bahremandpour et al [118] 2013) TheRomanbondage problem is NP-hard even for bipartite graphs

The exact value of 119887RM(119866) is known only for a few familyof graphs including the complete graphs cycles and paths

International Journal of Combinatorics 25

Theorem 153 (Jafari Rad and Volkmann [119] 2011) For anypositive integers 119899 and119898 with119898 ⩽ 119899

119887RM (119875119899) = 2 119894119891 119899 equiv 2 (mod 3) 1 119900119905ℎ119890119903119908119894119904119890

119887RM (119862119899) =

3 119894119891 119899 = 2 (mod 3) 2 119900119905ℎ119890119903119908119894119904119890

119887RM (119870119899) = lceil

119899

2rceil 119891119900119903 119899 ⩾ 3

119887RM (119870119898119899) =

1 119894119891 119898 = 1 119886119899119889 119899 = 1

4 119894119891 119898 = 119899 = 3

119898 119900119905ℎ119890119903119908119894119904119890

(63)

Lemma 154 (Cockayne et al [115] 2004) If 119866 is a graph oforder 119899 and contains vertices of degree 119899 minus 1 then 120574RM(119866) = 2

Using Lemma 154 the third conclusion in Theorem 153can be generalized to more general case which is similar toLemma 49

Theorem 155 (Jafari Rad and Volkmann [119] 2011) Let119866 bea graph with order 119899 ge 3 and 119905 the number of vertices of degree119899 minus 1 in 119866 If 119905 ⩾ 1 then 119887RM(119866) = lceil1199052rceil

Ebadi and PushpaLatha [120] conjectured that 119887RM(119866) ⩽119899 minus 1 for any graph 119866 of order 119899 ⩾ 3 Dehgardi et al[121] Akbari andQajar [122] independently showed that thisconjecture is true

Theorem 156 119887RM(119866) ⩽ 119899 minus 1 for any connected graph 119866 oforder 119899 ⩾ 3

For a complete 119905-partite graph we have the followingresult

Theorem 157 (Hu and Xu [123] 2011) Let 119866 = 11987011989811198982119898119905

bea complete 119905-partite graph with 119898

1= sdot sdot sdot = 119898

119894lt 119898

119894+1le sdot sdot sdot ⩽

119898119905 119905 ⩾ 2 and 119899 = sum119905

119895=1119898

119895 Then

119887RM (119866) =

lceil119894

2rceil 119894119891 119898

119894= 1 119886119899119889 119899 ⩾ 3

2 119894119891 119898119894= 2 119886119899119889 119894 = 1

119894 119894119891 119898119894= 2 119886119899119889 119894 ⩾ 2

4 119894119891 119898119894= 3 119886119899119889 119894 = 119905 = 2

119899 minus 1 119894119891 119898119894= 3 119886119899119889 119894 = 119905 ⩾ 3

119899 minus 119898119905

119894119891 119898119894⩾ 3 119886119899119889 119898

119905⩾ 4

(64)

Consider a complete 119905-partite graph 119870119905(3) for 119905 ⩾ 3

119887RM(119870119905(3)) = 119899 minus 1 by Theorem 157 where 119899 = 3119905

which shows that this upper bound of 119899 minus 1 on 119887RM(119866) inTheorem 156 is sharp This fact and 120574

119877(119870

119905(3)) = 4 give

a negative answer to the following two problems posed byDehgardi et al [121]Prove or Disprove for any connected graph 119866 of order 119899 ge 3(i) 119887RM(119866) = 119899 minus 1 if and only if 119866 cong 119870

3 (ii) 119887RM(119866) ⩽ 119899 minus

120574119877(119866) + 1Note that a complete 119905-partite graph 119870

119905(3) is (119899 minus 3)-

regular and 119887RM(119870119905(3)) = 119899 minus 1 for 119905 ⩾ 3 where 119899 = 3119905

Hu and Xu further determined that 119887119877(119866) = 119899 minus 2 for any

(119899 minus 3)-regular graph 119866 of order 119899 ⩽ 5 except 119870119905(3)

Theorem 158 (Hu and Xu [123] 2011) For any (119899minus3)-regulargraph 119866 of order 119899 other than119870

119905(3) 119887RM(119866) = 119899 minus 2 for 119899 ⩾ 5

For a tree 119879 with order 119899 ⩾ 3 Ebadi and PushpaLatha[120] and Jafari Rad and Volkmann [119] independentlyobtained an upper bound on 119887RM(119879)

Theorem 159 119887RM(119879) ⩽ 3 for any tree 119879 with order 119899 ⩾ 3

Theorem 160 (Bahremandpour et al [118] 2013) 119887RM(1198752 times119875119899) = 2 for 119899 ⩾ 2

Theorem 161 (Jafari Rad and Volkmann [119] 2011) Let119866 bea graph with order at least three

(a) If (119909 119910 119911) is a path of length 2 in 119866 then 119887RM(119866) ⩽119889119866(119909) + 119889

119866(119910) + 119889

119866(119911) minus 3 minus |119873

119866(119909) cap 119873

119866(119910)|

(b) If 119866 is connected then 119887RM(119866) ⩽ 120582(119866) + 2Δ(119866) minus 3

Theorem 161 (b) implies 119887RM(119866) ⩽ 120575(119866) + 2Δ(119866) minus 3 for aconnected graph 119866 Note that for a planar graph 119866 120575(119866) ⩽ 5moreover 120575(119866) ⩽ 3 if the girth at least 4 and 120575(119866) ⩽ 2 if thegirth at least 6 These two facts show that 119887RM(119866) ⩽ 2Δ(119866) +2 for a connected planar graph 119866 Jafari Rad and Volkmann[124] improved this bound

Theorem 162 (Jafari Rad and Volkmann [124] 2011) For anyconnected planar graph 119866 of order 119899 ⩾ 3 with girth 119892(119866)

119887RM (119866)

⩽

2Δ (119866)

Δ (119866) + 6

Δ (119866) + 5 119894119891 119866 119888119900119899119905119886119894119899119904 119899119900 V119890119903119905119894119888119890119904 119900119891 119889119890119892119903119890119890 119891119894V119890

Δ (119866) + 4 119894119891 119892 (119866) ⩾ 4

Δ (119866) + 3 119894119891 119892 (119866) ⩾ 5

Δ (119866) + 2 119894119891 119892 (119866) ⩾ 6

Δ (119866) + 1 119894119891 119892 (119866) ⩾ 8

(65)

According toTheorem 157 119887RM(119862119899) = 3 = Δ(119862

119899)+1 for a

cycle 119862119899of length 119899 ⩾ 8 with 119899 equiv 2 (mod 3) and therefore

the last upper bound inTheorem 162 is sharp at least for Δ =2

Combining the fact that every planar graph 119866 withminimum degree 5 contains an edge 119909119910 with 119889

119866(119909) = 5

and 119889119866(119910) isin 5 6 with Theorem 161 (a) Akbari et al [125]

obtained the following result

26 International Journal of Combinatorics

middot middot middot

middot middot middot

middot middot middot

middot middot middot

119866

Figure 9 The graph 119866 is constructed from 119866

Theorem 163 (Akbari et al [125] 2012) 119887RM(119866) ⩽ 15 forevery planar graph 119866

It remains open to show whether the bound inTheorem 163 is sharp or not Though finding a planargraph 119866 with 119887RM(119866) = 15 seems to be difficult Akbari etal [125] constructed an infinite family of planar graphs withRoman bondage number equal to 7 by proving the followingresult

Theorem 164 (Akbari et al [125] 2012) Let 119866 be a graph oforder 119899 and 119866 is the graph of order 5119899 obtained from 119866 byattaching the central vertex of a copy of a path 119875

5to each vertex

of119866 (see Figure 9) Then 120574RM(119866) = 4119899 and 119887RM(119866) = 120575(119866)+2

ByTheorem 164 infinitemany planar graphswith Romanbondage number 7 can be obtained by considering any planargraph 119866 with 120575(119866) = 5 (eg the icosahedron graph)

Conjecture 165 (Akbari et al [125] 2012) The Romanbondage number of every planar graph is at most 7

For general bounds the following observation is directlyobtained which is similar to Observation 1

Observation 4 Let 119866 be a graph of order 119899 with maximumdegree at least two Assume that 119867 is a spanning subgraphobtained from 119866 by removing 119896 edges If 120574RM(119867) = 120574RM(119866)then 119887

119877(119867) ⩽ 119887RM(119866) ⩽ 119887RM(119867) + 119896

Theorem 166 (Bahremandpour et al [118] 2013) For anyconnected graph 119866 of order 119899 if 119899 ⩾ 3 and 120574RM(119866) = 120574(119866) + 1then

119887RM (119866) ⩽ min 119887 (119866) 119899Δ (66)

where 119899Δis the number of vertices with maximum degree Δ in

119866

Observation 5 For any graph 119866 if 120574(119866) = 120574RM(119866) then119887RM(119866) ⩽ 119887(119866)

Theorem 167 (Bahremandpour et al [118] 2013) For everyRoman graph 119866

119887RM (119866) ⩾ 119887 (119866) (67)

The bound is sharp for cycles on 119899 vertices where 119899 equiv 0 (mod3)

The strict inequality in Theorem 167 can hold for exam-ple 119887(119862

3119896+2) = 2 lt 3 = 119887RM(1198623119896+2

) byTheorem 157A graph 119866 is called to be vertex Roman domination-

critical if 120574RM(119866 minus 119909) lt 120574RM(119866) for every vertex 119909 in 119866 If119866 has no isolated vertices then 120574RM(119866) ⩽ 2120574(119866) ⩽ 2120573(119866)If 120574RM(119866) = 2120573(119866) then 120574RM(119866) = 2120574(119866) and hence 119866 is aRoman graph In [30] Volkmann gave a lot of graphs with120574(119866) = 120573(119866) The following result is similar to Theorem 57

Theorem 168 (Bahremandpour et al [118] 2013) For anygraph 119866 if 120574RM(119866) = 2120573(119866) then

(a) 119887RM(119866) ⩾ 120575(119866)

(b) 119887RM(119866) ⩾ 120575(119866)+1 if119866 is a vertex Roman domination-critical graph

If 120574RM(119866) = 2 then obviously 119887RM(119866) ⩽ 120575(119866) So weassume 120574RM(119866) ⩾ 3 Dehgardi et al [121] proved 119887RM(119866) ⩽(120574RM(119866) minus 2)Δ(119866) + 1 and posed the following problem if 119866is a connected graph of order 119899 ⩾ 4 with 120574RM(119866) ⩾ 3 then

119887RM (119866) ⩽ (120574RM (119866) minus 2) Δ (119866) (68)

Theorem 161 (a) shows that the inequality (68) holds if120574RM(119866) ⩾ 5 Thus the bound in (68) is of interest only when120574RM(119866) is 3 or 4

Lemma 169 (Bahremandpour et al [118] 2013) For anynonempty graph 119866 of order 119899 ⩾ 3 120574RM(119866) = 3 if and onlyif Δ(119866) = 119899 minus 2

The following result shows that (68) holds for all graphs119866 of order 119899 ⩾ 4 with 120574RM(119866) = 3 or 4 which improvesTheorem 161 (b)

Theorem 170 (Bahremandpour et al [118] 2013) For anyconnected graph 119866 of order 119899 ⩾ 4

119887RM (119866) le Δ (119866) = 119899 minus 2 119894119891 120574RM (119866) = 3

Δ (119866) + 120575 (119866) minus 1 119894119891 120574RM (119866) = 4(69)

with the first equality if and only if 119866 cong 1198624

Recently Akbari and Qajar [122] have proved the follow-ing result

Theorem 171 (Akbari and Qajar [122]) For any connectedgraph 119866 of order 119899 ⩾ 3

119887RM (119866) ⩽ 119899 minus 120574RM (119866) + 5 (70)

International Journal of Combinatorics 27

We conclude this section with the following problems

Problem 8 Characterize all connected graphs 119866 of order 119899 ⩾3 for which 119887RM(119866) = 119899 minus 1

Problem 9 Prove or disprove if 119866 is a connected graph oforder 119899 ⩾ 3 then

119887RM (119866) ⩽ 119899 minus 120574RM (119866) + 3 (71)

Note that 120574119877(119870

119905(3)) = 4 and 119887RM(119870119905

(3)) = 119899minus1 where 119899 =3119905 for 119905 ⩾ 3 The upper bound on 119887RM(119866) given in Problem 9is sharp if Problem 9 has positive answer

95 Rainbow Bondage Numbers For a positive integer 119896 a119896-rainbow dominating function of a graph 119866 is a function 119891from the vertex-set 119881(119866) to the set of all subsets of the set1 2 119896 such that for any vertex 119909 in 119866 with 119891(119909) = 0 thecondition

⋃

119910isin119873119866(119909)

119891 (119910) = 1 2 119896 (72)

is fulfilledThe weight of a 119896-rainbow dominating function 119891on 119866 is the value

119891 (119866) = sum

119909isin119881(119866)

1003816100381610038161003816119891 (119909)1003816100381610038161003816 (73)

The 119896-rainbow domination number of a graph 119866 denoted by120574119903119896(119866) is the minimum weight of a 119896-rainbow dominating

function 119891 of 119866Note that 120574

1199031(119866) is the classical domination number 120574(119866)

The 119896-rainbow domination number is introduced by Bresaret al [126] Rainbow domination of a graph 119866 coincides withordinary domination of the Cartesian product of 119866 with thecomplete graph in particular 120574

119903119896(119866) = 120574(119866 times 119870

119896) for any

positive integer 119896 and any graph 119866 [126] Hartnell and Rall[127] proved that 120574(119879) lt 120574

1199032(119879) for any tree 119879 no graph has

120574 = 120574119903119896for 119896 ⩾ 3 for any 119896 ⩾ 2 and any graph 119866 of order 119899

min 119899 120574 (119866) + 119896 minus 2 ⩽ 120574119903119896(119866) ⩽ 119896120574 (119866) (74)

The introduction of rainbow domination is motivatedby the study of paired domination in Cartesian productsof graphs where certain upper bounds can be expressed interms of rainbow domination that is for any graph 119866 andany graph 119867 if 119881(119867) can be partitioned into 119896120574

119875-sets then

120574119875(119866 times 119867) ⩽ (1119896)120592(119867)120574

119903119896(119866) proved by Bresar et al [126]

Dehgardi et al [128] introduced the concept of 119896-rainbowbondage number of graphs Let 119866 be a graph with maximumdegree at least two The 119896-rainbow bondage number 119887

119903119896(119866)

of 119866 is the minimum cardinality of all sets 119861 sube 119864(119866) forwhich 120574

119903119896(119866 minus 119861) gt 120574

119903119896(119866) Since in the study of 119896-rainbow

bondage number the assumption Δ(119866) ⩾ 2 is necessarywe always assume that when we discuss 119887

119903119896(119866) all graphs

involved satisfy Δ(119866) ⩾ 2The 2-rainbow bondage number of some special families

of graphs has been determined

Theorem 172 (Dehgardi et al [128]) For any integer 119899 ⩾ 3

1198871199032(119875

119899) = 1 119891119900119903 119899 ⩾ 2

1198871199032(119862

119899) =

1 119894119891 119899 equiv 0 (mod 4) 2 119900119905ℎ119890119903119908119894119904119890

1198871199032(119870

119899119899) =

1 119894119891 119899 = 2

3 119894119891 119899 = 3

6 119894119891 119899 = 4

119898 119894119891 119899 ⩾ 5

(75)

Theorem 173 (Dehgardi et al [128]) For any connected graph119866 of order 119899 ⩾ 3 119887

1199032(119866) ⩽ 120582(119866) + 2Δ(119866) minus 3

Theorem 174 (Dehgardi et al [128]) For any tree 119879 of order119899 ⩾ 3 119887

1199032(119879) ⩽ 2

Theorem 175 (Dehgardi et al [128]) If 119866 is a planar graphwithmaximumdegree at least two then 119887

1199032(119866) ⩽ 15Moreover

if the girth 119892(119866) ⩾ 4 then 1198871199032(119866) ⩽ 11 and if 119892(119866) ⩾ 6 then

1198871199032(119866) ⩽ 9

Theorem 176 (Dehgardi et al [128]) For a complete bipartitegraph 119870

119901119902 if 2119896 + 1 ⩽ 119901 ⩽ 119902 then 119887

119903119896(119870

119901119902) = 119901

Theorem177 (Dehgardi et al [128]) For a complete graph119870119899

if 119899 ⩾ 119896 + 1 then 119887119903119896(119870

119899) = lceil119896119899(119896 + 1)rceil

The special case 119896 = 1 of Theorem 177 can be found inTheorem 1 The following is a simple observation similar toObservation 1

Observation 6 (Dehgardi et al [128]) Let 119866 be a graphwith maximum degree at least two 119867 a spanning subgraphobtained from 119866 by removing 119896 edges If 120574

119903119896(119867) = 120574

119903119896(119866)

then 119887119903119896(119867) ⩽ 119887

119903119896(119866) ⩽ 119887

119903119896(119867) + 119896

Theorem 178 (Dehgardi et al [128]) For every graph 119866 with120574119903119896(119866) = 119896120574(119866) 119887

119903119896(119866) ⩾ 119887(119866)

A graph 119866 is said to be vertex 119896-rainbow domination-critical if 120574

119903119896(119866 minus 119909) lt 120574

119903119896(119866) for every vertex 119909 in 119866 It is

clear that if 119866 has no isolated vertices then 120574119903119896(119866) ⩽ 119896120574(119866) ⩽

119896120573(119866) where 120573(119866) is the vertex-covering number of119866 Thusif 120574

119903119896(119866) = 119896120573(119866) then 120574

119903119896(119866) = 119896120574(119866)

Theorem 179 (Dehgardi et al [128]) For any graph 119866 if120574119903119896(119866) = 119896120573(119866) then

(a) 119887119903119896(119866) ⩾ 120575(119866)

(b) 119887119903119896(119866) ⩾ 120575(119866)+1 if119866 is vertex 119896-rainbow domination-

critical

As far as we know there are no more results on the 119896-rainbow bondage number of graphs in the literature

28 International Journal of Combinatorics

10 Results on Digraphs

Although domination has been extensively studied in undi-rected graphs it is natural to think of a dominating setas a one-way relationship between vertices of the graphIndeed among the earliest literatures on this subject vonNeumann and Morgenstern [129] used what is now calleddomination in digraphs to find solution (or kernels whichare independent dominating sets) for cooperative 119899-persongames Most likely the first formulation of domination byBerge [130] in 1958 was given in the context of digraphs andonly some years latter by Ore [131] in 1962 for undirectedgraphs Despite this history examination of domination andits variants in digraphs has been essentially overlooked (see[4] for an overview of the domination literature) Thus thereare few if any such results on domination for digraphs in theliterature

The bondage number and its related topics for undirectedgraph have become one of major areas both in theoreticaland applied researches However until recently Carlsonand Develin [42] Shan and Kang [132] and Huang andXu [133 134] studied the bondage number for digraphsindependently In this section we introduce their resultsfor general digraphs Results for some special digraphs suchas vertex-transitive digraphs are introduced in the nextsection

101 Upper Bounds for General Digraphs Let 119866 = (119881 119864)

be a digraph without loops and parallel edges A digraph 119866is called to be asymmetric if whenever (119909 119910) isin 119864(119866) then(119910 119909) notin 119864(119866) and to be symmetric if (119909 119910) isin 119864(119866) implies(119910 119909) isin 119864(119866) For a vertex 119909 of 119881(119866) the sets of out-neighbors and in-neighbors of 119909 are respectively defined as119873

+

119866(119909) = 119910 isin 119881(119866) (119909 119910) isin 119864(119866) and 119873minus

119866(119909) = 119909 isin

119881(119866) (119909 119910) isin 119864(119866) the out-degree and the in-degree of119909 are respectively defined as 119889+

119866(119909) = |119873

+

119866(119909)| and 119889minus

119866(119909) =

|119873+

119866(119909)| Denote themaximum and theminimum out-degree

(resp in-degree) of 119866 by Δ+(119866) and 120575+(119866) (resp Δminus(119866) and120575minus

(119866))The degree 119889119866(119909) of 119909 is defined as 119889+

119866(119909)+119889

minus

119866(119909) and

the maximum and the minimum degrees of119866 are denoted byΔ(119866) and 120575(119866) respectively that is Δ(119866) = max119889

119866(119909) 119909 isin

119881(119866) and 120575(119866) = min119889119866(119909) 119909 isin 119881(119866) Note that the

definitions here are different from the ones in the textbookon digraphs see for example Xu [1]

A subset 119878 of119881(119866) is called a dominating set if119881(119866) = 119878cup119873

+

119866(119878) where119873+

119866(119878) is the set of out-neighbors of 119878Then just

as for undirected graphs 120574(119866) is the minimum cardinalityof a dominating set and the bondage number 119887(119866) is thesmallest cardinality of a set 119861 of edges such that 120574(119866 minus 119861) gt120574(119866) if such a subset 119861 sube 119864(119866) exists Otherwise we put119887(119866) = infin

Some basic results for undirected graphs stated inSection 3 can be generalized to digraphs For exampleTheorem 18 is generalized by Carlson and Develin [42]Huang andXu [133] and Shan andKang [132] independentlyas follows

Theorem 180 Let 119866 be a digraph and (119909 119910) isin 119864(119866) Then119887(119866) ⩽ 119889

119866(119910) + 119889

minus

119866(119909) minus |119873

minus

119866(119909) cap 119873

minus

119866(119910)|

Corollary 181 For a digraph 119866 119887(119866) ⩽ 120575minus(119866) + Δ(119866)

Since 120575minus

(119866) ⩽ (12)Δ(119866) from Corollary 181Conjecture 53 is valid for digraphs

Corollary 182 For a digraph 119866 119887(119866) ⩽ (32)Δ(119866)

In the case of undirected graphs the bondage number of119866119899= 119870

119899times 119870

119899achieves this bound However it was shown

by Carlson and Develin in [42] that if we take the symmetricdigraph119866

119899 we haveΔ(119866

119899) = 4(119899minus1) 120574(119866

119899) = 119899 and 119887(119866

119899) ⩽

4(119899 minus 1) + 1 = Δ(119866119899) + 1 So this family of digraphs cannot

show that the bound in Corollary 182 is tightCorresponding to Conjecture 51 which is discredited

for undirected graphs and is valid for digraphs the sameconjecture for digraphs can be proposed as follows

Conjecture 183 (Carlson and Develin [42] 2006) 119887(119866) ⩽Δ(119866) + 1 for any digraph 119866

If Conjecture 183 is true then the following results showsthat this upper bound is tight since 119887(119870

119899times119870

119899) = Δ(119870

119899times119870

119899)+1

for a complete digraph 119870119899

In 2007 Shan and Kang [132] gave some tight upperbounds on the bondage numbers for some asymmetricdigraphs For example 119887(119879) ⩽ Δ(119879) for any asymmetricdirected tree 119879 119887(119866) ⩽ Δ(119866) for any asymmetric digraph 119866with order at least 4 and 120574(119866) ⩽ 2 For planar digraphs theyobtained the following results

Theorem 184 (Shan and Kang [132] 2007) Let 119866 be aasymmetric planar digraphThen 119887(119866) ⩽ Δ(119866)+2 and 119887(119866) ⩽Δ(119866) + 1 if Δ(119866) ⩾ 5 and 119889minus

119866(119909) ⩾ 3 for every vertex 119909 with

119889119866(119909) ⩾ 4

102 Results for Some Special Digraphs The exact values andbounds on 119887(119866) for some standard digraphs are determined

Theorem185 (Huang andXu [133] 2006) For a directed cycle119862119899and a directed path 119875

119899

119887 (119862119899) =

3 119894119891 119899 119894119904 119900119889119889

2 119894119891 119899 119894119904 119890V119890119899

119887 (119875119899) =

2 119894119891 119899 119894119904 119900119889119889

1 119894119891 119899 119894119904 119890V119890119899

(76)

For the de Bruijn digraph 119861(119889 119899) and the Kautz digraph119870(119889 119899)

119887 (119861 (119889 119899)) = 119889 119894119891 119899 119894119904 119900119889119889

119889 ⩽ 119887 (119861 (119889 119899)) ⩽ 2119889 119894119891 119899 119894119904 119890V119890119899

119887 (119870 (119889 119899)) = 119889 + 1

(77)

Like undirected graphs we can define the total domi-nation number and the total bondage number On the totalbondage numbers for some special digraphs the knownresults are as follows

International Journal of Combinatorics 29

Theorem 186 (Huang and Xu [134] 2007) For a directedcycle 119862

119899and a directed path 119875

119899 119887

119879(119875

119899) and 119887

119879(119862

119899) all do not

exist For a complete digraph 119870119899

119887119879(119870

119899) = infin 119894119891 119899 = 1 2

119887119879(119870

119899) = 3 119894119891 119899 = 3

119899 ⩽ 119887119879(119870

119899) ⩽ 2119899 minus 3 119894119891 119899 ⩾ 4

(78)

The extended de Bruijn digraph EB(119889 119899 1199021 119902

119901) and

the extended Kautz digraph EK(119889 119899 1199021 119902

119901) are intro-

duced by Shibata and Gonda [135] If 119901 = 1 then theyare the de Bruijn digraph B(119889 119899) and the Kautz digraphK(119889 119899) respectively Huang and Xu [134] determined theirtotal domination numbers In particular their total bondagenumbers for general cases are determined as follows

Theorem 187 (Huang andXu [134] 2007) If 119889 ⩾ 2 and 119902119894⩾ 2

for each 119894 = 1 2 119901 then

119887119879(EB(119889 119899 119902

1 119902

119901)) = 119889

119901

minus 1

119887119879(EK(119889 119899 119902

1 119902

119901)) = 119889

119901

(79)

In particular for the de Bruijn digraph B(119889 119899) and the Kautzdigraph K(119889 119899)

119887119879(B (119889 119899)) = 119889 minus 1 119887

119879(K (119889 119899)) = 119889 (80)

Zhang et al [136] determined the bondage number incomplete 119905-partite digraphs

Theorem 188 (Zhang et al [136] 2009) For a complete 119905-partite digraph 119870

11989911198992119899119905

where 1198991⩽ 119899

2⩽ sdot sdot sdot ⩽ 119899

119905

119887 (11987011989911198992119899119905

)

=

119898 119894119891 119899119898= 1 119899

119898+1⩾ 2 119891119900119903 119904119900119898119890

119898 (1 ⩽ 119898 lt 119905)

4119905 minus 3 119894119891 1198991= 119899

2= sdot sdot sdot = 119899

119905= 2

119905minus1

sum

119894=1

119899119894+ 2 (119905 minus 1) 119900119905ℎ119890119903119908119894119904119890

(81)

Since an undirected graph can be thought of as a sym-metric digraph any result for digraphs has an analogy forundirected graphs in general In view of this point studyingthe bondage number for digraphs is more significant thanfor undirected graphs Thus we should further study thebondage number of digraphs and try to generalize knownresults on the bondage number and related variants for undi-rected graphs to digraphs prove or disprove Conjecture 183In particular determine the exact values of 119887(B(119889 119899)) for aneven 119899 and 119887

119879(119870

119899) for 119899 ⩾ 4

11 Efficient Dominating Sets

A dominating set 119878 of a graph 119866 is called to be efficient if forevery vertex 119909 in119866 |119873

119866[119909]cap119878| = 1 if119866 is a undirected graph

or |119873minus

119866[119909] cap 119878| = 1 if 119866 is a directed graph The concept of an

efficient set was proposed by Bange et al [137] in 1988 In theliterature an efficient set is also called a perfect dominatingset (see Livingston and Stout [138] for an explanation)

From definition if 119878 is an efficient dominating set of agraph 119866 then 119878 is certainly an independent set and everyvertex not in 119878 is adjacent to exactly one vertex in 119878

It is also clear from definition that a dominating set 119878 isefficient if and only if N

119866[119878] = 119873

119866[119909] 119909 isin 119878 for the

undirected graph 119866 or N+

119866[119878] = 119873

+

119866[119909] 119909 isin 119878 for the

digraph119866 is a partition of119881(119866) where the induced subgraphby119873

119866[119909] or119873+

119866[119909] is a star or an out-star with the root 119909

The efficient domination has important applications inmany areas such as error-correcting codes and receivesmuch attention in the late years

The concept of efficient dominating sets is a measureof the efficiency of domination in graphs Unfortunately asshown in [137] not every graph has an efficient dominatingset andmoreover it is anNP-complete problem to determinewhether a given graph has an efficient dominating set Inaddition it was shown by Clark [139] in 1993 that for a widerange of 119901 almost every random undirected graph 119866 isin

G(120592 119901) has no efficient dominating sets This means thatundirected graphs possessing an efficient dominating set arerare However it is easy to show that every undirected graphhas an orientationwith an efficient dominating set (see Bangeet al [140])

In 1993 Barkauskas and Host [141] showed that deter-mining whether an arbitrary oriented graph has an efficientdominating set is NP-complete Even so the existence ofefficient dominating sets for some special classes of graphs hasbeen examined see for example Dejter and Serra [142] andLee [143] for Cayley graph Gu et al [144] formeshes and toriObradovic et al [145] Huang and Xu [36] Reji Kumar andMacGillivray [146] for circulant graphs Harary graphs andtori and Van Wieren et al [147] for cube-connected cycles

In this section we introduce results of the bondagenumber for some graphs with efficient dominating sets dueto Huang and Xu [36]

111 Results for General Graphs In this subsection we intro-duce some results on bondage numbers obtained by applyingefficient dominating sets We first state the two followinglemmas

Lemma 189 For any 119896-regular graph or digraph 119866 of order 119899120574(119866) ⩾ 119899(119896 + 1) with equality if and only if 119866 has an efficientdominating set In addition if 119866 has an efficient dominatingset then every efficient dominating set is certainly a 120574-set andvice versa

Let 119890 be an edge and 119878 a dominating set in 119866 We say that119890 supports 119878 if 119890 isin (119878 119878) where (119878 119878) = (119909 119910) isin 119864(119866) 119909 isin 119878 119910 isin 119878 Denote by 119904(119866) the minimum number of edgeswhich support all 120574-sets in 119866

Lemma 190 For any graph or digraph 119866 119887(119866) ⩾ 119904(119866) withequality if 119866 is regular and has an efficient dominating set

30 International Journal of Combinatorics

A graph 119866 is called to be vertex-transitive if its automor-phism group Aut(119866) acts transitively on its vertex-set 119881(119866)A vertex-transitive graph is regular Applying Lemmas 189and 190 Huang and Xu obtained some results on bondagenumbers for vertex-transitive graphs or digraphs

Theorem 191 (Huang and Xu [36] 2008) For any vertex-transitive graph or digraph 119866 of order 119899

119887 (119866) ⩾

lceil119899

2120574 (119866)rceil 119894119891 119866 119894119904 119906119899119889119894119903119890119888119905119890119889

lceil119899

120574 (119866)rceil 119894119891 119866 119894119904 119889119894119903119890119888119905119890119889

(82)

Theorem192 (Huang andXu [36] 2008) Let119866 be a 119896-regulargraph of order 119899 Then

119887(119866) ⩽ 119896 if119866 is undirected and 119899 ⩾ 120574(119866)(119896+1)minus119896+1119887(119866) ⩽ 119896+1+ℓ if119866 is directed and 119899 ⩾ 120574(119866)(119896+1)minusℓwith 0 ⩽ ℓ ⩽ 119896 minus 1

Next we introduce a better upper bound on 119887(119866) To thisaim we need the following concept which is a generalizationof the concept of the edge-covering of a graph 119866 For 1198811015840

sube

119881(119866) and 1198641015840 sube 119864(119866) we say that 1198641015840covers 1198811015840 and call 1198641015840an edge-covering for 1198811015840 if there exists an edge (119909 119910) isin 1198641015840 forany vertex 119909 isin 1198811015840 For 119910 isin 119881(119866) let 1205731015840[119910] be the minimumcardinality over all edge-coverings for119873minus

119866[119910]

Theorem 193 (Huang and Xu [36] 2008) If 119866 is a 119896-regulargraph with order 120574(119866)(119896 + 1) then 119887(119866) ⩽ 1205731015840[119910] for any 119910 isin119881(119866)

Theupper bound on 119887(119866) given inTheorem 193 is tight inview of 119887(119862

119899) for a cycle or a directed cycle 119862

119899(seeTheorems

1 and 185 resp)It is easy to see that for a 119896-regular graph 119866 lceil(119896 + 1)2rceil ⩽

1205731015840

[119910] ⩽ 119896 when 119866 is undirected and 1205731015840[119910] = 119896 + 1 when 119866 isdirected By this fact and Lemma 189 the following theoremis merely a simple combination of Theorems 191 and 193

Theorem 194 (Huang and Xu [36] 2008) Let 119866 be a vertex-transitive graph of degree 119896 If 119866 has an efficient dominatingset then

lceil119896 + 1

2rceil ⩽ 119887 (119866) ⩽ 119896 119894119891 119866 119894119904 119906119899119889119894119903119890119888119905119890119889

119887 (119866) = 119896 + 1 119894119891 119866 119894119904 119889119894119903119890119888119905119890119889

(83)

Theorem 195 (Huang and Xu [36] 2008) If 119866 is an undi-rected vertex-transitive cubic graph with order 4120574(119866) and girth119892(119866) ⩽ 5 then 119887(119866) = 2

The proof of Theorem 195 leads to a byproduct If 119892 =5 let (119906

1 119906

2 119906

3 119906

4 119906

5) be a cycle and let D

119894be the family

of efficient dominating sets containing 119906119894for each 119894 isin

1 2 3 4 5 ThenD119894capD

119895= 0 for 119894 = 1 2 5 Then 119866 has

at least 5119904 efficient dominating sets where 119904 = 119904(119866) But thereare only 4s (=ns120574) distinct efficient dominating sets in119866This

contradiction implies that an undirected vertex-transitivecubic graph with girth five has no efficient dominating setsBut a similar argument for 119892(119866) = 3 4 or 119892(119866) ⩾ 6 couldnot give any contradiction This is consistent with the resultthat CCC(119899) a vertex-transitive cubic graph with girth 119899 if3 ⩽ 119899 ⩽ 8 or girth 8 if 119899 ⩾ 9 has efficient dominating setsfor all 119899 ⩾ 3 except 119899 = 5 (see Theorem 199 in the followingsubsection)

112 Results for Cayley Graphs In this subsection we useTheorem 194 to determine the exact values or approximativevalues of bondage numbers for some special vertex-transitivegraphs by characterizing the existence of efficient dominatingsets in these graphs

Let Γ be a nontrivial finite group 119878 a nonempty subset ofΓ without the identity element of Γ A digraph 119866 is defined asfollows

119881 (119866) = Γ (119909 119910) isin 119864 (119866) lArrrArr 119909minus1

119910 isin 119878

for any 119909 119910 isin Γ(84)

is called a Cayley digraph of the group Γ with respect to119878 denoted by 119862

Γ(119878) If 119878minus1 = 119904minus1 119904 isin 119878 = 119878 then

119862Γ(119878) is symmetric and is called a Cayley undirected graph

a Cayley graph for short Cayley graphs or digraphs arecertainly vertex-transitive (see Xu [1])

A circulant graph 119866(119899 119878) of order 119899 is a Cayley graph119862119885119899

(119878) where 119885119899= 0 119899 minus 1 is the addition group of

order 119899 and 119878 is a nonempty subset of119885119899without the identity

element and hence is a vertex-transitive digraph of degree|119878| If 119878minus1 = 119878 then119866(119899 119878) is an undirected graph If 119878 = 1 119896where 2 ⩽ 119896 ⩽ 119899 minus 2 we write 119866(119899 1 119896) for 119866(119899 1 119896) or119866(119899 plusmn1 plusmn119896) and call it a double-loop circulant graph

Huang and Xu [36] showed that for directed 119866 =

119866(119899 1 119896) lceil1198993rceil ⩽ 120574(119866) ⩽ lceil1198992rceil and 119866 has an efficientdominating set if and only if 3 | 119899 and 119896 equiv 2 (mod 3) fordirected 119866 = 119866(119899 1 119896) and 119896 = 1198992 lceil1198995rceil ⩽ 120574(119866) ⩽ lceil1198993rceiland 119866 has an efficient dominating set if and only if 5 | 119899 and119896 equiv plusmn2 (mod 5) By Theorem 194 we obtain the bondagenumber of a double loop circulant graph if it has an efficientdominating set

Theorem 196 (Huang and Xu [36] 2008) Let 119866 be a double-loop circulant graph 119866(119899 1 119896) If 119866 is directed with 3 | 119899and 119896 equiv 2 (mod 3) or 119866 is undirected with 5 | 119899 and119896 equiv plusmn2 (mod 5) then 119887(119866) = 3

The 119898 times 119899 torus is the Cartesian product 119862119898times 119862

119899of

two cycles and is a Cayley graph 119862119885119898times119885119899

(119878) where 119878 =

(0 1) (1 0) for directed cycles and 119878 = (0 plusmn1) (plusmn1 0) forundirected cycles and hence is vertex-transitive Gu et al[144] showed that the undirected torus119862

119898times119862

119899has an efficient

dominating set if and only if both 119898 and 119899 are multiplesof 5 Huang and Xu [36] showed that the directed torus119862119898times 119862

119899has an efficient dominating set if and only if both

119898 and 119899 are multiples of 3 Moreover they found a necessarycondition for a dominating set containing the vertex (0 0)in 119862

119898times 119862

119899to be efficient and obtained the following

result

International Journal of Combinatorics 31

Theorem 197 (Huang and Xu [36] 2008) Let 119866 = 119862119898times 119862

119899

If 119866 is undirected and both119898 and 119899 are multiples of 5 or if 119866 isdirected and both119898 and 119899 are multiples of 3 then 119887(119866) = 3

The hypercube 119876119899

is the Cayley graph 119862Γ(119878)

where Γ = 1198852times sdot sdot sdot times 119885

2= (119885

2)119899 and 119878 =

100 sdot sdot sdot 0 010 sdot sdot sdot 0 00 sdot sdot sdot 01 Lee [143] showed that119876119899has an efficient dominating set if and only if 119899 = 2119898 minus 1

for a positive integer119898 ByTheorem 194 the following resultis obtained immediately

Theorem 198 (Huang and Xu [36] 2008) If 119899 = 2119898 minus 1 for apositive integer119898 then 2119898minus1

⩽ 119887(119876119899) ⩽ 2

119898

minus 1

Problem 10 Determine the exact value of 119887(119876119899) for 119899 = 2119898minus1

The 119899-dimensional cube-connected cycle denoted byCCC(119899) is constructed from the 119899-dimensional hypercube119876119899by replacing each vertex 119909 in 119876

119899with an undirected cycle

119862119899of length 119899 and linking the 119894th vertex of the 119862

119899to the 119894th

neighbor of 119909 It has been proved that CCC(119899) is a Cayleygraph and hence is a vertex-transitive graph with degree 3Van Wieren et al [147] proved that CCC(119899) has an efficientdominating set if and only if 119899 = 5 From Theorems 194 and195 the following result can be derived

Theorem 199 (Huang and Xu [36] 2008) Let 119866 = 119862119862119862(119899)be the 119899-dimensional cube-connected cycles with 119899 ⩾ 3 and119899 = 5 Then 120574(119866) = 1198992119899minus2 and 2 ⩽ 119887(119866) ⩽ 3 In addition119887(119862119862119862(3)) = 119887(119862119862119862(4)) = 2

Problem 11 Determine the exact value of 119887(CCC(119899)) for 119899 ⩾5

Acknowledgments

This work was supported by NNSF of China (Grants nos11071233 61272008) The author would like to express hisgratitude to the anonymous referees for their kind sugges-tions and comments on the original paper to ProfessorSheikholeslami and Professor Jafari Rad for sending thereferences [128] and [81] which resulted in this version

References

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[2] S THedetniemi andR C Laskar ldquoBibliography on dominationin graphs and some basic definitions of domination parame-tersrdquo Discrete Mathematics vol 86 no 1ndash3 pp 257ndash277 1990

[3] TW Haynes S T Hedetniemi and P J Slater Fundamentals ofDomination in Graphs vol 208 ofMonographs and Textbooks inPure and Applied Mathematics Marcel Dekker New York NYUSA 1998

[4] TWHaynes S THedetniemi and P J Slater EdsDominationin Graphs vol 209 of Monographs and Textbooks in Pure andAppliedMathematicsMarcelDekker NewYorkNYUSA 1998

[5] M R Garey andD S JohnsonComputers and Intractability WH Freeman and Company San Francisco Calif USA 1979

[6] H B Walikar and B D Acharya ldquoDomination critical graphsrdquoNational Academy Science Letters vol 2 pp 70ndash72 1979

[7] D Bauer F Harary J Nieminen and C L Suffel ldquoDominationalteration sets in graphsrdquo Discrete Mathematics vol 47 no 2-3pp 153ndash161 1983

[8] J F Fink M S Jacobson L F Kinch and J Roberts ldquoThebondage number of a graphrdquo Discrete Mathematics vol 86 no1ndash3 pp 47ndash57 1990

[9] F-T Hu and J-M Xu ldquoThe bondage number of (119899 minus 3)-regulargraph G of order nrdquo Ars Combinatoria to appear

[10] U Teschner ldquoNew results about the bondage number of agraphrdquoDiscreteMathematics vol 171 no 1ndash3 pp 249ndash259 1997

[11] B L Hartnell and D F Rall ldquoA bound on the size of a graphwith given order and bondage numberrdquo Discrete Mathematicsvol 197-198 pp 409ndash413 1999 16th British CombinatorialConference (London 1997)

[12] B L Hartnell and D F Rall ldquoA characterization of trees inwhich no edge is essential to the domination numberrdquo ArsCombinatoria vol 33 pp 65ndash76 1992

[13] J Topp and P D Vestergaard ldquo120572119896- and 120574

119896-stable graphsrdquo

Discrete Mathematics vol 212 no 1-2 pp 149ndash160 2000[14] A Meir and J W Moon ldquoRelations between packing and

covering numbers of a treerdquo Pacific Journal of Mathematics vol61 no 1 pp 225ndash233 1975

[15] B L Hartnell L K Joslashrgensen P D Vestergaard and CA Whitehead ldquoEdge stability of the k-domination numberof treesrdquo Bulletin of the Institute of Combinatorics and ItsApplications vol 22 pp 31ndash40 1998

[16] F-T Hu and J-M Xu ldquoOn the complexity of the bondage andreinforcement problemsrdquo Journal of Complexity vol 28 no 2pp 192ndash201 2012

[17] B L Hartnell and D F Rall ldquoBounds on the bondage numberof a graphrdquo Discrete Mathematics vol 128 no 1ndash3 pp 173ndash1771994

[18] Y-L Wang ldquoOn the bondage number of a graphrdquo DiscreteMathematics vol 159 no 1ndash3 pp 291ndash294 1996

[19] G H Fricke TW Haynes S M Hedetniemi S T Hedetniemiand R C Laskar ldquoExcellent treesrdquo Bulletin of the Institute ofCombinatorics and Its Applications vol 34 pp 27ndash38 2002

[20] V Samodivkin ldquoThe bondage number of graphs good and badverticesrdquoDiscussiones Mathematicae GraphTheory vol 28 no3 pp 453ndash462 2008

[21] J E Dunbar T W Haynes U Teschner and L VolkmannldquoBondage insensitivity and reinforcementrdquo in Domination inGraphs vol 209 of Monographs and Textbooks in Pure andAppliedMathematics pp 471ndash489 Dekker NewYork NY USA1998

[22] H Liu and L Sun ldquoThe bondage and connectivity of a graphrdquoDiscrete Mathematics vol 263 no 1ndash3 pp 289ndash293 2003

[23] A-H Esfahanian and S L Hakimi ldquoOn computing a con-ditional edge-connectivity of a graphrdquo Information ProcessingLetters vol 27 no 4 pp 195ndash199 1988

[24] R C Brigham P Z Chinn and R D Dutton ldquoVertexdomination-critical graphsrdquoNetworks vol 18 no 3 pp 173ndash1791988

[25] V Samodivkin ldquoDomination with respect to nondegenerateproperties bondage numberrdquoThe Australasian Journal of Com-binatorics vol 45 pp 217ndash226 2009

[26] U Teschner ldquoA new upper bound for the bondage numberof graphs with small domination numberrdquo The AustralasianJournal of Combinatorics vol 12 pp 27ndash35 1995

32 International Journal of Combinatorics

[27] J Fulman D Hanson and G MacGillivray ldquoVertex dom-ination-critical graphsrdquoNetworks vol 25 no 2 pp 41ndash43 1995

[28] U Teschner ldquoA counterexample to a conjecture on the bondagenumber of a graphrdquo Discrete Mathematics vol 122 no 1ndash3 pp393ndash395 1993

[29] U Teschner ldquoThe bondage number of a graph G can be muchgreater than Δ(G)rdquo Ars Combinatoria vol 43 pp 81ndash87 1996

[30] L Volkmann ldquoOn graphs with equal domination and coveringnumbersrdquoDiscrete AppliedMathematics vol 51 no 1-2 pp 211ndash217 1994

[31] L A Sanchis ldquoMaximum number of edges in connected graphswith a given domination numberrdquoDiscreteMathematics vol 87no 1 pp 65ndash72 1991

[32] S Klavzar and N Seifter ldquoDominating Cartesian products ofcyclesrdquoDiscrete AppliedMathematics vol 59 no 2 pp 129ndash1361995

[33] H K Kim ldquoA note on Vizingrsquos conjecture about the bondagenumberrdquo Korean Journal of Mathematics vol 10 no 2 pp 39ndash42 2003

[34] M Y Sohn X Yuan and H S Jeong ldquoThe bondage number of1198623times119862

119899rdquo Journal of the KoreanMathematical Society vol 44 no

6 pp 1213ndash1231 2007[35] L Kang M Y Sohn and H K Kim ldquoBondage number of the

discrete torus 119862119899times 119862

4rdquo Discrete Mathematics vol 303 no 1ndash3

pp 80ndash86 2005[36] J Huang and J-M Xu ldquoThe bondage numbers and efficient

dominations of vertex-transitive graphsrdquoDiscrete Mathematicsvol 308 no 4 pp 571ndash582 2008

[37] C J Xiang X Yuan andM Y Sohn ldquoDomination and bondagenumber of119862

3times119862

119899rdquoArs Combinatoria vol 97 pp 299ndash310 2010

[38] M S Jacobson and L F Kinch ldquoOn the domination numberof products of graphs Irdquo Ars Combinatoria vol 18 pp 33ndash441984

[39] Y-Q Li ldquoA note on the bondage number of a graphrdquo ChineseQuarterly Journal of Mathematics vol 9 no 4 pp 1ndash3 1994

[40] F-T Hu and J-M Xu ldquoBondage number of mesh networksrdquoFrontiers ofMathematics inChina vol 7 no 5 pp 813ndash826 2012

[41] R Frucht and F Harary ldquoOn the corona of two graphsrdquoAequationes Mathematicae vol 4 pp 322ndash325 1970

[42] K Carlson and M Develin ldquoOn the bondage number of planarand directed graphsrdquoDiscreteMathematics vol 306 no 8-9 pp820ndash826 2006

[43] U Teschner ldquoOn the bondage number of block graphsrdquo ArsCombinatoria vol 46 pp 25ndash32 1997

[44] J Huang and J-M Xu ldquoNote on conjectures of bondagenumbers of planar graphsrdquo Applied Mathematical Sciences vol6 no 65ndash68 pp 3277ndash3287 2012

[45] L Kang and J Yuan ldquoBondage number of planar graphsrdquoDiscrete Mathematics vol 222 no 1ndash3 pp 191ndash198 2000

[46] M Fischermann D Rautenbach and L Volkmann ldquoRemarkson the bondage number of planar graphsrdquo Discrete Mathemat-ics vol 260 no 1ndash3 pp 57ndash67 2003

[47] J Hou G Liu and J Wu ldquoBondage number of planar graphswithout small cyclesrdquoUtilitasMathematica vol 84 pp 189ndash1992011

[48] J Huang and J-M Xu ldquoThe bondage numbers of graphs withsmall crossing numbersrdquo Discrete Mathematics vol 307 no 15pp 1881ndash1897 2007

[49] Q Ma S Zhang and J Wang ldquoBondage number of 1-planargraphrdquo Applied Mathematics vol 1 no 2 pp 101ndash103 2010

[50] J Hou and G Liu ldquoA bound on the bondage number of toroidalgraphsrdquoDiscreteMathematics Algorithms and Applications vol4 no 3 Article ID 1250046 5 pages 2012

[51] Y-C Cao J-M Xu and X-R Xu ldquoOn bondage number oftoroidal graphsrdquo Journal of University of Science and Technologyof China vol 39 no 3 pp 225ndash228 2009

[52] Y-C Cao J Huang and J-M Xu ldquoThe bondage number ofgraphs with crossing number less than fourrdquo Ars Combinatoriato appear

[53] H K Kim ldquoOn the bondage numbers of some graphsrdquo KoreanJournal of Mathematics vol 11 no 2 pp 11ndash15 2004

[54] A Gagarin and V Zverovich ldquoUpper bounds for the bondagenumber of graphs on topological surfacesrdquo Discrete Mathemat-ics 2012

[55] J Huang ldquoAn improved upper bound for the bondage numberof graphs on surfacesrdquoDiscrete Mathematics vol 312 no 18 pp2776ndash2781 2012

[56] A Gagarin and V Zverovich ldquoThe bondage number of graphson topological surfaces and Teschnerrsquos conjecturerdquo DiscreteMathematics vol 313 no 6 pp 796ndash808 2013

[57] V Samodivkin ldquoNote on the bondage number of graphs ontopological surfacesrdquo httparxivorgabs12086203

[58] E J Cockayne R M Dawes and S T Hedetniemi ldquoTotaldomination in graphsrdquoNetworks vol 10 no 3 pp 211ndash219 1980

[59] R Laskar J Pfaff S M Hedetniemi and S T HedetniemildquoOn the algorithmic complexity of total dominationrdquo Society forIndustrial and Applied Mathematics vol 5 no 3 pp 420ndash4251984

[60] J Pfaff R C Laskar and S T Hedetniemi ldquoNP-completeness oftotal and connected domination and irredundance for bipartitegraphsrdquo Tech Rep 428 Department of Mathematical SciencesClemson University 1983

[61] M A Henning ldquoA survey of selected recent results on totaldomination in graphsrdquoDiscrete Mathematics vol 309 no 1 pp32ndash63 2009

[62] V R Kulli and D K Patwari ldquoThe total bondage number ofa graphrdquo in Advances in Graph Theory pp 227ndash235 VishwaGulbarga India 1991

[63] F-T Hu Y Lu and J-M Xu ldquoThe total bondage number ofgrid graphsrdquo Discrete Applied Mathematics vol 160 no 16-17pp 2408ndash2418 2012

[64] J Cao M Shi and B Wu ldquoResearch on the total bondagenumber of a spectial networkrdquo in Proceedings of the 3rdInternational Joint Conference on Computational Sciences andOptimization (CSO rsquo10) pp 456ndash458 May 2010

[65] N Sridharan MD Elias and V S A Subramanian ldquoTotalbondage number of a graphrdquo AKCE International Journal ofGraphs and Combinatorics vol 4 no 2 pp 203ndash209 2007

[66] N Jafari Rad and J Raczek ldquoSome progress on total bondage ingraphsrdquo Graphs and Combinatorics 2011

[67] T W Haynes and P J Slater ldquoPaired-domination and thepaired-domatic numberrdquoCongressusNumerantium vol 109 pp65ndash72 1995

[68] T W Haynes and P J Slater ldquoPaired-domination in graphsrdquoNetworks vol 32 no 3 pp 199ndash206 1998

[69] X Hou ldquoA characterization of 2120574 120574119875-treesrdquoDiscrete Mathemat-

ics vol 308 no 15 pp 3420ndash3426 2008[70] K E Proffitt T W Haynes and P J Slater ldquoPaired-domination

in grid graphsrdquo Congressus Numerantium vol 150 pp 161ndash1722001

International Journal of Combinatorics 33

[71] H Qiao L KangM Cardei andD-Z Du ldquoPaired-dominationof treesrdquo Journal of Global Optimization vol 25 no 1 pp 43ndash542003

[72] E Shan L Kang and M A Henning ldquoA characterizationof trees with equal total domination and paired-dominationnumbersrdquo The Australasian Journal of Combinatorics vol 30pp 31ndash39 2004

[73] J Raczek ldquoPaired bondage in treesrdquo Discrete Mathematics vol308 no 23 pp 5570ndash5575 2008

[74] J A Telle and A Proskurowski ldquoAlgorithms for vertex parti-tioning problems on partial k-treesrdquo SIAM Journal on DiscreteMathematics vol 10 no 4 pp 529ndash550 1997

[75] G S Domke J H Hattingh M A Henning and L R MarkusldquoRestrained domination in treesrdquoDiscreteMathematics vol 211no 1ndash3 pp 1ndash9 2000

[76] G S Domke J H Hattingh M A Henning and L R MarkusldquoRestrained domination in graphs with minimum degree twordquoJournal of Combinatorial Mathematics and Combinatorial Com-puting vol 35 pp 239ndash254 2000

[77] G S Domke J H Hattingh S T Hedetniemi R C Laskarand L R Markus ldquoRestrained domination in graphsrdquo DiscreteMathematics vol 203 no 1ndash3 pp 61ndash69 1999

[78] J H Hattingh and E J Joubert ldquoAn upper bound for therestrained domination number of a graph with minimumdegree at least two in terms of order and minimum degreerdquoDiscrete Applied Mathematics vol 157 no 13 pp 2846ndash28582009

[79] M A Henning ldquoGraphs with large restrained dominationnumberrdquo Discrete Mathematics vol 197-198 pp 415ndash429 199916th British Combinatorial Conference (London 1997)

[80] J H Hattingh and A R Plummer ldquoRestrained bondage ingraphsrdquo Discrete Mathematics vol 308 no 23 pp 5446ndash54532008

[81] N Jafari Rad R Hasni J Raczek and L Volkmann ldquoTotalrestrained bondage in graphsrdquoActaMathematica Sinica vol 29no 6 pp 1033ndash1042 2013

[82] J Cyman and J Raczek ldquoOn the total restrained dominationnumber of a graphrdquoThe Australasian Journal of Combinatoricsvol 36 pp 91ndash100 2006

[83] M A Henning ldquoGraphs with large total domination numberrdquoJournal of Graph Theory vol 35 no 1 pp 21ndash45 2000

[84] D-X Ma X-G Chen and L Sun ldquoOn total restraineddomination in graphsrdquoCzechoslovakMathematical Journal vol55 no 1 pp 165ndash173 2005

[85] B Zelinka ldquoRemarks on restrained domination and totalrestrained domination in graphsrdquo Czechoslovak MathematicalJournal vol 55 no 2 pp 393ndash396 2005

[86] W Goddard and M A Henning ldquoIndependent domination ingraphs a survey and recent resultsrdquo Discrete Mathematics vol313 no 7 pp 839ndash854 2013

[87] J H Zhang H L Liu and L Sun ldquoIndependence bondagenumber and reinforcement number of some graphsrdquo Transac-tions of Beijing Institute of Technology vol 23 no 2 pp 140ndash1422003

[88] K D Dharmalingam Studies in Graph Theorymdashequitabledomination and bottleneck domination [PhD thesis] MaduraiKamaraj University Madurai India 2006

[89] GDeepakND Soner andAAlwardi ldquoThe equitable bondagenumber of a graphrdquo Research Journal of Pure Algebra vol 1 no9 pp 209ndash212 2011

[90] E Sampathkumar and H B Walikar ldquoThe connected domina-tion number of a graphrdquo Journal of Mathematical and PhysicalSciences vol 13 no 6 pp 607ndash613 1979

[91] K White M Farber and W Pulleyblank ldquoSteiner trees con-nected domination and strongly chordal graphsrdquoNetworks vol15 no 1 pp 109ndash124 1985

[92] M Cadei M X Cheng X Cheng and D-Z Du ldquoConnecteddomination in Ad Hoc wireless networksrdquo in Proceedings ofthe 6th International Conference on Computer Science andInformatics (CSampI rsquo02) 2002

[93] J F Fink and M S Jacobson ldquon-domination in graphsrdquo inGraph Theory with Applications to Algorithms and ComputerScience Wiley-Interscience Publications pp 283ndash300 WileyNew York NY USA 1985

[94] M Chellali O Favaron A Hansberg and L Volkmann ldquok-domination and k-independence in graphs a surveyrdquo Graphsand Combinatorics vol 28 no 1 pp 1ndash55 2012

[95] Y Lu X Hou J-M Xu andN Li ldquoTrees with uniqueminimump-dominating setsrdquo Utilitas Mathematica vol 86 pp 193ndash2052011

[96] Y Lu and J-M Xu ldquoThe p-bondage number of treesrdquo Graphsand Combinatorics vol 27 no 1 pp 129ndash141 2011

[97] Y Lu and J-M Xu ldquoThe 2-domination and 2-bondage numbersof grid graphs A manuscriptrdquo httparxivorgabs12044514

[98] M Krzywkowski ldquoNon-isolating 2-bondage in graphsrdquo TheJournal of the Mathematical Society of Japan vol 64 no 4 2012

[99] M A Henning O R Oellermann and H C Swart ldquoBoundson distance domination parametersrdquo Journal of CombinatoricsInformation amp System Sciences vol 16 no 1 pp 11ndash18 1991

[100] F Tian and J-M Xu ldquoBounds for distance domination numbersof graphsrdquo Journal of University of Science and Technology ofChina vol 34 no 5 pp 529ndash534 2004

[101] F Tian and J-M Xu ldquoDistance domination-critical graphsrdquoApplied Mathematics Letters vol 21 no 4 pp 416ndash420 2008

[102] F Tian and J-M Xu ldquoA note on distance domination numbersof graphsrdquo The Australasian Journal of Combinatorics vol 43pp 181ndash190 2009

[103] F Tian and J-M Xu ldquoAverage distances and distance domina-tion numbersrdquo Discrete Applied Mathematics vol 157 no 5 pp1113ndash1127 2009

[104] Z-L Liu F Tian and J-M Xu ldquoProbabilistic analysis of upperbounds for 2-connected distance k-dominating sets in graphsrdquoTheoretical Computer Science vol 410 no 38ndash40 pp 3804ndash3813 2009

[105] M A Henning O R Oellermann and H C Swart ldquoDistancedomination critical graphsrdquo Journal of Combinatorial Mathe-matics and Combinatorial Computing vol 44 pp 33ndash45 2003

[106] T J Bean M A Henning and H C Swart ldquoOn the integrityof distance domination in graphsrdquo The Australasian Journal ofCombinatorics vol 10 pp 29ndash43 1994

[107] M Fischermann and L Volkmann ldquoA remark on a conjecturefor the (kp)-domination numberrdquoUtilitasMathematica vol 67pp 223ndash227 2005

[108] T Korneffel D Meierling and L Volkmann ldquoA remark on the(22)-domination numberrdquo Discussiones Mathematicae GraphTheory vol 28 no 2 pp 361ndash366 2008

[109] Y Lu X Hou and J-M Xu ldquoOn the (22)-domination numberof treesrdquo Discussiones Mathematicae Graph Theory vol 30 no2 pp 185ndash199 2010

34 International Journal of Combinatorics

[110] H Li and J-M Xu ldquo(dm)-domination numbers of m-connected graphsrdquo Rapports de Recherche 1130 LRI UniversiteParis-Sud 1997

[111] S M Hedetniemi S T Hedetniemi and T V Wimer ldquoLineartime resource allocation algorithms for treesrdquo Tech Rep URI-014 Department of Mathematical Sciences Clemson Univer-sity 1987

[112] M Farber ldquoDomination independent domination and dualityin strongly chordal graphsrdquo Discrete Applied Mathematics vol7 no 2 pp 115ndash130 1984

[113] G S Domke and R C Laskar ldquoThe bondage and reinforcementnumbers of 120574

119891for some graphsrdquoDiscrete Mathematics vol 167-

168 pp 249ndash259 1997[114] G S Domke S T Hedetniemi and R C Laskar ldquoFractional

packings coverings and irredundance in graphsrdquo vol 66 pp227ndash238 1988

[115] E J Cockayne P A Dreyer Jr S M Hedetniemi andS T Hedetniemi ldquoRoman domination in graphsrdquo DiscreteMathematics vol 278 no 1ndash3 pp 11ndash22 2004

[116] I Stewart ldquoDefend the roman empirerdquo Scientific American vol281 pp 136ndash139 1999

[117] C S ReVelle and K E Rosing ldquoDefendens imperiumromanum a classical problem in military strategyrdquoThe Ameri-can Mathematical Monthly vol 107 no 7 pp 585ndash594 2000

[118] A Bahremandpour F-T Hu S M Sheikholeslami and J-MXu ldquoOn the Roman bondage number of a graphrdquo DiscreteMathematics Algorithms and Applications vol 5 no 1 ArticleID 1350001 15 pages 2013

[119] N Jafari Rad and L Volkmann ldquoRoman bondage in graphsrdquoDiscussiones Mathematicae Graph Theory vol 31 no 4 pp763ndash773 2011

[120] K Ebadi and L PushpaLatha ldquoRoman bondage and Romanreinforcement numbers of a graphrdquo International Journal ofContemporary Mathematical Sciences vol 5 pp 1487ndash14972010

[121] N Dehgardi S M Sheikholeslami and L Volkman ldquoOn theRoman k-bondage number of a graphrdquo AKCE InternationalJournal of Graphs and Combinatorics vol 8 pp 169ndash180 2011

[122] S Akbari and S Qajar ldquoA note on Roman bondage number ofgraphsrdquo Ars Combinatoria to Appear

[123] F-T Hu and J-M Xu ldquoRoman bondage numbers of somegraphsrdquo httparxivorgabs11093933

[124] N Jafari Rad and L Volkmann ldquoOn the Roman bondagenumber of planar graphsrdquo Graphs and Combinatorics vol 27no 4 pp 531ndash538 2011

[125] S Akbari M Khatirinejad and S Qajar ldquoA note on the romanbondage number of planar graphsrdquo Graphs and Combinatorics2012

[126] B Bresar M A Henning and D F Rall ldquoRainbow dominationin graphsrdquo Taiwanese Journal of Mathematics vol 12 no 1 pp213ndash225 2008

[127] B L Hartnell and D F Rall ldquoOn dominating the Cartesianproduct of a graph and 120572krdquo Discussiones Mathematicae GraphTheory vol 24 no 3 pp 389ndash402 2004

[128] N Dehgardi S M Sheikholeslami and L Volkman ldquoThe k-rainbow bondage number of a graphrdquo to appear in DiscreteApplied Mathematics

[129] J von Neumann and O Morgenstern Theory of Games andEconomic Behavior Princeton University Press Princeton NJUSA 1944

[130] C BergeTheorıe des Graphes et ses Applications 1958[131] O Ore Theory of Graphs vol 38 American Mathematical

Society Colloquium Publications 1962[132] E Shan and L Kang ldquoBondage number in oriented graphsrdquoArs

Combinatoria vol 84 pp 319ndash331 2007[133] J Huang and J-M Xu ldquoThe bondage numbers of extended

de Bruijn and Kautz digraphsrdquo Computers amp Mathematics withApplications vol 51 no 6-7 pp 1137ndash1147 2006

[134] J Huang and J-M Xu ldquoThe total domination and total bondagenumbers of extended de Bruijn and Kautz digraphsrdquoComputersamp Mathematics with Applications vol 53 no 8 pp 1206ndash12132007

[135] Y Shibata and Y Gonda ldquoExtension of de Bruijn graph andKautz graphrdquo Computers amp Mathematics with Applications vol30 no 9 pp 51ndash61 1995

[136] X Zhang J Liu and JMeng ldquoThebondage number in completet-partite digraphsrdquo Information Processing Letters vol 109 no17 pp 997ndash1000 2009

[137] D W Bange A E Barkauskas and P J Slater ldquoEfficient dom-inating sets in graphsrdquo in Applications of Discrete Mathematicspp 189ndash199 SIAM Philadelphia Pa USA 1988

[138] M Livingston and Q F Stout ldquoPerfect dominating setsrdquoCongressus Numerantium vol 79 pp 187ndash203 1990

[139] L Clark ldquoPerfect domination in random graphsrdquo Journal ofCombinatorial Mathematics and Combinatorial Computing vol14 pp 173ndash182 1993

[140] D W Bange A E Barkauskas L H Host and L H ClarkldquoEfficient domination of the orientations of a graphrdquo DiscreteMathematics vol 178 no 1ndash3 pp 1ndash14 1998

[141] A E Barkauskas and L H Host ldquoFinding efficient dominatingsets in oriented graphsrdquo Congressus Numerantium vol 98 pp27ndash32 1993

[142] I J Dejter and O Serra ldquoEfficient dominating sets in CayleygraphsrdquoDiscrete AppliedMathematics vol 129 no 2-3 pp 319ndash328 2003

[143] J Lee ldquoIndependent perfect domination sets in Cayley graphsrdquoJournal of Graph Theory vol 37 no 4 pp 213ndash219 2001

[144] WGu X Jia and J Shen ldquoIndependent perfect domination setsin meshes tori and treesrdquo Unpublished

[145] N Obradovic J Peters and G Ruzic ldquoEfficient domination incirculant graphswith two chord lengthsrdquo Information ProcessingLetters vol 102 no 6 pp 253ndash258 2007

[146] K Reji Kumar and G MacGillivray ldquoEfficient domination incirculant graphsrdquo Discrete Mathematics vol 313 no 6 pp 767ndash771 2013

[147] D Van Wieren M Livingston and Q F Stout ldquoPerfectdominating sets on cube-connected cyclesrdquo Congressus Numer-antium vol 97 pp 51ndash70 1993

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 International Journal of Combinatorics

Theorem 8 (Teschner [10] 1997) A tree 119879 has bondagenumber 1 if and only if 119879 has a universal vertex or an edge119909119910 satisfying

(1) 119909 and 119910 are neither universal nor idle and

(2) all neighbors of 119909 and 119910 (except for 119909 and 119910) are idle

For a positive integer 119896 a subset 119868 sube 119881(119866) is called a 119896-independent set (also called a 119896-packing) if 119889

119866(119909 119910) gt 119896 for

any two distinct vertices 119909 and 119910 in 119868 When 119896 = 1 1-set is thenormal independent setThemaximumcardinality among all119896-independent sets is called the 119896-independence number (or119896-packing number) of 119866 denoted by 120572

119896(119866) A 119896-independent

set 119868 is called an 120572119896-set if |119868| = 120572

119896(119866) A graph 119866 is said to be

120572119896-stable if 120572

119896(119866) = 120572

119896(119866 minus 119890) for every edge 119890 of 119866 There are

two important results on 119896-independent sets

Proposition 9 (Topp and Vestergaard [13] 2000) A tree 119879 is120572119896-stable if and only if 119879 has a unique 120572

119896-set

Proposition 10 (Meir and Moon [14] 1975) 1205722(119866) ⩽ 120574(119866)

for any connected graph 119866 with equality for any tree

Hartnell et al [15] independently and Topp and Vester-gaard [13] also gave a constructive characterization of treeswith bondage number 2 and applying Proposition 10 pre-sented another characterization of those trees

Theorem 11 (Hartnell et al [15] 1998 Top and Vestergaard[13] 2000) 119887(119879) = 2 for a tree 119879 if and only if 119879 has a unique1205722-set

According to this characterization Hartnell et al [15]presented a linear algorithm for determining the bondagenumber of a tree

In this subsection we introduce three characterizationsfor trees with bondage number 1 or 2 The characterizationin Theorem 7 is constructive constructing all trees withbondage number 2 a natural and straightforward methodby a series of graph-operations The characterization inTheorem 8 is a little advisable by describing the inherentproperty of trees with bondage number 1 The characteriza-tion in Theorem 11 is wonderful by using a strong graph-theoretic concept 120572

119896-set In fact this characterization is a

byproduct of some results related to 120572119896-sets for trees It is

that this characterization closed the relation between twoconcepts the bondage number and the 119896-independent setand hence is of research value and important significance

23 Complexity for General Graphs Asmentioned above thebondage number of a tree can be determined by a linear timealgorithm According to this algorithm we can determinewithin polynomial time the domination number of any treeby removing each edge and verifyingwhether the dominationnumber is enlarged according to the known linear timealgorithm for domination numbers of trees

However it is impossible to find a polynomial timealgorithm for bondage numbers of general graphs If suchan algorithm 119860 exists then the domination number of any

nonempty undirected graph 119866 can be determined withinpolynomial time by repeatedly using 119860 Let 119866

0= 119866 and

119866119894+1= 119866

119894minus119861

119894 where 119861

119894is the minimum edge set of 119866

119894found

by 119860 such that 120574(119866119894minus 119861

119894) = 120574(119866

119894minus1) + 1 for each 119894 = 0 1

we can always find the minimum 119861119894whose removal from 119866

119894

enlarges the domination number until 119866119896= 119866

119896minus1minus 119861

119896minus1

is empty for some 119896 ⩾ 1 though 119861119896minus1

is not empty Then120574(119866) = 120574(119866

119896) minus 119896 = 120592(119866) minus 119896 As known to all if NP = 119875

the minimum dominating set problem is NP-complete andso polynomial time algorithms for the bondage number donot exist unless NP = 119875

In fact Hu and Xu [16] have recently shown that theproblem determining the bondage number of general graphsis NP-hard

Problem 1 Consider the decision problemBondage ProblemInstance a graph 119866 and a positive integer 119887 (⩽ 120576(119866))Question is 119887(119866) ⩽ 119887

Theorem 12 (Hu and Xu [16] 2012) The bondage problem isNP-hard

Thebasicway of the proof is to followGarey and Johnsonrsquostechniques for proving NP-hardness [5] by describing apolynomial transformation from the known NP-completeproblem 3-satisfiability problem

Theorem 12 shows that we are unable to find a polynomialtime algorithm to determine bondage numbers of generalgraphs unless NP = 119875 At the same time this result alsoshows that the following study on the bondage number is ofimportant significance

(i) Find approximation polynomial algorithms with per-formance ratio as small as possible

(ii) Find the lower and upper bounds with difference assmall as possible

(iii) Determine exact values for some graphs speciallywell-known networks

Unfortunately we cannot provewhether or not determin-ing the bondage number is NP-problem since for any subset119861 sub 119864(119866) it is not clear that there is a polynomial algorithmto verify 120574(119866 minus 119861) gt 120574(119866) Since the problem of determiningthe domination number is NP-complete we conjecture thatit is not in NPThis is a worthwhile task to be studied further

Motivated by the linear time algorithm of Hartnell et alto compute the bondage number of a tree we can makean attempt to consider whether there is a polynomial timealgorithm to compute the bondage number for some specialclasses of graphs such as planar graphs Cayley graphs orgraphs with some restrictions of graph-theoretical parame-ters such as degree diameter connectivity and dominationnumber

3 Upper Bounds

ByTheorem 12 since we cannot find a polynomial time algo-rithm for determining the exact values of bondage numbers

International Journal of Combinatorics 5

of general graphs it is weightily significative to establishsome sharp bounds on the bondage number of a graph Inthis section we survey several known upper bounds on thebondage number in terms of some other graph-theoreticalparameters

31 Most Basic Upper Bounds We start this subsection witha simple observation

Observation 1 (Teschner [10] 1997) Let 119867 be a spanningsubgraph obtained from a graph119866 by removing 119896 edgesThen119887(119866) ⩽ 119887(119867) + 119896

If we select a spanning subgraph 119867 such that 119887(119867) = 1thenObservation 1 yields some upper bounds on the bondagenumber of a graph 119866 For example take119867 = 119866 minus 119864

119909+ 119890 and

119896 = 119889119866(119909) minus 1 where 119890 isin 119864

119909 the following result can be

obtained

Theorem 13 (Bauer et al [7] 1983) If there exists at least onevertex 119909 in a graph 119866 such that 120574(119866 minus 119909) ⩾ 120574(119866) then 119887(119866) ⩽119889119866(119909) ⩽ Δ(119866)

The following early result obtained can be derived fromObservation 1 by taking119867 = 119866minus119864

119909119910and 119896 = 119889(119909)+119889(119910)minus2

Theorem 14 (Bauer et al [7] 1983 Fink et al [8] 1990)119887(119866) ⩽ 119889(119909) + 119889(119910) minus 1 for any two adjacent vertices 119909 and119910 in a graph 119866 that is

119887 (119866) ⩽ min119909119910isin119864(119866)

119889119866(119909) + 119889

119866(119910) minus 1 (4)

This theorem gives a natural corollary obtained by severalauthors

Corollary 15 (Bauer et al [7] 1983 Fink et al [8] 1990) If 119866is a graphwithout isolated vertices then 119887(119866) ⩽ Δ(119866)+120575(119866)minus1

In 1999 Hartnell and Rall [11] extended Theorem 14 tothe following more general case which can be also derivedfrom Observation 1 by taking119867 = 119866 minus 119864

119909minus 119864

119910+ (119909 119911 119910) if

119889119866(119909 119910) = 2 where (119909 119911 119910) is a path of length 2 in 119866

Theorem 16 (Hartnell and Rall [11] 1999) 119887(119866) ⩽ 119889(119909) +119889(119910)minus1 for any distinct two vertices 119909 and 119910 in a graph119866with119889119866(119909 119910) ⩽ 2 that is

119887 (119866) ⩽ min119889119866(119909119910)⩽2

119889119866(119909) + 119889

119866(119910) minus 1 (5)

Corollary 17 (Fink et al [8] 1990) If a vertex of a graph 119866 isadjacent with two ormore vertices of degree one then 119887(119866) = 1

We remark that the bounds stated in Corollary 15 andTheorem 16 are sharp As indicated byTheorem 1 one class ofgraphs in which the bondage number achieves these boundsis the class of cycles whose orders are congruent to 1 modulo3

On the other hand Hartnell and Rall [17] sharpened theupper bound in Theorem 14 as follows which can be alsoderived from Observation 1

1198792(119909 119910)

1198794(119909 119910)

1198793(119909 119910)

1198791(119909 119910)

119909

119910

Figure 2 Illustration of 119879119894(119909 119910) for 119894 = 1 2 3 4

Theorem 18 (Hartnell and Rall [17] 1994) 119887(119866) ⩽ 119889119866(119909) +

119889119866(119910) minus 1 minus |119873

119866(119909) cap 119873

119866(119910)| for any two adjacent vertices 119909

and 119910 in a graph 119866 that is

119887 (119866) ⩽ min119909119910isin119864(119866)

119889119866(119909) + 119889

119866(119910) minus 1 minus

1003816100381610038161003816119873119866(119909) cap 119873

119866(119910)1003816100381610038161003816

(6)

These results give simple but important upper bounds onthe bondage number of a graph and is also the foundationof almost all results on bondage numbers upper boundsobtained till now

By careful consideration of the nature of the edges fromthe neighbors of119909 and119910Wang [18] further refined the boundin Theorem 18 For any edge 119909119910 isin 119864(119866) 119873

119866(119910) contains the

following four subsets

(1) 1198791(119909 119910) = 119873

119866[119909] cap 119873

119866(119910)

(2) 1198792(119909 119910) = 119908 isin 119873

119866(119910) 119873

119866(119908) sube 119873

119866(119910) minus 119909

(3) 1198793(119909 119910) = 119908 isin 119873

119866(119910) 119873

119866(119908) sube 119873

119866(119911)minus119909 for some

119911 isin 119873119866(119909) cap 119873

119866(119910)

(4) 1198794(119909 119910) = 119873

119866(119910) (119879

1(119909 119910) cup 119879

2(119909 119910) cup 119879

3(119909 119910))

The illustrations of 1198791(119909 119910) 119879

2(119909 119910) 119879

3(119909 119910) and

1198794(119909 119910) are shown in Figure 2 (corresponding vertices

pointed by dashed arrows)

Theorem 19 (Wang [18] 1996) For any nonempty graph 119866

119887 (119866) ⩽ min119909119910isin119864(119866)

119889119866(119909) +

10038161003816100381610038161198794 (119909 119910)1003816100381610038161003816 (7)

The graph in Figure 2 shows that the upper bound givenin Theorem 19 is better than that in Theorems 16 and 18for the upper bounds obtained from these two theorems are119889119866(119909)+119889

119866(119910)minus1 = 11 and 119889

119866(119909)+119889

119866(119910)minus|119873

119866(119909)cap119873

119866(119910)| =

9 respectively while the upper bound given byTheorem 19 is119889119866(119909) + |119879

4(119909 119910)| = 6

The following result is also an improvement ofTheorem 14 in which 119905 = 2

6 International Journal of Combinatorics

Theorem 20 (Teschner [10] 1997) If 119866 contains a completesubgraph 119870

119905with 119905 ⩾ 2 then 119887(119866) ⩽ min

119909119910isin119864(119870119905)119889

119866(119909) +

119889119866(119910) minus 119905 + 1

Following Fricke et al [19] a vertex 119909 of a graph 119866 is 120574-good if 119909 belongs to some 120574-set of119866 and 120574-bad if 119909 belongs tono 120574-set of 119866 Let 119860(119866) be the set of 120574-good vertices and let119861(119866) be the set of 120574-bad vertices in 119866 Clearly 119860(119866) 119861(119866)is a partition of119881(119866) Note there exists some 119909 isin 119860 such that120574(119866 minus 119909) = 120574(119866) say one end-vertex of 119875

5 Samodivkin [20]

presented some sharp upper bounds for 119887(119866) in terms of 120574-good and 120574-bad vertices of 119866

Theorem 21 (Samodivkin [20] 2008) Let 119866 be a graph

(a) Let 119862(119866) = 119909 isin 119881(119866) 120574(119866minus119909) ⩾ 120574(119866) If 119862(119866) = 0then

119887 (119866) ⩽ min 119889119866(119909) + 120574 (119866) minus 120574 (119866 minus 119909) 119909 isin 119862 (119866)

(8)

(b) If 119861 = 0 then

119887 (119866) ⩽ min 1003816100381610038161003816119873119866(119909) cap 119860

1003816100381610038161003816 119909 isin 119861 (119866) (9)

Theorem 22 (Samodivkin [20] 2008) Let 119866 be a graph If119860+

(119866) = 119909 isin 119860(119866) 119889119866(119909) ⩾ 1 and 120574(119866 minus 119909) lt 120574(119866) = 0

then

119887 (119866) ⩽ min119909isin119860+(119866)119910isin119861(119866minus119909)

119889119866(119909) +

1003816100381610038161003816119873119866(119910) cap 119860 (119866 minus 119909)

1003816100381610038161003816

(10)

Proposition 23 (Samodivkin [20] 2008) Under the notationof Theorem 19 if 119909 isin 119860+

(119866) then (1198791(119909 119910) minus 119909) cup 119879

2(119909 119910) cup

1198793(119909 119910) sube 119873

119866(119910) 119861(119866 minus 119909)

By Proposition 23 if 119909 isin 119860+

(119866) then

119889119866(119909) + min

119910isin119873119866(119909)

10038161003816100381610038161198794 (119909 119910)

1003816100381610038161003816

⩾ 119889119866(119909) + min

119910isin119873119866(119909)

1003816100381610038161003816119873119866

(119910) cap 119860 (119866 minus 119909)1003816100381610038161003816

⩾ 119889119866(119909) + min

119910isin119861(119866minus119909)

1003816100381610038161003816119873119866

(119910) cap 119860 (119866 minus 119909)1003816100381610038161003816

(11)

Hence Theorem 19 can be seen to follow from Theorem 22Any graph 119866 with 119887(119866) achieving the upper bound inTheorem 19 can be used to show that the bound inTheorem 22 is sharp

Let 119905 ⩾ 2 be an integer Samodivkin [20] con-structed a very interesting graph 119866

119905to show that the upper

bound in Theorem 22 is better than the known bounds Let119867

0 119867

1 119867

2 119867

119905+1be mutually vertex-disjoint graphs such

that 1198670cong 119870

2 119867

119905+1cong 119870

119905+3and 119867

119894cong 119870

119905+3minus 119890 for each

119894 = 1 2 119905 Let 119881(1198670) = 119909 119910 119909

119905+1isin 119881(119867

119905+1) and

1198671

1198672

1198673

11990911

11990912

11990921 11990922

1199093

119909

119910

Figure 3 The graph 1198662

1199091198941 119909

1198942isin 119881(119867

119894) 119909

11989411199091198942notin 119864(119867

119894) for each 119894 = 1 2 119905 The

graph 119866119905is defined as follows

119881 (119866119905) =

119905+1

⋃

119896=0

119881 (119867119896)

119864 (119866119905)=(

119905+1

⋃

119896=0

119864 (119867119896))cup(

119905

⋃

119894=1

1199091199091198941 119909119909

1198942)cup119909119909

119905+1

(12)

Such a constructed graph119866119905is shown in Figure 3 when 119905 = 2

Observe that 120574(119866119905) = 119905 + 2 119860(119866

119905) = 119881(119866

119905) 119889

119866119905

(119909) =

2119905 + 2 119889119866119905

(119909119905+1) = 119905 + 3 119889

119866119905

(119910) = 1 and 119889119866119905

(119911) = 119905 + 2

for each 119911 isin 119881(119866119905minus 119909 119910 119909

119905+1) Moreover 120574(119866 minus 119910) lt 120574(119866)

and 120574(119866119905minus119911) = 120574(119866

119905) for any 119911 isin 119881(119866

119905) minus 119910 Hence each of

the bounds stated inTheorems 13ndash20 is greater than or equals119905 + 2

Consider the graph 119866119905minus 119909119910 Clearly 120574(119866

119905minus 119909119910) = 120574(119866

119905)

and

119861 (119866119905minus 119909119910) = 119861 (119866

119905minus 119910)

= 119909 cup 119881 (119867119905+1minus 119909

119905+1) cup (

119905

⋃

119896=1

1199091198961

1199091198962

)

(13)

Therefore 119873119866119905

(119909) cap 119866(119866119905minus 119910) = 119909

119905+1 which implies that

the upper bound stated in Theorem 22 is equal to 119889119866119905

(119910) +

|119909119905+1| = 2 Clearly 119887(119866

119905) = 2 and hence this bound is sharp

for 119866119905

From the graph 119866119905 we obtain the following statement

immediately

Proposition 24 For every integer 119905 ⩾ 2 there is a graph 119866such that the difference between any upper bound stated inTheorems 13ndash20 and the upper bound in Theorem 22 is equalto 119905

Although Theorem 22 supplies us with the upper boundthat is closer to 119887(119866) for some graph 119866 than what any one of

International Journal of Combinatorics 7

Theorems 13ndash20 provides it is not easy to determine the sets119860+

(119866) and 119861(119866) mentioned in Theorem 22 for an arbitrarygraph 119866 Thus the upper bound given in Theorem 22 is oftheoretical importance but not applied since until now wehave not found a new class of graphswhose bondage numbersare determined byTheorem 22

The above-mentioned upper bounds on the bondagenumber are involved in only degrees of two vertices Hartnelland Rall [11] established an upper bound on 119887(119866) in terms ofthe numbers of vertices and edges of 119866 with order 119899 For anyconnected graph 119866 let 120575(119866) represent the average degree ofvertices in 119866 that is 120575(119866) = (1119899)sum

119909isin119881119889119866(119909) Hartnell and

Rall first discovered the following proposition

Proposition 25 For any connected graph 119866 there exist twovertices 119909 and 119910 with distance at most two and with theproperty that 119889

119866(119909) + 119889

119866(119910) ⩽ 2120575(119866)

Using Proposition 25 and Theorem 16 Hartnell and Rallgave the following bound

Theorem 26 (Hartnell and Rall [11] 1999) 119887(119866) ⩽ 2120575(119866) minus 1for any connected graph 119866

Note that 2120576 = 119899120575(119866) for any graph 119866 with order 119899 andsize 120576 Theorem 26 implies the following bound in terms oforder 119899 and size 120576

Corollary 27 119887(119866) ⩽ (4120576119899) minus 1 for any connected graph 119866with order 119899 and size 120576

A lower bound on 120576 in terms of order 119899 and the bondagenumber is obtained from Corollary 28 immediately

Corollary 28 120576 ⩾ (1198994)(119887(119866) + 1) for any connected graph 119866with order 119899 and size 120576

Hartnell and Rall [11] gave some graphs 119866 with order 119899to show that for each value of 119887(119866) the lower bound on 120576(119866)given in the Corollary 28 is sharp for some values of 119899

32 Bounds Implied by Connectivity Use 120581(119866) and 120582(119866) todenote the vertex-connectivity and the edge-connectivity ofa connected graph 119866 respectively which are the minimumnumbers of vertices and edges whose removal result in 119866disconnected The famous Whitney inequality is stated as120581(119866) ⩽ 120582(119866) ⩽ 120575(119866) for any graph or digraph 119866 Corollary 15is improved by several authors as follows

Theorem 29 (Hartnell and Rall [17] 1994 Teschner [10]1997) If 119866 is a connected graph then 119887(119866) ⩽ 120582(119866)+Δ(119866)minus 1

The upper bound given in Theorem 29 can be attainedFor example a cycle 119862

3119896+1with 119896 ⩾ 1 119887(119862

3119896+1) = 3 by

Theorem 1 Since 120581(1198623119896+1) = 120582(119862

3119896+1) = 2 we have 120581(119862

3119896+1)+

120582(1198623119896+1) minus 1 = 2 + 2 minus 1 = 3 = 119887(119862

3119896+1)

Motivated byCorollary 15Theorems 29 and theWhitneyinequality Dunbar et al [21] naturally proposed the followingconjecture

Figure 4 A graph119867 with 120574(119867) = 3 and 119887(119867) = 5

Conjecture 30 If119866 is a connected graph then 119887(119866) ⩽ Δ(119866)+120581(119866) minus 1

However Liu and Sun [22] presented a counterexampleto this conjecture They first constructed a graph 119867 showedin Figure 4 with 120574(119867) = 3 and 119887(119867) = 5 Then let 119866 be thedisjoint union of two copies of119867 by identifying two verticesof degree two They proved that 119887(119866) ⩾ 5 Clearly 119866 is a 4-regular graph with 120581(119866) = 1 and 120582(119866) = 2 and so 119887(119866) ⩽ 5byTheorem 29 Thus 119887(119866) = 5 gt 4 = Δ(119866) + 120581(119866) minus 1

With a suspicion of the relationship between the bondagenumber and the vertex-connectivity of a graph the followingconjecture is proposed

Conjecture 31 (Liu and Sun [22] 2003) For any positiveinteger 119903 there exists a connected graph 119866 such that 119887(119866) ⩾Δ(119866) + 120581(119866) + 119903

To the knowledge of the author until now no results havebeen known about this conjecture

We conclude this subsection with the following remarksFromTheorem 29 if Conjecture 31 holds for some connectedgraph119866 then 120582(119866) gt 120581(119866)+119903 which implies that119866 is of largeedge-connectivity and small vertex-connectivity Use 120585(119866) todenote the minimum edge-degree of 119866 that is

120585 (119866) = min119909119910isin119864(119866)

119889119866(119909) + 119889

119866(119910) minus 2 (14)

Theorem 14 implies the following result

Proposition 32 119887(119866) ⩽ 120585(119866) + 1 for any graph 119866

Use 1205821015840(119866) to denote the restricted edge-connectivity ofa connected graph 119866 which is the minimum number ofedgeswhose removal result in119866 disconnected and no isolatedvertices

Proposition 33 (Esfahanian and Hakimi [23] 1988) If 119866 isneither 119870

1119899nor 119870

3 then 1205821015840(119866) ⩽ 120585(119866)

Combining Propositions 32 and 33 we propose a conjec-ture

Conjecture 34 If a connected 119866 is neither 1198701119899

nor 1198703 then

119887(119866) ⩽ 120575(119866) + 1205821015840

(119866) minus 1

8 International Journal of Combinatorics

A cycle 1198623119896+1

also satisfies 119887(1198623119896+1) = 3 = 120575(119862

3119896+1) +

1205821015840

(1198623119896+1) minus 1 since 1205821015840(119862

3119896+1) = 120575(119862

3119896+1) = 2 for any integer

119896 ⩾ 1For the graph119867 shown in Figure 41205821015840(119867) = 4 and 120575(119867) =

2 and so 119887(119867) = 5 = 120575(119867) + 1205821015840(119867) minus 1For the 4-regular graph119866 constructed by Liu and Sun [22]

obtained from the disjoint union of two copies of the graph119867 showed in Figure 4 by identifying two vertices of degreetwo we have 119887(119866) ⩾ 5 Clearly 1205821015840(119866) = 2 Thus 119887(119866) = 5 =120575(119866) + 120582

1015840

(119866) minus 1For the 4-regular graph 119866

119905constructed by Samodivkin

[20] see Figure 3 for 119905 = 2 119887(119866119905) = 2 Clearly 120575(119866

119905) = 1

and 1205821015840(119866) = 2 Thus 119887(119866) = 2 = 120575(119866) + 1205821015840(119866) minus 1These examples show that if Conjecture 34 is true then the

given upper bound is tight

33 Bounds Implied by Degree Sequence Now let us return toTheorem 16 from which Teschner [10] obtained some otherbounds in terms of the degree sequencesThe degree sequence120587(119866) of a graph 119866 with vertex-set 119881 = 119909

1 119909

2 119909

119899 is the

sequence 120587 = (1198891 119889

2 119889

119899) with 119889

1⩽ 119889

2⩽ sdot sdot sdot ⩽ 119889

119899 where

119889119894= 119889

119866(119909

119894) for each 119894 = 1 2 119899 The following result is

essentially a corollary of Theorem 16

Theorem35 (Teschner [10] 1997) Let119866 be a nonempty graphwith degree sequence120587(119866) If120572

2(119866) = 119905 then 119887(119866) ⩽ 119889

119905+119889

119905+1minus

1

CombiningTheorem 35with Proposition 10 (ie 1205722(119866) ⩽

120574(119866)) we have the following corollary

Corollary 36 (Teschner [10] 1997) Let 119866 be a nonemptygraph with the degree sequence 120587(119866) If 120574(119866) = 120574 then 119887(119866) ⩽119889120574+ 119889

120574+1minus 1

In [10] Teschner showed that these two bounds are sharpfor arbitrarily many graphs Let119867 = 119862

3119896+1+ 119909

11199094 119909

11199093119896minus1

where 1198623119896+1

is a cycle (1199091 119909

2 119909

3119896+1 119909

1) for any integer

119896 ⩾ 2 Then 120574(119867) = 119896 + 1 and so 119887(119867) ⩽ 2 + 2 minus 1 = 3by Corollary 36 Since119862

3119896+1is a spanning subgraph of119867 and

120574(119866) = 120574(119867) Observation 1 yields that 119887(119867) ⩾ 119887(1198623119896+1) = 3

and so 119887(119867) = 3Although various upper bounds have been established

as the above we find that the appearance of these boundsis essentially based upon the local structures of a graphprecisely speaking the structures of the neighborhoods oftwo vertices within distance 2 Even if these bounds can beachieved by some special graphs it is more often not the caseThe reason lies essentially in the definition of the bondagenumber which is theminimumvalue among all bondage setsan integral property of a graph While it easy to find upperbounds just by choosing some bondage set the gap betweenthe exact value of the bondage number and such a boundobtained only from local structures of a graph is often largeFor example a star 119870

1Δ however large Δ is 119887(119870

1Δ) = 1

Therefore one has been longing for better bounds upon someintegral parameters However as what we will see below it isdifficult to establish such upper bounds

34 Bounds in 120574-Critical Graphs A graph119866 is called a vertexdomination-critical graph (vc-graph or 120574-critical for short) if120574(119866 minus 119909) lt 120574(119866) for any 119909 isin 119881(119866) proposed by Brighamet al [24] in 1988 Several families of graphs are known to be120574-critical From definition it is clear that if 119866 is a 120574-criticalgraph then 120574(119866) ⩾ 2The class of 120574-critical graphs with 120574 = 2is characterized as follows

Proposition 37 (Brigham et al [24] 1988) A graph 119866 with120574(119866) = 2 is a 120574-critical graph if and only if 119866 is a completegraph 119870

2119905(119905 ⩾ 2) with a perfect matching removed

A more interesting family is composed of the 119899-criticalgraphs 119866

119898119899defined for 119898 119899 ⩾ 2 by the circulant undirected

graph 119866(119873 plusmn119878) where 119873 = (119898 + 1)(119899 minus 1) + 1 and119878 = 1 2 lfloor1198982rfloor in which 119881(119866(119873 plusmn119878)) = 119873 and119864(119866(119873 plusmn119878)) = 119894119895 |119895 minus 119894| equiv 119904 (mod 119873) 119904 isin 119878

The reason why 120574-critical graphs are of special interest inthis context is easy to see that they play an important role instudy of the bondage number For instance it immediatelyfollows from Theorem 13 that if 119887(119866) gt Δ(119866) then 119866 is a 120574-critical graphThe 120574-critical graphs are defined exactly in thisway In order to find graphs 119866 with a high bondage number(ie higher than Δ(119866)) and beyond its general upper boundsfor the bondage number we therefore have to look at 120574-critical graphs

The bondage numbers of some 120574-critical graphs havebeen examined by several authors see for example [1020 25 26] From Theorem 13 we know that the bondagenumber of a graph 119866 is bounded from above by Δ(119866)if 119866 is not a 120574-critical graph For 120574-critical graphs it ismore difficult to find an upper bound We will see that thebondage numbers of 120574-critical graphs in general are noteven bounded from above by Δ + 119888 for any fixed naturalnumber 119888

In this subsection we introduce some upper bounds forthe bondage number of a 120574-critical graph By Proposition 37we easily obtain the following result

Theorem 38 If 119866 is a 120574-critical graph with 120574(119866) = 2 then119887(119866) ⩽ Δ + 1

In Section 4 by Theorem 59 we will see that the equalityin Theorem 38 holds that is 119887(119866) = Δ + 1 if 119866 is a 120574-criticalgraph with 120574(119866) = 2

Theorem 39 (Teschner [10] 1997) Let119866 be a 120574-critical graphwith degree sequence 120587(119866) Then 119887(119866) ⩽ maxΔ(119866) + 1 119889

1+

1198892+ sdot sdot sdot + 119889

120574minus1 where 120574 = 120574(119866)

As we mentioned above if 119866 is a 120574-critical graph with120574(119866) = 2 then 119887(119866) = Δ + 1 which shows that thebound given in Theorem 39 can be attained for 120574 = 2However we have not known whether this bound is tightfor general 120574 ⩾ 3 Theorem 39 gives the following corollaryimmediately

Corollary 40 (Teschner [10] 1997) Let 119866 be a 120574-criticalgraph with degree sequence 120587(119866) If 120574(119866) = 3 then 119887(119866) ⩽maxΔ(119866) + 1 120575(119866) + 119889

2

International Journal of Combinatorics 9

From Theorem 38 and Corollary 40 we have 119887(119866) ⩽2Δ(119866) if 119866 is a 120574-critical graph with 120574(119866) ⩽ 3 The followingresult shows that this bound is not tight

Theorem 41 (Teschner [26] 1995) Let119866 be a 120574-critical graphgraph with 120574(119866) ⩽ 3 Then 119887(119866) ⩽ (32)Δ(119866)

Until now we have not known whether the bound givenin Theorem 41 is tight or not We state two conjectures on120574-critical graphs proposed by Samodivkin [20] The first ofthem is motivated byTheorems 22 and 19

Conjecture 42 (Samodivkin [20] 2008) For every connectednontrivial 120574-critical graph 119866

min119909isin119860+(119866)119910isin119861(119866minus119909)

119889119866(119909) +

1003816100381610038161003816119873119866(119910) cap 119860 (119866 minus 119909)

1003816100381610038161003816 ⩽3

2Δ (119866)

(15)

To state the second conjecture we need the followingresult on 120574-critical graphs

Proposition 43 If119866 is a 120574-critical graph then 120592(119866) ⩽ (Δ(119866)+1)(120574(119866)minus1)+1 moreover if the equality holds then119866 is regular

The upper bound in Proposition 43 is sharp in thesense that equality holds for the infinite class of 120574-criticalgraphs 119866

119898119899defined in the beginning of this subsection In

Proposition 43 the first result is due to Brigham et al [24] in1988 the second is due to Fulman et al [27] in 1995

For a 120574-critical graph 119866 with 120574 = 3 by Proposition 43120592(119866) ⩽ 2Δ(119866) + 3

Theorem 44 (Teschner [26] 1995) If 119866 is a 120574-critical graphwith 120574 = 3 and 120592(119866) = 2Δ(119866) + 3 then 119887(119866) ⩽ Δ + 1

We have not known yet whether the equality inTheorem 44holds or notHowever Samodivkin proposed thefollowing conjecture

Conjecture 45 (Samodivkin [20] 2008) If 119866 is a 120574-criticalgraph with (Δ(119866)+1)(120574minus1)+1 vertices then 119887(119866) = Δ(119866)+1

In general based on Theorem 41 Teschner proposed thefollowing conjecture

Conjecture 46 (Teschner [26] 1995) If119866 is a 120574-critical graphthen 119887(119866) ⩽ (32)Δ(119866) for 120574 ⩾ 4

We conclude this subsection with some remarks Graphswhich are minimal or critical with respect to a given prop-erty or parameter frequently play an important role in theinvestigation of that property or parameter Not only are suchgraphs of considerable interest in their own right but also aknowledge of their structure often aids in the developmentof the general theory In particular when investigating anyfinite structure a great number of results are proven byinduction Consequently it is desirable to learn as much aspossible about those graphs that are critical with respect toa given property or parameter so as to aid and abet suchinvestigations

In this subsection we survey some results on the bondagenumber for 120574-critical graphs Although these results are notvery perfect they provides a feasible method to approach thebondage number from different viewpoints In particular themethods given in Teschner [26] worth further explorationand development

The following proposition is maybe useful for us tofurther investigate the bondage number of a 120574-critical graph

Proposition 47 (Brigham et al [24] 1988) If 119866 has anonisolated vertex 119909 such that the subgraph induced by119873

119866(119909)

is a complete graph then 119866 is not 120574-critical

This simple fact shows that any 120574-critical graph containsno vertices of degree one

35 Bounds Implied by Domination In the preceding sub-section we introduced some upper bounds on the bondagenumbers for 120574-critical graphs by consideration of domina-tions In this subsection we introduce related results for ageneral graph with given domination number

Theorem 48 (Fink et al [8] 1990) For any connected graph119866 of order 119899 (⩾2)

(a) 119887(119866) ⩽ 119899 minus 1

(b) 119887(119866) ⩽ Δ(119866) + 1 if 120574(119866) = 2

(c) 119887(119866) ⩽ 119899 minus 120574(119866) + 1

While the upper bound of 119899minus1 on 119887(119866) is not particularlygood for many classes of graphs (eg trees and cycles) it isan attainable bound For example if 119866 is a complete 119905-partitegraph 119870

119905(2) for 119905 = 1198992 and 119899 ⩽ 4 an even integer then the

three bounds on 119887(119866) in Theorem 48 are sharpTeschner [26] consider a graphwith 120574(119866) = 1 or 120574(119866) = 2

The next result is almost trivial but useful

Lemma 49 Let 119866 be a graph with order 119899 and 120574(119866) = 1 119905 thenumber of vertices of degree 119899 minus 1 Then 119887(119866) = lceil1199052rceil

Since 119905 ⩽ Δ(119866) clearly Lemma 49 yields the followingresult immediately

Theorem 50 (Teschner [26] 1995) 119887(119866) ⩽ (12)Δ+1 for anygraph 119866 with 120574(119866) = 1

For a complete graph1198702119896+1

119887(1198702119896+1) = 119896+1 = (12)Δ+1

which shows that the upper bound given in Theorem 50 canbe attained For a graph119866with 120574(119866) = 2 byTheorem 59 laterthe upper bound given inTheorem 48 (b) can be also attainedby a 2-critical graph (see Proposition 37)

36 Two Conjectures In 1990 when Fink et al [8] introducedthe concept of the bondage number they proposed thefollowing conjecture

Conjecture 51 119887(119866) ⩽ Δ(119866) + 1 for any nonempty graph 119866

10 International Journal of Combinatorics

Although some results partially support Conjecture 51Teschner [28] in 1993 found a counterexample to this con-jecture the cartesian product 119870

3times 119870

3 which shows that

119887(119866) = Δ(119866) + 2If a graph 119866 is a counterexample to Conjecture 51 it

must be a 120574-critical graph by Theorem 13 That is why thevertex domination-critical graphs are of special interest in theliterature

Now we return to Conjecture 51 Hartnell and Rall [17]and Teschner [29] independently proved that 119887(119866) can bemuch greater than Δ(119866) by showing the following result

Theorem 52 (Hartnell and Rall [17] 1994 Teschner [29]1996) For an integer 119899 ⩾ 3 let 119866

119899be the cartesian product

119870119899times 119870

119899 Then 119887(119866

119899) = 3(119899 minus 1) = (32)Δ(119866

119899)

This theorem shows that there exist no upper boundsof the form 119887(119866) ⩽ Δ(119866) + 119888 for any integer 119888 Teschner[26] proved that 119887(119866) ⩽ (32)Δ(119866) for any graph 119866 with120574(119866) ⩽ 2 (see Theorems 50 and 48) and for 120574-critical graphswith 120574(119866) = 3 and proposed the following conjecture

Conjecture 53 (Teschner [26] 1995) 119887(119866) ⩽ (32)Δ(119866) forany graph 119866

We believe that this conjecture is true but so far no muchwork about it has been known

4 Lower Bounds

Since the bondage number is defined as the smallest numberof edges whose removal results in an increase of dominationnumber each constructive method that creates a concretebondage set leads to an upper bound on the bondage numberFor that reason it is hard to find lower bounds Neverthelessthere are still a few lower bounds

41 Bounds Implied by Subgraphs The first lower boundon the bondage number is gotten in terms of its spanningsubgraph

Theorem 54 (Teschner [10] 1997) Let 119867 be a spanningsubgraph of a nonempty graph119866 If 120574(119867) = 120574(119866) then 119887(119867) ⩽119887(119866)

By Theorem 1 119887(119862119899) = 3 if 119899 equiv 1 (mod 3) and 119887(119862

119899) =

2 otherwise 119887(119875119899) = 2 if 119899 equiv 1 (mod 3) and 119887(119875

119899) = 1

otherwise From these results and Theorem 54 we get thefollowing two corollaries

Corollary 55 If119866 is hamiltonianwith order 119899 ⩾ 3 and 120574(119866) =lceil1198993rceil then 119887(119866) ⩾ 2 and in addition 119887(119866) ⩾ 3 if 119899 equiv 1 (mod3)

Corollary 56 If 119866 with order 119899 ⩾ 2 has a hamiltonian pathand 120574(119866) = lceil1198993rceil then 119887(119866) ⩾ 2 if 119899 equiv 1 (mod 3)

42 Other Bounds The vertex covering number 120573(119866) of 119866 isthe minimum number of vertices that are incident with all

Figure 5 A 120574-critical graph with 120573 = 120574 = 4 120575 = 2 and 119887 = 3

edges in 119866 If 119866 has no isolated vertices then 120574(119866) ⩽ 120573(119866)clearly In [30] Volkmann gave a lot of graphs with 120573 = 120574

Theorem 57 (Teschner [10] 1997) Let119866 be a graph If 120574(119866) =120573(119866) then

(a) 119887(119866) ⩾ 120575(119866)

(b) 119887(119866) ⩾ 120575(119866) + 1 if 119866 is a 120574-critical graph

The graph shown in Figure 5 shows that the bound givenin Theorem 57(b) is sharp For the graph 119866 it is a 120574-criticalgraph and 120574(119866) = 120573(119866) = 4 By Theorem 57 we have 119887(119866) ⩾3 On the other hand 119887(119866) ⩽ 3 by Theorem 16 Thus 119887(119866) =3

Proposition 58 (Sanchis [31] 1991) Let 119866 be a graph 119866 oforder 119899 (⩾ 6) and size 120576 If 119866 has no isolated vertices and3 ⩽ 120574(119866) ⩽ 1198992 then 120576 ⩽ ( 119899minus120574(119866)+1

2)

Using the idea in the proof of Theorem 57 every upperbound for 120574(119866) can lead to a lower bound for 119887(119866) Inthis way Teschner [10] obtained another lower bound fromProposition 58

Theorem59 (Teschner [10] 1997) Let119866 be a graph119866 of order119899 (⩾6) and size 120576 If 2 ⩽ 120574(119866) ⩽ 1198992 minus 1 then

(a) 119887(119866) ⩾ min120575(119866) 120576 minus ( 119899minus120574(119866)2)

(b) 119887(119866) ⩾ min120575(119866) + 1 120576 minus ( 119899minus120574(119866)2) if 119866 is a 120574-critical

graph

The lower bound in Theorem 59(b) is sharp for a classof 120574-critical graphs with domination number 2 Let 119866 be thegraph obtained from complete graph119870

2119905(119905 ⩾ 2) by removing

a perfect matching By Proposition 37 119866 is a 120574-critical graphwith 120574(119866) = 2 Then 119887(119866) ⩾ 120575(119866) + 1 = Δ(119866) + 1 byTheorem 59 and 119887(119866) ⩽ Δ(119866) + 1 by Theorem 38 and so119887(119866) = Δ(119866) + 1 = 2119905 minus 1

As far as the author knows there are no more lowerbounds in the present literature In view of applications ofthe bondage number a network is vulnerable if its bondagenumber is small while it is stable if its bondage number islargeTherefore better lower bounds let us learn better aboutthe stability of networks from this point of view Thus it isof great significance to seek more lower bounds for variousclasses of graphs

International Journal of Combinatorics 11

5 Results on Graph Operations

Generally speaking it is quite difficult to determine the exactvalue of the bondage number for a given graph since itstrongly depends on the dominating number of the graphThus determining bondage numbers for some special graphsis interesting if the dominating numbers of those graphs areknown or can be easily determined In this section we willintroduce some known results on bondage numbers for somespecial classes of graphs obtained by some graph operations

51 Cartesian Product Graphs Let 1198661= (119881

1 119864

1) and 119866

2=

(1198812 119864

2) be two graphs The Cartesian product of 119866

1and 119866

2

is an undirected graph denoted by 1198661times 119866

2 where 119881(119866

1times

1198662) = 119881

1times 119881

2 two distinct vertices 119909

11199092and 119910

11199102 where

1199091 119910

1isin 119881(119866

1) and 119909

2 119910

2isin 119881(119866

2) are linked by an edge in

1198661times 119866

2if and only if either 119909

1= 119910

1and 119909

21199102isin 119864(119866

2) or

1199092= 119910

2and 119909

11199101isin 119864(119866

1) The Cartesian product is a very

effective method for constructing a larger graph from severalspecified small graphs

Theorem 60 (Dunbar et al [21] 1998) For 119899 ⩾ 3

119887 (119862119899times 119870

2) =

2 119894119891 119899 equiv 0 119900119903 1 (mod 4) 3 119894119891 119899 equiv 3 (mod 4) 4 119894119891 119899 equiv 2 (mod 4)

(16)

For the Cartesian product 119862119899times 119862

119898of two cycles 119862

119899and

119862119898 where 119899 ⩾ 4 and 119898 ⩾ 3 Klavzar and Seifter [32]

determined 120574(119862119899times 119862

119898) for some 119899 and 119898 For example

120574(119862119899times119862

4) = 119899 for 119899 ⩾ 3 and 120574(119862

119899times119862

3) = 119899minuslfloor1198994rfloor for 119899 ⩾ 4

Kim [33] determined 119887(1198623times 119862

3) = 6 and 119887(119862

4times 119862

4) = 4 For

a general 119899 ⩾ 4 the exact values of the bondage numbers of119862119899times 119862

3and 119862

119899times 119862

4are determined as follows

Theorem 61 (Sohn et al [34] 2007) For 119899 ⩾ 4

119887 (119862119899times 119862

3) =

2 119894119891 119899 equiv 0 (mod 4) 4 119894119891 119899 equiv 1 119900119903 2 (mod 4) 5 119894119891 119899 equiv 3 (mod 4)

(17)

Theorem62 (Kang et al [35] 2005) 119887(119862119899times119862

4) = 4 for 119899 ⩾ 4

For larger 119898 and 119899 Huang and Xu [36] obtained thefollowing result see Theorem 197 for more details

Theorem 63 (Huang and Xu [36] 2008) 119887(1198625119894times119862

5119895) = 3 for

any positive integers 119894 and 119895

Xiang et al [37] determined that for 119899 ⩾ 5

119887 (119862119899times 119862

5)

= 3 if 119899 equiv 0 or 1 (mod 5) = 4 if 119899 equiv 2 or 4 (mod 5) ⩽ 7 if 119899 equiv 3 (mod 5)

(18)

For the Cartesian product 119875119899times119875

119898of two paths 119875

119899and 119875

119898

Jacobson and Kinch [38] determined 120574(119875119899times119875

2) = lceil(119899+1)2rceil

120574(119875119899times 119875

3) = 119899 minus lfloor(119899 minus 1)4rfloor and 120574(119875

119899times 119875

4) = 119899 + 1 if 119899 =

1 2 3 5 6 9 and 119899 otherwiseThe bondage number of119875119899times119875

119898

for 119899 ⩾ 4 and 2 ⩽ 119898 ⩽ 4 is determined as follows (Li [39] alsodetermined 119887(119875

119899times 119875

2))

Theorem 64 (Hu and Xu [40] 2012) For 119899 ⩾ 4

119887 (119875119899times 119875

2) =

1 119894119891 119899 equiv 1 (mod 2)2 119894119891 119899 equiv 0 (mod 2)

119887 (119875119899times 119875

3) =

1 119894119891 119899 equiv 1 119900119903 2 (mod 4)2 119894119891 119899 equiv 0 119900119903 3 (mod 4)

119887 (119875119899times 119875

4) = 1 119891119900119903 119899 ⩾ 14

(19)

From the proof of Theorem 64 we find that if 119875119899=

0 1 119899 minus 1 and 119875119898= 0 1 119898 minus 1 then the removal

of the vertex (0 0) in 119875119899times119875

119898does not change the domination

number If119898 increases the effect of (0 0) for the dominationnumber will be smaller and smaller from the probabilityTherefore we expect it is possible that 120574(119875

119899times 119875

119898minus (0 0)) =

120574(119875119899times 119875

119898) for119898 ⩾ 5 and give the following conjecture

Conjecture 65 119887(119875119899times 119875

119898) ⩽ 2 for119898 ⩾ 5 and 119899 ⩾ 4

52 Block Graphs and Cactus Graphs In this subsection weintroduce some results for corona graphs block graphs andcactus graphs

The corona1198661∘ 119866

2 proposed by Frucht and Harary [41]

is the graph formed from a copy of 1198661and 120592(119866

1) copies of

1198662by joining the 119894th vertex of 119866

1to the 119894th copy of 119866

2 In

particular we are concernedwith the corona119867∘1198701 the graph

formed by adding a new vertex V119894 and a new edge 119906

119894V119894for

every vertex 119906119894in119867

Theorem 66 (Carlson and Develin [42] 2006) 119887(119867 ∘ 1198701) =

120575(119867) + 1

This is a very important result which is often used toconstruct a graph with required bondage number In otherwords this result implies that for any given positive integer 119887there is a graph 119866 such that 119887(119866) = 119887

A block graph is a graph whose blocks are completegraphs Each block in a cactus graph is either a cycle or a119870

2

If each block of a graph is either a complete graph or a cyclethen we call this graph a block-cactus graph

Theorem67 (Teschner [43] 1997) 119887(119866) le Δ(119866) for any blockgraph 119866

Teschner [43] characterized all block graphs with 119887(119866) =Δ(119866) In the same paper Teschner found that 120574-criticalgraphs are instrumental in determining bounds for thebondage number of cactus and block graphs and obtained thefollowing result

Theorem 68 (Teschner [43] 1997) 119887(119866) ⩽ 3 for anynontrivial cactus graph 119866

This bound can be achieved by 119867 ∘ 1198701where 119867 is

a nontrivial cactus graph without vertices of degree one

12 International Journal of Combinatorics

by Theorem 66 In 1998 Dunbar et al [21] proposed thefollowing problem

Problem 2 Characterize all cactus graphs with bondagenumber 3

Some upper bounds for block-cactus graphs were alsoobtained

Theorem 69 (Dunbar et al [21] 1998) Let 119866 be a connectedblock-cactus graph with at least two blocksThen 119887(119866) ⩽ Δ(119866)

Theorem 70 (Dunbar et al [21] 1998) Let 119866 be a connectedblock-cactus graph which is neither a cactus graph nor a blockgraph Then 119887(119866) ⩽ Δ(119866) minus 1

53 Edge-Deletion and Edge-Contraction In order to obtainfurther results of the bondage number we may consider howthe bondage number changes under some operations of agraph 119866 which remain the domination number unchangedor preserve some property of 119866 such as planarity Twosimple operations satisfying these requirements are the edge-deletion and the edge-contraction

Theorem 71 (Huang and Xu [44] 2012) Let 119866 be any graphand 119890 isin 119864(119866) Then 119887(119866 minus 119890) ⩾ 119887(119866) minus 1 Moreover 119887(119866 minus 119890) ⩽119887(119866) if 120574(119866 minus 119890) = 120574(119866)

FromTheorem 1 119887(119862119899minus 119890) = 119887(119875

119899) = 119887(119862

119899) minus 1 for any 119890

in119862119899 which shows that the lower bound on 119887(119866minus119890) is sharp

The upper bound can reach for any graph 119866 with 119887(119866) ⩾ 2Next we consider the effect of the edge-contraction on

the bondage number Given a graph 119866 the contraction of 119866by the edge 119890 = 119909119910 denoted by 119866119909119910 is the graph obtainedfrom 119866 by contracting two vertices 119909 and 119910 to a new vertexand then deleting all multiedges It is easy to observe that forany graph 119866 120574(119866) minus 1 ⩽ 120574(119866119909119910) ⩽ 120574(119866) for any edge 119909119910 of119866

Theorem 72 (Huang and Xu [44] 2012) Let 119866 be any graphand 119909119910 be any edge in119866 If119873

119866(119909)cap119873

119866(119910) = 0 and 120574(119866119909119910) =

120574(119866) then 119887(119866119909119910) ⩾ 119887(119866)

We can use examples 119862119899and 119870

119899to show that the

conditions ofTheorem 72 are necessary and the lower boundinTheorem 72 is tight Clearly 120574(119862

119899) = lceil1198993rceil and 120574(119870

119899) = 1

for any edge 119909119910 119862119899119909119910 = 119862

119899minus1and 119870

119899119909119910 = 119870

119899minus1 By

Theorem 1 if 119899 equiv 1 (mod 3) then 120574(119862119899119909119910) lt 120574(119866) and

119887(119862119899119909119910) = 2 lt 3 = 119887(119862

119899) Thus the result in Theorem 72

is generally invalid without the hypothesis 120574(119866119909119910) = 120574(119866)Furthermore the condition 119873

119866(119909) cap 119873

119866(119910) = 0 cannot be

omitted even if 120574(119866119909119910) = 120574(119866) since for odd 119899 119887(119870119899119909119910) =

lceil(119899 minus 1)2rceil lt lceil1198992rceil = 119887(119870119899) by Theorem 1

On the other hand 119887(119862119899119909119910) = 119887(119862

119899) = 2 if 119899 equiv 0 (mod

3) and 119887(119870119899119909119910) = 119887(119870

119899) if 119899 is even which shows that the

equality in 119887(119866119909119910) ⩾ 119887(119866) may hold Thus the bound inTheorem 72 is tight However provided all the conditions119887(119866119909119910) can be arbitrarily larger than 119887(119866) Given a graph119867let119866 be the graph formed from119867∘119870

1by adding a new vertex

119909 and joining it to an vertex 119910 of degree one in119867 ∘ 1198701 Then

119866119909119910 = 119867 ∘ 1198701 120574(119866) = 120574(119866119909119910) and 119873

119866(119909) cap 119873

119866(119910) = 0

But 119887(119866) = 1 since 120574(119866 minus 119909119910) = 120574(119866119909119910) + 1 and 119887(119866119909119910) =120575(119867) + 1 byTheorem 66The gap between 119887(119866) and 119887(119866119909119910)is 120575(119867)

6 Results on Planar Graphs

From Section 2 we have seen that the bondage numberfor a tree has been completely solved Moreover a lineartime algorithm to compute the bondage number of a tree isdesigned by Hartnell et al [15] It is quite natural to considerthe bondage number for a planar graph In this section westate some results and problems on the bondage number fora planar graph

61 Conjecture on Bondage Number As mentioned inSection 3 the bondage number can be much larger than themaximum degree But for a planar graph 119866 the bondagenumber 119887(119866) cannot exceed Δ(119866) too much It is clear that119887(119866) ⩽ Δ(119866) + 4 by Corollary 15 since 120575(119866) ⩽ 5 for anyplanar graph119866 In 1998Dunbar et al [21] posed the followingconjecture

Conjecture 73 If 119866 is a planar graph then 119887(119866) ⩽ Δ(119866) + 1

Because of its attraction it immediately became the focusof attention when this conjecture is proposed In fact themain aim concerning the research of the bondage number fora planar graph is focused on this conjecture

It has been mentioned in Theorems 1 and 60 that119887(119862

3119896+1) = 3 = Δ + 1 and 119887(119862

4119896+2times 119870

2) = 4 = Δ + 1 It

is known that 119887(1198706minus 119872) = 5 = Δ + 1 where119872 is a perfect

matching of the complete graph1198706These examples show that

if Conjecture 73 is true then the upper bound is sharp for2 ⩽ Δ ⩽ 4

Here we note that it is sufficient to prove this conjecturefor connected planar graphs since the bondage number of adisconnected graph is simply the minimum of the bondagenumbers of its components

The first paper attacking this conjecture is due to Kangand Yuan [45] which confirmed the conjecture for everyconnected planar graph 119866 with Δ ⩾ 7 The proofs are mainlybased onTheorems 16 and 18

Theorem 74 (Kang and Yuan [45] 2000) If 119866 is a connectedplanar graph then 119887(119866) ⩽ min8 Δ(119866) + 2

Obviously in view of Theorem 74 Conjecture 73 is truefor any connected planar graph with Δ ⩾ 7

62 Bounds Implied by Degree Conditions As we have seenfrom Theorem 74 to attack Conjecture 73 we only need toconsider connected planar graphs with maximum Δ ⩽ 6Thus studying the bondage number of planar graphs bydegree conditions is of significanceThe first result on boundsimplied by degree conditions is obtained by Kang and Yuan[45]

International Journal of Combinatorics 13

Theorem 75 (Kang and Yuan [45] 2000) If 119866 is a connectedplanar graph without vertices of degree 5 then 119887(119866) ⩽ 7

Fischermann et al [46] generalized Theorem 75 as fol-lows

Theorem 76 (Fischermann et al [46] 2003) Let 119866 be aconnected planar graph and 119883 the set of vertices of degree 5which have distance at least 3 to vertices of degrees 1 2 and3 If all vertices in 119883 not adjacent with vertices of degree 4are independent and not adjacent to vertices of degree 6 then119887(119866) ⩽ 7

Clearly if119866 has no vertices of degree 5 then119883 = 0 ThusTheorem 76 yields Theorem 75

Use 119899119894to denote the number of vertices of degree 119894 in 119866

for each 119894 = 1 2 Δ(119866) Using Theorem 18 Fischermannet al obtained the following two theorems

Theorem 77 (Fischermann et al [46] 2003) For any con-nected planar graph 119866

(1) 119887(119866) ⩽ 7 if 1198995lt 2119899

2+ 3119899

3+ 2119899

4+ 12

(2) 119887(119866) ⩽ 6 if 119866 contains no vertices of degrees 4 and 5

Theorem 78 (Fischermann et al [46] 2003) For any con-nected planar graph 119866 119887(119866) ⩽ Δ(119866) + 1 if

(1) Δ(119866) = 6 and every edge 119890 = 119909119910 with 119889119866(119909) = 5 and

119889119866(119910) = 6 is contained in at most one triangle

(2) Δ(119866) = 5 and no triangle contains an edge 119890 = 119909119910 with119889119866(119909) = 5 and 4 ⩽ 119889

119866(119910) ⩽ 5

63 Bounds Implied by Girth Conditions The girth 119892(119866) of agraph 119866 is the length of the shortest cycle in 119866 If 119866 has nocycles we define 119892(119866) = infin

Combining Theorem 74 with Theorem 78 we find thatif a planar graph contains no triangles and has maximumdegree Δ ⩾ 5 then Conjecture 73 holds This fact motivatedFischermann et alrsquos [46] attempt to attack Conjecture 73 bygirth restraints

Theorem 79 (Fischermann et al [46] 2003) For any con-nected planar graph 119866

119887 (119866) ⩽

6 119894119891 119892 (119866) ⩾ 4

5 119894119891 119892 (119866) ⩾ 5

4 119894119891 119892 (119866) ⩾ 6

3 119894119891 119892 (119866) ⩾ 8

(20)

The first result in Theorem 79 shows that Conjecture 73is valid for all connected planar graphs with 119892(119866) ⩾ 4 andΔ(119866) ⩾ 5 It is easy to verify that the second result inTheorem 79 implies that Conjecture 73 is valid for all not3-regular graphs of girth 119892(119866) ⩾ 5 which is stated in thefollowing corollary

Corollary 80 For any connected planar graph 119866 if 119866 is not3-regular and 119892(119866) ⩾ 5 then 119887(119866) ⩽ Δ(119866) + 1

The first result in Theorem 79 also implies that if 119866 isa connected planar graph with no cycles of length 3 then119887(119866) ⩽ 6 Hou et al [47] improved this result as follows

Theorem 81 (Hou et al [47] 2011) For any connected planargraph 119866 if 119866 contains no cycles of length 119894 (4 ⩽ 119894 ⩽ 6) then119887(119866) ⩽ 6

Since 119887(1198623119896+1) = 3 for 119896 ⩾ 3 (see Theorem 1) the

last bound in Theorem 79 is tight Whether other bounds inTheorem 79 are tight remains open In 2003 Fischermann etal [46] posed the following conjecture

Conjecture 82 For any connected planar graph 119866 119887(119866) ⩽ 7and

119887 (119866) ⩽ 5 119894119891 119892 (119866) ⩾ 4

4 119894119891 119892 (119866) ⩾ 5(21)

We conclude this subsection with a question on bondagenumbers of planar graphs

Question 1 (Fischermann et al [46] 2003) Is there a planargraph 119866 with 6 ⩽ 119887(119866) ⩽ 8

In 2006 Carlson andDevelin [42] showed that the corona119866 = 119867 ∘ 119870

1for a planar graph 119867 with 120575(119867) = 5 has the

bondage number 119887(119866) = 120575(119867) + 1 = 6 (see Theorem 66)Since the minimum degree of planar graphs is at most 5then 119887(119866) can attach 6 If we take 119867 as the graph of theicosahedron then 119866 = 119867 ∘ 119870

1is such an example The

question for the existence of planar graphs with bondagenumber 7 or 8 remains open

64 Comments on the Conjectures Conjecture 73 is true forall connected planar graphs with minimum degree 120575 ⩽ 2 byTheorem 16 or maximum degree Δ ⩾ 7 by Theorem 74 ornot 120574-critical planar graphs by Theorem 13 Thus to attackConjecture 73 we only need to consider connected criticalplanar graphs with degree-restriction 3 ⩽ 120575 ⩽ Δ ⩽ 6

Recalling and anatomizing the proofs of all results men-tioned in the preceding subsections on the bondage numberfor connected planar graphs we find that the proofs of theseresults strongly depend upon Theorem 16 or Theorem 18 Inother words a basic way used in the proofs is to find twovertices 119909 and 119910 with distance at most two in a consideredplanar graph 119866 such that

119889119866(119909) + 119889

119866(119910) or 119889

119866(119909) + 119889

119866(119910) minus

1003816100381610038161003816119873119866(119909) cap 119873

119866(119910)1003816100381610038161003816

(22)

which bounds 119887(119866) is as small as possible Very recentlyHuang and Xu [44] have considered the following parameter

119861 (119866) = min119909119910isin119881(119866)

119889119909119910 1 ⩽ 119889

119866(119909 119910) ⩽ 2

cup 119889119909119910minus 119899

119909119910 119889 (119909 119910) = 1

(23)

where 119889119909119910= 119889

119866(119909) + 119889

119866(119910) minus 1 and 119899

119909119910= |119873

119866(119909) cap 119873

119866(119910)|

14 International Journal of Combinatorics

It follows fromTheorems 16 and 18 that

119887 (119866) ⩽ 119861 (119866) (24)

The proofs given in Theorems 74 and 79 indeed imply thefollowing stronger results

Theorem 83 If 119866 is a connected planar graph then

119861 (119866) ⩽

min 8 Δ (119866) + 26 119894119891 119892 (119866) ⩾ 4

5 119894119891 119892 (119866) ⩾ 5

4 119894119891 119892 (119866) ⩾ 6

3 119894119891 119892 (119866) ⩾ 8

(25)

Thus using Theorem 16 or Theorem 18 if we can proveConjectures 73 and 82 then we can prove the followingstatement

Statement If 119866 is a connected planar graph then

119861 (119866) ⩽

Δ (119866) + 1

7

5 if 119892 (119866) ⩾ 44 if 119892 (119866) ⩾ 5

(26)

It follows from (24) that Statement implies Conjectures 73and 82 However Huang and Xu [44] gave examples to showthat none of conclusions in Statement is true As a result theystated the following conclusion

Theorem 84 (Huang and Xu [44] 2012) It is not possible toprove Conjectures 73 and 82 if they are right using Theorems16 and 18

Therefore a new method is needed to prove or disprovethese conjectures At the present time one cannot prove theseconjectures and may consider to disprove them If one ofthese conjectures is invalid then there exists a minimumcounterexample 119866 with respect to 120592(119866) + 120576(119866) In order toobtain a minimum counterexample one may consider twosimple operations satisfying these requirements the edge-deletion and the edge-contractionwhich decrease 120592(119866)+120576(119866)and preserve planarity However applying Theorems 71 and72 Huang and Xu [44] presented the following results

Theorem 85 (Huang and Xu [44] 2012) It is not possible toconstruct minimum counterexamples to Conjectures 73 and 82by the operation of an edge-deletion or an edge-contraction

7 Results on Crossing Number Restraints

It is quite natural to generalize the known results on thebondage number for planar graphs to formore general graphsin terms of graph-theoretical parameters In this section weconsider graphs with crossing number restraints

The crossing number cr(119866) of 119866 is the smallest numberof pairwise intersections of its edges when 119866 is drawn in theplane If cr(119866) = 0 then 119866 is a planar graph

71 General Methods Use 119899119894to denote the number of vertices

of degree 119894 in119866 for 119894 = 1 2 Δ(119866) Huang and Xu obtainedthe following results

Theorem 86 (Huang and Xu [48] 2007) For any connectedgraph 119866

119887 (119866) ⩽

6 119894119891 119892 (119866) ⩾ 4 119886119899119889 2cr (119866) lt 121198861

5 119894119891 119892 (119866) ⩾ 5 119886119899119889 6cr (119866) lt 161198862

4 119894119891 119892 (119866) ⩾ 6 119886119899119889 4cr (119866) lt 141198863

3 119894119891 119892 (119866) ⩾ 8 119886119899119889 6cr (119866) lt 161198864

(27)

where 1198861= 119899

1+ 2119899

2+ 2119899

3+sum

Δ

119894=8(119894 minus 7)119899

119894+ 8 119886

2= 3119899

1+ 6119899

2+

51198993+sum

Δ

119894=7(3119894minus18)119899

119894+20 119886

3= 119899

1+2119899

2+sum

Δ

119894=6(2119894minus10)119899

119894+12

and 1198864= sum

Δ

119894=5(3119894 minus 12)119899

119894+ 16

Simplifying the conditions in Theorem 86 we obtain thefollowing corollaries immediately

Corollary 87 (Huang and Xu [48] 2007) For any connectedgraph 119866

119887 (119866) ⩽

6 119894119891 119892 (119866) ⩾ 4 119886119899119889 cr (119866) ⩽ 3

5 119894119891 119892 (119866) ⩾ 5 119886119899119889 cr (119866) ⩽ 4

4 119894119891 119892 (119866) ⩾ 6 119886119899119889 cr (119866) ⩽ 2

3 119894119891 119892 (119866) ⩾ 8 119886119899119889 cr (119866) ⩽ 2

(28)

Corollary 88 (Huang and Xu [48] 2007) For any connectedgraph 119866

(a) 119887(119866) ⩽ 6 if 119866 is not 4-regular cr(119866) = 4 and 119892(119866) ⩾ 4

(b) 119887(119866) ⩽ Δ(119866) + 1 if 119866 is not 3-regular cr(119866) ⩽ 4 and119892(119866) ⩾ 5

(c) 119887(119866) ⩽ 4 if 119866 is not 3-regular cr(119866) = 3 and 119892(119866) ⩾ 6

(d) 119887(119866) ⩽ 3 if cr(119866) = 3 119892(119866) ⩾ 8 and Δ(119866) ⩾ 5

These corollaries generalize some known results for pla-nar graphs For example Corollary 87 contains Theorem 79Corollary 88 (b) contains Corollary 80

Theorem 89 (Huang and Xu [48] 2007) For any connectedgraph 119866 if cr(119866) ⩽ 119899

3(119866) + 119899

4(119866) + 3 then 119887(119866) ⩽ 8

Corollary 90 (Huang and Xu [48] 2007) For any connectedgraph 119866 if cr(119866) ⩽ 3 then 119887(119866) ⩽ 8

Perhaps being unaware of this result in 2010 Ma et al[49] proved that 119887(119866) ⩽ 12 for any graph 119866 with cr(119866) = 1

Theorem91 (Huang and Xu [48] 2007) Let119866 be a connectedgraph and 119868 = 119909 isin 119881(119866) 119889

119866(119909) = 5 119889

119866(119909 119910) ⩾ 3 if

International Journal of Combinatorics 15

119889119866(119910) ⩽ 3 and 119889

119866(119910) = 4 for every 119910 isin 119873

119866(119909) If 119868 is

independent no vertices adjacent to vertices of degree 6 and

cr (119866) lt max5119899

3+ |119868| minus 2119899

4+ 28

117119899

3+ 40

16 (29)

then 119887(119866) ⩽ 7

Corollary 92 (Huang and Xu [48] 2007) Let 119866 be aconnected graph with cr(119866) ⩽ 2 If 119868 = 119909 isin 119881(119866) 119889

119866(119909) =

5 119889119866(119910 119909) ⩾ 3 if 119889

119866(119910) ⩽ 3 and 119889

119866(119910) = 4 for every 119910 isin

119873119866(119909) is independent and no vertices adjacent to vertices of

degree 6 then 119887(119866) ⩽ 7

Theorem93 (Huang andXu [48] 2007) Let119866 be a connectedgraph If 119866 satisfies

(a) 5cr(119866) + 1198995lt 2119899

2+ 3119899

3+ 2119899

4+ 12 or

(b) 7cr(119866) + 21198995lt 3119899

2+ 4119899

4+ 24

then 119887(119866) ⩽ 7

Proposition 94 (Huang and Xu [48] 2007) Let 119866 be aconnected graph with no vertices of degrees four and five Ifcr(119866) ⩽ 2 then 119887(119866) ⩽ 6

Theorem 95 (Huang and Xu [48] 2007) If 119866 is a connectedgraph with cr(119866) ⩽ 4 and not 4-regular when cr(119866) = 4 then119887(119866) ⩽ Δ(119866) + 2

The above results generalize some known results forplanar graphs For example Corollary 90 and Theorem 95contain Theorem 74 the first condition in Theorem 93 con-tains the second condition inTheorem 77

From Corollary 88 and Theorem 95 we suggest the fol-lowing questions

Question 2 Is there a

(a) 4-regular graph 119866 with cr(119866) = 4 such that 119887(119866) ⩾ 7

(b) 3-regular graph 119866 with cr ⩽ 4 and 119892(119866) ⩾ 5 such that119887(119866) = 5

(c) 3-regular graph 119866 with cr = 3 and 119892(119866) = 6 or 7 suchthat 119887(119866) = 5

72 Carlson-Develin Methods In this subsection we intro-duce an elegant method presented by Carlson and Develin[42] to obtain some upper bounds for the bondage numberof a graph

Suppose that 119866 is a connected graph We say that 119866 hasgenus 120588 if 119866 can be embedded in a surface 119878 with 120588 handlessuch that edges are pairwise disjoint except possibly for end-vertices Let119866 be an embedding of119866 in surface 119878 and let120601(119866)denote the number of regions in 119866 The boundary of everyregion contains at least three edges and every edge is on theboundary of atmost two regions (the two regions are identicalwhen 119890 is a cut-edge) For any edge 119890 of 119866 let 1199031

119866(119890) and 1199032

119866(119890)

be the numbers of edges comprising the regions in 119866 whichthe edge 119890 borders It is clear that every 119890 = 119909119910 isin 119864(119866)

1199032

119866(119890) ⩾ 119903

1

119866(119890) ⩾ 4 if 1003816100381610038161003816119873119866

(119909) cap 119873119866(119910)1003816100381610038161003816 = 0

1199032

119866(119890) ⩾ 4 119903

1

119866(119890) ⩾ 3 if 1003816100381610038161003816119873119866

(119909) cap 119873119866(119910)1003816100381610038161003816 = 1

1199032

119866(119890) ⩾ 119903

1

119866(119890) ⩾ 3 if 1003816100381610038161003816119873119866

(119909) cap 119873119866(119910)1003816100381610038161003816 ⩾ 2

(30)

Following Carlson and Develin [42] for any edge 119890 = 119909119910 of119866 we define

119863119866(119890) =

1

119889119866(119909)+

1

119889119866(119910)

+1

1199031

119866(119890)+

1

1199032

119866(119890)minus 1 (31)

By the well-known Eulerrsquos formula

120592 (119866) minus 120576 (119866) + 120601 (119866) = 2 minus 2120588 (119866) (32)

it is easy to see that

sum

119890isin119864(119866)

119863119866(119890) = 120592 (119866) minus 120576 (119866) + 120601 (119866) = 2 minus 2120588 (119866) (33)

If 119866 is a planar graph that is 120588(119866) = 0 then

sum

119890isin119864(119866)

119863119866(119890) = 120592 (119866) minus 120576 (119866) + 120601 (119866) = 2 (34)

Combining these formulas withTheorem 18 Carlson andDevelin [42] gave a simple and intuitive proof ofTheorem 74and obtained the following result

Theorem 96 (Carlson and Develin [42] 2006) Let 119866 be aconnected graph which can be embedded on a torus Then119887(119866) ⩽ Δ(119866) + 3

Recently Hou and Liu [50] improved Theorem 96 asfollows

Theorem 97 (Hou and Liu [50] 2012) Let 119866 be a connectedgraph which can be embedded on a torus Then 119887(119866) ⩽ 9Moreover if Δ(119866) = 6 then 119887(119866) ⩽ 8

By the way Cao et al [51] generalized the result inTheorem 79 to a connected graph 119866 that can be embeddedon a torus that is

119887 (119866) le

6 if 119892 (119866) ⩾ 4 and 119866 is not 4-regular5 if 119892 (119866) ⩾ 54 if 119892 (119866) ⩾ 6 and 119866 is not 3-regular3 if 119892 (119866) ⩾ 8

(35)

Several authors used this method to obtain some resultson the bondage number For example Fischermann et al[46] used this method to prove the second conclusion inTheorem 78 Recently Cao et al [52] have used thismethod todeal with more general graphs with small crossing numbersFirst they found the following property

Lemma 98 120575(119866) ⩽ 5 for any graph 119866 with cr(119866) ⩽ 5

16 International Journal of Combinatorics

This result implies that 119887(119866) ⩽ Δ(119866) + 4 by Corollary 15for any graph 119866 with cr(119866) ⩽ 5 Cao et al established thefollowing relation in terms of maximum planar subgraphs

Lemma 99 (Cao et al [52]) Let 119866 be a connected graph withcrossing number cr(119866) For any maximum planar subgraph119867of 119866

sum

119890isin119864(119867)

119863119866(119890) ⩾ 2 minus

2cr (119866)120575 (119866)

(36)

Using Carlson-Develin method and combiningLemma 98 with Lemma 99 Cao et al proved the followingresults

Theorem 100 (Cao et al [52]) For any connected graph 119866119887(119866) ⩽ Δ(119866) + 2 if 119866 satisfies one of the following conditions

(a) cr(119866) ⩽ 3(b) cr(119866) = 4 and 119866 is not 4-regular(c) cr(119866) = 5 and 119866 contains no vertices of degree 4

Theorem101 (Cao et al [52]) Let119866 be a connected graphwithΔ(119866) = 5 and cr(119866) ⩽ 4 If no triangles contain two vertices ofdegree 5 then 119887(119866) ⩽ 6 = Δ(119866) + 1

Theorem 102 (Cao et al [52]) Let 119866 be a connected graphwith Δ(119866) ⩾ 6 and cr(119866) ⩽ 3 If Δ(119866) ⩾ 7 or if Δ(119866) = 6120575(119866) = 3 and every edge 119890 = 119909119910 with 119889

119866(119909) = 5 and 119889

119866(119910) = 6

is contained in atmost one triangle then 119887(119866) ⩽ min8 Δ(119866)+1

Using Carlson-Develin method it can be proved that119887(119866) ⩽ Δ(119866) + 2 if cr(119866) ⩽ 3 (see the first conclusion inTheorem 100) and not yet proved that 119887(119866) ⩽ 8 if cr(119866) ⩽3 although it has been proved by using other method (seeCorollary 90)

By using Carlson-Develin method Samodivkin [25]obtained some results on the bondage number for graphswithsome given properties

Kim [53] showed that 119887(119866) ⩽ Δ(119866) + 2 for a connectedgraph 119866 with genus 1 Δ(119866) ⩽ 5 and having a toroidalembedding of which at least one region is not 4-sidedRecently Gagarin and Zverovich [54] further have extendedCarlson-Develin ideas to establish a nice upper bound forarbitrary graphs that can be embedded on orientable ornonorientable topological surface

Theorem 103 (Gagarin and Zverovich [54] 2012) Let 119866 bea graph embeddable on an orientable surface of genus ℎ and anonorientable surface of genus 119896 Then 119887(119866) ⩽ minΔ(119866)+ℎ+2 Δ(119866) + 119896 + 1

This result generalizes the corresponding upper boundsin Theorems 74 and 96 for any orientable or nonori-entable topological surface By investigating the proof ofTheorem 103 Huang [55] found that the issue of the ori-entability can be avoided by using the Euler characteristic120594(= 120592(119866) minus 120576(119866) + 120601(119866)) instead of the genera ℎ and 119896 the

relations are 120594 = 2minus2ℎ and 120594 = 2minus119896 To obtain the best resultfromTheorem 103 onewants ℎ and 119896 as small as possible thisis equivalent to having 120594 as large as possible

According toTheorem 103 if119866 is planar (ℎ = 0 120594 = 2) orcan be embedded on the real projective plane (119896 = 1 120594 = 1)then 119887(119866) ⩽ Δ(119866) + 2 In all other cases Huang [55] had thefollowing improvement for Theorem 103 the proof is basedon the technique developed byCarlson-Develin andGagarin-Zverovich and includes some elementary calculus as a newingredient mainly the intermediate-value theorem and themean-value theorem

Theorem 104 (Huang [55] 2012) Let 119866 be a graph embed-dable on a surface whose Euler characteristic 120594 is as large aspossible If 120594 ⩽ 0 then 119887(119866) ⩽ Δ(119866)+lfloor119903rfloor where 119903 is the largestreal root of the following cubic equation in 119911

1199113

+ 21199112

+ (6120594 minus 7) 119911 + 18120594 minus 24 = 0 (37)

In addition if 120594 decreases then 119903 increases

The following result is asymptotically equivalent toTheorem 104

Theorem 105 (Huang [55] 2012) Let 119866 be a graph embed-dable on a surface whose Euler characteristic 120594 is as large aspossible If 120594 ⩽ 0 then 119887(119866) lt Δ(119866) + radic12 minus 6120594 + 12 orequivalently 119887(119866) ⩽ Δ(119866) + lceilradic12 minus 6120594 minus 12rceil

Also Gagarin andZverovich [54] indicated that the upperbound in Theorem 103 can be improved for larger values ofthe genera ℎ and 119896 by adjusting the proofs and stated thefollowing general conjecture

Conjecture 106 (Gagarin and Zverovich [54] 2012) For aconnected graph G of orientable genus ℎ and nonorientablegenus 119896

119887 (119866) ⩽ min 119888ℎ 119888

1015840

119896 Δ (119866) + 119900 (ℎ) Δ (119866) + 119900 (119896) (38)

where 119888ℎand 1198881015840

119896are constants depending respectively on the

orientable and nonorientable genera of 119866

In the recent paper Gagarin and Zverovich [56] providedconstant upper bounds for the bondage number of graphs ontopological surfaces which can be used as the first estimationfor the constants 119888

ℎand 1198881015840

119896of Conjecture 106

Theorem 107 (Gagarin and Zverovich [56] 2013) Let 119866 bea connected graph of order 119899 If 119866 can be embedded on thesurface of its orientable or nonorientable genus of the Eulercharacteristic 120594 then

(a) 120594 ⩾ 1 implies 119887(119866) ⩽ 10(b) 120594 ⩽ 0 and 119899 gt minus12120594 imply 119887(119866) ⩽ 11(c) 120594 ⩽ minus1 and 119899 ⩽ minus12120594 imply 119887(119866) ⩽ 11 +

3120594(radic17 minus 8120594 minus 3)(120594 minus 1) = 11 + 119874(radicminus120594)

Theorem 79 shows that a connected planar triangle-freegraph 119866 has 119887(119866) ⩽ 6 The following result generalizes to itall the other topological surfaces

International Journal of Combinatorics 17

Theorem 108 (Gagarin and Zverovich [56] 2013) Let 119866 be aconnected triangle-free graph of order 119899 If 119866 can be embeddedon the surface of its orientable or nonorientable genus of theEuler characteristic 120594 then

(a) 120594 ⩾ 1 implies 119887(119866) ⩽ 6(b) 120594 ⩽ 0 and 119899 gt minus8120594 imply 119887(119866) ⩽ 7(c) 120594 ⩽ minus1 and 119899 ⩽ minus8120594 imply 119887(119866) ⩽ 7minus4120594(1+radic2 minus 120594)

In [56] the authors also explicitly improved upperbounds in Theorem 103 and gave tight lower bounds forthe number of vertices of graphs 2-cell embeddable ontopological surfaces of a given genusThe interested reader isreferred to the original paper for details The following resultis interesting although its proof is easy

Theorem 109 (Samodivkin [57]) Let119866 be a graph with order119899 and girth 119892 If 119866 is embeddable on a surface whose Eulercharacteristic 120594 is as large as possible then

119887 (119866) ⩽ 3 +8

119892 minus 2minus

4120594119892

119899 (119892 minus 2) (39)

If 119866 is a planar graph with girth 119892 ⩾ 4 + 119894 for any 119894 isin0 1 2 then Theorem 109 leads to 119887(119866) ⩽ 6 minus 119894 which isthe result inTheorem 79 obtained by Fischermann et al [46]Also inmany cases the bound stated inTheorem 109 is betterthan those given byTheorems 103 and 105

8 Conditional Bondage Numbers

Since the concept of the bondage number is based upon thedomination all sorts of dominations which are variationsof the normal domination by adding restricted conditionsto the normal dominating set can develop a new ldquobondagenumberrdquo as long as a variation of domination number isgiven In this section we survey results on the bondagenumber under some restricted conditions

81 Total Bondage Numbers A dominating set 119878 of a graph119866 without isolated vertices is called total if the subgraphinduced by 119878 contains no isolated vertices The minimumcardinality of a total dominating set is called the totaldomination number of 119866 and denoted by 120574

119879(119866) It is clear

that 120574(119866) ⩽ 120574119879(119866) ⩽ 2120574(119866) for any graph 119866 without isolated

verticesThe total domination in graphs was introduced by Cock-

ayne et al [58] in 1980 Pfaff et al [59 60] in 1983 showedthat the problem determining total domination number forgeneral graphs is NP-complete even for bipartite graphs andchordal graphs Even now total domination in graphs hasbeen extensively studied in the literature In 2009 Henning[61] gave a survey of selected recent results on total domina-tion in graphs

The total bondage number of 119866 denoted by 119887119879(119866) is the

smallest cardinality of a subset 119861 sube 119864(119866) with the propertythat119866minus119861 contains no isolated vertices and 120574

119879(119866minus119861) gt 120574

119879(119866)

From definition 119887119879(119866)may not exist for some graphs for

example119866 = 1198701119899 We put 119887

119879(119866) = infin if 119887

119879(119866) does not exist

It is easy to see that 119887119879(119866) is finite for any connected graph 119866

other than 1198701 119870

2 119870

3 and 119870

1119899 In fact for such a graph 119866

since there is a path of length 3 in 119866 we can find 1198611sube 119864(119866)

such that 119866 minus 1198611is a spanning tree 119879 of 119866 containing a path

of length 3 So 120574119879(119879) = 120574

119879(119866minus119861

1) ⩾ 120574

119879(119866) For the tree119879 we

find 1198612sube 119864(119879) such that 120574

119879(119879 minus 119861

2) gt 120574

119879(119879) ⩾ 120574

119879(119866) Thus

we have 120574119879(119866minus119861

2minus119861

1) gt 120574

119879(119866) and so 119887

119879(119866) ⩽ |119861

1|+|119861

2| In

the following discussion we always assume that 119887119879(119866) exists

when a graph 119866 is mentionedIn 1991 Kulli and Patwari [62] first studied the total

bondage number of a graph and calculated the exact valuesof 119887

119879(119866) for some standard graphs

Theorem 110 (Kulli and Patwari [62] 1991) For 119899 ⩾ 4

119887119879(119862

119899) =

3 119894119891 119899 equiv 2 (mod 4) 2 119900119905ℎ119890119903119908119894119904119890

119887119879(119875

119899) =

2 119894119891 119899 equiv 2 (mod 4) 1 119900119905ℎ119890119903119908119894119904119890

119887119879(119870

119898119899) = 119898 119908119894119905ℎ 2 ⩽ 119898 ⩽ 119899

119887119879(119870

119899) =

4 119891119900119903 119899 = 4

2119899 minus 5 119891119900119903 119899 ⩾ 5

(40)

Recently Hu et al [63] have obtained some results on thetotal bondage number of the Cartesian product119875

119898times119875

119899of two

paths 119875119898and 119875

119899

Theorem 111 (Hu et al [63] 2012) For the Cartesian product119875119898times 119875

119899of two paths 119875

119898and 119875

119899

119887119879(119875

2times 119875

119899) =

1 119894119891 119899 equiv 0 (mod 3) 2 119894119891 119899 equiv 2 (mod 3) 3 119894119891 119899 equiv 1 (mod 3)

119887119879(119875

3times 119875

119899) = 1

119887119879(119875

4times 119875

119899)

= 1 119894119891 119899 equiv 1 (mod 5) = 2 119894119891 119899 equiv 4 (mod 5) ⩽ 3 119894119891 119899 equiv 2 (mod 5) ⩽ 4 119894119891 119899 equiv 0 3 (mod 5)

(41)

Generalized Petersen graphs are an important class ofcommonly used interconnection networks and have beenstudied recently By constructing a family of minimum totaldominating sets Cao et al [64] determined the total bondagenumber of the generalized Petersen graphs

FromTheorem 6 we know that 119887(119879) ⩽ 2 for a nontrivialtree 119879 But given any positive integer 119896 Sridharan et al [65]constructed a tree 119879 for which 119887

119879(119879) = 119896 Let119867

119896be the tree

obtained from a star1198701119896+1

by subdividing 119896 edges twice Thetree 119867

7is shown in Figure 6 It can be easily verified that

119887119879(119867

119896) = 119896 This fact can be stated as the following theorem

Theorem 112 (Sridharan et al [65] 2007) For any positiveinteger 119896 there exists a tree 119879 with 119887

119879(119879) = 119896

18 International Journal of Combinatorics

Figure 61198677

Combining Theorem 1 with Theorem 112 we have whatfollows

Corollary 113 (Sridharan et al [65] 2007) The bondagenumber and the total bondage number are unrelated even fortrees

However Sridharan et al [65] gave an upper bound onthe total bondage number of a tree in terms of its maximumdegree For any tree 119879 of order 119899 if 119879 =119870

1119899minus1 then 119887

119879(119879) =

minΔ(119879) (13)(119899 minus 1) Rad and Raczek [66] improved thisupper bound and gave a constructive characterization of acertain class of trees attaching the upper bound

Theorem 114 (Rad and Raczek [66] 2011) 119887119879(119879) ⩽ Δ(119879) minus 1

for any tree 119879 with maximum degree at least three

In general the decision problem for 119887119879(119866) is NP-complete

for any graph 119866 We state the decision problem for the totalbondage as follows

Problem 3 Consider the decision problemTotal Bondage ProblemInstance a graph 119866 and a positive integer 119887Question is 119887

119879(119866) ⩽ 119887

Theorem 115 (Hu and Xu [16] 2012) The total bondageproblem is NP-complete

Consequently in view of its computational hardnessit is significative to establish some sharp bounds on thetotal bondage number of a graph in terms of other graphicparameters In 1991 Kulli and Patwari [62] showed that119887119879(119866) ⩽ 2119899 minus 5 for any graph 119866 with order 119899 ⩾ 5 Sridharan

et al [65] improved this result as follows

Theorem 116 (Sridharan et al [65] 2007) For any graph 119866with order 119899 if 119899 ⩾ 5 then

119887119879(119866) ⩽

1

3(119899 minus 1) 119894119891 119866 119888119900119899119905119886119894119899119904 119899119900 119888119910119888119897119890119904

119899 minus 1 119894119891 119892 (119866) ⩾ 5

119899 minus 2 119894119891 119892 (119866) = 4

(42)

Rad and Raczek [66] also established some upperbounds on 119887

119879(119866) for a general graph 119866 In particular they

gave an upper bound on 119887119879(119866) in terms of the girth of

119866

Theorem 117 (Rad and Raczek [66] 2011) 119887119879(119866)⩽4Δ(119866) minus 5

for any graph 119866 with 119892(119866) ⩾ 4

82 Paired Bondage Numbers A dominating set 119878 of 119866 iscalled to be paired if the subgraph induced by 119878 containsa perfect matching The paired domination number of 119866denoted by 120574

119875(119866) is the minimum cardinality of a paired

dominating set of 119866 Clearly 120574119879(119866) ⩽ 120574

119875(119866) for every

connected graph 119866 with order at least two where 120574119879(119866) is

the total domination number of 119866 and 120574119875(119866) ⩽ 2120574(119866) for

any graph119866without isolated vertices Paired domination wasintroduced by Haynes and Slater [67 68] and further studiedin [69ndash72]

The paired bondage number of 119866 with 120575(119866) ⩾ 1 denotedby 119887P(119866) is the minimum cardinality among all sets of edges119861 sube 119864 such that 120575(119866 minus 119861) ⩾ 1 and 120574

119875(119866 minus 119861) gt 120574

119875(119866)

The concept of the paired bondage number was firstproposed by Raczek [73] in 2008The following observationsfollow immediately from the definition of the paired bondagenumber

Observation 2 Let 119866 be a graph with 120575(119866) ⩾ 1

(a) If 119867 is a subgraph of 119866 such that 120575(119867) ⩾ 1 then119887119875(119867) ⩽ 119887

119875(119866)

(b) If 119867 is a subgraph of 119866 such that 119887119875(119867) = 1 and 119896

is the number of edges removed to form119867 then 1 ⩽119887119875(119866) ⩽ 119896 + 1

(c) If 119909119910 isin 119864(119866) such that 119889119866(119909) 119889

119866(119910) gt 1 and 119909119910

belongs to each perfect matching of each minimumpaired dominating set of 119866 then 119887

119875(119866) = 1

Based on these observations 119887119875(119875

119899) ⩽ 2 In fact the

paired bondage number of a path has been determined

Theorem 118 (Raczek [73] 2008) For any positive integer 119896if 119899 ⩾ 2 then

119887119875(119875

119899) =

0 119894119891 119899 = 2 3 119900119903 5

1 119894119891 119899 = 4119896 4119896 + 3 119900119903 4119896 + 6

2 119900119905ℎ119890119903119908119894119904119890

(43)

For a cycle 119862119899of order 119899 ⩾ 3 since 120574

119875(119862

119899) = 120574

119875(119875

119899) from

Theorem 118 the following result holds immediately

Corollary 119 (Raczek [73] 2008) For any positive integer 119896if 119899 ⩾ 3 then

119887119875(119862

119899) =

0 119894119891 119899 = 3 119900119903 5

2 119894119891 119899 = 4119896 4119896 + 3 119900119903 4119896 + 6

3 119900119905ℎ119890119903119908119894119904119890

(44)

A wheel 119882119899 where 119899 ⩾ 4 is a graph with 119899 vertices

formed by connecting a single vertex to all vertices of a cycle119862119899minus1

Of course 120574119875(119882

119899) = 2

International Journal of Combinatorics 19

119896

Figure 7 A tree 119879 with 119887119875(119879) = 119896

Theorem 120 (Raczek [73] 2008) For positive integers 119899 and119898

119887119875(119870

119898119899) = 119898 119891119900119903 1 lt 119898 ⩽ 119899

119887119875(119882

119899) =

4 119894119891 119899 = 4

3 119894119891 119899 = 5

2 119900119905ℎ119890119903119908119894119904119890

(45)

Theorem 16 shows that if 119879 is a nontrivial tree then119887(119879) ⩽ 2 However no similar result exists for pairedbondage For any nonnegative integer 119896 let 119879

119896be a tree

obtained by subdividing all but one edge of a star 1198701119896+1

(asshown in Figure 7) It is easy to see that 119887

119875(119879

119896) = 119896 We

state this fact as the following theorem which is similar toTheorem 112

Theorem121 (Raczek [73] 2008) For any nonnegative integer119896 there exists a tree 119879 with 119887

119875(119879) = 119896

Consider the tree defined in Figure 5 for 119896 = 3 Then119887(119879

3) ⩽ 2 and 119887

119875(119879

3) = 3 On the other hand 119887(119875

4) = 2

and 119887119875(119875

4) = 1 Thus we have what follows which is similar

to Corollary 113

Corollary 122 The bondage number and the paired bondagenumber are unrelated even for trees

A constructive characterization of trees with 119887(119879) = 2is given by Hartnell and Rall in [12] Raczek [73] provideda constructive characterization of trees with 119887

119875(119879) = 0 In

order to state the characterization we define a labeling andthree simple operations on a tree119879 Let 119910 isin 119881(119879) and let ℓ(119910)be the label assigned to 119910

Operation T1 If ℓ(119910) = 119861 add a vertex 119909 and the

edge 119910119909 and let ℓ(119909) = 119860OperationT

2 If ℓ(119910) = 119862 add a path (119909

1 119909

2) and the

edge 1199101199091 and let ℓ(119909

1) = 119861 ℓ(119909

2) = 119860

Operation T3 If ℓ(119910) = 119861 add a path (119909

1 119909

2 119909

3)

and the edge 1199101199091 and let ℓ(119909

1) = 119862 ℓ(119909

2) = 119861 and

ℓ(1199093) = 119860

Let 1198752= (119906 V) with ℓ(119906) = 119860 and ℓ(V) = 119861 Let T be

the class of all trees obtained from the labeled 1198752by a finite

sequence of operationsT1 T

2 T

3

A tree 119879 in Figure 8 belongs to the familyTRaczek [73] obtained the following characterization of all

trees 119879 with 119887119875(119879) = 0

119860

119860

119860

119860

119860

119861 119862

119861 119862 119861

119861119860

Figure 8 A tree 119879 belong to the familyT

Theorem 123 (Raczek [73] 2008) For a tree 119879 119887119875(119879) = 0 if

and only if 119879 is inT

We state the decision problem for the paired bondage asfollows

Problem 4 Consider the decision problem

Paired Bondage ProblemInstance a graph 119866 and a positive integer 119887Question is 119887

119875(119866) ⩽ 119887

Conjecture 124 The paired bondage problem is NP-complete

83 Restrained Bondage Numbers A dominating set 119878 of 119866 iscalled to be restrained if the subgraph induced by 119878 containsno isolated vertices where 119878 = 119881(119866) 119878 The restraineddomination number of 119866 denoted by 120574

119877(119866) is the smallest

cardinality of a restrained dominating set of 119866 It is clear that120574119877(119866) exists and 120574(119866) ⩽ 120574

119877(119866) for any nonempty graph 119866

The concept of restrained domination was introducedby Telle and Proskurowski [74] in 1997 albeit indirectly asa vertex partitioning problem Concerning the study of therestrained domination numbers the reader is referred to [74ndash79]

In 2008 Hattingh and Plummer [80] defined therestrained bondage number 119887R(119866) of a nonempty graph 119866 tobe the minimum cardinality among all subsets 119861 sube 119864(119866) forwhich 120574

119877(119866 minus 119861) gt 120574

119877(119866)

For some simple graphs their restrained dominationnumbers can be easily determined and so restrained bondagenumbers have been also determined For example it is clearthat 120574

119877(119870

119899) = 1 Domke et al [77] showed that 120574

119877(119862

119899) =

119899 minus 2lfloor1198993rfloor for 119899 ⩾ 3 and 120574119877(119875

119899) = 119899 minus 2lfloor(119899 minus 1)3rfloor for 119899 ⩾ 1

Using these results Hattingh and Plummer [80] obtained therestricted bondage numbers for119870

119899 119862

119899 119875

119899 and119866 = 119870

11989911198992119899119905

as follows

Theorem 125 (Hattingh and Plummer [80] 2008) For 119899 ⩾ 3

119887R (119870119899) =

1 119894119891 119899 = 3

lceil119899

2rceil 119900119905ℎ119890119903119908119894119904119890

119887R (119862119899) =

1 119894119891 119899 equiv 0 (mod 3) 2 119900119905ℎ119890119903119908119894119904119890

119887R (119875119899) = 1 119899 ⩾ 4

20 International Journal of Combinatorics

119887R (119866)

=

lceil119898

2rceil 119894119891 119899

119898= 1 119886119899119889 119899

119898+1⩾ 2 (1 ⩽ 119898 lt 119905)

2119905 minus 2 119894119891 1198991= 119899

2= sdot sdot sdot = 119899

119905= 2 (119905 ⩾ 2)

2 119894119891 1198991= 2 119886119899119889 119899

2⩾ 3 (119905 = 2)

119905minus1

sum

119894=1

119899119894minus 1 119900119905ℎ119890119903119908119894119904119890

(46)

Theorem 126 (Hattingh and Plummer [80] 2008) For a tree119879 of order 119899 (⩾ 4) 119879 ≇ 119870

1119899minus1if and only if 119887R(119879) = 1

Theorem 126 shows that the restrained bondage numberof a tree can be computed in constant time However thedecision problem for 119887R(119866) is NP-complete even for bipartitegraphs

Problem 5 Consider the decision problem

Restrained Bondage Problem

Instance a graph 119866 and a positive integer 119887

Question is 119887R(119866) ⩽ 119887

Theorem 127 (Hattingh and Plummer [80] 2008) Therestrained bondage problem is NP-complete even for bipartitegraphs

Consequently in view of its computational hardnessit is significative to establish some sharp bounds on therestrained bondage number of a graph in terms of othergraphic parameters

Theorem 128 (Hattingh and Plummer [80] 2008) For anygraph 119866 with 120575(119866) ⩾ 2

119887R (119866) ⩽ min119909119910isin119864(119866)

119889119866(119909) + 119889

119866(119910) minus 2 (47)

Corollary 129 119887R(119866) ⩽ Δ(119866)+120575(119866)minus2 for any graph119866 with120575(119866) ⩾ 2

Notice that the bounds stated in Theorem 128 andCorollary 129 are sharp Indeed the class of cycles whoseorders are congruent to 1 2 (mod 3) have a restrained bondagenumber achieving these bounds (see Theorem 125)

Theorem 128 is an analogue of Theorem 14 A quitenatural problem is whether or not for restricted bondagenumber there are analogues of Theorem 16 Theorem 18Theorem 29 and so on Indeed we should consider all resultsfor the bondage number whether or not there are analoguesfor restricted bondage number

Theorem 130 (Hattingh and Plummer [80] 2008) For anygraph 119866 with 120574

119877(119866) = 2

119887R (119866) ⩽ Δ (119866) + 1 (48)

We now consider the relation between 119887119877(119866) and 119887(119866)

Theorem 131 (Hattingh and Plummer [80] 2008) If 120574119877(119866) =

120574(119866) for some graph 119866 then 119887R(119866) ⩽ 119887(119866)

Corollary 129 and Theorem 131 provide a possibility toattack Conjecture 73 by characterizing planar graphs with120574119877= 120574 and 119887R = 119887 when 3 ⩽ Δ ⩽ 6However we do not have 119887R(119866) = 119887(119866) for any graph

119866 even if 120574119877(119866) = 120574(119866) Observe that 120574

119877(119870

3) = 120574(119870

3) yet

119887119877(119870

3) = 1 and 119887(119870

3) = 2 We still may not claim that

119887119877(119866) = 119887(119866) even in the case that every 120574-set is a 120574

119877-set

The example 1198703again demonstrates this

Theorem 132 (Hattingh and Plummer [80] 2008) There isan infinite class of graphs inwhich each graph119866 satisfies 119887(119866) lt119887R(119866)

In fact such an infinite class of graphs can be obtained asfollows Let119867 be a connected graph BC(119867) a graph obtainedfrom 119867 by attaching ℓ (⩾ 2) vertices of degree 1 to eachvertex in 119867 and B = 119866 119866 = BC(119867) for some graph 119867such that 120575(119867) ⩾ 2 By Theorem 3 we can easily check that119887(119866) = 1 lt 2 ⩽ min120575(119867) ℓ = 119887R(119866) for any 119866 isin BMoreover there exists a graph 119866 isin B such that 119887R(119866) canbe much larger than 119887(119866) which is stated as the followingtheorem

Theorem 133 (Hattingh and Plummer [80] 2008) For eachpositive integer 119896 there is a graph119866 such that 119896 = 119887R(119866)minus119887(119866)

Combining Theorem 133 with Theorem 132 we have thefollowing corollary

Corollary 134 The bondage number and the restrainedbondage number are unrelated

Very recently Jafari Rad et al [81] have consideredthe total restrained bondage in graphs based on both totaldominating sets and restrained dominating sets and obtainedseveral properties exact values and bounds for the totalrestrained bondage number of a graph

A restrained dominating set 119878 of a graph 119866 without iso-lated vertices is called to be a total restrained dominating setif 119878 is a total dominating set The total restrained dominationnumber of119866 denoted by 120574TR (119866) is theminimum cardinalityof a total restrained dominating set of 119866 For referenceson this domination in graphs see [3 61 82ndash85] The totalrestrained bondage number 119887TR (119866) of a graph 119866 with noisolated vertex is the cardinality of a smallest set of edges 119861 sube119864(119866) for which119866minus119861 has no isolated vertex and 120574TR (119866minus119861) gt120574TR (119866) In the case that there is no such subset 119861 we define119887TR (119866) = infin

Ma et al [84] determined that 120574TR (119875119899) = 119899minus2lfloor(119899minus2)4rfloorfor 119899 ⩾ 3 120574TR (119862119899

) = 119899 minus 2lfloor1198994rfloor for 119899 ⩾ 3 120574TR (1198703) =

3 and 120574TR (119870119899) = 2 for 119899 = 3 and 120574TR (1198701119899

) = 1 + 119899

and 120574TR (119870119898119899) = 2 for 2 ⩽ 119898 ⩽ 119899 According to these

results Jafari Rad et al [81] determined the exact valuesof total restrained bondage numbers for the correspondinggraphs

International Journal of Combinatorics 21

Theorem 135 (Jafari Rad et al [81]) For an integer 119899

119887TR (119875119899) = infin 119894119891 2 ⩽ 119899 ⩽ 5

1 119894119891 119899 ⩾ 5

119887TR (119862119899) =

infin 119894119891 119899 = 3

1 119894119891 3 lt 119899 119899 equiv 0 1 (mod 4)2 119894119891 3 lt 119899 119899 equiv 2 3 (mod 4)

119887TR (119882119899) = 2 119891119900119903 119899 ⩾ 6

119887TR (119870119899) = 119899 minus 1 119891119900119903 119899 ⩾ 4

119887TR (119870119898119899) = 119898 minus 1 119891119900119903 2 ⩽ 119898 ⩽ 119899

(49)

119887TR (119870119899) = 119899minus1 shows the fact that for any integer 119899 ⩾ 3 there

exists a connected graph namely119870119899+1

such that 119887TR (119870119899+1) =

119899 that is the total restrained bondage number is unboundedfrom the above

A vertex is said to be a support vertex if it is adjacent toa vertex of degree one Let A be the family of all connectedgraphs such that119866 belongs toA if and only if every edge of119866is incident to a support vertex or 119866 is a cycle of length three

Proposition 136 (Cyman and Raczek [82] 2006) For aconnected graph119866 of order 119899 120574TR (119866) = 119899 if and only if119866 isin A

Hence if 119866 isin A then the removal of any subset of edgesof119866 cannot increase the total restrained domination numberof 119866 and thus 119887TR (119866) = infin When 119866 notin A a result similar toTheorem 6 is obtained

Theorem 137 (Jafari Rad et al [81]) For any tree 119879 notin A119887TR (119879) ⩽ 2 This bound is sharp

In the samepaper the authors provided a characterizationfor trees 119879 notin A with 119887TR (119879) = 1 or 2 The following result issimilar to Observation 1

Observation 3 (Jafari Rad et al [81]) If 119896 edges can beremoved from a graph 119866 to obtain a graph 119867 without anisolated vertex and with 119887TR (119867) = 119905 then 119887TR (119866) ⩽ 119896 + 119905

ApplyingTheorem 137 andObservation 3 Jafari Rad et alcharacterized all connected graphs 119866 with 119887TR (119866) = infin

Theorem 138 (Jafari Rad et al [81]) For a connected graph119866119887TR (119866) = infin if and only if 119866 isin A

84 Other Conditional Bondage Numbers A subset 119868 sube 119881(119866)is called an independent set if no two vertices in 119868 are adjacentin 119866 The maximum cardinality among all independent setsis called the independence number of 119866 denoted by 120572(119866)

A dominating set 119878 of a graph 119866 is called to be inde-pendent if 119878 is an independent set of 119866 The minimumcardinality among all independent dominating set is calledthe independence domination number of 119866 and denoted by120574119868(119866)

Since an independent dominating set is not only adominating set but also an independent set 120574(119866) ⩽ 120574

119868(119866) ⩽

120572(119866) for any graph 119866It is clear that a maximal independent set is certainly a

dominating set Thus an independent set is maximal if andonly if it is an independent dominating set and so 120574

119868(119866) is the

minimum cardinality among all maximal independent setsof 119866 This graph-theoretical invariant has been well studiedin the literature see for example Haynes et al [3] for earlyresults Goddard and Henning [86] for recent results onindependent domination in graphs

In 2003 Zhang et al [87] defined the independencebondage number 119887

119868(119866) of a nonempty graph 119866 to be the

minimum cardinality among all subsets 119861 sube 119864(119866) for which120574119868(119866 minus 119861) gt 120574

119868(119866) For some ordinary graphs their indepen-

dence domination numbers can be easily determined and soindependence bondage numbers have been also determinedClearly 119887

119868(119870

119899) = 1 if 119899 ⩾ 2

Theorem 139 (Zhang et al [87] 2003) For any integer 119899 ⩾ 4

119887119868(119862

119899) =

1 119894119891 119899 equiv 1 (mod 2) 2 119894119891 119899 equiv 0 (mod 2)

119887119868(119875

119899) =

1 119894119891 119899 equiv 0 (mod 2) 2 119894119891 119899 equiv 1 (mod 2)

119887119868(119870

119898119899) = 119899 119891119900119903 119898 ⩽ 119899

(50)

Apart from the above-mentioned results as far as we findno other results on the independence bondage number in thepresent literature we never knew much of any result on thisparameter for a tree

A subset 119878 of 119881(119866) is called an equitable dominatingset if for every 119910 isin 119881(119866 minus 119878) there exists a vertex 119909 isin 119878such that 119909119910 isin 119864(119866) and |119889

119866(119909) minus 119889

119866(119910)| ⩽ 1 proposed

by Dharmalingam [88] The minimum cardinality of such adominating set is called the equitable domination number andis denoted by 120574

119864(119866) Deepak et al [89] defined the equitable

bondage number 119887119864(119866) of a graph 119866 to be the cardinality of a

smallest set 119861 sube 119864(119866) of edges for which 120574119864(119866 minus 119861) gt 120574

119864(119866)

and determined 119887119864(119866) for some special graphs

Theorem 140 (Deepak et al [89] 2011) For any integer 119899 ⩾ 2

119887119864(119870

119899) = lceil

119899

2rceil

119887119864(119870

119898119899) = 119898 119891119900119903 0 ⩽ 119899 minus 119898 ⩽ 1

119887119864(119862

119899) =

3 119894119891 119899 equiv 1 (mod 3)2 119900119905ℎ119890119903119908119894119904119890

119887119864(119875

119899) =

2 119894119891 119899 equiv 1 (mod 3)1 119900119905ℎ119890119903119908119894119904119890

119887119864(119879) ⩽ 2 119891119900119903 119886119899119910 119899119900119899119905119903119894119907119894119886119897 119905119903119890119890 119879

119887119864(119866) ⩽ 119899 minus 1 119891119900119903 119886119899119910 119888119900119899119899119890119888119905119890119889 119892119903119886119901ℎ 119866 119900119891 119900119903119889119890119903 119899

(51)

22 International Journal of Combinatorics

As far as we know there are no more results on theequitable bondage number of graphs in the present literature

There are other variations of the normal domination byadding restricted conditions to the normal dominating setFor example the connected domination set of a graph119866 pro-posed by Sampathkumar and Walikar [90] is a dominatingset 119878 such that the subgraph induced by 119878 is connected Theproblem of finding a minimum connected domination set ina graph is equivalent to the problemof finding a spanning treewithmaximumnumber of vertices of degree one [91] and hassome important applications in wireless networks [92] thusit has received much research attention However for suchan important domination there is no result on its bondagenumber We believe that the topic is of significance for futureresearch

9 Generalized Bondage Numbers

There are various generalizations of the classical dominationsuch as 119901-domination distance domination fractional dom-ination Roman domination and rainbow domination Everysuch a generalization can lead to a corresponding bondageIn this section we introduce some of them

91 119901-Bondage Numbers In 1985 Fink and Jacobson [93]introduced the concept of 119901-domination Let 119901 be a positiveinteger A subset 119878 of 119881(119866) is a 119901-dominating set of 119866 if|119878 cap 119873

119866(119910)| ⩾ 119901 for every 119910 isin 119878 The 119901-domination number

120574119901(119866) is the minimum cardinality among all 119901-dominating

sets of 119866 Any 119901-dominating set of 119866 with cardinality 120574119901(119866)

is called a 120574119901-set of 119866 Note that a 120574

1-set is a classic minimum

dominating set Notice that every graph has a 119901-dominatingset since the vertex-set119881(119866) is such a setWe also note that a 1-dominating set is a dominating set and so 120574(119866) = 120574

1(119866) The

119901-domination number has received much research attentionsee a state-of-the-art survey article by Chellali et al [94]

It is clear from definition that every 119901-dominating setof a graph certainly contains all vertices of degree at most119901 minus 1 By this simple observation to avoid the trivial caseoccurrence we always assume that Δ(119866) ⩾ 119901 For 119901 ⩾ 2 Luet al [95] gave a constructive characterization of trees withunique minimum 119901-dominating sets

Recently Lu and Xu [96] have introduced the conceptto the 119901-bondage number of 119866 denoted by 119887

119901(119866) as the

minimum cardinality among all sets of edges 119861 sube 119864(119866) suchthat 120574

119901(119866 minus 119861) gt 120574

119901(119866) Clearly 119887

1(119866) = 119887(119866)

Lu and Xu [96] established a lower bound and an upperbound on 119887

119901(119879) for any integer 119901 ⩾ 2 and any tree 119879 with

Δ(119879) ⩾ 119901 and characterized all trees achieving the lowerbound and the upper bound respectively

Theorem 141 (Lu and Xu [96] 2011) For any integer 119901 ⩾ 2and any tree 119879 with Δ(119879) ⩾ 119901

1 ⩽ 119887119901(119879) ⩽ Δ (119879) minus 119901 + 1 (52)

Let 119878 be a given subset of 119881(119866) and for any 119909 isin 119878 let

119873119901(119909 119878 119866) = 119910 isin 119878 cap 119873

119866(119909)

1003816100381610038161003816119873119866(119910) cap 119878

1003816100381610038161003816 = 119901

(53)

Then all trees achieving the lower bound 1 can be character-ized as follows

Theorem 142 (Lu and Xu [96] 2011) Let 119879 be a tree withΔ(119879) ⩾ 119901 ⩾ 2 Then 119887

119901(119879) = 1 if and only if for any 120574

119901-set

119878 of 119879 there exists an edge 119909119910 isin (119878 119878) such that 119910 isin 119873119901(119909 119878 119879)

The notation 119878(119886 119887) denotes a double star obtained byadding an edge between the central vertices of two stars119870

1119886minus1

and1198701119887minus1

And the vertex with degree 119886 (resp 119887) in 119878(119886 119887) iscalled the 119871-central vertex (resp 119877-central vertex) of 119878(119886 119887)

To characterize all trees attaining the upper bound givenin Theorem 141 we define three types of operations on a tree119879 with Δ(119879) = Δ ⩾ 119901 + 1

Type 1 attach a pendant edge to a vertex 119910with 119889119879(119910) ⩽ 119901minus

2 in 119879

Type 2 attach a star 1198701Δminus1

to a vertex 119910 of 119879 by joining itscentral vertex to 119910 where 119910 in a 120574

119901-set of 119879 and

119889119879(119910) ⩽ Δ minus 1

Type 3 attach a double star 119878(119901 Δ minus 1) to a pendant vertex 119910of 119879 by coinciding its 119877-central vertex with 119910 wherethe unique neighbor of 119910 is in a 120574

119901-set of 119879

Let B = 119879 a tree obtained from 1198701Δ

or 119878(119901 Δ) by afinite sequence of operations of Types 1 2 and 3

Theorem 143 (Lu and Xu [96] 2011) A tree with the maxi-mum Δ ⩾ 119901 + 1 has 119901-bondage number Δ minus 119901 + 1 if and onlyif it belongs toB

Theorem 144 (Lu and Xu [97] 2012) Let 119866119898119899= 119875

119898times 119875

119899

Then

1198872(119866

2119899) = 1 119891119900119903 119899 ⩾ 2

1198872(119866

3119899) =

2 119894119891 119899 equiv 1 (mod 3) 1 119900119905ℎ119890119903119908119894119904119890

for 119899 ⩾ 2

1198872(119866

4119899) =

1 119894119891 119899 equiv 3 (mod 4) 2 119900119905ℎ119890119903119908119894119904119890

for 119899 ⩾ 7

(54)

Recently Krzywkowski [98] has proposed the conceptof the nonisolating 2-bondage of a graph The nonisolating2-bondage number of a graph 119866 denoted by 1198871015840

2(119866) is the

minimum cardinality among all sets of edges 119861 sube 119864(119866) suchthat 119866 minus 119861 has no isolated vertices and 120574

2(119866 minus 119861) gt 120574

2(119866) If

for every 119861 sube 119864(119866) either 1205742(119866 minus 119861) = 120574

2(119866) or 119866 minus 119861 has

isolated vertices then define 11988710158402(119866) = 0 and say that 119866 is a 2-

nonisolatedly strongly stable graph Krzywkowski presentedsome basic properties of nonisolating 2-bondage in graphsshowed that for every nonnegative integer 119896 there exists atree 119879 such 1198871015840

2(119879) = 119896 characterized all 2-non-isolatedly

strongly stable trees and determined 11988710158402(119866) for several special

graphs

International Journal of Combinatorics 23

Theorem 145 (Krzywkowski [98] 2012) For any positiveintegers 119899 and119898 with119898 ⩽ 119899

1198871015840

2(119870

119899) =

0 119894119891 119899 = 1 2 3

lfloor2119899

3rfloor 119900119905ℎ119890119903119908119894119904119890

1198871015840

2(119875

119899) =

0 119894119891 119899 = 1 2 3

1 119900119905ℎ119890119903119908119894119904119890

1198871015840

2(119862

119899) =

0 119894119891 119899 = 3

1 119894119891 119899 119894119904 119890V119890119899

2 119894119891 119899 119894119904 119900119889119889

1198871015840

2(119870

119898119899) =

3 119894119891 119898 = 119899 = 3

1 119894119891 119898 = 119899 = 4

2 119900119905ℎ119890119903119908119894119904119890

(55)

92 Distance Bondage Numbers A subset 119878 of vertices of agraph119866 is said to be a distance 119896-dominating set for119866 if everyvertex in 119866 not in 119878 is at distance at most 119896 from some vertexof 119878 The minimum cardinality of all distance 119896-dominatingsets is called the distance 119896-domination number of 119866 anddenoted by 120574

119896(119866) (do not confuse with the above-mentioned

120574119901(119866)) When 119896 = 1 a distance 1-dominating set is a normal

dominating set and so 1205741(119866) = 120574(119866) for any graph 119866 Thus

the distance 119896-domination is a generalization of the classicaldomination

The relation between 120574119896and 120572

119896(the 119896-independence

number defined in Section 22) for a tree obtained by Meirand Moon [14] who proved that 120574

119896(119879) = 120572

2119896(119879) for any tree

119879 (a special case for 119896 = 1 is stated in Proposition 10) Thefurther research results can be found in Henning et al [99]Tian and Xu [100ndash103] and Liu et al [104]

In 1998 Hartnell et al [15] defined the distance 119896-bondagenumber of 119866 denoted by 119887

119896(119866) to be the cardinality of a

smallest subset 119861 of edges of 119866 with the property that 120574119896(119866 minus

119861) gt 120574119896(119866) FromTheorem 6 it is clear that if119879 is a nontrivial

tree then 1 ⩽ 1198871(119879) ⩽ 2 Hartnell et al [15] generalized this

result to any integer 119896 ⩾ 1

Theorem 146 (Hartnell et al [15] 1998) For every nontrivialtree 119879 and positive integer 119896 1 ⩽ 119887

119896(119879) ⩽ 2

Hartnell et al [15] and Topp and Vestergaard [13] alsocharacterized the trees having distance 119896-bondage number 2In particular the class of trees for which 119887

1(119879) = 2 are just

those which have a unique maximum 2-independent set (seeTheorem 11)

Since when 119896 = 1 the distance 1-bondage number 1198871(119866)

is the classical bondage number 119887(119866) Theorem 12 gives theNP-hardness of deciding the distance 119896-bondage number ofgeneral graphs

Theorem 147 Given a nonempty undirected graph 119866 andpositive integers 119896 and 119887 with 119887 ⩽ 120576(119866) determining wetheror not 119887

119896(119866) ⩽ 119887 is NP-hard

For a vertex 119909 in119866 the open 119896-neighborhood119873119896(119909) of 119909 is

defined as119873119896(119909) = 119910 isin 119881(119866) 1 ⩽ 119889

119866(119909 119910) le 119896The closed

119896-neighborhood119873119896[119909] of 119909 in 119866 is defined as119873

119896(119909) cup 119909 Let

Δ119896(119866) = max 1003816100381610038161003816119873119896

(119909)1003816100381610038161003816 for any 119909 isin 119881 (119866) (56)

Clearly Δ1(119866) = Δ(119866) The 119896th power of a graph 119866 is the

graph119866119896 with vertex-set119881(119866119896

) = 119881(119866) and edge-set119864(119866119896

) =

119909119910 1 ⩽ 119889119866(119909 119910) ⩽ 119896 The following lemma holds directly

from the definition of 119866119896

Proposition 148 (Tian and Xu [101]) Δ(119866119896

) = Δ119896(119866) and

120574(119866119896

) = 120574119896(119866) for any graph 119866 and each 119896 ⩾ 1

A graph 119866 is 119896-distance domination-critical or 120574119896-critical

for short if 120574119896(119866 minus 119909) lt 120574

119896(119866) for every vertex 119909 in 119866

proposed by Henning et al [105]

Proposition 149 (Tian and Xu [101]) For each 119896 ⩾ 1 a graph119866 is 120574

119896-critical if and only if 119866119896 is 120574

119896-critical

By the above facts we suggest the following worthwhileproblem

Problem 6 Can we generalize the results on the bondage for119866 to 119866119896 In particular do the following propositions hold

(a) 119887(119866119896

) = 119887119896(119866) for any graph 119866 and each 119896 ⩾ 1

(b) 119887(119866119896

) ⩽ Δ119896(119866) if 119866 is not 120574

119896-critical

Let 119896 and 119901 be positive integers A subset 119878 of 119881(119866) isdefined to be a (119896 119901)-dominating set of 119866 if for any vertex119909 isin 119878 |119873

119896(119909) cap 119878| ⩾ 119901 The (119896 119901)-domination number of

119866 denoted by 120574119896119901(119866) is the minimum cardinality among

all (119896 119901)-dominating sets of 119866 Clearly for a graph 119866 a(1 1)-dominating set is a classical dominating set a (119896 1)-dominating set is a distance 119896-dominating set and a (1 119901)-dominating set is the above-mentioned 119901-dominating setThat is 120574

11(119866) = 120574(119866) 120574

1198961(119866) = 120574

119896(119866) and 120574

1119901(119866) = 120574

119901(119866)

The concept of (119896 119901)-domination in a graph 119866 is a gen-eralized domination which combines distance 119896-dominationand 119901-domination in 119866 So the investigation of (119896 119901)-domination of 119866 is more interesting and has received theattention of many researchers see for example [106ndash109]

More general Li and Xu [110] proposed the concept of the(ℓ 119908)-domination number of a119908-connected graph A subset119878 of 119881(119866) is defined to be an (ℓ 119908)-dominating set of 119866 iffor any vertex 119909 isin 119878 there are 119908 internally disjoint pathsof length at most ℓ from 119909 to some vertex in 119878 The (ℓ 119908)-domination number of119866 denoted by 120574

ℓ119908(119866) is theminimum

cardinality among all (ℓ 119908)-dominating sets of 119866 Clearly1205741198961(119866) = 120574

119896(119866) and 120574

1119901(119866) = 120574

119901(119866)

It is quite natural to propose the concept of bondage num-ber for (119896 119901)-domination or (ℓ 119908)-domination However asfar as we know none has proposed this concept until todayThis is a worthwhile topic for further research

24 International Journal of Combinatorics

93 Fractional Bondage Numbers If 120590 is a function mappingthe vertex-set 119881 into some set of real numbers then for anysubset 119878 sube 119881 let 120590(119878) = sum

119909isin119878120590(119909) Also let |120590| = 120590(119881)

A real-value function 120590 119881 rarr [0 1] is a dominatingfunction of a graph 119866 if for every 119909 isin 119881(119866) 120590(119873

119866[119909]) ⩾ 1

Thus if 119878 is a dominating set of 119866 120590 is a function where

120590 (119909) = 1 if 119909 isin 1198780 otherwise

(57)

then 120590 is a dominating function of119866 The fractional domina-tion number of 119866 denoted by 120574

119865(119866) is defined as follows

120574119865(119866) = min |120590| 120590 is a domination function of 119866

(58)

The 120574119865-bondage number of 119866 denoted by 119887

119865(119866) is

defined as the minimum cardinality of a subset 119861 sube 119864 whoseremoval results in 120574

119865(119866 minus 119861) gt 120574

119865(119866)

Hedetniemi et al [111] are the first to study fractionaldomination although Farber [112] introduced the idea indi-rectly The concept of the fractional domination numberwas proposed by Domke and Laskar [113] in 1997 Thefractional domination numbers for some ordinary graphs aredetermined

Proposition 150 (Domke et al [114] 1998 Domke and Laskar[113] 1997) (a) If 119866 is a 119896-regular graph with order 119899 then120574119865(119866) = 119899119896 + 1(b) 120574

119865(119862

119899) = 1198993 120574

119865(119875

119899) = lceil1198993rceil 120574

119865(119870

119899) = 1

(c) 120574119865(119866) = 1 if and only if Δ(119866) = 119899 minus 1

(d) 120574119865(119879) = 120574(119879) for any tree 119879

(e) 120574119865(119870

119899119898) = (119899(119898 + 1) + 119898(119899 + 1))(119899119898 minus 1) where

max119899119898 gt 1

The assertions (a)ndash(d) are due to Domke et al [114] andthe assertion (e) is due to Domke and Laskar [113] Accordingto these results Domke and Laskar [113] determined the 120574

119865-

bondage numbers for these graphs

Theorem 151 (Domke and Laskar [113] 1997) 119887119865(119870

119899) =

lceil1198992rceil 119887119865(119870

119899119898) = 1 wheremax119899119898 gt 1

119887119865(119862

119899) =

2 119894119891 119899 equiv 0 (mod 3) 1 119900119905ℎ119890119903119908119894119904119890

119887119865(119875

119899) =

2 119894119891 119899 equiv 1 (mod 3) 1 119900119905ℎ119890119903119908119894119904119890

(59)

It is easy to see that for any tree119879 119887119865(119879) ⩽ 2 In fact since

120574119865(119879) = 120574(119879) for any tree 119879 120574

119865(119879

1015840

) = 120574(1198791015840

) for any subgraph1198791015840 of 119879 It follows that

119887119865(119879) = min 119861 sube 119864 (119879) 120574

119865(119879 minus 119861) gt 120574

119865(119879)

= min 119861 sube 119864 (119879) 120574 (119879 minus 119861) gt 120574 (119879)

= 119887 (119879) ⩽ 2

(60)

Except for these no results on 119887119865(119866) have been known as

yet

94 Roman Bondage Numbers A Roman dominating func-tion on a graph 119866 is a labeling 119891 119881 rarr 0 1 2 such thatevery vertex with label 0 has at least one neighbor with label2 The weight of a Roman dominating function 119891 on 119866 is thevalue

119891 (119866) = sum

119906isin119881(119866)

119891 (119906) (61)

The minimum weight of a Roman dominating function ona graph 119866 is called the Roman domination number of 119866denoted by 120574RM (119866)

A Roman dominating function 119891 119881 rarr 0 1 2

can be represented by the ordered partition (1198810 119881

1 119881

2) (or

(119881119891

0 119881

119891

1 119881

119891

2) to refer to119891) of119881 where119881

119894= V isin 119881 | 119891(V) = 119894

In this representation its weight 119891(119866) = |1198811| + 2|119881

2| It is

clear that119881119891

1cup119881

119891

2is a dominating set of 119866 called the Roman

dominating set denoted by 119863119891

R = (1198811 1198812) Since 119881119891

1cup 119881

119891

2is

a dominating set when 119891 is a Roman dominating functionand since placing weight 2 at the vertices of a dominating setyields a Roman dominating function in [115] it is observedthat

120574 (119866) ⩽ 120574RM (119866) ⩽ 2120574 (119866) (62)

A graph 119866 is called to be Roman if 120574RM (119866) = 2120574(119866)The definition of the Roman domination function is

proposed implicitly by Stewart [116] and ReVelle and Rosing[117] Roman dominating numbers have been deeply studiedIn particular Bahremandpour et al [118] showed that theproblem determining the Roman domination number is NP-complete even for bipartite graphs

Let 119866 be a graph with maximum degree at least twoThe Roman bondage number 119887RM (119866) of 119866 is the minimumcardinality of all sets 1198641015840 sube 119864 for which 120574RM (119866 minus 119864

1015840

) gt

120574RM (119866) Since in study of Roman bondage number theassumption Δ(119866) ⩾ 2 is necessary we always assume thatwhen we discuss 119887RM (119866) all graphs involved satisfy Δ(119866) ⩾2 The Roman bondage number 119887RM (119866) is introduced byJafari Rad and Volkmann in [119]

Recently Bahremandpour et al [118] have shown that theproblem determining the Roman bondage number is NP-hard even for bipartite graphs

Problem 7 Consider the decision problem

Roman Bondage Problem

Instance a nonempty bipartite graph119866 and a positiveinteger 119896

Question is 119887RM (119866) ⩽ 119896

Theorem 152 (Bahremandpour et al [118] 2013) TheRomanbondage problem is NP-hard even for bipartite graphs

The exact value of 119887RM(119866) is known only for a few familyof graphs including the complete graphs cycles and paths

International Journal of Combinatorics 25

Theorem 153 (Jafari Rad and Volkmann [119] 2011) For anypositive integers 119899 and119898 with119898 ⩽ 119899

119887RM (119875119899) = 2 119894119891 119899 equiv 2 (mod 3) 1 119900119905ℎ119890119903119908119894119904119890

119887RM (119862119899) =

3 119894119891 119899 = 2 (mod 3) 2 119900119905ℎ119890119903119908119894119904119890

119887RM (119870119899) = lceil

119899

2rceil 119891119900119903 119899 ⩾ 3

119887RM (119870119898119899) =

1 119894119891 119898 = 1 119886119899119889 119899 = 1

4 119894119891 119898 = 119899 = 3

119898 119900119905ℎ119890119903119908119894119904119890

(63)

Lemma 154 (Cockayne et al [115] 2004) If 119866 is a graph oforder 119899 and contains vertices of degree 119899 minus 1 then 120574RM(119866) = 2

Using Lemma 154 the third conclusion in Theorem 153can be generalized to more general case which is similar toLemma 49

Theorem 155 (Jafari Rad and Volkmann [119] 2011) Let119866 bea graph with order 119899 ge 3 and 119905 the number of vertices of degree119899 minus 1 in 119866 If 119905 ⩾ 1 then 119887RM(119866) = lceil1199052rceil

Ebadi and PushpaLatha [120] conjectured that 119887RM(119866) ⩽119899 minus 1 for any graph 119866 of order 119899 ⩾ 3 Dehgardi et al[121] Akbari andQajar [122] independently showed that thisconjecture is true

Theorem 156 119887RM(119866) ⩽ 119899 minus 1 for any connected graph 119866 oforder 119899 ⩾ 3

For a complete 119905-partite graph we have the followingresult

Theorem 157 (Hu and Xu [123] 2011) Let 119866 = 11987011989811198982119898119905

bea complete 119905-partite graph with 119898

1= sdot sdot sdot = 119898

119894lt 119898

119894+1le sdot sdot sdot ⩽

119898119905 119905 ⩾ 2 and 119899 = sum119905

119895=1119898

119895 Then

119887RM (119866) =

lceil119894

2rceil 119894119891 119898

119894= 1 119886119899119889 119899 ⩾ 3

2 119894119891 119898119894= 2 119886119899119889 119894 = 1

119894 119894119891 119898119894= 2 119886119899119889 119894 ⩾ 2

4 119894119891 119898119894= 3 119886119899119889 119894 = 119905 = 2

119899 minus 1 119894119891 119898119894= 3 119886119899119889 119894 = 119905 ⩾ 3

119899 minus 119898119905

119894119891 119898119894⩾ 3 119886119899119889 119898

119905⩾ 4

(64)

Consider a complete 119905-partite graph 119870119905(3) for 119905 ⩾ 3

119887RM(119870119905(3)) = 119899 minus 1 by Theorem 157 where 119899 = 3119905

which shows that this upper bound of 119899 minus 1 on 119887RM(119866) inTheorem 156 is sharp This fact and 120574

119877(119870

119905(3)) = 4 give

a negative answer to the following two problems posed byDehgardi et al [121]Prove or Disprove for any connected graph 119866 of order 119899 ge 3(i) 119887RM(119866) = 119899 minus 1 if and only if 119866 cong 119870

3 (ii) 119887RM(119866) ⩽ 119899 minus

120574119877(119866) + 1Note that a complete 119905-partite graph 119870

119905(3) is (119899 minus 3)-

regular and 119887RM(119870119905(3)) = 119899 minus 1 for 119905 ⩾ 3 where 119899 = 3119905

Hu and Xu further determined that 119887119877(119866) = 119899 minus 2 for any

(119899 minus 3)-regular graph 119866 of order 119899 ⩽ 5 except 119870119905(3)

Theorem 158 (Hu and Xu [123] 2011) For any (119899minus3)-regulargraph 119866 of order 119899 other than119870

119905(3) 119887RM(119866) = 119899 minus 2 for 119899 ⩾ 5

For a tree 119879 with order 119899 ⩾ 3 Ebadi and PushpaLatha[120] and Jafari Rad and Volkmann [119] independentlyobtained an upper bound on 119887RM(119879)

Theorem 159 119887RM(119879) ⩽ 3 for any tree 119879 with order 119899 ⩾ 3

Theorem 160 (Bahremandpour et al [118] 2013) 119887RM(1198752 times119875119899) = 2 for 119899 ⩾ 2

Theorem 161 (Jafari Rad and Volkmann [119] 2011) Let119866 bea graph with order at least three

(a) If (119909 119910 119911) is a path of length 2 in 119866 then 119887RM(119866) ⩽119889119866(119909) + 119889

119866(119910) + 119889

119866(119911) minus 3 minus |119873

119866(119909) cap 119873

119866(119910)|

(b) If 119866 is connected then 119887RM(119866) ⩽ 120582(119866) + 2Δ(119866) minus 3

Theorem 161 (b) implies 119887RM(119866) ⩽ 120575(119866) + 2Δ(119866) minus 3 for aconnected graph 119866 Note that for a planar graph 119866 120575(119866) ⩽ 5moreover 120575(119866) ⩽ 3 if the girth at least 4 and 120575(119866) ⩽ 2 if thegirth at least 6 These two facts show that 119887RM(119866) ⩽ 2Δ(119866) +2 for a connected planar graph 119866 Jafari Rad and Volkmann[124] improved this bound

Theorem 162 (Jafari Rad and Volkmann [124] 2011) For anyconnected planar graph 119866 of order 119899 ⩾ 3 with girth 119892(119866)

119887RM (119866)

⩽

2Δ (119866)

Δ (119866) + 6

Δ (119866) + 5 119894119891 119866 119888119900119899119905119886119894119899119904 119899119900 V119890119903119905119894119888119890119904 119900119891 119889119890119892119903119890119890 119891119894V119890

Δ (119866) + 4 119894119891 119892 (119866) ⩾ 4

Δ (119866) + 3 119894119891 119892 (119866) ⩾ 5

Δ (119866) + 2 119894119891 119892 (119866) ⩾ 6

Δ (119866) + 1 119894119891 119892 (119866) ⩾ 8

(65)

According toTheorem 157 119887RM(119862119899) = 3 = Δ(119862

119899)+1 for a

cycle 119862119899of length 119899 ⩾ 8 with 119899 equiv 2 (mod 3) and therefore

the last upper bound inTheorem 162 is sharp at least for Δ =2

Combining the fact that every planar graph 119866 withminimum degree 5 contains an edge 119909119910 with 119889

119866(119909) = 5

and 119889119866(119910) isin 5 6 with Theorem 161 (a) Akbari et al [125]

obtained the following result

26 International Journal of Combinatorics

middot middot middot

middot middot middot

middot middot middot

middot middot middot

119866

Figure 9 The graph 119866 is constructed from 119866

Theorem 163 (Akbari et al [125] 2012) 119887RM(119866) ⩽ 15 forevery planar graph 119866

It remains open to show whether the bound inTheorem 163 is sharp or not Though finding a planargraph 119866 with 119887RM(119866) = 15 seems to be difficult Akbari etal [125] constructed an infinite family of planar graphs withRoman bondage number equal to 7 by proving the followingresult

Theorem 164 (Akbari et al [125] 2012) Let 119866 be a graph oforder 119899 and 119866 is the graph of order 5119899 obtained from 119866 byattaching the central vertex of a copy of a path 119875

5to each vertex

of119866 (see Figure 9) Then 120574RM(119866) = 4119899 and 119887RM(119866) = 120575(119866)+2

ByTheorem 164 infinitemany planar graphswith Romanbondage number 7 can be obtained by considering any planargraph 119866 with 120575(119866) = 5 (eg the icosahedron graph)

Conjecture 165 (Akbari et al [125] 2012) The Romanbondage number of every planar graph is at most 7

For general bounds the following observation is directlyobtained which is similar to Observation 1

Observation 4 Let 119866 be a graph of order 119899 with maximumdegree at least two Assume that 119867 is a spanning subgraphobtained from 119866 by removing 119896 edges If 120574RM(119867) = 120574RM(119866)then 119887

119877(119867) ⩽ 119887RM(119866) ⩽ 119887RM(119867) + 119896

Theorem 166 (Bahremandpour et al [118] 2013) For anyconnected graph 119866 of order 119899 if 119899 ⩾ 3 and 120574RM(119866) = 120574(119866) + 1then

119887RM (119866) ⩽ min 119887 (119866) 119899Δ (66)

where 119899Δis the number of vertices with maximum degree Δ in

119866

Observation 5 For any graph 119866 if 120574(119866) = 120574RM(119866) then119887RM(119866) ⩽ 119887(119866)

Theorem 167 (Bahremandpour et al [118] 2013) For everyRoman graph 119866

119887RM (119866) ⩾ 119887 (119866) (67)

The bound is sharp for cycles on 119899 vertices where 119899 equiv 0 (mod3)

The strict inequality in Theorem 167 can hold for exam-ple 119887(119862

3119896+2) = 2 lt 3 = 119887RM(1198623119896+2

) byTheorem 157A graph 119866 is called to be vertex Roman domination-

critical if 120574RM(119866 minus 119909) lt 120574RM(119866) for every vertex 119909 in 119866 If119866 has no isolated vertices then 120574RM(119866) ⩽ 2120574(119866) ⩽ 2120573(119866)If 120574RM(119866) = 2120573(119866) then 120574RM(119866) = 2120574(119866) and hence 119866 is aRoman graph In [30] Volkmann gave a lot of graphs with120574(119866) = 120573(119866) The following result is similar to Theorem 57

Theorem 168 (Bahremandpour et al [118] 2013) For anygraph 119866 if 120574RM(119866) = 2120573(119866) then

(a) 119887RM(119866) ⩾ 120575(119866)

(b) 119887RM(119866) ⩾ 120575(119866)+1 if119866 is a vertex Roman domination-critical graph

If 120574RM(119866) = 2 then obviously 119887RM(119866) ⩽ 120575(119866) So weassume 120574RM(119866) ⩾ 3 Dehgardi et al [121] proved 119887RM(119866) ⩽(120574RM(119866) minus 2)Δ(119866) + 1 and posed the following problem if 119866is a connected graph of order 119899 ⩾ 4 with 120574RM(119866) ⩾ 3 then

119887RM (119866) ⩽ (120574RM (119866) minus 2) Δ (119866) (68)

Theorem 161 (a) shows that the inequality (68) holds if120574RM(119866) ⩾ 5 Thus the bound in (68) is of interest only when120574RM(119866) is 3 or 4

Lemma 169 (Bahremandpour et al [118] 2013) For anynonempty graph 119866 of order 119899 ⩾ 3 120574RM(119866) = 3 if and onlyif Δ(119866) = 119899 minus 2

The following result shows that (68) holds for all graphs119866 of order 119899 ⩾ 4 with 120574RM(119866) = 3 or 4 which improvesTheorem 161 (b)

Theorem 170 (Bahremandpour et al [118] 2013) For anyconnected graph 119866 of order 119899 ⩾ 4

119887RM (119866) le Δ (119866) = 119899 minus 2 119894119891 120574RM (119866) = 3

Δ (119866) + 120575 (119866) minus 1 119894119891 120574RM (119866) = 4(69)

with the first equality if and only if 119866 cong 1198624

Recently Akbari and Qajar [122] have proved the follow-ing result

Theorem 171 (Akbari and Qajar [122]) For any connectedgraph 119866 of order 119899 ⩾ 3

119887RM (119866) ⩽ 119899 minus 120574RM (119866) + 5 (70)

International Journal of Combinatorics 27

We conclude this section with the following problems

Problem 8 Characterize all connected graphs 119866 of order 119899 ⩾3 for which 119887RM(119866) = 119899 minus 1

Problem 9 Prove or disprove if 119866 is a connected graph oforder 119899 ⩾ 3 then

119887RM (119866) ⩽ 119899 minus 120574RM (119866) + 3 (71)

Note that 120574119877(119870

119905(3)) = 4 and 119887RM(119870119905

(3)) = 119899minus1 where 119899 =3119905 for 119905 ⩾ 3 The upper bound on 119887RM(119866) given in Problem 9is sharp if Problem 9 has positive answer

95 Rainbow Bondage Numbers For a positive integer 119896 a119896-rainbow dominating function of a graph 119866 is a function 119891from the vertex-set 119881(119866) to the set of all subsets of the set1 2 119896 such that for any vertex 119909 in 119866 with 119891(119909) = 0 thecondition

⋃

119910isin119873119866(119909)

119891 (119910) = 1 2 119896 (72)

is fulfilledThe weight of a 119896-rainbow dominating function 119891on 119866 is the value

119891 (119866) = sum

119909isin119881(119866)

1003816100381610038161003816119891 (119909)1003816100381610038161003816 (73)

The 119896-rainbow domination number of a graph 119866 denoted by120574119903119896(119866) is the minimum weight of a 119896-rainbow dominating

function 119891 of 119866Note that 120574

1199031(119866) is the classical domination number 120574(119866)

The 119896-rainbow domination number is introduced by Bresaret al [126] Rainbow domination of a graph 119866 coincides withordinary domination of the Cartesian product of 119866 with thecomplete graph in particular 120574

119903119896(119866) = 120574(119866 times 119870

119896) for any

positive integer 119896 and any graph 119866 [126] Hartnell and Rall[127] proved that 120574(119879) lt 120574

1199032(119879) for any tree 119879 no graph has

120574 = 120574119903119896for 119896 ⩾ 3 for any 119896 ⩾ 2 and any graph 119866 of order 119899

min 119899 120574 (119866) + 119896 minus 2 ⩽ 120574119903119896(119866) ⩽ 119896120574 (119866) (74)

The introduction of rainbow domination is motivatedby the study of paired domination in Cartesian productsof graphs where certain upper bounds can be expressed interms of rainbow domination that is for any graph 119866 andany graph 119867 if 119881(119867) can be partitioned into 119896120574

119875-sets then

120574119875(119866 times 119867) ⩽ (1119896)120592(119867)120574

119903119896(119866) proved by Bresar et al [126]

Dehgardi et al [128] introduced the concept of 119896-rainbowbondage number of graphs Let 119866 be a graph with maximumdegree at least two The 119896-rainbow bondage number 119887

119903119896(119866)

of 119866 is the minimum cardinality of all sets 119861 sube 119864(119866) forwhich 120574

119903119896(119866 minus 119861) gt 120574

119903119896(119866) Since in the study of 119896-rainbow

bondage number the assumption Δ(119866) ⩾ 2 is necessarywe always assume that when we discuss 119887

119903119896(119866) all graphs

involved satisfy Δ(119866) ⩾ 2The 2-rainbow bondage number of some special families

of graphs has been determined

Theorem 172 (Dehgardi et al [128]) For any integer 119899 ⩾ 3

1198871199032(119875

119899) = 1 119891119900119903 119899 ⩾ 2

1198871199032(119862

119899) =

1 119894119891 119899 equiv 0 (mod 4) 2 119900119905ℎ119890119903119908119894119904119890

1198871199032(119870

119899119899) =

1 119894119891 119899 = 2

3 119894119891 119899 = 3

6 119894119891 119899 = 4

119898 119894119891 119899 ⩾ 5

(75)

Theorem 173 (Dehgardi et al [128]) For any connected graph119866 of order 119899 ⩾ 3 119887

1199032(119866) ⩽ 120582(119866) + 2Δ(119866) minus 3

Theorem 174 (Dehgardi et al [128]) For any tree 119879 of order119899 ⩾ 3 119887

1199032(119879) ⩽ 2

Theorem 175 (Dehgardi et al [128]) If 119866 is a planar graphwithmaximumdegree at least two then 119887

1199032(119866) ⩽ 15Moreover

if the girth 119892(119866) ⩾ 4 then 1198871199032(119866) ⩽ 11 and if 119892(119866) ⩾ 6 then

1198871199032(119866) ⩽ 9

Theorem 176 (Dehgardi et al [128]) For a complete bipartitegraph 119870

119901119902 if 2119896 + 1 ⩽ 119901 ⩽ 119902 then 119887

119903119896(119870

119901119902) = 119901

Theorem177 (Dehgardi et al [128]) For a complete graph119870119899

if 119899 ⩾ 119896 + 1 then 119887119903119896(119870

119899) = lceil119896119899(119896 + 1)rceil

The special case 119896 = 1 of Theorem 177 can be found inTheorem 1 The following is a simple observation similar toObservation 1

Observation 6 (Dehgardi et al [128]) Let 119866 be a graphwith maximum degree at least two 119867 a spanning subgraphobtained from 119866 by removing 119896 edges If 120574

119903119896(119867) = 120574

119903119896(119866)

then 119887119903119896(119867) ⩽ 119887

119903119896(119866) ⩽ 119887

119903119896(119867) + 119896

Theorem 178 (Dehgardi et al [128]) For every graph 119866 with120574119903119896(119866) = 119896120574(119866) 119887

119903119896(119866) ⩾ 119887(119866)

A graph 119866 is said to be vertex 119896-rainbow domination-critical if 120574

119903119896(119866 minus 119909) lt 120574

119903119896(119866) for every vertex 119909 in 119866 It is

clear that if 119866 has no isolated vertices then 120574119903119896(119866) ⩽ 119896120574(119866) ⩽

119896120573(119866) where 120573(119866) is the vertex-covering number of119866 Thusif 120574

119903119896(119866) = 119896120573(119866) then 120574

119903119896(119866) = 119896120574(119866)

Theorem 179 (Dehgardi et al [128]) For any graph 119866 if120574119903119896(119866) = 119896120573(119866) then

(a) 119887119903119896(119866) ⩾ 120575(119866)

(b) 119887119903119896(119866) ⩾ 120575(119866)+1 if119866 is vertex 119896-rainbow domination-

critical

As far as we know there are no more results on the 119896-rainbow bondage number of graphs in the literature

28 International Journal of Combinatorics

10 Results on Digraphs

Although domination has been extensively studied in undi-rected graphs it is natural to think of a dominating setas a one-way relationship between vertices of the graphIndeed among the earliest literatures on this subject vonNeumann and Morgenstern [129] used what is now calleddomination in digraphs to find solution (or kernels whichare independent dominating sets) for cooperative 119899-persongames Most likely the first formulation of domination byBerge [130] in 1958 was given in the context of digraphs andonly some years latter by Ore [131] in 1962 for undirectedgraphs Despite this history examination of domination andits variants in digraphs has been essentially overlooked (see[4] for an overview of the domination literature) Thus thereare few if any such results on domination for digraphs in theliterature

The bondage number and its related topics for undirectedgraph have become one of major areas both in theoreticaland applied researches However until recently Carlsonand Develin [42] Shan and Kang [132] and Huang andXu [133 134] studied the bondage number for digraphsindependently In this section we introduce their resultsfor general digraphs Results for some special digraphs suchas vertex-transitive digraphs are introduced in the nextsection

101 Upper Bounds for General Digraphs Let 119866 = (119881 119864)

be a digraph without loops and parallel edges A digraph 119866is called to be asymmetric if whenever (119909 119910) isin 119864(119866) then(119910 119909) notin 119864(119866) and to be symmetric if (119909 119910) isin 119864(119866) implies(119910 119909) isin 119864(119866) For a vertex 119909 of 119881(119866) the sets of out-neighbors and in-neighbors of 119909 are respectively defined as119873

+

119866(119909) = 119910 isin 119881(119866) (119909 119910) isin 119864(119866) and 119873minus

119866(119909) = 119909 isin

119881(119866) (119909 119910) isin 119864(119866) the out-degree and the in-degree of119909 are respectively defined as 119889+

119866(119909) = |119873

+

119866(119909)| and 119889minus

119866(119909) =

|119873+

119866(119909)| Denote themaximum and theminimum out-degree

(resp in-degree) of 119866 by Δ+(119866) and 120575+(119866) (resp Δminus(119866) and120575minus

(119866))The degree 119889119866(119909) of 119909 is defined as 119889+

119866(119909)+119889

minus

119866(119909) and

the maximum and the minimum degrees of119866 are denoted byΔ(119866) and 120575(119866) respectively that is Δ(119866) = max119889

119866(119909) 119909 isin

119881(119866) and 120575(119866) = min119889119866(119909) 119909 isin 119881(119866) Note that the

definitions here are different from the ones in the textbookon digraphs see for example Xu [1]

A subset 119878 of119881(119866) is called a dominating set if119881(119866) = 119878cup119873

+

119866(119878) where119873+

119866(119878) is the set of out-neighbors of 119878Then just

as for undirected graphs 120574(119866) is the minimum cardinalityof a dominating set and the bondage number 119887(119866) is thesmallest cardinality of a set 119861 of edges such that 120574(119866 minus 119861) gt120574(119866) if such a subset 119861 sube 119864(119866) exists Otherwise we put119887(119866) = infin

Some basic results for undirected graphs stated inSection 3 can be generalized to digraphs For exampleTheorem 18 is generalized by Carlson and Develin [42]Huang andXu [133] and Shan andKang [132] independentlyas follows

Theorem 180 Let 119866 be a digraph and (119909 119910) isin 119864(119866) Then119887(119866) ⩽ 119889

119866(119910) + 119889

minus

119866(119909) minus |119873

minus

119866(119909) cap 119873

minus

119866(119910)|

Corollary 181 For a digraph 119866 119887(119866) ⩽ 120575minus(119866) + Δ(119866)

Since 120575minus

(119866) ⩽ (12)Δ(119866) from Corollary 181Conjecture 53 is valid for digraphs

Corollary 182 For a digraph 119866 119887(119866) ⩽ (32)Δ(119866)

In the case of undirected graphs the bondage number of119866119899= 119870

119899times 119870

119899achieves this bound However it was shown

by Carlson and Develin in [42] that if we take the symmetricdigraph119866

119899 we haveΔ(119866

119899) = 4(119899minus1) 120574(119866

119899) = 119899 and 119887(119866

119899) ⩽

4(119899 minus 1) + 1 = Δ(119866119899) + 1 So this family of digraphs cannot

show that the bound in Corollary 182 is tightCorresponding to Conjecture 51 which is discredited

for undirected graphs and is valid for digraphs the sameconjecture for digraphs can be proposed as follows

Conjecture 183 (Carlson and Develin [42] 2006) 119887(119866) ⩽Δ(119866) + 1 for any digraph 119866

If Conjecture 183 is true then the following results showsthat this upper bound is tight since 119887(119870

119899times119870

119899) = Δ(119870

119899times119870

119899)+1

for a complete digraph 119870119899

In 2007 Shan and Kang [132] gave some tight upperbounds on the bondage numbers for some asymmetricdigraphs For example 119887(119879) ⩽ Δ(119879) for any asymmetricdirected tree 119879 119887(119866) ⩽ Δ(119866) for any asymmetric digraph 119866with order at least 4 and 120574(119866) ⩽ 2 For planar digraphs theyobtained the following results

Theorem 184 (Shan and Kang [132] 2007) Let 119866 be aasymmetric planar digraphThen 119887(119866) ⩽ Δ(119866)+2 and 119887(119866) ⩽Δ(119866) + 1 if Δ(119866) ⩾ 5 and 119889minus

119866(119909) ⩾ 3 for every vertex 119909 with

119889119866(119909) ⩾ 4

102 Results for Some Special Digraphs The exact values andbounds on 119887(119866) for some standard digraphs are determined

Theorem185 (Huang andXu [133] 2006) For a directed cycle119862119899and a directed path 119875

119899

119887 (119862119899) =

3 119894119891 119899 119894119904 119900119889119889

2 119894119891 119899 119894119904 119890V119890119899

119887 (119875119899) =

2 119894119891 119899 119894119904 119900119889119889

1 119894119891 119899 119894119904 119890V119890119899

(76)

For the de Bruijn digraph 119861(119889 119899) and the Kautz digraph119870(119889 119899)

119887 (119861 (119889 119899)) = 119889 119894119891 119899 119894119904 119900119889119889

119889 ⩽ 119887 (119861 (119889 119899)) ⩽ 2119889 119894119891 119899 119894119904 119890V119890119899

119887 (119870 (119889 119899)) = 119889 + 1

(77)

Like undirected graphs we can define the total domi-nation number and the total bondage number On the totalbondage numbers for some special digraphs the knownresults are as follows

International Journal of Combinatorics 29

Theorem 186 (Huang and Xu [134] 2007) For a directedcycle 119862

119899and a directed path 119875

119899 119887

119879(119875

119899) and 119887

119879(119862

119899) all do not

exist For a complete digraph 119870119899

119887119879(119870

119899) = infin 119894119891 119899 = 1 2

119887119879(119870

119899) = 3 119894119891 119899 = 3

119899 ⩽ 119887119879(119870

119899) ⩽ 2119899 minus 3 119894119891 119899 ⩾ 4

(78)

The extended de Bruijn digraph EB(119889 119899 1199021 119902

119901) and

the extended Kautz digraph EK(119889 119899 1199021 119902

119901) are intro-

duced by Shibata and Gonda [135] If 119901 = 1 then theyare the de Bruijn digraph B(119889 119899) and the Kautz digraphK(119889 119899) respectively Huang and Xu [134] determined theirtotal domination numbers In particular their total bondagenumbers for general cases are determined as follows

Theorem 187 (Huang andXu [134] 2007) If 119889 ⩾ 2 and 119902119894⩾ 2

for each 119894 = 1 2 119901 then

119887119879(EB(119889 119899 119902

1 119902

119901)) = 119889

119901

minus 1

119887119879(EK(119889 119899 119902

1 119902

119901)) = 119889

119901

(79)

In particular for the de Bruijn digraph B(119889 119899) and the Kautzdigraph K(119889 119899)

119887119879(B (119889 119899)) = 119889 minus 1 119887

119879(K (119889 119899)) = 119889 (80)

Zhang et al [136] determined the bondage number incomplete 119905-partite digraphs

Theorem 188 (Zhang et al [136] 2009) For a complete 119905-partite digraph 119870

11989911198992119899119905

where 1198991⩽ 119899

2⩽ sdot sdot sdot ⩽ 119899

119905

119887 (11987011989911198992119899119905

)

=

119898 119894119891 119899119898= 1 119899

119898+1⩾ 2 119891119900119903 119904119900119898119890

119898 (1 ⩽ 119898 lt 119905)

4119905 minus 3 119894119891 1198991= 119899

2= sdot sdot sdot = 119899

119905= 2

119905minus1

sum

119894=1

119899119894+ 2 (119905 minus 1) 119900119905ℎ119890119903119908119894119904119890

(81)

Since an undirected graph can be thought of as a sym-metric digraph any result for digraphs has an analogy forundirected graphs in general In view of this point studyingthe bondage number for digraphs is more significant thanfor undirected graphs Thus we should further study thebondage number of digraphs and try to generalize knownresults on the bondage number and related variants for undi-rected graphs to digraphs prove or disprove Conjecture 183In particular determine the exact values of 119887(B(119889 119899)) for aneven 119899 and 119887

119879(119870

119899) for 119899 ⩾ 4

11 Efficient Dominating Sets

A dominating set 119878 of a graph 119866 is called to be efficient if forevery vertex 119909 in119866 |119873

119866[119909]cap119878| = 1 if119866 is a undirected graph

or |119873minus

119866[119909] cap 119878| = 1 if 119866 is a directed graph The concept of an

efficient set was proposed by Bange et al [137] in 1988 In theliterature an efficient set is also called a perfect dominatingset (see Livingston and Stout [138] for an explanation)

From definition if 119878 is an efficient dominating set of agraph 119866 then 119878 is certainly an independent set and everyvertex not in 119878 is adjacent to exactly one vertex in 119878

It is also clear from definition that a dominating set 119878 isefficient if and only if N

119866[119878] = 119873

119866[119909] 119909 isin 119878 for the

undirected graph 119866 or N+

119866[119878] = 119873

+

119866[119909] 119909 isin 119878 for the

digraph119866 is a partition of119881(119866) where the induced subgraphby119873

119866[119909] or119873+

119866[119909] is a star or an out-star with the root 119909

The efficient domination has important applications inmany areas such as error-correcting codes and receivesmuch attention in the late years

The concept of efficient dominating sets is a measureof the efficiency of domination in graphs Unfortunately asshown in [137] not every graph has an efficient dominatingset andmoreover it is anNP-complete problem to determinewhether a given graph has an efficient dominating set Inaddition it was shown by Clark [139] in 1993 that for a widerange of 119901 almost every random undirected graph 119866 isin

G(120592 119901) has no efficient dominating sets This means thatundirected graphs possessing an efficient dominating set arerare However it is easy to show that every undirected graphhas an orientationwith an efficient dominating set (see Bangeet al [140])

In 1993 Barkauskas and Host [141] showed that deter-mining whether an arbitrary oriented graph has an efficientdominating set is NP-complete Even so the existence ofefficient dominating sets for some special classes of graphs hasbeen examined see for example Dejter and Serra [142] andLee [143] for Cayley graph Gu et al [144] formeshes and toriObradovic et al [145] Huang and Xu [36] Reji Kumar andMacGillivray [146] for circulant graphs Harary graphs andtori and Van Wieren et al [147] for cube-connected cycles

In this section we introduce results of the bondagenumber for some graphs with efficient dominating sets dueto Huang and Xu [36]

111 Results for General Graphs In this subsection we intro-duce some results on bondage numbers obtained by applyingefficient dominating sets We first state the two followinglemmas

Lemma 189 For any 119896-regular graph or digraph 119866 of order 119899120574(119866) ⩾ 119899(119896 + 1) with equality if and only if 119866 has an efficientdominating set In addition if 119866 has an efficient dominatingset then every efficient dominating set is certainly a 120574-set andvice versa

Let 119890 be an edge and 119878 a dominating set in 119866 We say that119890 supports 119878 if 119890 isin (119878 119878) where (119878 119878) = (119909 119910) isin 119864(119866) 119909 isin 119878 119910 isin 119878 Denote by 119904(119866) the minimum number of edgeswhich support all 120574-sets in 119866

Lemma 190 For any graph or digraph 119866 119887(119866) ⩾ 119904(119866) withequality if 119866 is regular and has an efficient dominating set

30 International Journal of Combinatorics

A graph 119866 is called to be vertex-transitive if its automor-phism group Aut(119866) acts transitively on its vertex-set 119881(119866)A vertex-transitive graph is regular Applying Lemmas 189and 190 Huang and Xu obtained some results on bondagenumbers for vertex-transitive graphs or digraphs

Theorem 191 (Huang and Xu [36] 2008) For any vertex-transitive graph or digraph 119866 of order 119899

119887 (119866) ⩾

lceil119899

2120574 (119866)rceil 119894119891 119866 119894119904 119906119899119889119894119903119890119888119905119890119889

lceil119899

120574 (119866)rceil 119894119891 119866 119894119904 119889119894119903119890119888119905119890119889

(82)

Theorem192 (Huang andXu [36] 2008) Let119866 be a 119896-regulargraph of order 119899 Then

119887(119866) ⩽ 119896 if119866 is undirected and 119899 ⩾ 120574(119866)(119896+1)minus119896+1119887(119866) ⩽ 119896+1+ℓ if119866 is directed and 119899 ⩾ 120574(119866)(119896+1)minusℓwith 0 ⩽ ℓ ⩽ 119896 minus 1

Next we introduce a better upper bound on 119887(119866) To thisaim we need the following concept which is a generalizationof the concept of the edge-covering of a graph 119866 For 1198811015840

sube

119881(119866) and 1198641015840 sube 119864(119866) we say that 1198641015840covers 1198811015840 and call 1198641015840an edge-covering for 1198811015840 if there exists an edge (119909 119910) isin 1198641015840 forany vertex 119909 isin 1198811015840 For 119910 isin 119881(119866) let 1205731015840[119910] be the minimumcardinality over all edge-coverings for119873minus

119866[119910]

Theorem 193 (Huang and Xu [36] 2008) If 119866 is a 119896-regulargraph with order 120574(119866)(119896 + 1) then 119887(119866) ⩽ 1205731015840[119910] for any 119910 isin119881(119866)

Theupper bound on 119887(119866) given inTheorem 193 is tight inview of 119887(119862

119899) for a cycle or a directed cycle 119862

119899(seeTheorems

1 and 185 resp)It is easy to see that for a 119896-regular graph 119866 lceil(119896 + 1)2rceil ⩽

1205731015840

[119910] ⩽ 119896 when 119866 is undirected and 1205731015840[119910] = 119896 + 1 when 119866 isdirected By this fact and Lemma 189 the following theoremis merely a simple combination of Theorems 191 and 193

Theorem 194 (Huang and Xu [36] 2008) Let 119866 be a vertex-transitive graph of degree 119896 If 119866 has an efficient dominatingset then

lceil119896 + 1

2rceil ⩽ 119887 (119866) ⩽ 119896 119894119891 119866 119894119904 119906119899119889119894119903119890119888119905119890119889

119887 (119866) = 119896 + 1 119894119891 119866 119894119904 119889119894119903119890119888119905119890119889

(83)

Theorem 195 (Huang and Xu [36] 2008) If 119866 is an undi-rected vertex-transitive cubic graph with order 4120574(119866) and girth119892(119866) ⩽ 5 then 119887(119866) = 2

The proof of Theorem 195 leads to a byproduct If 119892 =5 let (119906

1 119906

2 119906

3 119906

4 119906

5) be a cycle and let D

119894be the family

of efficient dominating sets containing 119906119894for each 119894 isin

1 2 3 4 5 ThenD119894capD

119895= 0 for 119894 = 1 2 5 Then 119866 has

at least 5119904 efficient dominating sets where 119904 = 119904(119866) But thereare only 4s (=ns120574) distinct efficient dominating sets in119866This

contradiction implies that an undirected vertex-transitivecubic graph with girth five has no efficient dominating setsBut a similar argument for 119892(119866) = 3 4 or 119892(119866) ⩾ 6 couldnot give any contradiction This is consistent with the resultthat CCC(119899) a vertex-transitive cubic graph with girth 119899 if3 ⩽ 119899 ⩽ 8 or girth 8 if 119899 ⩾ 9 has efficient dominating setsfor all 119899 ⩾ 3 except 119899 = 5 (see Theorem 199 in the followingsubsection)

112 Results for Cayley Graphs In this subsection we useTheorem 194 to determine the exact values or approximativevalues of bondage numbers for some special vertex-transitivegraphs by characterizing the existence of efficient dominatingsets in these graphs

Let Γ be a nontrivial finite group 119878 a nonempty subset ofΓ without the identity element of Γ A digraph 119866 is defined asfollows

119881 (119866) = Γ (119909 119910) isin 119864 (119866) lArrrArr 119909minus1

119910 isin 119878

for any 119909 119910 isin Γ(84)

is called a Cayley digraph of the group Γ with respect to119878 denoted by 119862

Γ(119878) If 119878minus1 = 119904minus1 119904 isin 119878 = 119878 then

119862Γ(119878) is symmetric and is called a Cayley undirected graph

a Cayley graph for short Cayley graphs or digraphs arecertainly vertex-transitive (see Xu [1])

A circulant graph 119866(119899 119878) of order 119899 is a Cayley graph119862119885119899

(119878) where 119885119899= 0 119899 minus 1 is the addition group of

order 119899 and 119878 is a nonempty subset of119885119899without the identity

element and hence is a vertex-transitive digraph of degree|119878| If 119878minus1 = 119878 then119866(119899 119878) is an undirected graph If 119878 = 1 119896where 2 ⩽ 119896 ⩽ 119899 minus 2 we write 119866(119899 1 119896) for 119866(119899 1 119896) or119866(119899 plusmn1 plusmn119896) and call it a double-loop circulant graph

Huang and Xu [36] showed that for directed 119866 =

119866(119899 1 119896) lceil1198993rceil ⩽ 120574(119866) ⩽ lceil1198992rceil and 119866 has an efficientdominating set if and only if 3 | 119899 and 119896 equiv 2 (mod 3) fordirected 119866 = 119866(119899 1 119896) and 119896 = 1198992 lceil1198995rceil ⩽ 120574(119866) ⩽ lceil1198993rceiland 119866 has an efficient dominating set if and only if 5 | 119899 and119896 equiv plusmn2 (mod 5) By Theorem 194 we obtain the bondagenumber of a double loop circulant graph if it has an efficientdominating set

Theorem 196 (Huang and Xu [36] 2008) Let 119866 be a double-loop circulant graph 119866(119899 1 119896) If 119866 is directed with 3 | 119899and 119896 equiv 2 (mod 3) or 119866 is undirected with 5 | 119899 and119896 equiv plusmn2 (mod 5) then 119887(119866) = 3

The 119898 times 119899 torus is the Cartesian product 119862119898times 119862

119899of

two cycles and is a Cayley graph 119862119885119898times119885119899

(119878) where 119878 =

(0 1) (1 0) for directed cycles and 119878 = (0 plusmn1) (plusmn1 0) forundirected cycles and hence is vertex-transitive Gu et al[144] showed that the undirected torus119862

119898times119862

119899has an efficient

dominating set if and only if both 119898 and 119899 are multiplesof 5 Huang and Xu [36] showed that the directed torus119862119898times 119862

119899has an efficient dominating set if and only if both

119898 and 119899 are multiples of 3 Moreover they found a necessarycondition for a dominating set containing the vertex (0 0)in 119862

119898times 119862

119899to be efficient and obtained the following

result

International Journal of Combinatorics 31

Theorem 197 (Huang and Xu [36] 2008) Let 119866 = 119862119898times 119862

119899

If 119866 is undirected and both119898 and 119899 are multiples of 5 or if 119866 isdirected and both119898 and 119899 are multiples of 3 then 119887(119866) = 3

The hypercube 119876119899

is the Cayley graph 119862Γ(119878)

where Γ = 1198852times sdot sdot sdot times 119885

2= (119885

2)119899 and 119878 =

100 sdot sdot sdot 0 010 sdot sdot sdot 0 00 sdot sdot sdot 01 Lee [143] showed that119876119899has an efficient dominating set if and only if 119899 = 2119898 minus 1

for a positive integer119898 ByTheorem 194 the following resultis obtained immediately

Theorem 198 (Huang and Xu [36] 2008) If 119899 = 2119898 minus 1 for apositive integer119898 then 2119898minus1

⩽ 119887(119876119899) ⩽ 2

119898

minus 1

Problem 10 Determine the exact value of 119887(119876119899) for 119899 = 2119898minus1

The 119899-dimensional cube-connected cycle denoted byCCC(119899) is constructed from the 119899-dimensional hypercube119876119899by replacing each vertex 119909 in 119876

119899with an undirected cycle

119862119899of length 119899 and linking the 119894th vertex of the 119862

119899to the 119894th

neighbor of 119909 It has been proved that CCC(119899) is a Cayleygraph and hence is a vertex-transitive graph with degree 3Van Wieren et al [147] proved that CCC(119899) has an efficientdominating set if and only if 119899 = 5 From Theorems 194 and195 the following result can be derived

Theorem 199 (Huang and Xu [36] 2008) Let 119866 = 119862119862119862(119899)be the 119899-dimensional cube-connected cycles with 119899 ⩾ 3 and119899 = 5 Then 120574(119866) = 1198992119899minus2 and 2 ⩽ 119887(119866) ⩽ 3 In addition119887(119862119862119862(3)) = 119887(119862119862119862(4)) = 2

Problem 11 Determine the exact value of 119887(CCC(119899)) for 119899 ⩾5

Acknowledgments

This work was supported by NNSF of China (Grants nos11071233 61272008) The author would like to express hisgratitude to the anonymous referees for their kind sugges-tions and comments on the original paper to ProfessorSheikholeslami and Professor Jafari Rad for sending thereferences [128] and [81] which resulted in this version

References

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[2] S THedetniemi andR C Laskar ldquoBibliography on dominationin graphs and some basic definitions of domination parame-tersrdquo Discrete Mathematics vol 86 no 1ndash3 pp 257ndash277 1990

[3] TW Haynes S T Hedetniemi and P J Slater Fundamentals ofDomination in Graphs vol 208 ofMonographs and Textbooks inPure and Applied Mathematics Marcel Dekker New York NYUSA 1998

[4] TWHaynes S THedetniemi and P J Slater EdsDominationin Graphs vol 209 of Monographs and Textbooks in Pure andAppliedMathematicsMarcelDekker NewYorkNYUSA 1998

[5] M R Garey andD S JohnsonComputers and Intractability WH Freeman and Company San Francisco Calif USA 1979

[6] H B Walikar and B D Acharya ldquoDomination critical graphsrdquoNational Academy Science Letters vol 2 pp 70ndash72 1979

[7] D Bauer F Harary J Nieminen and C L Suffel ldquoDominationalteration sets in graphsrdquo Discrete Mathematics vol 47 no 2-3pp 153ndash161 1983

[8] J F Fink M S Jacobson L F Kinch and J Roberts ldquoThebondage number of a graphrdquo Discrete Mathematics vol 86 no1ndash3 pp 47ndash57 1990

[9] F-T Hu and J-M Xu ldquoThe bondage number of (119899 minus 3)-regulargraph G of order nrdquo Ars Combinatoria to appear

[10] U Teschner ldquoNew results about the bondage number of agraphrdquoDiscreteMathematics vol 171 no 1ndash3 pp 249ndash259 1997

[11] B L Hartnell and D F Rall ldquoA bound on the size of a graphwith given order and bondage numberrdquo Discrete Mathematicsvol 197-198 pp 409ndash413 1999 16th British CombinatorialConference (London 1997)

[12] B L Hartnell and D F Rall ldquoA characterization of trees inwhich no edge is essential to the domination numberrdquo ArsCombinatoria vol 33 pp 65ndash76 1992

[13] J Topp and P D Vestergaard ldquo120572119896- and 120574

119896-stable graphsrdquo

Discrete Mathematics vol 212 no 1-2 pp 149ndash160 2000[14] A Meir and J W Moon ldquoRelations between packing and

covering numbers of a treerdquo Pacific Journal of Mathematics vol61 no 1 pp 225ndash233 1975

[15] B L Hartnell L K Joslashrgensen P D Vestergaard and CA Whitehead ldquoEdge stability of the k-domination numberof treesrdquo Bulletin of the Institute of Combinatorics and ItsApplications vol 22 pp 31ndash40 1998

[16] F-T Hu and J-M Xu ldquoOn the complexity of the bondage andreinforcement problemsrdquo Journal of Complexity vol 28 no 2pp 192ndash201 2012

[17] B L Hartnell and D F Rall ldquoBounds on the bondage numberof a graphrdquo Discrete Mathematics vol 128 no 1ndash3 pp 173ndash1771994

[18] Y-L Wang ldquoOn the bondage number of a graphrdquo DiscreteMathematics vol 159 no 1ndash3 pp 291ndash294 1996

[19] G H Fricke TW Haynes S M Hedetniemi S T Hedetniemiand R C Laskar ldquoExcellent treesrdquo Bulletin of the Institute ofCombinatorics and Its Applications vol 34 pp 27ndash38 2002

[20] V Samodivkin ldquoThe bondage number of graphs good and badverticesrdquoDiscussiones Mathematicae GraphTheory vol 28 no3 pp 453ndash462 2008

[21] J E Dunbar T W Haynes U Teschner and L VolkmannldquoBondage insensitivity and reinforcementrdquo in Domination inGraphs vol 209 of Monographs and Textbooks in Pure andAppliedMathematics pp 471ndash489 Dekker NewYork NY USA1998

[22] H Liu and L Sun ldquoThe bondage and connectivity of a graphrdquoDiscrete Mathematics vol 263 no 1ndash3 pp 289ndash293 2003

[23] A-H Esfahanian and S L Hakimi ldquoOn computing a con-ditional edge-connectivity of a graphrdquo Information ProcessingLetters vol 27 no 4 pp 195ndash199 1988

[24] R C Brigham P Z Chinn and R D Dutton ldquoVertexdomination-critical graphsrdquoNetworks vol 18 no 3 pp 173ndash1791988

[25] V Samodivkin ldquoDomination with respect to nondegenerateproperties bondage numberrdquoThe Australasian Journal of Com-binatorics vol 45 pp 217ndash226 2009

[26] U Teschner ldquoA new upper bound for the bondage numberof graphs with small domination numberrdquo The AustralasianJournal of Combinatorics vol 12 pp 27ndash35 1995

32 International Journal of Combinatorics

[27] J Fulman D Hanson and G MacGillivray ldquoVertex dom-ination-critical graphsrdquoNetworks vol 25 no 2 pp 41ndash43 1995

[28] U Teschner ldquoA counterexample to a conjecture on the bondagenumber of a graphrdquo Discrete Mathematics vol 122 no 1ndash3 pp393ndash395 1993

[29] U Teschner ldquoThe bondage number of a graph G can be muchgreater than Δ(G)rdquo Ars Combinatoria vol 43 pp 81ndash87 1996

[30] L Volkmann ldquoOn graphs with equal domination and coveringnumbersrdquoDiscrete AppliedMathematics vol 51 no 1-2 pp 211ndash217 1994

[31] L A Sanchis ldquoMaximum number of edges in connected graphswith a given domination numberrdquoDiscreteMathematics vol 87no 1 pp 65ndash72 1991

[32] S Klavzar and N Seifter ldquoDominating Cartesian products ofcyclesrdquoDiscrete AppliedMathematics vol 59 no 2 pp 129ndash1361995

[33] H K Kim ldquoA note on Vizingrsquos conjecture about the bondagenumberrdquo Korean Journal of Mathematics vol 10 no 2 pp 39ndash42 2003

[34] M Y Sohn X Yuan and H S Jeong ldquoThe bondage number of1198623times119862

119899rdquo Journal of the KoreanMathematical Society vol 44 no

6 pp 1213ndash1231 2007[35] L Kang M Y Sohn and H K Kim ldquoBondage number of the

discrete torus 119862119899times 119862

4rdquo Discrete Mathematics vol 303 no 1ndash3

pp 80ndash86 2005[36] J Huang and J-M Xu ldquoThe bondage numbers and efficient

dominations of vertex-transitive graphsrdquoDiscrete Mathematicsvol 308 no 4 pp 571ndash582 2008

[37] C J Xiang X Yuan andM Y Sohn ldquoDomination and bondagenumber of119862

3times119862

119899rdquoArs Combinatoria vol 97 pp 299ndash310 2010

[38] M S Jacobson and L F Kinch ldquoOn the domination numberof products of graphs Irdquo Ars Combinatoria vol 18 pp 33ndash441984

[39] Y-Q Li ldquoA note on the bondage number of a graphrdquo ChineseQuarterly Journal of Mathematics vol 9 no 4 pp 1ndash3 1994

[40] F-T Hu and J-M Xu ldquoBondage number of mesh networksrdquoFrontiers ofMathematics inChina vol 7 no 5 pp 813ndash826 2012

[41] R Frucht and F Harary ldquoOn the corona of two graphsrdquoAequationes Mathematicae vol 4 pp 322ndash325 1970

[42] K Carlson and M Develin ldquoOn the bondage number of planarand directed graphsrdquoDiscreteMathematics vol 306 no 8-9 pp820ndash826 2006

[43] U Teschner ldquoOn the bondage number of block graphsrdquo ArsCombinatoria vol 46 pp 25ndash32 1997

[44] J Huang and J-M Xu ldquoNote on conjectures of bondagenumbers of planar graphsrdquo Applied Mathematical Sciences vol6 no 65ndash68 pp 3277ndash3287 2012

[45] L Kang and J Yuan ldquoBondage number of planar graphsrdquoDiscrete Mathematics vol 222 no 1ndash3 pp 191ndash198 2000

[46] M Fischermann D Rautenbach and L Volkmann ldquoRemarkson the bondage number of planar graphsrdquo Discrete Mathemat-ics vol 260 no 1ndash3 pp 57ndash67 2003

[47] J Hou G Liu and J Wu ldquoBondage number of planar graphswithout small cyclesrdquoUtilitasMathematica vol 84 pp 189ndash1992011

[48] J Huang and J-M Xu ldquoThe bondage numbers of graphs withsmall crossing numbersrdquo Discrete Mathematics vol 307 no 15pp 1881ndash1897 2007

[49] Q Ma S Zhang and J Wang ldquoBondage number of 1-planargraphrdquo Applied Mathematics vol 1 no 2 pp 101ndash103 2010

[50] J Hou and G Liu ldquoA bound on the bondage number of toroidalgraphsrdquoDiscreteMathematics Algorithms and Applications vol4 no 3 Article ID 1250046 5 pages 2012

[51] Y-C Cao J-M Xu and X-R Xu ldquoOn bondage number oftoroidal graphsrdquo Journal of University of Science and Technologyof China vol 39 no 3 pp 225ndash228 2009

[52] Y-C Cao J Huang and J-M Xu ldquoThe bondage number ofgraphs with crossing number less than fourrdquo Ars Combinatoriato appear

[53] H K Kim ldquoOn the bondage numbers of some graphsrdquo KoreanJournal of Mathematics vol 11 no 2 pp 11ndash15 2004

[54] A Gagarin and V Zverovich ldquoUpper bounds for the bondagenumber of graphs on topological surfacesrdquo Discrete Mathemat-ics 2012

[55] J Huang ldquoAn improved upper bound for the bondage numberof graphs on surfacesrdquoDiscrete Mathematics vol 312 no 18 pp2776ndash2781 2012

[56] A Gagarin and V Zverovich ldquoThe bondage number of graphson topological surfaces and Teschnerrsquos conjecturerdquo DiscreteMathematics vol 313 no 6 pp 796ndash808 2013

[57] V Samodivkin ldquoNote on the bondage number of graphs ontopological surfacesrdquo httparxivorgabs12086203

[58] E J Cockayne R M Dawes and S T Hedetniemi ldquoTotaldomination in graphsrdquoNetworks vol 10 no 3 pp 211ndash219 1980

[59] R Laskar J Pfaff S M Hedetniemi and S T HedetniemildquoOn the algorithmic complexity of total dominationrdquo Society forIndustrial and Applied Mathematics vol 5 no 3 pp 420ndash4251984

[60] J Pfaff R C Laskar and S T Hedetniemi ldquoNP-completeness oftotal and connected domination and irredundance for bipartitegraphsrdquo Tech Rep 428 Department of Mathematical SciencesClemson University 1983

[61] M A Henning ldquoA survey of selected recent results on totaldomination in graphsrdquoDiscrete Mathematics vol 309 no 1 pp32ndash63 2009

[62] V R Kulli and D K Patwari ldquoThe total bondage number ofa graphrdquo in Advances in Graph Theory pp 227ndash235 VishwaGulbarga India 1991

[63] F-T Hu Y Lu and J-M Xu ldquoThe total bondage number ofgrid graphsrdquo Discrete Applied Mathematics vol 160 no 16-17pp 2408ndash2418 2012

[64] J Cao M Shi and B Wu ldquoResearch on the total bondagenumber of a spectial networkrdquo in Proceedings of the 3rdInternational Joint Conference on Computational Sciences andOptimization (CSO rsquo10) pp 456ndash458 May 2010

[65] N Sridharan MD Elias and V S A Subramanian ldquoTotalbondage number of a graphrdquo AKCE International Journal ofGraphs and Combinatorics vol 4 no 2 pp 203ndash209 2007

[66] N Jafari Rad and J Raczek ldquoSome progress on total bondage ingraphsrdquo Graphs and Combinatorics 2011

[67] T W Haynes and P J Slater ldquoPaired-domination and thepaired-domatic numberrdquoCongressusNumerantium vol 109 pp65ndash72 1995

[68] T W Haynes and P J Slater ldquoPaired-domination in graphsrdquoNetworks vol 32 no 3 pp 199ndash206 1998

[69] X Hou ldquoA characterization of 2120574 120574119875-treesrdquoDiscrete Mathemat-

ics vol 308 no 15 pp 3420ndash3426 2008[70] K E Proffitt T W Haynes and P J Slater ldquoPaired-domination

in grid graphsrdquo Congressus Numerantium vol 150 pp 161ndash1722001

International Journal of Combinatorics 33

[71] H Qiao L KangM Cardei andD-Z Du ldquoPaired-dominationof treesrdquo Journal of Global Optimization vol 25 no 1 pp 43ndash542003

[72] E Shan L Kang and M A Henning ldquoA characterizationof trees with equal total domination and paired-dominationnumbersrdquo The Australasian Journal of Combinatorics vol 30pp 31ndash39 2004

[73] J Raczek ldquoPaired bondage in treesrdquo Discrete Mathematics vol308 no 23 pp 5570ndash5575 2008

[74] J A Telle and A Proskurowski ldquoAlgorithms for vertex parti-tioning problems on partial k-treesrdquo SIAM Journal on DiscreteMathematics vol 10 no 4 pp 529ndash550 1997

[75] G S Domke J H Hattingh M A Henning and L R MarkusldquoRestrained domination in treesrdquoDiscreteMathematics vol 211no 1ndash3 pp 1ndash9 2000

[76] G S Domke J H Hattingh M A Henning and L R MarkusldquoRestrained domination in graphs with minimum degree twordquoJournal of Combinatorial Mathematics and Combinatorial Com-puting vol 35 pp 239ndash254 2000

[77] G S Domke J H Hattingh S T Hedetniemi R C Laskarand L R Markus ldquoRestrained domination in graphsrdquo DiscreteMathematics vol 203 no 1ndash3 pp 61ndash69 1999

[78] J H Hattingh and E J Joubert ldquoAn upper bound for therestrained domination number of a graph with minimumdegree at least two in terms of order and minimum degreerdquoDiscrete Applied Mathematics vol 157 no 13 pp 2846ndash28582009

[79] M A Henning ldquoGraphs with large restrained dominationnumberrdquo Discrete Mathematics vol 197-198 pp 415ndash429 199916th British Combinatorial Conference (London 1997)

[80] J H Hattingh and A R Plummer ldquoRestrained bondage ingraphsrdquo Discrete Mathematics vol 308 no 23 pp 5446ndash54532008

[81] N Jafari Rad R Hasni J Raczek and L Volkmann ldquoTotalrestrained bondage in graphsrdquoActaMathematica Sinica vol 29no 6 pp 1033ndash1042 2013

[82] J Cyman and J Raczek ldquoOn the total restrained dominationnumber of a graphrdquoThe Australasian Journal of Combinatoricsvol 36 pp 91ndash100 2006

[83] M A Henning ldquoGraphs with large total domination numberrdquoJournal of Graph Theory vol 35 no 1 pp 21ndash45 2000

[84] D-X Ma X-G Chen and L Sun ldquoOn total restraineddomination in graphsrdquoCzechoslovakMathematical Journal vol55 no 1 pp 165ndash173 2005

[85] B Zelinka ldquoRemarks on restrained domination and totalrestrained domination in graphsrdquo Czechoslovak MathematicalJournal vol 55 no 2 pp 393ndash396 2005

[86] W Goddard and M A Henning ldquoIndependent domination ingraphs a survey and recent resultsrdquo Discrete Mathematics vol313 no 7 pp 839ndash854 2013

[87] J H Zhang H L Liu and L Sun ldquoIndependence bondagenumber and reinforcement number of some graphsrdquo Transac-tions of Beijing Institute of Technology vol 23 no 2 pp 140ndash1422003

[88] K D Dharmalingam Studies in Graph Theorymdashequitabledomination and bottleneck domination [PhD thesis] MaduraiKamaraj University Madurai India 2006

[89] GDeepakND Soner andAAlwardi ldquoThe equitable bondagenumber of a graphrdquo Research Journal of Pure Algebra vol 1 no9 pp 209ndash212 2011

[90] E Sampathkumar and H B Walikar ldquoThe connected domina-tion number of a graphrdquo Journal of Mathematical and PhysicalSciences vol 13 no 6 pp 607ndash613 1979

[91] K White M Farber and W Pulleyblank ldquoSteiner trees con-nected domination and strongly chordal graphsrdquoNetworks vol15 no 1 pp 109ndash124 1985

[92] M Cadei M X Cheng X Cheng and D-Z Du ldquoConnecteddomination in Ad Hoc wireless networksrdquo in Proceedings ofthe 6th International Conference on Computer Science andInformatics (CSampI rsquo02) 2002

[93] J F Fink and M S Jacobson ldquon-domination in graphsrdquo inGraph Theory with Applications to Algorithms and ComputerScience Wiley-Interscience Publications pp 283ndash300 WileyNew York NY USA 1985

[94] M Chellali O Favaron A Hansberg and L Volkmann ldquok-domination and k-independence in graphs a surveyrdquo Graphsand Combinatorics vol 28 no 1 pp 1ndash55 2012

[95] Y Lu X Hou J-M Xu andN Li ldquoTrees with uniqueminimump-dominating setsrdquo Utilitas Mathematica vol 86 pp 193ndash2052011

[96] Y Lu and J-M Xu ldquoThe p-bondage number of treesrdquo Graphsand Combinatorics vol 27 no 1 pp 129ndash141 2011

[97] Y Lu and J-M Xu ldquoThe 2-domination and 2-bondage numbersof grid graphs A manuscriptrdquo httparxivorgabs12044514

[98] M Krzywkowski ldquoNon-isolating 2-bondage in graphsrdquo TheJournal of the Mathematical Society of Japan vol 64 no 4 2012

[99] M A Henning O R Oellermann and H C Swart ldquoBoundson distance domination parametersrdquo Journal of CombinatoricsInformation amp System Sciences vol 16 no 1 pp 11ndash18 1991

[100] F Tian and J-M Xu ldquoBounds for distance domination numbersof graphsrdquo Journal of University of Science and Technology ofChina vol 34 no 5 pp 529ndash534 2004

[101] F Tian and J-M Xu ldquoDistance domination-critical graphsrdquoApplied Mathematics Letters vol 21 no 4 pp 416ndash420 2008

[102] F Tian and J-M Xu ldquoA note on distance domination numbersof graphsrdquo The Australasian Journal of Combinatorics vol 43pp 181ndash190 2009

[103] F Tian and J-M Xu ldquoAverage distances and distance domina-tion numbersrdquo Discrete Applied Mathematics vol 157 no 5 pp1113ndash1127 2009

[104] Z-L Liu F Tian and J-M Xu ldquoProbabilistic analysis of upperbounds for 2-connected distance k-dominating sets in graphsrdquoTheoretical Computer Science vol 410 no 38ndash40 pp 3804ndash3813 2009

[105] M A Henning O R Oellermann and H C Swart ldquoDistancedomination critical graphsrdquo Journal of Combinatorial Mathe-matics and Combinatorial Computing vol 44 pp 33ndash45 2003

[106] T J Bean M A Henning and H C Swart ldquoOn the integrityof distance domination in graphsrdquo The Australasian Journal ofCombinatorics vol 10 pp 29ndash43 1994

[107] M Fischermann and L Volkmann ldquoA remark on a conjecturefor the (kp)-domination numberrdquoUtilitasMathematica vol 67pp 223ndash227 2005

[108] T Korneffel D Meierling and L Volkmann ldquoA remark on the(22)-domination numberrdquo Discussiones Mathematicae GraphTheory vol 28 no 2 pp 361ndash366 2008

[109] Y Lu X Hou and J-M Xu ldquoOn the (22)-domination numberof treesrdquo Discussiones Mathematicae Graph Theory vol 30 no2 pp 185ndash199 2010

34 International Journal of Combinatorics

[110] H Li and J-M Xu ldquo(dm)-domination numbers of m-connected graphsrdquo Rapports de Recherche 1130 LRI UniversiteParis-Sud 1997

[111] S M Hedetniemi S T Hedetniemi and T V Wimer ldquoLineartime resource allocation algorithms for treesrdquo Tech Rep URI-014 Department of Mathematical Sciences Clemson Univer-sity 1987

[112] M Farber ldquoDomination independent domination and dualityin strongly chordal graphsrdquo Discrete Applied Mathematics vol7 no 2 pp 115ndash130 1984

[113] G S Domke and R C Laskar ldquoThe bondage and reinforcementnumbers of 120574

119891for some graphsrdquoDiscrete Mathematics vol 167-

168 pp 249ndash259 1997[114] G S Domke S T Hedetniemi and R C Laskar ldquoFractional

packings coverings and irredundance in graphsrdquo vol 66 pp227ndash238 1988

[115] E J Cockayne P A Dreyer Jr S M Hedetniemi andS T Hedetniemi ldquoRoman domination in graphsrdquo DiscreteMathematics vol 278 no 1ndash3 pp 11ndash22 2004

[116] I Stewart ldquoDefend the roman empirerdquo Scientific American vol281 pp 136ndash139 1999

[117] C S ReVelle and K E Rosing ldquoDefendens imperiumromanum a classical problem in military strategyrdquoThe Ameri-can Mathematical Monthly vol 107 no 7 pp 585ndash594 2000

[118] A Bahremandpour F-T Hu S M Sheikholeslami and J-MXu ldquoOn the Roman bondage number of a graphrdquo DiscreteMathematics Algorithms and Applications vol 5 no 1 ArticleID 1350001 15 pages 2013

[119] N Jafari Rad and L Volkmann ldquoRoman bondage in graphsrdquoDiscussiones Mathematicae Graph Theory vol 31 no 4 pp763ndash773 2011

[120] K Ebadi and L PushpaLatha ldquoRoman bondage and Romanreinforcement numbers of a graphrdquo International Journal ofContemporary Mathematical Sciences vol 5 pp 1487ndash14972010

[121] N Dehgardi S M Sheikholeslami and L Volkman ldquoOn theRoman k-bondage number of a graphrdquo AKCE InternationalJournal of Graphs and Combinatorics vol 8 pp 169ndash180 2011

[122] S Akbari and S Qajar ldquoA note on Roman bondage number ofgraphsrdquo Ars Combinatoria to Appear

[123] F-T Hu and J-M Xu ldquoRoman bondage numbers of somegraphsrdquo httparxivorgabs11093933

[124] N Jafari Rad and L Volkmann ldquoOn the Roman bondagenumber of planar graphsrdquo Graphs and Combinatorics vol 27no 4 pp 531ndash538 2011

[125] S Akbari M Khatirinejad and S Qajar ldquoA note on the romanbondage number of planar graphsrdquo Graphs and Combinatorics2012

[126] B Bresar M A Henning and D F Rall ldquoRainbow dominationin graphsrdquo Taiwanese Journal of Mathematics vol 12 no 1 pp213ndash225 2008

[127] B L Hartnell and D F Rall ldquoOn dominating the Cartesianproduct of a graph and 120572krdquo Discussiones Mathematicae GraphTheory vol 24 no 3 pp 389ndash402 2004

[128] N Dehgardi S M Sheikholeslami and L Volkman ldquoThe k-rainbow bondage number of a graphrdquo to appear in DiscreteApplied Mathematics

[129] J von Neumann and O Morgenstern Theory of Games andEconomic Behavior Princeton University Press Princeton NJUSA 1944

[130] C BergeTheorıe des Graphes et ses Applications 1958[131] O Ore Theory of Graphs vol 38 American Mathematical

Society Colloquium Publications 1962[132] E Shan and L Kang ldquoBondage number in oriented graphsrdquoArs

Combinatoria vol 84 pp 319ndash331 2007[133] J Huang and J-M Xu ldquoThe bondage numbers of extended

de Bruijn and Kautz digraphsrdquo Computers amp Mathematics withApplications vol 51 no 6-7 pp 1137ndash1147 2006

[134] J Huang and J-M Xu ldquoThe total domination and total bondagenumbers of extended de Bruijn and Kautz digraphsrdquoComputersamp Mathematics with Applications vol 53 no 8 pp 1206ndash12132007

[135] Y Shibata and Y Gonda ldquoExtension of de Bruijn graph andKautz graphrdquo Computers amp Mathematics with Applications vol30 no 9 pp 51ndash61 1995

[136] X Zhang J Liu and JMeng ldquoThebondage number in completet-partite digraphsrdquo Information Processing Letters vol 109 no17 pp 997ndash1000 2009

[137] D W Bange A E Barkauskas and P J Slater ldquoEfficient dom-inating sets in graphsrdquo in Applications of Discrete Mathematicspp 189ndash199 SIAM Philadelphia Pa USA 1988

[138] M Livingston and Q F Stout ldquoPerfect dominating setsrdquoCongressus Numerantium vol 79 pp 187ndash203 1990

[139] L Clark ldquoPerfect domination in random graphsrdquo Journal ofCombinatorial Mathematics and Combinatorial Computing vol14 pp 173ndash182 1993

[140] D W Bange A E Barkauskas L H Host and L H ClarkldquoEfficient domination of the orientations of a graphrdquo DiscreteMathematics vol 178 no 1ndash3 pp 1ndash14 1998

[141] A E Barkauskas and L H Host ldquoFinding efficient dominatingsets in oriented graphsrdquo Congressus Numerantium vol 98 pp27ndash32 1993

[142] I J Dejter and O Serra ldquoEfficient dominating sets in CayleygraphsrdquoDiscrete AppliedMathematics vol 129 no 2-3 pp 319ndash328 2003

[143] J Lee ldquoIndependent perfect domination sets in Cayley graphsrdquoJournal of Graph Theory vol 37 no 4 pp 213ndash219 2001

[144] WGu X Jia and J Shen ldquoIndependent perfect domination setsin meshes tori and treesrdquo Unpublished

[145] N Obradovic J Peters and G Ruzic ldquoEfficient domination incirculant graphswith two chord lengthsrdquo Information ProcessingLetters vol 102 no 6 pp 253ndash258 2007

[146] K Reji Kumar and G MacGillivray ldquoEfficient domination incirculant graphsrdquo Discrete Mathematics vol 313 no 6 pp 767ndash771 2013

[147] D Van Wieren M Livingston and Q F Stout ldquoPerfectdominating sets on cube-connected cyclesrdquo Congressus Numer-antium vol 97 pp 51ndash70 1993

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

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Mathematical PhysicsAdvances in

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Operations ResearchAdvances in

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

International Journal of Combinatorics 5

of general graphs it is weightily significative to establishsome sharp bounds on the bondage number of a graph Inthis section we survey several known upper bounds on thebondage number in terms of some other graph-theoreticalparameters

31 Most Basic Upper Bounds We start this subsection witha simple observation

Observation 1 (Teschner [10] 1997) Let 119867 be a spanningsubgraph obtained from a graph119866 by removing 119896 edgesThen119887(119866) ⩽ 119887(119867) + 119896

If we select a spanning subgraph 119867 such that 119887(119867) = 1thenObservation 1 yields some upper bounds on the bondagenumber of a graph 119866 For example take119867 = 119866 minus 119864

119909+ 119890 and

119896 = 119889119866(119909) minus 1 where 119890 isin 119864

119909 the following result can be

obtained

Theorem 13 (Bauer et al [7] 1983) If there exists at least onevertex 119909 in a graph 119866 such that 120574(119866 minus 119909) ⩾ 120574(119866) then 119887(119866) ⩽119889119866(119909) ⩽ Δ(119866)

The following early result obtained can be derived fromObservation 1 by taking119867 = 119866minus119864

119909119910and 119896 = 119889(119909)+119889(119910)minus2

Theorem 14 (Bauer et al [7] 1983 Fink et al [8] 1990)119887(119866) ⩽ 119889(119909) + 119889(119910) minus 1 for any two adjacent vertices 119909 and119910 in a graph 119866 that is

119887 (119866) ⩽ min119909119910isin119864(119866)

119889119866(119909) + 119889

119866(119910) minus 1 (4)

This theorem gives a natural corollary obtained by severalauthors

Corollary 15 (Bauer et al [7] 1983 Fink et al [8] 1990) If 119866is a graphwithout isolated vertices then 119887(119866) ⩽ Δ(119866)+120575(119866)minus1

In 1999 Hartnell and Rall [11] extended Theorem 14 tothe following more general case which can be also derivedfrom Observation 1 by taking119867 = 119866 minus 119864

119909minus 119864

119910+ (119909 119911 119910) if

119889119866(119909 119910) = 2 where (119909 119911 119910) is a path of length 2 in 119866

Theorem 16 (Hartnell and Rall [11] 1999) 119887(119866) ⩽ 119889(119909) +119889(119910)minus1 for any distinct two vertices 119909 and 119910 in a graph119866with119889119866(119909 119910) ⩽ 2 that is

119887 (119866) ⩽ min119889119866(119909119910)⩽2

119889119866(119909) + 119889

119866(119910) minus 1 (5)

Corollary 17 (Fink et al [8] 1990) If a vertex of a graph 119866 isadjacent with two ormore vertices of degree one then 119887(119866) = 1

We remark that the bounds stated in Corollary 15 andTheorem 16 are sharp As indicated byTheorem 1 one class ofgraphs in which the bondage number achieves these boundsis the class of cycles whose orders are congruent to 1 modulo3

On the other hand Hartnell and Rall [17] sharpened theupper bound in Theorem 14 as follows which can be alsoderived from Observation 1

1198792(119909 119910)

1198794(119909 119910)

1198793(119909 119910)

1198791(119909 119910)

119909

119910

Figure 2 Illustration of 119879119894(119909 119910) for 119894 = 1 2 3 4

Theorem 18 (Hartnell and Rall [17] 1994) 119887(119866) ⩽ 119889119866(119909) +

119889119866(119910) minus 1 minus |119873

119866(119909) cap 119873

119866(119910)| for any two adjacent vertices 119909

and 119910 in a graph 119866 that is

119887 (119866) ⩽ min119909119910isin119864(119866)

119889119866(119909) + 119889

119866(119910) minus 1 minus

1003816100381610038161003816119873119866(119909) cap 119873

119866(119910)1003816100381610038161003816

(6)

These results give simple but important upper bounds onthe bondage number of a graph and is also the foundationof almost all results on bondage numbers upper boundsobtained till now

By careful consideration of the nature of the edges fromthe neighbors of119909 and119910Wang [18] further refined the boundin Theorem 18 For any edge 119909119910 isin 119864(119866) 119873

119866(119910) contains the

following four subsets

(1) 1198791(119909 119910) = 119873

119866[119909] cap 119873

119866(119910)

(2) 1198792(119909 119910) = 119908 isin 119873

119866(119910) 119873

119866(119908) sube 119873

119866(119910) minus 119909

(3) 1198793(119909 119910) = 119908 isin 119873

119866(119910) 119873

119866(119908) sube 119873

119866(119911)minus119909 for some

119911 isin 119873119866(119909) cap 119873

119866(119910)

(4) 1198794(119909 119910) = 119873

119866(119910) (119879

1(119909 119910) cup 119879

2(119909 119910) cup 119879

3(119909 119910))

The illustrations of 1198791(119909 119910) 119879

2(119909 119910) 119879

3(119909 119910) and

1198794(119909 119910) are shown in Figure 2 (corresponding vertices

pointed by dashed arrows)

Theorem 19 (Wang [18] 1996) For any nonempty graph 119866

119887 (119866) ⩽ min119909119910isin119864(119866)

119889119866(119909) +

10038161003816100381610038161198794 (119909 119910)1003816100381610038161003816 (7)

The graph in Figure 2 shows that the upper bound givenin Theorem 19 is better than that in Theorems 16 and 18for the upper bounds obtained from these two theorems are119889119866(119909)+119889

119866(119910)minus1 = 11 and 119889

119866(119909)+119889

119866(119910)minus|119873

119866(119909)cap119873

119866(119910)| =

9 respectively while the upper bound given byTheorem 19 is119889119866(119909) + |119879

4(119909 119910)| = 6

The following result is also an improvement ofTheorem 14 in which 119905 = 2

6 International Journal of Combinatorics

Theorem 20 (Teschner [10] 1997) If 119866 contains a completesubgraph 119870

119905with 119905 ⩾ 2 then 119887(119866) ⩽ min

119909119910isin119864(119870119905)119889

119866(119909) +

119889119866(119910) minus 119905 + 1

Following Fricke et al [19] a vertex 119909 of a graph 119866 is 120574-good if 119909 belongs to some 120574-set of119866 and 120574-bad if 119909 belongs tono 120574-set of 119866 Let 119860(119866) be the set of 120574-good vertices and let119861(119866) be the set of 120574-bad vertices in 119866 Clearly 119860(119866) 119861(119866)is a partition of119881(119866) Note there exists some 119909 isin 119860 such that120574(119866 minus 119909) = 120574(119866) say one end-vertex of 119875

5 Samodivkin [20]

presented some sharp upper bounds for 119887(119866) in terms of 120574-good and 120574-bad vertices of 119866

Theorem 21 (Samodivkin [20] 2008) Let 119866 be a graph

(a) Let 119862(119866) = 119909 isin 119881(119866) 120574(119866minus119909) ⩾ 120574(119866) If 119862(119866) = 0then

119887 (119866) ⩽ min 119889119866(119909) + 120574 (119866) minus 120574 (119866 minus 119909) 119909 isin 119862 (119866)

(8)

(b) If 119861 = 0 then

119887 (119866) ⩽ min 1003816100381610038161003816119873119866(119909) cap 119860

1003816100381610038161003816 119909 isin 119861 (119866) (9)

Theorem 22 (Samodivkin [20] 2008) Let 119866 be a graph If119860+

(119866) = 119909 isin 119860(119866) 119889119866(119909) ⩾ 1 and 120574(119866 minus 119909) lt 120574(119866) = 0

then

119887 (119866) ⩽ min119909isin119860+(119866)119910isin119861(119866minus119909)

119889119866(119909) +

1003816100381610038161003816119873119866(119910) cap 119860 (119866 minus 119909)

1003816100381610038161003816

(10)

Proposition 23 (Samodivkin [20] 2008) Under the notationof Theorem 19 if 119909 isin 119860+

(119866) then (1198791(119909 119910) minus 119909) cup 119879

2(119909 119910) cup

1198793(119909 119910) sube 119873

119866(119910) 119861(119866 minus 119909)

By Proposition 23 if 119909 isin 119860+

(119866) then

119889119866(119909) + min

119910isin119873119866(119909)

10038161003816100381610038161198794 (119909 119910)

1003816100381610038161003816

⩾ 119889119866(119909) + min

119910isin119873119866(119909)

1003816100381610038161003816119873119866

(119910) cap 119860 (119866 minus 119909)1003816100381610038161003816

⩾ 119889119866(119909) + min

119910isin119861(119866minus119909)

1003816100381610038161003816119873119866

(119910) cap 119860 (119866 minus 119909)1003816100381610038161003816

(11)

Hence Theorem 19 can be seen to follow from Theorem 22Any graph 119866 with 119887(119866) achieving the upper bound inTheorem 19 can be used to show that the bound inTheorem 22 is sharp

Let 119905 ⩾ 2 be an integer Samodivkin [20] con-structed a very interesting graph 119866

119905to show that the upper

bound in Theorem 22 is better than the known bounds Let119867

0 119867

1 119867

2 119867

119905+1be mutually vertex-disjoint graphs such

that 1198670cong 119870

2 119867

119905+1cong 119870

119905+3and 119867

119894cong 119870

119905+3minus 119890 for each

119894 = 1 2 119905 Let 119881(1198670) = 119909 119910 119909

119905+1isin 119881(119867

119905+1) and

1198671

1198672

1198673

11990911

11990912

11990921 11990922

1199093

119909

119910

Figure 3 The graph 1198662

1199091198941 119909

1198942isin 119881(119867

119894) 119909

11989411199091198942notin 119864(119867

119894) for each 119894 = 1 2 119905 The

graph 119866119905is defined as follows

119881 (119866119905) =

119905+1

⋃

119896=0

119881 (119867119896)

119864 (119866119905)=(

119905+1

⋃

119896=0

119864 (119867119896))cup(

119905

⋃

119894=1

1199091199091198941 119909119909

1198942)cup119909119909

119905+1

(12)

Such a constructed graph119866119905is shown in Figure 3 when 119905 = 2

Observe that 120574(119866119905) = 119905 + 2 119860(119866

119905) = 119881(119866

119905) 119889

119866119905

(119909) =

2119905 + 2 119889119866119905

(119909119905+1) = 119905 + 3 119889

119866119905

(119910) = 1 and 119889119866119905

(119911) = 119905 + 2

for each 119911 isin 119881(119866119905minus 119909 119910 119909

119905+1) Moreover 120574(119866 minus 119910) lt 120574(119866)

and 120574(119866119905minus119911) = 120574(119866

119905) for any 119911 isin 119881(119866

119905) minus 119910 Hence each of

the bounds stated inTheorems 13ndash20 is greater than or equals119905 + 2

Consider the graph 119866119905minus 119909119910 Clearly 120574(119866

119905minus 119909119910) = 120574(119866

119905)

and

119861 (119866119905minus 119909119910) = 119861 (119866

119905minus 119910)

= 119909 cup 119881 (119867119905+1minus 119909

119905+1) cup (

119905

⋃

119896=1

1199091198961

1199091198962

)

(13)

Therefore 119873119866119905

(119909) cap 119866(119866119905minus 119910) = 119909

119905+1 which implies that

the upper bound stated in Theorem 22 is equal to 119889119866119905

(119910) +

|119909119905+1| = 2 Clearly 119887(119866

119905) = 2 and hence this bound is sharp

for 119866119905

From the graph 119866119905 we obtain the following statement

immediately

Proposition 24 For every integer 119905 ⩾ 2 there is a graph 119866such that the difference between any upper bound stated inTheorems 13ndash20 and the upper bound in Theorem 22 is equalto 119905

Although Theorem 22 supplies us with the upper boundthat is closer to 119887(119866) for some graph 119866 than what any one of

International Journal of Combinatorics 7

Theorems 13ndash20 provides it is not easy to determine the sets119860+

(119866) and 119861(119866) mentioned in Theorem 22 for an arbitrarygraph 119866 Thus the upper bound given in Theorem 22 is oftheoretical importance but not applied since until now wehave not found a new class of graphswhose bondage numbersare determined byTheorem 22

The above-mentioned upper bounds on the bondagenumber are involved in only degrees of two vertices Hartnelland Rall [11] established an upper bound on 119887(119866) in terms ofthe numbers of vertices and edges of 119866 with order 119899 For anyconnected graph 119866 let 120575(119866) represent the average degree ofvertices in 119866 that is 120575(119866) = (1119899)sum

119909isin119881119889119866(119909) Hartnell and

Rall first discovered the following proposition

Proposition 25 For any connected graph 119866 there exist twovertices 119909 and 119910 with distance at most two and with theproperty that 119889

119866(119909) + 119889

119866(119910) ⩽ 2120575(119866)

Using Proposition 25 and Theorem 16 Hartnell and Rallgave the following bound

Theorem 26 (Hartnell and Rall [11] 1999) 119887(119866) ⩽ 2120575(119866) minus 1for any connected graph 119866

Note that 2120576 = 119899120575(119866) for any graph 119866 with order 119899 andsize 120576 Theorem 26 implies the following bound in terms oforder 119899 and size 120576

Corollary 27 119887(119866) ⩽ (4120576119899) minus 1 for any connected graph 119866with order 119899 and size 120576

A lower bound on 120576 in terms of order 119899 and the bondagenumber is obtained from Corollary 28 immediately

Corollary 28 120576 ⩾ (1198994)(119887(119866) + 1) for any connected graph 119866with order 119899 and size 120576

Hartnell and Rall [11] gave some graphs 119866 with order 119899to show that for each value of 119887(119866) the lower bound on 120576(119866)given in the Corollary 28 is sharp for some values of 119899

32 Bounds Implied by Connectivity Use 120581(119866) and 120582(119866) todenote the vertex-connectivity and the edge-connectivity ofa connected graph 119866 respectively which are the minimumnumbers of vertices and edges whose removal result in 119866disconnected The famous Whitney inequality is stated as120581(119866) ⩽ 120582(119866) ⩽ 120575(119866) for any graph or digraph 119866 Corollary 15is improved by several authors as follows

Theorem 29 (Hartnell and Rall [17] 1994 Teschner [10]1997) If 119866 is a connected graph then 119887(119866) ⩽ 120582(119866)+Δ(119866)minus 1

The upper bound given in Theorem 29 can be attainedFor example a cycle 119862

3119896+1with 119896 ⩾ 1 119887(119862

3119896+1) = 3 by

Theorem 1 Since 120581(1198623119896+1) = 120582(119862

3119896+1) = 2 we have 120581(119862

3119896+1)+

120582(1198623119896+1) minus 1 = 2 + 2 minus 1 = 3 = 119887(119862

3119896+1)

Motivated byCorollary 15Theorems 29 and theWhitneyinequality Dunbar et al [21] naturally proposed the followingconjecture

Figure 4 A graph119867 with 120574(119867) = 3 and 119887(119867) = 5

Conjecture 30 If119866 is a connected graph then 119887(119866) ⩽ Δ(119866)+120581(119866) minus 1

However Liu and Sun [22] presented a counterexampleto this conjecture They first constructed a graph 119867 showedin Figure 4 with 120574(119867) = 3 and 119887(119867) = 5 Then let 119866 be thedisjoint union of two copies of119867 by identifying two verticesof degree two They proved that 119887(119866) ⩾ 5 Clearly 119866 is a 4-regular graph with 120581(119866) = 1 and 120582(119866) = 2 and so 119887(119866) ⩽ 5byTheorem 29 Thus 119887(119866) = 5 gt 4 = Δ(119866) + 120581(119866) minus 1

With a suspicion of the relationship between the bondagenumber and the vertex-connectivity of a graph the followingconjecture is proposed

Conjecture 31 (Liu and Sun [22] 2003) For any positiveinteger 119903 there exists a connected graph 119866 such that 119887(119866) ⩾Δ(119866) + 120581(119866) + 119903

To the knowledge of the author until now no results havebeen known about this conjecture

We conclude this subsection with the following remarksFromTheorem 29 if Conjecture 31 holds for some connectedgraph119866 then 120582(119866) gt 120581(119866)+119903 which implies that119866 is of largeedge-connectivity and small vertex-connectivity Use 120585(119866) todenote the minimum edge-degree of 119866 that is

120585 (119866) = min119909119910isin119864(119866)

119889119866(119909) + 119889

119866(119910) minus 2 (14)

Theorem 14 implies the following result

Proposition 32 119887(119866) ⩽ 120585(119866) + 1 for any graph 119866

Use 1205821015840(119866) to denote the restricted edge-connectivity ofa connected graph 119866 which is the minimum number ofedgeswhose removal result in119866 disconnected and no isolatedvertices

Proposition 33 (Esfahanian and Hakimi [23] 1988) If 119866 isneither 119870

1119899nor 119870

3 then 1205821015840(119866) ⩽ 120585(119866)

Combining Propositions 32 and 33 we propose a conjec-ture

Conjecture 34 If a connected 119866 is neither 1198701119899

nor 1198703 then

119887(119866) ⩽ 120575(119866) + 1205821015840

(119866) minus 1

8 International Journal of Combinatorics

A cycle 1198623119896+1

also satisfies 119887(1198623119896+1) = 3 = 120575(119862

3119896+1) +

1205821015840

(1198623119896+1) minus 1 since 1205821015840(119862

3119896+1) = 120575(119862

3119896+1) = 2 for any integer

119896 ⩾ 1For the graph119867 shown in Figure 41205821015840(119867) = 4 and 120575(119867) =

2 and so 119887(119867) = 5 = 120575(119867) + 1205821015840(119867) minus 1For the 4-regular graph119866 constructed by Liu and Sun [22]

obtained from the disjoint union of two copies of the graph119867 showed in Figure 4 by identifying two vertices of degreetwo we have 119887(119866) ⩾ 5 Clearly 1205821015840(119866) = 2 Thus 119887(119866) = 5 =120575(119866) + 120582

1015840

(119866) minus 1For the 4-regular graph 119866

119905constructed by Samodivkin

[20] see Figure 3 for 119905 = 2 119887(119866119905) = 2 Clearly 120575(119866

119905) = 1

and 1205821015840(119866) = 2 Thus 119887(119866) = 2 = 120575(119866) + 1205821015840(119866) minus 1These examples show that if Conjecture 34 is true then the

given upper bound is tight

33 Bounds Implied by Degree Sequence Now let us return toTheorem 16 from which Teschner [10] obtained some otherbounds in terms of the degree sequencesThe degree sequence120587(119866) of a graph 119866 with vertex-set 119881 = 119909

1 119909

2 119909

119899 is the

sequence 120587 = (1198891 119889

2 119889

119899) with 119889

1⩽ 119889

2⩽ sdot sdot sdot ⩽ 119889

119899 where

119889119894= 119889

119866(119909

119894) for each 119894 = 1 2 119899 The following result is

essentially a corollary of Theorem 16

Theorem35 (Teschner [10] 1997) Let119866 be a nonempty graphwith degree sequence120587(119866) If120572

2(119866) = 119905 then 119887(119866) ⩽ 119889

119905+119889

119905+1minus

1

CombiningTheorem 35with Proposition 10 (ie 1205722(119866) ⩽

120574(119866)) we have the following corollary

Corollary 36 (Teschner [10] 1997) Let 119866 be a nonemptygraph with the degree sequence 120587(119866) If 120574(119866) = 120574 then 119887(119866) ⩽119889120574+ 119889

120574+1minus 1

In [10] Teschner showed that these two bounds are sharpfor arbitrarily many graphs Let119867 = 119862

3119896+1+ 119909

11199094 119909

11199093119896minus1

where 1198623119896+1

is a cycle (1199091 119909

2 119909

3119896+1 119909

1) for any integer

119896 ⩾ 2 Then 120574(119867) = 119896 + 1 and so 119887(119867) ⩽ 2 + 2 minus 1 = 3by Corollary 36 Since119862

3119896+1is a spanning subgraph of119867 and

120574(119866) = 120574(119867) Observation 1 yields that 119887(119867) ⩾ 119887(1198623119896+1) = 3

and so 119887(119867) = 3Although various upper bounds have been established

as the above we find that the appearance of these boundsis essentially based upon the local structures of a graphprecisely speaking the structures of the neighborhoods oftwo vertices within distance 2 Even if these bounds can beachieved by some special graphs it is more often not the caseThe reason lies essentially in the definition of the bondagenumber which is theminimumvalue among all bondage setsan integral property of a graph While it easy to find upperbounds just by choosing some bondage set the gap betweenthe exact value of the bondage number and such a boundobtained only from local structures of a graph is often largeFor example a star 119870

1Δ however large Δ is 119887(119870

1Δ) = 1

Therefore one has been longing for better bounds upon someintegral parameters However as what we will see below it isdifficult to establish such upper bounds

34 Bounds in 120574-Critical Graphs A graph119866 is called a vertexdomination-critical graph (vc-graph or 120574-critical for short) if120574(119866 minus 119909) lt 120574(119866) for any 119909 isin 119881(119866) proposed by Brighamet al [24] in 1988 Several families of graphs are known to be120574-critical From definition it is clear that if 119866 is a 120574-criticalgraph then 120574(119866) ⩾ 2The class of 120574-critical graphs with 120574 = 2is characterized as follows

Proposition 37 (Brigham et al [24] 1988) A graph 119866 with120574(119866) = 2 is a 120574-critical graph if and only if 119866 is a completegraph 119870

2119905(119905 ⩾ 2) with a perfect matching removed

A more interesting family is composed of the 119899-criticalgraphs 119866

119898119899defined for 119898 119899 ⩾ 2 by the circulant undirected

graph 119866(119873 plusmn119878) where 119873 = (119898 + 1)(119899 minus 1) + 1 and119878 = 1 2 lfloor1198982rfloor in which 119881(119866(119873 plusmn119878)) = 119873 and119864(119866(119873 plusmn119878)) = 119894119895 |119895 minus 119894| equiv 119904 (mod 119873) 119904 isin 119878

The reason why 120574-critical graphs are of special interest inthis context is easy to see that they play an important role instudy of the bondage number For instance it immediatelyfollows from Theorem 13 that if 119887(119866) gt Δ(119866) then 119866 is a 120574-critical graphThe 120574-critical graphs are defined exactly in thisway In order to find graphs 119866 with a high bondage number(ie higher than Δ(119866)) and beyond its general upper boundsfor the bondage number we therefore have to look at 120574-critical graphs

The bondage numbers of some 120574-critical graphs havebeen examined by several authors see for example [1020 25 26] From Theorem 13 we know that the bondagenumber of a graph 119866 is bounded from above by Δ(119866)if 119866 is not a 120574-critical graph For 120574-critical graphs it ismore difficult to find an upper bound We will see that thebondage numbers of 120574-critical graphs in general are noteven bounded from above by Δ + 119888 for any fixed naturalnumber 119888

In this subsection we introduce some upper bounds forthe bondage number of a 120574-critical graph By Proposition 37we easily obtain the following result

Theorem 38 If 119866 is a 120574-critical graph with 120574(119866) = 2 then119887(119866) ⩽ Δ + 1

In Section 4 by Theorem 59 we will see that the equalityin Theorem 38 holds that is 119887(119866) = Δ + 1 if 119866 is a 120574-criticalgraph with 120574(119866) = 2

Theorem 39 (Teschner [10] 1997) Let119866 be a 120574-critical graphwith degree sequence 120587(119866) Then 119887(119866) ⩽ maxΔ(119866) + 1 119889

1+

1198892+ sdot sdot sdot + 119889

120574minus1 where 120574 = 120574(119866)

As we mentioned above if 119866 is a 120574-critical graph with120574(119866) = 2 then 119887(119866) = Δ + 1 which shows that thebound given in Theorem 39 can be attained for 120574 = 2However we have not known whether this bound is tightfor general 120574 ⩾ 3 Theorem 39 gives the following corollaryimmediately

Corollary 40 (Teschner [10] 1997) Let 119866 be a 120574-criticalgraph with degree sequence 120587(119866) If 120574(119866) = 3 then 119887(119866) ⩽maxΔ(119866) + 1 120575(119866) + 119889

2

International Journal of Combinatorics 9

From Theorem 38 and Corollary 40 we have 119887(119866) ⩽2Δ(119866) if 119866 is a 120574-critical graph with 120574(119866) ⩽ 3 The followingresult shows that this bound is not tight

Theorem 41 (Teschner [26] 1995) Let119866 be a 120574-critical graphgraph with 120574(119866) ⩽ 3 Then 119887(119866) ⩽ (32)Δ(119866)

Until now we have not known whether the bound givenin Theorem 41 is tight or not We state two conjectures on120574-critical graphs proposed by Samodivkin [20] The first ofthem is motivated byTheorems 22 and 19

Conjecture 42 (Samodivkin [20] 2008) For every connectednontrivial 120574-critical graph 119866

min119909isin119860+(119866)119910isin119861(119866minus119909)

119889119866(119909) +

1003816100381610038161003816119873119866(119910) cap 119860 (119866 minus 119909)

1003816100381610038161003816 ⩽3

2Δ (119866)

(15)

To state the second conjecture we need the followingresult on 120574-critical graphs

Proposition 43 If119866 is a 120574-critical graph then 120592(119866) ⩽ (Δ(119866)+1)(120574(119866)minus1)+1 moreover if the equality holds then119866 is regular

The upper bound in Proposition 43 is sharp in thesense that equality holds for the infinite class of 120574-criticalgraphs 119866

119898119899defined in the beginning of this subsection In

Proposition 43 the first result is due to Brigham et al [24] in1988 the second is due to Fulman et al [27] in 1995

For a 120574-critical graph 119866 with 120574 = 3 by Proposition 43120592(119866) ⩽ 2Δ(119866) + 3

Theorem 44 (Teschner [26] 1995) If 119866 is a 120574-critical graphwith 120574 = 3 and 120592(119866) = 2Δ(119866) + 3 then 119887(119866) ⩽ Δ + 1

We have not known yet whether the equality inTheorem 44holds or notHowever Samodivkin proposed thefollowing conjecture

Conjecture 45 (Samodivkin [20] 2008) If 119866 is a 120574-criticalgraph with (Δ(119866)+1)(120574minus1)+1 vertices then 119887(119866) = Δ(119866)+1

In general based on Theorem 41 Teschner proposed thefollowing conjecture

Conjecture 46 (Teschner [26] 1995) If119866 is a 120574-critical graphthen 119887(119866) ⩽ (32)Δ(119866) for 120574 ⩾ 4

We conclude this subsection with some remarks Graphswhich are minimal or critical with respect to a given prop-erty or parameter frequently play an important role in theinvestigation of that property or parameter Not only are suchgraphs of considerable interest in their own right but also aknowledge of their structure often aids in the developmentof the general theory In particular when investigating anyfinite structure a great number of results are proven byinduction Consequently it is desirable to learn as much aspossible about those graphs that are critical with respect toa given property or parameter so as to aid and abet suchinvestigations

In this subsection we survey some results on the bondagenumber for 120574-critical graphs Although these results are notvery perfect they provides a feasible method to approach thebondage number from different viewpoints In particular themethods given in Teschner [26] worth further explorationand development

The following proposition is maybe useful for us tofurther investigate the bondage number of a 120574-critical graph

Proposition 47 (Brigham et al [24] 1988) If 119866 has anonisolated vertex 119909 such that the subgraph induced by119873

119866(119909)

is a complete graph then 119866 is not 120574-critical

This simple fact shows that any 120574-critical graph containsno vertices of degree one

35 Bounds Implied by Domination In the preceding sub-section we introduced some upper bounds on the bondagenumbers for 120574-critical graphs by consideration of domina-tions In this subsection we introduce related results for ageneral graph with given domination number

Theorem 48 (Fink et al [8] 1990) For any connected graph119866 of order 119899 (⩾2)

(a) 119887(119866) ⩽ 119899 minus 1

(b) 119887(119866) ⩽ Δ(119866) + 1 if 120574(119866) = 2

(c) 119887(119866) ⩽ 119899 minus 120574(119866) + 1

While the upper bound of 119899minus1 on 119887(119866) is not particularlygood for many classes of graphs (eg trees and cycles) it isan attainable bound For example if 119866 is a complete 119905-partitegraph 119870

119905(2) for 119905 = 1198992 and 119899 ⩽ 4 an even integer then the

three bounds on 119887(119866) in Theorem 48 are sharpTeschner [26] consider a graphwith 120574(119866) = 1 or 120574(119866) = 2

The next result is almost trivial but useful

Lemma 49 Let 119866 be a graph with order 119899 and 120574(119866) = 1 119905 thenumber of vertices of degree 119899 minus 1 Then 119887(119866) = lceil1199052rceil

Since 119905 ⩽ Δ(119866) clearly Lemma 49 yields the followingresult immediately

Theorem 50 (Teschner [26] 1995) 119887(119866) ⩽ (12)Δ+1 for anygraph 119866 with 120574(119866) = 1

For a complete graph1198702119896+1

119887(1198702119896+1) = 119896+1 = (12)Δ+1

which shows that the upper bound given in Theorem 50 canbe attained For a graph119866with 120574(119866) = 2 byTheorem 59 laterthe upper bound given inTheorem 48 (b) can be also attainedby a 2-critical graph (see Proposition 37)

36 Two Conjectures In 1990 when Fink et al [8] introducedthe concept of the bondage number they proposed thefollowing conjecture

Conjecture 51 119887(119866) ⩽ Δ(119866) + 1 for any nonempty graph 119866

10 International Journal of Combinatorics

Although some results partially support Conjecture 51Teschner [28] in 1993 found a counterexample to this con-jecture the cartesian product 119870

3times 119870

3 which shows that

119887(119866) = Δ(119866) + 2If a graph 119866 is a counterexample to Conjecture 51 it

must be a 120574-critical graph by Theorem 13 That is why thevertex domination-critical graphs are of special interest in theliterature

Now we return to Conjecture 51 Hartnell and Rall [17]and Teschner [29] independently proved that 119887(119866) can bemuch greater than Δ(119866) by showing the following result

Theorem 52 (Hartnell and Rall [17] 1994 Teschner [29]1996) For an integer 119899 ⩾ 3 let 119866

119899be the cartesian product

119870119899times 119870

119899 Then 119887(119866

119899) = 3(119899 minus 1) = (32)Δ(119866

119899)

This theorem shows that there exist no upper boundsof the form 119887(119866) ⩽ Δ(119866) + 119888 for any integer 119888 Teschner[26] proved that 119887(119866) ⩽ (32)Δ(119866) for any graph 119866 with120574(119866) ⩽ 2 (see Theorems 50 and 48) and for 120574-critical graphswith 120574(119866) = 3 and proposed the following conjecture

Conjecture 53 (Teschner [26] 1995) 119887(119866) ⩽ (32)Δ(119866) forany graph 119866

We believe that this conjecture is true but so far no muchwork about it has been known

4 Lower Bounds

Since the bondage number is defined as the smallest numberof edges whose removal results in an increase of dominationnumber each constructive method that creates a concretebondage set leads to an upper bound on the bondage numberFor that reason it is hard to find lower bounds Neverthelessthere are still a few lower bounds

41 Bounds Implied by Subgraphs The first lower boundon the bondage number is gotten in terms of its spanningsubgraph

Theorem 54 (Teschner [10] 1997) Let 119867 be a spanningsubgraph of a nonempty graph119866 If 120574(119867) = 120574(119866) then 119887(119867) ⩽119887(119866)

By Theorem 1 119887(119862119899) = 3 if 119899 equiv 1 (mod 3) and 119887(119862

119899) =

2 otherwise 119887(119875119899) = 2 if 119899 equiv 1 (mod 3) and 119887(119875

119899) = 1

otherwise From these results and Theorem 54 we get thefollowing two corollaries

Corollary 55 If119866 is hamiltonianwith order 119899 ⩾ 3 and 120574(119866) =lceil1198993rceil then 119887(119866) ⩾ 2 and in addition 119887(119866) ⩾ 3 if 119899 equiv 1 (mod3)

Corollary 56 If 119866 with order 119899 ⩾ 2 has a hamiltonian pathand 120574(119866) = lceil1198993rceil then 119887(119866) ⩾ 2 if 119899 equiv 1 (mod 3)

42 Other Bounds The vertex covering number 120573(119866) of 119866 isthe minimum number of vertices that are incident with all

Figure 5 A 120574-critical graph with 120573 = 120574 = 4 120575 = 2 and 119887 = 3

edges in 119866 If 119866 has no isolated vertices then 120574(119866) ⩽ 120573(119866)clearly In [30] Volkmann gave a lot of graphs with 120573 = 120574

Theorem 57 (Teschner [10] 1997) Let119866 be a graph If 120574(119866) =120573(119866) then

(a) 119887(119866) ⩾ 120575(119866)

(b) 119887(119866) ⩾ 120575(119866) + 1 if 119866 is a 120574-critical graph

The graph shown in Figure 5 shows that the bound givenin Theorem 57(b) is sharp For the graph 119866 it is a 120574-criticalgraph and 120574(119866) = 120573(119866) = 4 By Theorem 57 we have 119887(119866) ⩾3 On the other hand 119887(119866) ⩽ 3 by Theorem 16 Thus 119887(119866) =3

Proposition 58 (Sanchis [31] 1991) Let 119866 be a graph 119866 oforder 119899 (⩾ 6) and size 120576 If 119866 has no isolated vertices and3 ⩽ 120574(119866) ⩽ 1198992 then 120576 ⩽ ( 119899minus120574(119866)+1

2)

Using the idea in the proof of Theorem 57 every upperbound for 120574(119866) can lead to a lower bound for 119887(119866) Inthis way Teschner [10] obtained another lower bound fromProposition 58

Theorem59 (Teschner [10] 1997) Let119866 be a graph119866 of order119899 (⩾6) and size 120576 If 2 ⩽ 120574(119866) ⩽ 1198992 minus 1 then

(a) 119887(119866) ⩾ min120575(119866) 120576 minus ( 119899minus120574(119866)2)

(b) 119887(119866) ⩾ min120575(119866) + 1 120576 minus ( 119899minus120574(119866)2) if 119866 is a 120574-critical

graph

The lower bound in Theorem 59(b) is sharp for a classof 120574-critical graphs with domination number 2 Let 119866 be thegraph obtained from complete graph119870

2119905(119905 ⩾ 2) by removing

a perfect matching By Proposition 37 119866 is a 120574-critical graphwith 120574(119866) = 2 Then 119887(119866) ⩾ 120575(119866) + 1 = Δ(119866) + 1 byTheorem 59 and 119887(119866) ⩽ Δ(119866) + 1 by Theorem 38 and so119887(119866) = Δ(119866) + 1 = 2119905 minus 1

As far as the author knows there are no more lowerbounds in the present literature In view of applications ofthe bondage number a network is vulnerable if its bondagenumber is small while it is stable if its bondage number islargeTherefore better lower bounds let us learn better aboutthe stability of networks from this point of view Thus it isof great significance to seek more lower bounds for variousclasses of graphs

International Journal of Combinatorics 11

5 Results on Graph Operations

Generally speaking it is quite difficult to determine the exactvalue of the bondage number for a given graph since itstrongly depends on the dominating number of the graphThus determining bondage numbers for some special graphsis interesting if the dominating numbers of those graphs areknown or can be easily determined In this section we willintroduce some known results on bondage numbers for somespecial classes of graphs obtained by some graph operations

51 Cartesian Product Graphs Let 1198661= (119881

1 119864

1) and 119866

2=

(1198812 119864

2) be two graphs The Cartesian product of 119866

1and 119866

2

is an undirected graph denoted by 1198661times 119866

2 where 119881(119866

1times

1198662) = 119881

1times 119881

2 two distinct vertices 119909

11199092and 119910

11199102 where

1199091 119910

1isin 119881(119866

1) and 119909

2 119910

2isin 119881(119866

2) are linked by an edge in

1198661times 119866

2if and only if either 119909

1= 119910

1and 119909

21199102isin 119864(119866

2) or

1199092= 119910

2and 119909

11199101isin 119864(119866

1) The Cartesian product is a very

effective method for constructing a larger graph from severalspecified small graphs

Theorem 60 (Dunbar et al [21] 1998) For 119899 ⩾ 3

119887 (119862119899times 119870

2) =

2 119894119891 119899 equiv 0 119900119903 1 (mod 4) 3 119894119891 119899 equiv 3 (mod 4) 4 119894119891 119899 equiv 2 (mod 4)

(16)

For the Cartesian product 119862119899times 119862

119898of two cycles 119862

119899and

119862119898 where 119899 ⩾ 4 and 119898 ⩾ 3 Klavzar and Seifter [32]

determined 120574(119862119899times 119862

119898) for some 119899 and 119898 For example

120574(119862119899times119862

4) = 119899 for 119899 ⩾ 3 and 120574(119862

119899times119862

3) = 119899minuslfloor1198994rfloor for 119899 ⩾ 4

Kim [33] determined 119887(1198623times 119862

3) = 6 and 119887(119862

4times 119862

4) = 4 For

a general 119899 ⩾ 4 the exact values of the bondage numbers of119862119899times 119862

3and 119862

119899times 119862

4are determined as follows

Theorem 61 (Sohn et al [34] 2007) For 119899 ⩾ 4

119887 (119862119899times 119862

3) =

2 119894119891 119899 equiv 0 (mod 4) 4 119894119891 119899 equiv 1 119900119903 2 (mod 4) 5 119894119891 119899 equiv 3 (mod 4)

(17)

Theorem62 (Kang et al [35] 2005) 119887(119862119899times119862

4) = 4 for 119899 ⩾ 4

For larger 119898 and 119899 Huang and Xu [36] obtained thefollowing result see Theorem 197 for more details

Theorem 63 (Huang and Xu [36] 2008) 119887(1198625119894times119862

5119895) = 3 for

any positive integers 119894 and 119895

Xiang et al [37] determined that for 119899 ⩾ 5

119887 (119862119899times 119862

5)

= 3 if 119899 equiv 0 or 1 (mod 5) = 4 if 119899 equiv 2 or 4 (mod 5) ⩽ 7 if 119899 equiv 3 (mod 5)

(18)

For the Cartesian product 119875119899times119875

119898of two paths 119875

119899and 119875

119898

Jacobson and Kinch [38] determined 120574(119875119899times119875

2) = lceil(119899+1)2rceil

120574(119875119899times 119875

3) = 119899 minus lfloor(119899 minus 1)4rfloor and 120574(119875

119899times 119875

4) = 119899 + 1 if 119899 =

1 2 3 5 6 9 and 119899 otherwiseThe bondage number of119875119899times119875

119898

for 119899 ⩾ 4 and 2 ⩽ 119898 ⩽ 4 is determined as follows (Li [39] alsodetermined 119887(119875

119899times 119875

2))

Theorem 64 (Hu and Xu [40] 2012) For 119899 ⩾ 4

119887 (119875119899times 119875

2) =

1 119894119891 119899 equiv 1 (mod 2)2 119894119891 119899 equiv 0 (mod 2)

119887 (119875119899times 119875

3) =

1 119894119891 119899 equiv 1 119900119903 2 (mod 4)2 119894119891 119899 equiv 0 119900119903 3 (mod 4)

119887 (119875119899times 119875

4) = 1 119891119900119903 119899 ⩾ 14

(19)

From the proof of Theorem 64 we find that if 119875119899=

0 1 119899 minus 1 and 119875119898= 0 1 119898 minus 1 then the removal

of the vertex (0 0) in 119875119899times119875

119898does not change the domination

number If119898 increases the effect of (0 0) for the dominationnumber will be smaller and smaller from the probabilityTherefore we expect it is possible that 120574(119875

119899times 119875

119898minus (0 0)) =

120574(119875119899times 119875

119898) for119898 ⩾ 5 and give the following conjecture

Conjecture 65 119887(119875119899times 119875

119898) ⩽ 2 for119898 ⩾ 5 and 119899 ⩾ 4

52 Block Graphs and Cactus Graphs In this subsection weintroduce some results for corona graphs block graphs andcactus graphs

The corona1198661∘ 119866

2 proposed by Frucht and Harary [41]

is the graph formed from a copy of 1198661and 120592(119866

1) copies of

1198662by joining the 119894th vertex of 119866

1to the 119894th copy of 119866

2 In

particular we are concernedwith the corona119867∘1198701 the graph

formed by adding a new vertex V119894 and a new edge 119906

119894V119894for

every vertex 119906119894in119867

Theorem 66 (Carlson and Develin [42] 2006) 119887(119867 ∘ 1198701) =

120575(119867) + 1

This is a very important result which is often used toconstruct a graph with required bondage number In otherwords this result implies that for any given positive integer 119887there is a graph 119866 such that 119887(119866) = 119887

A block graph is a graph whose blocks are completegraphs Each block in a cactus graph is either a cycle or a119870

2

If each block of a graph is either a complete graph or a cyclethen we call this graph a block-cactus graph

Theorem67 (Teschner [43] 1997) 119887(119866) le Δ(119866) for any blockgraph 119866

Teschner [43] characterized all block graphs with 119887(119866) =Δ(119866) In the same paper Teschner found that 120574-criticalgraphs are instrumental in determining bounds for thebondage number of cactus and block graphs and obtained thefollowing result

Theorem 68 (Teschner [43] 1997) 119887(119866) ⩽ 3 for anynontrivial cactus graph 119866

This bound can be achieved by 119867 ∘ 1198701where 119867 is

a nontrivial cactus graph without vertices of degree one

12 International Journal of Combinatorics

by Theorem 66 In 1998 Dunbar et al [21] proposed thefollowing problem

Problem 2 Characterize all cactus graphs with bondagenumber 3

Some upper bounds for block-cactus graphs were alsoobtained

Theorem 69 (Dunbar et al [21] 1998) Let 119866 be a connectedblock-cactus graph with at least two blocksThen 119887(119866) ⩽ Δ(119866)

Theorem 70 (Dunbar et al [21] 1998) Let 119866 be a connectedblock-cactus graph which is neither a cactus graph nor a blockgraph Then 119887(119866) ⩽ Δ(119866) minus 1

53 Edge-Deletion and Edge-Contraction In order to obtainfurther results of the bondage number we may consider howthe bondage number changes under some operations of agraph 119866 which remain the domination number unchangedor preserve some property of 119866 such as planarity Twosimple operations satisfying these requirements are the edge-deletion and the edge-contraction

Theorem 71 (Huang and Xu [44] 2012) Let 119866 be any graphand 119890 isin 119864(119866) Then 119887(119866 minus 119890) ⩾ 119887(119866) minus 1 Moreover 119887(119866 minus 119890) ⩽119887(119866) if 120574(119866 minus 119890) = 120574(119866)

FromTheorem 1 119887(119862119899minus 119890) = 119887(119875

119899) = 119887(119862

119899) minus 1 for any 119890

in119862119899 which shows that the lower bound on 119887(119866minus119890) is sharp

The upper bound can reach for any graph 119866 with 119887(119866) ⩾ 2Next we consider the effect of the edge-contraction on

the bondage number Given a graph 119866 the contraction of 119866by the edge 119890 = 119909119910 denoted by 119866119909119910 is the graph obtainedfrom 119866 by contracting two vertices 119909 and 119910 to a new vertexand then deleting all multiedges It is easy to observe that forany graph 119866 120574(119866) minus 1 ⩽ 120574(119866119909119910) ⩽ 120574(119866) for any edge 119909119910 of119866

Theorem 72 (Huang and Xu [44] 2012) Let 119866 be any graphand 119909119910 be any edge in119866 If119873

119866(119909)cap119873

119866(119910) = 0 and 120574(119866119909119910) =

120574(119866) then 119887(119866119909119910) ⩾ 119887(119866)

We can use examples 119862119899and 119870

119899to show that the

conditions ofTheorem 72 are necessary and the lower boundinTheorem 72 is tight Clearly 120574(119862

119899) = lceil1198993rceil and 120574(119870

119899) = 1

for any edge 119909119910 119862119899119909119910 = 119862

119899minus1and 119870

119899119909119910 = 119870

119899minus1 By

Theorem 1 if 119899 equiv 1 (mod 3) then 120574(119862119899119909119910) lt 120574(119866) and

119887(119862119899119909119910) = 2 lt 3 = 119887(119862

119899) Thus the result in Theorem 72

is generally invalid without the hypothesis 120574(119866119909119910) = 120574(119866)Furthermore the condition 119873

119866(119909) cap 119873

119866(119910) = 0 cannot be

omitted even if 120574(119866119909119910) = 120574(119866) since for odd 119899 119887(119870119899119909119910) =

lceil(119899 minus 1)2rceil lt lceil1198992rceil = 119887(119870119899) by Theorem 1

On the other hand 119887(119862119899119909119910) = 119887(119862

119899) = 2 if 119899 equiv 0 (mod

3) and 119887(119870119899119909119910) = 119887(119870

119899) if 119899 is even which shows that the

equality in 119887(119866119909119910) ⩾ 119887(119866) may hold Thus the bound inTheorem 72 is tight However provided all the conditions119887(119866119909119910) can be arbitrarily larger than 119887(119866) Given a graph119867let119866 be the graph formed from119867∘119870

1by adding a new vertex

119909 and joining it to an vertex 119910 of degree one in119867 ∘ 1198701 Then

119866119909119910 = 119867 ∘ 1198701 120574(119866) = 120574(119866119909119910) and 119873

119866(119909) cap 119873

119866(119910) = 0

But 119887(119866) = 1 since 120574(119866 minus 119909119910) = 120574(119866119909119910) + 1 and 119887(119866119909119910) =120575(119867) + 1 byTheorem 66The gap between 119887(119866) and 119887(119866119909119910)is 120575(119867)

6 Results on Planar Graphs

From Section 2 we have seen that the bondage numberfor a tree has been completely solved Moreover a lineartime algorithm to compute the bondage number of a tree isdesigned by Hartnell et al [15] It is quite natural to considerthe bondage number for a planar graph In this section westate some results and problems on the bondage number fora planar graph

61 Conjecture on Bondage Number As mentioned inSection 3 the bondage number can be much larger than themaximum degree But for a planar graph 119866 the bondagenumber 119887(119866) cannot exceed Δ(119866) too much It is clear that119887(119866) ⩽ Δ(119866) + 4 by Corollary 15 since 120575(119866) ⩽ 5 for anyplanar graph119866 In 1998Dunbar et al [21] posed the followingconjecture

Conjecture 73 If 119866 is a planar graph then 119887(119866) ⩽ Δ(119866) + 1

Because of its attraction it immediately became the focusof attention when this conjecture is proposed In fact themain aim concerning the research of the bondage number fora planar graph is focused on this conjecture

It has been mentioned in Theorems 1 and 60 that119887(119862

3119896+1) = 3 = Δ + 1 and 119887(119862

4119896+2times 119870

2) = 4 = Δ + 1 It

is known that 119887(1198706minus 119872) = 5 = Δ + 1 where119872 is a perfect

matching of the complete graph1198706These examples show that

if Conjecture 73 is true then the upper bound is sharp for2 ⩽ Δ ⩽ 4

Here we note that it is sufficient to prove this conjecturefor connected planar graphs since the bondage number of adisconnected graph is simply the minimum of the bondagenumbers of its components

The first paper attacking this conjecture is due to Kangand Yuan [45] which confirmed the conjecture for everyconnected planar graph 119866 with Δ ⩾ 7 The proofs are mainlybased onTheorems 16 and 18

Theorem 74 (Kang and Yuan [45] 2000) If 119866 is a connectedplanar graph then 119887(119866) ⩽ min8 Δ(119866) + 2

Obviously in view of Theorem 74 Conjecture 73 is truefor any connected planar graph with Δ ⩾ 7

62 Bounds Implied by Degree Conditions As we have seenfrom Theorem 74 to attack Conjecture 73 we only need toconsider connected planar graphs with maximum Δ ⩽ 6Thus studying the bondage number of planar graphs bydegree conditions is of significanceThe first result on boundsimplied by degree conditions is obtained by Kang and Yuan[45]

International Journal of Combinatorics 13

Theorem 75 (Kang and Yuan [45] 2000) If 119866 is a connectedplanar graph without vertices of degree 5 then 119887(119866) ⩽ 7

Fischermann et al [46] generalized Theorem 75 as fol-lows

Theorem 76 (Fischermann et al [46] 2003) Let 119866 be aconnected planar graph and 119883 the set of vertices of degree 5which have distance at least 3 to vertices of degrees 1 2 and3 If all vertices in 119883 not adjacent with vertices of degree 4are independent and not adjacent to vertices of degree 6 then119887(119866) ⩽ 7

Clearly if119866 has no vertices of degree 5 then119883 = 0 ThusTheorem 76 yields Theorem 75

Use 119899119894to denote the number of vertices of degree 119894 in 119866

for each 119894 = 1 2 Δ(119866) Using Theorem 18 Fischermannet al obtained the following two theorems

Theorem 77 (Fischermann et al [46] 2003) For any con-nected planar graph 119866

(1) 119887(119866) ⩽ 7 if 1198995lt 2119899

2+ 3119899

3+ 2119899

4+ 12

(2) 119887(119866) ⩽ 6 if 119866 contains no vertices of degrees 4 and 5

Theorem 78 (Fischermann et al [46] 2003) For any con-nected planar graph 119866 119887(119866) ⩽ Δ(119866) + 1 if

(1) Δ(119866) = 6 and every edge 119890 = 119909119910 with 119889119866(119909) = 5 and

119889119866(119910) = 6 is contained in at most one triangle

(2) Δ(119866) = 5 and no triangle contains an edge 119890 = 119909119910 with119889119866(119909) = 5 and 4 ⩽ 119889

119866(119910) ⩽ 5

63 Bounds Implied by Girth Conditions The girth 119892(119866) of agraph 119866 is the length of the shortest cycle in 119866 If 119866 has nocycles we define 119892(119866) = infin

Combining Theorem 74 with Theorem 78 we find thatif a planar graph contains no triangles and has maximumdegree Δ ⩾ 5 then Conjecture 73 holds This fact motivatedFischermann et alrsquos [46] attempt to attack Conjecture 73 bygirth restraints

Theorem 79 (Fischermann et al [46] 2003) For any con-nected planar graph 119866

119887 (119866) ⩽

6 119894119891 119892 (119866) ⩾ 4

5 119894119891 119892 (119866) ⩾ 5

4 119894119891 119892 (119866) ⩾ 6

3 119894119891 119892 (119866) ⩾ 8

(20)

The first result in Theorem 79 shows that Conjecture 73is valid for all connected planar graphs with 119892(119866) ⩾ 4 andΔ(119866) ⩾ 5 It is easy to verify that the second result inTheorem 79 implies that Conjecture 73 is valid for all not3-regular graphs of girth 119892(119866) ⩾ 5 which is stated in thefollowing corollary

Corollary 80 For any connected planar graph 119866 if 119866 is not3-regular and 119892(119866) ⩾ 5 then 119887(119866) ⩽ Δ(119866) + 1

The first result in Theorem 79 also implies that if 119866 isa connected planar graph with no cycles of length 3 then119887(119866) ⩽ 6 Hou et al [47] improved this result as follows

Theorem 81 (Hou et al [47] 2011) For any connected planargraph 119866 if 119866 contains no cycles of length 119894 (4 ⩽ 119894 ⩽ 6) then119887(119866) ⩽ 6

Since 119887(1198623119896+1) = 3 for 119896 ⩾ 3 (see Theorem 1) the

last bound in Theorem 79 is tight Whether other bounds inTheorem 79 are tight remains open In 2003 Fischermann etal [46] posed the following conjecture

Conjecture 82 For any connected planar graph 119866 119887(119866) ⩽ 7and

119887 (119866) ⩽ 5 119894119891 119892 (119866) ⩾ 4

4 119894119891 119892 (119866) ⩾ 5(21)

We conclude this subsection with a question on bondagenumbers of planar graphs

Question 1 (Fischermann et al [46] 2003) Is there a planargraph 119866 with 6 ⩽ 119887(119866) ⩽ 8

In 2006 Carlson andDevelin [42] showed that the corona119866 = 119867 ∘ 119870

1for a planar graph 119867 with 120575(119867) = 5 has the

bondage number 119887(119866) = 120575(119867) + 1 = 6 (see Theorem 66)Since the minimum degree of planar graphs is at most 5then 119887(119866) can attach 6 If we take 119867 as the graph of theicosahedron then 119866 = 119867 ∘ 119870

1is such an example The

question for the existence of planar graphs with bondagenumber 7 or 8 remains open

64 Comments on the Conjectures Conjecture 73 is true forall connected planar graphs with minimum degree 120575 ⩽ 2 byTheorem 16 or maximum degree Δ ⩾ 7 by Theorem 74 ornot 120574-critical planar graphs by Theorem 13 Thus to attackConjecture 73 we only need to consider connected criticalplanar graphs with degree-restriction 3 ⩽ 120575 ⩽ Δ ⩽ 6

Recalling and anatomizing the proofs of all results men-tioned in the preceding subsections on the bondage numberfor connected planar graphs we find that the proofs of theseresults strongly depend upon Theorem 16 or Theorem 18 Inother words a basic way used in the proofs is to find twovertices 119909 and 119910 with distance at most two in a consideredplanar graph 119866 such that

119889119866(119909) + 119889

119866(119910) or 119889

119866(119909) + 119889

119866(119910) minus

1003816100381610038161003816119873119866(119909) cap 119873

119866(119910)1003816100381610038161003816

(22)

which bounds 119887(119866) is as small as possible Very recentlyHuang and Xu [44] have considered the following parameter

119861 (119866) = min119909119910isin119881(119866)

119889119909119910 1 ⩽ 119889

119866(119909 119910) ⩽ 2

cup 119889119909119910minus 119899

119909119910 119889 (119909 119910) = 1

(23)

where 119889119909119910= 119889

119866(119909) + 119889

119866(119910) minus 1 and 119899

119909119910= |119873

119866(119909) cap 119873

119866(119910)|

14 International Journal of Combinatorics

It follows fromTheorems 16 and 18 that

119887 (119866) ⩽ 119861 (119866) (24)

The proofs given in Theorems 74 and 79 indeed imply thefollowing stronger results

Theorem 83 If 119866 is a connected planar graph then

119861 (119866) ⩽

min 8 Δ (119866) + 26 119894119891 119892 (119866) ⩾ 4

5 119894119891 119892 (119866) ⩾ 5

4 119894119891 119892 (119866) ⩾ 6

3 119894119891 119892 (119866) ⩾ 8

(25)

Thus using Theorem 16 or Theorem 18 if we can proveConjectures 73 and 82 then we can prove the followingstatement

Statement If 119866 is a connected planar graph then

119861 (119866) ⩽

Δ (119866) + 1

7

5 if 119892 (119866) ⩾ 44 if 119892 (119866) ⩾ 5

(26)

It follows from (24) that Statement implies Conjectures 73and 82 However Huang and Xu [44] gave examples to showthat none of conclusions in Statement is true As a result theystated the following conclusion

Theorem 84 (Huang and Xu [44] 2012) It is not possible toprove Conjectures 73 and 82 if they are right using Theorems16 and 18

Therefore a new method is needed to prove or disprovethese conjectures At the present time one cannot prove theseconjectures and may consider to disprove them If one ofthese conjectures is invalid then there exists a minimumcounterexample 119866 with respect to 120592(119866) + 120576(119866) In order toobtain a minimum counterexample one may consider twosimple operations satisfying these requirements the edge-deletion and the edge-contractionwhich decrease 120592(119866)+120576(119866)and preserve planarity However applying Theorems 71 and72 Huang and Xu [44] presented the following results

Theorem 85 (Huang and Xu [44] 2012) It is not possible toconstruct minimum counterexamples to Conjectures 73 and 82by the operation of an edge-deletion or an edge-contraction

7 Results on Crossing Number Restraints

It is quite natural to generalize the known results on thebondage number for planar graphs to formore general graphsin terms of graph-theoretical parameters In this section weconsider graphs with crossing number restraints

The crossing number cr(119866) of 119866 is the smallest numberof pairwise intersections of its edges when 119866 is drawn in theplane If cr(119866) = 0 then 119866 is a planar graph

71 General Methods Use 119899119894to denote the number of vertices

of degree 119894 in119866 for 119894 = 1 2 Δ(119866) Huang and Xu obtainedthe following results

Theorem 86 (Huang and Xu [48] 2007) For any connectedgraph 119866

119887 (119866) ⩽

6 119894119891 119892 (119866) ⩾ 4 119886119899119889 2cr (119866) lt 121198861

5 119894119891 119892 (119866) ⩾ 5 119886119899119889 6cr (119866) lt 161198862

4 119894119891 119892 (119866) ⩾ 6 119886119899119889 4cr (119866) lt 141198863

3 119894119891 119892 (119866) ⩾ 8 119886119899119889 6cr (119866) lt 161198864

(27)

where 1198861= 119899

1+ 2119899

2+ 2119899

3+sum

Δ

119894=8(119894 minus 7)119899

119894+ 8 119886

2= 3119899

1+ 6119899

2+

51198993+sum

Δ

119894=7(3119894minus18)119899

119894+20 119886

3= 119899

1+2119899

2+sum

Δ

119894=6(2119894minus10)119899

119894+12

and 1198864= sum

Δ

119894=5(3119894 minus 12)119899

119894+ 16

Simplifying the conditions in Theorem 86 we obtain thefollowing corollaries immediately

Corollary 87 (Huang and Xu [48] 2007) For any connectedgraph 119866

119887 (119866) ⩽

6 119894119891 119892 (119866) ⩾ 4 119886119899119889 cr (119866) ⩽ 3

5 119894119891 119892 (119866) ⩾ 5 119886119899119889 cr (119866) ⩽ 4

4 119894119891 119892 (119866) ⩾ 6 119886119899119889 cr (119866) ⩽ 2

3 119894119891 119892 (119866) ⩾ 8 119886119899119889 cr (119866) ⩽ 2

(28)

Corollary 88 (Huang and Xu [48] 2007) For any connectedgraph 119866

(a) 119887(119866) ⩽ 6 if 119866 is not 4-regular cr(119866) = 4 and 119892(119866) ⩾ 4

(b) 119887(119866) ⩽ Δ(119866) + 1 if 119866 is not 3-regular cr(119866) ⩽ 4 and119892(119866) ⩾ 5

(c) 119887(119866) ⩽ 4 if 119866 is not 3-regular cr(119866) = 3 and 119892(119866) ⩾ 6

(d) 119887(119866) ⩽ 3 if cr(119866) = 3 119892(119866) ⩾ 8 and Δ(119866) ⩾ 5

These corollaries generalize some known results for pla-nar graphs For example Corollary 87 contains Theorem 79Corollary 88 (b) contains Corollary 80

Theorem 89 (Huang and Xu [48] 2007) For any connectedgraph 119866 if cr(119866) ⩽ 119899

3(119866) + 119899

4(119866) + 3 then 119887(119866) ⩽ 8

Corollary 90 (Huang and Xu [48] 2007) For any connectedgraph 119866 if cr(119866) ⩽ 3 then 119887(119866) ⩽ 8

Perhaps being unaware of this result in 2010 Ma et al[49] proved that 119887(119866) ⩽ 12 for any graph 119866 with cr(119866) = 1

Theorem91 (Huang and Xu [48] 2007) Let119866 be a connectedgraph and 119868 = 119909 isin 119881(119866) 119889

119866(119909) = 5 119889

119866(119909 119910) ⩾ 3 if

International Journal of Combinatorics 15

119889119866(119910) ⩽ 3 and 119889

119866(119910) = 4 for every 119910 isin 119873

119866(119909) If 119868 is

independent no vertices adjacent to vertices of degree 6 and

cr (119866) lt max5119899

3+ |119868| minus 2119899

4+ 28

117119899

3+ 40

16 (29)

then 119887(119866) ⩽ 7

Corollary 92 (Huang and Xu [48] 2007) Let 119866 be aconnected graph with cr(119866) ⩽ 2 If 119868 = 119909 isin 119881(119866) 119889

119866(119909) =

5 119889119866(119910 119909) ⩾ 3 if 119889

119866(119910) ⩽ 3 and 119889

119866(119910) = 4 for every 119910 isin

119873119866(119909) is independent and no vertices adjacent to vertices of

degree 6 then 119887(119866) ⩽ 7

Theorem93 (Huang andXu [48] 2007) Let119866 be a connectedgraph If 119866 satisfies

(a) 5cr(119866) + 1198995lt 2119899

2+ 3119899

3+ 2119899

4+ 12 or

(b) 7cr(119866) + 21198995lt 3119899

2+ 4119899

4+ 24

then 119887(119866) ⩽ 7

Proposition 94 (Huang and Xu [48] 2007) Let 119866 be aconnected graph with no vertices of degrees four and five Ifcr(119866) ⩽ 2 then 119887(119866) ⩽ 6

Theorem 95 (Huang and Xu [48] 2007) If 119866 is a connectedgraph with cr(119866) ⩽ 4 and not 4-regular when cr(119866) = 4 then119887(119866) ⩽ Δ(119866) + 2

The above results generalize some known results forplanar graphs For example Corollary 90 and Theorem 95contain Theorem 74 the first condition in Theorem 93 con-tains the second condition inTheorem 77

From Corollary 88 and Theorem 95 we suggest the fol-lowing questions

Question 2 Is there a

(a) 4-regular graph 119866 with cr(119866) = 4 such that 119887(119866) ⩾ 7

(b) 3-regular graph 119866 with cr ⩽ 4 and 119892(119866) ⩾ 5 such that119887(119866) = 5

(c) 3-regular graph 119866 with cr = 3 and 119892(119866) = 6 or 7 suchthat 119887(119866) = 5

72 Carlson-Develin Methods In this subsection we intro-duce an elegant method presented by Carlson and Develin[42] to obtain some upper bounds for the bondage numberof a graph

Suppose that 119866 is a connected graph We say that 119866 hasgenus 120588 if 119866 can be embedded in a surface 119878 with 120588 handlessuch that edges are pairwise disjoint except possibly for end-vertices Let119866 be an embedding of119866 in surface 119878 and let120601(119866)denote the number of regions in 119866 The boundary of everyregion contains at least three edges and every edge is on theboundary of atmost two regions (the two regions are identicalwhen 119890 is a cut-edge) For any edge 119890 of 119866 let 1199031

119866(119890) and 1199032

119866(119890)

be the numbers of edges comprising the regions in 119866 whichthe edge 119890 borders It is clear that every 119890 = 119909119910 isin 119864(119866)

1199032

119866(119890) ⩾ 119903

1

119866(119890) ⩾ 4 if 1003816100381610038161003816119873119866

(119909) cap 119873119866(119910)1003816100381610038161003816 = 0

1199032

119866(119890) ⩾ 4 119903

1

119866(119890) ⩾ 3 if 1003816100381610038161003816119873119866

(119909) cap 119873119866(119910)1003816100381610038161003816 = 1

1199032

119866(119890) ⩾ 119903

1

119866(119890) ⩾ 3 if 1003816100381610038161003816119873119866

(119909) cap 119873119866(119910)1003816100381610038161003816 ⩾ 2

(30)

Following Carlson and Develin [42] for any edge 119890 = 119909119910 of119866 we define

119863119866(119890) =

1

119889119866(119909)+

1

119889119866(119910)

+1

1199031

119866(119890)+

1

1199032

119866(119890)minus 1 (31)

By the well-known Eulerrsquos formula

120592 (119866) minus 120576 (119866) + 120601 (119866) = 2 minus 2120588 (119866) (32)

it is easy to see that

sum

119890isin119864(119866)

119863119866(119890) = 120592 (119866) minus 120576 (119866) + 120601 (119866) = 2 minus 2120588 (119866) (33)

If 119866 is a planar graph that is 120588(119866) = 0 then

sum

119890isin119864(119866)

119863119866(119890) = 120592 (119866) minus 120576 (119866) + 120601 (119866) = 2 (34)

Combining these formulas withTheorem 18 Carlson andDevelin [42] gave a simple and intuitive proof ofTheorem 74and obtained the following result

Theorem 96 (Carlson and Develin [42] 2006) Let 119866 be aconnected graph which can be embedded on a torus Then119887(119866) ⩽ Δ(119866) + 3

Recently Hou and Liu [50] improved Theorem 96 asfollows

Theorem 97 (Hou and Liu [50] 2012) Let 119866 be a connectedgraph which can be embedded on a torus Then 119887(119866) ⩽ 9Moreover if Δ(119866) = 6 then 119887(119866) ⩽ 8

By the way Cao et al [51] generalized the result inTheorem 79 to a connected graph 119866 that can be embeddedon a torus that is

119887 (119866) le

6 if 119892 (119866) ⩾ 4 and 119866 is not 4-regular5 if 119892 (119866) ⩾ 54 if 119892 (119866) ⩾ 6 and 119866 is not 3-regular3 if 119892 (119866) ⩾ 8

(35)

Several authors used this method to obtain some resultson the bondage number For example Fischermann et al[46] used this method to prove the second conclusion inTheorem 78 Recently Cao et al [52] have used thismethod todeal with more general graphs with small crossing numbersFirst they found the following property

Lemma 98 120575(119866) ⩽ 5 for any graph 119866 with cr(119866) ⩽ 5

16 International Journal of Combinatorics

This result implies that 119887(119866) ⩽ Δ(119866) + 4 by Corollary 15for any graph 119866 with cr(119866) ⩽ 5 Cao et al established thefollowing relation in terms of maximum planar subgraphs

Lemma 99 (Cao et al [52]) Let 119866 be a connected graph withcrossing number cr(119866) For any maximum planar subgraph119867of 119866

sum

119890isin119864(119867)

119863119866(119890) ⩾ 2 minus

2cr (119866)120575 (119866)

(36)

Using Carlson-Develin method and combiningLemma 98 with Lemma 99 Cao et al proved the followingresults

Theorem 100 (Cao et al [52]) For any connected graph 119866119887(119866) ⩽ Δ(119866) + 2 if 119866 satisfies one of the following conditions

(a) cr(119866) ⩽ 3(b) cr(119866) = 4 and 119866 is not 4-regular(c) cr(119866) = 5 and 119866 contains no vertices of degree 4

Theorem101 (Cao et al [52]) Let119866 be a connected graphwithΔ(119866) = 5 and cr(119866) ⩽ 4 If no triangles contain two vertices ofdegree 5 then 119887(119866) ⩽ 6 = Δ(119866) + 1

Theorem 102 (Cao et al [52]) Let 119866 be a connected graphwith Δ(119866) ⩾ 6 and cr(119866) ⩽ 3 If Δ(119866) ⩾ 7 or if Δ(119866) = 6120575(119866) = 3 and every edge 119890 = 119909119910 with 119889

119866(119909) = 5 and 119889

119866(119910) = 6

is contained in atmost one triangle then 119887(119866) ⩽ min8 Δ(119866)+1

Using Carlson-Develin method it can be proved that119887(119866) ⩽ Δ(119866) + 2 if cr(119866) ⩽ 3 (see the first conclusion inTheorem 100) and not yet proved that 119887(119866) ⩽ 8 if cr(119866) ⩽3 although it has been proved by using other method (seeCorollary 90)

By using Carlson-Develin method Samodivkin [25]obtained some results on the bondage number for graphswithsome given properties

Kim [53] showed that 119887(119866) ⩽ Δ(119866) + 2 for a connectedgraph 119866 with genus 1 Δ(119866) ⩽ 5 and having a toroidalembedding of which at least one region is not 4-sidedRecently Gagarin and Zverovich [54] further have extendedCarlson-Develin ideas to establish a nice upper bound forarbitrary graphs that can be embedded on orientable ornonorientable topological surface

Theorem 103 (Gagarin and Zverovich [54] 2012) Let 119866 bea graph embeddable on an orientable surface of genus ℎ and anonorientable surface of genus 119896 Then 119887(119866) ⩽ minΔ(119866)+ℎ+2 Δ(119866) + 119896 + 1

This result generalizes the corresponding upper boundsin Theorems 74 and 96 for any orientable or nonori-entable topological surface By investigating the proof ofTheorem 103 Huang [55] found that the issue of the ori-entability can be avoided by using the Euler characteristic120594(= 120592(119866) minus 120576(119866) + 120601(119866)) instead of the genera ℎ and 119896 the

relations are 120594 = 2minus2ℎ and 120594 = 2minus119896 To obtain the best resultfromTheorem 103 onewants ℎ and 119896 as small as possible thisis equivalent to having 120594 as large as possible

According toTheorem 103 if119866 is planar (ℎ = 0 120594 = 2) orcan be embedded on the real projective plane (119896 = 1 120594 = 1)then 119887(119866) ⩽ Δ(119866) + 2 In all other cases Huang [55] had thefollowing improvement for Theorem 103 the proof is basedon the technique developed byCarlson-Develin andGagarin-Zverovich and includes some elementary calculus as a newingredient mainly the intermediate-value theorem and themean-value theorem

Theorem 104 (Huang [55] 2012) Let 119866 be a graph embed-dable on a surface whose Euler characteristic 120594 is as large aspossible If 120594 ⩽ 0 then 119887(119866) ⩽ Δ(119866)+lfloor119903rfloor where 119903 is the largestreal root of the following cubic equation in 119911

1199113

+ 21199112

+ (6120594 minus 7) 119911 + 18120594 minus 24 = 0 (37)

In addition if 120594 decreases then 119903 increases

The following result is asymptotically equivalent toTheorem 104

Theorem 105 (Huang [55] 2012) Let 119866 be a graph embed-dable on a surface whose Euler characteristic 120594 is as large aspossible If 120594 ⩽ 0 then 119887(119866) lt Δ(119866) + radic12 minus 6120594 + 12 orequivalently 119887(119866) ⩽ Δ(119866) + lceilradic12 minus 6120594 minus 12rceil

Also Gagarin andZverovich [54] indicated that the upperbound in Theorem 103 can be improved for larger values ofthe genera ℎ and 119896 by adjusting the proofs and stated thefollowing general conjecture

Conjecture 106 (Gagarin and Zverovich [54] 2012) For aconnected graph G of orientable genus ℎ and nonorientablegenus 119896

119887 (119866) ⩽ min 119888ℎ 119888

1015840

119896 Δ (119866) + 119900 (ℎ) Δ (119866) + 119900 (119896) (38)

where 119888ℎand 1198881015840

119896are constants depending respectively on the

orientable and nonorientable genera of 119866

In the recent paper Gagarin and Zverovich [56] providedconstant upper bounds for the bondage number of graphs ontopological surfaces which can be used as the first estimationfor the constants 119888

ℎand 1198881015840

119896of Conjecture 106

Theorem 107 (Gagarin and Zverovich [56] 2013) Let 119866 bea connected graph of order 119899 If 119866 can be embedded on thesurface of its orientable or nonorientable genus of the Eulercharacteristic 120594 then

(a) 120594 ⩾ 1 implies 119887(119866) ⩽ 10(b) 120594 ⩽ 0 and 119899 gt minus12120594 imply 119887(119866) ⩽ 11(c) 120594 ⩽ minus1 and 119899 ⩽ minus12120594 imply 119887(119866) ⩽ 11 +

3120594(radic17 minus 8120594 minus 3)(120594 minus 1) = 11 + 119874(radicminus120594)

Theorem 79 shows that a connected planar triangle-freegraph 119866 has 119887(119866) ⩽ 6 The following result generalizes to itall the other topological surfaces

International Journal of Combinatorics 17

Theorem 108 (Gagarin and Zverovich [56] 2013) Let 119866 be aconnected triangle-free graph of order 119899 If 119866 can be embeddedon the surface of its orientable or nonorientable genus of theEuler characteristic 120594 then

(a) 120594 ⩾ 1 implies 119887(119866) ⩽ 6(b) 120594 ⩽ 0 and 119899 gt minus8120594 imply 119887(119866) ⩽ 7(c) 120594 ⩽ minus1 and 119899 ⩽ minus8120594 imply 119887(119866) ⩽ 7minus4120594(1+radic2 minus 120594)

In [56] the authors also explicitly improved upperbounds in Theorem 103 and gave tight lower bounds forthe number of vertices of graphs 2-cell embeddable ontopological surfaces of a given genusThe interested reader isreferred to the original paper for details The following resultis interesting although its proof is easy

Theorem 109 (Samodivkin [57]) Let119866 be a graph with order119899 and girth 119892 If 119866 is embeddable on a surface whose Eulercharacteristic 120594 is as large as possible then

119887 (119866) ⩽ 3 +8

119892 minus 2minus

4120594119892

119899 (119892 minus 2) (39)

If 119866 is a planar graph with girth 119892 ⩾ 4 + 119894 for any 119894 isin0 1 2 then Theorem 109 leads to 119887(119866) ⩽ 6 minus 119894 which isthe result inTheorem 79 obtained by Fischermann et al [46]Also inmany cases the bound stated inTheorem 109 is betterthan those given byTheorems 103 and 105

8 Conditional Bondage Numbers

Since the concept of the bondage number is based upon thedomination all sorts of dominations which are variationsof the normal domination by adding restricted conditionsto the normal dominating set can develop a new ldquobondagenumberrdquo as long as a variation of domination number isgiven In this section we survey results on the bondagenumber under some restricted conditions

81 Total Bondage Numbers A dominating set 119878 of a graph119866 without isolated vertices is called total if the subgraphinduced by 119878 contains no isolated vertices The minimumcardinality of a total dominating set is called the totaldomination number of 119866 and denoted by 120574

119879(119866) It is clear

that 120574(119866) ⩽ 120574119879(119866) ⩽ 2120574(119866) for any graph 119866 without isolated

verticesThe total domination in graphs was introduced by Cock-

ayne et al [58] in 1980 Pfaff et al [59 60] in 1983 showedthat the problem determining total domination number forgeneral graphs is NP-complete even for bipartite graphs andchordal graphs Even now total domination in graphs hasbeen extensively studied in the literature In 2009 Henning[61] gave a survey of selected recent results on total domina-tion in graphs

The total bondage number of 119866 denoted by 119887119879(119866) is the

smallest cardinality of a subset 119861 sube 119864(119866) with the propertythat119866minus119861 contains no isolated vertices and 120574

119879(119866minus119861) gt 120574

119879(119866)

From definition 119887119879(119866)may not exist for some graphs for

example119866 = 1198701119899 We put 119887

119879(119866) = infin if 119887

119879(119866) does not exist

It is easy to see that 119887119879(119866) is finite for any connected graph 119866

other than 1198701 119870

2 119870

3 and 119870

1119899 In fact for such a graph 119866

since there is a path of length 3 in 119866 we can find 1198611sube 119864(119866)

such that 119866 minus 1198611is a spanning tree 119879 of 119866 containing a path

of length 3 So 120574119879(119879) = 120574

119879(119866minus119861

1) ⩾ 120574

119879(119866) For the tree119879 we

find 1198612sube 119864(119879) such that 120574

119879(119879 minus 119861

2) gt 120574

119879(119879) ⩾ 120574

119879(119866) Thus

we have 120574119879(119866minus119861

2minus119861

1) gt 120574

119879(119866) and so 119887

119879(119866) ⩽ |119861

1|+|119861

2| In

the following discussion we always assume that 119887119879(119866) exists

when a graph 119866 is mentionedIn 1991 Kulli and Patwari [62] first studied the total

bondage number of a graph and calculated the exact valuesof 119887

119879(119866) for some standard graphs

Theorem 110 (Kulli and Patwari [62] 1991) For 119899 ⩾ 4

119887119879(119862

119899) =

3 119894119891 119899 equiv 2 (mod 4) 2 119900119905ℎ119890119903119908119894119904119890

119887119879(119875

119899) =

2 119894119891 119899 equiv 2 (mod 4) 1 119900119905ℎ119890119903119908119894119904119890

119887119879(119870

119898119899) = 119898 119908119894119905ℎ 2 ⩽ 119898 ⩽ 119899

119887119879(119870

119899) =

4 119891119900119903 119899 = 4

2119899 minus 5 119891119900119903 119899 ⩾ 5

(40)

Recently Hu et al [63] have obtained some results on thetotal bondage number of the Cartesian product119875

119898times119875

119899of two

paths 119875119898and 119875

119899

Theorem 111 (Hu et al [63] 2012) For the Cartesian product119875119898times 119875

119899of two paths 119875

119898and 119875

119899

119887119879(119875

2times 119875

119899) =

1 119894119891 119899 equiv 0 (mod 3) 2 119894119891 119899 equiv 2 (mod 3) 3 119894119891 119899 equiv 1 (mod 3)

119887119879(119875

3times 119875

119899) = 1

119887119879(119875

4times 119875

119899)

= 1 119894119891 119899 equiv 1 (mod 5) = 2 119894119891 119899 equiv 4 (mod 5) ⩽ 3 119894119891 119899 equiv 2 (mod 5) ⩽ 4 119894119891 119899 equiv 0 3 (mod 5)

(41)

Generalized Petersen graphs are an important class ofcommonly used interconnection networks and have beenstudied recently By constructing a family of minimum totaldominating sets Cao et al [64] determined the total bondagenumber of the generalized Petersen graphs

FromTheorem 6 we know that 119887(119879) ⩽ 2 for a nontrivialtree 119879 But given any positive integer 119896 Sridharan et al [65]constructed a tree 119879 for which 119887

119879(119879) = 119896 Let119867

119896be the tree

obtained from a star1198701119896+1

by subdividing 119896 edges twice Thetree 119867

7is shown in Figure 6 It can be easily verified that

119887119879(119867

119896) = 119896 This fact can be stated as the following theorem

Theorem 112 (Sridharan et al [65] 2007) For any positiveinteger 119896 there exists a tree 119879 with 119887

119879(119879) = 119896

18 International Journal of Combinatorics

Figure 61198677

Combining Theorem 1 with Theorem 112 we have whatfollows

Corollary 113 (Sridharan et al [65] 2007) The bondagenumber and the total bondage number are unrelated even fortrees

However Sridharan et al [65] gave an upper bound onthe total bondage number of a tree in terms of its maximumdegree For any tree 119879 of order 119899 if 119879 =119870

1119899minus1 then 119887

119879(119879) =

minΔ(119879) (13)(119899 minus 1) Rad and Raczek [66] improved thisupper bound and gave a constructive characterization of acertain class of trees attaching the upper bound

Theorem 114 (Rad and Raczek [66] 2011) 119887119879(119879) ⩽ Δ(119879) minus 1

for any tree 119879 with maximum degree at least three

In general the decision problem for 119887119879(119866) is NP-complete

for any graph 119866 We state the decision problem for the totalbondage as follows

Problem 3 Consider the decision problemTotal Bondage ProblemInstance a graph 119866 and a positive integer 119887Question is 119887

119879(119866) ⩽ 119887

Theorem 115 (Hu and Xu [16] 2012) The total bondageproblem is NP-complete

Consequently in view of its computational hardnessit is significative to establish some sharp bounds on thetotal bondage number of a graph in terms of other graphicparameters In 1991 Kulli and Patwari [62] showed that119887119879(119866) ⩽ 2119899 minus 5 for any graph 119866 with order 119899 ⩾ 5 Sridharan

et al [65] improved this result as follows

Theorem 116 (Sridharan et al [65] 2007) For any graph 119866with order 119899 if 119899 ⩾ 5 then

119887119879(119866) ⩽

1

3(119899 minus 1) 119894119891 119866 119888119900119899119905119886119894119899119904 119899119900 119888119910119888119897119890119904

119899 minus 1 119894119891 119892 (119866) ⩾ 5

119899 minus 2 119894119891 119892 (119866) = 4

(42)

Rad and Raczek [66] also established some upperbounds on 119887

119879(119866) for a general graph 119866 In particular they

gave an upper bound on 119887119879(119866) in terms of the girth of

119866

Theorem 117 (Rad and Raczek [66] 2011) 119887119879(119866)⩽4Δ(119866) minus 5

for any graph 119866 with 119892(119866) ⩾ 4

82 Paired Bondage Numbers A dominating set 119878 of 119866 iscalled to be paired if the subgraph induced by 119878 containsa perfect matching The paired domination number of 119866denoted by 120574

119875(119866) is the minimum cardinality of a paired

dominating set of 119866 Clearly 120574119879(119866) ⩽ 120574

119875(119866) for every

connected graph 119866 with order at least two where 120574119879(119866) is

the total domination number of 119866 and 120574119875(119866) ⩽ 2120574(119866) for

any graph119866without isolated vertices Paired domination wasintroduced by Haynes and Slater [67 68] and further studiedin [69ndash72]

The paired bondage number of 119866 with 120575(119866) ⩾ 1 denotedby 119887P(119866) is the minimum cardinality among all sets of edges119861 sube 119864 such that 120575(119866 minus 119861) ⩾ 1 and 120574

119875(119866 minus 119861) gt 120574

119875(119866)

The concept of the paired bondage number was firstproposed by Raczek [73] in 2008The following observationsfollow immediately from the definition of the paired bondagenumber

Observation 2 Let 119866 be a graph with 120575(119866) ⩾ 1

(a) If 119867 is a subgraph of 119866 such that 120575(119867) ⩾ 1 then119887119875(119867) ⩽ 119887

119875(119866)

(b) If 119867 is a subgraph of 119866 such that 119887119875(119867) = 1 and 119896

is the number of edges removed to form119867 then 1 ⩽119887119875(119866) ⩽ 119896 + 1

(c) If 119909119910 isin 119864(119866) such that 119889119866(119909) 119889

119866(119910) gt 1 and 119909119910

belongs to each perfect matching of each minimumpaired dominating set of 119866 then 119887

119875(119866) = 1

Based on these observations 119887119875(119875

119899) ⩽ 2 In fact the

paired bondage number of a path has been determined

Theorem 118 (Raczek [73] 2008) For any positive integer 119896if 119899 ⩾ 2 then

119887119875(119875

119899) =

0 119894119891 119899 = 2 3 119900119903 5

1 119894119891 119899 = 4119896 4119896 + 3 119900119903 4119896 + 6

2 119900119905ℎ119890119903119908119894119904119890

(43)

For a cycle 119862119899of order 119899 ⩾ 3 since 120574

119875(119862

119899) = 120574

119875(119875

119899) from

Theorem 118 the following result holds immediately

Corollary 119 (Raczek [73] 2008) For any positive integer 119896if 119899 ⩾ 3 then

119887119875(119862

119899) =

0 119894119891 119899 = 3 119900119903 5

2 119894119891 119899 = 4119896 4119896 + 3 119900119903 4119896 + 6

3 119900119905ℎ119890119903119908119894119904119890

(44)

A wheel 119882119899 where 119899 ⩾ 4 is a graph with 119899 vertices

formed by connecting a single vertex to all vertices of a cycle119862119899minus1

Of course 120574119875(119882

119899) = 2

International Journal of Combinatorics 19

119896

Figure 7 A tree 119879 with 119887119875(119879) = 119896

Theorem 120 (Raczek [73] 2008) For positive integers 119899 and119898

119887119875(119870

119898119899) = 119898 119891119900119903 1 lt 119898 ⩽ 119899

119887119875(119882

119899) =

4 119894119891 119899 = 4

3 119894119891 119899 = 5

2 119900119905ℎ119890119903119908119894119904119890

(45)

Theorem 16 shows that if 119879 is a nontrivial tree then119887(119879) ⩽ 2 However no similar result exists for pairedbondage For any nonnegative integer 119896 let 119879

119896be a tree

obtained by subdividing all but one edge of a star 1198701119896+1

(asshown in Figure 7) It is easy to see that 119887

119875(119879

119896) = 119896 We

state this fact as the following theorem which is similar toTheorem 112

Theorem121 (Raczek [73] 2008) For any nonnegative integer119896 there exists a tree 119879 with 119887

119875(119879) = 119896

Consider the tree defined in Figure 5 for 119896 = 3 Then119887(119879

3) ⩽ 2 and 119887

119875(119879

3) = 3 On the other hand 119887(119875

4) = 2

and 119887119875(119875

4) = 1 Thus we have what follows which is similar

to Corollary 113

Corollary 122 The bondage number and the paired bondagenumber are unrelated even for trees

A constructive characterization of trees with 119887(119879) = 2is given by Hartnell and Rall in [12] Raczek [73] provideda constructive characterization of trees with 119887

119875(119879) = 0 In

order to state the characterization we define a labeling andthree simple operations on a tree119879 Let 119910 isin 119881(119879) and let ℓ(119910)be the label assigned to 119910

Operation T1 If ℓ(119910) = 119861 add a vertex 119909 and the

edge 119910119909 and let ℓ(119909) = 119860OperationT

2 If ℓ(119910) = 119862 add a path (119909

1 119909

2) and the

edge 1199101199091 and let ℓ(119909

1) = 119861 ℓ(119909

2) = 119860

Operation T3 If ℓ(119910) = 119861 add a path (119909

1 119909

2 119909

3)

and the edge 1199101199091 and let ℓ(119909

1) = 119862 ℓ(119909

2) = 119861 and

ℓ(1199093) = 119860

Let 1198752= (119906 V) with ℓ(119906) = 119860 and ℓ(V) = 119861 Let T be

the class of all trees obtained from the labeled 1198752by a finite

sequence of operationsT1 T

2 T

3

A tree 119879 in Figure 8 belongs to the familyTRaczek [73] obtained the following characterization of all

trees 119879 with 119887119875(119879) = 0

119860

119860

119860

119860

119860

119861 119862

119861 119862 119861

119861119860

Figure 8 A tree 119879 belong to the familyT

Theorem 123 (Raczek [73] 2008) For a tree 119879 119887119875(119879) = 0 if

and only if 119879 is inT

We state the decision problem for the paired bondage asfollows

Problem 4 Consider the decision problem

Paired Bondage ProblemInstance a graph 119866 and a positive integer 119887Question is 119887

119875(119866) ⩽ 119887

Conjecture 124 The paired bondage problem is NP-complete

83 Restrained Bondage Numbers A dominating set 119878 of 119866 iscalled to be restrained if the subgraph induced by 119878 containsno isolated vertices where 119878 = 119881(119866) 119878 The restraineddomination number of 119866 denoted by 120574

119877(119866) is the smallest

cardinality of a restrained dominating set of 119866 It is clear that120574119877(119866) exists and 120574(119866) ⩽ 120574

119877(119866) for any nonempty graph 119866

The concept of restrained domination was introducedby Telle and Proskurowski [74] in 1997 albeit indirectly asa vertex partitioning problem Concerning the study of therestrained domination numbers the reader is referred to [74ndash79]

In 2008 Hattingh and Plummer [80] defined therestrained bondage number 119887R(119866) of a nonempty graph 119866 tobe the minimum cardinality among all subsets 119861 sube 119864(119866) forwhich 120574

119877(119866 minus 119861) gt 120574

119877(119866)

For some simple graphs their restrained dominationnumbers can be easily determined and so restrained bondagenumbers have been also determined For example it is clearthat 120574

119877(119870

119899) = 1 Domke et al [77] showed that 120574

119877(119862

119899) =

119899 minus 2lfloor1198993rfloor for 119899 ⩾ 3 and 120574119877(119875

119899) = 119899 minus 2lfloor(119899 minus 1)3rfloor for 119899 ⩾ 1

Using these results Hattingh and Plummer [80] obtained therestricted bondage numbers for119870

119899 119862

119899 119875

119899 and119866 = 119870

11989911198992119899119905

as follows

Theorem 125 (Hattingh and Plummer [80] 2008) For 119899 ⩾ 3

119887R (119870119899) =

1 119894119891 119899 = 3

lceil119899

2rceil 119900119905ℎ119890119903119908119894119904119890

119887R (119862119899) =

1 119894119891 119899 equiv 0 (mod 3) 2 119900119905ℎ119890119903119908119894119904119890

119887R (119875119899) = 1 119899 ⩾ 4

20 International Journal of Combinatorics

119887R (119866)

=

lceil119898

2rceil 119894119891 119899

119898= 1 119886119899119889 119899

119898+1⩾ 2 (1 ⩽ 119898 lt 119905)

2119905 minus 2 119894119891 1198991= 119899

2= sdot sdot sdot = 119899

119905= 2 (119905 ⩾ 2)

2 119894119891 1198991= 2 119886119899119889 119899

2⩾ 3 (119905 = 2)

119905minus1

sum

119894=1

119899119894minus 1 119900119905ℎ119890119903119908119894119904119890

(46)

Theorem 126 (Hattingh and Plummer [80] 2008) For a tree119879 of order 119899 (⩾ 4) 119879 ≇ 119870

1119899minus1if and only if 119887R(119879) = 1

Theorem 126 shows that the restrained bondage numberof a tree can be computed in constant time However thedecision problem for 119887R(119866) is NP-complete even for bipartitegraphs

Problem 5 Consider the decision problem

Restrained Bondage Problem

Instance a graph 119866 and a positive integer 119887

Question is 119887R(119866) ⩽ 119887

Theorem 127 (Hattingh and Plummer [80] 2008) Therestrained bondage problem is NP-complete even for bipartitegraphs

Consequently in view of its computational hardnessit is significative to establish some sharp bounds on therestrained bondage number of a graph in terms of othergraphic parameters

Theorem 128 (Hattingh and Plummer [80] 2008) For anygraph 119866 with 120575(119866) ⩾ 2

119887R (119866) ⩽ min119909119910isin119864(119866)

119889119866(119909) + 119889

119866(119910) minus 2 (47)

Corollary 129 119887R(119866) ⩽ Δ(119866)+120575(119866)minus2 for any graph119866 with120575(119866) ⩾ 2

Notice that the bounds stated in Theorem 128 andCorollary 129 are sharp Indeed the class of cycles whoseorders are congruent to 1 2 (mod 3) have a restrained bondagenumber achieving these bounds (see Theorem 125)

Theorem 128 is an analogue of Theorem 14 A quitenatural problem is whether or not for restricted bondagenumber there are analogues of Theorem 16 Theorem 18Theorem 29 and so on Indeed we should consider all resultsfor the bondage number whether or not there are analoguesfor restricted bondage number

Theorem 130 (Hattingh and Plummer [80] 2008) For anygraph 119866 with 120574

119877(119866) = 2

119887R (119866) ⩽ Δ (119866) + 1 (48)

We now consider the relation between 119887119877(119866) and 119887(119866)

Theorem 131 (Hattingh and Plummer [80] 2008) If 120574119877(119866) =

120574(119866) for some graph 119866 then 119887R(119866) ⩽ 119887(119866)

Corollary 129 and Theorem 131 provide a possibility toattack Conjecture 73 by characterizing planar graphs with120574119877= 120574 and 119887R = 119887 when 3 ⩽ Δ ⩽ 6However we do not have 119887R(119866) = 119887(119866) for any graph

119866 even if 120574119877(119866) = 120574(119866) Observe that 120574

119877(119870

3) = 120574(119870

3) yet

119887119877(119870

3) = 1 and 119887(119870

3) = 2 We still may not claim that

119887119877(119866) = 119887(119866) even in the case that every 120574-set is a 120574

119877-set

The example 1198703again demonstrates this

Theorem 132 (Hattingh and Plummer [80] 2008) There isan infinite class of graphs inwhich each graph119866 satisfies 119887(119866) lt119887R(119866)

In fact such an infinite class of graphs can be obtained asfollows Let119867 be a connected graph BC(119867) a graph obtainedfrom 119867 by attaching ℓ (⩾ 2) vertices of degree 1 to eachvertex in 119867 and B = 119866 119866 = BC(119867) for some graph 119867such that 120575(119867) ⩾ 2 By Theorem 3 we can easily check that119887(119866) = 1 lt 2 ⩽ min120575(119867) ℓ = 119887R(119866) for any 119866 isin BMoreover there exists a graph 119866 isin B such that 119887R(119866) canbe much larger than 119887(119866) which is stated as the followingtheorem

Theorem 133 (Hattingh and Plummer [80] 2008) For eachpositive integer 119896 there is a graph119866 such that 119896 = 119887R(119866)minus119887(119866)

Combining Theorem 133 with Theorem 132 we have thefollowing corollary

Corollary 134 The bondage number and the restrainedbondage number are unrelated

Very recently Jafari Rad et al [81] have consideredthe total restrained bondage in graphs based on both totaldominating sets and restrained dominating sets and obtainedseveral properties exact values and bounds for the totalrestrained bondage number of a graph

A restrained dominating set 119878 of a graph 119866 without iso-lated vertices is called to be a total restrained dominating setif 119878 is a total dominating set The total restrained dominationnumber of119866 denoted by 120574TR (119866) is theminimum cardinalityof a total restrained dominating set of 119866 For referenceson this domination in graphs see [3 61 82ndash85] The totalrestrained bondage number 119887TR (119866) of a graph 119866 with noisolated vertex is the cardinality of a smallest set of edges 119861 sube119864(119866) for which119866minus119861 has no isolated vertex and 120574TR (119866minus119861) gt120574TR (119866) In the case that there is no such subset 119861 we define119887TR (119866) = infin

Ma et al [84] determined that 120574TR (119875119899) = 119899minus2lfloor(119899minus2)4rfloorfor 119899 ⩾ 3 120574TR (119862119899

) = 119899 minus 2lfloor1198994rfloor for 119899 ⩾ 3 120574TR (1198703) =

3 and 120574TR (119870119899) = 2 for 119899 = 3 and 120574TR (1198701119899

) = 1 + 119899

and 120574TR (119870119898119899) = 2 for 2 ⩽ 119898 ⩽ 119899 According to these

results Jafari Rad et al [81] determined the exact valuesof total restrained bondage numbers for the correspondinggraphs

International Journal of Combinatorics 21

Theorem 135 (Jafari Rad et al [81]) For an integer 119899

119887TR (119875119899) = infin 119894119891 2 ⩽ 119899 ⩽ 5

1 119894119891 119899 ⩾ 5

119887TR (119862119899) =

infin 119894119891 119899 = 3

1 119894119891 3 lt 119899 119899 equiv 0 1 (mod 4)2 119894119891 3 lt 119899 119899 equiv 2 3 (mod 4)

119887TR (119882119899) = 2 119891119900119903 119899 ⩾ 6

119887TR (119870119899) = 119899 minus 1 119891119900119903 119899 ⩾ 4

119887TR (119870119898119899) = 119898 minus 1 119891119900119903 2 ⩽ 119898 ⩽ 119899

(49)

119887TR (119870119899) = 119899minus1 shows the fact that for any integer 119899 ⩾ 3 there

exists a connected graph namely119870119899+1

such that 119887TR (119870119899+1) =

119899 that is the total restrained bondage number is unboundedfrom the above

A vertex is said to be a support vertex if it is adjacent toa vertex of degree one Let A be the family of all connectedgraphs such that119866 belongs toA if and only if every edge of119866is incident to a support vertex or 119866 is a cycle of length three

Proposition 136 (Cyman and Raczek [82] 2006) For aconnected graph119866 of order 119899 120574TR (119866) = 119899 if and only if119866 isin A

Hence if 119866 isin A then the removal of any subset of edgesof119866 cannot increase the total restrained domination numberof 119866 and thus 119887TR (119866) = infin When 119866 notin A a result similar toTheorem 6 is obtained

Theorem 137 (Jafari Rad et al [81]) For any tree 119879 notin A119887TR (119879) ⩽ 2 This bound is sharp

In the samepaper the authors provided a characterizationfor trees 119879 notin A with 119887TR (119879) = 1 or 2 The following result issimilar to Observation 1

Observation 3 (Jafari Rad et al [81]) If 119896 edges can beremoved from a graph 119866 to obtain a graph 119867 without anisolated vertex and with 119887TR (119867) = 119905 then 119887TR (119866) ⩽ 119896 + 119905

ApplyingTheorem 137 andObservation 3 Jafari Rad et alcharacterized all connected graphs 119866 with 119887TR (119866) = infin

Theorem 138 (Jafari Rad et al [81]) For a connected graph119866119887TR (119866) = infin if and only if 119866 isin A

84 Other Conditional Bondage Numbers A subset 119868 sube 119881(119866)is called an independent set if no two vertices in 119868 are adjacentin 119866 The maximum cardinality among all independent setsis called the independence number of 119866 denoted by 120572(119866)

A dominating set 119878 of a graph 119866 is called to be inde-pendent if 119878 is an independent set of 119866 The minimumcardinality among all independent dominating set is calledthe independence domination number of 119866 and denoted by120574119868(119866)

Since an independent dominating set is not only adominating set but also an independent set 120574(119866) ⩽ 120574

119868(119866) ⩽

120572(119866) for any graph 119866It is clear that a maximal independent set is certainly a

dominating set Thus an independent set is maximal if andonly if it is an independent dominating set and so 120574

119868(119866) is the

minimum cardinality among all maximal independent setsof 119866 This graph-theoretical invariant has been well studiedin the literature see for example Haynes et al [3] for earlyresults Goddard and Henning [86] for recent results onindependent domination in graphs

In 2003 Zhang et al [87] defined the independencebondage number 119887

119868(119866) of a nonempty graph 119866 to be the

minimum cardinality among all subsets 119861 sube 119864(119866) for which120574119868(119866 minus 119861) gt 120574

119868(119866) For some ordinary graphs their indepen-

dence domination numbers can be easily determined and soindependence bondage numbers have been also determinedClearly 119887

119868(119870

119899) = 1 if 119899 ⩾ 2

Theorem 139 (Zhang et al [87] 2003) For any integer 119899 ⩾ 4

119887119868(119862

119899) =

1 119894119891 119899 equiv 1 (mod 2) 2 119894119891 119899 equiv 0 (mod 2)

119887119868(119875

119899) =

1 119894119891 119899 equiv 0 (mod 2) 2 119894119891 119899 equiv 1 (mod 2)

119887119868(119870

119898119899) = 119899 119891119900119903 119898 ⩽ 119899

(50)

Apart from the above-mentioned results as far as we findno other results on the independence bondage number in thepresent literature we never knew much of any result on thisparameter for a tree

A subset 119878 of 119881(119866) is called an equitable dominatingset if for every 119910 isin 119881(119866 minus 119878) there exists a vertex 119909 isin 119878such that 119909119910 isin 119864(119866) and |119889

119866(119909) minus 119889

119866(119910)| ⩽ 1 proposed

by Dharmalingam [88] The minimum cardinality of such adominating set is called the equitable domination number andis denoted by 120574

119864(119866) Deepak et al [89] defined the equitable

bondage number 119887119864(119866) of a graph 119866 to be the cardinality of a

smallest set 119861 sube 119864(119866) of edges for which 120574119864(119866 minus 119861) gt 120574

119864(119866)

and determined 119887119864(119866) for some special graphs

Theorem 140 (Deepak et al [89] 2011) For any integer 119899 ⩾ 2

119887119864(119870

119899) = lceil

119899

2rceil

119887119864(119870

119898119899) = 119898 119891119900119903 0 ⩽ 119899 minus 119898 ⩽ 1

119887119864(119862

119899) =

3 119894119891 119899 equiv 1 (mod 3)2 119900119905ℎ119890119903119908119894119904119890

119887119864(119875

119899) =

2 119894119891 119899 equiv 1 (mod 3)1 119900119905ℎ119890119903119908119894119904119890

119887119864(119879) ⩽ 2 119891119900119903 119886119899119910 119899119900119899119905119903119894119907119894119886119897 119905119903119890119890 119879

119887119864(119866) ⩽ 119899 minus 1 119891119900119903 119886119899119910 119888119900119899119899119890119888119905119890119889 119892119903119886119901ℎ 119866 119900119891 119900119903119889119890119903 119899

(51)

22 International Journal of Combinatorics

As far as we know there are no more results on theequitable bondage number of graphs in the present literature

There are other variations of the normal domination byadding restricted conditions to the normal dominating setFor example the connected domination set of a graph119866 pro-posed by Sampathkumar and Walikar [90] is a dominatingset 119878 such that the subgraph induced by 119878 is connected Theproblem of finding a minimum connected domination set ina graph is equivalent to the problemof finding a spanning treewithmaximumnumber of vertices of degree one [91] and hassome important applications in wireless networks [92] thusit has received much research attention However for suchan important domination there is no result on its bondagenumber We believe that the topic is of significance for futureresearch

9 Generalized Bondage Numbers

There are various generalizations of the classical dominationsuch as 119901-domination distance domination fractional dom-ination Roman domination and rainbow domination Everysuch a generalization can lead to a corresponding bondageIn this section we introduce some of them

91 119901-Bondage Numbers In 1985 Fink and Jacobson [93]introduced the concept of 119901-domination Let 119901 be a positiveinteger A subset 119878 of 119881(119866) is a 119901-dominating set of 119866 if|119878 cap 119873

119866(119910)| ⩾ 119901 for every 119910 isin 119878 The 119901-domination number

120574119901(119866) is the minimum cardinality among all 119901-dominating

sets of 119866 Any 119901-dominating set of 119866 with cardinality 120574119901(119866)

is called a 120574119901-set of 119866 Note that a 120574

1-set is a classic minimum

dominating set Notice that every graph has a 119901-dominatingset since the vertex-set119881(119866) is such a setWe also note that a 1-dominating set is a dominating set and so 120574(119866) = 120574

1(119866) The

119901-domination number has received much research attentionsee a state-of-the-art survey article by Chellali et al [94]

It is clear from definition that every 119901-dominating setof a graph certainly contains all vertices of degree at most119901 minus 1 By this simple observation to avoid the trivial caseoccurrence we always assume that Δ(119866) ⩾ 119901 For 119901 ⩾ 2 Luet al [95] gave a constructive characterization of trees withunique minimum 119901-dominating sets

Recently Lu and Xu [96] have introduced the conceptto the 119901-bondage number of 119866 denoted by 119887

119901(119866) as the

minimum cardinality among all sets of edges 119861 sube 119864(119866) suchthat 120574

119901(119866 minus 119861) gt 120574

119901(119866) Clearly 119887

1(119866) = 119887(119866)

Lu and Xu [96] established a lower bound and an upperbound on 119887

119901(119879) for any integer 119901 ⩾ 2 and any tree 119879 with

Δ(119879) ⩾ 119901 and characterized all trees achieving the lowerbound and the upper bound respectively

Theorem 141 (Lu and Xu [96] 2011) For any integer 119901 ⩾ 2and any tree 119879 with Δ(119879) ⩾ 119901

1 ⩽ 119887119901(119879) ⩽ Δ (119879) minus 119901 + 1 (52)

Let 119878 be a given subset of 119881(119866) and for any 119909 isin 119878 let

119873119901(119909 119878 119866) = 119910 isin 119878 cap 119873

119866(119909)

1003816100381610038161003816119873119866(119910) cap 119878

1003816100381610038161003816 = 119901

(53)

Then all trees achieving the lower bound 1 can be character-ized as follows

Theorem 142 (Lu and Xu [96] 2011) Let 119879 be a tree withΔ(119879) ⩾ 119901 ⩾ 2 Then 119887

119901(119879) = 1 if and only if for any 120574

119901-set

119878 of 119879 there exists an edge 119909119910 isin (119878 119878) such that 119910 isin 119873119901(119909 119878 119879)

The notation 119878(119886 119887) denotes a double star obtained byadding an edge between the central vertices of two stars119870

1119886minus1

and1198701119887minus1

And the vertex with degree 119886 (resp 119887) in 119878(119886 119887) iscalled the 119871-central vertex (resp 119877-central vertex) of 119878(119886 119887)

To characterize all trees attaining the upper bound givenin Theorem 141 we define three types of operations on a tree119879 with Δ(119879) = Δ ⩾ 119901 + 1

Type 1 attach a pendant edge to a vertex 119910with 119889119879(119910) ⩽ 119901minus

2 in 119879

Type 2 attach a star 1198701Δminus1

to a vertex 119910 of 119879 by joining itscentral vertex to 119910 where 119910 in a 120574

119901-set of 119879 and

119889119879(119910) ⩽ Δ minus 1

Type 3 attach a double star 119878(119901 Δ minus 1) to a pendant vertex 119910of 119879 by coinciding its 119877-central vertex with 119910 wherethe unique neighbor of 119910 is in a 120574

119901-set of 119879

Let B = 119879 a tree obtained from 1198701Δ

or 119878(119901 Δ) by afinite sequence of operations of Types 1 2 and 3

Theorem 143 (Lu and Xu [96] 2011) A tree with the maxi-mum Δ ⩾ 119901 + 1 has 119901-bondage number Δ minus 119901 + 1 if and onlyif it belongs toB

Theorem 144 (Lu and Xu [97] 2012) Let 119866119898119899= 119875

119898times 119875

119899

Then

1198872(119866

2119899) = 1 119891119900119903 119899 ⩾ 2

1198872(119866

3119899) =

2 119894119891 119899 equiv 1 (mod 3) 1 119900119905ℎ119890119903119908119894119904119890

for 119899 ⩾ 2

1198872(119866

4119899) =

1 119894119891 119899 equiv 3 (mod 4) 2 119900119905ℎ119890119903119908119894119904119890

for 119899 ⩾ 7

(54)

Recently Krzywkowski [98] has proposed the conceptof the nonisolating 2-bondage of a graph The nonisolating2-bondage number of a graph 119866 denoted by 1198871015840

2(119866) is the

minimum cardinality among all sets of edges 119861 sube 119864(119866) suchthat 119866 minus 119861 has no isolated vertices and 120574

2(119866 minus 119861) gt 120574

2(119866) If

for every 119861 sube 119864(119866) either 1205742(119866 minus 119861) = 120574

2(119866) or 119866 minus 119861 has

isolated vertices then define 11988710158402(119866) = 0 and say that 119866 is a 2-

nonisolatedly strongly stable graph Krzywkowski presentedsome basic properties of nonisolating 2-bondage in graphsshowed that for every nonnegative integer 119896 there exists atree 119879 such 1198871015840

2(119879) = 119896 characterized all 2-non-isolatedly

strongly stable trees and determined 11988710158402(119866) for several special

graphs

International Journal of Combinatorics 23

Theorem 145 (Krzywkowski [98] 2012) For any positiveintegers 119899 and119898 with119898 ⩽ 119899

1198871015840

2(119870

119899) =

0 119894119891 119899 = 1 2 3

lfloor2119899

3rfloor 119900119905ℎ119890119903119908119894119904119890

1198871015840

2(119875

119899) =

0 119894119891 119899 = 1 2 3

1 119900119905ℎ119890119903119908119894119904119890

1198871015840

2(119862

119899) =

0 119894119891 119899 = 3

1 119894119891 119899 119894119904 119890V119890119899

2 119894119891 119899 119894119904 119900119889119889

1198871015840

2(119870

119898119899) =

3 119894119891 119898 = 119899 = 3

1 119894119891 119898 = 119899 = 4

2 119900119905ℎ119890119903119908119894119904119890

(55)

92 Distance Bondage Numbers A subset 119878 of vertices of agraph119866 is said to be a distance 119896-dominating set for119866 if everyvertex in 119866 not in 119878 is at distance at most 119896 from some vertexof 119878 The minimum cardinality of all distance 119896-dominatingsets is called the distance 119896-domination number of 119866 anddenoted by 120574

119896(119866) (do not confuse with the above-mentioned

120574119901(119866)) When 119896 = 1 a distance 1-dominating set is a normal

dominating set and so 1205741(119866) = 120574(119866) for any graph 119866 Thus

the distance 119896-domination is a generalization of the classicaldomination

The relation between 120574119896and 120572

119896(the 119896-independence

number defined in Section 22) for a tree obtained by Meirand Moon [14] who proved that 120574

119896(119879) = 120572

2119896(119879) for any tree

119879 (a special case for 119896 = 1 is stated in Proposition 10) Thefurther research results can be found in Henning et al [99]Tian and Xu [100ndash103] and Liu et al [104]

In 1998 Hartnell et al [15] defined the distance 119896-bondagenumber of 119866 denoted by 119887

119896(119866) to be the cardinality of a

smallest subset 119861 of edges of 119866 with the property that 120574119896(119866 minus

119861) gt 120574119896(119866) FromTheorem 6 it is clear that if119879 is a nontrivial

tree then 1 ⩽ 1198871(119879) ⩽ 2 Hartnell et al [15] generalized this

result to any integer 119896 ⩾ 1

Theorem 146 (Hartnell et al [15] 1998) For every nontrivialtree 119879 and positive integer 119896 1 ⩽ 119887

119896(119879) ⩽ 2

Hartnell et al [15] and Topp and Vestergaard [13] alsocharacterized the trees having distance 119896-bondage number 2In particular the class of trees for which 119887

1(119879) = 2 are just

those which have a unique maximum 2-independent set (seeTheorem 11)

Since when 119896 = 1 the distance 1-bondage number 1198871(119866)

is the classical bondage number 119887(119866) Theorem 12 gives theNP-hardness of deciding the distance 119896-bondage number ofgeneral graphs

Theorem 147 Given a nonempty undirected graph 119866 andpositive integers 119896 and 119887 with 119887 ⩽ 120576(119866) determining wetheror not 119887

119896(119866) ⩽ 119887 is NP-hard

For a vertex 119909 in119866 the open 119896-neighborhood119873119896(119909) of 119909 is

defined as119873119896(119909) = 119910 isin 119881(119866) 1 ⩽ 119889

119866(119909 119910) le 119896The closed

119896-neighborhood119873119896[119909] of 119909 in 119866 is defined as119873

119896(119909) cup 119909 Let

Δ119896(119866) = max 1003816100381610038161003816119873119896

(119909)1003816100381610038161003816 for any 119909 isin 119881 (119866) (56)

Clearly Δ1(119866) = Δ(119866) The 119896th power of a graph 119866 is the

graph119866119896 with vertex-set119881(119866119896

) = 119881(119866) and edge-set119864(119866119896

) =

119909119910 1 ⩽ 119889119866(119909 119910) ⩽ 119896 The following lemma holds directly

from the definition of 119866119896

Proposition 148 (Tian and Xu [101]) Δ(119866119896

) = Δ119896(119866) and

120574(119866119896

) = 120574119896(119866) for any graph 119866 and each 119896 ⩾ 1

A graph 119866 is 119896-distance domination-critical or 120574119896-critical

for short if 120574119896(119866 minus 119909) lt 120574

119896(119866) for every vertex 119909 in 119866

proposed by Henning et al [105]

Proposition 149 (Tian and Xu [101]) For each 119896 ⩾ 1 a graph119866 is 120574

119896-critical if and only if 119866119896 is 120574

119896-critical

By the above facts we suggest the following worthwhileproblem

Problem 6 Can we generalize the results on the bondage for119866 to 119866119896 In particular do the following propositions hold

(a) 119887(119866119896

) = 119887119896(119866) for any graph 119866 and each 119896 ⩾ 1

(b) 119887(119866119896

) ⩽ Δ119896(119866) if 119866 is not 120574

119896-critical

Let 119896 and 119901 be positive integers A subset 119878 of 119881(119866) isdefined to be a (119896 119901)-dominating set of 119866 if for any vertex119909 isin 119878 |119873

119896(119909) cap 119878| ⩾ 119901 The (119896 119901)-domination number of

119866 denoted by 120574119896119901(119866) is the minimum cardinality among

all (119896 119901)-dominating sets of 119866 Clearly for a graph 119866 a(1 1)-dominating set is a classical dominating set a (119896 1)-dominating set is a distance 119896-dominating set and a (1 119901)-dominating set is the above-mentioned 119901-dominating setThat is 120574

11(119866) = 120574(119866) 120574

1198961(119866) = 120574

119896(119866) and 120574

1119901(119866) = 120574

119901(119866)

The concept of (119896 119901)-domination in a graph 119866 is a gen-eralized domination which combines distance 119896-dominationand 119901-domination in 119866 So the investigation of (119896 119901)-domination of 119866 is more interesting and has received theattention of many researchers see for example [106ndash109]

More general Li and Xu [110] proposed the concept of the(ℓ 119908)-domination number of a119908-connected graph A subset119878 of 119881(119866) is defined to be an (ℓ 119908)-dominating set of 119866 iffor any vertex 119909 isin 119878 there are 119908 internally disjoint pathsof length at most ℓ from 119909 to some vertex in 119878 The (ℓ 119908)-domination number of119866 denoted by 120574

ℓ119908(119866) is theminimum

cardinality among all (ℓ 119908)-dominating sets of 119866 Clearly1205741198961(119866) = 120574

119896(119866) and 120574

1119901(119866) = 120574

119901(119866)

It is quite natural to propose the concept of bondage num-ber for (119896 119901)-domination or (ℓ 119908)-domination However asfar as we know none has proposed this concept until todayThis is a worthwhile topic for further research

24 International Journal of Combinatorics

93 Fractional Bondage Numbers If 120590 is a function mappingthe vertex-set 119881 into some set of real numbers then for anysubset 119878 sube 119881 let 120590(119878) = sum

119909isin119878120590(119909) Also let |120590| = 120590(119881)

A real-value function 120590 119881 rarr [0 1] is a dominatingfunction of a graph 119866 if for every 119909 isin 119881(119866) 120590(119873

119866[119909]) ⩾ 1

Thus if 119878 is a dominating set of 119866 120590 is a function where

120590 (119909) = 1 if 119909 isin 1198780 otherwise

(57)

then 120590 is a dominating function of119866 The fractional domina-tion number of 119866 denoted by 120574

119865(119866) is defined as follows

120574119865(119866) = min |120590| 120590 is a domination function of 119866

(58)

The 120574119865-bondage number of 119866 denoted by 119887

119865(119866) is

defined as the minimum cardinality of a subset 119861 sube 119864 whoseremoval results in 120574

119865(119866 minus 119861) gt 120574

119865(119866)

Hedetniemi et al [111] are the first to study fractionaldomination although Farber [112] introduced the idea indi-rectly The concept of the fractional domination numberwas proposed by Domke and Laskar [113] in 1997 Thefractional domination numbers for some ordinary graphs aredetermined

Proposition 150 (Domke et al [114] 1998 Domke and Laskar[113] 1997) (a) If 119866 is a 119896-regular graph with order 119899 then120574119865(119866) = 119899119896 + 1(b) 120574

119865(119862

119899) = 1198993 120574

119865(119875

119899) = lceil1198993rceil 120574

119865(119870

119899) = 1

(c) 120574119865(119866) = 1 if and only if Δ(119866) = 119899 minus 1

(d) 120574119865(119879) = 120574(119879) for any tree 119879

(e) 120574119865(119870

119899119898) = (119899(119898 + 1) + 119898(119899 + 1))(119899119898 minus 1) where

max119899119898 gt 1

The assertions (a)ndash(d) are due to Domke et al [114] andthe assertion (e) is due to Domke and Laskar [113] Accordingto these results Domke and Laskar [113] determined the 120574

119865-

bondage numbers for these graphs

Theorem 151 (Domke and Laskar [113] 1997) 119887119865(119870

119899) =

lceil1198992rceil 119887119865(119870

119899119898) = 1 wheremax119899119898 gt 1

119887119865(119862

119899) =

2 119894119891 119899 equiv 0 (mod 3) 1 119900119905ℎ119890119903119908119894119904119890

119887119865(119875

119899) =

2 119894119891 119899 equiv 1 (mod 3) 1 119900119905ℎ119890119903119908119894119904119890

(59)

It is easy to see that for any tree119879 119887119865(119879) ⩽ 2 In fact since

120574119865(119879) = 120574(119879) for any tree 119879 120574

119865(119879

1015840

) = 120574(1198791015840

) for any subgraph1198791015840 of 119879 It follows that

119887119865(119879) = min 119861 sube 119864 (119879) 120574

119865(119879 minus 119861) gt 120574

119865(119879)

= min 119861 sube 119864 (119879) 120574 (119879 minus 119861) gt 120574 (119879)

= 119887 (119879) ⩽ 2

(60)

Except for these no results on 119887119865(119866) have been known as

yet

94 Roman Bondage Numbers A Roman dominating func-tion on a graph 119866 is a labeling 119891 119881 rarr 0 1 2 such thatevery vertex with label 0 has at least one neighbor with label2 The weight of a Roman dominating function 119891 on 119866 is thevalue

119891 (119866) = sum

119906isin119881(119866)

119891 (119906) (61)

The minimum weight of a Roman dominating function ona graph 119866 is called the Roman domination number of 119866denoted by 120574RM (119866)

A Roman dominating function 119891 119881 rarr 0 1 2

can be represented by the ordered partition (1198810 119881

1 119881

2) (or

(119881119891

0 119881

119891

1 119881

119891

2) to refer to119891) of119881 where119881

119894= V isin 119881 | 119891(V) = 119894

In this representation its weight 119891(119866) = |1198811| + 2|119881

2| It is

clear that119881119891

1cup119881

119891

2is a dominating set of 119866 called the Roman

dominating set denoted by 119863119891

R = (1198811 1198812) Since 119881119891

1cup 119881

119891

2is

a dominating set when 119891 is a Roman dominating functionand since placing weight 2 at the vertices of a dominating setyields a Roman dominating function in [115] it is observedthat

120574 (119866) ⩽ 120574RM (119866) ⩽ 2120574 (119866) (62)

A graph 119866 is called to be Roman if 120574RM (119866) = 2120574(119866)The definition of the Roman domination function is

proposed implicitly by Stewart [116] and ReVelle and Rosing[117] Roman dominating numbers have been deeply studiedIn particular Bahremandpour et al [118] showed that theproblem determining the Roman domination number is NP-complete even for bipartite graphs

Let 119866 be a graph with maximum degree at least twoThe Roman bondage number 119887RM (119866) of 119866 is the minimumcardinality of all sets 1198641015840 sube 119864 for which 120574RM (119866 minus 119864

1015840

) gt

120574RM (119866) Since in study of Roman bondage number theassumption Δ(119866) ⩾ 2 is necessary we always assume thatwhen we discuss 119887RM (119866) all graphs involved satisfy Δ(119866) ⩾2 The Roman bondage number 119887RM (119866) is introduced byJafari Rad and Volkmann in [119]

Recently Bahremandpour et al [118] have shown that theproblem determining the Roman bondage number is NP-hard even for bipartite graphs

Problem 7 Consider the decision problem

Roman Bondage Problem

Instance a nonempty bipartite graph119866 and a positiveinteger 119896

Question is 119887RM (119866) ⩽ 119896

Theorem 152 (Bahremandpour et al [118] 2013) TheRomanbondage problem is NP-hard even for bipartite graphs

The exact value of 119887RM(119866) is known only for a few familyof graphs including the complete graphs cycles and paths

International Journal of Combinatorics 25

Theorem 153 (Jafari Rad and Volkmann [119] 2011) For anypositive integers 119899 and119898 with119898 ⩽ 119899

119887RM (119875119899) = 2 119894119891 119899 equiv 2 (mod 3) 1 119900119905ℎ119890119903119908119894119904119890

119887RM (119862119899) =

3 119894119891 119899 = 2 (mod 3) 2 119900119905ℎ119890119903119908119894119904119890

119887RM (119870119899) = lceil

119899

2rceil 119891119900119903 119899 ⩾ 3

119887RM (119870119898119899) =

1 119894119891 119898 = 1 119886119899119889 119899 = 1

4 119894119891 119898 = 119899 = 3

119898 119900119905ℎ119890119903119908119894119904119890

(63)

Lemma 154 (Cockayne et al [115] 2004) If 119866 is a graph oforder 119899 and contains vertices of degree 119899 minus 1 then 120574RM(119866) = 2

Using Lemma 154 the third conclusion in Theorem 153can be generalized to more general case which is similar toLemma 49

Theorem 155 (Jafari Rad and Volkmann [119] 2011) Let119866 bea graph with order 119899 ge 3 and 119905 the number of vertices of degree119899 minus 1 in 119866 If 119905 ⩾ 1 then 119887RM(119866) = lceil1199052rceil

Ebadi and PushpaLatha [120] conjectured that 119887RM(119866) ⩽119899 minus 1 for any graph 119866 of order 119899 ⩾ 3 Dehgardi et al[121] Akbari andQajar [122] independently showed that thisconjecture is true

Theorem 156 119887RM(119866) ⩽ 119899 minus 1 for any connected graph 119866 oforder 119899 ⩾ 3

For a complete 119905-partite graph we have the followingresult

Theorem 157 (Hu and Xu [123] 2011) Let 119866 = 11987011989811198982119898119905

bea complete 119905-partite graph with 119898

1= sdot sdot sdot = 119898

119894lt 119898

119894+1le sdot sdot sdot ⩽

119898119905 119905 ⩾ 2 and 119899 = sum119905

119895=1119898

119895 Then

119887RM (119866) =

lceil119894

2rceil 119894119891 119898

119894= 1 119886119899119889 119899 ⩾ 3

2 119894119891 119898119894= 2 119886119899119889 119894 = 1

119894 119894119891 119898119894= 2 119886119899119889 119894 ⩾ 2

4 119894119891 119898119894= 3 119886119899119889 119894 = 119905 = 2

119899 minus 1 119894119891 119898119894= 3 119886119899119889 119894 = 119905 ⩾ 3

119899 minus 119898119905

119894119891 119898119894⩾ 3 119886119899119889 119898

119905⩾ 4

(64)

Consider a complete 119905-partite graph 119870119905(3) for 119905 ⩾ 3

119887RM(119870119905(3)) = 119899 minus 1 by Theorem 157 where 119899 = 3119905

which shows that this upper bound of 119899 minus 1 on 119887RM(119866) inTheorem 156 is sharp This fact and 120574

119877(119870

119905(3)) = 4 give

a negative answer to the following two problems posed byDehgardi et al [121]Prove or Disprove for any connected graph 119866 of order 119899 ge 3(i) 119887RM(119866) = 119899 minus 1 if and only if 119866 cong 119870

3 (ii) 119887RM(119866) ⩽ 119899 minus

120574119877(119866) + 1Note that a complete 119905-partite graph 119870

119905(3) is (119899 minus 3)-

regular and 119887RM(119870119905(3)) = 119899 minus 1 for 119905 ⩾ 3 where 119899 = 3119905

Hu and Xu further determined that 119887119877(119866) = 119899 minus 2 for any

(119899 minus 3)-regular graph 119866 of order 119899 ⩽ 5 except 119870119905(3)

Theorem 158 (Hu and Xu [123] 2011) For any (119899minus3)-regulargraph 119866 of order 119899 other than119870

119905(3) 119887RM(119866) = 119899 minus 2 for 119899 ⩾ 5

For a tree 119879 with order 119899 ⩾ 3 Ebadi and PushpaLatha[120] and Jafari Rad and Volkmann [119] independentlyobtained an upper bound on 119887RM(119879)

Theorem 159 119887RM(119879) ⩽ 3 for any tree 119879 with order 119899 ⩾ 3

Theorem 160 (Bahremandpour et al [118] 2013) 119887RM(1198752 times119875119899) = 2 for 119899 ⩾ 2

Theorem 161 (Jafari Rad and Volkmann [119] 2011) Let119866 bea graph with order at least three

(a) If (119909 119910 119911) is a path of length 2 in 119866 then 119887RM(119866) ⩽119889119866(119909) + 119889

119866(119910) + 119889

119866(119911) minus 3 minus |119873

119866(119909) cap 119873

119866(119910)|

(b) If 119866 is connected then 119887RM(119866) ⩽ 120582(119866) + 2Δ(119866) minus 3

Theorem 161 (b) implies 119887RM(119866) ⩽ 120575(119866) + 2Δ(119866) minus 3 for aconnected graph 119866 Note that for a planar graph 119866 120575(119866) ⩽ 5moreover 120575(119866) ⩽ 3 if the girth at least 4 and 120575(119866) ⩽ 2 if thegirth at least 6 These two facts show that 119887RM(119866) ⩽ 2Δ(119866) +2 for a connected planar graph 119866 Jafari Rad and Volkmann[124] improved this bound

Theorem 162 (Jafari Rad and Volkmann [124] 2011) For anyconnected planar graph 119866 of order 119899 ⩾ 3 with girth 119892(119866)

119887RM (119866)

⩽

2Δ (119866)

Δ (119866) + 6

Δ (119866) + 5 119894119891 119866 119888119900119899119905119886119894119899119904 119899119900 V119890119903119905119894119888119890119904 119900119891 119889119890119892119903119890119890 119891119894V119890

Δ (119866) + 4 119894119891 119892 (119866) ⩾ 4

Δ (119866) + 3 119894119891 119892 (119866) ⩾ 5

Δ (119866) + 2 119894119891 119892 (119866) ⩾ 6

Δ (119866) + 1 119894119891 119892 (119866) ⩾ 8

(65)

According toTheorem 157 119887RM(119862119899) = 3 = Δ(119862

119899)+1 for a

cycle 119862119899of length 119899 ⩾ 8 with 119899 equiv 2 (mod 3) and therefore

the last upper bound inTheorem 162 is sharp at least for Δ =2

Combining the fact that every planar graph 119866 withminimum degree 5 contains an edge 119909119910 with 119889

119866(119909) = 5

and 119889119866(119910) isin 5 6 with Theorem 161 (a) Akbari et al [125]

obtained the following result

26 International Journal of Combinatorics

middot middot middot

middot middot middot

middot middot middot

middot middot middot

119866

Figure 9 The graph 119866 is constructed from 119866

Theorem 163 (Akbari et al [125] 2012) 119887RM(119866) ⩽ 15 forevery planar graph 119866

It remains open to show whether the bound inTheorem 163 is sharp or not Though finding a planargraph 119866 with 119887RM(119866) = 15 seems to be difficult Akbari etal [125] constructed an infinite family of planar graphs withRoman bondage number equal to 7 by proving the followingresult

Theorem 164 (Akbari et al [125] 2012) Let 119866 be a graph oforder 119899 and 119866 is the graph of order 5119899 obtained from 119866 byattaching the central vertex of a copy of a path 119875

5to each vertex

of119866 (see Figure 9) Then 120574RM(119866) = 4119899 and 119887RM(119866) = 120575(119866)+2

ByTheorem 164 infinitemany planar graphswith Romanbondage number 7 can be obtained by considering any planargraph 119866 with 120575(119866) = 5 (eg the icosahedron graph)

Conjecture 165 (Akbari et al [125] 2012) The Romanbondage number of every planar graph is at most 7

For general bounds the following observation is directlyobtained which is similar to Observation 1

Observation 4 Let 119866 be a graph of order 119899 with maximumdegree at least two Assume that 119867 is a spanning subgraphobtained from 119866 by removing 119896 edges If 120574RM(119867) = 120574RM(119866)then 119887

119877(119867) ⩽ 119887RM(119866) ⩽ 119887RM(119867) + 119896

Theorem 166 (Bahremandpour et al [118] 2013) For anyconnected graph 119866 of order 119899 if 119899 ⩾ 3 and 120574RM(119866) = 120574(119866) + 1then

119887RM (119866) ⩽ min 119887 (119866) 119899Δ (66)

where 119899Δis the number of vertices with maximum degree Δ in

119866

Observation 5 For any graph 119866 if 120574(119866) = 120574RM(119866) then119887RM(119866) ⩽ 119887(119866)

Theorem 167 (Bahremandpour et al [118] 2013) For everyRoman graph 119866

119887RM (119866) ⩾ 119887 (119866) (67)

The bound is sharp for cycles on 119899 vertices where 119899 equiv 0 (mod3)

The strict inequality in Theorem 167 can hold for exam-ple 119887(119862

3119896+2) = 2 lt 3 = 119887RM(1198623119896+2

) byTheorem 157A graph 119866 is called to be vertex Roman domination-

critical if 120574RM(119866 minus 119909) lt 120574RM(119866) for every vertex 119909 in 119866 If119866 has no isolated vertices then 120574RM(119866) ⩽ 2120574(119866) ⩽ 2120573(119866)If 120574RM(119866) = 2120573(119866) then 120574RM(119866) = 2120574(119866) and hence 119866 is aRoman graph In [30] Volkmann gave a lot of graphs with120574(119866) = 120573(119866) The following result is similar to Theorem 57

Theorem 168 (Bahremandpour et al [118] 2013) For anygraph 119866 if 120574RM(119866) = 2120573(119866) then

(a) 119887RM(119866) ⩾ 120575(119866)

(b) 119887RM(119866) ⩾ 120575(119866)+1 if119866 is a vertex Roman domination-critical graph

If 120574RM(119866) = 2 then obviously 119887RM(119866) ⩽ 120575(119866) So weassume 120574RM(119866) ⩾ 3 Dehgardi et al [121] proved 119887RM(119866) ⩽(120574RM(119866) minus 2)Δ(119866) + 1 and posed the following problem if 119866is a connected graph of order 119899 ⩾ 4 with 120574RM(119866) ⩾ 3 then

119887RM (119866) ⩽ (120574RM (119866) minus 2) Δ (119866) (68)

Theorem 161 (a) shows that the inequality (68) holds if120574RM(119866) ⩾ 5 Thus the bound in (68) is of interest only when120574RM(119866) is 3 or 4

Lemma 169 (Bahremandpour et al [118] 2013) For anynonempty graph 119866 of order 119899 ⩾ 3 120574RM(119866) = 3 if and onlyif Δ(119866) = 119899 minus 2

The following result shows that (68) holds for all graphs119866 of order 119899 ⩾ 4 with 120574RM(119866) = 3 or 4 which improvesTheorem 161 (b)

Theorem 170 (Bahremandpour et al [118] 2013) For anyconnected graph 119866 of order 119899 ⩾ 4

119887RM (119866) le Δ (119866) = 119899 minus 2 119894119891 120574RM (119866) = 3

Δ (119866) + 120575 (119866) minus 1 119894119891 120574RM (119866) = 4(69)

with the first equality if and only if 119866 cong 1198624

Recently Akbari and Qajar [122] have proved the follow-ing result

Theorem 171 (Akbari and Qajar [122]) For any connectedgraph 119866 of order 119899 ⩾ 3

119887RM (119866) ⩽ 119899 minus 120574RM (119866) + 5 (70)

International Journal of Combinatorics 27

We conclude this section with the following problems

Problem 8 Characterize all connected graphs 119866 of order 119899 ⩾3 for which 119887RM(119866) = 119899 minus 1

Problem 9 Prove or disprove if 119866 is a connected graph oforder 119899 ⩾ 3 then

119887RM (119866) ⩽ 119899 minus 120574RM (119866) + 3 (71)

Note that 120574119877(119870

119905(3)) = 4 and 119887RM(119870119905

(3)) = 119899minus1 where 119899 =3119905 for 119905 ⩾ 3 The upper bound on 119887RM(119866) given in Problem 9is sharp if Problem 9 has positive answer

95 Rainbow Bondage Numbers For a positive integer 119896 a119896-rainbow dominating function of a graph 119866 is a function 119891from the vertex-set 119881(119866) to the set of all subsets of the set1 2 119896 such that for any vertex 119909 in 119866 with 119891(119909) = 0 thecondition

⋃

119910isin119873119866(119909)

119891 (119910) = 1 2 119896 (72)

is fulfilledThe weight of a 119896-rainbow dominating function 119891on 119866 is the value

119891 (119866) = sum

119909isin119881(119866)

1003816100381610038161003816119891 (119909)1003816100381610038161003816 (73)

The 119896-rainbow domination number of a graph 119866 denoted by120574119903119896(119866) is the minimum weight of a 119896-rainbow dominating

function 119891 of 119866Note that 120574

1199031(119866) is the classical domination number 120574(119866)

The 119896-rainbow domination number is introduced by Bresaret al [126] Rainbow domination of a graph 119866 coincides withordinary domination of the Cartesian product of 119866 with thecomplete graph in particular 120574

119903119896(119866) = 120574(119866 times 119870

119896) for any

positive integer 119896 and any graph 119866 [126] Hartnell and Rall[127] proved that 120574(119879) lt 120574

1199032(119879) for any tree 119879 no graph has

120574 = 120574119903119896for 119896 ⩾ 3 for any 119896 ⩾ 2 and any graph 119866 of order 119899

min 119899 120574 (119866) + 119896 minus 2 ⩽ 120574119903119896(119866) ⩽ 119896120574 (119866) (74)

The introduction of rainbow domination is motivatedby the study of paired domination in Cartesian productsof graphs where certain upper bounds can be expressed interms of rainbow domination that is for any graph 119866 andany graph 119867 if 119881(119867) can be partitioned into 119896120574

119875-sets then

120574119875(119866 times 119867) ⩽ (1119896)120592(119867)120574

119903119896(119866) proved by Bresar et al [126]

Dehgardi et al [128] introduced the concept of 119896-rainbowbondage number of graphs Let 119866 be a graph with maximumdegree at least two The 119896-rainbow bondage number 119887

119903119896(119866)

of 119866 is the minimum cardinality of all sets 119861 sube 119864(119866) forwhich 120574

119903119896(119866 minus 119861) gt 120574

119903119896(119866) Since in the study of 119896-rainbow

bondage number the assumption Δ(119866) ⩾ 2 is necessarywe always assume that when we discuss 119887

119903119896(119866) all graphs

involved satisfy Δ(119866) ⩾ 2The 2-rainbow bondage number of some special families

of graphs has been determined

Theorem 172 (Dehgardi et al [128]) For any integer 119899 ⩾ 3

1198871199032(119875

119899) = 1 119891119900119903 119899 ⩾ 2

1198871199032(119862

119899) =

1 119894119891 119899 equiv 0 (mod 4) 2 119900119905ℎ119890119903119908119894119904119890

1198871199032(119870

119899119899) =

1 119894119891 119899 = 2

3 119894119891 119899 = 3

6 119894119891 119899 = 4

119898 119894119891 119899 ⩾ 5

(75)

Theorem 173 (Dehgardi et al [128]) For any connected graph119866 of order 119899 ⩾ 3 119887

1199032(119866) ⩽ 120582(119866) + 2Δ(119866) minus 3

Theorem 174 (Dehgardi et al [128]) For any tree 119879 of order119899 ⩾ 3 119887

1199032(119879) ⩽ 2

Theorem 175 (Dehgardi et al [128]) If 119866 is a planar graphwithmaximumdegree at least two then 119887

1199032(119866) ⩽ 15Moreover

if the girth 119892(119866) ⩾ 4 then 1198871199032(119866) ⩽ 11 and if 119892(119866) ⩾ 6 then

1198871199032(119866) ⩽ 9

Theorem 176 (Dehgardi et al [128]) For a complete bipartitegraph 119870

119901119902 if 2119896 + 1 ⩽ 119901 ⩽ 119902 then 119887

119903119896(119870

119901119902) = 119901

Theorem177 (Dehgardi et al [128]) For a complete graph119870119899

if 119899 ⩾ 119896 + 1 then 119887119903119896(119870

119899) = lceil119896119899(119896 + 1)rceil

The special case 119896 = 1 of Theorem 177 can be found inTheorem 1 The following is a simple observation similar toObservation 1

Observation 6 (Dehgardi et al [128]) Let 119866 be a graphwith maximum degree at least two 119867 a spanning subgraphobtained from 119866 by removing 119896 edges If 120574

119903119896(119867) = 120574

119903119896(119866)

then 119887119903119896(119867) ⩽ 119887

119903119896(119866) ⩽ 119887

119903119896(119867) + 119896

Theorem 178 (Dehgardi et al [128]) For every graph 119866 with120574119903119896(119866) = 119896120574(119866) 119887

119903119896(119866) ⩾ 119887(119866)

A graph 119866 is said to be vertex 119896-rainbow domination-critical if 120574

119903119896(119866 minus 119909) lt 120574

119903119896(119866) for every vertex 119909 in 119866 It is

clear that if 119866 has no isolated vertices then 120574119903119896(119866) ⩽ 119896120574(119866) ⩽

119896120573(119866) where 120573(119866) is the vertex-covering number of119866 Thusif 120574

119903119896(119866) = 119896120573(119866) then 120574

119903119896(119866) = 119896120574(119866)

Theorem 179 (Dehgardi et al [128]) For any graph 119866 if120574119903119896(119866) = 119896120573(119866) then

(a) 119887119903119896(119866) ⩾ 120575(119866)

(b) 119887119903119896(119866) ⩾ 120575(119866)+1 if119866 is vertex 119896-rainbow domination-

critical

As far as we know there are no more results on the 119896-rainbow bondage number of graphs in the literature

28 International Journal of Combinatorics

10 Results on Digraphs

Although domination has been extensively studied in undi-rected graphs it is natural to think of a dominating setas a one-way relationship between vertices of the graphIndeed among the earliest literatures on this subject vonNeumann and Morgenstern [129] used what is now calleddomination in digraphs to find solution (or kernels whichare independent dominating sets) for cooperative 119899-persongames Most likely the first formulation of domination byBerge [130] in 1958 was given in the context of digraphs andonly some years latter by Ore [131] in 1962 for undirectedgraphs Despite this history examination of domination andits variants in digraphs has been essentially overlooked (see[4] for an overview of the domination literature) Thus thereare few if any such results on domination for digraphs in theliterature

The bondage number and its related topics for undirectedgraph have become one of major areas both in theoreticaland applied researches However until recently Carlsonand Develin [42] Shan and Kang [132] and Huang andXu [133 134] studied the bondage number for digraphsindependently In this section we introduce their resultsfor general digraphs Results for some special digraphs suchas vertex-transitive digraphs are introduced in the nextsection

101 Upper Bounds for General Digraphs Let 119866 = (119881 119864)

be a digraph without loops and parallel edges A digraph 119866is called to be asymmetric if whenever (119909 119910) isin 119864(119866) then(119910 119909) notin 119864(119866) and to be symmetric if (119909 119910) isin 119864(119866) implies(119910 119909) isin 119864(119866) For a vertex 119909 of 119881(119866) the sets of out-neighbors and in-neighbors of 119909 are respectively defined as119873

+

119866(119909) = 119910 isin 119881(119866) (119909 119910) isin 119864(119866) and 119873minus

119866(119909) = 119909 isin

119881(119866) (119909 119910) isin 119864(119866) the out-degree and the in-degree of119909 are respectively defined as 119889+

119866(119909) = |119873

+

119866(119909)| and 119889minus

119866(119909) =

|119873+

119866(119909)| Denote themaximum and theminimum out-degree

(resp in-degree) of 119866 by Δ+(119866) and 120575+(119866) (resp Δminus(119866) and120575minus

(119866))The degree 119889119866(119909) of 119909 is defined as 119889+

119866(119909)+119889

minus

119866(119909) and

the maximum and the minimum degrees of119866 are denoted byΔ(119866) and 120575(119866) respectively that is Δ(119866) = max119889

119866(119909) 119909 isin

119881(119866) and 120575(119866) = min119889119866(119909) 119909 isin 119881(119866) Note that the

definitions here are different from the ones in the textbookon digraphs see for example Xu [1]

A subset 119878 of119881(119866) is called a dominating set if119881(119866) = 119878cup119873

+

119866(119878) where119873+

119866(119878) is the set of out-neighbors of 119878Then just

as for undirected graphs 120574(119866) is the minimum cardinalityof a dominating set and the bondage number 119887(119866) is thesmallest cardinality of a set 119861 of edges such that 120574(119866 minus 119861) gt120574(119866) if such a subset 119861 sube 119864(119866) exists Otherwise we put119887(119866) = infin

Some basic results for undirected graphs stated inSection 3 can be generalized to digraphs For exampleTheorem 18 is generalized by Carlson and Develin [42]Huang andXu [133] and Shan andKang [132] independentlyas follows

Theorem 180 Let 119866 be a digraph and (119909 119910) isin 119864(119866) Then119887(119866) ⩽ 119889

119866(119910) + 119889

minus

119866(119909) minus |119873

minus

119866(119909) cap 119873

minus

119866(119910)|

Corollary 181 For a digraph 119866 119887(119866) ⩽ 120575minus(119866) + Δ(119866)

Since 120575minus

(119866) ⩽ (12)Δ(119866) from Corollary 181Conjecture 53 is valid for digraphs

Corollary 182 For a digraph 119866 119887(119866) ⩽ (32)Δ(119866)

In the case of undirected graphs the bondage number of119866119899= 119870

119899times 119870

119899achieves this bound However it was shown

by Carlson and Develin in [42] that if we take the symmetricdigraph119866

119899 we haveΔ(119866

119899) = 4(119899minus1) 120574(119866

119899) = 119899 and 119887(119866

119899) ⩽

4(119899 minus 1) + 1 = Δ(119866119899) + 1 So this family of digraphs cannot

show that the bound in Corollary 182 is tightCorresponding to Conjecture 51 which is discredited

for undirected graphs and is valid for digraphs the sameconjecture for digraphs can be proposed as follows

Conjecture 183 (Carlson and Develin [42] 2006) 119887(119866) ⩽Δ(119866) + 1 for any digraph 119866

If Conjecture 183 is true then the following results showsthat this upper bound is tight since 119887(119870

119899times119870

119899) = Δ(119870

119899times119870

119899)+1

for a complete digraph 119870119899

In 2007 Shan and Kang [132] gave some tight upperbounds on the bondage numbers for some asymmetricdigraphs For example 119887(119879) ⩽ Δ(119879) for any asymmetricdirected tree 119879 119887(119866) ⩽ Δ(119866) for any asymmetric digraph 119866with order at least 4 and 120574(119866) ⩽ 2 For planar digraphs theyobtained the following results

Theorem 184 (Shan and Kang [132] 2007) Let 119866 be aasymmetric planar digraphThen 119887(119866) ⩽ Δ(119866)+2 and 119887(119866) ⩽Δ(119866) + 1 if Δ(119866) ⩾ 5 and 119889minus

119866(119909) ⩾ 3 for every vertex 119909 with

119889119866(119909) ⩾ 4

102 Results for Some Special Digraphs The exact values andbounds on 119887(119866) for some standard digraphs are determined

Theorem185 (Huang andXu [133] 2006) For a directed cycle119862119899and a directed path 119875

119899

119887 (119862119899) =

3 119894119891 119899 119894119904 119900119889119889

2 119894119891 119899 119894119904 119890V119890119899

119887 (119875119899) =

2 119894119891 119899 119894119904 119900119889119889

1 119894119891 119899 119894119904 119890V119890119899

(76)

For the de Bruijn digraph 119861(119889 119899) and the Kautz digraph119870(119889 119899)

119887 (119861 (119889 119899)) = 119889 119894119891 119899 119894119904 119900119889119889

119889 ⩽ 119887 (119861 (119889 119899)) ⩽ 2119889 119894119891 119899 119894119904 119890V119890119899

119887 (119870 (119889 119899)) = 119889 + 1

(77)

Like undirected graphs we can define the total domi-nation number and the total bondage number On the totalbondage numbers for some special digraphs the knownresults are as follows

International Journal of Combinatorics 29

Theorem 186 (Huang and Xu [134] 2007) For a directedcycle 119862

119899and a directed path 119875

119899 119887

119879(119875

119899) and 119887

119879(119862

119899) all do not

exist For a complete digraph 119870119899

119887119879(119870

119899) = infin 119894119891 119899 = 1 2

119887119879(119870

119899) = 3 119894119891 119899 = 3

119899 ⩽ 119887119879(119870

119899) ⩽ 2119899 minus 3 119894119891 119899 ⩾ 4

(78)

The extended de Bruijn digraph EB(119889 119899 1199021 119902

119901) and

the extended Kautz digraph EK(119889 119899 1199021 119902

119901) are intro-

duced by Shibata and Gonda [135] If 119901 = 1 then theyare the de Bruijn digraph B(119889 119899) and the Kautz digraphK(119889 119899) respectively Huang and Xu [134] determined theirtotal domination numbers In particular their total bondagenumbers for general cases are determined as follows

Theorem 187 (Huang andXu [134] 2007) If 119889 ⩾ 2 and 119902119894⩾ 2

for each 119894 = 1 2 119901 then

119887119879(EB(119889 119899 119902

1 119902

119901)) = 119889

119901

minus 1

119887119879(EK(119889 119899 119902

1 119902

119901)) = 119889

119901

(79)

In particular for the de Bruijn digraph B(119889 119899) and the Kautzdigraph K(119889 119899)

119887119879(B (119889 119899)) = 119889 minus 1 119887

119879(K (119889 119899)) = 119889 (80)

Zhang et al [136] determined the bondage number incomplete 119905-partite digraphs

Theorem 188 (Zhang et al [136] 2009) For a complete 119905-partite digraph 119870

11989911198992119899119905

where 1198991⩽ 119899

2⩽ sdot sdot sdot ⩽ 119899

119905

119887 (11987011989911198992119899119905

)

=

119898 119894119891 119899119898= 1 119899

119898+1⩾ 2 119891119900119903 119904119900119898119890

119898 (1 ⩽ 119898 lt 119905)

4119905 minus 3 119894119891 1198991= 119899

2= sdot sdot sdot = 119899

119905= 2

119905minus1

sum

119894=1

119899119894+ 2 (119905 minus 1) 119900119905ℎ119890119903119908119894119904119890

(81)

Since an undirected graph can be thought of as a sym-metric digraph any result for digraphs has an analogy forundirected graphs in general In view of this point studyingthe bondage number for digraphs is more significant thanfor undirected graphs Thus we should further study thebondage number of digraphs and try to generalize knownresults on the bondage number and related variants for undi-rected graphs to digraphs prove or disprove Conjecture 183In particular determine the exact values of 119887(B(119889 119899)) for aneven 119899 and 119887

119879(119870

119899) for 119899 ⩾ 4

11 Efficient Dominating Sets

A dominating set 119878 of a graph 119866 is called to be efficient if forevery vertex 119909 in119866 |119873

119866[119909]cap119878| = 1 if119866 is a undirected graph

or |119873minus

119866[119909] cap 119878| = 1 if 119866 is a directed graph The concept of an

efficient set was proposed by Bange et al [137] in 1988 In theliterature an efficient set is also called a perfect dominatingset (see Livingston and Stout [138] for an explanation)

From definition if 119878 is an efficient dominating set of agraph 119866 then 119878 is certainly an independent set and everyvertex not in 119878 is adjacent to exactly one vertex in 119878

It is also clear from definition that a dominating set 119878 isefficient if and only if N

119866[119878] = 119873

119866[119909] 119909 isin 119878 for the

undirected graph 119866 or N+

119866[119878] = 119873

+

119866[119909] 119909 isin 119878 for the

digraph119866 is a partition of119881(119866) where the induced subgraphby119873

119866[119909] or119873+

119866[119909] is a star or an out-star with the root 119909

The efficient domination has important applications inmany areas such as error-correcting codes and receivesmuch attention in the late years

The concept of efficient dominating sets is a measureof the efficiency of domination in graphs Unfortunately asshown in [137] not every graph has an efficient dominatingset andmoreover it is anNP-complete problem to determinewhether a given graph has an efficient dominating set Inaddition it was shown by Clark [139] in 1993 that for a widerange of 119901 almost every random undirected graph 119866 isin

G(120592 119901) has no efficient dominating sets This means thatundirected graphs possessing an efficient dominating set arerare However it is easy to show that every undirected graphhas an orientationwith an efficient dominating set (see Bangeet al [140])

In 1993 Barkauskas and Host [141] showed that deter-mining whether an arbitrary oriented graph has an efficientdominating set is NP-complete Even so the existence ofefficient dominating sets for some special classes of graphs hasbeen examined see for example Dejter and Serra [142] andLee [143] for Cayley graph Gu et al [144] formeshes and toriObradovic et al [145] Huang and Xu [36] Reji Kumar andMacGillivray [146] for circulant graphs Harary graphs andtori and Van Wieren et al [147] for cube-connected cycles

In this section we introduce results of the bondagenumber for some graphs with efficient dominating sets dueto Huang and Xu [36]

111 Results for General Graphs In this subsection we intro-duce some results on bondage numbers obtained by applyingefficient dominating sets We first state the two followinglemmas

Lemma 189 For any 119896-regular graph or digraph 119866 of order 119899120574(119866) ⩾ 119899(119896 + 1) with equality if and only if 119866 has an efficientdominating set In addition if 119866 has an efficient dominatingset then every efficient dominating set is certainly a 120574-set andvice versa

Let 119890 be an edge and 119878 a dominating set in 119866 We say that119890 supports 119878 if 119890 isin (119878 119878) where (119878 119878) = (119909 119910) isin 119864(119866) 119909 isin 119878 119910 isin 119878 Denote by 119904(119866) the minimum number of edgeswhich support all 120574-sets in 119866

Lemma 190 For any graph or digraph 119866 119887(119866) ⩾ 119904(119866) withequality if 119866 is regular and has an efficient dominating set

30 International Journal of Combinatorics

A graph 119866 is called to be vertex-transitive if its automor-phism group Aut(119866) acts transitively on its vertex-set 119881(119866)A vertex-transitive graph is regular Applying Lemmas 189and 190 Huang and Xu obtained some results on bondagenumbers for vertex-transitive graphs or digraphs

Theorem 191 (Huang and Xu [36] 2008) For any vertex-transitive graph or digraph 119866 of order 119899

119887 (119866) ⩾

lceil119899

2120574 (119866)rceil 119894119891 119866 119894119904 119906119899119889119894119903119890119888119905119890119889

lceil119899

120574 (119866)rceil 119894119891 119866 119894119904 119889119894119903119890119888119905119890119889

(82)

Theorem192 (Huang andXu [36] 2008) Let119866 be a 119896-regulargraph of order 119899 Then

119887(119866) ⩽ 119896 if119866 is undirected and 119899 ⩾ 120574(119866)(119896+1)minus119896+1119887(119866) ⩽ 119896+1+ℓ if119866 is directed and 119899 ⩾ 120574(119866)(119896+1)minusℓwith 0 ⩽ ℓ ⩽ 119896 minus 1

Next we introduce a better upper bound on 119887(119866) To thisaim we need the following concept which is a generalizationof the concept of the edge-covering of a graph 119866 For 1198811015840

sube

119881(119866) and 1198641015840 sube 119864(119866) we say that 1198641015840covers 1198811015840 and call 1198641015840an edge-covering for 1198811015840 if there exists an edge (119909 119910) isin 1198641015840 forany vertex 119909 isin 1198811015840 For 119910 isin 119881(119866) let 1205731015840[119910] be the minimumcardinality over all edge-coverings for119873minus

119866[119910]

Theorem 193 (Huang and Xu [36] 2008) If 119866 is a 119896-regulargraph with order 120574(119866)(119896 + 1) then 119887(119866) ⩽ 1205731015840[119910] for any 119910 isin119881(119866)

Theupper bound on 119887(119866) given inTheorem 193 is tight inview of 119887(119862

119899) for a cycle or a directed cycle 119862

119899(seeTheorems

1 and 185 resp)It is easy to see that for a 119896-regular graph 119866 lceil(119896 + 1)2rceil ⩽

1205731015840

[119910] ⩽ 119896 when 119866 is undirected and 1205731015840[119910] = 119896 + 1 when 119866 isdirected By this fact and Lemma 189 the following theoremis merely a simple combination of Theorems 191 and 193

Theorem 194 (Huang and Xu [36] 2008) Let 119866 be a vertex-transitive graph of degree 119896 If 119866 has an efficient dominatingset then

lceil119896 + 1

2rceil ⩽ 119887 (119866) ⩽ 119896 119894119891 119866 119894119904 119906119899119889119894119903119890119888119905119890119889

119887 (119866) = 119896 + 1 119894119891 119866 119894119904 119889119894119903119890119888119905119890119889

(83)

Theorem 195 (Huang and Xu [36] 2008) If 119866 is an undi-rected vertex-transitive cubic graph with order 4120574(119866) and girth119892(119866) ⩽ 5 then 119887(119866) = 2

The proof of Theorem 195 leads to a byproduct If 119892 =5 let (119906

1 119906

2 119906

3 119906

4 119906

5) be a cycle and let D

119894be the family

of efficient dominating sets containing 119906119894for each 119894 isin

1 2 3 4 5 ThenD119894capD

119895= 0 for 119894 = 1 2 5 Then 119866 has

at least 5119904 efficient dominating sets where 119904 = 119904(119866) But thereare only 4s (=ns120574) distinct efficient dominating sets in119866This

contradiction implies that an undirected vertex-transitivecubic graph with girth five has no efficient dominating setsBut a similar argument for 119892(119866) = 3 4 or 119892(119866) ⩾ 6 couldnot give any contradiction This is consistent with the resultthat CCC(119899) a vertex-transitive cubic graph with girth 119899 if3 ⩽ 119899 ⩽ 8 or girth 8 if 119899 ⩾ 9 has efficient dominating setsfor all 119899 ⩾ 3 except 119899 = 5 (see Theorem 199 in the followingsubsection)

112 Results for Cayley Graphs In this subsection we useTheorem 194 to determine the exact values or approximativevalues of bondage numbers for some special vertex-transitivegraphs by characterizing the existence of efficient dominatingsets in these graphs

Let Γ be a nontrivial finite group 119878 a nonempty subset ofΓ without the identity element of Γ A digraph 119866 is defined asfollows

119881 (119866) = Γ (119909 119910) isin 119864 (119866) lArrrArr 119909minus1

119910 isin 119878

for any 119909 119910 isin Γ(84)

is called a Cayley digraph of the group Γ with respect to119878 denoted by 119862

Γ(119878) If 119878minus1 = 119904minus1 119904 isin 119878 = 119878 then

119862Γ(119878) is symmetric and is called a Cayley undirected graph

a Cayley graph for short Cayley graphs or digraphs arecertainly vertex-transitive (see Xu [1])

A circulant graph 119866(119899 119878) of order 119899 is a Cayley graph119862119885119899

(119878) where 119885119899= 0 119899 minus 1 is the addition group of

order 119899 and 119878 is a nonempty subset of119885119899without the identity

element and hence is a vertex-transitive digraph of degree|119878| If 119878minus1 = 119878 then119866(119899 119878) is an undirected graph If 119878 = 1 119896where 2 ⩽ 119896 ⩽ 119899 minus 2 we write 119866(119899 1 119896) for 119866(119899 1 119896) or119866(119899 plusmn1 plusmn119896) and call it a double-loop circulant graph

Huang and Xu [36] showed that for directed 119866 =

119866(119899 1 119896) lceil1198993rceil ⩽ 120574(119866) ⩽ lceil1198992rceil and 119866 has an efficientdominating set if and only if 3 | 119899 and 119896 equiv 2 (mod 3) fordirected 119866 = 119866(119899 1 119896) and 119896 = 1198992 lceil1198995rceil ⩽ 120574(119866) ⩽ lceil1198993rceiland 119866 has an efficient dominating set if and only if 5 | 119899 and119896 equiv plusmn2 (mod 5) By Theorem 194 we obtain the bondagenumber of a double loop circulant graph if it has an efficientdominating set

Theorem 196 (Huang and Xu [36] 2008) Let 119866 be a double-loop circulant graph 119866(119899 1 119896) If 119866 is directed with 3 | 119899and 119896 equiv 2 (mod 3) or 119866 is undirected with 5 | 119899 and119896 equiv plusmn2 (mod 5) then 119887(119866) = 3

The 119898 times 119899 torus is the Cartesian product 119862119898times 119862

119899of

two cycles and is a Cayley graph 119862119885119898times119885119899

(119878) where 119878 =

(0 1) (1 0) for directed cycles and 119878 = (0 plusmn1) (plusmn1 0) forundirected cycles and hence is vertex-transitive Gu et al[144] showed that the undirected torus119862

119898times119862

119899has an efficient

dominating set if and only if both 119898 and 119899 are multiplesof 5 Huang and Xu [36] showed that the directed torus119862119898times 119862

119899has an efficient dominating set if and only if both

119898 and 119899 are multiples of 3 Moreover they found a necessarycondition for a dominating set containing the vertex (0 0)in 119862

119898times 119862

119899to be efficient and obtained the following

result

International Journal of Combinatorics 31

Theorem 197 (Huang and Xu [36] 2008) Let 119866 = 119862119898times 119862

119899

If 119866 is undirected and both119898 and 119899 are multiples of 5 or if 119866 isdirected and both119898 and 119899 are multiples of 3 then 119887(119866) = 3

The hypercube 119876119899

is the Cayley graph 119862Γ(119878)

where Γ = 1198852times sdot sdot sdot times 119885

2= (119885

2)119899 and 119878 =

100 sdot sdot sdot 0 010 sdot sdot sdot 0 00 sdot sdot sdot 01 Lee [143] showed that119876119899has an efficient dominating set if and only if 119899 = 2119898 minus 1

for a positive integer119898 ByTheorem 194 the following resultis obtained immediately

Theorem 198 (Huang and Xu [36] 2008) If 119899 = 2119898 minus 1 for apositive integer119898 then 2119898minus1

⩽ 119887(119876119899) ⩽ 2

119898

minus 1

Problem 10 Determine the exact value of 119887(119876119899) for 119899 = 2119898minus1

The 119899-dimensional cube-connected cycle denoted byCCC(119899) is constructed from the 119899-dimensional hypercube119876119899by replacing each vertex 119909 in 119876

119899with an undirected cycle

119862119899of length 119899 and linking the 119894th vertex of the 119862

119899to the 119894th

neighbor of 119909 It has been proved that CCC(119899) is a Cayleygraph and hence is a vertex-transitive graph with degree 3Van Wieren et al [147] proved that CCC(119899) has an efficientdominating set if and only if 119899 = 5 From Theorems 194 and195 the following result can be derived

Theorem 199 (Huang and Xu [36] 2008) Let 119866 = 119862119862119862(119899)be the 119899-dimensional cube-connected cycles with 119899 ⩾ 3 and119899 = 5 Then 120574(119866) = 1198992119899minus2 and 2 ⩽ 119887(119866) ⩽ 3 In addition119887(119862119862119862(3)) = 119887(119862119862119862(4)) = 2

Problem 11 Determine the exact value of 119887(CCC(119899)) for 119899 ⩾5

Acknowledgments

This work was supported by NNSF of China (Grants nos11071233 61272008) The author would like to express hisgratitude to the anonymous referees for their kind sugges-tions and comments on the original paper to ProfessorSheikholeslami and Professor Jafari Rad for sending thereferences [128] and [81] which resulted in this version

References

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[2] S THedetniemi andR C Laskar ldquoBibliography on dominationin graphs and some basic definitions of domination parame-tersrdquo Discrete Mathematics vol 86 no 1ndash3 pp 257ndash277 1990

[3] TW Haynes S T Hedetniemi and P J Slater Fundamentals ofDomination in Graphs vol 208 ofMonographs and Textbooks inPure and Applied Mathematics Marcel Dekker New York NYUSA 1998

[4] TWHaynes S THedetniemi and P J Slater EdsDominationin Graphs vol 209 of Monographs and Textbooks in Pure andAppliedMathematicsMarcelDekker NewYorkNYUSA 1998

[5] M R Garey andD S JohnsonComputers and Intractability WH Freeman and Company San Francisco Calif USA 1979

[6] H B Walikar and B D Acharya ldquoDomination critical graphsrdquoNational Academy Science Letters vol 2 pp 70ndash72 1979

[7] D Bauer F Harary J Nieminen and C L Suffel ldquoDominationalteration sets in graphsrdquo Discrete Mathematics vol 47 no 2-3pp 153ndash161 1983

[8] J F Fink M S Jacobson L F Kinch and J Roberts ldquoThebondage number of a graphrdquo Discrete Mathematics vol 86 no1ndash3 pp 47ndash57 1990

[9] F-T Hu and J-M Xu ldquoThe bondage number of (119899 minus 3)-regulargraph G of order nrdquo Ars Combinatoria to appear

[10] U Teschner ldquoNew results about the bondage number of agraphrdquoDiscreteMathematics vol 171 no 1ndash3 pp 249ndash259 1997

[11] B L Hartnell and D F Rall ldquoA bound on the size of a graphwith given order and bondage numberrdquo Discrete Mathematicsvol 197-198 pp 409ndash413 1999 16th British CombinatorialConference (London 1997)

[12] B L Hartnell and D F Rall ldquoA characterization of trees inwhich no edge is essential to the domination numberrdquo ArsCombinatoria vol 33 pp 65ndash76 1992

[13] J Topp and P D Vestergaard ldquo120572119896- and 120574

119896-stable graphsrdquo

Discrete Mathematics vol 212 no 1-2 pp 149ndash160 2000[14] A Meir and J W Moon ldquoRelations between packing and

covering numbers of a treerdquo Pacific Journal of Mathematics vol61 no 1 pp 225ndash233 1975

[15] B L Hartnell L K Joslashrgensen P D Vestergaard and CA Whitehead ldquoEdge stability of the k-domination numberof treesrdquo Bulletin of the Institute of Combinatorics and ItsApplications vol 22 pp 31ndash40 1998

[16] F-T Hu and J-M Xu ldquoOn the complexity of the bondage andreinforcement problemsrdquo Journal of Complexity vol 28 no 2pp 192ndash201 2012

[17] B L Hartnell and D F Rall ldquoBounds on the bondage numberof a graphrdquo Discrete Mathematics vol 128 no 1ndash3 pp 173ndash1771994

[18] Y-L Wang ldquoOn the bondage number of a graphrdquo DiscreteMathematics vol 159 no 1ndash3 pp 291ndash294 1996

[19] G H Fricke TW Haynes S M Hedetniemi S T Hedetniemiand R C Laskar ldquoExcellent treesrdquo Bulletin of the Institute ofCombinatorics and Its Applications vol 34 pp 27ndash38 2002

[20] V Samodivkin ldquoThe bondage number of graphs good and badverticesrdquoDiscussiones Mathematicae GraphTheory vol 28 no3 pp 453ndash462 2008

[21] J E Dunbar T W Haynes U Teschner and L VolkmannldquoBondage insensitivity and reinforcementrdquo in Domination inGraphs vol 209 of Monographs and Textbooks in Pure andAppliedMathematics pp 471ndash489 Dekker NewYork NY USA1998

[22] H Liu and L Sun ldquoThe bondage and connectivity of a graphrdquoDiscrete Mathematics vol 263 no 1ndash3 pp 289ndash293 2003

[23] A-H Esfahanian and S L Hakimi ldquoOn computing a con-ditional edge-connectivity of a graphrdquo Information ProcessingLetters vol 27 no 4 pp 195ndash199 1988

[24] R C Brigham P Z Chinn and R D Dutton ldquoVertexdomination-critical graphsrdquoNetworks vol 18 no 3 pp 173ndash1791988

[25] V Samodivkin ldquoDomination with respect to nondegenerateproperties bondage numberrdquoThe Australasian Journal of Com-binatorics vol 45 pp 217ndash226 2009

[26] U Teschner ldquoA new upper bound for the bondage numberof graphs with small domination numberrdquo The AustralasianJournal of Combinatorics vol 12 pp 27ndash35 1995

32 International Journal of Combinatorics

[27] J Fulman D Hanson and G MacGillivray ldquoVertex dom-ination-critical graphsrdquoNetworks vol 25 no 2 pp 41ndash43 1995

[28] U Teschner ldquoA counterexample to a conjecture on the bondagenumber of a graphrdquo Discrete Mathematics vol 122 no 1ndash3 pp393ndash395 1993

[29] U Teschner ldquoThe bondage number of a graph G can be muchgreater than Δ(G)rdquo Ars Combinatoria vol 43 pp 81ndash87 1996

[30] L Volkmann ldquoOn graphs with equal domination and coveringnumbersrdquoDiscrete AppliedMathematics vol 51 no 1-2 pp 211ndash217 1994

[31] L A Sanchis ldquoMaximum number of edges in connected graphswith a given domination numberrdquoDiscreteMathematics vol 87no 1 pp 65ndash72 1991

[32] S Klavzar and N Seifter ldquoDominating Cartesian products ofcyclesrdquoDiscrete AppliedMathematics vol 59 no 2 pp 129ndash1361995

[33] H K Kim ldquoA note on Vizingrsquos conjecture about the bondagenumberrdquo Korean Journal of Mathematics vol 10 no 2 pp 39ndash42 2003

[34] M Y Sohn X Yuan and H S Jeong ldquoThe bondage number of1198623times119862

119899rdquo Journal of the KoreanMathematical Society vol 44 no

6 pp 1213ndash1231 2007[35] L Kang M Y Sohn and H K Kim ldquoBondage number of the

discrete torus 119862119899times 119862

4rdquo Discrete Mathematics vol 303 no 1ndash3

pp 80ndash86 2005[36] J Huang and J-M Xu ldquoThe bondage numbers and efficient

dominations of vertex-transitive graphsrdquoDiscrete Mathematicsvol 308 no 4 pp 571ndash582 2008

[37] C J Xiang X Yuan andM Y Sohn ldquoDomination and bondagenumber of119862

3times119862

119899rdquoArs Combinatoria vol 97 pp 299ndash310 2010

[38] M S Jacobson and L F Kinch ldquoOn the domination numberof products of graphs Irdquo Ars Combinatoria vol 18 pp 33ndash441984

[39] Y-Q Li ldquoA note on the bondage number of a graphrdquo ChineseQuarterly Journal of Mathematics vol 9 no 4 pp 1ndash3 1994

[40] F-T Hu and J-M Xu ldquoBondage number of mesh networksrdquoFrontiers ofMathematics inChina vol 7 no 5 pp 813ndash826 2012

[41] R Frucht and F Harary ldquoOn the corona of two graphsrdquoAequationes Mathematicae vol 4 pp 322ndash325 1970

[42] K Carlson and M Develin ldquoOn the bondage number of planarand directed graphsrdquoDiscreteMathematics vol 306 no 8-9 pp820ndash826 2006

[43] U Teschner ldquoOn the bondage number of block graphsrdquo ArsCombinatoria vol 46 pp 25ndash32 1997

[44] J Huang and J-M Xu ldquoNote on conjectures of bondagenumbers of planar graphsrdquo Applied Mathematical Sciences vol6 no 65ndash68 pp 3277ndash3287 2012

[45] L Kang and J Yuan ldquoBondage number of planar graphsrdquoDiscrete Mathematics vol 222 no 1ndash3 pp 191ndash198 2000

[46] M Fischermann D Rautenbach and L Volkmann ldquoRemarkson the bondage number of planar graphsrdquo Discrete Mathemat-ics vol 260 no 1ndash3 pp 57ndash67 2003

[47] J Hou G Liu and J Wu ldquoBondage number of planar graphswithout small cyclesrdquoUtilitasMathematica vol 84 pp 189ndash1992011

[48] J Huang and J-M Xu ldquoThe bondage numbers of graphs withsmall crossing numbersrdquo Discrete Mathematics vol 307 no 15pp 1881ndash1897 2007

[49] Q Ma S Zhang and J Wang ldquoBondage number of 1-planargraphrdquo Applied Mathematics vol 1 no 2 pp 101ndash103 2010

[50] J Hou and G Liu ldquoA bound on the bondage number of toroidalgraphsrdquoDiscreteMathematics Algorithms and Applications vol4 no 3 Article ID 1250046 5 pages 2012

[51] Y-C Cao J-M Xu and X-R Xu ldquoOn bondage number oftoroidal graphsrdquo Journal of University of Science and Technologyof China vol 39 no 3 pp 225ndash228 2009

[52] Y-C Cao J Huang and J-M Xu ldquoThe bondage number ofgraphs with crossing number less than fourrdquo Ars Combinatoriato appear

[53] H K Kim ldquoOn the bondage numbers of some graphsrdquo KoreanJournal of Mathematics vol 11 no 2 pp 11ndash15 2004

[54] A Gagarin and V Zverovich ldquoUpper bounds for the bondagenumber of graphs on topological surfacesrdquo Discrete Mathemat-ics 2012

[55] J Huang ldquoAn improved upper bound for the bondage numberof graphs on surfacesrdquoDiscrete Mathematics vol 312 no 18 pp2776ndash2781 2012

[56] A Gagarin and V Zverovich ldquoThe bondage number of graphson topological surfaces and Teschnerrsquos conjecturerdquo DiscreteMathematics vol 313 no 6 pp 796ndash808 2013

[57] V Samodivkin ldquoNote on the bondage number of graphs ontopological surfacesrdquo httparxivorgabs12086203

[58] E J Cockayne R M Dawes and S T Hedetniemi ldquoTotaldomination in graphsrdquoNetworks vol 10 no 3 pp 211ndash219 1980

[59] R Laskar J Pfaff S M Hedetniemi and S T HedetniemildquoOn the algorithmic complexity of total dominationrdquo Society forIndustrial and Applied Mathematics vol 5 no 3 pp 420ndash4251984

[60] J Pfaff R C Laskar and S T Hedetniemi ldquoNP-completeness oftotal and connected domination and irredundance for bipartitegraphsrdquo Tech Rep 428 Department of Mathematical SciencesClemson University 1983

[61] M A Henning ldquoA survey of selected recent results on totaldomination in graphsrdquoDiscrete Mathematics vol 309 no 1 pp32ndash63 2009

[62] V R Kulli and D K Patwari ldquoThe total bondage number ofa graphrdquo in Advances in Graph Theory pp 227ndash235 VishwaGulbarga India 1991

[63] F-T Hu Y Lu and J-M Xu ldquoThe total bondage number ofgrid graphsrdquo Discrete Applied Mathematics vol 160 no 16-17pp 2408ndash2418 2012

[64] J Cao M Shi and B Wu ldquoResearch on the total bondagenumber of a spectial networkrdquo in Proceedings of the 3rdInternational Joint Conference on Computational Sciences andOptimization (CSO rsquo10) pp 456ndash458 May 2010

[65] N Sridharan MD Elias and V S A Subramanian ldquoTotalbondage number of a graphrdquo AKCE International Journal ofGraphs and Combinatorics vol 4 no 2 pp 203ndash209 2007

[66] N Jafari Rad and J Raczek ldquoSome progress on total bondage ingraphsrdquo Graphs and Combinatorics 2011

[67] T W Haynes and P J Slater ldquoPaired-domination and thepaired-domatic numberrdquoCongressusNumerantium vol 109 pp65ndash72 1995

[68] T W Haynes and P J Slater ldquoPaired-domination in graphsrdquoNetworks vol 32 no 3 pp 199ndash206 1998

[69] X Hou ldquoA characterization of 2120574 120574119875-treesrdquoDiscrete Mathemat-

ics vol 308 no 15 pp 3420ndash3426 2008[70] K E Proffitt T W Haynes and P J Slater ldquoPaired-domination

in grid graphsrdquo Congressus Numerantium vol 150 pp 161ndash1722001

International Journal of Combinatorics 33

[71] H Qiao L KangM Cardei andD-Z Du ldquoPaired-dominationof treesrdquo Journal of Global Optimization vol 25 no 1 pp 43ndash542003

[72] E Shan L Kang and M A Henning ldquoA characterizationof trees with equal total domination and paired-dominationnumbersrdquo The Australasian Journal of Combinatorics vol 30pp 31ndash39 2004

[73] J Raczek ldquoPaired bondage in treesrdquo Discrete Mathematics vol308 no 23 pp 5570ndash5575 2008

[74] J A Telle and A Proskurowski ldquoAlgorithms for vertex parti-tioning problems on partial k-treesrdquo SIAM Journal on DiscreteMathematics vol 10 no 4 pp 529ndash550 1997

[75] G S Domke J H Hattingh M A Henning and L R MarkusldquoRestrained domination in treesrdquoDiscreteMathematics vol 211no 1ndash3 pp 1ndash9 2000

[76] G S Domke J H Hattingh M A Henning and L R MarkusldquoRestrained domination in graphs with minimum degree twordquoJournal of Combinatorial Mathematics and Combinatorial Com-puting vol 35 pp 239ndash254 2000

[77] G S Domke J H Hattingh S T Hedetniemi R C Laskarand L R Markus ldquoRestrained domination in graphsrdquo DiscreteMathematics vol 203 no 1ndash3 pp 61ndash69 1999

[78] J H Hattingh and E J Joubert ldquoAn upper bound for therestrained domination number of a graph with minimumdegree at least two in terms of order and minimum degreerdquoDiscrete Applied Mathematics vol 157 no 13 pp 2846ndash28582009

[79] M A Henning ldquoGraphs with large restrained dominationnumberrdquo Discrete Mathematics vol 197-198 pp 415ndash429 199916th British Combinatorial Conference (London 1997)

[80] J H Hattingh and A R Plummer ldquoRestrained bondage ingraphsrdquo Discrete Mathematics vol 308 no 23 pp 5446ndash54532008

[81] N Jafari Rad R Hasni J Raczek and L Volkmann ldquoTotalrestrained bondage in graphsrdquoActaMathematica Sinica vol 29no 6 pp 1033ndash1042 2013

[82] J Cyman and J Raczek ldquoOn the total restrained dominationnumber of a graphrdquoThe Australasian Journal of Combinatoricsvol 36 pp 91ndash100 2006

[83] M A Henning ldquoGraphs with large total domination numberrdquoJournal of Graph Theory vol 35 no 1 pp 21ndash45 2000

[84] D-X Ma X-G Chen and L Sun ldquoOn total restraineddomination in graphsrdquoCzechoslovakMathematical Journal vol55 no 1 pp 165ndash173 2005

[85] B Zelinka ldquoRemarks on restrained domination and totalrestrained domination in graphsrdquo Czechoslovak MathematicalJournal vol 55 no 2 pp 393ndash396 2005

[86] W Goddard and M A Henning ldquoIndependent domination ingraphs a survey and recent resultsrdquo Discrete Mathematics vol313 no 7 pp 839ndash854 2013

[87] J H Zhang H L Liu and L Sun ldquoIndependence bondagenumber and reinforcement number of some graphsrdquo Transac-tions of Beijing Institute of Technology vol 23 no 2 pp 140ndash1422003

[88] K D Dharmalingam Studies in Graph Theorymdashequitabledomination and bottleneck domination [PhD thesis] MaduraiKamaraj University Madurai India 2006

[89] GDeepakND Soner andAAlwardi ldquoThe equitable bondagenumber of a graphrdquo Research Journal of Pure Algebra vol 1 no9 pp 209ndash212 2011

[90] E Sampathkumar and H B Walikar ldquoThe connected domina-tion number of a graphrdquo Journal of Mathematical and PhysicalSciences vol 13 no 6 pp 607ndash613 1979

[91] K White M Farber and W Pulleyblank ldquoSteiner trees con-nected domination and strongly chordal graphsrdquoNetworks vol15 no 1 pp 109ndash124 1985

[92] M Cadei M X Cheng X Cheng and D-Z Du ldquoConnecteddomination in Ad Hoc wireless networksrdquo in Proceedings ofthe 6th International Conference on Computer Science andInformatics (CSampI rsquo02) 2002

[93] J F Fink and M S Jacobson ldquon-domination in graphsrdquo inGraph Theory with Applications to Algorithms and ComputerScience Wiley-Interscience Publications pp 283ndash300 WileyNew York NY USA 1985

[94] M Chellali O Favaron A Hansberg and L Volkmann ldquok-domination and k-independence in graphs a surveyrdquo Graphsand Combinatorics vol 28 no 1 pp 1ndash55 2012

[95] Y Lu X Hou J-M Xu andN Li ldquoTrees with uniqueminimump-dominating setsrdquo Utilitas Mathematica vol 86 pp 193ndash2052011

[96] Y Lu and J-M Xu ldquoThe p-bondage number of treesrdquo Graphsand Combinatorics vol 27 no 1 pp 129ndash141 2011

[97] Y Lu and J-M Xu ldquoThe 2-domination and 2-bondage numbersof grid graphs A manuscriptrdquo httparxivorgabs12044514

[98] M Krzywkowski ldquoNon-isolating 2-bondage in graphsrdquo TheJournal of the Mathematical Society of Japan vol 64 no 4 2012

[99] M A Henning O R Oellermann and H C Swart ldquoBoundson distance domination parametersrdquo Journal of CombinatoricsInformation amp System Sciences vol 16 no 1 pp 11ndash18 1991

[100] F Tian and J-M Xu ldquoBounds for distance domination numbersof graphsrdquo Journal of University of Science and Technology ofChina vol 34 no 5 pp 529ndash534 2004

[101] F Tian and J-M Xu ldquoDistance domination-critical graphsrdquoApplied Mathematics Letters vol 21 no 4 pp 416ndash420 2008

[102] F Tian and J-M Xu ldquoA note on distance domination numbersof graphsrdquo The Australasian Journal of Combinatorics vol 43pp 181ndash190 2009

[103] F Tian and J-M Xu ldquoAverage distances and distance domina-tion numbersrdquo Discrete Applied Mathematics vol 157 no 5 pp1113ndash1127 2009

[104] Z-L Liu F Tian and J-M Xu ldquoProbabilistic analysis of upperbounds for 2-connected distance k-dominating sets in graphsrdquoTheoretical Computer Science vol 410 no 38ndash40 pp 3804ndash3813 2009

[105] M A Henning O R Oellermann and H C Swart ldquoDistancedomination critical graphsrdquo Journal of Combinatorial Mathe-matics and Combinatorial Computing vol 44 pp 33ndash45 2003

[106] T J Bean M A Henning and H C Swart ldquoOn the integrityof distance domination in graphsrdquo The Australasian Journal ofCombinatorics vol 10 pp 29ndash43 1994

[107] M Fischermann and L Volkmann ldquoA remark on a conjecturefor the (kp)-domination numberrdquoUtilitasMathematica vol 67pp 223ndash227 2005

[108] T Korneffel D Meierling and L Volkmann ldquoA remark on the(22)-domination numberrdquo Discussiones Mathematicae GraphTheory vol 28 no 2 pp 361ndash366 2008

[109] Y Lu X Hou and J-M Xu ldquoOn the (22)-domination numberof treesrdquo Discussiones Mathematicae Graph Theory vol 30 no2 pp 185ndash199 2010

34 International Journal of Combinatorics

[110] H Li and J-M Xu ldquo(dm)-domination numbers of m-connected graphsrdquo Rapports de Recherche 1130 LRI UniversiteParis-Sud 1997

[111] S M Hedetniemi S T Hedetniemi and T V Wimer ldquoLineartime resource allocation algorithms for treesrdquo Tech Rep URI-014 Department of Mathematical Sciences Clemson Univer-sity 1987

[112] M Farber ldquoDomination independent domination and dualityin strongly chordal graphsrdquo Discrete Applied Mathematics vol7 no 2 pp 115ndash130 1984

[113] G S Domke and R C Laskar ldquoThe bondage and reinforcementnumbers of 120574

119891for some graphsrdquoDiscrete Mathematics vol 167-

168 pp 249ndash259 1997[114] G S Domke S T Hedetniemi and R C Laskar ldquoFractional

packings coverings and irredundance in graphsrdquo vol 66 pp227ndash238 1988

[115] E J Cockayne P A Dreyer Jr S M Hedetniemi andS T Hedetniemi ldquoRoman domination in graphsrdquo DiscreteMathematics vol 278 no 1ndash3 pp 11ndash22 2004

[116] I Stewart ldquoDefend the roman empirerdquo Scientific American vol281 pp 136ndash139 1999

[117] C S ReVelle and K E Rosing ldquoDefendens imperiumromanum a classical problem in military strategyrdquoThe Ameri-can Mathematical Monthly vol 107 no 7 pp 585ndash594 2000

[118] A Bahremandpour F-T Hu S M Sheikholeslami and J-MXu ldquoOn the Roman bondage number of a graphrdquo DiscreteMathematics Algorithms and Applications vol 5 no 1 ArticleID 1350001 15 pages 2013

[119] N Jafari Rad and L Volkmann ldquoRoman bondage in graphsrdquoDiscussiones Mathematicae Graph Theory vol 31 no 4 pp763ndash773 2011

[120] K Ebadi and L PushpaLatha ldquoRoman bondage and Romanreinforcement numbers of a graphrdquo International Journal ofContemporary Mathematical Sciences vol 5 pp 1487ndash14972010

[121] N Dehgardi S M Sheikholeslami and L Volkman ldquoOn theRoman k-bondage number of a graphrdquo AKCE InternationalJournal of Graphs and Combinatorics vol 8 pp 169ndash180 2011

[122] S Akbari and S Qajar ldquoA note on Roman bondage number ofgraphsrdquo Ars Combinatoria to Appear

[123] F-T Hu and J-M Xu ldquoRoman bondage numbers of somegraphsrdquo httparxivorgabs11093933

[124] N Jafari Rad and L Volkmann ldquoOn the Roman bondagenumber of planar graphsrdquo Graphs and Combinatorics vol 27no 4 pp 531ndash538 2011

[125] S Akbari M Khatirinejad and S Qajar ldquoA note on the romanbondage number of planar graphsrdquo Graphs and Combinatorics2012

[126] B Bresar M A Henning and D F Rall ldquoRainbow dominationin graphsrdquo Taiwanese Journal of Mathematics vol 12 no 1 pp213ndash225 2008

[127] B L Hartnell and D F Rall ldquoOn dominating the Cartesianproduct of a graph and 120572krdquo Discussiones Mathematicae GraphTheory vol 24 no 3 pp 389ndash402 2004

[128] N Dehgardi S M Sheikholeslami and L Volkman ldquoThe k-rainbow bondage number of a graphrdquo to appear in DiscreteApplied Mathematics

[129] J von Neumann and O Morgenstern Theory of Games andEconomic Behavior Princeton University Press Princeton NJUSA 1944

[130] C BergeTheorıe des Graphes et ses Applications 1958[131] O Ore Theory of Graphs vol 38 American Mathematical

Society Colloquium Publications 1962[132] E Shan and L Kang ldquoBondage number in oriented graphsrdquoArs

Combinatoria vol 84 pp 319ndash331 2007[133] J Huang and J-M Xu ldquoThe bondage numbers of extended

de Bruijn and Kautz digraphsrdquo Computers amp Mathematics withApplications vol 51 no 6-7 pp 1137ndash1147 2006

[134] J Huang and J-M Xu ldquoThe total domination and total bondagenumbers of extended de Bruijn and Kautz digraphsrdquoComputersamp Mathematics with Applications vol 53 no 8 pp 1206ndash12132007

[135] Y Shibata and Y Gonda ldquoExtension of de Bruijn graph andKautz graphrdquo Computers amp Mathematics with Applications vol30 no 9 pp 51ndash61 1995

[136] X Zhang J Liu and JMeng ldquoThebondage number in completet-partite digraphsrdquo Information Processing Letters vol 109 no17 pp 997ndash1000 2009

[137] D W Bange A E Barkauskas and P J Slater ldquoEfficient dom-inating sets in graphsrdquo in Applications of Discrete Mathematicspp 189ndash199 SIAM Philadelphia Pa USA 1988

[138] M Livingston and Q F Stout ldquoPerfect dominating setsrdquoCongressus Numerantium vol 79 pp 187ndash203 1990

[139] L Clark ldquoPerfect domination in random graphsrdquo Journal ofCombinatorial Mathematics and Combinatorial Computing vol14 pp 173ndash182 1993

[140] D W Bange A E Barkauskas L H Host and L H ClarkldquoEfficient domination of the orientations of a graphrdquo DiscreteMathematics vol 178 no 1ndash3 pp 1ndash14 1998

[141] A E Barkauskas and L H Host ldquoFinding efficient dominatingsets in oriented graphsrdquo Congressus Numerantium vol 98 pp27ndash32 1993

[142] I J Dejter and O Serra ldquoEfficient dominating sets in CayleygraphsrdquoDiscrete AppliedMathematics vol 129 no 2-3 pp 319ndash328 2003

[143] J Lee ldquoIndependent perfect domination sets in Cayley graphsrdquoJournal of Graph Theory vol 37 no 4 pp 213ndash219 2001

[144] WGu X Jia and J Shen ldquoIndependent perfect domination setsin meshes tori and treesrdquo Unpublished

[145] N Obradovic J Peters and G Ruzic ldquoEfficient domination incirculant graphswith two chord lengthsrdquo Information ProcessingLetters vol 102 no 6 pp 253ndash258 2007

[146] K Reji Kumar and G MacGillivray ldquoEfficient domination incirculant graphsrdquo Discrete Mathematics vol 313 no 6 pp 767ndash771 2013

[147] D Van Wieren M Livingston and Q F Stout ldquoPerfectdominating sets on cube-connected cyclesrdquo Congressus Numer-antium vol 97 pp 51ndash70 1993

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 International Journal of Combinatorics

Theorem 20 (Teschner [10] 1997) If 119866 contains a completesubgraph 119870

119905with 119905 ⩾ 2 then 119887(119866) ⩽ min

119909119910isin119864(119870119905)119889

119866(119909) +

119889119866(119910) minus 119905 + 1

Following Fricke et al [19] a vertex 119909 of a graph 119866 is 120574-good if 119909 belongs to some 120574-set of119866 and 120574-bad if 119909 belongs tono 120574-set of 119866 Let 119860(119866) be the set of 120574-good vertices and let119861(119866) be the set of 120574-bad vertices in 119866 Clearly 119860(119866) 119861(119866)is a partition of119881(119866) Note there exists some 119909 isin 119860 such that120574(119866 minus 119909) = 120574(119866) say one end-vertex of 119875

5 Samodivkin [20]

presented some sharp upper bounds for 119887(119866) in terms of 120574-good and 120574-bad vertices of 119866

Theorem 21 (Samodivkin [20] 2008) Let 119866 be a graph

(a) Let 119862(119866) = 119909 isin 119881(119866) 120574(119866minus119909) ⩾ 120574(119866) If 119862(119866) = 0then

119887 (119866) ⩽ min 119889119866(119909) + 120574 (119866) minus 120574 (119866 minus 119909) 119909 isin 119862 (119866)

(8)

(b) If 119861 = 0 then

119887 (119866) ⩽ min 1003816100381610038161003816119873119866(119909) cap 119860

1003816100381610038161003816 119909 isin 119861 (119866) (9)

Theorem 22 (Samodivkin [20] 2008) Let 119866 be a graph If119860+

(119866) = 119909 isin 119860(119866) 119889119866(119909) ⩾ 1 and 120574(119866 minus 119909) lt 120574(119866) = 0

then

119887 (119866) ⩽ min119909isin119860+(119866)119910isin119861(119866minus119909)

119889119866(119909) +

1003816100381610038161003816119873119866(119910) cap 119860 (119866 minus 119909)

1003816100381610038161003816

(10)

Proposition 23 (Samodivkin [20] 2008) Under the notationof Theorem 19 if 119909 isin 119860+

(119866) then (1198791(119909 119910) minus 119909) cup 119879

2(119909 119910) cup

1198793(119909 119910) sube 119873

119866(119910) 119861(119866 minus 119909)

By Proposition 23 if 119909 isin 119860+

(119866) then

119889119866(119909) + min

119910isin119873119866(119909)

10038161003816100381610038161198794 (119909 119910)

1003816100381610038161003816

⩾ 119889119866(119909) + min

119910isin119873119866(119909)

1003816100381610038161003816119873119866

(119910) cap 119860 (119866 minus 119909)1003816100381610038161003816

⩾ 119889119866(119909) + min

119910isin119861(119866minus119909)

1003816100381610038161003816119873119866

(119910) cap 119860 (119866 minus 119909)1003816100381610038161003816

(11)

Hence Theorem 19 can be seen to follow from Theorem 22Any graph 119866 with 119887(119866) achieving the upper bound inTheorem 19 can be used to show that the bound inTheorem 22 is sharp

Let 119905 ⩾ 2 be an integer Samodivkin [20] con-structed a very interesting graph 119866

119905to show that the upper

bound in Theorem 22 is better than the known bounds Let119867

0 119867

1 119867

2 119867

119905+1be mutually vertex-disjoint graphs such

that 1198670cong 119870

2 119867

119905+1cong 119870

119905+3and 119867

119894cong 119870

119905+3minus 119890 for each

119894 = 1 2 119905 Let 119881(1198670) = 119909 119910 119909

119905+1isin 119881(119867

119905+1) and

1198671

1198672

1198673

11990911

11990912

11990921 11990922

1199093

119909

119910

Figure 3 The graph 1198662

1199091198941 119909

1198942isin 119881(119867

119894) 119909

11989411199091198942notin 119864(119867

119894) for each 119894 = 1 2 119905 The

graph 119866119905is defined as follows

119881 (119866119905) =

119905+1

⋃

119896=0

119881 (119867119896)

119864 (119866119905)=(

119905+1

⋃

119896=0

119864 (119867119896))cup(

119905

⋃

119894=1

1199091199091198941 119909119909

1198942)cup119909119909

119905+1

(12)

Such a constructed graph119866119905is shown in Figure 3 when 119905 = 2

Observe that 120574(119866119905) = 119905 + 2 119860(119866

119905) = 119881(119866

119905) 119889

119866119905

(119909) =

2119905 + 2 119889119866119905

(119909119905+1) = 119905 + 3 119889

119866119905

(119910) = 1 and 119889119866119905

(119911) = 119905 + 2

for each 119911 isin 119881(119866119905minus 119909 119910 119909

119905+1) Moreover 120574(119866 minus 119910) lt 120574(119866)

and 120574(119866119905minus119911) = 120574(119866

119905) for any 119911 isin 119881(119866

119905) minus 119910 Hence each of

the bounds stated inTheorems 13ndash20 is greater than or equals119905 + 2

Consider the graph 119866119905minus 119909119910 Clearly 120574(119866

119905minus 119909119910) = 120574(119866

119905)

and

119861 (119866119905minus 119909119910) = 119861 (119866

119905minus 119910)

= 119909 cup 119881 (119867119905+1minus 119909

119905+1) cup (

119905

⋃

119896=1

1199091198961

1199091198962

)

(13)

Therefore 119873119866119905

(119909) cap 119866(119866119905minus 119910) = 119909

119905+1 which implies that

the upper bound stated in Theorem 22 is equal to 119889119866119905

(119910) +

|119909119905+1| = 2 Clearly 119887(119866

119905) = 2 and hence this bound is sharp

for 119866119905

From the graph 119866119905 we obtain the following statement

immediately

Proposition 24 For every integer 119905 ⩾ 2 there is a graph 119866such that the difference between any upper bound stated inTheorems 13ndash20 and the upper bound in Theorem 22 is equalto 119905

Although Theorem 22 supplies us with the upper boundthat is closer to 119887(119866) for some graph 119866 than what any one of

International Journal of Combinatorics 7

Theorems 13ndash20 provides it is not easy to determine the sets119860+

(119866) and 119861(119866) mentioned in Theorem 22 for an arbitrarygraph 119866 Thus the upper bound given in Theorem 22 is oftheoretical importance but not applied since until now wehave not found a new class of graphswhose bondage numbersare determined byTheorem 22

The above-mentioned upper bounds on the bondagenumber are involved in only degrees of two vertices Hartnelland Rall [11] established an upper bound on 119887(119866) in terms ofthe numbers of vertices and edges of 119866 with order 119899 For anyconnected graph 119866 let 120575(119866) represent the average degree ofvertices in 119866 that is 120575(119866) = (1119899)sum

119909isin119881119889119866(119909) Hartnell and

Rall first discovered the following proposition

Proposition 25 For any connected graph 119866 there exist twovertices 119909 and 119910 with distance at most two and with theproperty that 119889

119866(119909) + 119889

119866(119910) ⩽ 2120575(119866)

Using Proposition 25 and Theorem 16 Hartnell and Rallgave the following bound

Theorem 26 (Hartnell and Rall [11] 1999) 119887(119866) ⩽ 2120575(119866) minus 1for any connected graph 119866

Note that 2120576 = 119899120575(119866) for any graph 119866 with order 119899 andsize 120576 Theorem 26 implies the following bound in terms oforder 119899 and size 120576

Corollary 27 119887(119866) ⩽ (4120576119899) minus 1 for any connected graph 119866with order 119899 and size 120576

A lower bound on 120576 in terms of order 119899 and the bondagenumber is obtained from Corollary 28 immediately

Corollary 28 120576 ⩾ (1198994)(119887(119866) + 1) for any connected graph 119866with order 119899 and size 120576

Hartnell and Rall [11] gave some graphs 119866 with order 119899to show that for each value of 119887(119866) the lower bound on 120576(119866)given in the Corollary 28 is sharp for some values of 119899

32 Bounds Implied by Connectivity Use 120581(119866) and 120582(119866) todenote the vertex-connectivity and the edge-connectivity ofa connected graph 119866 respectively which are the minimumnumbers of vertices and edges whose removal result in 119866disconnected The famous Whitney inequality is stated as120581(119866) ⩽ 120582(119866) ⩽ 120575(119866) for any graph or digraph 119866 Corollary 15is improved by several authors as follows

Theorem 29 (Hartnell and Rall [17] 1994 Teschner [10]1997) If 119866 is a connected graph then 119887(119866) ⩽ 120582(119866)+Δ(119866)minus 1

The upper bound given in Theorem 29 can be attainedFor example a cycle 119862

3119896+1with 119896 ⩾ 1 119887(119862

3119896+1) = 3 by

Theorem 1 Since 120581(1198623119896+1) = 120582(119862

3119896+1) = 2 we have 120581(119862

3119896+1)+

120582(1198623119896+1) minus 1 = 2 + 2 minus 1 = 3 = 119887(119862

3119896+1)

Motivated byCorollary 15Theorems 29 and theWhitneyinequality Dunbar et al [21] naturally proposed the followingconjecture

Figure 4 A graph119867 with 120574(119867) = 3 and 119887(119867) = 5

Conjecture 30 If119866 is a connected graph then 119887(119866) ⩽ Δ(119866)+120581(119866) minus 1

However Liu and Sun [22] presented a counterexampleto this conjecture They first constructed a graph 119867 showedin Figure 4 with 120574(119867) = 3 and 119887(119867) = 5 Then let 119866 be thedisjoint union of two copies of119867 by identifying two verticesof degree two They proved that 119887(119866) ⩾ 5 Clearly 119866 is a 4-regular graph with 120581(119866) = 1 and 120582(119866) = 2 and so 119887(119866) ⩽ 5byTheorem 29 Thus 119887(119866) = 5 gt 4 = Δ(119866) + 120581(119866) minus 1

With a suspicion of the relationship between the bondagenumber and the vertex-connectivity of a graph the followingconjecture is proposed

Conjecture 31 (Liu and Sun [22] 2003) For any positiveinteger 119903 there exists a connected graph 119866 such that 119887(119866) ⩾Δ(119866) + 120581(119866) + 119903

To the knowledge of the author until now no results havebeen known about this conjecture

We conclude this subsection with the following remarksFromTheorem 29 if Conjecture 31 holds for some connectedgraph119866 then 120582(119866) gt 120581(119866)+119903 which implies that119866 is of largeedge-connectivity and small vertex-connectivity Use 120585(119866) todenote the minimum edge-degree of 119866 that is

120585 (119866) = min119909119910isin119864(119866)

119889119866(119909) + 119889

119866(119910) minus 2 (14)

Theorem 14 implies the following result

Proposition 32 119887(119866) ⩽ 120585(119866) + 1 for any graph 119866

Use 1205821015840(119866) to denote the restricted edge-connectivity ofa connected graph 119866 which is the minimum number ofedgeswhose removal result in119866 disconnected and no isolatedvertices

Proposition 33 (Esfahanian and Hakimi [23] 1988) If 119866 isneither 119870

1119899nor 119870

3 then 1205821015840(119866) ⩽ 120585(119866)

Combining Propositions 32 and 33 we propose a conjec-ture

Conjecture 34 If a connected 119866 is neither 1198701119899

nor 1198703 then

119887(119866) ⩽ 120575(119866) + 1205821015840

(119866) minus 1

8 International Journal of Combinatorics

A cycle 1198623119896+1

also satisfies 119887(1198623119896+1) = 3 = 120575(119862

3119896+1) +

1205821015840

(1198623119896+1) minus 1 since 1205821015840(119862

3119896+1) = 120575(119862

3119896+1) = 2 for any integer

119896 ⩾ 1For the graph119867 shown in Figure 41205821015840(119867) = 4 and 120575(119867) =

2 and so 119887(119867) = 5 = 120575(119867) + 1205821015840(119867) minus 1For the 4-regular graph119866 constructed by Liu and Sun [22]

obtained from the disjoint union of two copies of the graph119867 showed in Figure 4 by identifying two vertices of degreetwo we have 119887(119866) ⩾ 5 Clearly 1205821015840(119866) = 2 Thus 119887(119866) = 5 =120575(119866) + 120582

1015840

(119866) minus 1For the 4-regular graph 119866

119905constructed by Samodivkin

[20] see Figure 3 for 119905 = 2 119887(119866119905) = 2 Clearly 120575(119866

119905) = 1

and 1205821015840(119866) = 2 Thus 119887(119866) = 2 = 120575(119866) + 1205821015840(119866) minus 1These examples show that if Conjecture 34 is true then the

given upper bound is tight

33 Bounds Implied by Degree Sequence Now let us return toTheorem 16 from which Teschner [10] obtained some otherbounds in terms of the degree sequencesThe degree sequence120587(119866) of a graph 119866 with vertex-set 119881 = 119909

1 119909

2 119909

119899 is the

sequence 120587 = (1198891 119889

2 119889

119899) with 119889

1⩽ 119889

2⩽ sdot sdot sdot ⩽ 119889

119899 where

119889119894= 119889

119866(119909

119894) for each 119894 = 1 2 119899 The following result is

essentially a corollary of Theorem 16

Theorem35 (Teschner [10] 1997) Let119866 be a nonempty graphwith degree sequence120587(119866) If120572

2(119866) = 119905 then 119887(119866) ⩽ 119889

119905+119889

119905+1minus

1

CombiningTheorem 35with Proposition 10 (ie 1205722(119866) ⩽

120574(119866)) we have the following corollary

Corollary 36 (Teschner [10] 1997) Let 119866 be a nonemptygraph with the degree sequence 120587(119866) If 120574(119866) = 120574 then 119887(119866) ⩽119889120574+ 119889

120574+1minus 1

In [10] Teschner showed that these two bounds are sharpfor arbitrarily many graphs Let119867 = 119862

3119896+1+ 119909

11199094 119909

11199093119896minus1

where 1198623119896+1

is a cycle (1199091 119909

2 119909

3119896+1 119909

1) for any integer

119896 ⩾ 2 Then 120574(119867) = 119896 + 1 and so 119887(119867) ⩽ 2 + 2 minus 1 = 3by Corollary 36 Since119862

3119896+1is a spanning subgraph of119867 and

120574(119866) = 120574(119867) Observation 1 yields that 119887(119867) ⩾ 119887(1198623119896+1) = 3

and so 119887(119867) = 3Although various upper bounds have been established

as the above we find that the appearance of these boundsis essentially based upon the local structures of a graphprecisely speaking the structures of the neighborhoods oftwo vertices within distance 2 Even if these bounds can beachieved by some special graphs it is more often not the caseThe reason lies essentially in the definition of the bondagenumber which is theminimumvalue among all bondage setsan integral property of a graph While it easy to find upperbounds just by choosing some bondage set the gap betweenthe exact value of the bondage number and such a boundobtained only from local structures of a graph is often largeFor example a star 119870

1Δ however large Δ is 119887(119870

1Δ) = 1

Therefore one has been longing for better bounds upon someintegral parameters However as what we will see below it isdifficult to establish such upper bounds

34 Bounds in 120574-Critical Graphs A graph119866 is called a vertexdomination-critical graph (vc-graph or 120574-critical for short) if120574(119866 minus 119909) lt 120574(119866) for any 119909 isin 119881(119866) proposed by Brighamet al [24] in 1988 Several families of graphs are known to be120574-critical From definition it is clear that if 119866 is a 120574-criticalgraph then 120574(119866) ⩾ 2The class of 120574-critical graphs with 120574 = 2is characterized as follows

Proposition 37 (Brigham et al [24] 1988) A graph 119866 with120574(119866) = 2 is a 120574-critical graph if and only if 119866 is a completegraph 119870

2119905(119905 ⩾ 2) with a perfect matching removed

A more interesting family is composed of the 119899-criticalgraphs 119866

119898119899defined for 119898 119899 ⩾ 2 by the circulant undirected

graph 119866(119873 plusmn119878) where 119873 = (119898 + 1)(119899 minus 1) + 1 and119878 = 1 2 lfloor1198982rfloor in which 119881(119866(119873 plusmn119878)) = 119873 and119864(119866(119873 plusmn119878)) = 119894119895 |119895 minus 119894| equiv 119904 (mod 119873) 119904 isin 119878

The reason why 120574-critical graphs are of special interest inthis context is easy to see that they play an important role instudy of the bondage number For instance it immediatelyfollows from Theorem 13 that if 119887(119866) gt Δ(119866) then 119866 is a 120574-critical graphThe 120574-critical graphs are defined exactly in thisway In order to find graphs 119866 with a high bondage number(ie higher than Δ(119866)) and beyond its general upper boundsfor the bondage number we therefore have to look at 120574-critical graphs

The bondage numbers of some 120574-critical graphs havebeen examined by several authors see for example [1020 25 26] From Theorem 13 we know that the bondagenumber of a graph 119866 is bounded from above by Δ(119866)if 119866 is not a 120574-critical graph For 120574-critical graphs it ismore difficult to find an upper bound We will see that thebondage numbers of 120574-critical graphs in general are noteven bounded from above by Δ + 119888 for any fixed naturalnumber 119888

In this subsection we introduce some upper bounds forthe bondage number of a 120574-critical graph By Proposition 37we easily obtain the following result

Theorem 38 If 119866 is a 120574-critical graph with 120574(119866) = 2 then119887(119866) ⩽ Δ + 1

In Section 4 by Theorem 59 we will see that the equalityin Theorem 38 holds that is 119887(119866) = Δ + 1 if 119866 is a 120574-criticalgraph with 120574(119866) = 2

Theorem 39 (Teschner [10] 1997) Let119866 be a 120574-critical graphwith degree sequence 120587(119866) Then 119887(119866) ⩽ maxΔ(119866) + 1 119889

1+

1198892+ sdot sdot sdot + 119889

120574minus1 where 120574 = 120574(119866)

As we mentioned above if 119866 is a 120574-critical graph with120574(119866) = 2 then 119887(119866) = Δ + 1 which shows that thebound given in Theorem 39 can be attained for 120574 = 2However we have not known whether this bound is tightfor general 120574 ⩾ 3 Theorem 39 gives the following corollaryimmediately

Corollary 40 (Teschner [10] 1997) Let 119866 be a 120574-criticalgraph with degree sequence 120587(119866) If 120574(119866) = 3 then 119887(119866) ⩽maxΔ(119866) + 1 120575(119866) + 119889

2

International Journal of Combinatorics 9

From Theorem 38 and Corollary 40 we have 119887(119866) ⩽2Δ(119866) if 119866 is a 120574-critical graph with 120574(119866) ⩽ 3 The followingresult shows that this bound is not tight

Theorem 41 (Teschner [26] 1995) Let119866 be a 120574-critical graphgraph with 120574(119866) ⩽ 3 Then 119887(119866) ⩽ (32)Δ(119866)

Until now we have not known whether the bound givenin Theorem 41 is tight or not We state two conjectures on120574-critical graphs proposed by Samodivkin [20] The first ofthem is motivated byTheorems 22 and 19

Conjecture 42 (Samodivkin [20] 2008) For every connectednontrivial 120574-critical graph 119866

min119909isin119860+(119866)119910isin119861(119866minus119909)

119889119866(119909) +

1003816100381610038161003816119873119866(119910) cap 119860 (119866 minus 119909)

1003816100381610038161003816 ⩽3

2Δ (119866)

(15)

To state the second conjecture we need the followingresult on 120574-critical graphs

Proposition 43 If119866 is a 120574-critical graph then 120592(119866) ⩽ (Δ(119866)+1)(120574(119866)minus1)+1 moreover if the equality holds then119866 is regular

The upper bound in Proposition 43 is sharp in thesense that equality holds for the infinite class of 120574-criticalgraphs 119866

119898119899defined in the beginning of this subsection In

Proposition 43 the first result is due to Brigham et al [24] in1988 the second is due to Fulman et al [27] in 1995

For a 120574-critical graph 119866 with 120574 = 3 by Proposition 43120592(119866) ⩽ 2Δ(119866) + 3

Theorem 44 (Teschner [26] 1995) If 119866 is a 120574-critical graphwith 120574 = 3 and 120592(119866) = 2Δ(119866) + 3 then 119887(119866) ⩽ Δ + 1

We have not known yet whether the equality inTheorem 44holds or notHowever Samodivkin proposed thefollowing conjecture

Conjecture 45 (Samodivkin [20] 2008) If 119866 is a 120574-criticalgraph with (Δ(119866)+1)(120574minus1)+1 vertices then 119887(119866) = Δ(119866)+1

In general based on Theorem 41 Teschner proposed thefollowing conjecture

Conjecture 46 (Teschner [26] 1995) If119866 is a 120574-critical graphthen 119887(119866) ⩽ (32)Δ(119866) for 120574 ⩾ 4

We conclude this subsection with some remarks Graphswhich are minimal or critical with respect to a given prop-erty or parameter frequently play an important role in theinvestigation of that property or parameter Not only are suchgraphs of considerable interest in their own right but also aknowledge of their structure often aids in the developmentof the general theory In particular when investigating anyfinite structure a great number of results are proven byinduction Consequently it is desirable to learn as much aspossible about those graphs that are critical with respect toa given property or parameter so as to aid and abet suchinvestigations

In this subsection we survey some results on the bondagenumber for 120574-critical graphs Although these results are notvery perfect they provides a feasible method to approach thebondage number from different viewpoints In particular themethods given in Teschner [26] worth further explorationand development

The following proposition is maybe useful for us tofurther investigate the bondage number of a 120574-critical graph

Proposition 47 (Brigham et al [24] 1988) If 119866 has anonisolated vertex 119909 such that the subgraph induced by119873

119866(119909)

is a complete graph then 119866 is not 120574-critical

This simple fact shows that any 120574-critical graph containsno vertices of degree one

35 Bounds Implied by Domination In the preceding sub-section we introduced some upper bounds on the bondagenumbers for 120574-critical graphs by consideration of domina-tions In this subsection we introduce related results for ageneral graph with given domination number

Theorem 48 (Fink et al [8] 1990) For any connected graph119866 of order 119899 (⩾2)

(a) 119887(119866) ⩽ 119899 minus 1

(b) 119887(119866) ⩽ Δ(119866) + 1 if 120574(119866) = 2

(c) 119887(119866) ⩽ 119899 minus 120574(119866) + 1

While the upper bound of 119899minus1 on 119887(119866) is not particularlygood for many classes of graphs (eg trees and cycles) it isan attainable bound For example if 119866 is a complete 119905-partitegraph 119870

119905(2) for 119905 = 1198992 and 119899 ⩽ 4 an even integer then the

three bounds on 119887(119866) in Theorem 48 are sharpTeschner [26] consider a graphwith 120574(119866) = 1 or 120574(119866) = 2

The next result is almost trivial but useful

Lemma 49 Let 119866 be a graph with order 119899 and 120574(119866) = 1 119905 thenumber of vertices of degree 119899 minus 1 Then 119887(119866) = lceil1199052rceil

Since 119905 ⩽ Δ(119866) clearly Lemma 49 yields the followingresult immediately

Theorem 50 (Teschner [26] 1995) 119887(119866) ⩽ (12)Δ+1 for anygraph 119866 with 120574(119866) = 1

For a complete graph1198702119896+1

119887(1198702119896+1) = 119896+1 = (12)Δ+1

which shows that the upper bound given in Theorem 50 canbe attained For a graph119866with 120574(119866) = 2 byTheorem 59 laterthe upper bound given inTheorem 48 (b) can be also attainedby a 2-critical graph (see Proposition 37)

36 Two Conjectures In 1990 when Fink et al [8] introducedthe concept of the bondage number they proposed thefollowing conjecture

Conjecture 51 119887(119866) ⩽ Δ(119866) + 1 for any nonempty graph 119866

10 International Journal of Combinatorics

Although some results partially support Conjecture 51Teschner [28] in 1993 found a counterexample to this con-jecture the cartesian product 119870

3times 119870

3 which shows that

119887(119866) = Δ(119866) + 2If a graph 119866 is a counterexample to Conjecture 51 it

must be a 120574-critical graph by Theorem 13 That is why thevertex domination-critical graphs are of special interest in theliterature

Now we return to Conjecture 51 Hartnell and Rall [17]and Teschner [29] independently proved that 119887(119866) can bemuch greater than Δ(119866) by showing the following result

Theorem 52 (Hartnell and Rall [17] 1994 Teschner [29]1996) For an integer 119899 ⩾ 3 let 119866

119899be the cartesian product

119870119899times 119870

119899 Then 119887(119866

119899) = 3(119899 minus 1) = (32)Δ(119866

119899)

This theorem shows that there exist no upper boundsof the form 119887(119866) ⩽ Δ(119866) + 119888 for any integer 119888 Teschner[26] proved that 119887(119866) ⩽ (32)Δ(119866) for any graph 119866 with120574(119866) ⩽ 2 (see Theorems 50 and 48) and for 120574-critical graphswith 120574(119866) = 3 and proposed the following conjecture

Conjecture 53 (Teschner [26] 1995) 119887(119866) ⩽ (32)Δ(119866) forany graph 119866

We believe that this conjecture is true but so far no muchwork about it has been known

4 Lower Bounds

Since the bondage number is defined as the smallest numberof edges whose removal results in an increase of dominationnumber each constructive method that creates a concretebondage set leads to an upper bound on the bondage numberFor that reason it is hard to find lower bounds Neverthelessthere are still a few lower bounds

41 Bounds Implied by Subgraphs The first lower boundon the bondage number is gotten in terms of its spanningsubgraph

Theorem 54 (Teschner [10] 1997) Let 119867 be a spanningsubgraph of a nonempty graph119866 If 120574(119867) = 120574(119866) then 119887(119867) ⩽119887(119866)

By Theorem 1 119887(119862119899) = 3 if 119899 equiv 1 (mod 3) and 119887(119862

119899) =

2 otherwise 119887(119875119899) = 2 if 119899 equiv 1 (mod 3) and 119887(119875

119899) = 1

otherwise From these results and Theorem 54 we get thefollowing two corollaries

Corollary 55 If119866 is hamiltonianwith order 119899 ⩾ 3 and 120574(119866) =lceil1198993rceil then 119887(119866) ⩾ 2 and in addition 119887(119866) ⩾ 3 if 119899 equiv 1 (mod3)

Corollary 56 If 119866 with order 119899 ⩾ 2 has a hamiltonian pathand 120574(119866) = lceil1198993rceil then 119887(119866) ⩾ 2 if 119899 equiv 1 (mod 3)

42 Other Bounds The vertex covering number 120573(119866) of 119866 isthe minimum number of vertices that are incident with all

Figure 5 A 120574-critical graph with 120573 = 120574 = 4 120575 = 2 and 119887 = 3

edges in 119866 If 119866 has no isolated vertices then 120574(119866) ⩽ 120573(119866)clearly In [30] Volkmann gave a lot of graphs with 120573 = 120574

Theorem 57 (Teschner [10] 1997) Let119866 be a graph If 120574(119866) =120573(119866) then

(a) 119887(119866) ⩾ 120575(119866)

(b) 119887(119866) ⩾ 120575(119866) + 1 if 119866 is a 120574-critical graph

The graph shown in Figure 5 shows that the bound givenin Theorem 57(b) is sharp For the graph 119866 it is a 120574-criticalgraph and 120574(119866) = 120573(119866) = 4 By Theorem 57 we have 119887(119866) ⩾3 On the other hand 119887(119866) ⩽ 3 by Theorem 16 Thus 119887(119866) =3

Proposition 58 (Sanchis [31] 1991) Let 119866 be a graph 119866 oforder 119899 (⩾ 6) and size 120576 If 119866 has no isolated vertices and3 ⩽ 120574(119866) ⩽ 1198992 then 120576 ⩽ ( 119899minus120574(119866)+1

2)

Using the idea in the proof of Theorem 57 every upperbound for 120574(119866) can lead to a lower bound for 119887(119866) Inthis way Teschner [10] obtained another lower bound fromProposition 58

Theorem59 (Teschner [10] 1997) Let119866 be a graph119866 of order119899 (⩾6) and size 120576 If 2 ⩽ 120574(119866) ⩽ 1198992 minus 1 then

(a) 119887(119866) ⩾ min120575(119866) 120576 minus ( 119899minus120574(119866)2)

(b) 119887(119866) ⩾ min120575(119866) + 1 120576 minus ( 119899minus120574(119866)2) if 119866 is a 120574-critical

graph

The lower bound in Theorem 59(b) is sharp for a classof 120574-critical graphs with domination number 2 Let 119866 be thegraph obtained from complete graph119870

2119905(119905 ⩾ 2) by removing

a perfect matching By Proposition 37 119866 is a 120574-critical graphwith 120574(119866) = 2 Then 119887(119866) ⩾ 120575(119866) + 1 = Δ(119866) + 1 byTheorem 59 and 119887(119866) ⩽ Δ(119866) + 1 by Theorem 38 and so119887(119866) = Δ(119866) + 1 = 2119905 minus 1

As far as the author knows there are no more lowerbounds in the present literature In view of applications ofthe bondage number a network is vulnerable if its bondagenumber is small while it is stable if its bondage number islargeTherefore better lower bounds let us learn better aboutthe stability of networks from this point of view Thus it isof great significance to seek more lower bounds for variousclasses of graphs

International Journal of Combinatorics 11

5 Results on Graph Operations

Generally speaking it is quite difficult to determine the exactvalue of the bondage number for a given graph since itstrongly depends on the dominating number of the graphThus determining bondage numbers for some special graphsis interesting if the dominating numbers of those graphs areknown or can be easily determined In this section we willintroduce some known results on bondage numbers for somespecial classes of graphs obtained by some graph operations

51 Cartesian Product Graphs Let 1198661= (119881

1 119864

1) and 119866

2=

(1198812 119864

2) be two graphs The Cartesian product of 119866

1and 119866

2

is an undirected graph denoted by 1198661times 119866

2 where 119881(119866

1times

1198662) = 119881

1times 119881

2 two distinct vertices 119909

11199092and 119910

11199102 where

1199091 119910

1isin 119881(119866

1) and 119909

2 119910

2isin 119881(119866

2) are linked by an edge in

1198661times 119866

2if and only if either 119909

1= 119910

1and 119909

21199102isin 119864(119866

2) or

1199092= 119910

2and 119909

11199101isin 119864(119866

1) The Cartesian product is a very

effective method for constructing a larger graph from severalspecified small graphs

Theorem 60 (Dunbar et al [21] 1998) For 119899 ⩾ 3

119887 (119862119899times 119870

2) =

2 119894119891 119899 equiv 0 119900119903 1 (mod 4) 3 119894119891 119899 equiv 3 (mod 4) 4 119894119891 119899 equiv 2 (mod 4)

(16)

For the Cartesian product 119862119899times 119862

119898of two cycles 119862

119899and

119862119898 where 119899 ⩾ 4 and 119898 ⩾ 3 Klavzar and Seifter [32]

determined 120574(119862119899times 119862

119898) for some 119899 and 119898 For example

120574(119862119899times119862

4) = 119899 for 119899 ⩾ 3 and 120574(119862

119899times119862

3) = 119899minuslfloor1198994rfloor for 119899 ⩾ 4

Kim [33] determined 119887(1198623times 119862

3) = 6 and 119887(119862

4times 119862

4) = 4 For

a general 119899 ⩾ 4 the exact values of the bondage numbers of119862119899times 119862

3and 119862

119899times 119862

4are determined as follows

Theorem 61 (Sohn et al [34] 2007) For 119899 ⩾ 4

119887 (119862119899times 119862

3) =

2 119894119891 119899 equiv 0 (mod 4) 4 119894119891 119899 equiv 1 119900119903 2 (mod 4) 5 119894119891 119899 equiv 3 (mod 4)

(17)

Theorem62 (Kang et al [35] 2005) 119887(119862119899times119862

4) = 4 for 119899 ⩾ 4

For larger 119898 and 119899 Huang and Xu [36] obtained thefollowing result see Theorem 197 for more details

Theorem 63 (Huang and Xu [36] 2008) 119887(1198625119894times119862

5119895) = 3 for

any positive integers 119894 and 119895

Xiang et al [37] determined that for 119899 ⩾ 5

119887 (119862119899times 119862

5)

= 3 if 119899 equiv 0 or 1 (mod 5) = 4 if 119899 equiv 2 or 4 (mod 5) ⩽ 7 if 119899 equiv 3 (mod 5)

(18)

For the Cartesian product 119875119899times119875

119898of two paths 119875

119899and 119875

119898

Jacobson and Kinch [38] determined 120574(119875119899times119875

2) = lceil(119899+1)2rceil

120574(119875119899times 119875

3) = 119899 minus lfloor(119899 minus 1)4rfloor and 120574(119875

119899times 119875

4) = 119899 + 1 if 119899 =

1 2 3 5 6 9 and 119899 otherwiseThe bondage number of119875119899times119875

119898

for 119899 ⩾ 4 and 2 ⩽ 119898 ⩽ 4 is determined as follows (Li [39] alsodetermined 119887(119875

119899times 119875

2))

Theorem 64 (Hu and Xu [40] 2012) For 119899 ⩾ 4

119887 (119875119899times 119875

2) =

1 119894119891 119899 equiv 1 (mod 2)2 119894119891 119899 equiv 0 (mod 2)

119887 (119875119899times 119875

3) =

1 119894119891 119899 equiv 1 119900119903 2 (mod 4)2 119894119891 119899 equiv 0 119900119903 3 (mod 4)

119887 (119875119899times 119875

4) = 1 119891119900119903 119899 ⩾ 14

(19)

From the proof of Theorem 64 we find that if 119875119899=

0 1 119899 minus 1 and 119875119898= 0 1 119898 minus 1 then the removal

of the vertex (0 0) in 119875119899times119875

119898does not change the domination

number If119898 increases the effect of (0 0) for the dominationnumber will be smaller and smaller from the probabilityTherefore we expect it is possible that 120574(119875

119899times 119875

119898minus (0 0)) =

120574(119875119899times 119875

119898) for119898 ⩾ 5 and give the following conjecture

Conjecture 65 119887(119875119899times 119875

119898) ⩽ 2 for119898 ⩾ 5 and 119899 ⩾ 4

52 Block Graphs and Cactus Graphs In this subsection weintroduce some results for corona graphs block graphs andcactus graphs

The corona1198661∘ 119866

2 proposed by Frucht and Harary [41]

is the graph formed from a copy of 1198661and 120592(119866

1) copies of

1198662by joining the 119894th vertex of 119866

1to the 119894th copy of 119866

2 In

particular we are concernedwith the corona119867∘1198701 the graph

formed by adding a new vertex V119894 and a new edge 119906

119894V119894for

every vertex 119906119894in119867

Theorem 66 (Carlson and Develin [42] 2006) 119887(119867 ∘ 1198701) =

120575(119867) + 1

This is a very important result which is often used toconstruct a graph with required bondage number In otherwords this result implies that for any given positive integer 119887there is a graph 119866 such that 119887(119866) = 119887

A block graph is a graph whose blocks are completegraphs Each block in a cactus graph is either a cycle or a119870

2

If each block of a graph is either a complete graph or a cyclethen we call this graph a block-cactus graph

Theorem67 (Teschner [43] 1997) 119887(119866) le Δ(119866) for any blockgraph 119866

Teschner [43] characterized all block graphs with 119887(119866) =Δ(119866) In the same paper Teschner found that 120574-criticalgraphs are instrumental in determining bounds for thebondage number of cactus and block graphs and obtained thefollowing result

Theorem 68 (Teschner [43] 1997) 119887(119866) ⩽ 3 for anynontrivial cactus graph 119866

This bound can be achieved by 119867 ∘ 1198701where 119867 is

a nontrivial cactus graph without vertices of degree one

12 International Journal of Combinatorics

by Theorem 66 In 1998 Dunbar et al [21] proposed thefollowing problem

Problem 2 Characterize all cactus graphs with bondagenumber 3

Some upper bounds for block-cactus graphs were alsoobtained

Theorem 69 (Dunbar et al [21] 1998) Let 119866 be a connectedblock-cactus graph with at least two blocksThen 119887(119866) ⩽ Δ(119866)

Theorem 70 (Dunbar et al [21] 1998) Let 119866 be a connectedblock-cactus graph which is neither a cactus graph nor a blockgraph Then 119887(119866) ⩽ Δ(119866) minus 1

53 Edge-Deletion and Edge-Contraction In order to obtainfurther results of the bondage number we may consider howthe bondage number changes under some operations of agraph 119866 which remain the domination number unchangedor preserve some property of 119866 such as planarity Twosimple operations satisfying these requirements are the edge-deletion and the edge-contraction

Theorem 71 (Huang and Xu [44] 2012) Let 119866 be any graphand 119890 isin 119864(119866) Then 119887(119866 minus 119890) ⩾ 119887(119866) minus 1 Moreover 119887(119866 minus 119890) ⩽119887(119866) if 120574(119866 minus 119890) = 120574(119866)

FromTheorem 1 119887(119862119899minus 119890) = 119887(119875

119899) = 119887(119862

119899) minus 1 for any 119890

in119862119899 which shows that the lower bound on 119887(119866minus119890) is sharp

The upper bound can reach for any graph 119866 with 119887(119866) ⩾ 2Next we consider the effect of the edge-contraction on

the bondage number Given a graph 119866 the contraction of 119866by the edge 119890 = 119909119910 denoted by 119866119909119910 is the graph obtainedfrom 119866 by contracting two vertices 119909 and 119910 to a new vertexand then deleting all multiedges It is easy to observe that forany graph 119866 120574(119866) minus 1 ⩽ 120574(119866119909119910) ⩽ 120574(119866) for any edge 119909119910 of119866

Theorem 72 (Huang and Xu [44] 2012) Let 119866 be any graphand 119909119910 be any edge in119866 If119873

119866(119909)cap119873

119866(119910) = 0 and 120574(119866119909119910) =

120574(119866) then 119887(119866119909119910) ⩾ 119887(119866)

We can use examples 119862119899and 119870

119899to show that the

conditions ofTheorem 72 are necessary and the lower boundinTheorem 72 is tight Clearly 120574(119862

119899) = lceil1198993rceil and 120574(119870

119899) = 1

for any edge 119909119910 119862119899119909119910 = 119862

119899minus1and 119870

119899119909119910 = 119870

119899minus1 By

Theorem 1 if 119899 equiv 1 (mod 3) then 120574(119862119899119909119910) lt 120574(119866) and

119887(119862119899119909119910) = 2 lt 3 = 119887(119862

119899) Thus the result in Theorem 72

is generally invalid without the hypothesis 120574(119866119909119910) = 120574(119866)Furthermore the condition 119873

119866(119909) cap 119873

119866(119910) = 0 cannot be

omitted even if 120574(119866119909119910) = 120574(119866) since for odd 119899 119887(119870119899119909119910) =

lceil(119899 minus 1)2rceil lt lceil1198992rceil = 119887(119870119899) by Theorem 1

On the other hand 119887(119862119899119909119910) = 119887(119862

119899) = 2 if 119899 equiv 0 (mod

3) and 119887(119870119899119909119910) = 119887(119870

119899) if 119899 is even which shows that the

equality in 119887(119866119909119910) ⩾ 119887(119866) may hold Thus the bound inTheorem 72 is tight However provided all the conditions119887(119866119909119910) can be arbitrarily larger than 119887(119866) Given a graph119867let119866 be the graph formed from119867∘119870

1by adding a new vertex

119909 and joining it to an vertex 119910 of degree one in119867 ∘ 1198701 Then

119866119909119910 = 119867 ∘ 1198701 120574(119866) = 120574(119866119909119910) and 119873

119866(119909) cap 119873

119866(119910) = 0

But 119887(119866) = 1 since 120574(119866 minus 119909119910) = 120574(119866119909119910) + 1 and 119887(119866119909119910) =120575(119867) + 1 byTheorem 66The gap between 119887(119866) and 119887(119866119909119910)is 120575(119867)

6 Results on Planar Graphs

From Section 2 we have seen that the bondage numberfor a tree has been completely solved Moreover a lineartime algorithm to compute the bondage number of a tree isdesigned by Hartnell et al [15] It is quite natural to considerthe bondage number for a planar graph In this section westate some results and problems on the bondage number fora planar graph

61 Conjecture on Bondage Number As mentioned inSection 3 the bondage number can be much larger than themaximum degree But for a planar graph 119866 the bondagenumber 119887(119866) cannot exceed Δ(119866) too much It is clear that119887(119866) ⩽ Δ(119866) + 4 by Corollary 15 since 120575(119866) ⩽ 5 for anyplanar graph119866 In 1998Dunbar et al [21] posed the followingconjecture

Conjecture 73 If 119866 is a planar graph then 119887(119866) ⩽ Δ(119866) + 1

Because of its attraction it immediately became the focusof attention when this conjecture is proposed In fact themain aim concerning the research of the bondage number fora planar graph is focused on this conjecture

It has been mentioned in Theorems 1 and 60 that119887(119862

3119896+1) = 3 = Δ + 1 and 119887(119862

4119896+2times 119870

2) = 4 = Δ + 1 It

is known that 119887(1198706minus 119872) = 5 = Δ + 1 where119872 is a perfect

matching of the complete graph1198706These examples show that

if Conjecture 73 is true then the upper bound is sharp for2 ⩽ Δ ⩽ 4

Here we note that it is sufficient to prove this conjecturefor connected planar graphs since the bondage number of adisconnected graph is simply the minimum of the bondagenumbers of its components

The first paper attacking this conjecture is due to Kangand Yuan [45] which confirmed the conjecture for everyconnected planar graph 119866 with Δ ⩾ 7 The proofs are mainlybased onTheorems 16 and 18

Theorem 74 (Kang and Yuan [45] 2000) If 119866 is a connectedplanar graph then 119887(119866) ⩽ min8 Δ(119866) + 2

Obviously in view of Theorem 74 Conjecture 73 is truefor any connected planar graph with Δ ⩾ 7

62 Bounds Implied by Degree Conditions As we have seenfrom Theorem 74 to attack Conjecture 73 we only need toconsider connected planar graphs with maximum Δ ⩽ 6Thus studying the bondage number of planar graphs bydegree conditions is of significanceThe first result on boundsimplied by degree conditions is obtained by Kang and Yuan[45]

International Journal of Combinatorics 13

Theorem 75 (Kang and Yuan [45] 2000) If 119866 is a connectedplanar graph without vertices of degree 5 then 119887(119866) ⩽ 7

Fischermann et al [46] generalized Theorem 75 as fol-lows

Theorem 76 (Fischermann et al [46] 2003) Let 119866 be aconnected planar graph and 119883 the set of vertices of degree 5which have distance at least 3 to vertices of degrees 1 2 and3 If all vertices in 119883 not adjacent with vertices of degree 4are independent and not adjacent to vertices of degree 6 then119887(119866) ⩽ 7

Clearly if119866 has no vertices of degree 5 then119883 = 0 ThusTheorem 76 yields Theorem 75

Use 119899119894to denote the number of vertices of degree 119894 in 119866

for each 119894 = 1 2 Δ(119866) Using Theorem 18 Fischermannet al obtained the following two theorems

Theorem 77 (Fischermann et al [46] 2003) For any con-nected planar graph 119866

(1) 119887(119866) ⩽ 7 if 1198995lt 2119899

2+ 3119899

3+ 2119899

4+ 12

(2) 119887(119866) ⩽ 6 if 119866 contains no vertices of degrees 4 and 5

Theorem 78 (Fischermann et al [46] 2003) For any con-nected planar graph 119866 119887(119866) ⩽ Δ(119866) + 1 if

(1) Δ(119866) = 6 and every edge 119890 = 119909119910 with 119889119866(119909) = 5 and

119889119866(119910) = 6 is contained in at most one triangle

(2) Δ(119866) = 5 and no triangle contains an edge 119890 = 119909119910 with119889119866(119909) = 5 and 4 ⩽ 119889

119866(119910) ⩽ 5

63 Bounds Implied by Girth Conditions The girth 119892(119866) of agraph 119866 is the length of the shortest cycle in 119866 If 119866 has nocycles we define 119892(119866) = infin

Combining Theorem 74 with Theorem 78 we find thatif a planar graph contains no triangles and has maximumdegree Δ ⩾ 5 then Conjecture 73 holds This fact motivatedFischermann et alrsquos [46] attempt to attack Conjecture 73 bygirth restraints

Theorem 79 (Fischermann et al [46] 2003) For any con-nected planar graph 119866

119887 (119866) ⩽

6 119894119891 119892 (119866) ⩾ 4

5 119894119891 119892 (119866) ⩾ 5

4 119894119891 119892 (119866) ⩾ 6

3 119894119891 119892 (119866) ⩾ 8

(20)

The first result in Theorem 79 shows that Conjecture 73is valid for all connected planar graphs with 119892(119866) ⩾ 4 andΔ(119866) ⩾ 5 It is easy to verify that the second result inTheorem 79 implies that Conjecture 73 is valid for all not3-regular graphs of girth 119892(119866) ⩾ 5 which is stated in thefollowing corollary

Corollary 80 For any connected planar graph 119866 if 119866 is not3-regular and 119892(119866) ⩾ 5 then 119887(119866) ⩽ Δ(119866) + 1

The first result in Theorem 79 also implies that if 119866 isa connected planar graph with no cycles of length 3 then119887(119866) ⩽ 6 Hou et al [47] improved this result as follows

Theorem 81 (Hou et al [47] 2011) For any connected planargraph 119866 if 119866 contains no cycles of length 119894 (4 ⩽ 119894 ⩽ 6) then119887(119866) ⩽ 6

Since 119887(1198623119896+1) = 3 for 119896 ⩾ 3 (see Theorem 1) the

last bound in Theorem 79 is tight Whether other bounds inTheorem 79 are tight remains open In 2003 Fischermann etal [46] posed the following conjecture

Conjecture 82 For any connected planar graph 119866 119887(119866) ⩽ 7and

119887 (119866) ⩽ 5 119894119891 119892 (119866) ⩾ 4

4 119894119891 119892 (119866) ⩾ 5(21)

We conclude this subsection with a question on bondagenumbers of planar graphs

Question 1 (Fischermann et al [46] 2003) Is there a planargraph 119866 with 6 ⩽ 119887(119866) ⩽ 8

In 2006 Carlson andDevelin [42] showed that the corona119866 = 119867 ∘ 119870

1for a planar graph 119867 with 120575(119867) = 5 has the

bondage number 119887(119866) = 120575(119867) + 1 = 6 (see Theorem 66)Since the minimum degree of planar graphs is at most 5then 119887(119866) can attach 6 If we take 119867 as the graph of theicosahedron then 119866 = 119867 ∘ 119870

1is such an example The

question for the existence of planar graphs with bondagenumber 7 or 8 remains open

64 Comments on the Conjectures Conjecture 73 is true forall connected planar graphs with minimum degree 120575 ⩽ 2 byTheorem 16 or maximum degree Δ ⩾ 7 by Theorem 74 ornot 120574-critical planar graphs by Theorem 13 Thus to attackConjecture 73 we only need to consider connected criticalplanar graphs with degree-restriction 3 ⩽ 120575 ⩽ Δ ⩽ 6

Recalling and anatomizing the proofs of all results men-tioned in the preceding subsections on the bondage numberfor connected planar graphs we find that the proofs of theseresults strongly depend upon Theorem 16 or Theorem 18 Inother words a basic way used in the proofs is to find twovertices 119909 and 119910 with distance at most two in a consideredplanar graph 119866 such that

119889119866(119909) + 119889

119866(119910) or 119889

119866(119909) + 119889

119866(119910) minus

1003816100381610038161003816119873119866(119909) cap 119873

119866(119910)1003816100381610038161003816

(22)

which bounds 119887(119866) is as small as possible Very recentlyHuang and Xu [44] have considered the following parameter

119861 (119866) = min119909119910isin119881(119866)

119889119909119910 1 ⩽ 119889

119866(119909 119910) ⩽ 2

cup 119889119909119910minus 119899

119909119910 119889 (119909 119910) = 1

(23)

where 119889119909119910= 119889

119866(119909) + 119889

119866(119910) minus 1 and 119899

119909119910= |119873

119866(119909) cap 119873

119866(119910)|

14 International Journal of Combinatorics

It follows fromTheorems 16 and 18 that

119887 (119866) ⩽ 119861 (119866) (24)

The proofs given in Theorems 74 and 79 indeed imply thefollowing stronger results

Theorem 83 If 119866 is a connected planar graph then

119861 (119866) ⩽

min 8 Δ (119866) + 26 119894119891 119892 (119866) ⩾ 4

5 119894119891 119892 (119866) ⩾ 5

4 119894119891 119892 (119866) ⩾ 6

3 119894119891 119892 (119866) ⩾ 8

(25)

Thus using Theorem 16 or Theorem 18 if we can proveConjectures 73 and 82 then we can prove the followingstatement

Statement If 119866 is a connected planar graph then

119861 (119866) ⩽

Δ (119866) + 1

7

5 if 119892 (119866) ⩾ 44 if 119892 (119866) ⩾ 5

(26)

It follows from (24) that Statement implies Conjectures 73and 82 However Huang and Xu [44] gave examples to showthat none of conclusions in Statement is true As a result theystated the following conclusion

Theorem 84 (Huang and Xu [44] 2012) It is not possible toprove Conjectures 73 and 82 if they are right using Theorems16 and 18

Therefore a new method is needed to prove or disprovethese conjectures At the present time one cannot prove theseconjectures and may consider to disprove them If one ofthese conjectures is invalid then there exists a minimumcounterexample 119866 with respect to 120592(119866) + 120576(119866) In order toobtain a minimum counterexample one may consider twosimple operations satisfying these requirements the edge-deletion and the edge-contractionwhich decrease 120592(119866)+120576(119866)and preserve planarity However applying Theorems 71 and72 Huang and Xu [44] presented the following results

Theorem 85 (Huang and Xu [44] 2012) It is not possible toconstruct minimum counterexamples to Conjectures 73 and 82by the operation of an edge-deletion or an edge-contraction

7 Results on Crossing Number Restraints

It is quite natural to generalize the known results on thebondage number for planar graphs to formore general graphsin terms of graph-theoretical parameters In this section weconsider graphs with crossing number restraints

The crossing number cr(119866) of 119866 is the smallest numberof pairwise intersections of its edges when 119866 is drawn in theplane If cr(119866) = 0 then 119866 is a planar graph

71 General Methods Use 119899119894to denote the number of vertices

of degree 119894 in119866 for 119894 = 1 2 Δ(119866) Huang and Xu obtainedthe following results

Theorem 86 (Huang and Xu [48] 2007) For any connectedgraph 119866

119887 (119866) ⩽

6 119894119891 119892 (119866) ⩾ 4 119886119899119889 2cr (119866) lt 121198861

5 119894119891 119892 (119866) ⩾ 5 119886119899119889 6cr (119866) lt 161198862

4 119894119891 119892 (119866) ⩾ 6 119886119899119889 4cr (119866) lt 141198863

3 119894119891 119892 (119866) ⩾ 8 119886119899119889 6cr (119866) lt 161198864

(27)

where 1198861= 119899

1+ 2119899

2+ 2119899

3+sum

Δ

119894=8(119894 minus 7)119899

119894+ 8 119886

2= 3119899

1+ 6119899

2+

51198993+sum

Δ

119894=7(3119894minus18)119899

119894+20 119886

3= 119899

1+2119899

2+sum

Δ

119894=6(2119894minus10)119899

119894+12

and 1198864= sum

Δ

119894=5(3119894 minus 12)119899

119894+ 16

Simplifying the conditions in Theorem 86 we obtain thefollowing corollaries immediately

Corollary 87 (Huang and Xu [48] 2007) For any connectedgraph 119866

119887 (119866) ⩽

6 119894119891 119892 (119866) ⩾ 4 119886119899119889 cr (119866) ⩽ 3

5 119894119891 119892 (119866) ⩾ 5 119886119899119889 cr (119866) ⩽ 4

4 119894119891 119892 (119866) ⩾ 6 119886119899119889 cr (119866) ⩽ 2

3 119894119891 119892 (119866) ⩾ 8 119886119899119889 cr (119866) ⩽ 2

(28)

Corollary 88 (Huang and Xu [48] 2007) For any connectedgraph 119866

(a) 119887(119866) ⩽ 6 if 119866 is not 4-regular cr(119866) = 4 and 119892(119866) ⩾ 4

(b) 119887(119866) ⩽ Δ(119866) + 1 if 119866 is not 3-regular cr(119866) ⩽ 4 and119892(119866) ⩾ 5

(c) 119887(119866) ⩽ 4 if 119866 is not 3-regular cr(119866) = 3 and 119892(119866) ⩾ 6

(d) 119887(119866) ⩽ 3 if cr(119866) = 3 119892(119866) ⩾ 8 and Δ(119866) ⩾ 5

These corollaries generalize some known results for pla-nar graphs For example Corollary 87 contains Theorem 79Corollary 88 (b) contains Corollary 80

Theorem 89 (Huang and Xu [48] 2007) For any connectedgraph 119866 if cr(119866) ⩽ 119899

3(119866) + 119899

4(119866) + 3 then 119887(119866) ⩽ 8

Corollary 90 (Huang and Xu [48] 2007) For any connectedgraph 119866 if cr(119866) ⩽ 3 then 119887(119866) ⩽ 8

Perhaps being unaware of this result in 2010 Ma et al[49] proved that 119887(119866) ⩽ 12 for any graph 119866 with cr(119866) = 1

Theorem91 (Huang and Xu [48] 2007) Let119866 be a connectedgraph and 119868 = 119909 isin 119881(119866) 119889

119866(119909) = 5 119889

119866(119909 119910) ⩾ 3 if

International Journal of Combinatorics 15

119889119866(119910) ⩽ 3 and 119889

119866(119910) = 4 for every 119910 isin 119873

119866(119909) If 119868 is

independent no vertices adjacent to vertices of degree 6 and

cr (119866) lt max5119899

3+ |119868| minus 2119899

4+ 28

117119899

3+ 40

16 (29)

then 119887(119866) ⩽ 7

Corollary 92 (Huang and Xu [48] 2007) Let 119866 be aconnected graph with cr(119866) ⩽ 2 If 119868 = 119909 isin 119881(119866) 119889

119866(119909) =

5 119889119866(119910 119909) ⩾ 3 if 119889

119866(119910) ⩽ 3 and 119889

119866(119910) = 4 for every 119910 isin

119873119866(119909) is independent and no vertices adjacent to vertices of

degree 6 then 119887(119866) ⩽ 7

Theorem93 (Huang andXu [48] 2007) Let119866 be a connectedgraph If 119866 satisfies

(a) 5cr(119866) + 1198995lt 2119899

2+ 3119899

3+ 2119899

4+ 12 or

(b) 7cr(119866) + 21198995lt 3119899

2+ 4119899

4+ 24

then 119887(119866) ⩽ 7

Proposition 94 (Huang and Xu [48] 2007) Let 119866 be aconnected graph with no vertices of degrees four and five Ifcr(119866) ⩽ 2 then 119887(119866) ⩽ 6

Theorem 95 (Huang and Xu [48] 2007) If 119866 is a connectedgraph with cr(119866) ⩽ 4 and not 4-regular when cr(119866) = 4 then119887(119866) ⩽ Δ(119866) + 2

The above results generalize some known results forplanar graphs For example Corollary 90 and Theorem 95contain Theorem 74 the first condition in Theorem 93 con-tains the second condition inTheorem 77

From Corollary 88 and Theorem 95 we suggest the fol-lowing questions

Question 2 Is there a

(a) 4-regular graph 119866 with cr(119866) = 4 such that 119887(119866) ⩾ 7

(b) 3-regular graph 119866 with cr ⩽ 4 and 119892(119866) ⩾ 5 such that119887(119866) = 5

(c) 3-regular graph 119866 with cr = 3 and 119892(119866) = 6 or 7 suchthat 119887(119866) = 5

72 Carlson-Develin Methods In this subsection we intro-duce an elegant method presented by Carlson and Develin[42] to obtain some upper bounds for the bondage numberof a graph

Suppose that 119866 is a connected graph We say that 119866 hasgenus 120588 if 119866 can be embedded in a surface 119878 with 120588 handlessuch that edges are pairwise disjoint except possibly for end-vertices Let119866 be an embedding of119866 in surface 119878 and let120601(119866)denote the number of regions in 119866 The boundary of everyregion contains at least three edges and every edge is on theboundary of atmost two regions (the two regions are identicalwhen 119890 is a cut-edge) For any edge 119890 of 119866 let 1199031

119866(119890) and 1199032

119866(119890)

be the numbers of edges comprising the regions in 119866 whichthe edge 119890 borders It is clear that every 119890 = 119909119910 isin 119864(119866)

1199032

119866(119890) ⩾ 119903

1

119866(119890) ⩾ 4 if 1003816100381610038161003816119873119866

(119909) cap 119873119866(119910)1003816100381610038161003816 = 0

1199032

119866(119890) ⩾ 4 119903

1

119866(119890) ⩾ 3 if 1003816100381610038161003816119873119866

(119909) cap 119873119866(119910)1003816100381610038161003816 = 1

1199032

119866(119890) ⩾ 119903

1

119866(119890) ⩾ 3 if 1003816100381610038161003816119873119866

(119909) cap 119873119866(119910)1003816100381610038161003816 ⩾ 2

(30)

Following Carlson and Develin [42] for any edge 119890 = 119909119910 of119866 we define

119863119866(119890) =

1

119889119866(119909)+

1

119889119866(119910)

+1

1199031

119866(119890)+

1

1199032

119866(119890)minus 1 (31)

By the well-known Eulerrsquos formula

120592 (119866) minus 120576 (119866) + 120601 (119866) = 2 minus 2120588 (119866) (32)

it is easy to see that

sum

119890isin119864(119866)

119863119866(119890) = 120592 (119866) minus 120576 (119866) + 120601 (119866) = 2 minus 2120588 (119866) (33)

If 119866 is a planar graph that is 120588(119866) = 0 then

sum

119890isin119864(119866)

119863119866(119890) = 120592 (119866) minus 120576 (119866) + 120601 (119866) = 2 (34)

Combining these formulas withTheorem 18 Carlson andDevelin [42] gave a simple and intuitive proof ofTheorem 74and obtained the following result

Theorem 96 (Carlson and Develin [42] 2006) Let 119866 be aconnected graph which can be embedded on a torus Then119887(119866) ⩽ Δ(119866) + 3

Recently Hou and Liu [50] improved Theorem 96 asfollows

Theorem 97 (Hou and Liu [50] 2012) Let 119866 be a connectedgraph which can be embedded on a torus Then 119887(119866) ⩽ 9Moreover if Δ(119866) = 6 then 119887(119866) ⩽ 8

By the way Cao et al [51] generalized the result inTheorem 79 to a connected graph 119866 that can be embeddedon a torus that is

119887 (119866) le

6 if 119892 (119866) ⩾ 4 and 119866 is not 4-regular5 if 119892 (119866) ⩾ 54 if 119892 (119866) ⩾ 6 and 119866 is not 3-regular3 if 119892 (119866) ⩾ 8

(35)

Several authors used this method to obtain some resultson the bondage number For example Fischermann et al[46] used this method to prove the second conclusion inTheorem 78 Recently Cao et al [52] have used thismethod todeal with more general graphs with small crossing numbersFirst they found the following property

Lemma 98 120575(119866) ⩽ 5 for any graph 119866 with cr(119866) ⩽ 5

16 International Journal of Combinatorics

This result implies that 119887(119866) ⩽ Δ(119866) + 4 by Corollary 15for any graph 119866 with cr(119866) ⩽ 5 Cao et al established thefollowing relation in terms of maximum planar subgraphs

Lemma 99 (Cao et al [52]) Let 119866 be a connected graph withcrossing number cr(119866) For any maximum planar subgraph119867of 119866

sum

119890isin119864(119867)

119863119866(119890) ⩾ 2 minus

2cr (119866)120575 (119866)

(36)

Using Carlson-Develin method and combiningLemma 98 with Lemma 99 Cao et al proved the followingresults

Theorem 100 (Cao et al [52]) For any connected graph 119866119887(119866) ⩽ Δ(119866) + 2 if 119866 satisfies one of the following conditions

(a) cr(119866) ⩽ 3(b) cr(119866) = 4 and 119866 is not 4-regular(c) cr(119866) = 5 and 119866 contains no vertices of degree 4

Theorem101 (Cao et al [52]) Let119866 be a connected graphwithΔ(119866) = 5 and cr(119866) ⩽ 4 If no triangles contain two vertices ofdegree 5 then 119887(119866) ⩽ 6 = Δ(119866) + 1

Theorem 102 (Cao et al [52]) Let 119866 be a connected graphwith Δ(119866) ⩾ 6 and cr(119866) ⩽ 3 If Δ(119866) ⩾ 7 or if Δ(119866) = 6120575(119866) = 3 and every edge 119890 = 119909119910 with 119889

119866(119909) = 5 and 119889

119866(119910) = 6

is contained in atmost one triangle then 119887(119866) ⩽ min8 Δ(119866)+1

Using Carlson-Develin method it can be proved that119887(119866) ⩽ Δ(119866) + 2 if cr(119866) ⩽ 3 (see the first conclusion inTheorem 100) and not yet proved that 119887(119866) ⩽ 8 if cr(119866) ⩽3 although it has been proved by using other method (seeCorollary 90)

By using Carlson-Develin method Samodivkin [25]obtained some results on the bondage number for graphswithsome given properties

Kim [53] showed that 119887(119866) ⩽ Δ(119866) + 2 for a connectedgraph 119866 with genus 1 Δ(119866) ⩽ 5 and having a toroidalembedding of which at least one region is not 4-sidedRecently Gagarin and Zverovich [54] further have extendedCarlson-Develin ideas to establish a nice upper bound forarbitrary graphs that can be embedded on orientable ornonorientable topological surface

Theorem 103 (Gagarin and Zverovich [54] 2012) Let 119866 bea graph embeddable on an orientable surface of genus ℎ and anonorientable surface of genus 119896 Then 119887(119866) ⩽ minΔ(119866)+ℎ+2 Δ(119866) + 119896 + 1

This result generalizes the corresponding upper boundsin Theorems 74 and 96 for any orientable or nonori-entable topological surface By investigating the proof ofTheorem 103 Huang [55] found that the issue of the ori-entability can be avoided by using the Euler characteristic120594(= 120592(119866) minus 120576(119866) + 120601(119866)) instead of the genera ℎ and 119896 the

relations are 120594 = 2minus2ℎ and 120594 = 2minus119896 To obtain the best resultfromTheorem 103 onewants ℎ and 119896 as small as possible thisis equivalent to having 120594 as large as possible

According toTheorem 103 if119866 is planar (ℎ = 0 120594 = 2) orcan be embedded on the real projective plane (119896 = 1 120594 = 1)then 119887(119866) ⩽ Δ(119866) + 2 In all other cases Huang [55] had thefollowing improvement for Theorem 103 the proof is basedon the technique developed byCarlson-Develin andGagarin-Zverovich and includes some elementary calculus as a newingredient mainly the intermediate-value theorem and themean-value theorem

Theorem 104 (Huang [55] 2012) Let 119866 be a graph embed-dable on a surface whose Euler characteristic 120594 is as large aspossible If 120594 ⩽ 0 then 119887(119866) ⩽ Δ(119866)+lfloor119903rfloor where 119903 is the largestreal root of the following cubic equation in 119911

1199113

+ 21199112

+ (6120594 minus 7) 119911 + 18120594 minus 24 = 0 (37)

In addition if 120594 decreases then 119903 increases

The following result is asymptotically equivalent toTheorem 104

Theorem 105 (Huang [55] 2012) Let 119866 be a graph embed-dable on a surface whose Euler characteristic 120594 is as large aspossible If 120594 ⩽ 0 then 119887(119866) lt Δ(119866) + radic12 minus 6120594 + 12 orequivalently 119887(119866) ⩽ Δ(119866) + lceilradic12 minus 6120594 minus 12rceil

Also Gagarin andZverovich [54] indicated that the upperbound in Theorem 103 can be improved for larger values ofthe genera ℎ and 119896 by adjusting the proofs and stated thefollowing general conjecture

Conjecture 106 (Gagarin and Zverovich [54] 2012) For aconnected graph G of orientable genus ℎ and nonorientablegenus 119896

119887 (119866) ⩽ min 119888ℎ 119888

1015840

119896 Δ (119866) + 119900 (ℎ) Δ (119866) + 119900 (119896) (38)

where 119888ℎand 1198881015840

119896are constants depending respectively on the

orientable and nonorientable genera of 119866

In the recent paper Gagarin and Zverovich [56] providedconstant upper bounds for the bondage number of graphs ontopological surfaces which can be used as the first estimationfor the constants 119888

ℎand 1198881015840

119896of Conjecture 106

Theorem 107 (Gagarin and Zverovich [56] 2013) Let 119866 bea connected graph of order 119899 If 119866 can be embedded on thesurface of its orientable or nonorientable genus of the Eulercharacteristic 120594 then

(a) 120594 ⩾ 1 implies 119887(119866) ⩽ 10(b) 120594 ⩽ 0 and 119899 gt minus12120594 imply 119887(119866) ⩽ 11(c) 120594 ⩽ minus1 and 119899 ⩽ minus12120594 imply 119887(119866) ⩽ 11 +

3120594(radic17 minus 8120594 minus 3)(120594 minus 1) = 11 + 119874(radicminus120594)

Theorem 79 shows that a connected planar triangle-freegraph 119866 has 119887(119866) ⩽ 6 The following result generalizes to itall the other topological surfaces

International Journal of Combinatorics 17

Theorem 108 (Gagarin and Zverovich [56] 2013) Let 119866 be aconnected triangle-free graph of order 119899 If 119866 can be embeddedon the surface of its orientable or nonorientable genus of theEuler characteristic 120594 then

(a) 120594 ⩾ 1 implies 119887(119866) ⩽ 6(b) 120594 ⩽ 0 and 119899 gt minus8120594 imply 119887(119866) ⩽ 7(c) 120594 ⩽ minus1 and 119899 ⩽ minus8120594 imply 119887(119866) ⩽ 7minus4120594(1+radic2 minus 120594)

In [56] the authors also explicitly improved upperbounds in Theorem 103 and gave tight lower bounds forthe number of vertices of graphs 2-cell embeddable ontopological surfaces of a given genusThe interested reader isreferred to the original paper for details The following resultis interesting although its proof is easy

Theorem 109 (Samodivkin [57]) Let119866 be a graph with order119899 and girth 119892 If 119866 is embeddable on a surface whose Eulercharacteristic 120594 is as large as possible then

119887 (119866) ⩽ 3 +8

119892 minus 2minus

4120594119892

119899 (119892 minus 2) (39)

If 119866 is a planar graph with girth 119892 ⩾ 4 + 119894 for any 119894 isin0 1 2 then Theorem 109 leads to 119887(119866) ⩽ 6 minus 119894 which isthe result inTheorem 79 obtained by Fischermann et al [46]Also inmany cases the bound stated inTheorem 109 is betterthan those given byTheorems 103 and 105

8 Conditional Bondage Numbers

Since the concept of the bondage number is based upon thedomination all sorts of dominations which are variationsof the normal domination by adding restricted conditionsto the normal dominating set can develop a new ldquobondagenumberrdquo as long as a variation of domination number isgiven In this section we survey results on the bondagenumber under some restricted conditions

81 Total Bondage Numbers A dominating set 119878 of a graph119866 without isolated vertices is called total if the subgraphinduced by 119878 contains no isolated vertices The minimumcardinality of a total dominating set is called the totaldomination number of 119866 and denoted by 120574

119879(119866) It is clear

that 120574(119866) ⩽ 120574119879(119866) ⩽ 2120574(119866) for any graph 119866 without isolated

verticesThe total domination in graphs was introduced by Cock-

ayne et al [58] in 1980 Pfaff et al [59 60] in 1983 showedthat the problem determining total domination number forgeneral graphs is NP-complete even for bipartite graphs andchordal graphs Even now total domination in graphs hasbeen extensively studied in the literature In 2009 Henning[61] gave a survey of selected recent results on total domina-tion in graphs

The total bondage number of 119866 denoted by 119887119879(119866) is the

smallest cardinality of a subset 119861 sube 119864(119866) with the propertythat119866minus119861 contains no isolated vertices and 120574

119879(119866minus119861) gt 120574

119879(119866)

From definition 119887119879(119866)may not exist for some graphs for

example119866 = 1198701119899 We put 119887

119879(119866) = infin if 119887

119879(119866) does not exist

It is easy to see that 119887119879(119866) is finite for any connected graph 119866

other than 1198701 119870

2 119870

3 and 119870

1119899 In fact for such a graph 119866

since there is a path of length 3 in 119866 we can find 1198611sube 119864(119866)

such that 119866 minus 1198611is a spanning tree 119879 of 119866 containing a path

of length 3 So 120574119879(119879) = 120574

119879(119866minus119861

1) ⩾ 120574

119879(119866) For the tree119879 we

find 1198612sube 119864(119879) such that 120574

119879(119879 minus 119861

2) gt 120574

119879(119879) ⩾ 120574

119879(119866) Thus

we have 120574119879(119866minus119861

2minus119861

1) gt 120574

119879(119866) and so 119887

119879(119866) ⩽ |119861

1|+|119861

2| In

the following discussion we always assume that 119887119879(119866) exists

when a graph 119866 is mentionedIn 1991 Kulli and Patwari [62] first studied the total

bondage number of a graph and calculated the exact valuesof 119887

119879(119866) for some standard graphs

Theorem 110 (Kulli and Patwari [62] 1991) For 119899 ⩾ 4

119887119879(119862

119899) =

3 119894119891 119899 equiv 2 (mod 4) 2 119900119905ℎ119890119903119908119894119904119890

119887119879(119875

119899) =

2 119894119891 119899 equiv 2 (mod 4) 1 119900119905ℎ119890119903119908119894119904119890

119887119879(119870

119898119899) = 119898 119908119894119905ℎ 2 ⩽ 119898 ⩽ 119899

119887119879(119870

119899) =

4 119891119900119903 119899 = 4

2119899 minus 5 119891119900119903 119899 ⩾ 5

(40)

Recently Hu et al [63] have obtained some results on thetotal bondage number of the Cartesian product119875

119898times119875

119899of two

paths 119875119898and 119875

119899

Theorem 111 (Hu et al [63] 2012) For the Cartesian product119875119898times 119875

119899of two paths 119875

119898and 119875

119899

119887119879(119875

2times 119875

119899) =

1 119894119891 119899 equiv 0 (mod 3) 2 119894119891 119899 equiv 2 (mod 3) 3 119894119891 119899 equiv 1 (mod 3)

119887119879(119875

3times 119875

119899) = 1

119887119879(119875

4times 119875

119899)

= 1 119894119891 119899 equiv 1 (mod 5) = 2 119894119891 119899 equiv 4 (mod 5) ⩽ 3 119894119891 119899 equiv 2 (mod 5) ⩽ 4 119894119891 119899 equiv 0 3 (mod 5)

(41)

Generalized Petersen graphs are an important class ofcommonly used interconnection networks and have beenstudied recently By constructing a family of minimum totaldominating sets Cao et al [64] determined the total bondagenumber of the generalized Petersen graphs

FromTheorem 6 we know that 119887(119879) ⩽ 2 for a nontrivialtree 119879 But given any positive integer 119896 Sridharan et al [65]constructed a tree 119879 for which 119887

119879(119879) = 119896 Let119867

119896be the tree

obtained from a star1198701119896+1

by subdividing 119896 edges twice Thetree 119867

7is shown in Figure 6 It can be easily verified that

119887119879(119867

119896) = 119896 This fact can be stated as the following theorem

Theorem 112 (Sridharan et al [65] 2007) For any positiveinteger 119896 there exists a tree 119879 with 119887

119879(119879) = 119896

18 International Journal of Combinatorics

Figure 61198677

Combining Theorem 1 with Theorem 112 we have whatfollows

Corollary 113 (Sridharan et al [65] 2007) The bondagenumber and the total bondage number are unrelated even fortrees

However Sridharan et al [65] gave an upper bound onthe total bondage number of a tree in terms of its maximumdegree For any tree 119879 of order 119899 if 119879 =119870

1119899minus1 then 119887

119879(119879) =

minΔ(119879) (13)(119899 minus 1) Rad and Raczek [66] improved thisupper bound and gave a constructive characterization of acertain class of trees attaching the upper bound

Theorem 114 (Rad and Raczek [66] 2011) 119887119879(119879) ⩽ Δ(119879) minus 1

for any tree 119879 with maximum degree at least three

In general the decision problem for 119887119879(119866) is NP-complete

for any graph 119866 We state the decision problem for the totalbondage as follows

Problem 3 Consider the decision problemTotal Bondage ProblemInstance a graph 119866 and a positive integer 119887Question is 119887

119879(119866) ⩽ 119887

Theorem 115 (Hu and Xu [16] 2012) The total bondageproblem is NP-complete

Consequently in view of its computational hardnessit is significative to establish some sharp bounds on thetotal bondage number of a graph in terms of other graphicparameters In 1991 Kulli and Patwari [62] showed that119887119879(119866) ⩽ 2119899 minus 5 for any graph 119866 with order 119899 ⩾ 5 Sridharan

et al [65] improved this result as follows

Theorem 116 (Sridharan et al [65] 2007) For any graph 119866with order 119899 if 119899 ⩾ 5 then

119887119879(119866) ⩽

1

3(119899 minus 1) 119894119891 119866 119888119900119899119905119886119894119899119904 119899119900 119888119910119888119897119890119904

119899 minus 1 119894119891 119892 (119866) ⩾ 5

119899 minus 2 119894119891 119892 (119866) = 4

(42)

Rad and Raczek [66] also established some upperbounds on 119887

119879(119866) for a general graph 119866 In particular they

gave an upper bound on 119887119879(119866) in terms of the girth of

119866

Theorem 117 (Rad and Raczek [66] 2011) 119887119879(119866)⩽4Δ(119866) minus 5

for any graph 119866 with 119892(119866) ⩾ 4

82 Paired Bondage Numbers A dominating set 119878 of 119866 iscalled to be paired if the subgraph induced by 119878 containsa perfect matching The paired domination number of 119866denoted by 120574

119875(119866) is the minimum cardinality of a paired

dominating set of 119866 Clearly 120574119879(119866) ⩽ 120574

119875(119866) for every

connected graph 119866 with order at least two where 120574119879(119866) is

the total domination number of 119866 and 120574119875(119866) ⩽ 2120574(119866) for

any graph119866without isolated vertices Paired domination wasintroduced by Haynes and Slater [67 68] and further studiedin [69ndash72]

The paired bondage number of 119866 with 120575(119866) ⩾ 1 denotedby 119887P(119866) is the minimum cardinality among all sets of edges119861 sube 119864 such that 120575(119866 minus 119861) ⩾ 1 and 120574

119875(119866 minus 119861) gt 120574

119875(119866)

The concept of the paired bondage number was firstproposed by Raczek [73] in 2008The following observationsfollow immediately from the definition of the paired bondagenumber

Observation 2 Let 119866 be a graph with 120575(119866) ⩾ 1

(a) If 119867 is a subgraph of 119866 such that 120575(119867) ⩾ 1 then119887119875(119867) ⩽ 119887

119875(119866)

(b) If 119867 is a subgraph of 119866 such that 119887119875(119867) = 1 and 119896

is the number of edges removed to form119867 then 1 ⩽119887119875(119866) ⩽ 119896 + 1

(c) If 119909119910 isin 119864(119866) such that 119889119866(119909) 119889

119866(119910) gt 1 and 119909119910

belongs to each perfect matching of each minimumpaired dominating set of 119866 then 119887

119875(119866) = 1

Based on these observations 119887119875(119875

119899) ⩽ 2 In fact the

paired bondage number of a path has been determined

Theorem 118 (Raczek [73] 2008) For any positive integer 119896if 119899 ⩾ 2 then

119887119875(119875

119899) =

0 119894119891 119899 = 2 3 119900119903 5

1 119894119891 119899 = 4119896 4119896 + 3 119900119903 4119896 + 6

2 119900119905ℎ119890119903119908119894119904119890

(43)

For a cycle 119862119899of order 119899 ⩾ 3 since 120574

119875(119862

119899) = 120574

119875(119875

119899) from

Theorem 118 the following result holds immediately

Corollary 119 (Raczek [73] 2008) For any positive integer 119896if 119899 ⩾ 3 then

119887119875(119862

119899) =

0 119894119891 119899 = 3 119900119903 5

2 119894119891 119899 = 4119896 4119896 + 3 119900119903 4119896 + 6

3 119900119905ℎ119890119903119908119894119904119890

(44)

A wheel 119882119899 where 119899 ⩾ 4 is a graph with 119899 vertices

formed by connecting a single vertex to all vertices of a cycle119862119899minus1

Of course 120574119875(119882

119899) = 2

International Journal of Combinatorics 19

119896

Figure 7 A tree 119879 with 119887119875(119879) = 119896

Theorem 120 (Raczek [73] 2008) For positive integers 119899 and119898

119887119875(119870

119898119899) = 119898 119891119900119903 1 lt 119898 ⩽ 119899

119887119875(119882

119899) =

4 119894119891 119899 = 4

3 119894119891 119899 = 5

2 119900119905ℎ119890119903119908119894119904119890

(45)

Theorem 16 shows that if 119879 is a nontrivial tree then119887(119879) ⩽ 2 However no similar result exists for pairedbondage For any nonnegative integer 119896 let 119879

119896be a tree

obtained by subdividing all but one edge of a star 1198701119896+1

(asshown in Figure 7) It is easy to see that 119887

119875(119879

119896) = 119896 We

state this fact as the following theorem which is similar toTheorem 112

Theorem121 (Raczek [73] 2008) For any nonnegative integer119896 there exists a tree 119879 with 119887

119875(119879) = 119896

Consider the tree defined in Figure 5 for 119896 = 3 Then119887(119879

3) ⩽ 2 and 119887

119875(119879

3) = 3 On the other hand 119887(119875

4) = 2

and 119887119875(119875

4) = 1 Thus we have what follows which is similar

to Corollary 113

Corollary 122 The bondage number and the paired bondagenumber are unrelated even for trees

A constructive characterization of trees with 119887(119879) = 2is given by Hartnell and Rall in [12] Raczek [73] provideda constructive characterization of trees with 119887

119875(119879) = 0 In

order to state the characterization we define a labeling andthree simple operations on a tree119879 Let 119910 isin 119881(119879) and let ℓ(119910)be the label assigned to 119910

Operation T1 If ℓ(119910) = 119861 add a vertex 119909 and the

edge 119910119909 and let ℓ(119909) = 119860OperationT

2 If ℓ(119910) = 119862 add a path (119909

1 119909

2) and the

edge 1199101199091 and let ℓ(119909

1) = 119861 ℓ(119909

2) = 119860

Operation T3 If ℓ(119910) = 119861 add a path (119909

1 119909

2 119909

3)

and the edge 1199101199091 and let ℓ(119909

1) = 119862 ℓ(119909

2) = 119861 and

ℓ(1199093) = 119860

Let 1198752= (119906 V) with ℓ(119906) = 119860 and ℓ(V) = 119861 Let T be

the class of all trees obtained from the labeled 1198752by a finite

sequence of operationsT1 T

2 T

3

A tree 119879 in Figure 8 belongs to the familyTRaczek [73] obtained the following characterization of all

trees 119879 with 119887119875(119879) = 0

119860

119860

119860

119860

119860

119861 119862

119861 119862 119861

119861119860

Figure 8 A tree 119879 belong to the familyT

Theorem 123 (Raczek [73] 2008) For a tree 119879 119887119875(119879) = 0 if

and only if 119879 is inT

We state the decision problem for the paired bondage asfollows

Problem 4 Consider the decision problem

Paired Bondage ProblemInstance a graph 119866 and a positive integer 119887Question is 119887

119875(119866) ⩽ 119887

Conjecture 124 The paired bondage problem is NP-complete

83 Restrained Bondage Numbers A dominating set 119878 of 119866 iscalled to be restrained if the subgraph induced by 119878 containsno isolated vertices where 119878 = 119881(119866) 119878 The restraineddomination number of 119866 denoted by 120574

119877(119866) is the smallest

cardinality of a restrained dominating set of 119866 It is clear that120574119877(119866) exists and 120574(119866) ⩽ 120574

119877(119866) for any nonempty graph 119866

The concept of restrained domination was introducedby Telle and Proskurowski [74] in 1997 albeit indirectly asa vertex partitioning problem Concerning the study of therestrained domination numbers the reader is referred to [74ndash79]

In 2008 Hattingh and Plummer [80] defined therestrained bondage number 119887R(119866) of a nonempty graph 119866 tobe the minimum cardinality among all subsets 119861 sube 119864(119866) forwhich 120574

119877(119866 minus 119861) gt 120574

119877(119866)

For some simple graphs their restrained dominationnumbers can be easily determined and so restrained bondagenumbers have been also determined For example it is clearthat 120574

119877(119870

119899) = 1 Domke et al [77] showed that 120574

119877(119862

119899) =

119899 minus 2lfloor1198993rfloor for 119899 ⩾ 3 and 120574119877(119875

119899) = 119899 minus 2lfloor(119899 minus 1)3rfloor for 119899 ⩾ 1

Using these results Hattingh and Plummer [80] obtained therestricted bondage numbers for119870

119899 119862

119899 119875

119899 and119866 = 119870

11989911198992119899119905

as follows

Theorem 125 (Hattingh and Plummer [80] 2008) For 119899 ⩾ 3

119887R (119870119899) =

1 119894119891 119899 = 3

lceil119899

2rceil 119900119905ℎ119890119903119908119894119904119890

119887R (119862119899) =

1 119894119891 119899 equiv 0 (mod 3) 2 119900119905ℎ119890119903119908119894119904119890

119887R (119875119899) = 1 119899 ⩾ 4

20 International Journal of Combinatorics

119887R (119866)

=

lceil119898

2rceil 119894119891 119899

119898= 1 119886119899119889 119899

119898+1⩾ 2 (1 ⩽ 119898 lt 119905)

2119905 minus 2 119894119891 1198991= 119899

2= sdot sdot sdot = 119899

119905= 2 (119905 ⩾ 2)

2 119894119891 1198991= 2 119886119899119889 119899

2⩾ 3 (119905 = 2)

119905minus1

sum

119894=1

119899119894minus 1 119900119905ℎ119890119903119908119894119904119890

(46)

Theorem 126 (Hattingh and Plummer [80] 2008) For a tree119879 of order 119899 (⩾ 4) 119879 ≇ 119870

1119899minus1if and only if 119887R(119879) = 1

Theorem 126 shows that the restrained bondage numberof a tree can be computed in constant time However thedecision problem for 119887R(119866) is NP-complete even for bipartitegraphs

Problem 5 Consider the decision problem

Restrained Bondage Problem

Instance a graph 119866 and a positive integer 119887

Question is 119887R(119866) ⩽ 119887

Theorem 127 (Hattingh and Plummer [80] 2008) Therestrained bondage problem is NP-complete even for bipartitegraphs

Consequently in view of its computational hardnessit is significative to establish some sharp bounds on therestrained bondage number of a graph in terms of othergraphic parameters

Theorem 128 (Hattingh and Plummer [80] 2008) For anygraph 119866 with 120575(119866) ⩾ 2

119887R (119866) ⩽ min119909119910isin119864(119866)

119889119866(119909) + 119889

119866(119910) minus 2 (47)

Corollary 129 119887R(119866) ⩽ Δ(119866)+120575(119866)minus2 for any graph119866 with120575(119866) ⩾ 2

Notice that the bounds stated in Theorem 128 andCorollary 129 are sharp Indeed the class of cycles whoseorders are congruent to 1 2 (mod 3) have a restrained bondagenumber achieving these bounds (see Theorem 125)

Theorem 128 is an analogue of Theorem 14 A quitenatural problem is whether or not for restricted bondagenumber there are analogues of Theorem 16 Theorem 18Theorem 29 and so on Indeed we should consider all resultsfor the bondage number whether or not there are analoguesfor restricted bondage number

Theorem 130 (Hattingh and Plummer [80] 2008) For anygraph 119866 with 120574

119877(119866) = 2

119887R (119866) ⩽ Δ (119866) + 1 (48)

We now consider the relation between 119887119877(119866) and 119887(119866)

Theorem 131 (Hattingh and Plummer [80] 2008) If 120574119877(119866) =

120574(119866) for some graph 119866 then 119887R(119866) ⩽ 119887(119866)

Corollary 129 and Theorem 131 provide a possibility toattack Conjecture 73 by characterizing planar graphs with120574119877= 120574 and 119887R = 119887 when 3 ⩽ Δ ⩽ 6However we do not have 119887R(119866) = 119887(119866) for any graph

119866 even if 120574119877(119866) = 120574(119866) Observe that 120574

119877(119870

3) = 120574(119870

3) yet

119887119877(119870

3) = 1 and 119887(119870

3) = 2 We still may not claim that

119887119877(119866) = 119887(119866) even in the case that every 120574-set is a 120574

119877-set

The example 1198703again demonstrates this

Theorem 132 (Hattingh and Plummer [80] 2008) There isan infinite class of graphs inwhich each graph119866 satisfies 119887(119866) lt119887R(119866)

In fact such an infinite class of graphs can be obtained asfollows Let119867 be a connected graph BC(119867) a graph obtainedfrom 119867 by attaching ℓ (⩾ 2) vertices of degree 1 to eachvertex in 119867 and B = 119866 119866 = BC(119867) for some graph 119867such that 120575(119867) ⩾ 2 By Theorem 3 we can easily check that119887(119866) = 1 lt 2 ⩽ min120575(119867) ℓ = 119887R(119866) for any 119866 isin BMoreover there exists a graph 119866 isin B such that 119887R(119866) canbe much larger than 119887(119866) which is stated as the followingtheorem

Theorem 133 (Hattingh and Plummer [80] 2008) For eachpositive integer 119896 there is a graph119866 such that 119896 = 119887R(119866)minus119887(119866)

Combining Theorem 133 with Theorem 132 we have thefollowing corollary

Corollary 134 The bondage number and the restrainedbondage number are unrelated

Very recently Jafari Rad et al [81] have consideredthe total restrained bondage in graphs based on both totaldominating sets and restrained dominating sets and obtainedseveral properties exact values and bounds for the totalrestrained bondage number of a graph

A restrained dominating set 119878 of a graph 119866 without iso-lated vertices is called to be a total restrained dominating setif 119878 is a total dominating set The total restrained dominationnumber of119866 denoted by 120574TR (119866) is theminimum cardinalityof a total restrained dominating set of 119866 For referenceson this domination in graphs see [3 61 82ndash85] The totalrestrained bondage number 119887TR (119866) of a graph 119866 with noisolated vertex is the cardinality of a smallest set of edges 119861 sube119864(119866) for which119866minus119861 has no isolated vertex and 120574TR (119866minus119861) gt120574TR (119866) In the case that there is no such subset 119861 we define119887TR (119866) = infin

Ma et al [84] determined that 120574TR (119875119899) = 119899minus2lfloor(119899minus2)4rfloorfor 119899 ⩾ 3 120574TR (119862119899

) = 119899 minus 2lfloor1198994rfloor for 119899 ⩾ 3 120574TR (1198703) =

3 and 120574TR (119870119899) = 2 for 119899 = 3 and 120574TR (1198701119899

) = 1 + 119899

and 120574TR (119870119898119899) = 2 for 2 ⩽ 119898 ⩽ 119899 According to these

results Jafari Rad et al [81] determined the exact valuesof total restrained bondage numbers for the correspondinggraphs

International Journal of Combinatorics 21

Theorem 135 (Jafari Rad et al [81]) For an integer 119899

119887TR (119875119899) = infin 119894119891 2 ⩽ 119899 ⩽ 5

1 119894119891 119899 ⩾ 5

119887TR (119862119899) =

infin 119894119891 119899 = 3

1 119894119891 3 lt 119899 119899 equiv 0 1 (mod 4)2 119894119891 3 lt 119899 119899 equiv 2 3 (mod 4)

119887TR (119882119899) = 2 119891119900119903 119899 ⩾ 6

119887TR (119870119899) = 119899 minus 1 119891119900119903 119899 ⩾ 4

119887TR (119870119898119899) = 119898 minus 1 119891119900119903 2 ⩽ 119898 ⩽ 119899

(49)

119887TR (119870119899) = 119899minus1 shows the fact that for any integer 119899 ⩾ 3 there

exists a connected graph namely119870119899+1

such that 119887TR (119870119899+1) =

119899 that is the total restrained bondage number is unboundedfrom the above

A vertex is said to be a support vertex if it is adjacent toa vertex of degree one Let A be the family of all connectedgraphs such that119866 belongs toA if and only if every edge of119866is incident to a support vertex or 119866 is a cycle of length three

Proposition 136 (Cyman and Raczek [82] 2006) For aconnected graph119866 of order 119899 120574TR (119866) = 119899 if and only if119866 isin A

Hence if 119866 isin A then the removal of any subset of edgesof119866 cannot increase the total restrained domination numberof 119866 and thus 119887TR (119866) = infin When 119866 notin A a result similar toTheorem 6 is obtained

Theorem 137 (Jafari Rad et al [81]) For any tree 119879 notin A119887TR (119879) ⩽ 2 This bound is sharp

In the samepaper the authors provided a characterizationfor trees 119879 notin A with 119887TR (119879) = 1 or 2 The following result issimilar to Observation 1

Observation 3 (Jafari Rad et al [81]) If 119896 edges can beremoved from a graph 119866 to obtain a graph 119867 without anisolated vertex and with 119887TR (119867) = 119905 then 119887TR (119866) ⩽ 119896 + 119905

ApplyingTheorem 137 andObservation 3 Jafari Rad et alcharacterized all connected graphs 119866 with 119887TR (119866) = infin

Theorem 138 (Jafari Rad et al [81]) For a connected graph119866119887TR (119866) = infin if and only if 119866 isin A

84 Other Conditional Bondage Numbers A subset 119868 sube 119881(119866)is called an independent set if no two vertices in 119868 are adjacentin 119866 The maximum cardinality among all independent setsis called the independence number of 119866 denoted by 120572(119866)

A dominating set 119878 of a graph 119866 is called to be inde-pendent if 119878 is an independent set of 119866 The minimumcardinality among all independent dominating set is calledthe independence domination number of 119866 and denoted by120574119868(119866)

Since an independent dominating set is not only adominating set but also an independent set 120574(119866) ⩽ 120574

119868(119866) ⩽

120572(119866) for any graph 119866It is clear that a maximal independent set is certainly a

dominating set Thus an independent set is maximal if andonly if it is an independent dominating set and so 120574

119868(119866) is the

minimum cardinality among all maximal independent setsof 119866 This graph-theoretical invariant has been well studiedin the literature see for example Haynes et al [3] for earlyresults Goddard and Henning [86] for recent results onindependent domination in graphs

In 2003 Zhang et al [87] defined the independencebondage number 119887

119868(119866) of a nonempty graph 119866 to be the

minimum cardinality among all subsets 119861 sube 119864(119866) for which120574119868(119866 minus 119861) gt 120574

119868(119866) For some ordinary graphs their indepen-

dence domination numbers can be easily determined and soindependence bondage numbers have been also determinedClearly 119887

119868(119870

119899) = 1 if 119899 ⩾ 2

Theorem 139 (Zhang et al [87] 2003) For any integer 119899 ⩾ 4

119887119868(119862

119899) =

1 119894119891 119899 equiv 1 (mod 2) 2 119894119891 119899 equiv 0 (mod 2)

119887119868(119875

119899) =

1 119894119891 119899 equiv 0 (mod 2) 2 119894119891 119899 equiv 1 (mod 2)

119887119868(119870

119898119899) = 119899 119891119900119903 119898 ⩽ 119899

(50)

Apart from the above-mentioned results as far as we findno other results on the independence bondage number in thepresent literature we never knew much of any result on thisparameter for a tree

A subset 119878 of 119881(119866) is called an equitable dominatingset if for every 119910 isin 119881(119866 minus 119878) there exists a vertex 119909 isin 119878such that 119909119910 isin 119864(119866) and |119889

119866(119909) minus 119889

119866(119910)| ⩽ 1 proposed

by Dharmalingam [88] The minimum cardinality of such adominating set is called the equitable domination number andis denoted by 120574

119864(119866) Deepak et al [89] defined the equitable

bondage number 119887119864(119866) of a graph 119866 to be the cardinality of a

smallest set 119861 sube 119864(119866) of edges for which 120574119864(119866 minus 119861) gt 120574

119864(119866)

and determined 119887119864(119866) for some special graphs

Theorem 140 (Deepak et al [89] 2011) For any integer 119899 ⩾ 2

119887119864(119870

119899) = lceil

119899

2rceil

119887119864(119870

119898119899) = 119898 119891119900119903 0 ⩽ 119899 minus 119898 ⩽ 1

119887119864(119862

119899) =

3 119894119891 119899 equiv 1 (mod 3)2 119900119905ℎ119890119903119908119894119904119890

119887119864(119875

119899) =

2 119894119891 119899 equiv 1 (mod 3)1 119900119905ℎ119890119903119908119894119904119890

119887119864(119879) ⩽ 2 119891119900119903 119886119899119910 119899119900119899119905119903119894119907119894119886119897 119905119903119890119890 119879

119887119864(119866) ⩽ 119899 minus 1 119891119900119903 119886119899119910 119888119900119899119899119890119888119905119890119889 119892119903119886119901ℎ 119866 119900119891 119900119903119889119890119903 119899

(51)

22 International Journal of Combinatorics

As far as we know there are no more results on theequitable bondage number of graphs in the present literature

There are other variations of the normal domination byadding restricted conditions to the normal dominating setFor example the connected domination set of a graph119866 pro-posed by Sampathkumar and Walikar [90] is a dominatingset 119878 such that the subgraph induced by 119878 is connected Theproblem of finding a minimum connected domination set ina graph is equivalent to the problemof finding a spanning treewithmaximumnumber of vertices of degree one [91] and hassome important applications in wireless networks [92] thusit has received much research attention However for suchan important domination there is no result on its bondagenumber We believe that the topic is of significance for futureresearch

9 Generalized Bondage Numbers

There are various generalizations of the classical dominationsuch as 119901-domination distance domination fractional dom-ination Roman domination and rainbow domination Everysuch a generalization can lead to a corresponding bondageIn this section we introduce some of them

91 119901-Bondage Numbers In 1985 Fink and Jacobson [93]introduced the concept of 119901-domination Let 119901 be a positiveinteger A subset 119878 of 119881(119866) is a 119901-dominating set of 119866 if|119878 cap 119873

119866(119910)| ⩾ 119901 for every 119910 isin 119878 The 119901-domination number

120574119901(119866) is the minimum cardinality among all 119901-dominating

sets of 119866 Any 119901-dominating set of 119866 with cardinality 120574119901(119866)

is called a 120574119901-set of 119866 Note that a 120574

1-set is a classic minimum

dominating set Notice that every graph has a 119901-dominatingset since the vertex-set119881(119866) is such a setWe also note that a 1-dominating set is a dominating set and so 120574(119866) = 120574

1(119866) The

119901-domination number has received much research attentionsee a state-of-the-art survey article by Chellali et al [94]

It is clear from definition that every 119901-dominating setof a graph certainly contains all vertices of degree at most119901 minus 1 By this simple observation to avoid the trivial caseoccurrence we always assume that Δ(119866) ⩾ 119901 For 119901 ⩾ 2 Luet al [95] gave a constructive characterization of trees withunique minimum 119901-dominating sets

Recently Lu and Xu [96] have introduced the conceptto the 119901-bondage number of 119866 denoted by 119887

119901(119866) as the

minimum cardinality among all sets of edges 119861 sube 119864(119866) suchthat 120574

119901(119866 minus 119861) gt 120574

119901(119866) Clearly 119887

1(119866) = 119887(119866)

Lu and Xu [96] established a lower bound and an upperbound on 119887

119901(119879) for any integer 119901 ⩾ 2 and any tree 119879 with

Δ(119879) ⩾ 119901 and characterized all trees achieving the lowerbound and the upper bound respectively

Theorem 141 (Lu and Xu [96] 2011) For any integer 119901 ⩾ 2and any tree 119879 with Δ(119879) ⩾ 119901

1 ⩽ 119887119901(119879) ⩽ Δ (119879) minus 119901 + 1 (52)

Let 119878 be a given subset of 119881(119866) and for any 119909 isin 119878 let

119873119901(119909 119878 119866) = 119910 isin 119878 cap 119873

119866(119909)

1003816100381610038161003816119873119866(119910) cap 119878

1003816100381610038161003816 = 119901

(53)

Then all trees achieving the lower bound 1 can be character-ized as follows

Theorem 142 (Lu and Xu [96] 2011) Let 119879 be a tree withΔ(119879) ⩾ 119901 ⩾ 2 Then 119887

119901(119879) = 1 if and only if for any 120574

119901-set

119878 of 119879 there exists an edge 119909119910 isin (119878 119878) such that 119910 isin 119873119901(119909 119878 119879)

The notation 119878(119886 119887) denotes a double star obtained byadding an edge between the central vertices of two stars119870

1119886minus1

and1198701119887minus1

And the vertex with degree 119886 (resp 119887) in 119878(119886 119887) iscalled the 119871-central vertex (resp 119877-central vertex) of 119878(119886 119887)

To characterize all trees attaining the upper bound givenin Theorem 141 we define three types of operations on a tree119879 with Δ(119879) = Δ ⩾ 119901 + 1

Type 1 attach a pendant edge to a vertex 119910with 119889119879(119910) ⩽ 119901minus

2 in 119879

Type 2 attach a star 1198701Δminus1

to a vertex 119910 of 119879 by joining itscentral vertex to 119910 where 119910 in a 120574

119901-set of 119879 and

119889119879(119910) ⩽ Δ minus 1

Type 3 attach a double star 119878(119901 Δ minus 1) to a pendant vertex 119910of 119879 by coinciding its 119877-central vertex with 119910 wherethe unique neighbor of 119910 is in a 120574

119901-set of 119879

Let B = 119879 a tree obtained from 1198701Δ

or 119878(119901 Δ) by afinite sequence of operations of Types 1 2 and 3

Theorem 143 (Lu and Xu [96] 2011) A tree with the maxi-mum Δ ⩾ 119901 + 1 has 119901-bondage number Δ minus 119901 + 1 if and onlyif it belongs toB

Theorem 144 (Lu and Xu [97] 2012) Let 119866119898119899= 119875

119898times 119875

119899

Then

1198872(119866

2119899) = 1 119891119900119903 119899 ⩾ 2

1198872(119866

3119899) =

2 119894119891 119899 equiv 1 (mod 3) 1 119900119905ℎ119890119903119908119894119904119890

for 119899 ⩾ 2

1198872(119866

4119899) =

1 119894119891 119899 equiv 3 (mod 4) 2 119900119905ℎ119890119903119908119894119904119890

for 119899 ⩾ 7

(54)

Recently Krzywkowski [98] has proposed the conceptof the nonisolating 2-bondage of a graph The nonisolating2-bondage number of a graph 119866 denoted by 1198871015840

2(119866) is the

minimum cardinality among all sets of edges 119861 sube 119864(119866) suchthat 119866 minus 119861 has no isolated vertices and 120574

2(119866 minus 119861) gt 120574

2(119866) If

for every 119861 sube 119864(119866) either 1205742(119866 minus 119861) = 120574

2(119866) or 119866 minus 119861 has

isolated vertices then define 11988710158402(119866) = 0 and say that 119866 is a 2-

nonisolatedly strongly stable graph Krzywkowski presentedsome basic properties of nonisolating 2-bondage in graphsshowed that for every nonnegative integer 119896 there exists atree 119879 such 1198871015840

2(119879) = 119896 characterized all 2-non-isolatedly

strongly stable trees and determined 11988710158402(119866) for several special

graphs

International Journal of Combinatorics 23

Theorem 145 (Krzywkowski [98] 2012) For any positiveintegers 119899 and119898 with119898 ⩽ 119899

1198871015840

2(119870

119899) =

0 119894119891 119899 = 1 2 3

lfloor2119899

3rfloor 119900119905ℎ119890119903119908119894119904119890

1198871015840

2(119875

119899) =

0 119894119891 119899 = 1 2 3

1 119900119905ℎ119890119903119908119894119904119890

1198871015840

2(119862

119899) =

0 119894119891 119899 = 3

1 119894119891 119899 119894119904 119890V119890119899

2 119894119891 119899 119894119904 119900119889119889

1198871015840

2(119870

119898119899) =

3 119894119891 119898 = 119899 = 3

1 119894119891 119898 = 119899 = 4

2 119900119905ℎ119890119903119908119894119904119890

(55)

92 Distance Bondage Numbers A subset 119878 of vertices of agraph119866 is said to be a distance 119896-dominating set for119866 if everyvertex in 119866 not in 119878 is at distance at most 119896 from some vertexof 119878 The minimum cardinality of all distance 119896-dominatingsets is called the distance 119896-domination number of 119866 anddenoted by 120574

119896(119866) (do not confuse with the above-mentioned

120574119901(119866)) When 119896 = 1 a distance 1-dominating set is a normal

dominating set and so 1205741(119866) = 120574(119866) for any graph 119866 Thus

the distance 119896-domination is a generalization of the classicaldomination

The relation between 120574119896and 120572

119896(the 119896-independence

number defined in Section 22) for a tree obtained by Meirand Moon [14] who proved that 120574

119896(119879) = 120572

2119896(119879) for any tree

119879 (a special case for 119896 = 1 is stated in Proposition 10) Thefurther research results can be found in Henning et al [99]Tian and Xu [100ndash103] and Liu et al [104]

In 1998 Hartnell et al [15] defined the distance 119896-bondagenumber of 119866 denoted by 119887

119896(119866) to be the cardinality of a

smallest subset 119861 of edges of 119866 with the property that 120574119896(119866 minus

119861) gt 120574119896(119866) FromTheorem 6 it is clear that if119879 is a nontrivial

tree then 1 ⩽ 1198871(119879) ⩽ 2 Hartnell et al [15] generalized this

result to any integer 119896 ⩾ 1

Theorem 146 (Hartnell et al [15] 1998) For every nontrivialtree 119879 and positive integer 119896 1 ⩽ 119887

119896(119879) ⩽ 2

Hartnell et al [15] and Topp and Vestergaard [13] alsocharacterized the trees having distance 119896-bondage number 2In particular the class of trees for which 119887

1(119879) = 2 are just

those which have a unique maximum 2-independent set (seeTheorem 11)

Since when 119896 = 1 the distance 1-bondage number 1198871(119866)

is the classical bondage number 119887(119866) Theorem 12 gives theNP-hardness of deciding the distance 119896-bondage number ofgeneral graphs

Theorem 147 Given a nonempty undirected graph 119866 andpositive integers 119896 and 119887 with 119887 ⩽ 120576(119866) determining wetheror not 119887

119896(119866) ⩽ 119887 is NP-hard

For a vertex 119909 in119866 the open 119896-neighborhood119873119896(119909) of 119909 is

defined as119873119896(119909) = 119910 isin 119881(119866) 1 ⩽ 119889

119866(119909 119910) le 119896The closed

119896-neighborhood119873119896[119909] of 119909 in 119866 is defined as119873

119896(119909) cup 119909 Let

Δ119896(119866) = max 1003816100381610038161003816119873119896

(119909)1003816100381610038161003816 for any 119909 isin 119881 (119866) (56)

Clearly Δ1(119866) = Δ(119866) The 119896th power of a graph 119866 is the

graph119866119896 with vertex-set119881(119866119896

) = 119881(119866) and edge-set119864(119866119896

) =

119909119910 1 ⩽ 119889119866(119909 119910) ⩽ 119896 The following lemma holds directly

from the definition of 119866119896

Proposition 148 (Tian and Xu [101]) Δ(119866119896

) = Δ119896(119866) and

120574(119866119896

) = 120574119896(119866) for any graph 119866 and each 119896 ⩾ 1

A graph 119866 is 119896-distance domination-critical or 120574119896-critical

for short if 120574119896(119866 minus 119909) lt 120574

119896(119866) for every vertex 119909 in 119866

proposed by Henning et al [105]

Proposition 149 (Tian and Xu [101]) For each 119896 ⩾ 1 a graph119866 is 120574

119896-critical if and only if 119866119896 is 120574

119896-critical

By the above facts we suggest the following worthwhileproblem

Problem 6 Can we generalize the results on the bondage for119866 to 119866119896 In particular do the following propositions hold

(a) 119887(119866119896

) = 119887119896(119866) for any graph 119866 and each 119896 ⩾ 1

(b) 119887(119866119896

) ⩽ Δ119896(119866) if 119866 is not 120574

119896-critical

Let 119896 and 119901 be positive integers A subset 119878 of 119881(119866) isdefined to be a (119896 119901)-dominating set of 119866 if for any vertex119909 isin 119878 |119873

119896(119909) cap 119878| ⩾ 119901 The (119896 119901)-domination number of

119866 denoted by 120574119896119901(119866) is the minimum cardinality among

all (119896 119901)-dominating sets of 119866 Clearly for a graph 119866 a(1 1)-dominating set is a classical dominating set a (119896 1)-dominating set is a distance 119896-dominating set and a (1 119901)-dominating set is the above-mentioned 119901-dominating setThat is 120574

11(119866) = 120574(119866) 120574

1198961(119866) = 120574

119896(119866) and 120574

1119901(119866) = 120574

119901(119866)

The concept of (119896 119901)-domination in a graph 119866 is a gen-eralized domination which combines distance 119896-dominationand 119901-domination in 119866 So the investigation of (119896 119901)-domination of 119866 is more interesting and has received theattention of many researchers see for example [106ndash109]

More general Li and Xu [110] proposed the concept of the(ℓ 119908)-domination number of a119908-connected graph A subset119878 of 119881(119866) is defined to be an (ℓ 119908)-dominating set of 119866 iffor any vertex 119909 isin 119878 there are 119908 internally disjoint pathsof length at most ℓ from 119909 to some vertex in 119878 The (ℓ 119908)-domination number of119866 denoted by 120574

ℓ119908(119866) is theminimum

cardinality among all (ℓ 119908)-dominating sets of 119866 Clearly1205741198961(119866) = 120574

119896(119866) and 120574

1119901(119866) = 120574

119901(119866)

It is quite natural to propose the concept of bondage num-ber for (119896 119901)-domination or (ℓ 119908)-domination However asfar as we know none has proposed this concept until todayThis is a worthwhile topic for further research

24 International Journal of Combinatorics

93 Fractional Bondage Numbers If 120590 is a function mappingthe vertex-set 119881 into some set of real numbers then for anysubset 119878 sube 119881 let 120590(119878) = sum

119909isin119878120590(119909) Also let |120590| = 120590(119881)

A real-value function 120590 119881 rarr [0 1] is a dominatingfunction of a graph 119866 if for every 119909 isin 119881(119866) 120590(119873

119866[119909]) ⩾ 1

Thus if 119878 is a dominating set of 119866 120590 is a function where

120590 (119909) = 1 if 119909 isin 1198780 otherwise

(57)

then 120590 is a dominating function of119866 The fractional domina-tion number of 119866 denoted by 120574

119865(119866) is defined as follows

120574119865(119866) = min |120590| 120590 is a domination function of 119866

(58)

The 120574119865-bondage number of 119866 denoted by 119887

119865(119866) is

defined as the minimum cardinality of a subset 119861 sube 119864 whoseremoval results in 120574

119865(119866 minus 119861) gt 120574

119865(119866)

Hedetniemi et al [111] are the first to study fractionaldomination although Farber [112] introduced the idea indi-rectly The concept of the fractional domination numberwas proposed by Domke and Laskar [113] in 1997 Thefractional domination numbers for some ordinary graphs aredetermined

Proposition 150 (Domke et al [114] 1998 Domke and Laskar[113] 1997) (a) If 119866 is a 119896-regular graph with order 119899 then120574119865(119866) = 119899119896 + 1(b) 120574

119865(119862

119899) = 1198993 120574

119865(119875

119899) = lceil1198993rceil 120574

119865(119870

119899) = 1

(c) 120574119865(119866) = 1 if and only if Δ(119866) = 119899 minus 1

(d) 120574119865(119879) = 120574(119879) for any tree 119879

(e) 120574119865(119870

119899119898) = (119899(119898 + 1) + 119898(119899 + 1))(119899119898 minus 1) where

max119899119898 gt 1

The assertions (a)ndash(d) are due to Domke et al [114] andthe assertion (e) is due to Domke and Laskar [113] Accordingto these results Domke and Laskar [113] determined the 120574

119865-

bondage numbers for these graphs

Theorem 151 (Domke and Laskar [113] 1997) 119887119865(119870

119899) =

lceil1198992rceil 119887119865(119870

119899119898) = 1 wheremax119899119898 gt 1

119887119865(119862

119899) =

2 119894119891 119899 equiv 0 (mod 3) 1 119900119905ℎ119890119903119908119894119904119890

119887119865(119875

119899) =

2 119894119891 119899 equiv 1 (mod 3) 1 119900119905ℎ119890119903119908119894119904119890

(59)

It is easy to see that for any tree119879 119887119865(119879) ⩽ 2 In fact since

120574119865(119879) = 120574(119879) for any tree 119879 120574

119865(119879

1015840

) = 120574(1198791015840

) for any subgraph1198791015840 of 119879 It follows that

119887119865(119879) = min 119861 sube 119864 (119879) 120574

119865(119879 minus 119861) gt 120574

119865(119879)

= min 119861 sube 119864 (119879) 120574 (119879 minus 119861) gt 120574 (119879)

= 119887 (119879) ⩽ 2

(60)

Except for these no results on 119887119865(119866) have been known as

yet

94 Roman Bondage Numbers A Roman dominating func-tion on a graph 119866 is a labeling 119891 119881 rarr 0 1 2 such thatevery vertex with label 0 has at least one neighbor with label2 The weight of a Roman dominating function 119891 on 119866 is thevalue

119891 (119866) = sum

119906isin119881(119866)

119891 (119906) (61)

The minimum weight of a Roman dominating function ona graph 119866 is called the Roman domination number of 119866denoted by 120574RM (119866)

A Roman dominating function 119891 119881 rarr 0 1 2

can be represented by the ordered partition (1198810 119881

1 119881

2) (or

(119881119891

0 119881

119891

1 119881

119891

2) to refer to119891) of119881 where119881

119894= V isin 119881 | 119891(V) = 119894

In this representation its weight 119891(119866) = |1198811| + 2|119881

2| It is

clear that119881119891

1cup119881

119891

2is a dominating set of 119866 called the Roman

dominating set denoted by 119863119891

R = (1198811 1198812) Since 119881119891

1cup 119881

119891

2is

a dominating set when 119891 is a Roman dominating functionand since placing weight 2 at the vertices of a dominating setyields a Roman dominating function in [115] it is observedthat

120574 (119866) ⩽ 120574RM (119866) ⩽ 2120574 (119866) (62)

A graph 119866 is called to be Roman if 120574RM (119866) = 2120574(119866)The definition of the Roman domination function is

proposed implicitly by Stewart [116] and ReVelle and Rosing[117] Roman dominating numbers have been deeply studiedIn particular Bahremandpour et al [118] showed that theproblem determining the Roman domination number is NP-complete even for bipartite graphs

Let 119866 be a graph with maximum degree at least twoThe Roman bondage number 119887RM (119866) of 119866 is the minimumcardinality of all sets 1198641015840 sube 119864 for which 120574RM (119866 minus 119864

1015840

) gt

120574RM (119866) Since in study of Roman bondage number theassumption Δ(119866) ⩾ 2 is necessary we always assume thatwhen we discuss 119887RM (119866) all graphs involved satisfy Δ(119866) ⩾2 The Roman bondage number 119887RM (119866) is introduced byJafari Rad and Volkmann in [119]

Recently Bahremandpour et al [118] have shown that theproblem determining the Roman bondage number is NP-hard even for bipartite graphs

Problem 7 Consider the decision problem

Roman Bondage Problem

Instance a nonempty bipartite graph119866 and a positiveinteger 119896

Question is 119887RM (119866) ⩽ 119896

Theorem 152 (Bahremandpour et al [118] 2013) TheRomanbondage problem is NP-hard even for bipartite graphs

The exact value of 119887RM(119866) is known only for a few familyof graphs including the complete graphs cycles and paths

International Journal of Combinatorics 25

Theorem 153 (Jafari Rad and Volkmann [119] 2011) For anypositive integers 119899 and119898 with119898 ⩽ 119899

119887RM (119875119899) = 2 119894119891 119899 equiv 2 (mod 3) 1 119900119905ℎ119890119903119908119894119904119890

119887RM (119862119899) =

3 119894119891 119899 = 2 (mod 3) 2 119900119905ℎ119890119903119908119894119904119890

119887RM (119870119899) = lceil

119899

2rceil 119891119900119903 119899 ⩾ 3

119887RM (119870119898119899) =

1 119894119891 119898 = 1 119886119899119889 119899 = 1

4 119894119891 119898 = 119899 = 3

119898 119900119905ℎ119890119903119908119894119904119890

(63)

Lemma 154 (Cockayne et al [115] 2004) If 119866 is a graph oforder 119899 and contains vertices of degree 119899 minus 1 then 120574RM(119866) = 2

Using Lemma 154 the third conclusion in Theorem 153can be generalized to more general case which is similar toLemma 49

Theorem 155 (Jafari Rad and Volkmann [119] 2011) Let119866 bea graph with order 119899 ge 3 and 119905 the number of vertices of degree119899 minus 1 in 119866 If 119905 ⩾ 1 then 119887RM(119866) = lceil1199052rceil

Ebadi and PushpaLatha [120] conjectured that 119887RM(119866) ⩽119899 minus 1 for any graph 119866 of order 119899 ⩾ 3 Dehgardi et al[121] Akbari andQajar [122] independently showed that thisconjecture is true

Theorem 156 119887RM(119866) ⩽ 119899 minus 1 for any connected graph 119866 oforder 119899 ⩾ 3

For a complete 119905-partite graph we have the followingresult

Theorem 157 (Hu and Xu [123] 2011) Let 119866 = 11987011989811198982119898119905

bea complete 119905-partite graph with 119898

1= sdot sdot sdot = 119898

119894lt 119898

119894+1le sdot sdot sdot ⩽

119898119905 119905 ⩾ 2 and 119899 = sum119905

119895=1119898

119895 Then

119887RM (119866) =

lceil119894

2rceil 119894119891 119898

119894= 1 119886119899119889 119899 ⩾ 3

2 119894119891 119898119894= 2 119886119899119889 119894 = 1

119894 119894119891 119898119894= 2 119886119899119889 119894 ⩾ 2

4 119894119891 119898119894= 3 119886119899119889 119894 = 119905 = 2

119899 minus 1 119894119891 119898119894= 3 119886119899119889 119894 = 119905 ⩾ 3

119899 minus 119898119905

119894119891 119898119894⩾ 3 119886119899119889 119898

119905⩾ 4

(64)

Consider a complete 119905-partite graph 119870119905(3) for 119905 ⩾ 3

119887RM(119870119905(3)) = 119899 minus 1 by Theorem 157 where 119899 = 3119905

which shows that this upper bound of 119899 minus 1 on 119887RM(119866) inTheorem 156 is sharp This fact and 120574

119877(119870

119905(3)) = 4 give

a negative answer to the following two problems posed byDehgardi et al [121]Prove or Disprove for any connected graph 119866 of order 119899 ge 3(i) 119887RM(119866) = 119899 minus 1 if and only if 119866 cong 119870

3 (ii) 119887RM(119866) ⩽ 119899 minus

120574119877(119866) + 1Note that a complete 119905-partite graph 119870

119905(3) is (119899 minus 3)-

regular and 119887RM(119870119905(3)) = 119899 minus 1 for 119905 ⩾ 3 where 119899 = 3119905

Hu and Xu further determined that 119887119877(119866) = 119899 minus 2 for any

(119899 minus 3)-regular graph 119866 of order 119899 ⩽ 5 except 119870119905(3)

Theorem 158 (Hu and Xu [123] 2011) For any (119899minus3)-regulargraph 119866 of order 119899 other than119870

119905(3) 119887RM(119866) = 119899 minus 2 for 119899 ⩾ 5

For a tree 119879 with order 119899 ⩾ 3 Ebadi and PushpaLatha[120] and Jafari Rad and Volkmann [119] independentlyobtained an upper bound on 119887RM(119879)

Theorem 159 119887RM(119879) ⩽ 3 for any tree 119879 with order 119899 ⩾ 3

Theorem 160 (Bahremandpour et al [118] 2013) 119887RM(1198752 times119875119899) = 2 for 119899 ⩾ 2

Theorem 161 (Jafari Rad and Volkmann [119] 2011) Let119866 bea graph with order at least three

(a) If (119909 119910 119911) is a path of length 2 in 119866 then 119887RM(119866) ⩽119889119866(119909) + 119889

119866(119910) + 119889

119866(119911) minus 3 minus |119873

119866(119909) cap 119873

119866(119910)|

(b) If 119866 is connected then 119887RM(119866) ⩽ 120582(119866) + 2Δ(119866) minus 3

Theorem 161 (b) implies 119887RM(119866) ⩽ 120575(119866) + 2Δ(119866) minus 3 for aconnected graph 119866 Note that for a planar graph 119866 120575(119866) ⩽ 5moreover 120575(119866) ⩽ 3 if the girth at least 4 and 120575(119866) ⩽ 2 if thegirth at least 6 These two facts show that 119887RM(119866) ⩽ 2Δ(119866) +2 for a connected planar graph 119866 Jafari Rad and Volkmann[124] improved this bound

Theorem 162 (Jafari Rad and Volkmann [124] 2011) For anyconnected planar graph 119866 of order 119899 ⩾ 3 with girth 119892(119866)

119887RM (119866)

⩽

2Δ (119866)

Δ (119866) + 6

Δ (119866) + 5 119894119891 119866 119888119900119899119905119886119894119899119904 119899119900 V119890119903119905119894119888119890119904 119900119891 119889119890119892119903119890119890 119891119894V119890

Δ (119866) + 4 119894119891 119892 (119866) ⩾ 4

Δ (119866) + 3 119894119891 119892 (119866) ⩾ 5

Δ (119866) + 2 119894119891 119892 (119866) ⩾ 6

Δ (119866) + 1 119894119891 119892 (119866) ⩾ 8

(65)

According toTheorem 157 119887RM(119862119899) = 3 = Δ(119862

119899)+1 for a

cycle 119862119899of length 119899 ⩾ 8 with 119899 equiv 2 (mod 3) and therefore

the last upper bound inTheorem 162 is sharp at least for Δ =2

Combining the fact that every planar graph 119866 withminimum degree 5 contains an edge 119909119910 with 119889

119866(119909) = 5

and 119889119866(119910) isin 5 6 with Theorem 161 (a) Akbari et al [125]

obtained the following result

26 International Journal of Combinatorics

middot middot middot

middot middot middot

middot middot middot

middot middot middot

119866

Figure 9 The graph 119866 is constructed from 119866

Theorem 163 (Akbari et al [125] 2012) 119887RM(119866) ⩽ 15 forevery planar graph 119866

It remains open to show whether the bound inTheorem 163 is sharp or not Though finding a planargraph 119866 with 119887RM(119866) = 15 seems to be difficult Akbari etal [125] constructed an infinite family of planar graphs withRoman bondage number equal to 7 by proving the followingresult

Theorem 164 (Akbari et al [125] 2012) Let 119866 be a graph oforder 119899 and 119866 is the graph of order 5119899 obtained from 119866 byattaching the central vertex of a copy of a path 119875

5to each vertex

of119866 (see Figure 9) Then 120574RM(119866) = 4119899 and 119887RM(119866) = 120575(119866)+2

ByTheorem 164 infinitemany planar graphswith Romanbondage number 7 can be obtained by considering any planargraph 119866 with 120575(119866) = 5 (eg the icosahedron graph)

Conjecture 165 (Akbari et al [125] 2012) The Romanbondage number of every planar graph is at most 7

For general bounds the following observation is directlyobtained which is similar to Observation 1

Observation 4 Let 119866 be a graph of order 119899 with maximumdegree at least two Assume that 119867 is a spanning subgraphobtained from 119866 by removing 119896 edges If 120574RM(119867) = 120574RM(119866)then 119887

119877(119867) ⩽ 119887RM(119866) ⩽ 119887RM(119867) + 119896

Theorem 166 (Bahremandpour et al [118] 2013) For anyconnected graph 119866 of order 119899 if 119899 ⩾ 3 and 120574RM(119866) = 120574(119866) + 1then

119887RM (119866) ⩽ min 119887 (119866) 119899Δ (66)

where 119899Δis the number of vertices with maximum degree Δ in

119866

Observation 5 For any graph 119866 if 120574(119866) = 120574RM(119866) then119887RM(119866) ⩽ 119887(119866)

Theorem 167 (Bahremandpour et al [118] 2013) For everyRoman graph 119866

119887RM (119866) ⩾ 119887 (119866) (67)

The bound is sharp for cycles on 119899 vertices where 119899 equiv 0 (mod3)

The strict inequality in Theorem 167 can hold for exam-ple 119887(119862

3119896+2) = 2 lt 3 = 119887RM(1198623119896+2

) byTheorem 157A graph 119866 is called to be vertex Roman domination-

critical if 120574RM(119866 minus 119909) lt 120574RM(119866) for every vertex 119909 in 119866 If119866 has no isolated vertices then 120574RM(119866) ⩽ 2120574(119866) ⩽ 2120573(119866)If 120574RM(119866) = 2120573(119866) then 120574RM(119866) = 2120574(119866) and hence 119866 is aRoman graph In [30] Volkmann gave a lot of graphs with120574(119866) = 120573(119866) The following result is similar to Theorem 57

Theorem 168 (Bahremandpour et al [118] 2013) For anygraph 119866 if 120574RM(119866) = 2120573(119866) then

(a) 119887RM(119866) ⩾ 120575(119866)

(b) 119887RM(119866) ⩾ 120575(119866)+1 if119866 is a vertex Roman domination-critical graph

If 120574RM(119866) = 2 then obviously 119887RM(119866) ⩽ 120575(119866) So weassume 120574RM(119866) ⩾ 3 Dehgardi et al [121] proved 119887RM(119866) ⩽(120574RM(119866) minus 2)Δ(119866) + 1 and posed the following problem if 119866is a connected graph of order 119899 ⩾ 4 with 120574RM(119866) ⩾ 3 then

119887RM (119866) ⩽ (120574RM (119866) minus 2) Δ (119866) (68)

Theorem 161 (a) shows that the inequality (68) holds if120574RM(119866) ⩾ 5 Thus the bound in (68) is of interest only when120574RM(119866) is 3 or 4

Lemma 169 (Bahremandpour et al [118] 2013) For anynonempty graph 119866 of order 119899 ⩾ 3 120574RM(119866) = 3 if and onlyif Δ(119866) = 119899 minus 2

The following result shows that (68) holds for all graphs119866 of order 119899 ⩾ 4 with 120574RM(119866) = 3 or 4 which improvesTheorem 161 (b)

Theorem 170 (Bahremandpour et al [118] 2013) For anyconnected graph 119866 of order 119899 ⩾ 4

119887RM (119866) le Δ (119866) = 119899 minus 2 119894119891 120574RM (119866) = 3

Δ (119866) + 120575 (119866) minus 1 119894119891 120574RM (119866) = 4(69)

with the first equality if and only if 119866 cong 1198624

Recently Akbari and Qajar [122] have proved the follow-ing result

Theorem 171 (Akbari and Qajar [122]) For any connectedgraph 119866 of order 119899 ⩾ 3

119887RM (119866) ⩽ 119899 minus 120574RM (119866) + 5 (70)

International Journal of Combinatorics 27

We conclude this section with the following problems

Problem 8 Characterize all connected graphs 119866 of order 119899 ⩾3 for which 119887RM(119866) = 119899 minus 1

Problem 9 Prove or disprove if 119866 is a connected graph oforder 119899 ⩾ 3 then

119887RM (119866) ⩽ 119899 minus 120574RM (119866) + 3 (71)

Note that 120574119877(119870

119905(3)) = 4 and 119887RM(119870119905

(3)) = 119899minus1 where 119899 =3119905 for 119905 ⩾ 3 The upper bound on 119887RM(119866) given in Problem 9is sharp if Problem 9 has positive answer

95 Rainbow Bondage Numbers For a positive integer 119896 a119896-rainbow dominating function of a graph 119866 is a function 119891from the vertex-set 119881(119866) to the set of all subsets of the set1 2 119896 such that for any vertex 119909 in 119866 with 119891(119909) = 0 thecondition

⋃

119910isin119873119866(119909)

119891 (119910) = 1 2 119896 (72)

is fulfilledThe weight of a 119896-rainbow dominating function 119891on 119866 is the value

119891 (119866) = sum

119909isin119881(119866)

1003816100381610038161003816119891 (119909)1003816100381610038161003816 (73)

The 119896-rainbow domination number of a graph 119866 denoted by120574119903119896(119866) is the minimum weight of a 119896-rainbow dominating

function 119891 of 119866Note that 120574

1199031(119866) is the classical domination number 120574(119866)

The 119896-rainbow domination number is introduced by Bresaret al [126] Rainbow domination of a graph 119866 coincides withordinary domination of the Cartesian product of 119866 with thecomplete graph in particular 120574

119903119896(119866) = 120574(119866 times 119870

119896) for any

positive integer 119896 and any graph 119866 [126] Hartnell and Rall[127] proved that 120574(119879) lt 120574

1199032(119879) for any tree 119879 no graph has

120574 = 120574119903119896for 119896 ⩾ 3 for any 119896 ⩾ 2 and any graph 119866 of order 119899

min 119899 120574 (119866) + 119896 minus 2 ⩽ 120574119903119896(119866) ⩽ 119896120574 (119866) (74)

The introduction of rainbow domination is motivatedby the study of paired domination in Cartesian productsof graphs where certain upper bounds can be expressed interms of rainbow domination that is for any graph 119866 andany graph 119867 if 119881(119867) can be partitioned into 119896120574

119875-sets then

120574119875(119866 times 119867) ⩽ (1119896)120592(119867)120574

119903119896(119866) proved by Bresar et al [126]

Dehgardi et al [128] introduced the concept of 119896-rainbowbondage number of graphs Let 119866 be a graph with maximumdegree at least two The 119896-rainbow bondage number 119887

119903119896(119866)

of 119866 is the minimum cardinality of all sets 119861 sube 119864(119866) forwhich 120574

119903119896(119866 minus 119861) gt 120574

119903119896(119866) Since in the study of 119896-rainbow

bondage number the assumption Δ(119866) ⩾ 2 is necessarywe always assume that when we discuss 119887

119903119896(119866) all graphs

involved satisfy Δ(119866) ⩾ 2The 2-rainbow bondage number of some special families

of graphs has been determined

Theorem 172 (Dehgardi et al [128]) For any integer 119899 ⩾ 3

1198871199032(119875

119899) = 1 119891119900119903 119899 ⩾ 2

1198871199032(119862

119899) =

1 119894119891 119899 equiv 0 (mod 4) 2 119900119905ℎ119890119903119908119894119904119890

1198871199032(119870

119899119899) =

1 119894119891 119899 = 2

3 119894119891 119899 = 3

6 119894119891 119899 = 4

119898 119894119891 119899 ⩾ 5

(75)

Theorem 173 (Dehgardi et al [128]) For any connected graph119866 of order 119899 ⩾ 3 119887

1199032(119866) ⩽ 120582(119866) + 2Δ(119866) minus 3

Theorem 174 (Dehgardi et al [128]) For any tree 119879 of order119899 ⩾ 3 119887

1199032(119879) ⩽ 2

Theorem 175 (Dehgardi et al [128]) If 119866 is a planar graphwithmaximumdegree at least two then 119887

1199032(119866) ⩽ 15Moreover

if the girth 119892(119866) ⩾ 4 then 1198871199032(119866) ⩽ 11 and if 119892(119866) ⩾ 6 then

1198871199032(119866) ⩽ 9

Theorem 176 (Dehgardi et al [128]) For a complete bipartitegraph 119870

119901119902 if 2119896 + 1 ⩽ 119901 ⩽ 119902 then 119887

119903119896(119870

119901119902) = 119901

Theorem177 (Dehgardi et al [128]) For a complete graph119870119899

if 119899 ⩾ 119896 + 1 then 119887119903119896(119870

119899) = lceil119896119899(119896 + 1)rceil

The special case 119896 = 1 of Theorem 177 can be found inTheorem 1 The following is a simple observation similar toObservation 1

Observation 6 (Dehgardi et al [128]) Let 119866 be a graphwith maximum degree at least two 119867 a spanning subgraphobtained from 119866 by removing 119896 edges If 120574

119903119896(119867) = 120574

119903119896(119866)

then 119887119903119896(119867) ⩽ 119887

119903119896(119866) ⩽ 119887

119903119896(119867) + 119896

Theorem 178 (Dehgardi et al [128]) For every graph 119866 with120574119903119896(119866) = 119896120574(119866) 119887

119903119896(119866) ⩾ 119887(119866)

A graph 119866 is said to be vertex 119896-rainbow domination-critical if 120574

119903119896(119866 minus 119909) lt 120574

119903119896(119866) for every vertex 119909 in 119866 It is

clear that if 119866 has no isolated vertices then 120574119903119896(119866) ⩽ 119896120574(119866) ⩽

119896120573(119866) where 120573(119866) is the vertex-covering number of119866 Thusif 120574

119903119896(119866) = 119896120573(119866) then 120574

119903119896(119866) = 119896120574(119866)

Theorem 179 (Dehgardi et al [128]) For any graph 119866 if120574119903119896(119866) = 119896120573(119866) then

(a) 119887119903119896(119866) ⩾ 120575(119866)

(b) 119887119903119896(119866) ⩾ 120575(119866)+1 if119866 is vertex 119896-rainbow domination-

critical

As far as we know there are no more results on the 119896-rainbow bondage number of graphs in the literature

28 International Journal of Combinatorics

10 Results on Digraphs

Although domination has been extensively studied in undi-rected graphs it is natural to think of a dominating setas a one-way relationship between vertices of the graphIndeed among the earliest literatures on this subject vonNeumann and Morgenstern [129] used what is now calleddomination in digraphs to find solution (or kernels whichare independent dominating sets) for cooperative 119899-persongames Most likely the first formulation of domination byBerge [130] in 1958 was given in the context of digraphs andonly some years latter by Ore [131] in 1962 for undirectedgraphs Despite this history examination of domination andits variants in digraphs has been essentially overlooked (see[4] for an overview of the domination literature) Thus thereare few if any such results on domination for digraphs in theliterature

The bondage number and its related topics for undirectedgraph have become one of major areas both in theoreticaland applied researches However until recently Carlsonand Develin [42] Shan and Kang [132] and Huang andXu [133 134] studied the bondage number for digraphsindependently In this section we introduce their resultsfor general digraphs Results for some special digraphs suchas vertex-transitive digraphs are introduced in the nextsection

101 Upper Bounds for General Digraphs Let 119866 = (119881 119864)

be a digraph without loops and parallel edges A digraph 119866is called to be asymmetric if whenever (119909 119910) isin 119864(119866) then(119910 119909) notin 119864(119866) and to be symmetric if (119909 119910) isin 119864(119866) implies(119910 119909) isin 119864(119866) For a vertex 119909 of 119881(119866) the sets of out-neighbors and in-neighbors of 119909 are respectively defined as119873

+

119866(119909) = 119910 isin 119881(119866) (119909 119910) isin 119864(119866) and 119873minus

119866(119909) = 119909 isin

119881(119866) (119909 119910) isin 119864(119866) the out-degree and the in-degree of119909 are respectively defined as 119889+

119866(119909) = |119873

+

119866(119909)| and 119889minus

119866(119909) =

|119873+

119866(119909)| Denote themaximum and theminimum out-degree

(resp in-degree) of 119866 by Δ+(119866) and 120575+(119866) (resp Δminus(119866) and120575minus

(119866))The degree 119889119866(119909) of 119909 is defined as 119889+

119866(119909)+119889

minus

119866(119909) and

the maximum and the minimum degrees of119866 are denoted byΔ(119866) and 120575(119866) respectively that is Δ(119866) = max119889

119866(119909) 119909 isin

119881(119866) and 120575(119866) = min119889119866(119909) 119909 isin 119881(119866) Note that the

definitions here are different from the ones in the textbookon digraphs see for example Xu [1]

A subset 119878 of119881(119866) is called a dominating set if119881(119866) = 119878cup119873

+

119866(119878) where119873+

119866(119878) is the set of out-neighbors of 119878Then just

as for undirected graphs 120574(119866) is the minimum cardinalityof a dominating set and the bondage number 119887(119866) is thesmallest cardinality of a set 119861 of edges such that 120574(119866 minus 119861) gt120574(119866) if such a subset 119861 sube 119864(119866) exists Otherwise we put119887(119866) = infin

Some basic results for undirected graphs stated inSection 3 can be generalized to digraphs For exampleTheorem 18 is generalized by Carlson and Develin [42]Huang andXu [133] and Shan andKang [132] independentlyas follows

Theorem 180 Let 119866 be a digraph and (119909 119910) isin 119864(119866) Then119887(119866) ⩽ 119889

119866(119910) + 119889

minus

119866(119909) minus |119873

minus

119866(119909) cap 119873

minus

119866(119910)|

Corollary 181 For a digraph 119866 119887(119866) ⩽ 120575minus(119866) + Δ(119866)

Since 120575minus

(119866) ⩽ (12)Δ(119866) from Corollary 181Conjecture 53 is valid for digraphs

Corollary 182 For a digraph 119866 119887(119866) ⩽ (32)Δ(119866)

In the case of undirected graphs the bondage number of119866119899= 119870

119899times 119870

119899achieves this bound However it was shown

by Carlson and Develin in [42] that if we take the symmetricdigraph119866

119899 we haveΔ(119866

119899) = 4(119899minus1) 120574(119866

119899) = 119899 and 119887(119866

119899) ⩽

4(119899 minus 1) + 1 = Δ(119866119899) + 1 So this family of digraphs cannot

show that the bound in Corollary 182 is tightCorresponding to Conjecture 51 which is discredited

for undirected graphs and is valid for digraphs the sameconjecture for digraphs can be proposed as follows

Conjecture 183 (Carlson and Develin [42] 2006) 119887(119866) ⩽Δ(119866) + 1 for any digraph 119866

If Conjecture 183 is true then the following results showsthat this upper bound is tight since 119887(119870

119899times119870

119899) = Δ(119870

119899times119870

119899)+1

for a complete digraph 119870119899

In 2007 Shan and Kang [132] gave some tight upperbounds on the bondage numbers for some asymmetricdigraphs For example 119887(119879) ⩽ Δ(119879) for any asymmetricdirected tree 119879 119887(119866) ⩽ Δ(119866) for any asymmetric digraph 119866with order at least 4 and 120574(119866) ⩽ 2 For planar digraphs theyobtained the following results

Theorem 184 (Shan and Kang [132] 2007) Let 119866 be aasymmetric planar digraphThen 119887(119866) ⩽ Δ(119866)+2 and 119887(119866) ⩽Δ(119866) + 1 if Δ(119866) ⩾ 5 and 119889minus

119866(119909) ⩾ 3 for every vertex 119909 with

119889119866(119909) ⩾ 4

102 Results for Some Special Digraphs The exact values andbounds on 119887(119866) for some standard digraphs are determined

Theorem185 (Huang andXu [133] 2006) For a directed cycle119862119899and a directed path 119875

119899

119887 (119862119899) =

3 119894119891 119899 119894119904 119900119889119889

2 119894119891 119899 119894119904 119890V119890119899

119887 (119875119899) =

2 119894119891 119899 119894119904 119900119889119889

1 119894119891 119899 119894119904 119890V119890119899

(76)

For the de Bruijn digraph 119861(119889 119899) and the Kautz digraph119870(119889 119899)

119887 (119861 (119889 119899)) = 119889 119894119891 119899 119894119904 119900119889119889

119889 ⩽ 119887 (119861 (119889 119899)) ⩽ 2119889 119894119891 119899 119894119904 119890V119890119899

119887 (119870 (119889 119899)) = 119889 + 1

(77)

Like undirected graphs we can define the total domi-nation number and the total bondage number On the totalbondage numbers for some special digraphs the knownresults are as follows

International Journal of Combinatorics 29

Theorem 186 (Huang and Xu [134] 2007) For a directedcycle 119862

119899and a directed path 119875

119899 119887

119879(119875

119899) and 119887

119879(119862

119899) all do not

exist For a complete digraph 119870119899

119887119879(119870

119899) = infin 119894119891 119899 = 1 2

119887119879(119870

119899) = 3 119894119891 119899 = 3

119899 ⩽ 119887119879(119870

119899) ⩽ 2119899 minus 3 119894119891 119899 ⩾ 4

(78)

The extended de Bruijn digraph EB(119889 119899 1199021 119902

119901) and

the extended Kautz digraph EK(119889 119899 1199021 119902

119901) are intro-

duced by Shibata and Gonda [135] If 119901 = 1 then theyare the de Bruijn digraph B(119889 119899) and the Kautz digraphK(119889 119899) respectively Huang and Xu [134] determined theirtotal domination numbers In particular their total bondagenumbers for general cases are determined as follows

Theorem 187 (Huang andXu [134] 2007) If 119889 ⩾ 2 and 119902119894⩾ 2

for each 119894 = 1 2 119901 then

119887119879(EB(119889 119899 119902

1 119902

119901)) = 119889

119901

minus 1

119887119879(EK(119889 119899 119902

1 119902

119901)) = 119889

119901

(79)

In particular for the de Bruijn digraph B(119889 119899) and the Kautzdigraph K(119889 119899)

119887119879(B (119889 119899)) = 119889 minus 1 119887

119879(K (119889 119899)) = 119889 (80)

Zhang et al [136] determined the bondage number incomplete 119905-partite digraphs

Theorem 188 (Zhang et al [136] 2009) For a complete 119905-partite digraph 119870

11989911198992119899119905

where 1198991⩽ 119899

2⩽ sdot sdot sdot ⩽ 119899

119905

119887 (11987011989911198992119899119905

)

=

119898 119894119891 119899119898= 1 119899

119898+1⩾ 2 119891119900119903 119904119900119898119890

119898 (1 ⩽ 119898 lt 119905)

4119905 minus 3 119894119891 1198991= 119899

2= sdot sdot sdot = 119899

119905= 2

119905minus1

sum

119894=1

119899119894+ 2 (119905 minus 1) 119900119905ℎ119890119903119908119894119904119890

(81)

Since an undirected graph can be thought of as a sym-metric digraph any result for digraphs has an analogy forundirected graphs in general In view of this point studyingthe bondage number for digraphs is more significant thanfor undirected graphs Thus we should further study thebondage number of digraphs and try to generalize knownresults on the bondage number and related variants for undi-rected graphs to digraphs prove or disprove Conjecture 183In particular determine the exact values of 119887(B(119889 119899)) for aneven 119899 and 119887

119879(119870

119899) for 119899 ⩾ 4

11 Efficient Dominating Sets

A dominating set 119878 of a graph 119866 is called to be efficient if forevery vertex 119909 in119866 |119873

119866[119909]cap119878| = 1 if119866 is a undirected graph

or |119873minus

119866[119909] cap 119878| = 1 if 119866 is a directed graph The concept of an

efficient set was proposed by Bange et al [137] in 1988 In theliterature an efficient set is also called a perfect dominatingset (see Livingston and Stout [138] for an explanation)

From definition if 119878 is an efficient dominating set of agraph 119866 then 119878 is certainly an independent set and everyvertex not in 119878 is adjacent to exactly one vertex in 119878

It is also clear from definition that a dominating set 119878 isefficient if and only if N

119866[119878] = 119873

119866[119909] 119909 isin 119878 for the

undirected graph 119866 or N+

119866[119878] = 119873

+

119866[119909] 119909 isin 119878 for the

digraph119866 is a partition of119881(119866) where the induced subgraphby119873

119866[119909] or119873+

119866[119909] is a star or an out-star with the root 119909

The efficient domination has important applications inmany areas such as error-correcting codes and receivesmuch attention in the late years

The concept of efficient dominating sets is a measureof the efficiency of domination in graphs Unfortunately asshown in [137] not every graph has an efficient dominatingset andmoreover it is anNP-complete problem to determinewhether a given graph has an efficient dominating set Inaddition it was shown by Clark [139] in 1993 that for a widerange of 119901 almost every random undirected graph 119866 isin

G(120592 119901) has no efficient dominating sets This means thatundirected graphs possessing an efficient dominating set arerare However it is easy to show that every undirected graphhas an orientationwith an efficient dominating set (see Bangeet al [140])

In 1993 Barkauskas and Host [141] showed that deter-mining whether an arbitrary oriented graph has an efficientdominating set is NP-complete Even so the existence ofefficient dominating sets for some special classes of graphs hasbeen examined see for example Dejter and Serra [142] andLee [143] for Cayley graph Gu et al [144] formeshes and toriObradovic et al [145] Huang and Xu [36] Reji Kumar andMacGillivray [146] for circulant graphs Harary graphs andtori and Van Wieren et al [147] for cube-connected cycles

In this section we introduce results of the bondagenumber for some graphs with efficient dominating sets dueto Huang and Xu [36]

111 Results for General Graphs In this subsection we intro-duce some results on bondage numbers obtained by applyingefficient dominating sets We first state the two followinglemmas

Lemma 189 For any 119896-regular graph or digraph 119866 of order 119899120574(119866) ⩾ 119899(119896 + 1) with equality if and only if 119866 has an efficientdominating set In addition if 119866 has an efficient dominatingset then every efficient dominating set is certainly a 120574-set andvice versa

Let 119890 be an edge and 119878 a dominating set in 119866 We say that119890 supports 119878 if 119890 isin (119878 119878) where (119878 119878) = (119909 119910) isin 119864(119866) 119909 isin 119878 119910 isin 119878 Denote by 119904(119866) the minimum number of edgeswhich support all 120574-sets in 119866

Lemma 190 For any graph or digraph 119866 119887(119866) ⩾ 119904(119866) withequality if 119866 is regular and has an efficient dominating set

30 International Journal of Combinatorics

A graph 119866 is called to be vertex-transitive if its automor-phism group Aut(119866) acts transitively on its vertex-set 119881(119866)A vertex-transitive graph is regular Applying Lemmas 189and 190 Huang and Xu obtained some results on bondagenumbers for vertex-transitive graphs or digraphs

Theorem 191 (Huang and Xu [36] 2008) For any vertex-transitive graph or digraph 119866 of order 119899

119887 (119866) ⩾

lceil119899

2120574 (119866)rceil 119894119891 119866 119894119904 119906119899119889119894119903119890119888119905119890119889

lceil119899

120574 (119866)rceil 119894119891 119866 119894119904 119889119894119903119890119888119905119890119889

(82)

Theorem192 (Huang andXu [36] 2008) Let119866 be a 119896-regulargraph of order 119899 Then

119887(119866) ⩽ 119896 if119866 is undirected and 119899 ⩾ 120574(119866)(119896+1)minus119896+1119887(119866) ⩽ 119896+1+ℓ if119866 is directed and 119899 ⩾ 120574(119866)(119896+1)minusℓwith 0 ⩽ ℓ ⩽ 119896 minus 1

Next we introduce a better upper bound on 119887(119866) To thisaim we need the following concept which is a generalizationof the concept of the edge-covering of a graph 119866 For 1198811015840

sube

119881(119866) and 1198641015840 sube 119864(119866) we say that 1198641015840covers 1198811015840 and call 1198641015840an edge-covering for 1198811015840 if there exists an edge (119909 119910) isin 1198641015840 forany vertex 119909 isin 1198811015840 For 119910 isin 119881(119866) let 1205731015840[119910] be the minimumcardinality over all edge-coverings for119873minus

119866[119910]

Theorem 193 (Huang and Xu [36] 2008) If 119866 is a 119896-regulargraph with order 120574(119866)(119896 + 1) then 119887(119866) ⩽ 1205731015840[119910] for any 119910 isin119881(119866)

Theupper bound on 119887(119866) given inTheorem 193 is tight inview of 119887(119862

119899) for a cycle or a directed cycle 119862

119899(seeTheorems

1 and 185 resp)It is easy to see that for a 119896-regular graph 119866 lceil(119896 + 1)2rceil ⩽

1205731015840

[119910] ⩽ 119896 when 119866 is undirected and 1205731015840[119910] = 119896 + 1 when 119866 isdirected By this fact and Lemma 189 the following theoremis merely a simple combination of Theorems 191 and 193

Theorem 194 (Huang and Xu [36] 2008) Let 119866 be a vertex-transitive graph of degree 119896 If 119866 has an efficient dominatingset then

lceil119896 + 1

2rceil ⩽ 119887 (119866) ⩽ 119896 119894119891 119866 119894119904 119906119899119889119894119903119890119888119905119890119889

119887 (119866) = 119896 + 1 119894119891 119866 119894119904 119889119894119903119890119888119905119890119889

(83)

Theorem 195 (Huang and Xu [36] 2008) If 119866 is an undi-rected vertex-transitive cubic graph with order 4120574(119866) and girth119892(119866) ⩽ 5 then 119887(119866) = 2

The proof of Theorem 195 leads to a byproduct If 119892 =5 let (119906

1 119906

2 119906

3 119906

4 119906

5) be a cycle and let D

119894be the family

of efficient dominating sets containing 119906119894for each 119894 isin

1 2 3 4 5 ThenD119894capD

119895= 0 for 119894 = 1 2 5 Then 119866 has

at least 5119904 efficient dominating sets where 119904 = 119904(119866) But thereare only 4s (=ns120574) distinct efficient dominating sets in119866This

contradiction implies that an undirected vertex-transitivecubic graph with girth five has no efficient dominating setsBut a similar argument for 119892(119866) = 3 4 or 119892(119866) ⩾ 6 couldnot give any contradiction This is consistent with the resultthat CCC(119899) a vertex-transitive cubic graph with girth 119899 if3 ⩽ 119899 ⩽ 8 or girth 8 if 119899 ⩾ 9 has efficient dominating setsfor all 119899 ⩾ 3 except 119899 = 5 (see Theorem 199 in the followingsubsection)

112 Results for Cayley Graphs In this subsection we useTheorem 194 to determine the exact values or approximativevalues of bondage numbers for some special vertex-transitivegraphs by characterizing the existence of efficient dominatingsets in these graphs

Let Γ be a nontrivial finite group 119878 a nonempty subset ofΓ without the identity element of Γ A digraph 119866 is defined asfollows

119881 (119866) = Γ (119909 119910) isin 119864 (119866) lArrrArr 119909minus1

119910 isin 119878

for any 119909 119910 isin Γ(84)

is called a Cayley digraph of the group Γ with respect to119878 denoted by 119862

Γ(119878) If 119878minus1 = 119904minus1 119904 isin 119878 = 119878 then

119862Γ(119878) is symmetric and is called a Cayley undirected graph

a Cayley graph for short Cayley graphs or digraphs arecertainly vertex-transitive (see Xu [1])

A circulant graph 119866(119899 119878) of order 119899 is a Cayley graph119862119885119899

(119878) where 119885119899= 0 119899 minus 1 is the addition group of

order 119899 and 119878 is a nonempty subset of119885119899without the identity

element and hence is a vertex-transitive digraph of degree|119878| If 119878minus1 = 119878 then119866(119899 119878) is an undirected graph If 119878 = 1 119896where 2 ⩽ 119896 ⩽ 119899 minus 2 we write 119866(119899 1 119896) for 119866(119899 1 119896) or119866(119899 plusmn1 plusmn119896) and call it a double-loop circulant graph

Huang and Xu [36] showed that for directed 119866 =

119866(119899 1 119896) lceil1198993rceil ⩽ 120574(119866) ⩽ lceil1198992rceil and 119866 has an efficientdominating set if and only if 3 | 119899 and 119896 equiv 2 (mod 3) fordirected 119866 = 119866(119899 1 119896) and 119896 = 1198992 lceil1198995rceil ⩽ 120574(119866) ⩽ lceil1198993rceiland 119866 has an efficient dominating set if and only if 5 | 119899 and119896 equiv plusmn2 (mod 5) By Theorem 194 we obtain the bondagenumber of a double loop circulant graph if it has an efficientdominating set

Theorem 196 (Huang and Xu [36] 2008) Let 119866 be a double-loop circulant graph 119866(119899 1 119896) If 119866 is directed with 3 | 119899and 119896 equiv 2 (mod 3) or 119866 is undirected with 5 | 119899 and119896 equiv plusmn2 (mod 5) then 119887(119866) = 3

The 119898 times 119899 torus is the Cartesian product 119862119898times 119862

119899of

two cycles and is a Cayley graph 119862119885119898times119885119899

(119878) where 119878 =

(0 1) (1 0) for directed cycles and 119878 = (0 plusmn1) (plusmn1 0) forundirected cycles and hence is vertex-transitive Gu et al[144] showed that the undirected torus119862

119898times119862

119899has an efficient

dominating set if and only if both 119898 and 119899 are multiplesof 5 Huang and Xu [36] showed that the directed torus119862119898times 119862

119899has an efficient dominating set if and only if both

119898 and 119899 are multiples of 3 Moreover they found a necessarycondition for a dominating set containing the vertex (0 0)in 119862

119898times 119862

119899to be efficient and obtained the following

result

International Journal of Combinatorics 31

Theorem 197 (Huang and Xu [36] 2008) Let 119866 = 119862119898times 119862

119899

If 119866 is undirected and both119898 and 119899 are multiples of 5 or if 119866 isdirected and both119898 and 119899 are multiples of 3 then 119887(119866) = 3

The hypercube 119876119899

is the Cayley graph 119862Γ(119878)

where Γ = 1198852times sdot sdot sdot times 119885

2= (119885

2)119899 and 119878 =

100 sdot sdot sdot 0 010 sdot sdot sdot 0 00 sdot sdot sdot 01 Lee [143] showed that119876119899has an efficient dominating set if and only if 119899 = 2119898 minus 1

for a positive integer119898 ByTheorem 194 the following resultis obtained immediately

Theorem 198 (Huang and Xu [36] 2008) If 119899 = 2119898 minus 1 for apositive integer119898 then 2119898minus1

⩽ 119887(119876119899) ⩽ 2

119898

minus 1

Problem 10 Determine the exact value of 119887(119876119899) for 119899 = 2119898minus1

The 119899-dimensional cube-connected cycle denoted byCCC(119899) is constructed from the 119899-dimensional hypercube119876119899by replacing each vertex 119909 in 119876

119899with an undirected cycle

119862119899of length 119899 and linking the 119894th vertex of the 119862

119899to the 119894th

neighbor of 119909 It has been proved that CCC(119899) is a Cayleygraph and hence is a vertex-transitive graph with degree 3Van Wieren et al [147] proved that CCC(119899) has an efficientdominating set if and only if 119899 = 5 From Theorems 194 and195 the following result can be derived

Theorem 199 (Huang and Xu [36] 2008) Let 119866 = 119862119862119862(119899)be the 119899-dimensional cube-connected cycles with 119899 ⩾ 3 and119899 = 5 Then 120574(119866) = 1198992119899minus2 and 2 ⩽ 119887(119866) ⩽ 3 In addition119887(119862119862119862(3)) = 119887(119862119862119862(4)) = 2

Problem 11 Determine the exact value of 119887(CCC(119899)) for 119899 ⩾5

Acknowledgments

This work was supported by NNSF of China (Grants nos11071233 61272008) The author would like to express hisgratitude to the anonymous referees for their kind sugges-tions and comments on the original paper to ProfessorSheikholeslami and Professor Jafari Rad for sending thereferences [128] and [81] which resulted in this version

References

[1] J-M Xu Theory and Application of Graphs vol 10 of NetworkTheory and Applications Kluwer Academic Dordrecht TheNetherlands 2003

[2] S THedetniemi andR C Laskar ldquoBibliography on dominationin graphs and some basic definitions of domination parame-tersrdquo Discrete Mathematics vol 86 no 1ndash3 pp 257ndash277 1990

[3] TW Haynes S T Hedetniemi and P J Slater Fundamentals ofDomination in Graphs vol 208 ofMonographs and Textbooks inPure and Applied Mathematics Marcel Dekker New York NYUSA 1998

[4] TWHaynes S THedetniemi and P J Slater EdsDominationin Graphs vol 209 of Monographs and Textbooks in Pure andAppliedMathematicsMarcelDekker NewYorkNYUSA 1998

[5] M R Garey andD S JohnsonComputers and Intractability WH Freeman and Company San Francisco Calif USA 1979

[6] H B Walikar and B D Acharya ldquoDomination critical graphsrdquoNational Academy Science Letters vol 2 pp 70ndash72 1979

[7] D Bauer F Harary J Nieminen and C L Suffel ldquoDominationalteration sets in graphsrdquo Discrete Mathematics vol 47 no 2-3pp 153ndash161 1983

[8] J F Fink M S Jacobson L F Kinch and J Roberts ldquoThebondage number of a graphrdquo Discrete Mathematics vol 86 no1ndash3 pp 47ndash57 1990

[9] F-T Hu and J-M Xu ldquoThe bondage number of (119899 minus 3)-regulargraph G of order nrdquo Ars Combinatoria to appear

[10] U Teschner ldquoNew results about the bondage number of agraphrdquoDiscreteMathematics vol 171 no 1ndash3 pp 249ndash259 1997

[11] B L Hartnell and D F Rall ldquoA bound on the size of a graphwith given order and bondage numberrdquo Discrete Mathematicsvol 197-198 pp 409ndash413 1999 16th British CombinatorialConference (London 1997)

[12] B L Hartnell and D F Rall ldquoA characterization of trees inwhich no edge is essential to the domination numberrdquo ArsCombinatoria vol 33 pp 65ndash76 1992

[13] J Topp and P D Vestergaard ldquo120572119896- and 120574

119896-stable graphsrdquo

Discrete Mathematics vol 212 no 1-2 pp 149ndash160 2000[14] A Meir and J W Moon ldquoRelations between packing and

covering numbers of a treerdquo Pacific Journal of Mathematics vol61 no 1 pp 225ndash233 1975

[15] B L Hartnell L K Joslashrgensen P D Vestergaard and CA Whitehead ldquoEdge stability of the k-domination numberof treesrdquo Bulletin of the Institute of Combinatorics and ItsApplications vol 22 pp 31ndash40 1998

[16] F-T Hu and J-M Xu ldquoOn the complexity of the bondage andreinforcement problemsrdquo Journal of Complexity vol 28 no 2pp 192ndash201 2012

[17] B L Hartnell and D F Rall ldquoBounds on the bondage numberof a graphrdquo Discrete Mathematics vol 128 no 1ndash3 pp 173ndash1771994

[18] Y-L Wang ldquoOn the bondage number of a graphrdquo DiscreteMathematics vol 159 no 1ndash3 pp 291ndash294 1996

[19] G H Fricke TW Haynes S M Hedetniemi S T Hedetniemiand R C Laskar ldquoExcellent treesrdquo Bulletin of the Institute ofCombinatorics and Its Applications vol 34 pp 27ndash38 2002

[20] V Samodivkin ldquoThe bondage number of graphs good and badverticesrdquoDiscussiones Mathematicae GraphTheory vol 28 no3 pp 453ndash462 2008

[21] J E Dunbar T W Haynes U Teschner and L VolkmannldquoBondage insensitivity and reinforcementrdquo in Domination inGraphs vol 209 of Monographs and Textbooks in Pure andAppliedMathematics pp 471ndash489 Dekker NewYork NY USA1998

[22] H Liu and L Sun ldquoThe bondage and connectivity of a graphrdquoDiscrete Mathematics vol 263 no 1ndash3 pp 289ndash293 2003

[23] A-H Esfahanian and S L Hakimi ldquoOn computing a con-ditional edge-connectivity of a graphrdquo Information ProcessingLetters vol 27 no 4 pp 195ndash199 1988

[24] R C Brigham P Z Chinn and R D Dutton ldquoVertexdomination-critical graphsrdquoNetworks vol 18 no 3 pp 173ndash1791988

[25] V Samodivkin ldquoDomination with respect to nondegenerateproperties bondage numberrdquoThe Australasian Journal of Com-binatorics vol 45 pp 217ndash226 2009

[26] U Teschner ldquoA new upper bound for the bondage numberof graphs with small domination numberrdquo The AustralasianJournal of Combinatorics vol 12 pp 27ndash35 1995

32 International Journal of Combinatorics

[27] J Fulman D Hanson and G MacGillivray ldquoVertex dom-ination-critical graphsrdquoNetworks vol 25 no 2 pp 41ndash43 1995

[28] U Teschner ldquoA counterexample to a conjecture on the bondagenumber of a graphrdquo Discrete Mathematics vol 122 no 1ndash3 pp393ndash395 1993

[29] U Teschner ldquoThe bondage number of a graph G can be muchgreater than Δ(G)rdquo Ars Combinatoria vol 43 pp 81ndash87 1996

[30] L Volkmann ldquoOn graphs with equal domination and coveringnumbersrdquoDiscrete AppliedMathematics vol 51 no 1-2 pp 211ndash217 1994

[31] L A Sanchis ldquoMaximum number of edges in connected graphswith a given domination numberrdquoDiscreteMathematics vol 87no 1 pp 65ndash72 1991

[32] S Klavzar and N Seifter ldquoDominating Cartesian products ofcyclesrdquoDiscrete AppliedMathematics vol 59 no 2 pp 129ndash1361995

[33] H K Kim ldquoA note on Vizingrsquos conjecture about the bondagenumberrdquo Korean Journal of Mathematics vol 10 no 2 pp 39ndash42 2003

[34] M Y Sohn X Yuan and H S Jeong ldquoThe bondage number of1198623times119862

119899rdquo Journal of the KoreanMathematical Society vol 44 no

6 pp 1213ndash1231 2007[35] L Kang M Y Sohn and H K Kim ldquoBondage number of the

discrete torus 119862119899times 119862

4rdquo Discrete Mathematics vol 303 no 1ndash3

pp 80ndash86 2005[36] J Huang and J-M Xu ldquoThe bondage numbers and efficient

dominations of vertex-transitive graphsrdquoDiscrete Mathematicsvol 308 no 4 pp 571ndash582 2008

[37] C J Xiang X Yuan andM Y Sohn ldquoDomination and bondagenumber of119862

3times119862

119899rdquoArs Combinatoria vol 97 pp 299ndash310 2010

[38] M S Jacobson and L F Kinch ldquoOn the domination numberof products of graphs Irdquo Ars Combinatoria vol 18 pp 33ndash441984

[39] Y-Q Li ldquoA note on the bondage number of a graphrdquo ChineseQuarterly Journal of Mathematics vol 9 no 4 pp 1ndash3 1994

[40] F-T Hu and J-M Xu ldquoBondage number of mesh networksrdquoFrontiers ofMathematics inChina vol 7 no 5 pp 813ndash826 2012

[41] R Frucht and F Harary ldquoOn the corona of two graphsrdquoAequationes Mathematicae vol 4 pp 322ndash325 1970

[42] K Carlson and M Develin ldquoOn the bondage number of planarand directed graphsrdquoDiscreteMathematics vol 306 no 8-9 pp820ndash826 2006

[43] U Teschner ldquoOn the bondage number of block graphsrdquo ArsCombinatoria vol 46 pp 25ndash32 1997

[44] J Huang and J-M Xu ldquoNote on conjectures of bondagenumbers of planar graphsrdquo Applied Mathematical Sciences vol6 no 65ndash68 pp 3277ndash3287 2012

[45] L Kang and J Yuan ldquoBondage number of planar graphsrdquoDiscrete Mathematics vol 222 no 1ndash3 pp 191ndash198 2000

[46] M Fischermann D Rautenbach and L Volkmann ldquoRemarkson the bondage number of planar graphsrdquo Discrete Mathemat-ics vol 260 no 1ndash3 pp 57ndash67 2003

[47] J Hou G Liu and J Wu ldquoBondage number of planar graphswithout small cyclesrdquoUtilitasMathematica vol 84 pp 189ndash1992011

[48] J Huang and J-M Xu ldquoThe bondage numbers of graphs withsmall crossing numbersrdquo Discrete Mathematics vol 307 no 15pp 1881ndash1897 2007

[49] Q Ma S Zhang and J Wang ldquoBondage number of 1-planargraphrdquo Applied Mathematics vol 1 no 2 pp 101ndash103 2010

[50] J Hou and G Liu ldquoA bound on the bondage number of toroidalgraphsrdquoDiscreteMathematics Algorithms and Applications vol4 no 3 Article ID 1250046 5 pages 2012

[51] Y-C Cao J-M Xu and X-R Xu ldquoOn bondage number oftoroidal graphsrdquo Journal of University of Science and Technologyof China vol 39 no 3 pp 225ndash228 2009

[52] Y-C Cao J Huang and J-M Xu ldquoThe bondage number ofgraphs with crossing number less than fourrdquo Ars Combinatoriato appear

[53] H K Kim ldquoOn the bondage numbers of some graphsrdquo KoreanJournal of Mathematics vol 11 no 2 pp 11ndash15 2004

[54] A Gagarin and V Zverovich ldquoUpper bounds for the bondagenumber of graphs on topological surfacesrdquo Discrete Mathemat-ics 2012

[55] J Huang ldquoAn improved upper bound for the bondage numberof graphs on surfacesrdquoDiscrete Mathematics vol 312 no 18 pp2776ndash2781 2012

[56] A Gagarin and V Zverovich ldquoThe bondage number of graphson topological surfaces and Teschnerrsquos conjecturerdquo DiscreteMathematics vol 313 no 6 pp 796ndash808 2013

[57] V Samodivkin ldquoNote on the bondage number of graphs ontopological surfacesrdquo httparxivorgabs12086203

[58] E J Cockayne R M Dawes and S T Hedetniemi ldquoTotaldomination in graphsrdquoNetworks vol 10 no 3 pp 211ndash219 1980

[59] R Laskar J Pfaff S M Hedetniemi and S T HedetniemildquoOn the algorithmic complexity of total dominationrdquo Society forIndustrial and Applied Mathematics vol 5 no 3 pp 420ndash4251984

[60] J Pfaff R C Laskar and S T Hedetniemi ldquoNP-completeness oftotal and connected domination and irredundance for bipartitegraphsrdquo Tech Rep 428 Department of Mathematical SciencesClemson University 1983

[61] M A Henning ldquoA survey of selected recent results on totaldomination in graphsrdquoDiscrete Mathematics vol 309 no 1 pp32ndash63 2009

[62] V R Kulli and D K Patwari ldquoThe total bondage number ofa graphrdquo in Advances in Graph Theory pp 227ndash235 VishwaGulbarga India 1991

[63] F-T Hu Y Lu and J-M Xu ldquoThe total bondage number ofgrid graphsrdquo Discrete Applied Mathematics vol 160 no 16-17pp 2408ndash2418 2012

[64] J Cao M Shi and B Wu ldquoResearch on the total bondagenumber of a spectial networkrdquo in Proceedings of the 3rdInternational Joint Conference on Computational Sciences andOptimization (CSO rsquo10) pp 456ndash458 May 2010

[65] N Sridharan MD Elias and V S A Subramanian ldquoTotalbondage number of a graphrdquo AKCE International Journal ofGraphs and Combinatorics vol 4 no 2 pp 203ndash209 2007

[66] N Jafari Rad and J Raczek ldquoSome progress on total bondage ingraphsrdquo Graphs and Combinatorics 2011

[67] T W Haynes and P J Slater ldquoPaired-domination and thepaired-domatic numberrdquoCongressusNumerantium vol 109 pp65ndash72 1995

[68] T W Haynes and P J Slater ldquoPaired-domination in graphsrdquoNetworks vol 32 no 3 pp 199ndash206 1998

[69] X Hou ldquoA characterization of 2120574 120574119875-treesrdquoDiscrete Mathemat-

ics vol 308 no 15 pp 3420ndash3426 2008[70] K E Proffitt T W Haynes and P J Slater ldquoPaired-domination

in grid graphsrdquo Congressus Numerantium vol 150 pp 161ndash1722001

International Journal of Combinatorics 33

[71] H Qiao L KangM Cardei andD-Z Du ldquoPaired-dominationof treesrdquo Journal of Global Optimization vol 25 no 1 pp 43ndash542003

[72] E Shan L Kang and M A Henning ldquoA characterizationof trees with equal total domination and paired-dominationnumbersrdquo The Australasian Journal of Combinatorics vol 30pp 31ndash39 2004

[73] J Raczek ldquoPaired bondage in treesrdquo Discrete Mathematics vol308 no 23 pp 5570ndash5575 2008

[74] J A Telle and A Proskurowski ldquoAlgorithms for vertex parti-tioning problems on partial k-treesrdquo SIAM Journal on DiscreteMathematics vol 10 no 4 pp 529ndash550 1997

[75] G S Domke J H Hattingh M A Henning and L R MarkusldquoRestrained domination in treesrdquoDiscreteMathematics vol 211no 1ndash3 pp 1ndash9 2000

[76] G S Domke J H Hattingh M A Henning and L R MarkusldquoRestrained domination in graphs with minimum degree twordquoJournal of Combinatorial Mathematics and Combinatorial Com-puting vol 35 pp 239ndash254 2000

[77] G S Domke J H Hattingh S T Hedetniemi R C Laskarand L R Markus ldquoRestrained domination in graphsrdquo DiscreteMathematics vol 203 no 1ndash3 pp 61ndash69 1999

[78] J H Hattingh and E J Joubert ldquoAn upper bound for therestrained domination number of a graph with minimumdegree at least two in terms of order and minimum degreerdquoDiscrete Applied Mathematics vol 157 no 13 pp 2846ndash28582009

[79] M A Henning ldquoGraphs with large restrained dominationnumberrdquo Discrete Mathematics vol 197-198 pp 415ndash429 199916th British Combinatorial Conference (London 1997)

[80] J H Hattingh and A R Plummer ldquoRestrained bondage ingraphsrdquo Discrete Mathematics vol 308 no 23 pp 5446ndash54532008

[81] N Jafari Rad R Hasni J Raczek and L Volkmann ldquoTotalrestrained bondage in graphsrdquoActaMathematica Sinica vol 29no 6 pp 1033ndash1042 2013

[82] J Cyman and J Raczek ldquoOn the total restrained dominationnumber of a graphrdquoThe Australasian Journal of Combinatoricsvol 36 pp 91ndash100 2006

[83] M A Henning ldquoGraphs with large total domination numberrdquoJournal of Graph Theory vol 35 no 1 pp 21ndash45 2000

[84] D-X Ma X-G Chen and L Sun ldquoOn total restraineddomination in graphsrdquoCzechoslovakMathematical Journal vol55 no 1 pp 165ndash173 2005

[85] B Zelinka ldquoRemarks on restrained domination and totalrestrained domination in graphsrdquo Czechoslovak MathematicalJournal vol 55 no 2 pp 393ndash396 2005

[86] W Goddard and M A Henning ldquoIndependent domination ingraphs a survey and recent resultsrdquo Discrete Mathematics vol313 no 7 pp 839ndash854 2013

[87] J H Zhang H L Liu and L Sun ldquoIndependence bondagenumber and reinforcement number of some graphsrdquo Transac-tions of Beijing Institute of Technology vol 23 no 2 pp 140ndash1422003

[88] K D Dharmalingam Studies in Graph Theorymdashequitabledomination and bottleneck domination [PhD thesis] MaduraiKamaraj University Madurai India 2006

[89] GDeepakND Soner andAAlwardi ldquoThe equitable bondagenumber of a graphrdquo Research Journal of Pure Algebra vol 1 no9 pp 209ndash212 2011

[90] E Sampathkumar and H B Walikar ldquoThe connected domina-tion number of a graphrdquo Journal of Mathematical and PhysicalSciences vol 13 no 6 pp 607ndash613 1979

[91] K White M Farber and W Pulleyblank ldquoSteiner trees con-nected domination and strongly chordal graphsrdquoNetworks vol15 no 1 pp 109ndash124 1985

[92] M Cadei M X Cheng X Cheng and D-Z Du ldquoConnecteddomination in Ad Hoc wireless networksrdquo in Proceedings ofthe 6th International Conference on Computer Science andInformatics (CSampI rsquo02) 2002

[93] J F Fink and M S Jacobson ldquon-domination in graphsrdquo inGraph Theory with Applications to Algorithms and ComputerScience Wiley-Interscience Publications pp 283ndash300 WileyNew York NY USA 1985

[94] M Chellali O Favaron A Hansberg and L Volkmann ldquok-domination and k-independence in graphs a surveyrdquo Graphsand Combinatorics vol 28 no 1 pp 1ndash55 2012

[95] Y Lu X Hou J-M Xu andN Li ldquoTrees with uniqueminimump-dominating setsrdquo Utilitas Mathematica vol 86 pp 193ndash2052011

[96] Y Lu and J-M Xu ldquoThe p-bondage number of treesrdquo Graphsand Combinatorics vol 27 no 1 pp 129ndash141 2011

[97] Y Lu and J-M Xu ldquoThe 2-domination and 2-bondage numbersof grid graphs A manuscriptrdquo httparxivorgabs12044514

[98] M Krzywkowski ldquoNon-isolating 2-bondage in graphsrdquo TheJournal of the Mathematical Society of Japan vol 64 no 4 2012

[99] M A Henning O R Oellermann and H C Swart ldquoBoundson distance domination parametersrdquo Journal of CombinatoricsInformation amp System Sciences vol 16 no 1 pp 11ndash18 1991

[100] F Tian and J-M Xu ldquoBounds for distance domination numbersof graphsrdquo Journal of University of Science and Technology ofChina vol 34 no 5 pp 529ndash534 2004

[101] F Tian and J-M Xu ldquoDistance domination-critical graphsrdquoApplied Mathematics Letters vol 21 no 4 pp 416ndash420 2008

[102] F Tian and J-M Xu ldquoA note on distance domination numbersof graphsrdquo The Australasian Journal of Combinatorics vol 43pp 181ndash190 2009

[103] F Tian and J-M Xu ldquoAverage distances and distance domina-tion numbersrdquo Discrete Applied Mathematics vol 157 no 5 pp1113ndash1127 2009

[104] Z-L Liu F Tian and J-M Xu ldquoProbabilistic analysis of upperbounds for 2-connected distance k-dominating sets in graphsrdquoTheoretical Computer Science vol 410 no 38ndash40 pp 3804ndash3813 2009

[105] M A Henning O R Oellermann and H C Swart ldquoDistancedomination critical graphsrdquo Journal of Combinatorial Mathe-matics and Combinatorial Computing vol 44 pp 33ndash45 2003

[106] T J Bean M A Henning and H C Swart ldquoOn the integrityof distance domination in graphsrdquo The Australasian Journal ofCombinatorics vol 10 pp 29ndash43 1994

[107] M Fischermann and L Volkmann ldquoA remark on a conjecturefor the (kp)-domination numberrdquoUtilitasMathematica vol 67pp 223ndash227 2005

[108] T Korneffel D Meierling and L Volkmann ldquoA remark on the(22)-domination numberrdquo Discussiones Mathematicae GraphTheory vol 28 no 2 pp 361ndash366 2008

[109] Y Lu X Hou and J-M Xu ldquoOn the (22)-domination numberof treesrdquo Discussiones Mathematicae Graph Theory vol 30 no2 pp 185ndash199 2010

34 International Journal of Combinatorics

[110] H Li and J-M Xu ldquo(dm)-domination numbers of m-connected graphsrdquo Rapports de Recherche 1130 LRI UniversiteParis-Sud 1997

[111] S M Hedetniemi S T Hedetniemi and T V Wimer ldquoLineartime resource allocation algorithms for treesrdquo Tech Rep URI-014 Department of Mathematical Sciences Clemson Univer-sity 1987

[112] M Farber ldquoDomination independent domination and dualityin strongly chordal graphsrdquo Discrete Applied Mathematics vol7 no 2 pp 115ndash130 1984

[113] G S Domke and R C Laskar ldquoThe bondage and reinforcementnumbers of 120574

119891for some graphsrdquoDiscrete Mathematics vol 167-

168 pp 249ndash259 1997[114] G S Domke S T Hedetniemi and R C Laskar ldquoFractional

packings coverings and irredundance in graphsrdquo vol 66 pp227ndash238 1988

[115] E J Cockayne P A Dreyer Jr S M Hedetniemi andS T Hedetniemi ldquoRoman domination in graphsrdquo DiscreteMathematics vol 278 no 1ndash3 pp 11ndash22 2004

[116] I Stewart ldquoDefend the roman empirerdquo Scientific American vol281 pp 136ndash139 1999

[117] C S ReVelle and K E Rosing ldquoDefendens imperiumromanum a classical problem in military strategyrdquoThe Ameri-can Mathematical Monthly vol 107 no 7 pp 585ndash594 2000

[118] A Bahremandpour F-T Hu S M Sheikholeslami and J-MXu ldquoOn the Roman bondage number of a graphrdquo DiscreteMathematics Algorithms and Applications vol 5 no 1 ArticleID 1350001 15 pages 2013

[119] N Jafari Rad and L Volkmann ldquoRoman bondage in graphsrdquoDiscussiones Mathematicae Graph Theory vol 31 no 4 pp763ndash773 2011

[120] K Ebadi and L PushpaLatha ldquoRoman bondage and Romanreinforcement numbers of a graphrdquo International Journal ofContemporary Mathematical Sciences vol 5 pp 1487ndash14972010

[121] N Dehgardi S M Sheikholeslami and L Volkman ldquoOn theRoman k-bondage number of a graphrdquo AKCE InternationalJournal of Graphs and Combinatorics vol 8 pp 169ndash180 2011

[122] S Akbari and S Qajar ldquoA note on Roman bondage number ofgraphsrdquo Ars Combinatoria to Appear

[123] F-T Hu and J-M Xu ldquoRoman bondage numbers of somegraphsrdquo httparxivorgabs11093933

[124] N Jafari Rad and L Volkmann ldquoOn the Roman bondagenumber of planar graphsrdquo Graphs and Combinatorics vol 27no 4 pp 531ndash538 2011

[125] S Akbari M Khatirinejad and S Qajar ldquoA note on the romanbondage number of planar graphsrdquo Graphs and Combinatorics2012

[126] B Bresar M A Henning and D F Rall ldquoRainbow dominationin graphsrdquo Taiwanese Journal of Mathematics vol 12 no 1 pp213ndash225 2008

[127] B L Hartnell and D F Rall ldquoOn dominating the Cartesianproduct of a graph and 120572krdquo Discussiones Mathematicae GraphTheory vol 24 no 3 pp 389ndash402 2004

[128] N Dehgardi S M Sheikholeslami and L Volkman ldquoThe k-rainbow bondage number of a graphrdquo to appear in DiscreteApplied Mathematics

[129] J von Neumann and O Morgenstern Theory of Games andEconomic Behavior Princeton University Press Princeton NJUSA 1944

[130] C BergeTheorıe des Graphes et ses Applications 1958[131] O Ore Theory of Graphs vol 38 American Mathematical

Society Colloquium Publications 1962[132] E Shan and L Kang ldquoBondage number in oriented graphsrdquoArs

Combinatoria vol 84 pp 319ndash331 2007[133] J Huang and J-M Xu ldquoThe bondage numbers of extended

de Bruijn and Kautz digraphsrdquo Computers amp Mathematics withApplications vol 51 no 6-7 pp 1137ndash1147 2006

[134] J Huang and J-M Xu ldquoThe total domination and total bondagenumbers of extended de Bruijn and Kautz digraphsrdquoComputersamp Mathematics with Applications vol 53 no 8 pp 1206ndash12132007

[135] Y Shibata and Y Gonda ldquoExtension of de Bruijn graph andKautz graphrdquo Computers amp Mathematics with Applications vol30 no 9 pp 51ndash61 1995

[136] X Zhang J Liu and JMeng ldquoThebondage number in completet-partite digraphsrdquo Information Processing Letters vol 109 no17 pp 997ndash1000 2009

[137] D W Bange A E Barkauskas and P J Slater ldquoEfficient dom-inating sets in graphsrdquo in Applications of Discrete Mathematicspp 189ndash199 SIAM Philadelphia Pa USA 1988

[138] M Livingston and Q F Stout ldquoPerfect dominating setsrdquoCongressus Numerantium vol 79 pp 187ndash203 1990

[139] L Clark ldquoPerfect domination in random graphsrdquo Journal ofCombinatorial Mathematics and Combinatorial Computing vol14 pp 173ndash182 1993

[140] D W Bange A E Barkauskas L H Host and L H ClarkldquoEfficient domination of the orientations of a graphrdquo DiscreteMathematics vol 178 no 1ndash3 pp 1ndash14 1998

[141] A E Barkauskas and L H Host ldquoFinding efficient dominatingsets in oriented graphsrdquo Congressus Numerantium vol 98 pp27ndash32 1993

[142] I J Dejter and O Serra ldquoEfficient dominating sets in CayleygraphsrdquoDiscrete AppliedMathematics vol 129 no 2-3 pp 319ndash328 2003

[143] J Lee ldquoIndependent perfect domination sets in Cayley graphsrdquoJournal of Graph Theory vol 37 no 4 pp 213ndash219 2001

[144] WGu X Jia and J Shen ldquoIndependent perfect domination setsin meshes tori and treesrdquo Unpublished

[145] N Obradovic J Peters and G Ruzic ldquoEfficient domination incirculant graphswith two chord lengthsrdquo Information ProcessingLetters vol 102 no 6 pp 253ndash258 2007

[146] K Reji Kumar and G MacGillivray ldquoEfficient domination incirculant graphsrdquo Discrete Mathematics vol 313 no 6 pp 767ndash771 2013

[147] D Van Wieren M Livingston and Q F Stout ldquoPerfectdominating sets on cube-connected cyclesrdquo Congressus Numer-antium vol 97 pp 51ndash70 1993

Submit your manuscripts athttpwwwhindawicom

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

International Journal of Combinatorics 7

Theorems 13ndash20 provides it is not easy to determine the sets119860+

(119866) and 119861(119866) mentioned in Theorem 22 for an arbitrarygraph 119866 Thus the upper bound given in Theorem 22 is oftheoretical importance but not applied since until now wehave not found a new class of graphswhose bondage numbersare determined byTheorem 22

The above-mentioned upper bounds on the bondagenumber are involved in only degrees of two vertices Hartnelland Rall [11] established an upper bound on 119887(119866) in terms ofthe numbers of vertices and edges of 119866 with order 119899 For anyconnected graph 119866 let 120575(119866) represent the average degree ofvertices in 119866 that is 120575(119866) = (1119899)sum

119909isin119881119889119866(119909) Hartnell and

Rall first discovered the following proposition

Proposition 25 For any connected graph 119866 there exist twovertices 119909 and 119910 with distance at most two and with theproperty that 119889

119866(119909) + 119889

119866(119910) ⩽ 2120575(119866)

Using Proposition 25 and Theorem 16 Hartnell and Rallgave the following bound

Theorem 26 (Hartnell and Rall [11] 1999) 119887(119866) ⩽ 2120575(119866) minus 1for any connected graph 119866

Note that 2120576 = 119899120575(119866) for any graph 119866 with order 119899 andsize 120576 Theorem 26 implies the following bound in terms oforder 119899 and size 120576

Corollary 27 119887(119866) ⩽ (4120576119899) minus 1 for any connected graph 119866with order 119899 and size 120576

A lower bound on 120576 in terms of order 119899 and the bondagenumber is obtained from Corollary 28 immediately

Corollary 28 120576 ⩾ (1198994)(119887(119866) + 1) for any connected graph 119866with order 119899 and size 120576

Hartnell and Rall [11] gave some graphs 119866 with order 119899to show that for each value of 119887(119866) the lower bound on 120576(119866)given in the Corollary 28 is sharp for some values of 119899

32 Bounds Implied by Connectivity Use 120581(119866) and 120582(119866) todenote the vertex-connectivity and the edge-connectivity ofa connected graph 119866 respectively which are the minimumnumbers of vertices and edges whose removal result in 119866disconnected The famous Whitney inequality is stated as120581(119866) ⩽ 120582(119866) ⩽ 120575(119866) for any graph or digraph 119866 Corollary 15is improved by several authors as follows

Theorem 29 (Hartnell and Rall [17] 1994 Teschner [10]1997) If 119866 is a connected graph then 119887(119866) ⩽ 120582(119866)+Δ(119866)minus 1

The upper bound given in Theorem 29 can be attainedFor example a cycle 119862

3119896+1with 119896 ⩾ 1 119887(119862

3119896+1) = 3 by

Theorem 1 Since 120581(1198623119896+1) = 120582(119862

3119896+1) = 2 we have 120581(119862

3119896+1)+

120582(1198623119896+1) minus 1 = 2 + 2 minus 1 = 3 = 119887(119862

3119896+1)

Motivated byCorollary 15Theorems 29 and theWhitneyinequality Dunbar et al [21] naturally proposed the followingconjecture

Figure 4 A graph119867 with 120574(119867) = 3 and 119887(119867) = 5

Conjecture 30 If119866 is a connected graph then 119887(119866) ⩽ Δ(119866)+120581(119866) minus 1

However Liu and Sun [22] presented a counterexampleto this conjecture They first constructed a graph 119867 showedin Figure 4 with 120574(119867) = 3 and 119887(119867) = 5 Then let 119866 be thedisjoint union of two copies of119867 by identifying two verticesof degree two They proved that 119887(119866) ⩾ 5 Clearly 119866 is a 4-regular graph with 120581(119866) = 1 and 120582(119866) = 2 and so 119887(119866) ⩽ 5byTheorem 29 Thus 119887(119866) = 5 gt 4 = Δ(119866) + 120581(119866) minus 1

With a suspicion of the relationship between the bondagenumber and the vertex-connectivity of a graph the followingconjecture is proposed

Conjecture 31 (Liu and Sun [22] 2003) For any positiveinteger 119903 there exists a connected graph 119866 such that 119887(119866) ⩾Δ(119866) + 120581(119866) + 119903

To the knowledge of the author until now no results havebeen known about this conjecture

We conclude this subsection with the following remarksFromTheorem 29 if Conjecture 31 holds for some connectedgraph119866 then 120582(119866) gt 120581(119866)+119903 which implies that119866 is of largeedge-connectivity and small vertex-connectivity Use 120585(119866) todenote the minimum edge-degree of 119866 that is

120585 (119866) = min119909119910isin119864(119866)

119889119866(119909) + 119889

119866(119910) minus 2 (14)

Theorem 14 implies the following result

Proposition 32 119887(119866) ⩽ 120585(119866) + 1 for any graph 119866

Use 1205821015840(119866) to denote the restricted edge-connectivity ofa connected graph 119866 which is the minimum number ofedgeswhose removal result in119866 disconnected and no isolatedvertices

Proposition 33 (Esfahanian and Hakimi [23] 1988) If 119866 isneither 119870

1119899nor 119870

3 then 1205821015840(119866) ⩽ 120585(119866)

Combining Propositions 32 and 33 we propose a conjec-ture

Conjecture 34 If a connected 119866 is neither 1198701119899

nor 1198703 then

119887(119866) ⩽ 120575(119866) + 1205821015840

(119866) minus 1

8 International Journal of Combinatorics

A cycle 1198623119896+1

also satisfies 119887(1198623119896+1) = 3 = 120575(119862

3119896+1) +

1205821015840

(1198623119896+1) minus 1 since 1205821015840(119862

3119896+1) = 120575(119862

3119896+1) = 2 for any integer

119896 ⩾ 1For the graph119867 shown in Figure 41205821015840(119867) = 4 and 120575(119867) =

2 and so 119887(119867) = 5 = 120575(119867) + 1205821015840(119867) minus 1For the 4-regular graph119866 constructed by Liu and Sun [22]

obtained from the disjoint union of two copies of the graph119867 showed in Figure 4 by identifying two vertices of degreetwo we have 119887(119866) ⩾ 5 Clearly 1205821015840(119866) = 2 Thus 119887(119866) = 5 =120575(119866) + 120582

1015840

(119866) minus 1For the 4-regular graph 119866

119905constructed by Samodivkin

[20] see Figure 3 for 119905 = 2 119887(119866119905) = 2 Clearly 120575(119866

119905) = 1

and 1205821015840(119866) = 2 Thus 119887(119866) = 2 = 120575(119866) + 1205821015840(119866) minus 1These examples show that if Conjecture 34 is true then the

given upper bound is tight

33 Bounds Implied by Degree Sequence Now let us return toTheorem 16 from which Teschner [10] obtained some otherbounds in terms of the degree sequencesThe degree sequence120587(119866) of a graph 119866 with vertex-set 119881 = 119909

1 119909

2 119909

119899 is the

sequence 120587 = (1198891 119889

2 119889

119899) with 119889

1⩽ 119889

2⩽ sdot sdot sdot ⩽ 119889

119899 where

119889119894= 119889

119866(119909

119894) for each 119894 = 1 2 119899 The following result is

essentially a corollary of Theorem 16

Theorem35 (Teschner [10] 1997) Let119866 be a nonempty graphwith degree sequence120587(119866) If120572

2(119866) = 119905 then 119887(119866) ⩽ 119889

119905+119889

119905+1minus

1

CombiningTheorem 35with Proposition 10 (ie 1205722(119866) ⩽

120574(119866)) we have the following corollary

Corollary 36 (Teschner [10] 1997) Let 119866 be a nonemptygraph with the degree sequence 120587(119866) If 120574(119866) = 120574 then 119887(119866) ⩽119889120574+ 119889

120574+1minus 1

In [10] Teschner showed that these two bounds are sharpfor arbitrarily many graphs Let119867 = 119862

3119896+1+ 119909

11199094 119909

11199093119896minus1

where 1198623119896+1

is a cycle (1199091 119909

2 119909

3119896+1 119909

1) for any integer

119896 ⩾ 2 Then 120574(119867) = 119896 + 1 and so 119887(119867) ⩽ 2 + 2 minus 1 = 3by Corollary 36 Since119862

3119896+1is a spanning subgraph of119867 and

120574(119866) = 120574(119867) Observation 1 yields that 119887(119867) ⩾ 119887(1198623119896+1) = 3

and so 119887(119867) = 3Although various upper bounds have been established

as the above we find that the appearance of these boundsis essentially based upon the local structures of a graphprecisely speaking the structures of the neighborhoods oftwo vertices within distance 2 Even if these bounds can beachieved by some special graphs it is more often not the caseThe reason lies essentially in the definition of the bondagenumber which is theminimumvalue among all bondage setsan integral property of a graph While it easy to find upperbounds just by choosing some bondage set the gap betweenthe exact value of the bondage number and such a boundobtained only from local structures of a graph is often largeFor example a star 119870

1Δ however large Δ is 119887(119870

1Δ) = 1

Therefore one has been longing for better bounds upon someintegral parameters However as what we will see below it isdifficult to establish such upper bounds

34 Bounds in 120574-Critical Graphs A graph119866 is called a vertexdomination-critical graph (vc-graph or 120574-critical for short) if120574(119866 minus 119909) lt 120574(119866) for any 119909 isin 119881(119866) proposed by Brighamet al [24] in 1988 Several families of graphs are known to be120574-critical From definition it is clear that if 119866 is a 120574-criticalgraph then 120574(119866) ⩾ 2The class of 120574-critical graphs with 120574 = 2is characterized as follows

Proposition 37 (Brigham et al [24] 1988) A graph 119866 with120574(119866) = 2 is a 120574-critical graph if and only if 119866 is a completegraph 119870

2119905(119905 ⩾ 2) with a perfect matching removed

A more interesting family is composed of the 119899-criticalgraphs 119866

119898119899defined for 119898 119899 ⩾ 2 by the circulant undirected

graph 119866(119873 plusmn119878) where 119873 = (119898 + 1)(119899 minus 1) + 1 and119878 = 1 2 lfloor1198982rfloor in which 119881(119866(119873 plusmn119878)) = 119873 and119864(119866(119873 plusmn119878)) = 119894119895 |119895 minus 119894| equiv 119904 (mod 119873) 119904 isin 119878

The reason why 120574-critical graphs are of special interest inthis context is easy to see that they play an important role instudy of the bondage number For instance it immediatelyfollows from Theorem 13 that if 119887(119866) gt Δ(119866) then 119866 is a 120574-critical graphThe 120574-critical graphs are defined exactly in thisway In order to find graphs 119866 with a high bondage number(ie higher than Δ(119866)) and beyond its general upper boundsfor the bondage number we therefore have to look at 120574-critical graphs

The bondage numbers of some 120574-critical graphs havebeen examined by several authors see for example [1020 25 26] From Theorem 13 we know that the bondagenumber of a graph 119866 is bounded from above by Δ(119866)if 119866 is not a 120574-critical graph For 120574-critical graphs it ismore difficult to find an upper bound We will see that thebondage numbers of 120574-critical graphs in general are noteven bounded from above by Δ + 119888 for any fixed naturalnumber 119888

In this subsection we introduce some upper bounds forthe bondage number of a 120574-critical graph By Proposition 37we easily obtain the following result

Theorem 38 If 119866 is a 120574-critical graph with 120574(119866) = 2 then119887(119866) ⩽ Δ + 1

In Section 4 by Theorem 59 we will see that the equalityin Theorem 38 holds that is 119887(119866) = Δ + 1 if 119866 is a 120574-criticalgraph with 120574(119866) = 2

Theorem 39 (Teschner [10] 1997) Let119866 be a 120574-critical graphwith degree sequence 120587(119866) Then 119887(119866) ⩽ maxΔ(119866) + 1 119889

1+

1198892+ sdot sdot sdot + 119889

120574minus1 where 120574 = 120574(119866)

As we mentioned above if 119866 is a 120574-critical graph with120574(119866) = 2 then 119887(119866) = Δ + 1 which shows that thebound given in Theorem 39 can be attained for 120574 = 2However we have not known whether this bound is tightfor general 120574 ⩾ 3 Theorem 39 gives the following corollaryimmediately

Corollary 40 (Teschner [10] 1997) Let 119866 be a 120574-criticalgraph with degree sequence 120587(119866) If 120574(119866) = 3 then 119887(119866) ⩽maxΔ(119866) + 1 120575(119866) + 119889

2

International Journal of Combinatorics 9

From Theorem 38 and Corollary 40 we have 119887(119866) ⩽2Δ(119866) if 119866 is a 120574-critical graph with 120574(119866) ⩽ 3 The followingresult shows that this bound is not tight

Theorem 41 (Teschner [26] 1995) Let119866 be a 120574-critical graphgraph with 120574(119866) ⩽ 3 Then 119887(119866) ⩽ (32)Δ(119866)

Until now we have not known whether the bound givenin Theorem 41 is tight or not We state two conjectures on120574-critical graphs proposed by Samodivkin [20] The first ofthem is motivated byTheorems 22 and 19

Conjecture 42 (Samodivkin [20] 2008) For every connectednontrivial 120574-critical graph 119866

min119909isin119860+(119866)119910isin119861(119866minus119909)

119889119866(119909) +

1003816100381610038161003816119873119866(119910) cap 119860 (119866 minus 119909)

1003816100381610038161003816 ⩽3

2Δ (119866)

(15)

To state the second conjecture we need the followingresult on 120574-critical graphs

Proposition 43 If119866 is a 120574-critical graph then 120592(119866) ⩽ (Δ(119866)+1)(120574(119866)minus1)+1 moreover if the equality holds then119866 is regular

The upper bound in Proposition 43 is sharp in thesense that equality holds for the infinite class of 120574-criticalgraphs 119866

119898119899defined in the beginning of this subsection In

Proposition 43 the first result is due to Brigham et al [24] in1988 the second is due to Fulman et al [27] in 1995

For a 120574-critical graph 119866 with 120574 = 3 by Proposition 43120592(119866) ⩽ 2Δ(119866) + 3

Theorem 44 (Teschner [26] 1995) If 119866 is a 120574-critical graphwith 120574 = 3 and 120592(119866) = 2Δ(119866) + 3 then 119887(119866) ⩽ Δ + 1

We have not known yet whether the equality inTheorem 44holds or notHowever Samodivkin proposed thefollowing conjecture

Conjecture 45 (Samodivkin [20] 2008) If 119866 is a 120574-criticalgraph with (Δ(119866)+1)(120574minus1)+1 vertices then 119887(119866) = Δ(119866)+1

In general based on Theorem 41 Teschner proposed thefollowing conjecture

Conjecture 46 (Teschner [26] 1995) If119866 is a 120574-critical graphthen 119887(119866) ⩽ (32)Δ(119866) for 120574 ⩾ 4

We conclude this subsection with some remarks Graphswhich are minimal or critical with respect to a given prop-erty or parameter frequently play an important role in theinvestigation of that property or parameter Not only are suchgraphs of considerable interest in their own right but also aknowledge of their structure often aids in the developmentof the general theory In particular when investigating anyfinite structure a great number of results are proven byinduction Consequently it is desirable to learn as much aspossible about those graphs that are critical with respect toa given property or parameter so as to aid and abet suchinvestigations

In this subsection we survey some results on the bondagenumber for 120574-critical graphs Although these results are notvery perfect they provides a feasible method to approach thebondage number from different viewpoints In particular themethods given in Teschner [26] worth further explorationand development

The following proposition is maybe useful for us tofurther investigate the bondage number of a 120574-critical graph

Proposition 47 (Brigham et al [24] 1988) If 119866 has anonisolated vertex 119909 such that the subgraph induced by119873

119866(119909)

is a complete graph then 119866 is not 120574-critical

This simple fact shows that any 120574-critical graph containsno vertices of degree one

35 Bounds Implied by Domination In the preceding sub-section we introduced some upper bounds on the bondagenumbers for 120574-critical graphs by consideration of domina-tions In this subsection we introduce related results for ageneral graph with given domination number

Theorem 48 (Fink et al [8] 1990) For any connected graph119866 of order 119899 (⩾2)

(a) 119887(119866) ⩽ 119899 minus 1

(b) 119887(119866) ⩽ Δ(119866) + 1 if 120574(119866) = 2

(c) 119887(119866) ⩽ 119899 minus 120574(119866) + 1

While the upper bound of 119899minus1 on 119887(119866) is not particularlygood for many classes of graphs (eg trees and cycles) it isan attainable bound For example if 119866 is a complete 119905-partitegraph 119870

119905(2) for 119905 = 1198992 and 119899 ⩽ 4 an even integer then the

three bounds on 119887(119866) in Theorem 48 are sharpTeschner [26] consider a graphwith 120574(119866) = 1 or 120574(119866) = 2

The next result is almost trivial but useful

Lemma 49 Let 119866 be a graph with order 119899 and 120574(119866) = 1 119905 thenumber of vertices of degree 119899 minus 1 Then 119887(119866) = lceil1199052rceil

Since 119905 ⩽ Δ(119866) clearly Lemma 49 yields the followingresult immediately

Theorem 50 (Teschner [26] 1995) 119887(119866) ⩽ (12)Δ+1 for anygraph 119866 with 120574(119866) = 1

For a complete graph1198702119896+1

119887(1198702119896+1) = 119896+1 = (12)Δ+1

which shows that the upper bound given in Theorem 50 canbe attained For a graph119866with 120574(119866) = 2 byTheorem 59 laterthe upper bound given inTheorem 48 (b) can be also attainedby a 2-critical graph (see Proposition 37)

36 Two Conjectures In 1990 when Fink et al [8] introducedthe concept of the bondage number they proposed thefollowing conjecture

Conjecture 51 119887(119866) ⩽ Δ(119866) + 1 for any nonempty graph 119866

10 International Journal of Combinatorics

Although some results partially support Conjecture 51Teschner [28] in 1993 found a counterexample to this con-jecture the cartesian product 119870

3times 119870

3 which shows that

119887(119866) = Δ(119866) + 2If a graph 119866 is a counterexample to Conjecture 51 it

must be a 120574-critical graph by Theorem 13 That is why thevertex domination-critical graphs are of special interest in theliterature

Now we return to Conjecture 51 Hartnell and Rall [17]and Teschner [29] independently proved that 119887(119866) can bemuch greater than Δ(119866) by showing the following result

Theorem 52 (Hartnell and Rall [17] 1994 Teschner [29]1996) For an integer 119899 ⩾ 3 let 119866

119899be the cartesian product

119870119899times 119870

119899 Then 119887(119866

119899) = 3(119899 minus 1) = (32)Δ(119866

119899)

This theorem shows that there exist no upper boundsof the form 119887(119866) ⩽ Δ(119866) + 119888 for any integer 119888 Teschner[26] proved that 119887(119866) ⩽ (32)Δ(119866) for any graph 119866 with120574(119866) ⩽ 2 (see Theorems 50 and 48) and for 120574-critical graphswith 120574(119866) = 3 and proposed the following conjecture

Conjecture 53 (Teschner [26] 1995) 119887(119866) ⩽ (32)Δ(119866) forany graph 119866

We believe that this conjecture is true but so far no muchwork about it has been known

4 Lower Bounds

Since the bondage number is defined as the smallest numberof edges whose removal results in an increase of dominationnumber each constructive method that creates a concretebondage set leads to an upper bound on the bondage numberFor that reason it is hard to find lower bounds Neverthelessthere are still a few lower bounds

41 Bounds Implied by Subgraphs The first lower boundon the bondage number is gotten in terms of its spanningsubgraph

Theorem 54 (Teschner [10] 1997) Let 119867 be a spanningsubgraph of a nonempty graph119866 If 120574(119867) = 120574(119866) then 119887(119867) ⩽119887(119866)

By Theorem 1 119887(119862119899) = 3 if 119899 equiv 1 (mod 3) and 119887(119862

119899) =

2 otherwise 119887(119875119899) = 2 if 119899 equiv 1 (mod 3) and 119887(119875

119899) = 1

otherwise From these results and Theorem 54 we get thefollowing two corollaries

Corollary 55 If119866 is hamiltonianwith order 119899 ⩾ 3 and 120574(119866) =lceil1198993rceil then 119887(119866) ⩾ 2 and in addition 119887(119866) ⩾ 3 if 119899 equiv 1 (mod3)

Corollary 56 If 119866 with order 119899 ⩾ 2 has a hamiltonian pathand 120574(119866) = lceil1198993rceil then 119887(119866) ⩾ 2 if 119899 equiv 1 (mod 3)

42 Other Bounds The vertex covering number 120573(119866) of 119866 isthe minimum number of vertices that are incident with all

Figure 5 A 120574-critical graph with 120573 = 120574 = 4 120575 = 2 and 119887 = 3

edges in 119866 If 119866 has no isolated vertices then 120574(119866) ⩽ 120573(119866)clearly In [30] Volkmann gave a lot of graphs with 120573 = 120574

Theorem 57 (Teschner [10] 1997) Let119866 be a graph If 120574(119866) =120573(119866) then

(a) 119887(119866) ⩾ 120575(119866)

(b) 119887(119866) ⩾ 120575(119866) + 1 if 119866 is a 120574-critical graph

The graph shown in Figure 5 shows that the bound givenin Theorem 57(b) is sharp For the graph 119866 it is a 120574-criticalgraph and 120574(119866) = 120573(119866) = 4 By Theorem 57 we have 119887(119866) ⩾3 On the other hand 119887(119866) ⩽ 3 by Theorem 16 Thus 119887(119866) =3

Proposition 58 (Sanchis [31] 1991) Let 119866 be a graph 119866 oforder 119899 (⩾ 6) and size 120576 If 119866 has no isolated vertices and3 ⩽ 120574(119866) ⩽ 1198992 then 120576 ⩽ ( 119899minus120574(119866)+1

2)

Using the idea in the proof of Theorem 57 every upperbound for 120574(119866) can lead to a lower bound for 119887(119866) Inthis way Teschner [10] obtained another lower bound fromProposition 58

Theorem59 (Teschner [10] 1997) Let119866 be a graph119866 of order119899 (⩾6) and size 120576 If 2 ⩽ 120574(119866) ⩽ 1198992 minus 1 then

(a) 119887(119866) ⩾ min120575(119866) 120576 minus ( 119899minus120574(119866)2)

(b) 119887(119866) ⩾ min120575(119866) + 1 120576 minus ( 119899minus120574(119866)2) if 119866 is a 120574-critical

graph

The lower bound in Theorem 59(b) is sharp for a classof 120574-critical graphs with domination number 2 Let 119866 be thegraph obtained from complete graph119870

2119905(119905 ⩾ 2) by removing

a perfect matching By Proposition 37 119866 is a 120574-critical graphwith 120574(119866) = 2 Then 119887(119866) ⩾ 120575(119866) + 1 = Δ(119866) + 1 byTheorem 59 and 119887(119866) ⩽ Δ(119866) + 1 by Theorem 38 and so119887(119866) = Δ(119866) + 1 = 2119905 minus 1

As far as the author knows there are no more lowerbounds in the present literature In view of applications ofthe bondage number a network is vulnerable if its bondagenumber is small while it is stable if its bondage number islargeTherefore better lower bounds let us learn better aboutthe stability of networks from this point of view Thus it isof great significance to seek more lower bounds for variousclasses of graphs

International Journal of Combinatorics 11

5 Results on Graph Operations

Generally speaking it is quite difficult to determine the exactvalue of the bondage number for a given graph since itstrongly depends on the dominating number of the graphThus determining bondage numbers for some special graphsis interesting if the dominating numbers of those graphs areknown or can be easily determined In this section we willintroduce some known results on bondage numbers for somespecial classes of graphs obtained by some graph operations

51 Cartesian Product Graphs Let 1198661= (119881

1 119864

1) and 119866

2=

(1198812 119864

2) be two graphs The Cartesian product of 119866

1and 119866

2

is an undirected graph denoted by 1198661times 119866

2 where 119881(119866

1times

1198662) = 119881

1times 119881

2 two distinct vertices 119909

11199092and 119910

11199102 where

1199091 119910

1isin 119881(119866

1) and 119909

2 119910

2isin 119881(119866

2) are linked by an edge in

1198661times 119866

2if and only if either 119909

1= 119910

1and 119909

21199102isin 119864(119866

2) or

1199092= 119910

2and 119909

11199101isin 119864(119866

1) The Cartesian product is a very

effective method for constructing a larger graph from severalspecified small graphs

Theorem 60 (Dunbar et al [21] 1998) For 119899 ⩾ 3

119887 (119862119899times 119870

2) =

2 119894119891 119899 equiv 0 119900119903 1 (mod 4) 3 119894119891 119899 equiv 3 (mod 4) 4 119894119891 119899 equiv 2 (mod 4)

(16)

For the Cartesian product 119862119899times 119862

119898of two cycles 119862

119899and

119862119898 where 119899 ⩾ 4 and 119898 ⩾ 3 Klavzar and Seifter [32]

determined 120574(119862119899times 119862

119898) for some 119899 and 119898 For example

120574(119862119899times119862

4) = 119899 for 119899 ⩾ 3 and 120574(119862

119899times119862

3) = 119899minuslfloor1198994rfloor for 119899 ⩾ 4

Kim [33] determined 119887(1198623times 119862

3) = 6 and 119887(119862

4times 119862

4) = 4 For

a general 119899 ⩾ 4 the exact values of the bondage numbers of119862119899times 119862

3and 119862

119899times 119862

4are determined as follows

Theorem 61 (Sohn et al [34] 2007) For 119899 ⩾ 4

119887 (119862119899times 119862

3) =

2 119894119891 119899 equiv 0 (mod 4) 4 119894119891 119899 equiv 1 119900119903 2 (mod 4) 5 119894119891 119899 equiv 3 (mod 4)

(17)

Theorem62 (Kang et al [35] 2005) 119887(119862119899times119862

4) = 4 for 119899 ⩾ 4

For larger 119898 and 119899 Huang and Xu [36] obtained thefollowing result see Theorem 197 for more details

Theorem 63 (Huang and Xu [36] 2008) 119887(1198625119894times119862

5119895) = 3 for

any positive integers 119894 and 119895

Xiang et al [37] determined that for 119899 ⩾ 5

119887 (119862119899times 119862

5)

= 3 if 119899 equiv 0 or 1 (mod 5) = 4 if 119899 equiv 2 or 4 (mod 5) ⩽ 7 if 119899 equiv 3 (mod 5)

(18)

For the Cartesian product 119875119899times119875

119898of two paths 119875

119899and 119875

119898

Jacobson and Kinch [38] determined 120574(119875119899times119875

2) = lceil(119899+1)2rceil

120574(119875119899times 119875

3) = 119899 minus lfloor(119899 minus 1)4rfloor and 120574(119875

119899times 119875

4) = 119899 + 1 if 119899 =

1 2 3 5 6 9 and 119899 otherwiseThe bondage number of119875119899times119875

119898

for 119899 ⩾ 4 and 2 ⩽ 119898 ⩽ 4 is determined as follows (Li [39] alsodetermined 119887(119875

119899times 119875

2))

Theorem 64 (Hu and Xu [40] 2012) For 119899 ⩾ 4

119887 (119875119899times 119875

2) =

1 119894119891 119899 equiv 1 (mod 2)2 119894119891 119899 equiv 0 (mod 2)

119887 (119875119899times 119875

3) =

1 119894119891 119899 equiv 1 119900119903 2 (mod 4)2 119894119891 119899 equiv 0 119900119903 3 (mod 4)

119887 (119875119899times 119875

4) = 1 119891119900119903 119899 ⩾ 14

(19)

From the proof of Theorem 64 we find that if 119875119899=

0 1 119899 minus 1 and 119875119898= 0 1 119898 minus 1 then the removal

of the vertex (0 0) in 119875119899times119875

119898does not change the domination

number If119898 increases the effect of (0 0) for the dominationnumber will be smaller and smaller from the probabilityTherefore we expect it is possible that 120574(119875

119899times 119875

119898minus (0 0)) =

120574(119875119899times 119875

119898) for119898 ⩾ 5 and give the following conjecture

Conjecture 65 119887(119875119899times 119875

119898) ⩽ 2 for119898 ⩾ 5 and 119899 ⩾ 4

52 Block Graphs and Cactus Graphs In this subsection weintroduce some results for corona graphs block graphs andcactus graphs

The corona1198661∘ 119866

2 proposed by Frucht and Harary [41]

is the graph formed from a copy of 1198661and 120592(119866

1) copies of

1198662by joining the 119894th vertex of 119866

1to the 119894th copy of 119866

2 In

particular we are concernedwith the corona119867∘1198701 the graph

formed by adding a new vertex V119894 and a new edge 119906

119894V119894for

every vertex 119906119894in119867

Theorem 66 (Carlson and Develin [42] 2006) 119887(119867 ∘ 1198701) =

120575(119867) + 1

This is a very important result which is often used toconstruct a graph with required bondage number In otherwords this result implies that for any given positive integer 119887there is a graph 119866 such that 119887(119866) = 119887

A block graph is a graph whose blocks are completegraphs Each block in a cactus graph is either a cycle or a119870

2

If each block of a graph is either a complete graph or a cyclethen we call this graph a block-cactus graph

Theorem67 (Teschner [43] 1997) 119887(119866) le Δ(119866) for any blockgraph 119866

Teschner [43] characterized all block graphs with 119887(119866) =Δ(119866) In the same paper Teschner found that 120574-criticalgraphs are instrumental in determining bounds for thebondage number of cactus and block graphs and obtained thefollowing result

Theorem 68 (Teschner [43] 1997) 119887(119866) ⩽ 3 for anynontrivial cactus graph 119866

This bound can be achieved by 119867 ∘ 1198701where 119867 is

a nontrivial cactus graph without vertices of degree one

12 International Journal of Combinatorics

by Theorem 66 In 1998 Dunbar et al [21] proposed thefollowing problem

Problem 2 Characterize all cactus graphs with bondagenumber 3

Some upper bounds for block-cactus graphs were alsoobtained

Theorem 69 (Dunbar et al [21] 1998) Let 119866 be a connectedblock-cactus graph with at least two blocksThen 119887(119866) ⩽ Δ(119866)

Theorem 70 (Dunbar et al [21] 1998) Let 119866 be a connectedblock-cactus graph which is neither a cactus graph nor a blockgraph Then 119887(119866) ⩽ Δ(119866) minus 1

53 Edge-Deletion and Edge-Contraction In order to obtainfurther results of the bondage number we may consider howthe bondage number changes under some operations of agraph 119866 which remain the domination number unchangedor preserve some property of 119866 such as planarity Twosimple operations satisfying these requirements are the edge-deletion and the edge-contraction

Theorem 71 (Huang and Xu [44] 2012) Let 119866 be any graphand 119890 isin 119864(119866) Then 119887(119866 minus 119890) ⩾ 119887(119866) minus 1 Moreover 119887(119866 minus 119890) ⩽119887(119866) if 120574(119866 minus 119890) = 120574(119866)

FromTheorem 1 119887(119862119899minus 119890) = 119887(119875

119899) = 119887(119862

119899) minus 1 for any 119890

in119862119899 which shows that the lower bound on 119887(119866minus119890) is sharp

The upper bound can reach for any graph 119866 with 119887(119866) ⩾ 2Next we consider the effect of the edge-contraction on

the bondage number Given a graph 119866 the contraction of 119866by the edge 119890 = 119909119910 denoted by 119866119909119910 is the graph obtainedfrom 119866 by contracting two vertices 119909 and 119910 to a new vertexand then deleting all multiedges It is easy to observe that forany graph 119866 120574(119866) minus 1 ⩽ 120574(119866119909119910) ⩽ 120574(119866) for any edge 119909119910 of119866

Theorem 72 (Huang and Xu [44] 2012) Let 119866 be any graphand 119909119910 be any edge in119866 If119873

119866(119909)cap119873

119866(119910) = 0 and 120574(119866119909119910) =

120574(119866) then 119887(119866119909119910) ⩾ 119887(119866)

We can use examples 119862119899and 119870

119899to show that the

conditions ofTheorem 72 are necessary and the lower boundinTheorem 72 is tight Clearly 120574(119862

119899) = lceil1198993rceil and 120574(119870

119899) = 1

for any edge 119909119910 119862119899119909119910 = 119862

119899minus1and 119870

119899119909119910 = 119870

119899minus1 By

Theorem 1 if 119899 equiv 1 (mod 3) then 120574(119862119899119909119910) lt 120574(119866) and

119887(119862119899119909119910) = 2 lt 3 = 119887(119862

119899) Thus the result in Theorem 72

is generally invalid without the hypothesis 120574(119866119909119910) = 120574(119866)Furthermore the condition 119873

119866(119909) cap 119873

119866(119910) = 0 cannot be

omitted even if 120574(119866119909119910) = 120574(119866) since for odd 119899 119887(119870119899119909119910) =

lceil(119899 minus 1)2rceil lt lceil1198992rceil = 119887(119870119899) by Theorem 1

On the other hand 119887(119862119899119909119910) = 119887(119862

119899) = 2 if 119899 equiv 0 (mod

3) and 119887(119870119899119909119910) = 119887(119870

119899) if 119899 is even which shows that the

equality in 119887(119866119909119910) ⩾ 119887(119866) may hold Thus the bound inTheorem 72 is tight However provided all the conditions119887(119866119909119910) can be arbitrarily larger than 119887(119866) Given a graph119867let119866 be the graph formed from119867∘119870

1by adding a new vertex

119909 and joining it to an vertex 119910 of degree one in119867 ∘ 1198701 Then

119866119909119910 = 119867 ∘ 1198701 120574(119866) = 120574(119866119909119910) and 119873

119866(119909) cap 119873

119866(119910) = 0

But 119887(119866) = 1 since 120574(119866 minus 119909119910) = 120574(119866119909119910) + 1 and 119887(119866119909119910) =120575(119867) + 1 byTheorem 66The gap between 119887(119866) and 119887(119866119909119910)is 120575(119867)

6 Results on Planar Graphs

From Section 2 we have seen that the bondage numberfor a tree has been completely solved Moreover a lineartime algorithm to compute the bondage number of a tree isdesigned by Hartnell et al [15] It is quite natural to considerthe bondage number for a planar graph In this section westate some results and problems on the bondage number fora planar graph

61 Conjecture on Bondage Number As mentioned inSection 3 the bondage number can be much larger than themaximum degree But for a planar graph 119866 the bondagenumber 119887(119866) cannot exceed Δ(119866) too much It is clear that119887(119866) ⩽ Δ(119866) + 4 by Corollary 15 since 120575(119866) ⩽ 5 for anyplanar graph119866 In 1998Dunbar et al [21] posed the followingconjecture

Conjecture 73 If 119866 is a planar graph then 119887(119866) ⩽ Δ(119866) + 1

Because of its attraction it immediately became the focusof attention when this conjecture is proposed In fact themain aim concerning the research of the bondage number fora planar graph is focused on this conjecture

It has been mentioned in Theorems 1 and 60 that119887(119862

3119896+1) = 3 = Δ + 1 and 119887(119862

4119896+2times 119870

2) = 4 = Δ + 1 It

is known that 119887(1198706minus 119872) = 5 = Δ + 1 where119872 is a perfect

matching of the complete graph1198706These examples show that

if Conjecture 73 is true then the upper bound is sharp for2 ⩽ Δ ⩽ 4

Here we note that it is sufficient to prove this conjecturefor connected planar graphs since the bondage number of adisconnected graph is simply the minimum of the bondagenumbers of its components

The first paper attacking this conjecture is due to Kangand Yuan [45] which confirmed the conjecture for everyconnected planar graph 119866 with Δ ⩾ 7 The proofs are mainlybased onTheorems 16 and 18

Theorem 74 (Kang and Yuan [45] 2000) If 119866 is a connectedplanar graph then 119887(119866) ⩽ min8 Δ(119866) + 2

Obviously in view of Theorem 74 Conjecture 73 is truefor any connected planar graph with Δ ⩾ 7

62 Bounds Implied by Degree Conditions As we have seenfrom Theorem 74 to attack Conjecture 73 we only need toconsider connected planar graphs with maximum Δ ⩽ 6Thus studying the bondage number of planar graphs bydegree conditions is of significanceThe first result on boundsimplied by degree conditions is obtained by Kang and Yuan[45]

International Journal of Combinatorics 13

Theorem 75 (Kang and Yuan [45] 2000) If 119866 is a connectedplanar graph without vertices of degree 5 then 119887(119866) ⩽ 7

Fischermann et al [46] generalized Theorem 75 as fol-lows

Theorem 76 (Fischermann et al [46] 2003) Let 119866 be aconnected planar graph and 119883 the set of vertices of degree 5which have distance at least 3 to vertices of degrees 1 2 and3 If all vertices in 119883 not adjacent with vertices of degree 4are independent and not adjacent to vertices of degree 6 then119887(119866) ⩽ 7

Clearly if119866 has no vertices of degree 5 then119883 = 0 ThusTheorem 76 yields Theorem 75

Use 119899119894to denote the number of vertices of degree 119894 in 119866

for each 119894 = 1 2 Δ(119866) Using Theorem 18 Fischermannet al obtained the following two theorems

Theorem 77 (Fischermann et al [46] 2003) For any con-nected planar graph 119866

(1) 119887(119866) ⩽ 7 if 1198995lt 2119899

2+ 3119899

3+ 2119899

4+ 12

(2) 119887(119866) ⩽ 6 if 119866 contains no vertices of degrees 4 and 5

Theorem 78 (Fischermann et al [46] 2003) For any con-nected planar graph 119866 119887(119866) ⩽ Δ(119866) + 1 if

(1) Δ(119866) = 6 and every edge 119890 = 119909119910 with 119889119866(119909) = 5 and

119889119866(119910) = 6 is contained in at most one triangle

(2) Δ(119866) = 5 and no triangle contains an edge 119890 = 119909119910 with119889119866(119909) = 5 and 4 ⩽ 119889

119866(119910) ⩽ 5

63 Bounds Implied by Girth Conditions The girth 119892(119866) of agraph 119866 is the length of the shortest cycle in 119866 If 119866 has nocycles we define 119892(119866) = infin

Combining Theorem 74 with Theorem 78 we find thatif a planar graph contains no triangles and has maximumdegree Δ ⩾ 5 then Conjecture 73 holds This fact motivatedFischermann et alrsquos [46] attempt to attack Conjecture 73 bygirth restraints

Theorem 79 (Fischermann et al [46] 2003) For any con-nected planar graph 119866

119887 (119866) ⩽

6 119894119891 119892 (119866) ⩾ 4

5 119894119891 119892 (119866) ⩾ 5

4 119894119891 119892 (119866) ⩾ 6

3 119894119891 119892 (119866) ⩾ 8

(20)

The first result in Theorem 79 shows that Conjecture 73is valid for all connected planar graphs with 119892(119866) ⩾ 4 andΔ(119866) ⩾ 5 It is easy to verify that the second result inTheorem 79 implies that Conjecture 73 is valid for all not3-regular graphs of girth 119892(119866) ⩾ 5 which is stated in thefollowing corollary

Corollary 80 For any connected planar graph 119866 if 119866 is not3-regular and 119892(119866) ⩾ 5 then 119887(119866) ⩽ Δ(119866) + 1

The first result in Theorem 79 also implies that if 119866 isa connected planar graph with no cycles of length 3 then119887(119866) ⩽ 6 Hou et al [47] improved this result as follows

Theorem 81 (Hou et al [47] 2011) For any connected planargraph 119866 if 119866 contains no cycles of length 119894 (4 ⩽ 119894 ⩽ 6) then119887(119866) ⩽ 6

Since 119887(1198623119896+1) = 3 for 119896 ⩾ 3 (see Theorem 1) the

last bound in Theorem 79 is tight Whether other bounds inTheorem 79 are tight remains open In 2003 Fischermann etal [46] posed the following conjecture

Conjecture 82 For any connected planar graph 119866 119887(119866) ⩽ 7and

119887 (119866) ⩽ 5 119894119891 119892 (119866) ⩾ 4

4 119894119891 119892 (119866) ⩾ 5(21)

We conclude this subsection with a question on bondagenumbers of planar graphs

Question 1 (Fischermann et al [46] 2003) Is there a planargraph 119866 with 6 ⩽ 119887(119866) ⩽ 8

In 2006 Carlson andDevelin [42] showed that the corona119866 = 119867 ∘ 119870

1for a planar graph 119867 with 120575(119867) = 5 has the

bondage number 119887(119866) = 120575(119867) + 1 = 6 (see Theorem 66)Since the minimum degree of planar graphs is at most 5then 119887(119866) can attach 6 If we take 119867 as the graph of theicosahedron then 119866 = 119867 ∘ 119870

1is such an example The

question for the existence of planar graphs with bondagenumber 7 or 8 remains open

64 Comments on the Conjectures Conjecture 73 is true forall connected planar graphs with minimum degree 120575 ⩽ 2 byTheorem 16 or maximum degree Δ ⩾ 7 by Theorem 74 ornot 120574-critical planar graphs by Theorem 13 Thus to attackConjecture 73 we only need to consider connected criticalplanar graphs with degree-restriction 3 ⩽ 120575 ⩽ Δ ⩽ 6

Recalling and anatomizing the proofs of all results men-tioned in the preceding subsections on the bondage numberfor connected planar graphs we find that the proofs of theseresults strongly depend upon Theorem 16 or Theorem 18 Inother words a basic way used in the proofs is to find twovertices 119909 and 119910 with distance at most two in a consideredplanar graph 119866 such that

119889119866(119909) + 119889

119866(119910) or 119889

119866(119909) + 119889

119866(119910) minus

1003816100381610038161003816119873119866(119909) cap 119873

119866(119910)1003816100381610038161003816

(22)

which bounds 119887(119866) is as small as possible Very recentlyHuang and Xu [44] have considered the following parameter

119861 (119866) = min119909119910isin119881(119866)

119889119909119910 1 ⩽ 119889

119866(119909 119910) ⩽ 2

cup 119889119909119910minus 119899

119909119910 119889 (119909 119910) = 1

(23)

where 119889119909119910= 119889

119866(119909) + 119889

119866(119910) minus 1 and 119899

119909119910= |119873

119866(119909) cap 119873

119866(119910)|

14 International Journal of Combinatorics

It follows fromTheorems 16 and 18 that

119887 (119866) ⩽ 119861 (119866) (24)

The proofs given in Theorems 74 and 79 indeed imply thefollowing stronger results

Theorem 83 If 119866 is a connected planar graph then

119861 (119866) ⩽

min 8 Δ (119866) + 26 119894119891 119892 (119866) ⩾ 4

5 119894119891 119892 (119866) ⩾ 5

4 119894119891 119892 (119866) ⩾ 6

3 119894119891 119892 (119866) ⩾ 8

(25)

Thus using Theorem 16 or Theorem 18 if we can proveConjectures 73 and 82 then we can prove the followingstatement

Statement If 119866 is a connected planar graph then

119861 (119866) ⩽

Δ (119866) + 1

7

5 if 119892 (119866) ⩾ 44 if 119892 (119866) ⩾ 5

(26)

It follows from (24) that Statement implies Conjectures 73and 82 However Huang and Xu [44] gave examples to showthat none of conclusions in Statement is true As a result theystated the following conclusion

Theorem 84 (Huang and Xu [44] 2012) It is not possible toprove Conjectures 73 and 82 if they are right using Theorems16 and 18

Therefore a new method is needed to prove or disprovethese conjectures At the present time one cannot prove theseconjectures and may consider to disprove them If one ofthese conjectures is invalid then there exists a minimumcounterexample 119866 with respect to 120592(119866) + 120576(119866) In order toobtain a minimum counterexample one may consider twosimple operations satisfying these requirements the edge-deletion and the edge-contractionwhich decrease 120592(119866)+120576(119866)and preserve planarity However applying Theorems 71 and72 Huang and Xu [44] presented the following results

Theorem 85 (Huang and Xu [44] 2012) It is not possible toconstruct minimum counterexamples to Conjectures 73 and 82by the operation of an edge-deletion or an edge-contraction

7 Results on Crossing Number Restraints

It is quite natural to generalize the known results on thebondage number for planar graphs to formore general graphsin terms of graph-theoretical parameters In this section weconsider graphs with crossing number restraints

The crossing number cr(119866) of 119866 is the smallest numberof pairwise intersections of its edges when 119866 is drawn in theplane If cr(119866) = 0 then 119866 is a planar graph

71 General Methods Use 119899119894to denote the number of vertices

of degree 119894 in119866 for 119894 = 1 2 Δ(119866) Huang and Xu obtainedthe following results

Theorem 86 (Huang and Xu [48] 2007) For any connectedgraph 119866

119887 (119866) ⩽

6 119894119891 119892 (119866) ⩾ 4 119886119899119889 2cr (119866) lt 121198861

5 119894119891 119892 (119866) ⩾ 5 119886119899119889 6cr (119866) lt 161198862

4 119894119891 119892 (119866) ⩾ 6 119886119899119889 4cr (119866) lt 141198863

3 119894119891 119892 (119866) ⩾ 8 119886119899119889 6cr (119866) lt 161198864

(27)

where 1198861= 119899

1+ 2119899

2+ 2119899

3+sum

Δ

119894=8(119894 minus 7)119899

119894+ 8 119886

2= 3119899

1+ 6119899

2+

51198993+sum

Δ

119894=7(3119894minus18)119899

119894+20 119886

3= 119899

1+2119899

2+sum

Δ

119894=6(2119894minus10)119899

119894+12

and 1198864= sum

Δ

119894=5(3119894 minus 12)119899

119894+ 16

Simplifying the conditions in Theorem 86 we obtain thefollowing corollaries immediately

Corollary 87 (Huang and Xu [48] 2007) For any connectedgraph 119866

119887 (119866) ⩽

6 119894119891 119892 (119866) ⩾ 4 119886119899119889 cr (119866) ⩽ 3

5 119894119891 119892 (119866) ⩾ 5 119886119899119889 cr (119866) ⩽ 4

4 119894119891 119892 (119866) ⩾ 6 119886119899119889 cr (119866) ⩽ 2

3 119894119891 119892 (119866) ⩾ 8 119886119899119889 cr (119866) ⩽ 2

(28)

Corollary 88 (Huang and Xu [48] 2007) For any connectedgraph 119866

(a) 119887(119866) ⩽ 6 if 119866 is not 4-regular cr(119866) = 4 and 119892(119866) ⩾ 4

(b) 119887(119866) ⩽ Δ(119866) + 1 if 119866 is not 3-regular cr(119866) ⩽ 4 and119892(119866) ⩾ 5

(c) 119887(119866) ⩽ 4 if 119866 is not 3-regular cr(119866) = 3 and 119892(119866) ⩾ 6

(d) 119887(119866) ⩽ 3 if cr(119866) = 3 119892(119866) ⩾ 8 and Δ(119866) ⩾ 5

These corollaries generalize some known results for pla-nar graphs For example Corollary 87 contains Theorem 79Corollary 88 (b) contains Corollary 80

Theorem 89 (Huang and Xu [48] 2007) For any connectedgraph 119866 if cr(119866) ⩽ 119899

3(119866) + 119899

4(119866) + 3 then 119887(119866) ⩽ 8

Corollary 90 (Huang and Xu [48] 2007) For any connectedgraph 119866 if cr(119866) ⩽ 3 then 119887(119866) ⩽ 8

Perhaps being unaware of this result in 2010 Ma et al[49] proved that 119887(119866) ⩽ 12 for any graph 119866 with cr(119866) = 1

Theorem91 (Huang and Xu [48] 2007) Let119866 be a connectedgraph and 119868 = 119909 isin 119881(119866) 119889

119866(119909) = 5 119889

119866(119909 119910) ⩾ 3 if

International Journal of Combinatorics 15

119889119866(119910) ⩽ 3 and 119889

119866(119910) = 4 for every 119910 isin 119873

119866(119909) If 119868 is

independent no vertices adjacent to vertices of degree 6 and

cr (119866) lt max5119899

3+ |119868| minus 2119899

4+ 28

117119899

3+ 40

16 (29)

then 119887(119866) ⩽ 7

Corollary 92 (Huang and Xu [48] 2007) Let 119866 be aconnected graph with cr(119866) ⩽ 2 If 119868 = 119909 isin 119881(119866) 119889

119866(119909) =

5 119889119866(119910 119909) ⩾ 3 if 119889

119866(119910) ⩽ 3 and 119889

119866(119910) = 4 for every 119910 isin

119873119866(119909) is independent and no vertices adjacent to vertices of

degree 6 then 119887(119866) ⩽ 7

Theorem93 (Huang andXu [48] 2007) Let119866 be a connectedgraph If 119866 satisfies

(a) 5cr(119866) + 1198995lt 2119899

2+ 3119899

3+ 2119899

4+ 12 or

(b) 7cr(119866) + 21198995lt 3119899

2+ 4119899

4+ 24

then 119887(119866) ⩽ 7

Proposition 94 (Huang and Xu [48] 2007) Let 119866 be aconnected graph with no vertices of degrees four and five Ifcr(119866) ⩽ 2 then 119887(119866) ⩽ 6

Theorem 95 (Huang and Xu [48] 2007) If 119866 is a connectedgraph with cr(119866) ⩽ 4 and not 4-regular when cr(119866) = 4 then119887(119866) ⩽ Δ(119866) + 2

The above results generalize some known results forplanar graphs For example Corollary 90 and Theorem 95contain Theorem 74 the first condition in Theorem 93 con-tains the second condition inTheorem 77

From Corollary 88 and Theorem 95 we suggest the fol-lowing questions

Question 2 Is there a

(a) 4-regular graph 119866 with cr(119866) = 4 such that 119887(119866) ⩾ 7

(b) 3-regular graph 119866 with cr ⩽ 4 and 119892(119866) ⩾ 5 such that119887(119866) = 5

(c) 3-regular graph 119866 with cr = 3 and 119892(119866) = 6 or 7 suchthat 119887(119866) = 5

72 Carlson-Develin Methods In this subsection we intro-duce an elegant method presented by Carlson and Develin[42] to obtain some upper bounds for the bondage numberof a graph

Suppose that 119866 is a connected graph We say that 119866 hasgenus 120588 if 119866 can be embedded in a surface 119878 with 120588 handlessuch that edges are pairwise disjoint except possibly for end-vertices Let119866 be an embedding of119866 in surface 119878 and let120601(119866)denote the number of regions in 119866 The boundary of everyregion contains at least three edges and every edge is on theboundary of atmost two regions (the two regions are identicalwhen 119890 is a cut-edge) For any edge 119890 of 119866 let 1199031

119866(119890) and 1199032

119866(119890)

be the numbers of edges comprising the regions in 119866 whichthe edge 119890 borders It is clear that every 119890 = 119909119910 isin 119864(119866)

1199032

119866(119890) ⩾ 119903

1

119866(119890) ⩾ 4 if 1003816100381610038161003816119873119866

(119909) cap 119873119866(119910)1003816100381610038161003816 = 0

1199032

119866(119890) ⩾ 4 119903

1

119866(119890) ⩾ 3 if 1003816100381610038161003816119873119866

(119909) cap 119873119866(119910)1003816100381610038161003816 = 1

1199032

119866(119890) ⩾ 119903

1

119866(119890) ⩾ 3 if 1003816100381610038161003816119873119866

(119909) cap 119873119866(119910)1003816100381610038161003816 ⩾ 2

(30)

Following Carlson and Develin [42] for any edge 119890 = 119909119910 of119866 we define

119863119866(119890) =

1

119889119866(119909)+

1

119889119866(119910)

+1

1199031

119866(119890)+

1

1199032

119866(119890)minus 1 (31)

By the well-known Eulerrsquos formula

120592 (119866) minus 120576 (119866) + 120601 (119866) = 2 minus 2120588 (119866) (32)

it is easy to see that

sum

119890isin119864(119866)

119863119866(119890) = 120592 (119866) minus 120576 (119866) + 120601 (119866) = 2 minus 2120588 (119866) (33)

If 119866 is a planar graph that is 120588(119866) = 0 then

sum

119890isin119864(119866)

119863119866(119890) = 120592 (119866) minus 120576 (119866) + 120601 (119866) = 2 (34)

Combining these formulas withTheorem 18 Carlson andDevelin [42] gave a simple and intuitive proof ofTheorem 74and obtained the following result

Theorem 96 (Carlson and Develin [42] 2006) Let 119866 be aconnected graph which can be embedded on a torus Then119887(119866) ⩽ Δ(119866) + 3

Recently Hou and Liu [50] improved Theorem 96 asfollows

Theorem 97 (Hou and Liu [50] 2012) Let 119866 be a connectedgraph which can be embedded on a torus Then 119887(119866) ⩽ 9Moreover if Δ(119866) = 6 then 119887(119866) ⩽ 8

By the way Cao et al [51] generalized the result inTheorem 79 to a connected graph 119866 that can be embeddedon a torus that is

119887 (119866) le

6 if 119892 (119866) ⩾ 4 and 119866 is not 4-regular5 if 119892 (119866) ⩾ 54 if 119892 (119866) ⩾ 6 and 119866 is not 3-regular3 if 119892 (119866) ⩾ 8

(35)

Several authors used this method to obtain some resultson the bondage number For example Fischermann et al[46] used this method to prove the second conclusion inTheorem 78 Recently Cao et al [52] have used thismethod todeal with more general graphs with small crossing numbersFirst they found the following property

Lemma 98 120575(119866) ⩽ 5 for any graph 119866 with cr(119866) ⩽ 5

16 International Journal of Combinatorics

This result implies that 119887(119866) ⩽ Δ(119866) + 4 by Corollary 15for any graph 119866 with cr(119866) ⩽ 5 Cao et al established thefollowing relation in terms of maximum planar subgraphs

Lemma 99 (Cao et al [52]) Let 119866 be a connected graph withcrossing number cr(119866) For any maximum planar subgraph119867of 119866

sum

119890isin119864(119867)

119863119866(119890) ⩾ 2 minus

2cr (119866)120575 (119866)

(36)

Using Carlson-Develin method and combiningLemma 98 with Lemma 99 Cao et al proved the followingresults

Theorem 100 (Cao et al [52]) For any connected graph 119866119887(119866) ⩽ Δ(119866) + 2 if 119866 satisfies one of the following conditions

(a) cr(119866) ⩽ 3(b) cr(119866) = 4 and 119866 is not 4-regular(c) cr(119866) = 5 and 119866 contains no vertices of degree 4

Theorem101 (Cao et al [52]) Let119866 be a connected graphwithΔ(119866) = 5 and cr(119866) ⩽ 4 If no triangles contain two vertices ofdegree 5 then 119887(119866) ⩽ 6 = Δ(119866) + 1

Theorem 102 (Cao et al [52]) Let 119866 be a connected graphwith Δ(119866) ⩾ 6 and cr(119866) ⩽ 3 If Δ(119866) ⩾ 7 or if Δ(119866) = 6120575(119866) = 3 and every edge 119890 = 119909119910 with 119889

119866(119909) = 5 and 119889

119866(119910) = 6

is contained in atmost one triangle then 119887(119866) ⩽ min8 Δ(119866)+1

Using Carlson-Develin method it can be proved that119887(119866) ⩽ Δ(119866) + 2 if cr(119866) ⩽ 3 (see the first conclusion inTheorem 100) and not yet proved that 119887(119866) ⩽ 8 if cr(119866) ⩽3 although it has been proved by using other method (seeCorollary 90)

By using Carlson-Develin method Samodivkin [25]obtained some results on the bondage number for graphswithsome given properties

Kim [53] showed that 119887(119866) ⩽ Δ(119866) + 2 for a connectedgraph 119866 with genus 1 Δ(119866) ⩽ 5 and having a toroidalembedding of which at least one region is not 4-sidedRecently Gagarin and Zverovich [54] further have extendedCarlson-Develin ideas to establish a nice upper bound forarbitrary graphs that can be embedded on orientable ornonorientable topological surface

Theorem 103 (Gagarin and Zverovich [54] 2012) Let 119866 bea graph embeddable on an orientable surface of genus ℎ and anonorientable surface of genus 119896 Then 119887(119866) ⩽ minΔ(119866)+ℎ+2 Δ(119866) + 119896 + 1

This result generalizes the corresponding upper boundsin Theorems 74 and 96 for any orientable or nonori-entable topological surface By investigating the proof ofTheorem 103 Huang [55] found that the issue of the ori-entability can be avoided by using the Euler characteristic120594(= 120592(119866) minus 120576(119866) + 120601(119866)) instead of the genera ℎ and 119896 the

relations are 120594 = 2minus2ℎ and 120594 = 2minus119896 To obtain the best resultfromTheorem 103 onewants ℎ and 119896 as small as possible thisis equivalent to having 120594 as large as possible

According toTheorem 103 if119866 is planar (ℎ = 0 120594 = 2) orcan be embedded on the real projective plane (119896 = 1 120594 = 1)then 119887(119866) ⩽ Δ(119866) + 2 In all other cases Huang [55] had thefollowing improvement for Theorem 103 the proof is basedon the technique developed byCarlson-Develin andGagarin-Zverovich and includes some elementary calculus as a newingredient mainly the intermediate-value theorem and themean-value theorem

Theorem 104 (Huang [55] 2012) Let 119866 be a graph embed-dable on a surface whose Euler characteristic 120594 is as large aspossible If 120594 ⩽ 0 then 119887(119866) ⩽ Δ(119866)+lfloor119903rfloor where 119903 is the largestreal root of the following cubic equation in 119911

1199113

+ 21199112

+ (6120594 minus 7) 119911 + 18120594 minus 24 = 0 (37)

In addition if 120594 decreases then 119903 increases

The following result is asymptotically equivalent toTheorem 104

Theorem 105 (Huang [55] 2012) Let 119866 be a graph embed-dable on a surface whose Euler characteristic 120594 is as large aspossible If 120594 ⩽ 0 then 119887(119866) lt Δ(119866) + radic12 minus 6120594 + 12 orequivalently 119887(119866) ⩽ Δ(119866) + lceilradic12 minus 6120594 minus 12rceil

Also Gagarin andZverovich [54] indicated that the upperbound in Theorem 103 can be improved for larger values ofthe genera ℎ and 119896 by adjusting the proofs and stated thefollowing general conjecture

Conjecture 106 (Gagarin and Zverovich [54] 2012) For aconnected graph G of orientable genus ℎ and nonorientablegenus 119896

119887 (119866) ⩽ min 119888ℎ 119888

1015840

119896 Δ (119866) + 119900 (ℎ) Δ (119866) + 119900 (119896) (38)

where 119888ℎand 1198881015840

119896are constants depending respectively on the

orientable and nonorientable genera of 119866

In the recent paper Gagarin and Zverovich [56] providedconstant upper bounds for the bondage number of graphs ontopological surfaces which can be used as the first estimationfor the constants 119888

ℎand 1198881015840

119896of Conjecture 106

Theorem 107 (Gagarin and Zverovich [56] 2013) Let 119866 bea connected graph of order 119899 If 119866 can be embedded on thesurface of its orientable or nonorientable genus of the Eulercharacteristic 120594 then

(a) 120594 ⩾ 1 implies 119887(119866) ⩽ 10(b) 120594 ⩽ 0 and 119899 gt minus12120594 imply 119887(119866) ⩽ 11(c) 120594 ⩽ minus1 and 119899 ⩽ minus12120594 imply 119887(119866) ⩽ 11 +

3120594(radic17 minus 8120594 minus 3)(120594 minus 1) = 11 + 119874(radicminus120594)

Theorem 79 shows that a connected planar triangle-freegraph 119866 has 119887(119866) ⩽ 6 The following result generalizes to itall the other topological surfaces

International Journal of Combinatorics 17

Theorem 108 (Gagarin and Zverovich [56] 2013) Let 119866 be aconnected triangle-free graph of order 119899 If 119866 can be embeddedon the surface of its orientable or nonorientable genus of theEuler characteristic 120594 then

(a) 120594 ⩾ 1 implies 119887(119866) ⩽ 6(b) 120594 ⩽ 0 and 119899 gt minus8120594 imply 119887(119866) ⩽ 7(c) 120594 ⩽ minus1 and 119899 ⩽ minus8120594 imply 119887(119866) ⩽ 7minus4120594(1+radic2 minus 120594)

In [56] the authors also explicitly improved upperbounds in Theorem 103 and gave tight lower bounds forthe number of vertices of graphs 2-cell embeddable ontopological surfaces of a given genusThe interested reader isreferred to the original paper for details The following resultis interesting although its proof is easy

Theorem 109 (Samodivkin [57]) Let119866 be a graph with order119899 and girth 119892 If 119866 is embeddable on a surface whose Eulercharacteristic 120594 is as large as possible then

119887 (119866) ⩽ 3 +8

119892 minus 2minus

4120594119892

119899 (119892 minus 2) (39)

If 119866 is a planar graph with girth 119892 ⩾ 4 + 119894 for any 119894 isin0 1 2 then Theorem 109 leads to 119887(119866) ⩽ 6 minus 119894 which isthe result inTheorem 79 obtained by Fischermann et al [46]Also inmany cases the bound stated inTheorem 109 is betterthan those given byTheorems 103 and 105

8 Conditional Bondage Numbers

Since the concept of the bondage number is based upon thedomination all sorts of dominations which are variationsof the normal domination by adding restricted conditionsto the normal dominating set can develop a new ldquobondagenumberrdquo as long as a variation of domination number isgiven In this section we survey results on the bondagenumber under some restricted conditions

81 Total Bondage Numbers A dominating set 119878 of a graph119866 without isolated vertices is called total if the subgraphinduced by 119878 contains no isolated vertices The minimumcardinality of a total dominating set is called the totaldomination number of 119866 and denoted by 120574

119879(119866) It is clear

that 120574(119866) ⩽ 120574119879(119866) ⩽ 2120574(119866) for any graph 119866 without isolated

verticesThe total domination in graphs was introduced by Cock-

ayne et al [58] in 1980 Pfaff et al [59 60] in 1983 showedthat the problem determining total domination number forgeneral graphs is NP-complete even for bipartite graphs andchordal graphs Even now total domination in graphs hasbeen extensively studied in the literature In 2009 Henning[61] gave a survey of selected recent results on total domina-tion in graphs

The total bondage number of 119866 denoted by 119887119879(119866) is the

smallest cardinality of a subset 119861 sube 119864(119866) with the propertythat119866minus119861 contains no isolated vertices and 120574

119879(119866minus119861) gt 120574

119879(119866)

From definition 119887119879(119866)may not exist for some graphs for

example119866 = 1198701119899 We put 119887

119879(119866) = infin if 119887

119879(119866) does not exist

It is easy to see that 119887119879(119866) is finite for any connected graph 119866

other than 1198701 119870

2 119870

3 and 119870

1119899 In fact for such a graph 119866

since there is a path of length 3 in 119866 we can find 1198611sube 119864(119866)

such that 119866 minus 1198611is a spanning tree 119879 of 119866 containing a path

of length 3 So 120574119879(119879) = 120574

119879(119866minus119861

1) ⩾ 120574

119879(119866) For the tree119879 we

find 1198612sube 119864(119879) such that 120574

119879(119879 minus 119861

2) gt 120574

119879(119879) ⩾ 120574

119879(119866) Thus

we have 120574119879(119866minus119861

2minus119861

1) gt 120574

119879(119866) and so 119887

119879(119866) ⩽ |119861

1|+|119861

2| In

the following discussion we always assume that 119887119879(119866) exists

when a graph 119866 is mentionedIn 1991 Kulli and Patwari [62] first studied the total

bondage number of a graph and calculated the exact valuesof 119887

119879(119866) for some standard graphs

Theorem 110 (Kulli and Patwari [62] 1991) For 119899 ⩾ 4

119887119879(119862

119899) =

3 119894119891 119899 equiv 2 (mod 4) 2 119900119905ℎ119890119903119908119894119904119890

119887119879(119875

119899) =

2 119894119891 119899 equiv 2 (mod 4) 1 119900119905ℎ119890119903119908119894119904119890

119887119879(119870

119898119899) = 119898 119908119894119905ℎ 2 ⩽ 119898 ⩽ 119899

119887119879(119870

119899) =

4 119891119900119903 119899 = 4

2119899 minus 5 119891119900119903 119899 ⩾ 5

(40)

Recently Hu et al [63] have obtained some results on thetotal bondage number of the Cartesian product119875

119898times119875

119899of two

paths 119875119898and 119875

119899

Theorem 111 (Hu et al [63] 2012) For the Cartesian product119875119898times 119875

119899of two paths 119875

119898and 119875

119899

119887119879(119875

2times 119875

119899) =

1 119894119891 119899 equiv 0 (mod 3) 2 119894119891 119899 equiv 2 (mod 3) 3 119894119891 119899 equiv 1 (mod 3)

119887119879(119875

3times 119875

119899) = 1

119887119879(119875

4times 119875

119899)

= 1 119894119891 119899 equiv 1 (mod 5) = 2 119894119891 119899 equiv 4 (mod 5) ⩽ 3 119894119891 119899 equiv 2 (mod 5) ⩽ 4 119894119891 119899 equiv 0 3 (mod 5)

(41)

Generalized Petersen graphs are an important class ofcommonly used interconnection networks and have beenstudied recently By constructing a family of minimum totaldominating sets Cao et al [64] determined the total bondagenumber of the generalized Petersen graphs

FromTheorem 6 we know that 119887(119879) ⩽ 2 for a nontrivialtree 119879 But given any positive integer 119896 Sridharan et al [65]constructed a tree 119879 for which 119887

119879(119879) = 119896 Let119867

119896be the tree

obtained from a star1198701119896+1

by subdividing 119896 edges twice Thetree 119867

7is shown in Figure 6 It can be easily verified that

119887119879(119867

119896) = 119896 This fact can be stated as the following theorem

Theorem 112 (Sridharan et al [65] 2007) For any positiveinteger 119896 there exists a tree 119879 with 119887

119879(119879) = 119896

18 International Journal of Combinatorics

Figure 61198677

Combining Theorem 1 with Theorem 112 we have whatfollows

Corollary 113 (Sridharan et al [65] 2007) The bondagenumber and the total bondage number are unrelated even fortrees

However Sridharan et al [65] gave an upper bound onthe total bondage number of a tree in terms of its maximumdegree For any tree 119879 of order 119899 if 119879 =119870

1119899minus1 then 119887

119879(119879) =

minΔ(119879) (13)(119899 minus 1) Rad and Raczek [66] improved thisupper bound and gave a constructive characterization of acertain class of trees attaching the upper bound

Theorem 114 (Rad and Raczek [66] 2011) 119887119879(119879) ⩽ Δ(119879) minus 1

for any tree 119879 with maximum degree at least three

In general the decision problem for 119887119879(119866) is NP-complete

for any graph 119866 We state the decision problem for the totalbondage as follows

Problem 3 Consider the decision problemTotal Bondage ProblemInstance a graph 119866 and a positive integer 119887Question is 119887

119879(119866) ⩽ 119887

Theorem 115 (Hu and Xu [16] 2012) The total bondageproblem is NP-complete

Consequently in view of its computational hardnessit is significative to establish some sharp bounds on thetotal bondage number of a graph in terms of other graphicparameters In 1991 Kulli and Patwari [62] showed that119887119879(119866) ⩽ 2119899 minus 5 for any graph 119866 with order 119899 ⩾ 5 Sridharan

et al [65] improved this result as follows

Theorem 116 (Sridharan et al [65] 2007) For any graph 119866with order 119899 if 119899 ⩾ 5 then

119887119879(119866) ⩽

1

3(119899 minus 1) 119894119891 119866 119888119900119899119905119886119894119899119904 119899119900 119888119910119888119897119890119904

119899 minus 1 119894119891 119892 (119866) ⩾ 5

119899 minus 2 119894119891 119892 (119866) = 4

(42)

Rad and Raczek [66] also established some upperbounds on 119887

119879(119866) for a general graph 119866 In particular they

gave an upper bound on 119887119879(119866) in terms of the girth of

119866

Theorem 117 (Rad and Raczek [66] 2011) 119887119879(119866)⩽4Δ(119866) minus 5

for any graph 119866 with 119892(119866) ⩾ 4

82 Paired Bondage Numbers A dominating set 119878 of 119866 iscalled to be paired if the subgraph induced by 119878 containsa perfect matching The paired domination number of 119866denoted by 120574

119875(119866) is the minimum cardinality of a paired

dominating set of 119866 Clearly 120574119879(119866) ⩽ 120574

119875(119866) for every

connected graph 119866 with order at least two where 120574119879(119866) is

the total domination number of 119866 and 120574119875(119866) ⩽ 2120574(119866) for

any graph119866without isolated vertices Paired domination wasintroduced by Haynes and Slater [67 68] and further studiedin [69ndash72]

The paired bondage number of 119866 with 120575(119866) ⩾ 1 denotedby 119887P(119866) is the minimum cardinality among all sets of edges119861 sube 119864 such that 120575(119866 minus 119861) ⩾ 1 and 120574

119875(119866 minus 119861) gt 120574

119875(119866)

The concept of the paired bondage number was firstproposed by Raczek [73] in 2008The following observationsfollow immediately from the definition of the paired bondagenumber

Observation 2 Let 119866 be a graph with 120575(119866) ⩾ 1

(a) If 119867 is a subgraph of 119866 such that 120575(119867) ⩾ 1 then119887119875(119867) ⩽ 119887

119875(119866)

(b) If 119867 is a subgraph of 119866 such that 119887119875(119867) = 1 and 119896

is the number of edges removed to form119867 then 1 ⩽119887119875(119866) ⩽ 119896 + 1

(c) If 119909119910 isin 119864(119866) such that 119889119866(119909) 119889

119866(119910) gt 1 and 119909119910

belongs to each perfect matching of each minimumpaired dominating set of 119866 then 119887

119875(119866) = 1

Based on these observations 119887119875(119875

119899) ⩽ 2 In fact the

paired bondage number of a path has been determined

Theorem 118 (Raczek [73] 2008) For any positive integer 119896if 119899 ⩾ 2 then

119887119875(119875

119899) =

0 119894119891 119899 = 2 3 119900119903 5

1 119894119891 119899 = 4119896 4119896 + 3 119900119903 4119896 + 6

2 119900119905ℎ119890119903119908119894119904119890

(43)

For a cycle 119862119899of order 119899 ⩾ 3 since 120574

119875(119862

119899) = 120574

119875(119875

119899) from

Theorem 118 the following result holds immediately

Corollary 119 (Raczek [73] 2008) For any positive integer 119896if 119899 ⩾ 3 then

119887119875(119862

119899) =

0 119894119891 119899 = 3 119900119903 5

2 119894119891 119899 = 4119896 4119896 + 3 119900119903 4119896 + 6

3 119900119905ℎ119890119903119908119894119904119890

(44)

A wheel 119882119899 where 119899 ⩾ 4 is a graph with 119899 vertices

formed by connecting a single vertex to all vertices of a cycle119862119899minus1

Of course 120574119875(119882

119899) = 2

International Journal of Combinatorics 19

119896

Figure 7 A tree 119879 with 119887119875(119879) = 119896

Theorem 120 (Raczek [73] 2008) For positive integers 119899 and119898

119887119875(119870

119898119899) = 119898 119891119900119903 1 lt 119898 ⩽ 119899

119887119875(119882

119899) =

4 119894119891 119899 = 4

3 119894119891 119899 = 5

2 119900119905ℎ119890119903119908119894119904119890

(45)

Theorem 16 shows that if 119879 is a nontrivial tree then119887(119879) ⩽ 2 However no similar result exists for pairedbondage For any nonnegative integer 119896 let 119879

119896be a tree

obtained by subdividing all but one edge of a star 1198701119896+1

(asshown in Figure 7) It is easy to see that 119887

119875(119879

119896) = 119896 We

state this fact as the following theorem which is similar toTheorem 112

Theorem121 (Raczek [73] 2008) For any nonnegative integer119896 there exists a tree 119879 with 119887

119875(119879) = 119896

Consider the tree defined in Figure 5 for 119896 = 3 Then119887(119879

3) ⩽ 2 and 119887

119875(119879

3) = 3 On the other hand 119887(119875

4) = 2

and 119887119875(119875

4) = 1 Thus we have what follows which is similar

to Corollary 113

Corollary 122 The bondage number and the paired bondagenumber are unrelated even for trees

A constructive characterization of trees with 119887(119879) = 2is given by Hartnell and Rall in [12] Raczek [73] provideda constructive characterization of trees with 119887

119875(119879) = 0 In

order to state the characterization we define a labeling andthree simple operations on a tree119879 Let 119910 isin 119881(119879) and let ℓ(119910)be the label assigned to 119910

Operation T1 If ℓ(119910) = 119861 add a vertex 119909 and the

edge 119910119909 and let ℓ(119909) = 119860OperationT

2 If ℓ(119910) = 119862 add a path (119909

1 119909

2) and the

edge 1199101199091 and let ℓ(119909

1) = 119861 ℓ(119909

2) = 119860

Operation T3 If ℓ(119910) = 119861 add a path (119909

1 119909

2 119909

3)

and the edge 1199101199091 and let ℓ(119909

1) = 119862 ℓ(119909

2) = 119861 and

ℓ(1199093) = 119860

Let 1198752= (119906 V) with ℓ(119906) = 119860 and ℓ(V) = 119861 Let T be

the class of all trees obtained from the labeled 1198752by a finite

sequence of operationsT1 T

2 T

3

A tree 119879 in Figure 8 belongs to the familyTRaczek [73] obtained the following characterization of all

trees 119879 with 119887119875(119879) = 0

119860

119860

119860

119860

119860

119861 119862

119861 119862 119861

119861119860

Figure 8 A tree 119879 belong to the familyT

Theorem 123 (Raczek [73] 2008) For a tree 119879 119887119875(119879) = 0 if

and only if 119879 is inT

We state the decision problem for the paired bondage asfollows

Problem 4 Consider the decision problem

Paired Bondage ProblemInstance a graph 119866 and a positive integer 119887Question is 119887

119875(119866) ⩽ 119887

Conjecture 124 The paired bondage problem is NP-complete

83 Restrained Bondage Numbers A dominating set 119878 of 119866 iscalled to be restrained if the subgraph induced by 119878 containsno isolated vertices where 119878 = 119881(119866) 119878 The restraineddomination number of 119866 denoted by 120574

119877(119866) is the smallest

cardinality of a restrained dominating set of 119866 It is clear that120574119877(119866) exists and 120574(119866) ⩽ 120574

119877(119866) for any nonempty graph 119866

The concept of restrained domination was introducedby Telle and Proskurowski [74] in 1997 albeit indirectly asa vertex partitioning problem Concerning the study of therestrained domination numbers the reader is referred to [74ndash79]

In 2008 Hattingh and Plummer [80] defined therestrained bondage number 119887R(119866) of a nonempty graph 119866 tobe the minimum cardinality among all subsets 119861 sube 119864(119866) forwhich 120574

119877(119866 minus 119861) gt 120574

119877(119866)

For some simple graphs their restrained dominationnumbers can be easily determined and so restrained bondagenumbers have been also determined For example it is clearthat 120574

119877(119870

119899) = 1 Domke et al [77] showed that 120574

119877(119862

119899) =

119899 minus 2lfloor1198993rfloor for 119899 ⩾ 3 and 120574119877(119875

119899) = 119899 minus 2lfloor(119899 minus 1)3rfloor for 119899 ⩾ 1

Using these results Hattingh and Plummer [80] obtained therestricted bondage numbers for119870

119899 119862

119899 119875

119899 and119866 = 119870

11989911198992119899119905

as follows

Theorem 125 (Hattingh and Plummer [80] 2008) For 119899 ⩾ 3

119887R (119870119899) =

1 119894119891 119899 = 3

lceil119899

2rceil 119900119905ℎ119890119903119908119894119904119890

119887R (119862119899) =

1 119894119891 119899 equiv 0 (mod 3) 2 119900119905ℎ119890119903119908119894119904119890

119887R (119875119899) = 1 119899 ⩾ 4

20 International Journal of Combinatorics

119887R (119866)

=

lceil119898

2rceil 119894119891 119899

119898= 1 119886119899119889 119899

119898+1⩾ 2 (1 ⩽ 119898 lt 119905)

2119905 minus 2 119894119891 1198991= 119899

2= sdot sdot sdot = 119899

119905= 2 (119905 ⩾ 2)

2 119894119891 1198991= 2 119886119899119889 119899

2⩾ 3 (119905 = 2)

119905minus1

sum

119894=1

119899119894minus 1 119900119905ℎ119890119903119908119894119904119890

(46)

Theorem 126 (Hattingh and Plummer [80] 2008) For a tree119879 of order 119899 (⩾ 4) 119879 ≇ 119870

1119899minus1if and only if 119887R(119879) = 1

Theorem 126 shows that the restrained bondage numberof a tree can be computed in constant time However thedecision problem for 119887R(119866) is NP-complete even for bipartitegraphs

Problem 5 Consider the decision problem

Restrained Bondage Problem

Instance a graph 119866 and a positive integer 119887

Question is 119887R(119866) ⩽ 119887

Theorem 127 (Hattingh and Plummer [80] 2008) Therestrained bondage problem is NP-complete even for bipartitegraphs

Consequently in view of its computational hardnessit is significative to establish some sharp bounds on therestrained bondage number of a graph in terms of othergraphic parameters

Theorem 128 (Hattingh and Plummer [80] 2008) For anygraph 119866 with 120575(119866) ⩾ 2

119887R (119866) ⩽ min119909119910isin119864(119866)

119889119866(119909) + 119889

119866(119910) minus 2 (47)

Corollary 129 119887R(119866) ⩽ Δ(119866)+120575(119866)minus2 for any graph119866 with120575(119866) ⩾ 2

Notice that the bounds stated in Theorem 128 andCorollary 129 are sharp Indeed the class of cycles whoseorders are congruent to 1 2 (mod 3) have a restrained bondagenumber achieving these bounds (see Theorem 125)

Theorem 128 is an analogue of Theorem 14 A quitenatural problem is whether or not for restricted bondagenumber there are analogues of Theorem 16 Theorem 18Theorem 29 and so on Indeed we should consider all resultsfor the bondage number whether or not there are analoguesfor restricted bondage number

Theorem 130 (Hattingh and Plummer [80] 2008) For anygraph 119866 with 120574

119877(119866) = 2

119887R (119866) ⩽ Δ (119866) + 1 (48)

We now consider the relation between 119887119877(119866) and 119887(119866)

Theorem 131 (Hattingh and Plummer [80] 2008) If 120574119877(119866) =

120574(119866) for some graph 119866 then 119887R(119866) ⩽ 119887(119866)

Corollary 129 and Theorem 131 provide a possibility toattack Conjecture 73 by characterizing planar graphs with120574119877= 120574 and 119887R = 119887 when 3 ⩽ Δ ⩽ 6However we do not have 119887R(119866) = 119887(119866) for any graph

119866 even if 120574119877(119866) = 120574(119866) Observe that 120574

119877(119870

3) = 120574(119870

3) yet

119887119877(119870

3) = 1 and 119887(119870

3) = 2 We still may not claim that

119887119877(119866) = 119887(119866) even in the case that every 120574-set is a 120574

119877-set

The example 1198703again demonstrates this

Theorem 132 (Hattingh and Plummer [80] 2008) There isan infinite class of graphs inwhich each graph119866 satisfies 119887(119866) lt119887R(119866)

In fact such an infinite class of graphs can be obtained asfollows Let119867 be a connected graph BC(119867) a graph obtainedfrom 119867 by attaching ℓ (⩾ 2) vertices of degree 1 to eachvertex in 119867 and B = 119866 119866 = BC(119867) for some graph 119867such that 120575(119867) ⩾ 2 By Theorem 3 we can easily check that119887(119866) = 1 lt 2 ⩽ min120575(119867) ℓ = 119887R(119866) for any 119866 isin BMoreover there exists a graph 119866 isin B such that 119887R(119866) canbe much larger than 119887(119866) which is stated as the followingtheorem

Theorem 133 (Hattingh and Plummer [80] 2008) For eachpositive integer 119896 there is a graph119866 such that 119896 = 119887R(119866)minus119887(119866)

Combining Theorem 133 with Theorem 132 we have thefollowing corollary

Corollary 134 The bondage number and the restrainedbondage number are unrelated

Very recently Jafari Rad et al [81] have consideredthe total restrained bondage in graphs based on both totaldominating sets and restrained dominating sets and obtainedseveral properties exact values and bounds for the totalrestrained bondage number of a graph

A restrained dominating set 119878 of a graph 119866 without iso-lated vertices is called to be a total restrained dominating setif 119878 is a total dominating set The total restrained dominationnumber of119866 denoted by 120574TR (119866) is theminimum cardinalityof a total restrained dominating set of 119866 For referenceson this domination in graphs see [3 61 82ndash85] The totalrestrained bondage number 119887TR (119866) of a graph 119866 with noisolated vertex is the cardinality of a smallest set of edges 119861 sube119864(119866) for which119866minus119861 has no isolated vertex and 120574TR (119866minus119861) gt120574TR (119866) In the case that there is no such subset 119861 we define119887TR (119866) = infin

Ma et al [84] determined that 120574TR (119875119899) = 119899minus2lfloor(119899minus2)4rfloorfor 119899 ⩾ 3 120574TR (119862119899

) = 119899 minus 2lfloor1198994rfloor for 119899 ⩾ 3 120574TR (1198703) =

3 and 120574TR (119870119899) = 2 for 119899 = 3 and 120574TR (1198701119899

) = 1 + 119899

and 120574TR (119870119898119899) = 2 for 2 ⩽ 119898 ⩽ 119899 According to these

results Jafari Rad et al [81] determined the exact valuesof total restrained bondage numbers for the correspondinggraphs

International Journal of Combinatorics 21

Theorem 135 (Jafari Rad et al [81]) For an integer 119899

119887TR (119875119899) = infin 119894119891 2 ⩽ 119899 ⩽ 5

1 119894119891 119899 ⩾ 5

119887TR (119862119899) =

infin 119894119891 119899 = 3

1 119894119891 3 lt 119899 119899 equiv 0 1 (mod 4)2 119894119891 3 lt 119899 119899 equiv 2 3 (mod 4)

119887TR (119882119899) = 2 119891119900119903 119899 ⩾ 6

119887TR (119870119899) = 119899 minus 1 119891119900119903 119899 ⩾ 4

119887TR (119870119898119899) = 119898 minus 1 119891119900119903 2 ⩽ 119898 ⩽ 119899

(49)

119887TR (119870119899) = 119899minus1 shows the fact that for any integer 119899 ⩾ 3 there

exists a connected graph namely119870119899+1

such that 119887TR (119870119899+1) =

119899 that is the total restrained bondage number is unboundedfrom the above

A vertex is said to be a support vertex if it is adjacent toa vertex of degree one Let A be the family of all connectedgraphs such that119866 belongs toA if and only if every edge of119866is incident to a support vertex or 119866 is a cycle of length three

Proposition 136 (Cyman and Raczek [82] 2006) For aconnected graph119866 of order 119899 120574TR (119866) = 119899 if and only if119866 isin A

Hence if 119866 isin A then the removal of any subset of edgesof119866 cannot increase the total restrained domination numberof 119866 and thus 119887TR (119866) = infin When 119866 notin A a result similar toTheorem 6 is obtained

Theorem 137 (Jafari Rad et al [81]) For any tree 119879 notin A119887TR (119879) ⩽ 2 This bound is sharp

In the samepaper the authors provided a characterizationfor trees 119879 notin A with 119887TR (119879) = 1 or 2 The following result issimilar to Observation 1

Observation 3 (Jafari Rad et al [81]) If 119896 edges can beremoved from a graph 119866 to obtain a graph 119867 without anisolated vertex and with 119887TR (119867) = 119905 then 119887TR (119866) ⩽ 119896 + 119905

ApplyingTheorem 137 andObservation 3 Jafari Rad et alcharacterized all connected graphs 119866 with 119887TR (119866) = infin

Theorem 138 (Jafari Rad et al [81]) For a connected graph119866119887TR (119866) = infin if and only if 119866 isin A

84 Other Conditional Bondage Numbers A subset 119868 sube 119881(119866)is called an independent set if no two vertices in 119868 are adjacentin 119866 The maximum cardinality among all independent setsis called the independence number of 119866 denoted by 120572(119866)

A dominating set 119878 of a graph 119866 is called to be inde-pendent if 119878 is an independent set of 119866 The minimumcardinality among all independent dominating set is calledthe independence domination number of 119866 and denoted by120574119868(119866)

Since an independent dominating set is not only adominating set but also an independent set 120574(119866) ⩽ 120574

119868(119866) ⩽

120572(119866) for any graph 119866It is clear that a maximal independent set is certainly a

dominating set Thus an independent set is maximal if andonly if it is an independent dominating set and so 120574

119868(119866) is the

minimum cardinality among all maximal independent setsof 119866 This graph-theoretical invariant has been well studiedin the literature see for example Haynes et al [3] for earlyresults Goddard and Henning [86] for recent results onindependent domination in graphs

In 2003 Zhang et al [87] defined the independencebondage number 119887

119868(119866) of a nonempty graph 119866 to be the

minimum cardinality among all subsets 119861 sube 119864(119866) for which120574119868(119866 minus 119861) gt 120574

119868(119866) For some ordinary graphs their indepen-

dence domination numbers can be easily determined and soindependence bondage numbers have been also determinedClearly 119887

119868(119870

119899) = 1 if 119899 ⩾ 2

Theorem 139 (Zhang et al [87] 2003) For any integer 119899 ⩾ 4

119887119868(119862

119899) =

1 119894119891 119899 equiv 1 (mod 2) 2 119894119891 119899 equiv 0 (mod 2)

119887119868(119875

119899) =

1 119894119891 119899 equiv 0 (mod 2) 2 119894119891 119899 equiv 1 (mod 2)

119887119868(119870

119898119899) = 119899 119891119900119903 119898 ⩽ 119899

(50)

Apart from the above-mentioned results as far as we findno other results on the independence bondage number in thepresent literature we never knew much of any result on thisparameter for a tree

A subset 119878 of 119881(119866) is called an equitable dominatingset if for every 119910 isin 119881(119866 minus 119878) there exists a vertex 119909 isin 119878such that 119909119910 isin 119864(119866) and |119889

119866(119909) minus 119889

119866(119910)| ⩽ 1 proposed

by Dharmalingam [88] The minimum cardinality of such adominating set is called the equitable domination number andis denoted by 120574

119864(119866) Deepak et al [89] defined the equitable

bondage number 119887119864(119866) of a graph 119866 to be the cardinality of a

smallest set 119861 sube 119864(119866) of edges for which 120574119864(119866 minus 119861) gt 120574

119864(119866)

and determined 119887119864(119866) for some special graphs

Theorem 140 (Deepak et al [89] 2011) For any integer 119899 ⩾ 2

119887119864(119870

119899) = lceil

119899

2rceil

119887119864(119870

119898119899) = 119898 119891119900119903 0 ⩽ 119899 minus 119898 ⩽ 1

119887119864(119862

119899) =

3 119894119891 119899 equiv 1 (mod 3)2 119900119905ℎ119890119903119908119894119904119890

119887119864(119875

119899) =

2 119894119891 119899 equiv 1 (mod 3)1 119900119905ℎ119890119903119908119894119904119890

119887119864(119879) ⩽ 2 119891119900119903 119886119899119910 119899119900119899119905119903119894119907119894119886119897 119905119903119890119890 119879

119887119864(119866) ⩽ 119899 minus 1 119891119900119903 119886119899119910 119888119900119899119899119890119888119905119890119889 119892119903119886119901ℎ 119866 119900119891 119900119903119889119890119903 119899

(51)

22 International Journal of Combinatorics

As far as we know there are no more results on theequitable bondage number of graphs in the present literature

There are other variations of the normal domination byadding restricted conditions to the normal dominating setFor example the connected domination set of a graph119866 pro-posed by Sampathkumar and Walikar [90] is a dominatingset 119878 such that the subgraph induced by 119878 is connected Theproblem of finding a minimum connected domination set ina graph is equivalent to the problemof finding a spanning treewithmaximumnumber of vertices of degree one [91] and hassome important applications in wireless networks [92] thusit has received much research attention However for suchan important domination there is no result on its bondagenumber We believe that the topic is of significance for futureresearch

9 Generalized Bondage Numbers

There are various generalizations of the classical dominationsuch as 119901-domination distance domination fractional dom-ination Roman domination and rainbow domination Everysuch a generalization can lead to a corresponding bondageIn this section we introduce some of them

91 119901-Bondage Numbers In 1985 Fink and Jacobson [93]introduced the concept of 119901-domination Let 119901 be a positiveinteger A subset 119878 of 119881(119866) is a 119901-dominating set of 119866 if|119878 cap 119873

119866(119910)| ⩾ 119901 for every 119910 isin 119878 The 119901-domination number

120574119901(119866) is the minimum cardinality among all 119901-dominating

sets of 119866 Any 119901-dominating set of 119866 with cardinality 120574119901(119866)

is called a 120574119901-set of 119866 Note that a 120574

1-set is a classic minimum

dominating set Notice that every graph has a 119901-dominatingset since the vertex-set119881(119866) is such a setWe also note that a 1-dominating set is a dominating set and so 120574(119866) = 120574

1(119866) The

119901-domination number has received much research attentionsee a state-of-the-art survey article by Chellali et al [94]

It is clear from definition that every 119901-dominating setof a graph certainly contains all vertices of degree at most119901 minus 1 By this simple observation to avoid the trivial caseoccurrence we always assume that Δ(119866) ⩾ 119901 For 119901 ⩾ 2 Luet al [95] gave a constructive characterization of trees withunique minimum 119901-dominating sets

Recently Lu and Xu [96] have introduced the conceptto the 119901-bondage number of 119866 denoted by 119887

119901(119866) as the

minimum cardinality among all sets of edges 119861 sube 119864(119866) suchthat 120574

119901(119866 minus 119861) gt 120574

119901(119866) Clearly 119887

1(119866) = 119887(119866)

Lu and Xu [96] established a lower bound and an upperbound on 119887

119901(119879) for any integer 119901 ⩾ 2 and any tree 119879 with

Δ(119879) ⩾ 119901 and characterized all trees achieving the lowerbound and the upper bound respectively

Theorem 141 (Lu and Xu [96] 2011) For any integer 119901 ⩾ 2and any tree 119879 with Δ(119879) ⩾ 119901

1 ⩽ 119887119901(119879) ⩽ Δ (119879) minus 119901 + 1 (52)

Let 119878 be a given subset of 119881(119866) and for any 119909 isin 119878 let

119873119901(119909 119878 119866) = 119910 isin 119878 cap 119873

119866(119909)

1003816100381610038161003816119873119866(119910) cap 119878

1003816100381610038161003816 = 119901

(53)

Then all trees achieving the lower bound 1 can be character-ized as follows

Theorem 142 (Lu and Xu [96] 2011) Let 119879 be a tree withΔ(119879) ⩾ 119901 ⩾ 2 Then 119887

119901(119879) = 1 if and only if for any 120574

119901-set

119878 of 119879 there exists an edge 119909119910 isin (119878 119878) such that 119910 isin 119873119901(119909 119878 119879)

The notation 119878(119886 119887) denotes a double star obtained byadding an edge between the central vertices of two stars119870

1119886minus1

and1198701119887minus1

And the vertex with degree 119886 (resp 119887) in 119878(119886 119887) iscalled the 119871-central vertex (resp 119877-central vertex) of 119878(119886 119887)

To characterize all trees attaining the upper bound givenin Theorem 141 we define three types of operations on a tree119879 with Δ(119879) = Δ ⩾ 119901 + 1

Type 1 attach a pendant edge to a vertex 119910with 119889119879(119910) ⩽ 119901minus

2 in 119879

Type 2 attach a star 1198701Δminus1

to a vertex 119910 of 119879 by joining itscentral vertex to 119910 where 119910 in a 120574

119901-set of 119879 and

119889119879(119910) ⩽ Δ minus 1

Type 3 attach a double star 119878(119901 Δ minus 1) to a pendant vertex 119910of 119879 by coinciding its 119877-central vertex with 119910 wherethe unique neighbor of 119910 is in a 120574

119901-set of 119879

Let B = 119879 a tree obtained from 1198701Δ

or 119878(119901 Δ) by afinite sequence of operations of Types 1 2 and 3

Theorem 143 (Lu and Xu [96] 2011) A tree with the maxi-mum Δ ⩾ 119901 + 1 has 119901-bondage number Δ minus 119901 + 1 if and onlyif it belongs toB

Theorem 144 (Lu and Xu [97] 2012) Let 119866119898119899= 119875

119898times 119875

119899

Then

1198872(119866

2119899) = 1 119891119900119903 119899 ⩾ 2

1198872(119866

3119899) =

2 119894119891 119899 equiv 1 (mod 3) 1 119900119905ℎ119890119903119908119894119904119890

for 119899 ⩾ 2

1198872(119866

4119899) =

1 119894119891 119899 equiv 3 (mod 4) 2 119900119905ℎ119890119903119908119894119904119890

for 119899 ⩾ 7

(54)

Recently Krzywkowski [98] has proposed the conceptof the nonisolating 2-bondage of a graph The nonisolating2-bondage number of a graph 119866 denoted by 1198871015840

2(119866) is the

minimum cardinality among all sets of edges 119861 sube 119864(119866) suchthat 119866 minus 119861 has no isolated vertices and 120574

2(119866 minus 119861) gt 120574

2(119866) If

for every 119861 sube 119864(119866) either 1205742(119866 minus 119861) = 120574

2(119866) or 119866 minus 119861 has

isolated vertices then define 11988710158402(119866) = 0 and say that 119866 is a 2-

nonisolatedly strongly stable graph Krzywkowski presentedsome basic properties of nonisolating 2-bondage in graphsshowed that for every nonnegative integer 119896 there exists atree 119879 such 1198871015840

2(119879) = 119896 characterized all 2-non-isolatedly

strongly stable trees and determined 11988710158402(119866) for several special

graphs

International Journal of Combinatorics 23

Theorem 145 (Krzywkowski [98] 2012) For any positiveintegers 119899 and119898 with119898 ⩽ 119899

1198871015840

2(119870

119899) =

0 119894119891 119899 = 1 2 3

lfloor2119899

3rfloor 119900119905ℎ119890119903119908119894119904119890

1198871015840

2(119875

119899) =

0 119894119891 119899 = 1 2 3

1 119900119905ℎ119890119903119908119894119904119890

1198871015840

2(119862

119899) =

0 119894119891 119899 = 3

1 119894119891 119899 119894119904 119890V119890119899

2 119894119891 119899 119894119904 119900119889119889

1198871015840

2(119870

119898119899) =

3 119894119891 119898 = 119899 = 3

1 119894119891 119898 = 119899 = 4

2 119900119905ℎ119890119903119908119894119904119890

(55)

92 Distance Bondage Numbers A subset 119878 of vertices of agraph119866 is said to be a distance 119896-dominating set for119866 if everyvertex in 119866 not in 119878 is at distance at most 119896 from some vertexof 119878 The minimum cardinality of all distance 119896-dominatingsets is called the distance 119896-domination number of 119866 anddenoted by 120574

119896(119866) (do not confuse with the above-mentioned

120574119901(119866)) When 119896 = 1 a distance 1-dominating set is a normal

dominating set and so 1205741(119866) = 120574(119866) for any graph 119866 Thus

the distance 119896-domination is a generalization of the classicaldomination

The relation between 120574119896and 120572

119896(the 119896-independence

number defined in Section 22) for a tree obtained by Meirand Moon [14] who proved that 120574

119896(119879) = 120572

2119896(119879) for any tree

119879 (a special case for 119896 = 1 is stated in Proposition 10) Thefurther research results can be found in Henning et al [99]Tian and Xu [100ndash103] and Liu et al [104]

In 1998 Hartnell et al [15] defined the distance 119896-bondagenumber of 119866 denoted by 119887

119896(119866) to be the cardinality of a

smallest subset 119861 of edges of 119866 with the property that 120574119896(119866 minus

119861) gt 120574119896(119866) FromTheorem 6 it is clear that if119879 is a nontrivial

tree then 1 ⩽ 1198871(119879) ⩽ 2 Hartnell et al [15] generalized this

result to any integer 119896 ⩾ 1

Theorem 146 (Hartnell et al [15] 1998) For every nontrivialtree 119879 and positive integer 119896 1 ⩽ 119887

119896(119879) ⩽ 2

Hartnell et al [15] and Topp and Vestergaard [13] alsocharacterized the trees having distance 119896-bondage number 2In particular the class of trees for which 119887

1(119879) = 2 are just

those which have a unique maximum 2-independent set (seeTheorem 11)

Since when 119896 = 1 the distance 1-bondage number 1198871(119866)

is the classical bondage number 119887(119866) Theorem 12 gives theNP-hardness of deciding the distance 119896-bondage number ofgeneral graphs

Theorem 147 Given a nonempty undirected graph 119866 andpositive integers 119896 and 119887 with 119887 ⩽ 120576(119866) determining wetheror not 119887

119896(119866) ⩽ 119887 is NP-hard

For a vertex 119909 in119866 the open 119896-neighborhood119873119896(119909) of 119909 is

defined as119873119896(119909) = 119910 isin 119881(119866) 1 ⩽ 119889

119866(119909 119910) le 119896The closed

119896-neighborhood119873119896[119909] of 119909 in 119866 is defined as119873

119896(119909) cup 119909 Let

Δ119896(119866) = max 1003816100381610038161003816119873119896

(119909)1003816100381610038161003816 for any 119909 isin 119881 (119866) (56)

Clearly Δ1(119866) = Δ(119866) The 119896th power of a graph 119866 is the

graph119866119896 with vertex-set119881(119866119896

) = 119881(119866) and edge-set119864(119866119896

) =

119909119910 1 ⩽ 119889119866(119909 119910) ⩽ 119896 The following lemma holds directly

from the definition of 119866119896

Proposition 148 (Tian and Xu [101]) Δ(119866119896

) = Δ119896(119866) and

120574(119866119896

) = 120574119896(119866) for any graph 119866 and each 119896 ⩾ 1

A graph 119866 is 119896-distance domination-critical or 120574119896-critical

for short if 120574119896(119866 minus 119909) lt 120574

119896(119866) for every vertex 119909 in 119866

proposed by Henning et al [105]

Proposition 149 (Tian and Xu [101]) For each 119896 ⩾ 1 a graph119866 is 120574

119896-critical if and only if 119866119896 is 120574

119896-critical

By the above facts we suggest the following worthwhileproblem

Problem 6 Can we generalize the results on the bondage for119866 to 119866119896 In particular do the following propositions hold

(a) 119887(119866119896

) = 119887119896(119866) for any graph 119866 and each 119896 ⩾ 1

(b) 119887(119866119896

) ⩽ Δ119896(119866) if 119866 is not 120574

119896-critical

Let 119896 and 119901 be positive integers A subset 119878 of 119881(119866) isdefined to be a (119896 119901)-dominating set of 119866 if for any vertex119909 isin 119878 |119873

119896(119909) cap 119878| ⩾ 119901 The (119896 119901)-domination number of

119866 denoted by 120574119896119901(119866) is the minimum cardinality among

all (119896 119901)-dominating sets of 119866 Clearly for a graph 119866 a(1 1)-dominating set is a classical dominating set a (119896 1)-dominating set is a distance 119896-dominating set and a (1 119901)-dominating set is the above-mentioned 119901-dominating setThat is 120574

11(119866) = 120574(119866) 120574

1198961(119866) = 120574

119896(119866) and 120574

1119901(119866) = 120574

119901(119866)

The concept of (119896 119901)-domination in a graph 119866 is a gen-eralized domination which combines distance 119896-dominationand 119901-domination in 119866 So the investigation of (119896 119901)-domination of 119866 is more interesting and has received theattention of many researchers see for example [106ndash109]

More general Li and Xu [110] proposed the concept of the(ℓ 119908)-domination number of a119908-connected graph A subset119878 of 119881(119866) is defined to be an (ℓ 119908)-dominating set of 119866 iffor any vertex 119909 isin 119878 there are 119908 internally disjoint pathsof length at most ℓ from 119909 to some vertex in 119878 The (ℓ 119908)-domination number of119866 denoted by 120574

ℓ119908(119866) is theminimum

cardinality among all (ℓ 119908)-dominating sets of 119866 Clearly1205741198961(119866) = 120574

119896(119866) and 120574

1119901(119866) = 120574

119901(119866)

It is quite natural to propose the concept of bondage num-ber for (119896 119901)-domination or (ℓ 119908)-domination However asfar as we know none has proposed this concept until todayThis is a worthwhile topic for further research

24 International Journal of Combinatorics

93 Fractional Bondage Numbers If 120590 is a function mappingthe vertex-set 119881 into some set of real numbers then for anysubset 119878 sube 119881 let 120590(119878) = sum

119909isin119878120590(119909) Also let |120590| = 120590(119881)

A real-value function 120590 119881 rarr [0 1] is a dominatingfunction of a graph 119866 if for every 119909 isin 119881(119866) 120590(119873

119866[119909]) ⩾ 1

Thus if 119878 is a dominating set of 119866 120590 is a function where

120590 (119909) = 1 if 119909 isin 1198780 otherwise

(57)

then 120590 is a dominating function of119866 The fractional domina-tion number of 119866 denoted by 120574

119865(119866) is defined as follows

120574119865(119866) = min |120590| 120590 is a domination function of 119866

(58)

The 120574119865-bondage number of 119866 denoted by 119887

119865(119866) is

defined as the minimum cardinality of a subset 119861 sube 119864 whoseremoval results in 120574

119865(119866 minus 119861) gt 120574

119865(119866)

Hedetniemi et al [111] are the first to study fractionaldomination although Farber [112] introduced the idea indi-rectly The concept of the fractional domination numberwas proposed by Domke and Laskar [113] in 1997 Thefractional domination numbers for some ordinary graphs aredetermined

Proposition 150 (Domke et al [114] 1998 Domke and Laskar[113] 1997) (a) If 119866 is a 119896-regular graph with order 119899 then120574119865(119866) = 119899119896 + 1(b) 120574

119865(119862

119899) = 1198993 120574

119865(119875

119899) = lceil1198993rceil 120574

119865(119870

119899) = 1

(c) 120574119865(119866) = 1 if and only if Δ(119866) = 119899 minus 1

(d) 120574119865(119879) = 120574(119879) for any tree 119879

(e) 120574119865(119870

119899119898) = (119899(119898 + 1) + 119898(119899 + 1))(119899119898 minus 1) where

max119899119898 gt 1

The assertions (a)ndash(d) are due to Domke et al [114] andthe assertion (e) is due to Domke and Laskar [113] Accordingto these results Domke and Laskar [113] determined the 120574

119865-

bondage numbers for these graphs

Theorem 151 (Domke and Laskar [113] 1997) 119887119865(119870

119899) =

lceil1198992rceil 119887119865(119870

119899119898) = 1 wheremax119899119898 gt 1

119887119865(119862

119899) =

2 119894119891 119899 equiv 0 (mod 3) 1 119900119905ℎ119890119903119908119894119904119890

119887119865(119875

119899) =

2 119894119891 119899 equiv 1 (mod 3) 1 119900119905ℎ119890119903119908119894119904119890

(59)

It is easy to see that for any tree119879 119887119865(119879) ⩽ 2 In fact since

120574119865(119879) = 120574(119879) for any tree 119879 120574

119865(119879

1015840

) = 120574(1198791015840

) for any subgraph1198791015840 of 119879 It follows that

119887119865(119879) = min 119861 sube 119864 (119879) 120574

119865(119879 minus 119861) gt 120574

119865(119879)

= min 119861 sube 119864 (119879) 120574 (119879 minus 119861) gt 120574 (119879)

= 119887 (119879) ⩽ 2

(60)

Except for these no results on 119887119865(119866) have been known as

yet

94 Roman Bondage Numbers A Roman dominating func-tion on a graph 119866 is a labeling 119891 119881 rarr 0 1 2 such thatevery vertex with label 0 has at least one neighbor with label2 The weight of a Roman dominating function 119891 on 119866 is thevalue

119891 (119866) = sum

119906isin119881(119866)

119891 (119906) (61)

The minimum weight of a Roman dominating function ona graph 119866 is called the Roman domination number of 119866denoted by 120574RM (119866)

A Roman dominating function 119891 119881 rarr 0 1 2

can be represented by the ordered partition (1198810 119881

1 119881

2) (or

(119881119891

0 119881

119891

1 119881

119891

2) to refer to119891) of119881 where119881

119894= V isin 119881 | 119891(V) = 119894

In this representation its weight 119891(119866) = |1198811| + 2|119881

2| It is

clear that119881119891

1cup119881

119891

2is a dominating set of 119866 called the Roman

dominating set denoted by 119863119891

R = (1198811 1198812) Since 119881119891

1cup 119881

119891

2is

a dominating set when 119891 is a Roman dominating functionand since placing weight 2 at the vertices of a dominating setyields a Roman dominating function in [115] it is observedthat

120574 (119866) ⩽ 120574RM (119866) ⩽ 2120574 (119866) (62)

A graph 119866 is called to be Roman if 120574RM (119866) = 2120574(119866)The definition of the Roman domination function is

proposed implicitly by Stewart [116] and ReVelle and Rosing[117] Roman dominating numbers have been deeply studiedIn particular Bahremandpour et al [118] showed that theproblem determining the Roman domination number is NP-complete even for bipartite graphs

Let 119866 be a graph with maximum degree at least twoThe Roman bondage number 119887RM (119866) of 119866 is the minimumcardinality of all sets 1198641015840 sube 119864 for which 120574RM (119866 minus 119864

1015840

) gt

120574RM (119866) Since in study of Roman bondage number theassumption Δ(119866) ⩾ 2 is necessary we always assume thatwhen we discuss 119887RM (119866) all graphs involved satisfy Δ(119866) ⩾2 The Roman bondage number 119887RM (119866) is introduced byJafari Rad and Volkmann in [119]

Recently Bahremandpour et al [118] have shown that theproblem determining the Roman bondage number is NP-hard even for bipartite graphs

Problem 7 Consider the decision problem

Roman Bondage Problem

Instance a nonempty bipartite graph119866 and a positiveinteger 119896

Question is 119887RM (119866) ⩽ 119896

Theorem 152 (Bahremandpour et al [118] 2013) TheRomanbondage problem is NP-hard even for bipartite graphs

The exact value of 119887RM(119866) is known only for a few familyof graphs including the complete graphs cycles and paths

International Journal of Combinatorics 25

Theorem 153 (Jafari Rad and Volkmann [119] 2011) For anypositive integers 119899 and119898 with119898 ⩽ 119899

119887RM (119875119899) = 2 119894119891 119899 equiv 2 (mod 3) 1 119900119905ℎ119890119903119908119894119904119890

119887RM (119862119899) =

3 119894119891 119899 = 2 (mod 3) 2 119900119905ℎ119890119903119908119894119904119890

119887RM (119870119899) = lceil

119899

2rceil 119891119900119903 119899 ⩾ 3

119887RM (119870119898119899) =

1 119894119891 119898 = 1 119886119899119889 119899 = 1

4 119894119891 119898 = 119899 = 3

119898 119900119905ℎ119890119903119908119894119904119890

(63)

Lemma 154 (Cockayne et al [115] 2004) If 119866 is a graph oforder 119899 and contains vertices of degree 119899 minus 1 then 120574RM(119866) = 2

Using Lemma 154 the third conclusion in Theorem 153can be generalized to more general case which is similar toLemma 49

Theorem 155 (Jafari Rad and Volkmann [119] 2011) Let119866 bea graph with order 119899 ge 3 and 119905 the number of vertices of degree119899 minus 1 in 119866 If 119905 ⩾ 1 then 119887RM(119866) = lceil1199052rceil

Ebadi and PushpaLatha [120] conjectured that 119887RM(119866) ⩽119899 minus 1 for any graph 119866 of order 119899 ⩾ 3 Dehgardi et al[121] Akbari andQajar [122] independently showed that thisconjecture is true

Theorem 156 119887RM(119866) ⩽ 119899 minus 1 for any connected graph 119866 oforder 119899 ⩾ 3

For a complete 119905-partite graph we have the followingresult

Theorem 157 (Hu and Xu [123] 2011) Let 119866 = 11987011989811198982119898119905

bea complete 119905-partite graph with 119898

1= sdot sdot sdot = 119898

119894lt 119898

119894+1le sdot sdot sdot ⩽

119898119905 119905 ⩾ 2 and 119899 = sum119905

119895=1119898

119895 Then

119887RM (119866) =

lceil119894

2rceil 119894119891 119898

119894= 1 119886119899119889 119899 ⩾ 3

2 119894119891 119898119894= 2 119886119899119889 119894 = 1

119894 119894119891 119898119894= 2 119886119899119889 119894 ⩾ 2

4 119894119891 119898119894= 3 119886119899119889 119894 = 119905 = 2

119899 minus 1 119894119891 119898119894= 3 119886119899119889 119894 = 119905 ⩾ 3

119899 minus 119898119905

119894119891 119898119894⩾ 3 119886119899119889 119898

119905⩾ 4

(64)

Consider a complete 119905-partite graph 119870119905(3) for 119905 ⩾ 3

119887RM(119870119905(3)) = 119899 minus 1 by Theorem 157 where 119899 = 3119905

which shows that this upper bound of 119899 minus 1 on 119887RM(119866) inTheorem 156 is sharp This fact and 120574

119877(119870

119905(3)) = 4 give

a negative answer to the following two problems posed byDehgardi et al [121]Prove or Disprove for any connected graph 119866 of order 119899 ge 3(i) 119887RM(119866) = 119899 minus 1 if and only if 119866 cong 119870

3 (ii) 119887RM(119866) ⩽ 119899 minus

120574119877(119866) + 1Note that a complete 119905-partite graph 119870

119905(3) is (119899 minus 3)-

regular and 119887RM(119870119905(3)) = 119899 minus 1 for 119905 ⩾ 3 where 119899 = 3119905

Hu and Xu further determined that 119887119877(119866) = 119899 minus 2 for any

(119899 minus 3)-regular graph 119866 of order 119899 ⩽ 5 except 119870119905(3)

Theorem 158 (Hu and Xu [123] 2011) For any (119899minus3)-regulargraph 119866 of order 119899 other than119870

119905(3) 119887RM(119866) = 119899 minus 2 for 119899 ⩾ 5

For a tree 119879 with order 119899 ⩾ 3 Ebadi and PushpaLatha[120] and Jafari Rad and Volkmann [119] independentlyobtained an upper bound on 119887RM(119879)

Theorem 159 119887RM(119879) ⩽ 3 for any tree 119879 with order 119899 ⩾ 3

Theorem 160 (Bahremandpour et al [118] 2013) 119887RM(1198752 times119875119899) = 2 for 119899 ⩾ 2

Theorem 161 (Jafari Rad and Volkmann [119] 2011) Let119866 bea graph with order at least three

(a) If (119909 119910 119911) is a path of length 2 in 119866 then 119887RM(119866) ⩽119889119866(119909) + 119889

119866(119910) + 119889

119866(119911) minus 3 minus |119873

119866(119909) cap 119873

119866(119910)|

(b) If 119866 is connected then 119887RM(119866) ⩽ 120582(119866) + 2Δ(119866) minus 3

Theorem 161 (b) implies 119887RM(119866) ⩽ 120575(119866) + 2Δ(119866) minus 3 for aconnected graph 119866 Note that for a planar graph 119866 120575(119866) ⩽ 5moreover 120575(119866) ⩽ 3 if the girth at least 4 and 120575(119866) ⩽ 2 if thegirth at least 6 These two facts show that 119887RM(119866) ⩽ 2Δ(119866) +2 for a connected planar graph 119866 Jafari Rad and Volkmann[124] improved this bound

Theorem 162 (Jafari Rad and Volkmann [124] 2011) For anyconnected planar graph 119866 of order 119899 ⩾ 3 with girth 119892(119866)

119887RM (119866)

⩽

2Δ (119866)

Δ (119866) + 6

Δ (119866) + 5 119894119891 119866 119888119900119899119905119886119894119899119904 119899119900 V119890119903119905119894119888119890119904 119900119891 119889119890119892119903119890119890 119891119894V119890

Δ (119866) + 4 119894119891 119892 (119866) ⩾ 4

Δ (119866) + 3 119894119891 119892 (119866) ⩾ 5

Δ (119866) + 2 119894119891 119892 (119866) ⩾ 6

Δ (119866) + 1 119894119891 119892 (119866) ⩾ 8

(65)

According toTheorem 157 119887RM(119862119899) = 3 = Δ(119862

119899)+1 for a

cycle 119862119899of length 119899 ⩾ 8 with 119899 equiv 2 (mod 3) and therefore

the last upper bound inTheorem 162 is sharp at least for Δ =2

Combining the fact that every planar graph 119866 withminimum degree 5 contains an edge 119909119910 with 119889

119866(119909) = 5

and 119889119866(119910) isin 5 6 with Theorem 161 (a) Akbari et al [125]

obtained the following result

26 International Journal of Combinatorics

middot middot middot

middot middot middot

middot middot middot

middot middot middot

119866

Figure 9 The graph 119866 is constructed from 119866

Theorem 163 (Akbari et al [125] 2012) 119887RM(119866) ⩽ 15 forevery planar graph 119866

It remains open to show whether the bound inTheorem 163 is sharp or not Though finding a planargraph 119866 with 119887RM(119866) = 15 seems to be difficult Akbari etal [125] constructed an infinite family of planar graphs withRoman bondage number equal to 7 by proving the followingresult

Theorem 164 (Akbari et al [125] 2012) Let 119866 be a graph oforder 119899 and 119866 is the graph of order 5119899 obtained from 119866 byattaching the central vertex of a copy of a path 119875

5to each vertex

of119866 (see Figure 9) Then 120574RM(119866) = 4119899 and 119887RM(119866) = 120575(119866)+2

ByTheorem 164 infinitemany planar graphswith Romanbondage number 7 can be obtained by considering any planargraph 119866 with 120575(119866) = 5 (eg the icosahedron graph)

Conjecture 165 (Akbari et al [125] 2012) The Romanbondage number of every planar graph is at most 7

For general bounds the following observation is directlyobtained which is similar to Observation 1

Observation 4 Let 119866 be a graph of order 119899 with maximumdegree at least two Assume that 119867 is a spanning subgraphobtained from 119866 by removing 119896 edges If 120574RM(119867) = 120574RM(119866)then 119887

119877(119867) ⩽ 119887RM(119866) ⩽ 119887RM(119867) + 119896

Theorem 166 (Bahremandpour et al [118] 2013) For anyconnected graph 119866 of order 119899 if 119899 ⩾ 3 and 120574RM(119866) = 120574(119866) + 1then

119887RM (119866) ⩽ min 119887 (119866) 119899Δ (66)

where 119899Δis the number of vertices with maximum degree Δ in

119866

Observation 5 For any graph 119866 if 120574(119866) = 120574RM(119866) then119887RM(119866) ⩽ 119887(119866)

Theorem 167 (Bahremandpour et al [118] 2013) For everyRoman graph 119866

119887RM (119866) ⩾ 119887 (119866) (67)

The bound is sharp for cycles on 119899 vertices where 119899 equiv 0 (mod3)

The strict inequality in Theorem 167 can hold for exam-ple 119887(119862

3119896+2) = 2 lt 3 = 119887RM(1198623119896+2

) byTheorem 157A graph 119866 is called to be vertex Roman domination-

critical if 120574RM(119866 minus 119909) lt 120574RM(119866) for every vertex 119909 in 119866 If119866 has no isolated vertices then 120574RM(119866) ⩽ 2120574(119866) ⩽ 2120573(119866)If 120574RM(119866) = 2120573(119866) then 120574RM(119866) = 2120574(119866) and hence 119866 is aRoman graph In [30] Volkmann gave a lot of graphs with120574(119866) = 120573(119866) The following result is similar to Theorem 57

Theorem 168 (Bahremandpour et al [118] 2013) For anygraph 119866 if 120574RM(119866) = 2120573(119866) then

(a) 119887RM(119866) ⩾ 120575(119866)

(b) 119887RM(119866) ⩾ 120575(119866)+1 if119866 is a vertex Roman domination-critical graph

If 120574RM(119866) = 2 then obviously 119887RM(119866) ⩽ 120575(119866) So weassume 120574RM(119866) ⩾ 3 Dehgardi et al [121] proved 119887RM(119866) ⩽(120574RM(119866) minus 2)Δ(119866) + 1 and posed the following problem if 119866is a connected graph of order 119899 ⩾ 4 with 120574RM(119866) ⩾ 3 then

119887RM (119866) ⩽ (120574RM (119866) minus 2) Δ (119866) (68)

Theorem 161 (a) shows that the inequality (68) holds if120574RM(119866) ⩾ 5 Thus the bound in (68) is of interest only when120574RM(119866) is 3 or 4

Lemma 169 (Bahremandpour et al [118] 2013) For anynonempty graph 119866 of order 119899 ⩾ 3 120574RM(119866) = 3 if and onlyif Δ(119866) = 119899 minus 2

The following result shows that (68) holds for all graphs119866 of order 119899 ⩾ 4 with 120574RM(119866) = 3 or 4 which improvesTheorem 161 (b)

Theorem 170 (Bahremandpour et al [118] 2013) For anyconnected graph 119866 of order 119899 ⩾ 4

119887RM (119866) le Δ (119866) = 119899 minus 2 119894119891 120574RM (119866) = 3

Δ (119866) + 120575 (119866) minus 1 119894119891 120574RM (119866) = 4(69)

with the first equality if and only if 119866 cong 1198624

Recently Akbari and Qajar [122] have proved the follow-ing result

Theorem 171 (Akbari and Qajar [122]) For any connectedgraph 119866 of order 119899 ⩾ 3

119887RM (119866) ⩽ 119899 minus 120574RM (119866) + 5 (70)

International Journal of Combinatorics 27

We conclude this section with the following problems

Problem 8 Characterize all connected graphs 119866 of order 119899 ⩾3 for which 119887RM(119866) = 119899 minus 1

Problem 9 Prove or disprove if 119866 is a connected graph oforder 119899 ⩾ 3 then

119887RM (119866) ⩽ 119899 minus 120574RM (119866) + 3 (71)

Note that 120574119877(119870

119905(3)) = 4 and 119887RM(119870119905

(3)) = 119899minus1 where 119899 =3119905 for 119905 ⩾ 3 The upper bound on 119887RM(119866) given in Problem 9is sharp if Problem 9 has positive answer

95 Rainbow Bondage Numbers For a positive integer 119896 a119896-rainbow dominating function of a graph 119866 is a function 119891from the vertex-set 119881(119866) to the set of all subsets of the set1 2 119896 such that for any vertex 119909 in 119866 with 119891(119909) = 0 thecondition

⋃

119910isin119873119866(119909)

119891 (119910) = 1 2 119896 (72)

is fulfilledThe weight of a 119896-rainbow dominating function 119891on 119866 is the value

119891 (119866) = sum

119909isin119881(119866)

1003816100381610038161003816119891 (119909)1003816100381610038161003816 (73)

The 119896-rainbow domination number of a graph 119866 denoted by120574119903119896(119866) is the minimum weight of a 119896-rainbow dominating

function 119891 of 119866Note that 120574

1199031(119866) is the classical domination number 120574(119866)

The 119896-rainbow domination number is introduced by Bresaret al [126] Rainbow domination of a graph 119866 coincides withordinary domination of the Cartesian product of 119866 with thecomplete graph in particular 120574

119903119896(119866) = 120574(119866 times 119870

119896) for any

positive integer 119896 and any graph 119866 [126] Hartnell and Rall[127] proved that 120574(119879) lt 120574

1199032(119879) for any tree 119879 no graph has

120574 = 120574119903119896for 119896 ⩾ 3 for any 119896 ⩾ 2 and any graph 119866 of order 119899

min 119899 120574 (119866) + 119896 minus 2 ⩽ 120574119903119896(119866) ⩽ 119896120574 (119866) (74)

The introduction of rainbow domination is motivatedby the study of paired domination in Cartesian productsof graphs where certain upper bounds can be expressed interms of rainbow domination that is for any graph 119866 andany graph 119867 if 119881(119867) can be partitioned into 119896120574

119875-sets then

120574119875(119866 times 119867) ⩽ (1119896)120592(119867)120574

119903119896(119866) proved by Bresar et al [126]

Dehgardi et al [128] introduced the concept of 119896-rainbowbondage number of graphs Let 119866 be a graph with maximumdegree at least two The 119896-rainbow bondage number 119887

119903119896(119866)

of 119866 is the minimum cardinality of all sets 119861 sube 119864(119866) forwhich 120574

119903119896(119866 minus 119861) gt 120574

119903119896(119866) Since in the study of 119896-rainbow

bondage number the assumption Δ(119866) ⩾ 2 is necessarywe always assume that when we discuss 119887

119903119896(119866) all graphs

involved satisfy Δ(119866) ⩾ 2The 2-rainbow bondage number of some special families

of graphs has been determined

Theorem 172 (Dehgardi et al [128]) For any integer 119899 ⩾ 3

1198871199032(119875

119899) = 1 119891119900119903 119899 ⩾ 2

1198871199032(119862

119899) =

1 119894119891 119899 equiv 0 (mod 4) 2 119900119905ℎ119890119903119908119894119904119890

1198871199032(119870

119899119899) =

1 119894119891 119899 = 2

3 119894119891 119899 = 3

6 119894119891 119899 = 4

119898 119894119891 119899 ⩾ 5

(75)

Theorem 173 (Dehgardi et al [128]) For any connected graph119866 of order 119899 ⩾ 3 119887

1199032(119866) ⩽ 120582(119866) + 2Δ(119866) minus 3

Theorem 174 (Dehgardi et al [128]) For any tree 119879 of order119899 ⩾ 3 119887

1199032(119879) ⩽ 2

Theorem 175 (Dehgardi et al [128]) If 119866 is a planar graphwithmaximumdegree at least two then 119887

1199032(119866) ⩽ 15Moreover

if the girth 119892(119866) ⩾ 4 then 1198871199032(119866) ⩽ 11 and if 119892(119866) ⩾ 6 then

1198871199032(119866) ⩽ 9

Theorem 176 (Dehgardi et al [128]) For a complete bipartitegraph 119870

119901119902 if 2119896 + 1 ⩽ 119901 ⩽ 119902 then 119887

119903119896(119870

119901119902) = 119901

Theorem177 (Dehgardi et al [128]) For a complete graph119870119899

if 119899 ⩾ 119896 + 1 then 119887119903119896(119870

119899) = lceil119896119899(119896 + 1)rceil

The special case 119896 = 1 of Theorem 177 can be found inTheorem 1 The following is a simple observation similar toObservation 1

Observation 6 (Dehgardi et al [128]) Let 119866 be a graphwith maximum degree at least two 119867 a spanning subgraphobtained from 119866 by removing 119896 edges If 120574

119903119896(119867) = 120574

119903119896(119866)

then 119887119903119896(119867) ⩽ 119887

119903119896(119866) ⩽ 119887

119903119896(119867) + 119896

Theorem 178 (Dehgardi et al [128]) For every graph 119866 with120574119903119896(119866) = 119896120574(119866) 119887

119903119896(119866) ⩾ 119887(119866)

A graph 119866 is said to be vertex 119896-rainbow domination-critical if 120574

119903119896(119866 minus 119909) lt 120574

119903119896(119866) for every vertex 119909 in 119866 It is

clear that if 119866 has no isolated vertices then 120574119903119896(119866) ⩽ 119896120574(119866) ⩽

119896120573(119866) where 120573(119866) is the vertex-covering number of119866 Thusif 120574

119903119896(119866) = 119896120573(119866) then 120574

119903119896(119866) = 119896120574(119866)

Theorem 179 (Dehgardi et al [128]) For any graph 119866 if120574119903119896(119866) = 119896120573(119866) then

(a) 119887119903119896(119866) ⩾ 120575(119866)

(b) 119887119903119896(119866) ⩾ 120575(119866)+1 if119866 is vertex 119896-rainbow domination-

critical

As far as we know there are no more results on the 119896-rainbow bondage number of graphs in the literature

28 International Journal of Combinatorics

10 Results on Digraphs

Although domination has been extensively studied in undi-rected graphs it is natural to think of a dominating setas a one-way relationship between vertices of the graphIndeed among the earliest literatures on this subject vonNeumann and Morgenstern [129] used what is now calleddomination in digraphs to find solution (or kernels whichare independent dominating sets) for cooperative 119899-persongames Most likely the first formulation of domination byBerge [130] in 1958 was given in the context of digraphs andonly some years latter by Ore [131] in 1962 for undirectedgraphs Despite this history examination of domination andits variants in digraphs has been essentially overlooked (see[4] for an overview of the domination literature) Thus thereare few if any such results on domination for digraphs in theliterature

The bondage number and its related topics for undirectedgraph have become one of major areas both in theoreticaland applied researches However until recently Carlsonand Develin [42] Shan and Kang [132] and Huang andXu [133 134] studied the bondage number for digraphsindependently In this section we introduce their resultsfor general digraphs Results for some special digraphs suchas vertex-transitive digraphs are introduced in the nextsection

101 Upper Bounds for General Digraphs Let 119866 = (119881 119864)

be a digraph without loops and parallel edges A digraph 119866is called to be asymmetric if whenever (119909 119910) isin 119864(119866) then(119910 119909) notin 119864(119866) and to be symmetric if (119909 119910) isin 119864(119866) implies(119910 119909) isin 119864(119866) For a vertex 119909 of 119881(119866) the sets of out-neighbors and in-neighbors of 119909 are respectively defined as119873

+

119866(119909) = 119910 isin 119881(119866) (119909 119910) isin 119864(119866) and 119873minus

119866(119909) = 119909 isin

119881(119866) (119909 119910) isin 119864(119866) the out-degree and the in-degree of119909 are respectively defined as 119889+

119866(119909) = |119873

+

119866(119909)| and 119889minus

119866(119909) =

|119873+

119866(119909)| Denote themaximum and theminimum out-degree

(resp in-degree) of 119866 by Δ+(119866) and 120575+(119866) (resp Δminus(119866) and120575minus

(119866))The degree 119889119866(119909) of 119909 is defined as 119889+

119866(119909)+119889

minus

119866(119909) and

the maximum and the minimum degrees of119866 are denoted byΔ(119866) and 120575(119866) respectively that is Δ(119866) = max119889

119866(119909) 119909 isin

119881(119866) and 120575(119866) = min119889119866(119909) 119909 isin 119881(119866) Note that the

definitions here are different from the ones in the textbookon digraphs see for example Xu [1]

A subset 119878 of119881(119866) is called a dominating set if119881(119866) = 119878cup119873

+

119866(119878) where119873+

119866(119878) is the set of out-neighbors of 119878Then just

as for undirected graphs 120574(119866) is the minimum cardinalityof a dominating set and the bondage number 119887(119866) is thesmallest cardinality of a set 119861 of edges such that 120574(119866 minus 119861) gt120574(119866) if such a subset 119861 sube 119864(119866) exists Otherwise we put119887(119866) = infin

Some basic results for undirected graphs stated inSection 3 can be generalized to digraphs For exampleTheorem 18 is generalized by Carlson and Develin [42]Huang andXu [133] and Shan andKang [132] independentlyas follows

Theorem 180 Let 119866 be a digraph and (119909 119910) isin 119864(119866) Then119887(119866) ⩽ 119889

119866(119910) + 119889

minus

119866(119909) minus |119873

minus

119866(119909) cap 119873

minus

119866(119910)|

Corollary 181 For a digraph 119866 119887(119866) ⩽ 120575minus(119866) + Δ(119866)

Since 120575minus

(119866) ⩽ (12)Δ(119866) from Corollary 181Conjecture 53 is valid for digraphs

Corollary 182 For a digraph 119866 119887(119866) ⩽ (32)Δ(119866)

In the case of undirected graphs the bondage number of119866119899= 119870

119899times 119870

119899achieves this bound However it was shown

by Carlson and Develin in [42] that if we take the symmetricdigraph119866

119899 we haveΔ(119866

119899) = 4(119899minus1) 120574(119866

119899) = 119899 and 119887(119866

119899) ⩽

4(119899 minus 1) + 1 = Δ(119866119899) + 1 So this family of digraphs cannot

show that the bound in Corollary 182 is tightCorresponding to Conjecture 51 which is discredited

for undirected graphs and is valid for digraphs the sameconjecture for digraphs can be proposed as follows

Conjecture 183 (Carlson and Develin [42] 2006) 119887(119866) ⩽Δ(119866) + 1 for any digraph 119866

If Conjecture 183 is true then the following results showsthat this upper bound is tight since 119887(119870

119899times119870

119899) = Δ(119870

119899times119870

119899)+1

for a complete digraph 119870119899

In 2007 Shan and Kang [132] gave some tight upperbounds on the bondage numbers for some asymmetricdigraphs For example 119887(119879) ⩽ Δ(119879) for any asymmetricdirected tree 119879 119887(119866) ⩽ Δ(119866) for any asymmetric digraph 119866with order at least 4 and 120574(119866) ⩽ 2 For planar digraphs theyobtained the following results

Theorem 184 (Shan and Kang [132] 2007) Let 119866 be aasymmetric planar digraphThen 119887(119866) ⩽ Δ(119866)+2 and 119887(119866) ⩽Δ(119866) + 1 if Δ(119866) ⩾ 5 and 119889minus

119866(119909) ⩾ 3 for every vertex 119909 with

119889119866(119909) ⩾ 4

102 Results for Some Special Digraphs The exact values andbounds on 119887(119866) for some standard digraphs are determined

Theorem185 (Huang andXu [133] 2006) For a directed cycle119862119899and a directed path 119875

119899

119887 (119862119899) =

3 119894119891 119899 119894119904 119900119889119889

2 119894119891 119899 119894119904 119890V119890119899

119887 (119875119899) =

2 119894119891 119899 119894119904 119900119889119889

1 119894119891 119899 119894119904 119890V119890119899

(76)

For the de Bruijn digraph 119861(119889 119899) and the Kautz digraph119870(119889 119899)

119887 (119861 (119889 119899)) = 119889 119894119891 119899 119894119904 119900119889119889

119889 ⩽ 119887 (119861 (119889 119899)) ⩽ 2119889 119894119891 119899 119894119904 119890V119890119899

119887 (119870 (119889 119899)) = 119889 + 1

(77)

Like undirected graphs we can define the total domi-nation number and the total bondage number On the totalbondage numbers for some special digraphs the knownresults are as follows

International Journal of Combinatorics 29

Theorem 186 (Huang and Xu [134] 2007) For a directedcycle 119862

119899and a directed path 119875

119899 119887

119879(119875

119899) and 119887

119879(119862

119899) all do not

exist For a complete digraph 119870119899

119887119879(119870

119899) = infin 119894119891 119899 = 1 2

119887119879(119870

119899) = 3 119894119891 119899 = 3

119899 ⩽ 119887119879(119870

119899) ⩽ 2119899 minus 3 119894119891 119899 ⩾ 4

(78)

The extended de Bruijn digraph EB(119889 119899 1199021 119902

119901) and

the extended Kautz digraph EK(119889 119899 1199021 119902

119901) are intro-

duced by Shibata and Gonda [135] If 119901 = 1 then theyare the de Bruijn digraph B(119889 119899) and the Kautz digraphK(119889 119899) respectively Huang and Xu [134] determined theirtotal domination numbers In particular their total bondagenumbers for general cases are determined as follows

Theorem 187 (Huang andXu [134] 2007) If 119889 ⩾ 2 and 119902119894⩾ 2

for each 119894 = 1 2 119901 then

119887119879(EB(119889 119899 119902

1 119902

119901)) = 119889

119901

minus 1

119887119879(EK(119889 119899 119902

1 119902

119901)) = 119889

119901

(79)

In particular for the de Bruijn digraph B(119889 119899) and the Kautzdigraph K(119889 119899)

119887119879(B (119889 119899)) = 119889 minus 1 119887

119879(K (119889 119899)) = 119889 (80)

Zhang et al [136] determined the bondage number incomplete 119905-partite digraphs

Theorem 188 (Zhang et al [136] 2009) For a complete 119905-partite digraph 119870

11989911198992119899119905

where 1198991⩽ 119899

2⩽ sdot sdot sdot ⩽ 119899

119905

119887 (11987011989911198992119899119905

)

=

119898 119894119891 119899119898= 1 119899

119898+1⩾ 2 119891119900119903 119904119900119898119890

119898 (1 ⩽ 119898 lt 119905)

4119905 minus 3 119894119891 1198991= 119899

2= sdot sdot sdot = 119899

119905= 2

119905minus1

sum

119894=1

119899119894+ 2 (119905 minus 1) 119900119905ℎ119890119903119908119894119904119890

(81)

Since an undirected graph can be thought of as a sym-metric digraph any result for digraphs has an analogy forundirected graphs in general In view of this point studyingthe bondage number for digraphs is more significant thanfor undirected graphs Thus we should further study thebondage number of digraphs and try to generalize knownresults on the bondage number and related variants for undi-rected graphs to digraphs prove or disprove Conjecture 183In particular determine the exact values of 119887(B(119889 119899)) for aneven 119899 and 119887

119879(119870

119899) for 119899 ⩾ 4

11 Efficient Dominating Sets

A dominating set 119878 of a graph 119866 is called to be efficient if forevery vertex 119909 in119866 |119873

119866[119909]cap119878| = 1 if119866 is a undirected graph

or |119873minus

119866[119909] cap 119878| = 1 if 119866 is a directed graph The concept of an

efficient set was proposed by Bange et al [137] in 1988 In theliterature an efficient set is also called a perfect dominatingset (see Livingston and Stout [138] for an explanation)

From definition if 119878 is an efficient dominating set of agraph 119866 then 119878 is certainly an independent set and everyvertex not in 119878 is adjacent to exactly one vertex in 119878

It is also clear from definition that a dominating set 119878 isefficient if and only if N

119866[119878] = 119873

119866[119909] 119909 isin 119878 for the

undirected graph 119866 or N+

119866[119878] = 119873

+

119866[119909] 119909 isin 119878 for the

digraph119866 is a partition of119881(119866) where the induced subgraphby119873

119866[119909] or119873+

119866[119909] is a star or an out-star with the root 119909

The efficient domination has important applications inmany areas such as error-correcting codes and receivesmuch attention in the late years

The concept of efficient dominating sets is a measureof the efficiency of domination in graphs Unfortunately asshown in [137] not every graph has an efficient dominatingset andmoreover it is anNP-complete problem to determinewhether a given graph has an efficient dominating set Inaddition it was shown by Clark [139] in 1993 that for a widerange of 119901 almost every random undirected graph 119866 isin

G(120592 119901) has no efficient dominating sets This means thatundirected graphs possessing an efficient dominating set arerare However it is easy to show that every undirected graphhas an orientationwith an efficient dominating set (see Bangeet al [140])

In 1993 Barkauskas and Host [141] showed that deter-mining whether an arbitrary oriented graph has an efficientdominating set is NP-complete Even so the existence ofefficient dominating sets for some special classes of graphs hasbeen examined see for example Dejter and Serra [142] andLee [143] for Cayley graph Gu et al [144] formeshes and toriObradovic et al [145] Huang and Xu [36] Reji Kumar andMacGillivray [146] for circulant graphs Harary graphs andtori and Van Wieren et al [147] for cube-connected cycles

In this section we introduce results of the bondagenumber for some graphs with efficient dominating sets dueto Huang and Xu [36]

111 Results for General Graphs In this subsection we intro-duce some results on bondage numbers obtained by applyingefficient dominating sets We first state the two followinglemmas

Lemma 189 For any 119896-regular graph or digraph 119866 of order 119899120574(119866) ⩾ 119899(119896 + 1) with equality if and only if 119866 has an efficientdominating set In addition if 119866 has an efficient dominatingset then every efficient dominating set is certainly a 120574-set andvice versa

Let 119890 be an edge and 119878 a dominating set in 119866 We say that119890 supports 119878 if 119890 isin (119878 119878) where (119878 119878) = (119909 119910) isin 119864(119866) 119909 isin 119878 119910 isin 119878 Denote by 119904(119866) the minimum number of edgeswhich support all 120574-sets in 119866

Lemma 190 For any graph or digraph 119866 119887(119866) ⩾ 119904(119866) withequality if 119866 is regular and has an efficient dominating set

30 International Journal of Combinatorics

A graph 119866 is called to be vertex-transitive if its automor-phism group Aut(119866) acts transitively on its vertex-set 119881(119866)A vertex-transitive graph is regular Applying Lemmas 189and 190 Huang and Xu obtained some results on bondagenumbers for vertex-transitive graphs or digraphs

Theorem 191 (Huang and Xu [36] 2008) For any vertex-transitive graph or digraph 119866 of order 119899

119887 (119866) ⩾

lceil119899

2120574 (119866)rceil 119894119891 119866 119894119904 119906119899119889119894119903119890119888119905119890119889

lceil119899

120574 (119866)rceil 119894119891 119866 119894119904 119889119894119903119890119888119905119890119889

(82)

Theorem192 (Huang andXu [36] 2008) Let119866 be a 119896-regulargraph of order 119899 Then

119887(119866) ⩽ 119896 if119866 is undirected and 119899 ⩾ 120574(119866)(119896+1)minus119896+1119887(119866) ⩽ 119896+1+ℓ if119866 is directed and 119899 ⩾ 120574(119866)(119896+1)minusℓwith 0 ⩽ ℓ ⩽ 119896 minus 1

Next we introduce a better upper bound on 119887(119866) To thisaim we need the following concept which is a generalizationof the concept of the edge-covering of a graph 119866 For 1198811015840

sube

119881(119866) and 1198641015840 sube 119864(119866) we say that 1198641015840covers 1198811015840 and call 1198641015840an edge-covering for 1198811015840 if there exists an edge (119909 119910) isin 1198641015840 forany vertex 119909 isin 1198811015840 For 119910 isin 119881(119866) let 1205731015840[119910] be the minimumcardinality over all edge-coverings for119873minus

119866[119910]

Theorem 193 (Huang and Xu [36] 2008) If 119866 is a 119896-regulargraph with order 120574(119866)(119896 + 1) then 119887(119866) ⩽ 1205731015840[119910] for any 119910 isin119881(119866)

Theupper bound on 119887(119866) given inTheorem 193 is tight inview of 119887(119862

119899) for a cycle or a directed cycle 119862

119899(seeTheorems

1 and 185 resp)It is easy to see that for a 119896-regular graph 119866 lceil(119896 + 1)2rceil ⩽

1205731015840

[119910] ⩽ 119896 when 119866 is undirected and 1205731015840[119910] = 119896 + 1 when 119866 isdirected By this fact and Lemma 189 the following theoremis merely a simple combination of Theorems 191 and 193

Theorem 194 (Huang and Xu [36] 2008) Let 119866 be a vertex-transitive graph of degree 119896 If 119866 has an efficient dominatingset then

lceil119896 + 1

2rceil ⩽ 119887 (119866) ⩽ 119896 119894119891 119866 119894119904 119906119899119889119894119903119890119888119905119890119889

119887 (119866) = 119896 + 1 119894119891 119866 119894119904 119889119894119903119890119888119905119890119889

(83)

Theorem 195 (Huang and Xu [36] 2008) If 119866 is an undi-rected vertex-transitive cubic graph with order 4120574(119866) and girth119892(119866) ⩽ 5 then 119887(119866) = 2

The proof of Theorem 195 leads to a byproduct If 119892 =5 let (119906

1 119906

2 119906

3 119906

4 119906

5) be a cycle and let D

119894be the family

of efficient dominating sets containing 119906119894for each 119894 isin

1 2 3 4 5 ThenD119894capD

119895= 0 for 119894 = 1 2 5 Then 119866 has

at least 5119904 efficient dominating sets where 119904 = 119904(119866) But thereare only 4s (=ns120574) distinct efficient dominating sets in119866This

contradiction implies that an undirected vertex-transitivecubic graph with girth five has no efficient dominating setsBut a similar argument for 119892(119866) = 3 4 or 119892(119866) ⩾ 6 couldnot give any contradiction This is consistent with the resultthat CCC(119899) a vertex-transitive cubic graph with girth 119899 if3 ⩽ 119899 ⩽ 8 or girth 8 if 119899 ⩾ 9 has efficient dominating setsfor all 119899 ⩾ 3 except 119899 = 5 (see Theorem 199 in the followingsubsection)

112 Results for Cayley Graphs In this subsection we useTheorem 194 to determine the exact values or approximativevalues of bondage numbers for some special vertex-transitivegraphs by characterizing the existence of efficient dominatingsets in these graphs

Let Γ be a nontrivial finite group 119878 a nonempty subset ofΓ without the identity element of Γ A digraph 119866 is defined asfollows

119881 (119866) = Γ (119909 119910) isin 119864 (119866) lArrrArr 119909minus1

119910 isin 119878

for any 119909 119910 isin Γ(84)

is called a Cayley digraph of the group Γ with respect to119878 denoted by 119862

Γ(119878) If 119878minus1 = 119904minus1 119904 isin 119878 = 119878 then

119862Γ(119878) is symmetric and is called a Cayley undirected graph

a Cayley graph for short Cayley graphs or digraphs arecertainly vertex-transitive (see Xu [1])

A circulant graph 119866(119899 119878) of order 119899 is a Cayley graph119862119885119899

(119878) where 119885119899= 0 119899 minus 1 is the addition group of

order 119899 and 119878 is a nonempty subset of119885119899without the identity

element and hence is a vertex-transitive digraph of degree|119878| If 119878minus1 = 119878 then119866(119899 119878) is an undirected graph If 119878 = 1 119896where 2 ⩽ 119896 ⩽ 119899 minus 2 we write 119866(119899 1 119896) for 119866(119899 1 119896) or119866(119899 plusmn1 plusmn119896) and call it a double-loop circulant graph

Huang and Xu [36] showed that for directed 119866 =

119866(119899 1 119896) lceil1198993rceil ⩽ 120574(119866) ⩽ lceil1198992rceil and 119866 has an efficientdominating set if and only if 3 | 119899 and 119896 equiv 2 (mod 3) fordirected 119866 = 119866(119899 1 119896) and 119896 = 1198992 lceil1198995rceil ⩽ 120574(119866) ⩽ lceil1198993rceiland 119866 has an efficient dominating set if and only if 5 | 119899 and119896 equiv plusmn2 (mod 5) By Theorem 194 we obtain the bondagenumber of a double loop circulant graph if it has an efficientdominating set

Theorem 196 (Huang and Xu [36] 2008) Let 119866 be a double-loop circulant graph 119866(119899 1 119896) If 119866 is directed with 3 | 119899and 119896 equiv 2 (mod 3) or 119866 is undirected with 5 | 119899 and119896 equiv plusmn2 (mod 5) then 119887(119866) = 3

The 119898 times 119899 torus is the Cartesian product 119862119898times 119862

119899of

two cycles and is a Cayley graph 119862119885119898times119885119899

(119878) where 119878 =

(0 1) (1 0) for directed cycles and 119878 = (0 plusmn1) (plusmn1 0) forundirected cycles and hence is vertex-transitive Gu et al[144] showed that the undirected torus119862

119898times119862

119899has an efficient

dominating set if and only if both 119898 and 119899 are multiplesof 5 Huang and Xu [36] showed that the directed torus119862119898times 119862

119899has an efficient dominating set if and only if both

119898 and 119899 are multiples of 3 Moreover they found a necessarycondition for a dominating set containing the vertex (0 0)in 119862

119898times 119862

119899to be efficient and obtained the following

result

International Journal of Combinatorics 31

Theorem 197 (Huang and Xu [36] 2008) Let 119866 = 119862119898times 119862

119899

If 119866 is undirected and both119898 and 119899 are multiples of 5 or if 119866 isdirected and both119898 and 119899 are multiples of 3 then 119887(119866) = 3

The hypercube 119876119899

is the Cayley graph 119862Γ(119878)

where Γ = 1198852times sdot sdot sdot times 119885

2= (119885

2)119899 and 119878 =

100 sdot sdot sdot 0 010 sdot sdot sdot 0 00 sdot sdot sdot 01 Lee [143] showed that119876119899has an efficient dominating set if and only if 119899 = 2119898 minus 1

for a positive integer119898 ByTheorem 194 the following resultis obtained immediately

Theorem 198 (Huang and Xu [36] 2008) If 119899 = 2119898 minus 1 for apositive integer119898 then 2119898minus1

⩽ 119887(119876119899) ⩽ 2

119898

minus 1

Problem 10 Determine the exact value of 119887(119876119899) for 119899 = 2119898minus1

The 119899-dimensional cube-connected cycle denoted byCCC(119899) is constructed from the 119899-dimensional hypercube119876119899by replacing each vertex 119909 in 119876

119899with an undirected cycle

119862119899of length 119899 and linking the 119894th vertex of the 119862

119899to the 119894th

neighbor of 119909 It has been proved that CCC(119899) is a Cayleygraph and hence is a vertex-transitive graph with degree 3Van Wieren et al [147] proved that CCC(119899) has an efficientdominating set if and only if 119899 = 5 From Theorems 194 and195 the following result can be derived

Theorem 199 (Huang and Xu [36] 2008) Let 119866 = 119862119862119862(119899)be the 119899-dimensional cube-connected cycles with 119899 ⩾ 3 and119899 = 5 Then 120574(119866) = 1198992119899minus2 and 2 ⩽ 119887(119866) ⩽ 3 In addition119887(119862119862119862(3)) = 119887(119862119862119862(4)) = 2

Problem 11 Determine the exact value of 119887(CCC(119899)) for 119899 ⩾5

Acknowledgments

This work was supported by NNSF of China (Grants nos11071233 61272008) The author would like to express hisgratitude to the anonymous referees for their kind sugges-tions and comments on the original paper to ProfessorSheikholeslami and Professor Jafari Rad for sending thereferences [128] and [81] which resulted in this version

References

[1] J-M Xu Theory and Application of Graphs vol 10 of NetworkTheory and Applications Kluwer Academic Dordrecht TheNetherlands 2003

[2] S THedetniemi andR C Laskar ldquoBibliography on dominationin graphs and some basic definitions of domination parame-tersrdquo Discrete Mathematics vol 86 no 1ndash3 pp 257ndash277 1990

[3] TW Haynes S T Hedetniemi and P J Slater Fundamentals ofDomination in Graphs vol 208 ofMonographs and Textbooks inPure and Applied Mathematics Marcel Dekker New York NYUSA 1998

[4] TWHaynes S THedetniemi and P J Slater EdsDominationin Graphs vol 209 of Monographs and Textbooks in Pure andAppliedMathematicsMarcelDekker NewYorkNYUSA 1998

[5] M R Garey andD S JohnsonComputers and Intractability WH Freeman and Company San Francisco Calif USA 1979

[6] H B Walikar and B D Acharya ldquoDomination critical graphsrdquoNational Academy Science Letters vol 2 pp 70ndash72 1979

[7] D Bauer F Harary J Nieminen and C L Suffel ldquoDominationalteration sets in graphsrdquo Discrete Mathematics vol 47 no 2-3pp 153ndash161 1983

[8] J F Fink M S Jacobson L F Kinch and J Roberts ldquoThebondage number of a graphrdquo Discrete Mathematics vol 86 no1ndash3 pp 47ndash57 1990

[9] F-T Hu and J-M Xu ldquoThe bondage number of (119899 minus 3)-regulargraph G of order nrdquo Ars Combinatoria to appear

[10] U Teschner ldquoNew results about the bondage number of agraphrdquoDiscreteMathematics vol 171 no 1ndash3 pp 249ndash259 1997

[11] B L Hartnell and D F Rall ldquoA bound on the size of a graphwith given order and bondage numberrdquo Discrete Mathematicsvol 197-198 pp 409ndash413 1999 16th British CombinatorialConference (London 1997)

[12] B L Hartnell and D F Rall ldquoA characterization of trees inwhich no edge is essential to the domination numberrdquo ArsCombinatoria vol 33 pp 65ndash76 1992

[13] J Topp and P D Vestergaard ldquo120572119896- and 120574

119896-stable graphsrdquo

Discrete Mathematics vol 212 no 1-2 pp 149ndash160 2000[14] A Meir and J W Moon ldquoRelations between packing and

covering numbers of a treerdquo Pacific Journal of Mathematics vol61 no 1 pp 225ndash233 1975

[15] B L Hartnell L K Joslashrgensen P D Vestergaard and CA Whitehead ldquoEdge stability of the k-domination numberof treesrdquo Bulletin of the Institute of Combinatorics and ItsApplications vol 22 pp 31ndash40 1998

[16] F-T Hu and J-M Xu ldquoOn the complexity of the bondage andreinforcement problemsrdquo Journal of Complexity vol 28 no 2pp 192ndash201 2012

[17] B L Hartnell and D F Rall ldquoBounds on the bondage numberof a graphrdquo Discrete Mathematics vol 128 no 1ndash3 pp 173ndash1771994

[18] Y-L Wang ldquoOn the bondage number of a graphrdquo DiscreteMathematics vol 159 no 1ndash3 pp 291ndash294 1996

[19] G H Fricke TW Haynes S M Hedetniemi S T Hedetniemiand R C Laskar ldquoExcellent treesrdquo Bulletin of the Institute ofCombinatorics and Its Applications vol 34 pp 27ndash38 2002

[20] V Samodivkin ldquoThe bondage number of graphs good and badverticesrdquoDiscussiones Mathematicae GraphTheory vol 28 no3 pp 453ndash462 2008

[21] J E Dunbar T W Haynes U Teschner and L VolkmannldquoBondage insensitivity and reinforcementrdquo in Domination inGraphs vol 209 of Monographs and Textbooks in Pure andAppliedMathematics pp 471ndash489 Dekker NewYork NY USA1998

[22] H Liu and L Sun ldquoThe bondage and connectivity of a graphrdquoDiscrete Mathematics vol 263 no 1ndash3 pp 289ndash293 2003

[23] A-H Esfahanian and S L Hakimi ldquoOn computing a con-ditional edge-connectivity of a graphrdquo Information ProcessingLetters vol 27 no 4 pp 195ndash199 1988

[24] R C Brigham P Z Chinn and R D Dutton ldquoVertexdomination-critical graphsrdquoNetworks vol 18 no 3 pp 173ndash1791988

[25] V Samodivkin ldquoDomination with respect to nondegenerateproperties bondage numberrdquoThe Australasian Journal of Com-binatorics vol 45 pp 217ndash226 2009

[26] U Teschner ldquoA new upper bound for the bondage numberof graphs with small domination numberrdquo The AustralasianJournal of Combinatorics vol 12 pp 27ndash35 1995

32 International Journal of Combinatorics

[27] J Fulman D Hanson and G MacGillivray ldquoVertex dom-ination-critical graphsrdquoNetworks vol 25 no 2 pp 41ndash43 1995

[28] U Teschner ldquoA counterexample to a conjecture on the bondagenumber of a graphrdquo Discrete Mathematics vol 122 no 1ndash3 pp393ndash395 1993

[29] U Teschner ldquoThe bondage number of a graph G can be muchgreater than Δ(G)rdquo Ars Combinatoria vol 43 pp 81ndash87 1996

[30] L Volkmann ldquoOn graphs with equal domination and coveringnumbersrdquoDiscrete AppliedMathematics vol 51 no 1-2 pp 211ndash217 1994

[31] L A Sanchis ldquoMaximum number of edges in connected graphswith a given domination numberrdquoDiscreteMathematics vol 87no 1 pp 65ndash72 1991

[32] S Klavzar and N Seifter ldquoDominating Cartesian products ofcyclesrdquoDiscrete AppliedMathematics vol 59 no 2 pp 129ndash1361995

[33] H K Kim ldquoA note on Vizingrsquos conjecture about the bondagenumberrdquo Korean Journal of Mathematics vol 10 no 2 pp 39ndash42 2003

[34] M Y Sohn X Yuan and H S Jeong ldquoThe bondage number of1198623times119862

119899rdquo Journal of the KoreanMathematical Society vol 44 no

6 pp 1213ndash1231 2007[35] L Kang M Y Sohn and H K Kim ldquoBondage number of the

discrete torus 119862119899times 119862

4rdquo Discrete Mathematics vol 303 no 1ndash3

pp 80ndash86 2005[36] J Huang and J-M Xu ldquoThe bondage numbers and efficient

dominations of vertex-transitive graphsrdquoDiscrete Mathematicsvol 308 no 4 pp 571ndash582 2008

[37] C J Xiang X Yuan andM Y Sohn ldquoDomination and bondagenumber of119862

3times119862

119899rdquoArs Combinatoria vol 97 pp 299ndash310 2010

[38] M S Jacobson and L F Kinch ldquoOn the domination numberof products of graphs Irdquo Ars Combinatoria vol 18 pp 33ndash441984

[39] Y-Q Li ldquoA note on the bondage number of a graphrdquo ChineseQuarterly Journal of Mathematics vol 9 no 4 pp 1ndash3 1994

[40] F-T Hu and J-M Xu ldquoBondage number of mesh networksrdquoFrontiers ofMathematics inChina vol 7 no 5 pp 813ndash826 2012

[41] R Frucht and F Harary ldquoOn the corona of two graphsrdquoAequationes Mathematicae vol 4 pp 322ndash325 1970

[42] K Carlson and M Develin ldquoOn the bondage number of planarand directed graphsrdquoDiscreteMathematics vol 306 no 8-9 pp820ndash826 2006

[43] U Teschner ldquoOn the bondage number of block graphsrdquo ArsCombinatoria vol 46 pp 25ndash32 1997

[44] J Huang and J-M Xu ldquoNote on conjectures of bondagenumbers of planar graphsrdquo Applied Mathematical Sciences vol6 no 65ndash68 pp 3277ndash3287 2012

[45] L Kang and J Yuan ldquoBondage number of planar graphsrdquoDiscrete Mathematics vol 222 no 1ndash3 pp 191ndash198 2000

[46] M Fischermann D Rautenbach and L Volkmann ldquoRemarkson the bondage number of planar graphsrdquo Discrete Mathemat-ics vol 260 no 1ndash3 pp 57ndash67 2003

[47] J Hou G Liu and J Wu ldquoBondage number of planar graphswithout small cyclesrdquoUtilitasMathematica vol 84 pp 189ndash1992011

[48] J Huang and J-M Xu ldquoThe bondage numbers of graphs withsmall crossing numbersrdquo Discrete Mathematics vol 307 no 15pp 1881ndash1897 2007

[49] Q Ma S Zhang and J Wang ldquoBondage number of 1-planargraphrdquo Applied Mathematics vol 1 no 2 pp 101ndash103 2010

[50] J Hou and G Liu ldquoA bound on the bondage number of toroidalgraphsrdquoDiscreteMathematics Algorithms and Applications vol4 no 3 Article ID 1250046 5 pages 2012

[51] Y-C Cao J-M Xu and X-R Xu ldquoOn bondage number oftoroidal graphsrdquo Journal of University of Science and Technologyof China vol 39 no 3 pp 225ndash228 2009

[52] Y-C Cao J Huang and J-M Xu ldquoThe bondage number ofgraphs with crossing number less than fourrdquo Ars Combinatoriato appear

[53] H K Kim ldquoOn the bondage numbers of some graphsrdquo KoreanJournal of Mathematics vol 11 no 2 pp 11ndash15 2004

[54] A Gagarin and V Zverovich ldquoUpper bounds for the bondagenumber of graphs on topological surfacesrdquo Discrete Mathemat-ics 2012

[55] J Huang ldquoAn improved upper bound for the bondage numberof graphs on surfacesrdquoDiscrete Mathematics vol 312 no 18 pp2776ndash2781 2012

[56] A Gagarin and V Zverovich ldquoThe bondage number of graphson topological surfaces and Teschnerrsquos conjecturerdquo DiscreteMathematics vol 313 no 6 pp 796ndash808 2013

[57] V Samodivkin ldquoNote on the bondage number of graphs ontopological surfacesrdquo httparxivorgabs12086203

[58] E J Cockayne R M Dawes and S T Hedetniemi ldquoTotaldomination in graphsrdquoNetworks vol 10 no 3 pp 211ndash219 1980

[59] R Laskar J Pfaff S M Hedetniemi and S T HedetniemildquoOn the algorithmic complexity of total dominationrdquo Society forIndustrial and Applied Mathematics vol 5 no 3 pp 420ndash4251984

[60] J Pfaff R C Laskar and S T Hedetniemi ldquoNP-completeness oftotal and connected domination and irredundance for bipartitegraphsrdquo Tech Rep 428 Department of Mathematical SciencesClemson University 1983

[61] M A Henning ldquoA survey of selected recent results on totaldomination in graphsrdquoDiscrete Mathematics vol 309 no 1 pp32ndash63 2009

[62] V R Kulli and D K Patwari ldquoThe total bondage number ofa graphrdquo in Advances in Graph Theory pp 227ndash235 VishwaGulbarga India 1991

[63] F-T Hu Y Lu and J-M Xu ldquoThe total bondage number ofgrid graphsrdquo Discrete Applied Mathematics vol 160 no 16-17pp 2408ndash2418 2012

[64] J Cao M Shi and B Wu ldquoResearch on the total bondagenumber of a spectial networkrdquo in Proceedings of the 3rdInternational Joint Conference on Computational Sciences andOptimization (CSO rsquo10) pp 456ndash458 May 2010

[65] N Sridharan MD Elias and V S A Subramanian ldquoTotalbondage number of a graphrdquo AKCE International Journal ofGraphs and Combinatorics vol 4 no 2 pp 203ndash209 2007

[66] N Jafari Rad and J Raczek ldquoSome progress on total bondage ingraphsrdquo Graphs and Combinatorics 2011

[67] T W Haynes and P J Slater ldquoPaired-domination and thepaired-domatic numberrdquoCongressusNumerantium vol 109 pp65ndash72 1995

[68] T W Haynes and P J Slater ldquoPaired-domination in graphsrdquoNetworks vol 32 no 3 pp 199ndash206 1998

[69] X Hou ldquoA characterization of 2120574 120574119875-treesrdquoDiscrete Mathemat-

ics vol 308 no 15 pp 3420ndash3426 2008[70] K E Proffitt T W Haynes and P J Slater ldquoPaired-domination

in grid graphsrdquo Congressus Numerantium vol 150 pp 161ndash1722001

International Journal of Combinatorics 33

[71] H Qiao L KangM Cardei andD-Z Du ldquoPaired-dominationof treesrdquo Journal of Global Optimization vol 25 no 1 pp 43ndash542003

[72] E Shan L Kang and M A Henning ldquoA characterizationof trees with equal total domination and paired-dominationnumbersrdquo The Australasian Journal of Combinatorics vol 30pp 31ndash39 2004

[73] J Raczek ldquoPaired bondage in treesrdquo Discrete Mathematics vol308 no 23 pp 5570ndash5575 2008

[74] J A Telle and A Proskurowski ldquoAlgorithms for vertex parti-tioning problems on partial k-treesrdquo SIAM Journal on DiscreteMathematics vol 10 no 4 pp 529ndash550 1997

[75] G S Domke J H Hattingh M A Henning and L R MarkusldquoRestrained domination in treesrdquoDiscreteMathematics vol 211no 1ndash3 pp 1ndash9 2000

[76] G S Domke J H Hattingh M A Henning and L R MarkusldquoRestrained domination in graphs with minimum degree twordquoJournal of Combinatorial Mathematics and Combinatorial Com-puting vol 35 pp 239ndash254 2000

[77] G S Domke J H Hattingh S T Hedetniemi R C Laskarand L R Markus ldquoRestrained domination in graphsrdquo DiscreteMathematics vol 203 no 1ndash3 pp 61ndash69 1999

[78] J H Hattingh and E J Joubert ldquoAn upper bound for therestrained domination number of a graph with minimumdegree at least two in terms of order and minimum degreerdquoDiscrete Applied Mathematics vol 157 no 13 pp 2846ndash28582009

[79] M A Henning ldquoGraphs with large restrained dominationnumberrdquo Discrete Mathematics vol 197-198 pp 415ndash429 199916th British Combinatorial Conference (London 1997)

[80] J H Hattingh and A R Plummer ldquoRestrained bondage ingraphsrdquo Discrete Mathematics vol 308 no 23 pp 5446ndash54532008

[81] N Jafari Rad R Hasni J Raczek and L Volkmann ldquoTotalrestrained bondage in graphsrdquoActaMathematica Sinica vol 29no 6 pp 1033ndash1042 2013

[82] J Cyman and J Raczek ldquoOn the total restrained dominationnumber of a graphrdquoThe Australasian Journal of Combinatoricsvol 36 pp 91ndash100 2006

[83] M A Henning ldquoGraphs with large total domination numberrdquoJournal of Graph Theory vol 35 no 1 pp 21ndash45 2000

[84] D-X Ma X-G Chen and L Sun ldquoOn total restraineddomination in graphsrdquoCzechoslovakMathematical Journal vol55 no 1 pp 165ndash173 2005

[85] B Zelinka ldquoRemarks on restrained domination and totalrestrained domination in graphsrdquo Czechoslovak MathematicalJournal vol 55 no 2 pp 393ndash396 2005

[86] W Goddard and M A Henning ldquoIndependent domination ingraphs a survey and recent resultsrdquo Discrete Mathematics vol313 no 7 pp 839ndash854 2013

[87] J H Zhang H L Liu and L Sun ldquoIndependence bondagenumber and reinforcement number of some graphsrdquo Transac-tions of Beijing Institute of Technology vol 23 no 2 pp 140ndash1422003

[88] K D Dharmalingam Studies in Graph Theorymdashequitabledomination and bottleneck domination [PhD thesis] MaduraiKamaraj University Madurai India 2006

[89] GDeepakND Soner andAAlwardi ldquoThe equitable bondagenumber of a graphrdquo Research Journal of Pure Algebra vol 1 no9 pp 209ndash212 2011

[90] E Sampathkumar and H B Walikar ldquoThe connected domina-tion number of a graphrdquo Journal of Mathematical and PhysicalSciences vol 13 no 6 pp 607ndash613 1979

[91] K White M Farber and W Pulleyblank ldquoSteiner trees con-nected domination and strongly chordal graphsrdquoNetworks vol15 no 1 pp 109ndash124 1985

[92] M Cadei M X Cheng X Cheng and D-Z Du ldquoConnecteddomination in Ad Hoc wireless networksrdquo in Proceedings ofthe 6th International Conference on Computer Science andInformatics (CSampI rsquo02) 2002

[93] J F Fink and M S Jacobson ldquon-domination in graphsrdquo inGraph Theory with Applications to Algorithms and ComputerScience Wiley-Interscience Publications pp 283ndash300 WileyNew York NY USA 1985

[94] M Chellali O Favaron A Hansberg and L Volkmann ldquok-domination and k-independence in graphs a surveyrdquo Graphsand Combinatorics vol 28 no 1 pp 1ndash55 2012

[95] Y Lu X Hou J-M Xu andN Li ldquoTrees with uniqueminimump-dominating setsrdquo Utilitas Mathematica vol 86 pp 193ndash2052011

[96] Y Lu and J-M Xu ldquoThe p-bondage number of treesrdquo Graphsand Combinatorics vol 27 no 1 pp 129ndash141 2011

[97] Y Lu and J-M Xu ldquoThe 2-domination and 2-bondage numbersof grid graphs A manuscriptrdquo httparxivorgabs12044514

[98] M Krzywkowski ldquoNon-isolating 2-bondage in graphsrdquo TheJournal of the Mathematical Society of Japan vol 64 no 4 2012

[99] M A Henning O R Oellermann and H C Swart ldquoBoundson distance domination parametersrdquo Journal of CombinatoricsInformation amp System Sciences vol 16 no 1 pp 11ndash18 1991

[100] F Tian and J-M Xu ldquoBounds for distance domination numbersof graphsrdquo Journal of University of Science and Technology ofChina vol 34 no 5 pp 529ndash534 2004

[101] F Tian and J-M Xu ldquoDistance domination-critical graphsrdquoApplied Mathematics Letters vol 21 no 4 pp 416ndash420 2008

[102] F Tian and J-M Xu ldquoA note on distance domination numbersof graphsrdquo The Australasian Journal of Combinatorics vol 43pp 181ndash190 2009

[103] F Tian and J-M Xu ldquoAverage distances and distance domina-tion numbersrdquo Discrete Applied Mathematics vol 157 no 5 pp1113ndash1127 2009

[104] Z-L Liu F Tian and J-M Xu ldquoProbabilistic analysis of upperbounds for 2-connected distance k-dominating sets in graphsrdquoTheoretical Computer Science vol 410 no 38ndash40 pp 3804ndash3813 2009

[105] M A Henning O R Oellermann and H C Swart ldquoDistancedomination critical graphsrdquo Journal of Combinatorial Mathe-matics and Combinatorial Computing vol 44 pp 33ndash45 2003

[106] T J Bean M A Henning and H C Swart ldquoOn the integrityof distance domination in graphsrdquo The Australasian Journal ofCombinatorics vol 10 pp 29ndash43 1994

[107] M Fischermann and L Volkmann ldquoA remark on a conjecturefor the (kp)-domination numberrdquoUtilitasMathematica vol 67pp 223ndash227 2005

[108] T Korneffel D Meierling and L Volkmann ldquoA remark on the(22)-domination numberrdquo Discussiones Mathematicae GraphTheory vol 28 no 2 pp 361ndash366 2008

[109] Y Lu X Hou and J-M Xu ldquoOn the (22)-domination numberof treesrdquo Discussiones Mathematicae Graph Theory vol 30 no2 pp 185ndash199 2010

34 International Journal of Combinatorics

[110] H Li and J-M Xu ldquo(dm)-domination numbers of m-connected graphsrdquo Rapports de Recherche 1130 LRI UniversiteParis-Sud 1997

[111] S M Hedetniemi S T Hedetniemi and T V Wimer ldquoLineartime resource allocation algorithms for treesrdquo Tech Rep URI-014 Department of Mathematical Sciences Clemson Univer-sity 1987

[112] M Farber ldquoDomination independent domination and dualityin strongly chordal graphsrdquo Discrete Applied Mathematics vol7 no 2 pp 115ndash130 1984

[113] G S Domke and R C Laskar ldquoThe bondage and reinforcementnumbers of 120574

119891for some graphsrdquoDiscrete Mathematics vol 167-

168 pp 249ndash259 1997[114] G S Domke S T Hedetniemi and R C Laskar ldquoFractional

packings coverings and irredundance in graphsrdquo vol 66 pp227ndash238 1988

[115] E J Cockayne P A Dreyer Jr S M Hedetniemi andS T Hedetniemi ldquoRoman domination in graphsrdquo DiscreteMathematics vol 278 no 1ndash3 pp 11ndash22 2004

[116] I Stewart ldquoDefend the roman empirerdquo Scientific American vol281 pp 136ndash139 1999

[117] C S ReVelle and K E Rosing ldquoDefendens imperiumromanum a classical problem in military strategyrdquoThe Ameri-can Mathematical Monthly vol 107 no 7 pp 585ndash594 2000

[118] A Bahremandpour F-T Hu S M Sheikholeslami and J-MXu ldquoOn the Roman bondage number of a graphrdquo DiscreteMathematics Algorithms and Applications vol 5 no 1 ArticleID 1350001 15 pages 2013

[119] N Jafari Rad and L Volkmann ldquoRoman bondage in graphsrdquoDiscussiones Mathematicae Graph Theory vol 31 no 4 pp763ndash773 2011

[120] K Ebadi and L PushpaLatha ldquoRoman bondage and Romanreinforcement numbers of a graphrdquo International Journal ofContemporary Mathematical Sciences vol 5 pp 1487ndash14972010

[121] N Dehgardi S M Sheikholeslami and L Volkman ldquoOn theRoman k-bondage number of a graphrdquo AKCE InternationalJournal of Graphs and Combinatorics vol 8 pp 169ndash180 2011

[122] S Akbari and S Qajar ldquoA note on Roman bondage number ofgraphsrdquo Ars Combinatoria to Appear

[123] F-T Hu and J-M Xu ldquoRoman bondage numbers of somegraphsrdquo httparxivorgabs11093933

[124] N Jafari Rad and L Volkmann ldquoOn the Roman bondagenumber of planar graphsrdquo Graphs and Combinatorics vol 27no 4 pp 531ndash538 2011

[125] S Akbari M Khatirinejad and S Qajar ldquoA note on the romanbondage number of planar graphsrdquo Graphs and Combinatorics2012

[126] B Bresar M A Henning and D F Rall ldquoRainbow dominationin graphsrdquo Taiwanese Journal of Mathematics vol 12 no 1 pp213ndash225 2008

[127] B L Hartnell and D F Rall ldquoOn dominating the Cartesianproduct of a graph and 120572krdquo Discussiones Mathematicae GraphTheory vol 24 no 3 pp 389ndash402 2004

[128] N Dehgardi S M Sheikholeslami and L Volkman ldquoThe k-rainbow bondage number of a graphrdquo to appear in DiscreteApplied Mathematics

[129] J von Neumann and O Morgenstern Theory of Games andEconomic Behavior Princeton University Press Princeton NJUSA 1944

[130] C BergeTheorıe des Graphes et ses Applications 1958[131] O Ore Theory of Graphs vol 38 American Mathematical

Society Colloquium Publications 1962[132] E Shan and L Kang ldquoBondage number in oriented graphsrdquoArs

Combinatoria vol 84 pp 319ndash331 2007[133] J Huang and J-M Xu ldquoThe bondage numbers of extended

de Bruijn and Kautz digraphsrdquo Computers amp Mathematics withApplications vol 51 no 6-7 pp 1137ndash1147 2006

[134] J Huang and J-M Xu ldquoThe total domination and total bondagenumbers of extended de Bruijn and Kautz digraphsrdquoComputersamp Mathematics with Applications vol 53 no 8 pp 1206ndash12132007

[135] Y Shibata and Y Gonda ldquoExtension of de Bruijn graph andKautz graphrdquo Computers amp Mathematics with Applications vol30 no 9 pp 51ndash61 1995

[136] X Zhang J Liu and JMeng ldquoThebondage number in completet-partite digraphsrdquo Information Processing Letters vol 109 no17 pp 997ndash1000 2009

[137] D W Bange A E Barkauskas and P J Slater ldquoEfficient dom-inating sets in graphsrdquo in Applications of Discrete Mathematicspp 189ndash199 SIAM Philadelphia Pa USA 1988

[138] M Livingston and Q F Stout ldquoPerfect dominating setsrdquoCongressus Numerantium vol 79 pp 187ndash203 1990

[139] L Clark ldquoPerfect domination in random graphsrdquo Journal ofCombinatorial Mathematics and Combinatorial Computing vol14 pp 173ndash182 1993

[140] D W Bange A E Barkauskas L H Host and L H ClarkldquoEfficient domination of the orientations of a graphrdquo DiscreteMathematics vol 178 no 1ndash3 pp 1ndash14 1998

[141] A E Barkauskas and L H Host ldquoFinding efficient dominatingsets in oriented graphsrdquo Congressus Numerantium vol 98 pp27ndash32 1993

[142] I J Dejter and O Serra ldquoEfficient dominating sets in CayleygraphsrdquoDiscrete AppliedMathematics vol 129 no 2-3 pp 319ndash328 2003

[143] J Lee ldquoIndependent perfect domination sets in Cayley graphsrdquoJournal of Graph Theory vol 37 no 4 pp 213ndash219 2001

[144] WGu X Jia and J Shen ldquoIndependent perfect domination setsin meshes tori and treesrdquo Unpublished

[145] N Obradovic J Peters and G Ruzic ldquoEfficient domination incirculant graphswith two chord lengthsrdquo Information ProcessingLetters vol 102 no 6 pp 253ndash258 2007

[146] K Reji Kumar and G MacGillivray ldquoEfficient domination incirculant graphsrdquo Discrete Mathematics vol 313 no 6 pp 767ndash771 2013

[147] D Van Wieren M Livingston and Q F Stout ldquoPerfectdominating sets on cube-connected cyclesrdquo Congressus Numer-antium vol 97 pp 51ndash70 1993

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 International Journal of Combinatorics

A cycle 1198623119896+1

also satisfies 119887(1198623119896+1) = 3 = 120575(119862

3119896+1) +

1205821015840

(1198623119896+1) minus 1 since 1205821015840(119862

3119896+1) = 120575(119862

3119896+1) = 2 for any integer

119896 ⩾ 1For the graph119867 shown in Figure 41205821015840(119867) = 4 and 120575(119867) =

2 and so 119887(119867) = 5 = 120575(119867) + 1205821015840(119867) minus 1For the 4-regular graph119866 constructed by Liu and Sun [22]

obtained from the disjoint union of two copies of the graph119867 showed in Figure 4 by identifying two vertices of degreetwo we have 119887(119866) ⩾ 5 Clearly 1205821015840(119866) = 2 Thus 119887(119866) = 5 =120575(119866) + 120582

1015840

(119866) minus 1For the 4-regular graph 119866

119905constructed by Samodivkin

[20] see Figure 3 for 119905 = 2 119887(119866119905) = 2 Clearly 120575(119866

119905) = 1

and 1205821015840(119866) = 2 Thus 119887(119866) = 2 = 120575(119866) + 1205821015840(119866) minus 1These examples show that if Conjecture 34 is true then the

given upper bound is tight

33 Bounds Implied by Degree Sequence Now let us return toTheorem 16 from which Teschner [10] obtained some otherbounds in terms of the degree sequencesThe degree sequence120587(119866) of a graph 119866 with vertex-set 119881 = 119909

1 119909

2 119909

119899 is the

sequence 120587 = (1198891 119889

2 119889

119899) with 119889

1⩽ 119889

2⩽ sdot sdot sdot ⩽ 119889

119899 where

119889119894= 119889

119866(119909

119894) for each 119894 = 1 2 119899 The following result is

essentially a corollary of Theorem 16

Theorem35 (Teschner [10] 1997) Let119866 be a nonempty graphwith degree sequence120587(119866) If120572

2(119866) = 119905 then 119887(119866) ⩽ 119889

119905+119889

119905+1minus

1

CombiningTheorem 35with Proposition 10 (ie 1205722(119866) ⩽

120574(119866)) we have the following corollary

Corollary 36 (Teschner [10] 1997) Let 119866 be a nonemptygraph with the degree sequence 120587(119866) If 120574(119866) = 120574 then 119887(119866) ⩽119889120574+ 119889

120574+1minus 1

In [10] Teschner showed that these two bounds are sharpfor arbitrarily many graphs Let119867 = 119862

3119896+1+ 119909

11199094 119909

11199093119896minus1

where 1198623119896+1

is a cycle (1199091 119909

2 119909

3119896+1 119909

1) for any integer

119896 ⩾ 2 Then 120574(119867) = 119896 + 1 and so 119887(119867) ⩽ 2 + 2 minus 1 = 3by Corollary 36 Since119862

3119896+1is a spanning subgraph of119867 and

120574(119866) = 120574(119867) Observation 1 yields that 119887(119867) ⩾ 119887(1198623119896+1) = 3

and so 119887(119867) = 3Although various upper bounds have been established

as the above we find that the appearance of these boundsis essentially based upon the local structures of a graphprecisely speaking the structures of the neighborhoods oftwo vertices within distance 2 Even if these bounds can beachieved by some special graphs it is more often not the caseThe reason lies essentially in the definition of the bondagenumber which is theminimumvalue among all bondage setsan integral property of a graph While it easy to find upperbounds just by choosing some bondage set the gap betweenthe exact value of the bondage number and such a boundobtained only from local structures of a graph is often largeFor example a star 119870

1Δ however large Δ is 119887(119870

1Δ) = 1

Therefore one has been longing for better bounds upon someintegral parameters However as what we will see below it isdifficult to establish such upper bounds

34 Bounds in 120574-Critical Graphs A graph119866 is called a vertexdomination-critical graph (vc-graph or 120574-critical for short) if120574(119866 minus 119909) lt 120574(119866) for any 119909 isin 119881(119866) proposed by Brighamet al [24] in 1988 Several families of graphs are known to be120574-critical From definition it is clear that if 119866 is a 120574-criticalgraph then 120574(119866) ⩾ 2The class of 120574-critical graphs with 120574 = 2is characterized as follows

Proposition 37 (Brigham et al [24] 1988) A graph 119866 with120574(119866) = 2 is a 120574-critical graph if and only if 119866 is a completegraph 119870

2119905(119905 ⩾ 2) with a perfect matching removed

A more interesting family is composed of the 119899-criticalgraphs 119866

119898119899defined for 119898 119899 ⩾ 2 by the circulant undirected

graph 119866(119873 plusmn119878) where 119873 = (119898 + 1)(119899 minus 1) + 1 and119878 = 1 2 lfloor1198982rfloor in which 119881(119866(119873 plusmn119878)) = 119873 and119864(119866(119873 plusmn119878)) = 119894119895 |119895 minus 119894| equiv 119904 (mod 119873) 119904 isin 119878

The reason why 120574-critical graphs are of special interest inthis context is easy to see that they play an important role instudy of the bondage number For instance it immediatelyfollows from Theorem 13 that if 119887(119866) gt Δ(119866) then 119866 is a 120574-critical graphThe 120574-critical graphs are defined exactly in thisway In order to find graphs 119866 with a high bondage number(ie higher than Δ(119866)) and beyond its general upper boundsfor the bondage number we therefore have to look at 120574-critical graphs

The bondage numbers of some 120574-critical graphs havebeen examined by several authors see for example [1020 25 26] From Theorem 13 we know that the bondagenumber of a graph 119866 is bounded from above by Δ(119866)if 119866 is not a 120574-critical graph For 120574-critical graphs it ismore difficult to find an upper bound We will see that thebondage numbers of 120574-critical graphs in general are noteven bounded from above by Δ + 119888 for any fixed naturalnumber 119888

In this subsection we introduce some upper bounds forthe bondage number of a 120574-critical graph By Proposition 37we easily obtain the following result

Theorem 38 If 119866 is a 120574-critical graph with 120574(119866) = 2 then119887(119866) ⩽ Δ + 1

In Section 4 by Theorem 59 we will see that the equalityin Theorem 38 holds that is 119887(119866) = Δ + 1 if 119866 is a 120574-criticalgraph with 120574(119866) = 2

Theorem 39 (Teschner [10] 1997) Let119866 be a 120574-critical graphwith degree sequence 120587(119866) Then 119887(119866) ⩽ maxΔ(119866) + 1 119889

1+

1198892+ sdot sdot sdot + 119889

120574minus1 where 120574 = 120574(119866)

As we mentioned above if 119866 is a 120574-critical graph with120574(119866) = 2 then 119887(119866) = Δ + 1 which shows that thebound given in Theorem 39 can be attained for 120574 = 2However we have not known whether this bound is tightfor general 120574 ⩾ 3 Theorem 39 gives the following corollaryimmediately

Corollary 40 (Teschner [10] 1997) Let 119866 be a 120574-criticalgraph with degree sequence 120587(119866) If 120574(119866) = 3 then 119887(119866) ⩽maxΔ(119866) + 1 120575(119866) + 119889

2

International Journal of Combinatorics 9

From Theorem 38 and Corollary 40 we have 119887(119866) ⩽2Δ(119866) if 119866 is a 120574-critical graph with 120574(119866) ⩽ 3 The followingresult shows that this bound is not tight

Theorem 41 (Teschner [26] 1995) Let119866 be a 120574-critical graphgraph with 120574(119866) ⩽ 3 Then 119887(119866) ⩽ (32)Δ(119866)

Until now we have not known whether the bound givenin Theorem 41 is tight or not We state two conjectures on120574-critical graphs proposed by Samodivkin [20] The first ofthem is motivated byTheorems 22 and 19

Conjecture 42 (Samodivkin [20] 2008) For every connectednontrivial 120574-critical graph 119866

min119909isin119860+(119866)119910isin119861(119866minus119909)

119889119866(119909) +

1003816100381610038161003816119873119866(119910) cap 119860 (119866 minus 119909)

1003816100381610038161003816 ⩽3

2Δ (119866)

(15)

To state the second conjecture we need the followingresult on 120574-critical graphs

Proposition 43 If119866 is a 120574-critical graph then 120592(119866) ⩽ (Δ(119866)+1)(120574(119866)minus1)+1 moreover if the equality holds then119866 is regular

The upper bound in Proposition 43 is sharp in thesense that equality holds for the infinite class of 120574-criticalgraphs 119866

119898119899defined in the beginning of this subsection In

Proposition 43 the first result is due to Brigham et al [24] in1988 the second is due to Fulman et al [27] in 1995

For a 120574-critical graph 119866 with 120574 = 3 by Proposition 43120592(119866) ⩽ 2Δ(119866) + 3

Theorem 44 (Teschner [26] 1995) If 119866 is a 120574-critical graphwith 120574 = 3 and 120592(119866) = 2Δ(119866) + 3 then 119887(119866) ⩽ Δ + 1

We have not known yet whether the equality inTheorem 44holds or notHowever Samodivkin proposed thefollowing conjecture

Conjecture 45 (Samodivkin [20] 2008) If 119866 is a 120574-criticalgraph with (Δ(119866)+1)(120574minus1)+1 vertices then 119887(119866) = Δ(119866)+1

In general based on Theorem 41 Teschner proposed thefollowing conjecture

Conjecture 46 (Teschner [26] 1995) If119866 is a 120574-critical graphthen 119887(119866) ⩽ (32)Δ(119866) for 120574 ⩾ 4

We conclude this subsection with some remarks Graphswhich are minimal or critical with respect to a given prop-erty or parameter frequently play an important role in theinvestigation of that property or parameter Not only are suchgraphs of considerable interest in their own right but also aknowledge of their structure often aids in the developmentof the general theory In particular when investigating anyfinite structure a great number of results are proven byinduction Consequently it is desirable to learn as much aspossible about those graphs that are critical with respect toa given property or parameter so as to aid and abet suchinvestigations

In this subsection we survey some results on the bondagenumber for 120574-critical graphs Although these results are notvery perfect they provides a feasible method to approach thebondage number from different viewpoints In particular themethods given in Teschner [26] worth further explorationand development

The following proposition is maybe useful for us tofurther investigate the bondage number of a 120574-critical graph

Proposition 47 (Brigham et al [24] 1988) If 119866 has anonisolated vertex 119909 such that the subgraph induced by119873

119866(119909)

is a complete graph then 119866 is not 120574-critical

This simple fact shows that any 120574-critical graph containsno vertices of degree one

35 Bounds Implied by Domination In the preceding sub-section we introduced some upper bounds on the bondagenumbers for 120574-critical graphs by consideration of domina-tions In this subsection we introduce related results for ageneral graph with given domination number

Theorem 48 (Fink et al [8] 1990) For any connected graph119866 of order 119899 (⩾2)

(a) 119887(119866) ⩽ 119899 minus 1

(b) 119887(119866) ⩽ Δ(119866) + 1 if 120574(119866) = 2

(c) 119887(119866) ⩽ 119899 minus 120574(119866) + 1

While the upper bound of 119899minus1 on 119887(119866) is not particularlygood for many classes of graphs (eg trees and cycles) it isan attainable bound For example if 119866 is a complete 119905-partitegraph 119870

119905(2) for 119905 = 1198992 and 119899 ⩽ 4 an even integer then the

three bounds on 119887(119866) in Theorem 48 are sharpTeschner [26] consider a graphwith 120574(119866) = 1 or 120574(119866) = 2

The next result is almost trivial but useful

Lemma 49 Let 119866 be a graph with order 119899 and 120574(119866) = 1 119905 thenumber of vertices of degree 119899 minus 1 Then 119887(119866) = lceil1199052rceil

Since 119905 ⩽ Δ(119866) clearly Lemma 49 yields the followingresult immediately

Theorem 50 (Teschner [26] 1995) 119887(119866) ⩽ (12)Δ+1 for anygraph 119866 with 120574(119866) = 1

For a complete graph1198702119896+1

119887(1198702119896+1) = 119896+1 = (12)Δ+1

which shows that the upper bound given in Theorem 50 canbe attained For a graph119866with 120574(119866) = 2 byTheorem 59 laterthe upper bound given inTheorem 48 (b) can be also attainedby a 2-critical graph (see Proposition 37)

36 Two Conjectures In 1990 when Fink et al [8] introducedthe concept of the bondage number they proposed thefollowing conjecture

Conjecture 51 119887(119866) ⩽ Δ(119866) + 1 for any nonempty graph 119866

10 International Journal of Combinatorics

Although some results partially support Conjecture 51Teschner [28] in 1993 found a counterexample to this con-jecture the cartesian product 119870

3times 119870

3 which shows that

119887(119866) = Δ(119866) + 2If a graph 119866 is a counterexample to Conjecture 51 it

must be a 120574-critical graph by Theorem 13 That is why thevertex domination-critical graphs are of special interest in theliterature

Now we return to Conjecture 51 Hartnell and Rall [17]and Teschner [29] independently proved that 119887(119866) can bemuch greater than Δ(119866) by showing the following result

Theorem 52 (Hartnell and Rall [17] 1994 Teschner [29]1996) For an integer 119899 ⩾ 3 let 119866

119899be the cartesian product

119870119899times 119870

119899 Then 119887(119866

119899) = 3(119899 minus 1) = (32)Δ(119866

119899)

This theorem shows that there exist no upper boundsof the form 119887(119866) ⩽ Δ(119866) + 119888 for any integer 119888 Teschner[26] proved that 119887(119866) ⩽ (32)Δ(119866) for any graph 119866 with120574(119866) ⩽ 2 (see Theorems 50 and 48) and for 120574-critical graphswith 120574(119866) = 3 and proposed the following conjecture

Conjecture 53 (Teschner [26] 1995) 119887(119866) ⩽ (32)Δ(119866) forany graph 119866

We believe that this conjecture is true but so far no muchwork about it has been known

4 Lower Bounds

Since the bondage number is defined as the smallest numberof edges whose removal results in an increase of dominationnumber each constructive method that creates a concretebondage set leads to an upper bound on the bondage numberFor that reason it is hard to find lower bounds Neverthelessthere are still a few lower bounds

41 Bounds Implied by Subgraphs The first lower boundon the bondage number is gotten in terms of its spanningsubgraph

Theorem 54 (Teschner [10] 1997) Let 119867 be a spanningsubgraph of a nonempty graph119866 If 120574(119867) = 120574(119866) then 119887(119867) ⩽119887(119866)

By Theorem 1 119887(119862119899) = 3 if 119899 equiv 1 (mod 3) and 119887(119862

119899) =

2 otherwise 119887(119875119899) = 2 if 119899 equiv 1 (mod 3) and 119887(119875

119899) = 1

otherwise From these results and Theorem 54 we get thefollowing two corollaries

Corollary 55 If119866 is hamiltonianwith order 119899 ⩾ 3 and 120574(119866) =lceil1198993rceil then 119887(119866) ⩾ 2 and in addition 119887(119866) ⩾ 3 if 119899 equiv 1 (mod3)

Corollary 56 If 119866 with order 119899 ⩾ 2 has a hamiltonian pathand 120574(119866) = lceil1198993rceil then 119887(119866) ⩾ 2 if 119899 equiv 1 (mod 3)

42 Other Bounds The vertex covering number 120573(119866) of 119866 isthe minimum number of vertices that are incident with all

Figure 5 A 120574-critical graph with 120573 = 120574 = 4 120575 = 2 and 119887 = 3

edges in 119866 If 119866 has no isolated vertices then 120574(119866) ⩽ 120573(119866)clearly In [30] Volkmann gave a lot of graphs with 120573 = 120574

Theorem 57 (Teschner [10] 1997) Let119866 be a graph If 120574(119866) =120573(119866) then

(a) 119887(119866) ⩾ 120575(119866)

(b) 119887(119866) ⩾ 120575(119866) + 1 if 119866 is a 120574-critical graph

The graph shown in Figure 5 shows that the bound givenin Theorem 57(b) is sharp For the graph 119866 it is a 120574-criticalgraph and 120574(119866) = 120573(119866) = 4 By Theorem 57 we have 119887(119866) ⩾3 On the other hand 119887(119866) ⩽ 3 by Theorem 16 Thus 119887(119866) =3

Proposition 58 (Sanchis [31] 1991) Let 119866 be a graph 119866 oforder 119899 (⩾ 6) and size 120576 If 119866 has no isolated vertices and3 ⩽ 120574(119866) ⩽ 1198992 then 120576 ⩽ ( 119899minus120574(119866)+1

2)

Using the idea in the proof of Theorem 57 every upperbound for 120574(119866) can lead to a lower bound for 119887(119866) Inthis way Teschner [10] obtained another lower bound fromProposition 58

Theorem59 (Teschner [10] 1997) Let119866 be a graph119866 of order119899 (⩾6) and size 120576 If 2 ⩽ 120574(119866) ⩽ 1198992 minus 1 then

(a) 119887(119866) ⩾ min120575(119866) 120576 minus ( 119899minus120574(119866)2)

(b) 119887(119866) ⩾ min120575(119866) + 1 120576 minus ( 119899minus120574(119866)2) if 119866 is a 120574-critical

graph

The lower bound in Theorem 59(b) is sharp for a classof 120574-critical graphs with domination number 2 Let 119866 be thegraph obtained from complete graph119870

2119905(119905 ⩾ 2) by removing

a perfect matching By Proposition 37 119866 is a 120574-critical graphwith 120574(119866) = 2 Then 119887(119866) ⩾ 120575(119866) + 1 = Δ(119866) + 1 byTheorem 59 and 119887(119866) ⩽ Δ(119866) + 1 by Theorem 38 and so119887(119866) = Δ(119866) + 1 = 2119905 minus 1

As far as the author knows there are no more lowerbounds in the present literature In view of applications ofthe bondage number a network is vulnerable if its bondagenumber is small while it is stable if its bondage number islargeTherefore better lower bounds let us learn better aboutthe stability of networks from this point of view Thus it isof great significance to seek more lower bounds for variousclasses of graphs

International Journal of Combinatorics 11

5 Results on Graph Operations

Generally speaking it is quite difficult to determine the exactvalue of the bondage number for a given graph since itstrongly depends on the dominating number of the graphThus determining bondage numbers for some special graphsis interesting if the dominating numbers of those graphs areknown or can be easily determined In this section we willintroduce some known results on bondage numbers for somespecial classes of graphs obtained by some graph operations

51 Cartesian Product Graphs Let 1198661= (119881

1 119864

1) and 119866

2=

(1198812 119864

2) be two graphs The Cartesian product of 119866

1and 119866

2

is an undirected graph denoted by 1198661times 119866

2 where 119881(119866

1times

1198662) = 119881

1times 119881

2 two distinct vertices 119909

11199092and 119910

11199102 where

1199091 119910

1isin 119881(119866

1) and 119909

2 119910

2isin 119881(119866

2) are linked by an edge in

1198661times 119866

2if and only if either 119909

1= 119910

1and 119909

21199102isin 119864(119866

2) or

1199092= 119910

2and 119909

11199101isin 119864(119866

1) The Cartesian product is a very

effective method for constructing a larger graph from severalspecified small graphs

Theorem 60 (Dunbar et al [21] 1998) For 119899 ⩾ 3

119887 (119862119899times 119870

2) =

2 119894119891 119899 equiv 0 119900119903 1 (mod 4) 3 119894119891 119899 equiv 3 (mod 4) 4 119894119891 119899 equiv 2 (mod 4)

(16)

For the Cartesian product 119862119899times 119862

119898of two cycles 119862

119899and

119862119898 where 119899 ⩾ 4 and 119898 ⩾ 3 Klavzar and Seifter [32]

determined 120574(119862119899times 119862

119898) for some 119899 and 119898 For example

120574(119862119899times119862

4) = 119899 for 119899 ⩾ 3 and 120574(119862

119899times119862

3) = 119899minuslfloor1198994rfloor for 119899 ⩾ 4

Kim [33] determined 119887(1198623times 119862

3) = 6 and 119887(119862

4times 119862

4) = 4 For

a general 119899 ⩾ 4 the exact values of the bondage numbers of119862119899times 119862

3and 119862

119899times 119862

4are determined as follows

Theorem 61 (Sohn et al [34] 2007) For 119899 ⩾ 4

119887 (119862119899times 119862

3) =

2 119894119891 119899 equiv 0 (mod 4) 4 119894119891 119899 equiv 1 119900119903 2 (mod 4) 5 119894119891 119899 equiv 3 (mod 4)

(17)

Theorem62 (Kang et al [35] 2005) 119887(119862119899times119862

4) = 4 for 119899 ⩾ 4

For larger 119898 and 119899 Huang and Xu [36] obtained thefollowing result see Theorem 197 for more details

Theorem 63 (Huang and Xu [36] 2008) 119887(1198625119894times119862

5119895) = 3 for

any positive integers 119894 and 119895

Xiang et al [37] determined that for 119899 ⩾ 5

119887 (119862119899times 119862

5)

= 3 if 119899 equiv 0 or 1 (mod 5) = 4 if 119899 equiv 2 or 4 (mod 5) ⩽ 7 if 119899 equiv 3 (mod 5)

(18)

For the Cartesian product 119875119899times119875

119898of two paths 119875

119899and 119875

119898

Jacobson and Kinch [38] determined 120574(119875119899times119875

2) = lceil(119899+1)2rceil

120574(119875119899times 119875

3) = 119899 minus lfloor(119899 minus 1)4rfloor and 120574(119875

119899times 119875

4) = 119899 + 1 if 119899 =

1 2 3 5 6 9 and 119899 otherwiseThe bondage number of119875119899times119875

119898

for 119899 ⩾ 4 and 2 ⩽ 119898 ⩽ 4 is determined as follows (Li [39] alsodetermined 119887(119875

119899times 119875

2))

Theorem 64 (Hu and Xu [40] 2012) For 119899 ⩾ 4

119887 (119875119899times 119875

2) =

1 119894119891 119899 equiv 1 (mod 2)2 119894119891 119899 equiv 0 (mod 2)

119887 (119875119899times 119875

3) =

1 119894119891 119899 equiv 1 119900119903 2 (mod 4)2 119894119891 119899 equiv 0 119900119903 3 (mod 4)

119887 (119875119899times 119875

4) = 1 119891119900119903 119899 ⩾ 14

(19)

From the proof of Theorem 64 we find that if 119875119899=

0 1 119899 minus 1 and 119875119898= 0 1 119898 minus 1 then the removal

of the vertex (0 0) in 119875119899times119875

119898does not change the domination

number If119898 increases the effect of (0 0) for the dominationnumber will be smaller and smaller from the probabilityTherefore we expect it is possible that 120574(119875

119899times 119875

119898minus (0 0)) =

120574(119875119899times 119875

119898) for119898 ⩾ 5 and give the following conjecture

Conjecture 65 119887(119875119899times 119875

119898) ⩽ 2 for119898 ⩾ 5 and 119899 ⩾ 4

52 Block Graphs and Cactus Graphs In this subsection weintroduce some results for corona graphs block graphs andcactus graphs

The corona1198661∘ 119866

2 proposed by Frucht and Harary [41]

is the graph formed from a copy of 1198661and 120592(119866

1) copies of

1198662by joining the 119894th vertex of 119866

1to the 119894th copy of 119866

2 In

particular we are concernedwith the corona119867∘1198701 the graph

formed by adding a new vertex V119894 and a new edge 119906

119894V119894for

every vertex 119906119894in119867

Theorem 66 (Carlson and Develin [42] 2006) 119887(119867 ∘ 1198701) =

120575(119867) + 1

This is a very important result which is often used toconstruct a graph with required bondage number In otherwords this result implies that for any given positive integer 119887there is a graph 119866 such that 119887(119866) = 119887

A block graph is a graph whose blocks are completegraphs Each block in a cactus graph is either a cycle or a119870

2

If each block of a graph is either a complete graph or a cyclethen we call this graph a block-cactus graph

Theorem67 (Teschner [43] 1997) 119887(119866) le Δ(119866) for any blockgraph 119866

Teschner [43] characterized all block graphs with 119887(119866) =Δ(119866) In the same paper Teschner found that 120574-criticalgraphs are instrumental in determining bounds for thebondage number of cactus and block graphs and obtained thefollowing result

Theorem 68 (Teschner [43] 1997) 119887(119866) ⩽ 3 for anynontrivial cactus graph 119866

This bound can be achieved by 119867 ∘ 1198701where 119867 is

a nontrivial cactus graph without vertices of degree one

12 International Journal of Combinatorics

by Theorem 66 In 1998 Dunbar et al [21] proposed thefollowing problem

Problem 2 Characterize all cactus graphs with bondagenumber 3

Some upper bounds for block-cactus graphs were alsoobtained

Theorem 69 (Dunbar et al [21] 1998) Let 119866 be a connectedblock-cactus graph with at least two blocksThen 119887(119866) ⩽ Δ(119866)

Theorem 70 (Dunbar et al [21] 1998) Let 119866 be a connectedblock-cactus graph which is neither a cactus graph nor a blockgraph Then 119887(119866) ⩽ Δ(119866) minus 1

53 Edge-Deletion and Edge-Contraction In order to obtainfurther results of the bondage number we may consider howthe bondage number changes under some operations of agraph 119866 which remain the domination number unchangedor preserve some property of 119866 such as planarity Twosimple operations satisfying these requirements are the edge-deletion and the edge-contraction

Theorem 71 (Huang and Xu [44] 2012) Let 119866 be any graphand 119890 isin 119864(119866) Then 119887(119866 minus 119890) ⩾ 119887(119866) minus 1 Moreover 119887(119866 minus 119890) ⩽119887(119866) if 120574(119866 minus 119890) = 120574(119866)

FromTheorem 1 119887(119862119899minus 119890) = 119887(119875

119899) = 119887(119862

119899) minus 1 for any 119890

in119862119899 which shows that the lower bound on 119887(119866minus119890) is sharp

The upper bound can reach for any graph 119866 with 119887(119866) ⩾ 2Next we consider the effect of the edge-contraction on

the bondage number Given a graph 119866 the contraction of 119866by the edge 119890 = 119909119910 denoted by 119866119909119910 is the graph obtainedfrom 119866 by contracting two vertices 119909 and 119910 to a new vertexand then deleting all multiedges It is easy to observe that forany graph 119866 120574(119866) minus 1 ⩽ 120574(119866119909119910) ⩽ 120574(119866) for any edge 119909119910 of119866

Theorem 72 (Huang and Xu [44] 2012) Let 119866 be any graphand 119909119910 be any edge in119866 If119873

119866(119909)cap119873

119866(119910) = 0 and 120574(119866119909119910) =

120574(119866) then 119887(119866119909119910) ⩾ 119887(119866)

We can use examples 119862119899and 119870

119899to show that the

conditions ofTheorem 72 are necessary and the lower boundinTheorem 72 is tight Clearly 120574(119862

119899) = lceil1198993rceil and 120574(119870

119899) = 1

for any edge 119909119910 119862119899119909119910 = 119862

119899minus1and 119870

119899119909119910 = 119870

119899minus1 By

Theorem 1 if 119899 equiv 1 (mod 3) then 120574(119862119899119909119910) lt 120574(119866) and

119887(119862119899119909119910) = 2 lt 3 = 119887(119862

119899) Thus the result in Theorem 72

is generally invalid without the hypothesis 120574(119866119909119910) = 120574(119866)Furthermore the condition 119873

119866(119909) cap 119873

119866(119910) = 0 cannot be

omitted even if 120574(119866119909119910) = 120574(119866) since for odd 119899 119887(119870119899119909119910) =

lceil(119899 minus 1)2rceil lt lceil1198992rceil = 119887(119870119899) by Theorem 1

On the other hand 119887(119862119899119909119910) = 119887(119862

119899) = 2 if 119899 equiv 0 (mod

3) and 119887(119870119899119909119910) = 119887(119870

119899) if 119899 is even which shows that the

equality in 119887(119866119909119910) ⩾ 119887(119866) may hold Thus the bound inTheorem 72 is tight However provided all the conditions119887(119866119909119910) can be arbitrarily larger than 119887(119866) Given a graph119867let119866 be the graph formed from119867∘119870

1by adding a new vertex

119909 and joining it to an vertex 119910 of degree one in119867 ∘ 1198701 Then

119866119909119910 = 119867 ∘ 1198701 120574(119866) = 120574(119866119909119910) and 119873

119866(119909) cap 119873

119866(119910) = 0

But 119887(119866) = 1 since 120574(119866 minus 119909119910) = 120574(119866119909119910) + 1 and 119887(119866119909119910) =120575(119867) + 1 byTheorem 66The gap between 119887(119866) and 119887(119866119909119910)is 120575(119867)

6 Results on Planar Graphs

From Section 2 we have seen that the bondage numberfor a tree has been completely solved Moreover a lineartime algorithm to compute the bondage number of a tree isdesigned by Hartnell et al [15] It is quite natural to considerthe bondage number for a planar graph In this section westate some results and problems on the bondage number fora planar graph

61 Conjecture on Bondage Number As mentioned inSection 3 the bondage number can be much larger than themaximum degree But for a planar graph 119866 the bondagenumber 119887(119866) cannot exceed Δ(119866) too much It is clear that119887(119866) ⩽ Δ(119866) + 4 by Corollary 15 since 120575(119866) ⩽ 5 for anyplanar graph119866 In 1998Dunbar et al [21] posed the followingconjecture

Conjecture 73 If 119866 is a planar graph then 119887(119866) ⩽ Δ(119866) + 1

Because of its attraction it immediately became the focusof attention when this conjecture is proposed In fact themain aim concerning the research of the bondage number fora planar graph is focused on this conjecture

It has been mentioned in Theorems 1 and 60 that119887(119862

3119896+1) = 3 = Δ + 1 and 119887(119862

4119896+2times 119870

2) = 4 = Δ + 1 It

is known that 119887(1198706minus 119872) = 5 = Δ + 1 where119872 is a perfect

matching of the complete graph1198706These examples show that

if Conjecture 73 is true then the upper bound is sharp for2 ⩽ Δ ⩽ 4

Here we note that it is sufficient to prove this conjecturefor connected planar graphs since the bondage number of adisconnected graph is simply the minimum of the bondagenumbers of its components

The first paper attacking this conjecture is due to Kangand Yuan [45] which confirmed the conjecture for everyconnected planar graph 119866 with Δ ⩾ 7 The proofs are mainlybased onTheorems 16 and 18

Theorem 74 (Kang and Yuan [45] 2000) If 119866 is a connectedplanar graph then 119887(119866) ⩽ min8 Δ(119866) + 2

Obviously in view of Theorem 74 Conjecture 73 is truefor any connected planar graph with Δ ⩾ 7

62 Bounds Implied by Degree Conditions As we have seenfrom Theorem 74 to attack Conjecture 73 we only need toconsider connected planar graphs with maximum Δ ⩽ 6Thus studying the bondage number of planar graphs bydegree conditions is of significanceThe first result on boundsimplied by degree conditions is obtained by Kang and Yuan[45]

International Journal of Combinatorics 13

Theorem 75 (Kang and Yuan [45] 2000) If 119866 is a connectedplanar graph without vertices of degree 5 then 119887(119866) ⩽ 7

Fischermann et al [46] generalized Theorem 75 as fol-lows

Theorem 76 (Fischermann et al [46] 2003) Let 119866 be aconnected planar graph and 119883 the set of vertices of degree 5which have distance at least 3 to vertices of degrees 1 2 and3 If all vertices in 119883 not adjacent with vertices of degree 4are independent and not adjacent to vertices of degree 6 then119887(119866) ⩽ 7

Clearly if119866 has no vertices of degree 5 then119883 = 0 ThusTheorem 76 yields Theorem 75

Use 119899119894to denote the number of vertices of degree 119894 in 119866

for each 119894 = 1 2 Δ(119866) Using Theorem 18 Fischermannet al obtained the following two theorems

Theorem 77 (Fischermann et al [46] 2003) For any con-nected planar graph 119866

(1) 119887(119866) ⩽ 7 if 1198995lt 2119899

2+ 3119899

3+ 2119899

4+ 12

(2) 119887(119866) ⩽ 6 if 119866 contains no vertices of degrees 4 and 5

Theorem 78 (Fischermann et al [46] 2003) For any con-nected planar graph 119866 119887(119866) ⩽ Δ(119866) + 1 if

(1) Δ(119866) = 6 and every edge 119890 = 119909119910 with 119889119866(119909) = 5 and

119889119866(119910) = 6 is contained in at most one triangle

(2) Δ(119866) = 5 and no triangle contains an edge 119890 = 119909119910 with119889119866(119909) = 5 and 4 ⩽ 119889

119866(119910) ⩽ 5

63 Bounds Implied by Girth Conditions The girth 119892(119866) of agraph 119866 is the length of the shortest cycle in 119866 If 119866 has nocycles we define 119892(119866) = infin

Combining Theorem 74 with Theorem 78 we find thatif a planar graph contains no triangles and has maximumdegree Δ ⩾ 5 then Conjecture 73 holds This fact motivatedFischermann et alrsquos [46] attempt to attack Conjecture 73 bygirth restraints

Theorem 79 (Fischermann et al [46] 2003) For any con-nected planar graph 119866

119887 (119866) ⩽

6 119894119891 119892 (119866) ⩾ 4

5 119894119891 119892 (119866) ⩾ 5

4 119894119891 119892 (119866) ⩾ 6

3 119894119891 119892 (119866) ⩾ 8

(20)

The first result in Theorem 79 shows that Conjecture 73is valid for all connected planar graphs with 119892(119866) ⩾ 4 andΔ(119866) ⩾ 5 It is easy to verify that the second result inTheorem 79 implies that Conjecture 73 is valid for all not3-regular graphs of girth 119892(119866) ⩾ 5 which is stated in thefollowing corollary

Corollary 80 For any connected planar graph 119866 if 119866 is not3-regular and 119892(119866) ⩾ 5 then 119887(119866) ⩽ Δ(119866) + 1

The first result in Theorem 79 also implies that if 119866 isa connected planar graph with no cycles of length 3 then119887(119866) ⩽ 6 Hou et al [47] improved this result as follows

Theorem 81 (Hou et al [47] 2011) For any connected planargraph 119866 if 119866 contains no cycles of length 119894 (4 ⩽ 119894 ⩽ 6) then119887(119866) ⩽ 6

Since 119887(1198623119896+1) = 3 for 119896 ⩾ 3 (see Theorem 1) the

last bound in Theorem 79 is tight Whether other bounds inTheorem 79 are tight remains open In 2003 Fischermann etal [46] posed the following conjecture

Conjecture 82 For any connected planar graph 119866 119887(119866) ⩽ 7and

119887 (119866) ⩽ 5 119894119891 119892 (119866) ⩾ 4

4 119894119891 119892 (119866) ⩾ 5(21)

We conclude this subsection with a question on bondagenumbers of planar graphs

Question 1 (Fischermann et al [46] 2003) Is there a planargraph 119866 with 6 ⩽ 119887(119866) ⩽ 8

In 2006 Carlson andDevelin [42] showed that the corona119866 = 119867 ∘ 119870

1for a planar graph 119867 with 120575(119867) = 5 has the

bondage number 119887(119866) = 120575(119867) + 1 = 6 (see Theorem 66)Since the minimum degree of planar graphs is at most 5then 119887(119866) can attach 6 If we take 119867 as the graph of theicosahedron then 119866 = 119867 ∘ 119870

1is such an example The

question for the existence of planar graphs with bondagenumber 7 or 8 remains open

64 Comments on the Conjectures Conjecture 73 is true forall connected planar graphs with minimum degree 120575 ⩽ 2 byTheorem 16 or maximum degree Δ ⩾ 7 by Theorem 74 ornot 120574-critical planar graphs by Theorem 13 Thus to attackConjecture 73 we only need to consider connected criticalplanar graphs with degree-restriction 3 ⩽ 120575 ⩽ Δ ⩽ 6

Recalling and anatomizing the proofs of all results men-tioned in the preceding subsections on the bondage numberfor connected planar graphs we find that the proofs of theseresults strongly depend upon Theorem 16 or Theorem 18 Inother words a basic way used in the proofs is to find twovertices 119909 and 119910 with distance at most two in a consideredplanar graph 119866 such that

119889119866(119909) + 119889

119866(119910) or 119889

119866(119909) + 119889

119866(119910) minus

1003816100381610038161003816119873119866(119909) cap 119873

119866(119910)1003816100381610038161003816

(22)

which bounds 119887(119866) is as small as possible Very recentlyHuang and Xu [44] have considered the following parameter

119861 (119866) = min119909119910isin119881(119866)

119889119909119910 1 ⩽ 119889

119866(119909 119910) ⩽ 2

cup 119889119909119910minus 119899

119909119910 119889 (119909 119910) = 1

(23)

where 119889119909119910= 119889

119866(119909) + 119889

119866(119910) minus 1 and 119899

119909119910= |119873

119866(119909) cap 119873

119866(119910)|

14 International Journal of Combinatorics

It follows fromTheorems 16 and 18 that

119887 (119866) ⩽ 119861 (119866) (24)

The proofs given in Theorems 74 and 79 indeed imply thefollowing stronger results

Theorem 83 If 119866 is a connected planar graph then

119861 (119866) ⩽

min 8 Δ (119866) + 26 119894119891 119892 (119866) ⩾ 4

5 119894119891 119892 (119866) ⩾ 5

4 119894119891 119892 (119866) ⩾ 6

3 119894119891 119892 (119866) ⩾ 8

(25)

Thus using Theorem 16 or Theorem 18 if we can proveConjectures 73 and 82 then we can prove the followingstatement

Statement If 119866 is a connected planar graph then

119861 (119866) ⩽

Δ (119866) + 1

7

5 if 119892 (119866) ⩾ 44 if 119892 (119866) ⩾ 5

(26)

It follows from (24) that Statement implies Conjectures 73and 82 However Huang and Xu [44] gave examples to showthat none of conclusions in Statement is true As a result theystated the following conclusion

Theorem 84 (Huang and Xu [44] 2012) It is not possible toprove Conjectures 73 and 82 if they are right using Theorems16 and 18

Therefore a new method is needed to prove or disprovethese conjectures At the present time one cannot prove theseconjectures and may consider to disprove them If one ofthese conjectures is invalid then there exists a minimumcounterexample 119866 with respect to 120592(119866) + 120576(119866) In order toobtain a minimum counterexample one may consider twosimple operations satisfying these requirements the edge-deletion and the edge-contractionwhich decrease 120592(119866)+120576(119866)and preserve planarity However applying Theorems 71 and72 Huang and Xu [44] presented the following results

Theorem 85 (Huang and Xu [44] 2012) It is not possible toconstruct minimum counterexamples to Conjectures 73 and 82by the operation of an edge-deletion or an edge-contraction

7 Results on Crossing Number Restraints

It is quite natural to generalize the known results on thebondage number for planar graphs to formore general graphsin terms of graph-theoretical parameters In this section weconsider graphs with crossing number restraints

The crossing number cr(119866) of 119866 is the smallest numberof pairwise intersections of its edges when 119866 is drawn in theplane If cr(119866) = 0 then 119866 is a planar graph

71 General Methods Use 119899119894to denote the number of vertices

of degree 119894 in119866 for 119894 = 1 2 Δ(119866) Huang and Xu obtainedthe following results

Theorem 86 (Huang and Xu [48] 2007) For any connectedgraph 119866

119887 (119866) ⩽

6 119894119891 119892 (119866) ⩾ 4 119886119899119889 2cr (119866) lt 121198861

5 119894119891 119892 (119866) ⩾ 5 119886119899119889 6cr (119866) lt 161198862

4 119894119891 119892 (119866) ⩾ 6 119886119899119889 4cr (119866) lt 141198863

3 119894119891 119892 (119866) ⩾ 8 119886119899119889 6cr (119866) lt 161198864

(27)

where 1198861= 119899

1+ 2119899

2+ 2119899

3+sum

Δ

119894=8(119894 minus 7)119899

119894+ 8 119886

2= 3119899

1+ 6119899

2+

51198993+sum

Δ

119894=7(3119894minus18)119899

119894+20 119886

3= 119899

1+2119899

2+sum

Δ

119894=6(2119894minus10)119899

119894+12

and 1198864= sum

Δ

119894=5(3119894 minus 12)119899

119894+ 16

Simplifying the conditions in Theorem 86 we obtain thefollowing corollaries immediately

Corollary 87 (Huang and Xu [48] 2007) For any connectedgraph 119866

119887 (119866) ⩽

6 119894119891 119892 (119866) ⩾ 4 119886119899119889 cr (119866) ⩽ 3

5 119894119891 119892 (119866) ⩾ 5 119886119899119889 cr (119866) ⩽ 4

4 119894119891 119892 (119866) ⩾ 6 119886119899119889 cr (119866) ⩽ 2

3 119894119891 119892 (119866) ⩾ 8 119886119899119889 cr (119866) ⩽ 2

(28)

Corollary 88 (Huang and Xu [48] 2007) For any connectedgraph 119866

(a) 119887(119866) ⩽ 6 if 119866 is not 4-regular cr(119866) = 4 and 119892(119866) ⩾ 4

(b) 119887(119866) ⩽ Δ(119866) + 1 if 119866 is not 3-regular cr(119866) ⩽ 4 and119892(119866) ⩾ 5

(c) 119887(119866) ⩽ 4 if 119866 is not 3-regular cr(119866) = 3 and 119892(119866) ⩾ 6

(d) 119887(119866) ⩽ 3 if cr(119866) = 3 119892(119866) ⩾ 8 and Δ(119866) ⩾ 5

These corollaries generalize some known results for pla-nar graphs For example Corollary 87 contains Theorem 79Corollary 88 (b) contains Corollary 80

Theorem 89 (Huang and Xu [48] 2007) For any connectedgraph 119866 if cr(119866) ⩽ 119899

3(119866) + 119899

4(119866) + 3 then 119887(119866) ⩽ 8

Corollary 90 (Huang and Xu [48] 2007) For any connectedgraph 119866 if cr(119866) ⩽ 3 then 119887(119866) ⩽ 8

Perhaps being unaware of this result in 2010 Ma et al[49] proved that 119887(119866) ⩽ 12 for any graph 119866 with cr(119866) = 1

Theorem91 (Huang and Xu [48] 2007) Let119866 be a connectedgraph and 119868 = 119909 isin 119881(119866) 119889

119866(119909) = 5 119889

119866(119909 119910) ⩾ 3 if

International Journal of Combinatorics 15

119889119866(119910) ⩽ 3 and 119889

119866(119910) = 4 for every 119910 isin 119873

119866(119909) If 119868 is

independent no vertices adjacent to vertices of degree 6 and

cr (119866) lt max5119899

3+ |119868| minus 2119899

4+ 28

117119899

3+ 40

16 (29)

then 119887(119866) ⩽ 7

Corollary 92 (Huang and Xu [48] 2007) Let 119866 be aconnected graph with cr(119866) ⩽ 2 If 119868 = 119909 isin 119881(119866) 119889

119866(119909) =

5 119889119866(119910 119909) ⩾ 3 if 119889

119866(119910) ⩽ 3 and 119889

119866(119910) = 4 for every 119910 isin

119873119866(119909) is independent and no vertices adjacent to vertices of

degree 6 then 119887(119866) ⩽ 7

Theorem93 (Huang andXu [48] 2007) Let119866 be a connectedgraph If 119866 satisfies

(a) 5cr(119866) + 1198995lt 2119899

2+ 3119899

3+ 2119899

4+ 12 or

(b) 7cr(119866) + 21198995lt 3119899

2+ 4119899

4+ 24

then 119887(119866) ⩽ 7

Proposition 94 (Huang and Xu [48] 2007) Let 119866 be aconnected graph with no vertices of degrees four and five Ifcr(119866) ⩽ 2 then 119887(119866) ⩽ 6

Theorem 95 (Huang and Xu [48] 2007) If 119866 is a connectedgraph with cr(119866) ⩽ 4 and not 4-regular when cr(119866) = 4 then119887(119866) ⩽ Δ(119866) + 2

The above results generalize some known results forplanar graphs For example Corollary 90 and Theorem 95contain Theorem 74 the first condition in Theorem 93 con-tains the second condition inTheorem 77

From Corollary 88 and Theorem 95 we suggest the fol-lowing questions

Question 2 Is there a

(a) 4-regular graph 119866 with cr(119866) = 4 such that 119887(119866) ⩾ 7

(b) 3-regular graph 119866 with cr ⩽ 4 and 119892(119866) ⩾ 5 such that119887(119866) = 5

(c) 3-regular graph 119866 with cr = 3 and 119892(119866) = 6 or 7 suchthat 119887(119866) = 5

72 Carlson-Develin Methods In this subsection we intro-duce an elegant method presented by Carlson and Develin[42] to obtain some upper bounds for the bondage numberof a graph

Suppose that 119866 is a connected graph We say that 119866 hasgenus 120588 if 119866 can be embedded in a surface 119878 with 120588 handlessuch that edges are pairwise disjoint except possibly for end-vertices Let119866 be an embedding of119866 in surface 119878 and let120601(119866)denote the number of regions in 119866 The boundary of everyregion contains at least three edges and every edge is on theboundary of atmost two regions (the two regions are identicalwhen 119890 is a cut-edge) For any edge 119890 of 119866 let 1199031

119866(119890) and 1199032

119866(119890)

be the numbers of edges comprising the regions in 119866 whichthe edge 119890 borders It is clear that every 119890 = 119909119910 isin 119864(119866)

1199032

119866(119890) ⩾ 119903

1

119866(119890) ⩾ 4 if 1003816100381610038161003816119873119866

(119909) cap 119873119866(119910)1003816100381610038161003816 = 0

1199032

119866(119890) ⩾ 4 119903

1

119866(119890) ⩾ 3 if 1003816100381610038161003816119873119866

(119909) cap 119873119866(119910)1003816100381610038161003816 = 1

1199032

119866(119890) ⩾ 119903

1

119866(119890) ⩾ 3 if 1003816100381610038161003816119873119866

(119909) cap 119873119866(119910)1003816100381610038161003816 ⩾ 2

(30)

Following Carlson and Develin [42] for any edge 119890 = 119909119910 of119866 we define

119863119866(119890) =

1

119889119866(119909)+

1

119889119866(119910)

+1

1199031

119866(119890)+

1

1199032

119866(119890)minus 1 (31)

By the well-known Eulerrsquos formula

120592 (119866) minus 120576 (119866) + 120601 (119866) = 2 minus 2120588 (119866) (32)

it is easy to see that

sum

119890isin119864(119866)

119863119866(119890) = 120592 (119866) minus 120576 (119866) + 120601 (119866) = 2 minus 2120588 (119866) (33)

If 119866 is a planar graph that is 120588(119866) = 0 then

sum

119890isin119864(119866)

119863119866(119890) = 120592 (119866) minus 120576 (119866) + 120601 (119866) = 2 (34)

Combining these formulas withTheorem 18 Carlson andDevelin [42] gave a simple and intuitive proof ofTheorem 74and obtained the following result

Theorem 96 (Carlson and Develin [42] 2006) Let 119866 be aconnected graph which can be embedded on a torus Then119887(119866) ⩽ Δ(119866) + 3

Recently Hou and Liu [50] improved Theorem 96 asfollows

Theorem 97 (Hou and Liu [50] 2012) Let 119866 be a connectedgraph which can be embedded on a torus Then 119887(119866) ⩽ 9Moreover if Δ(119866) = 6 then 119887(119866) ⩽ 8

By the way Cao et al [51] generalized the result inTheorem 79 to a connected graph 119866 that can be embeddedon a torus that is

119887 (119866) le

6 if 119892 (119866) ⩾ 4 and 119866 is not 4-regular5 if 119892 (119866) ⩾ 54 if 119892 (119866) ⩾ 6 and 119866 is not 3-regular3 if 119892 (119866) ⩾ 8

(35)

Several authors used this method to obtain some resultson the bondage number For example Fischermann et al[46] used this method to prove the second conclusion inTheorem 78 Recently Cao et al [52] have used thismethod todeal with more general graphs with small crossing numbersFirst they found the following property

Lemma 98 120575(119866) ⩽ 5 for any graph 119866 with cr(119866) ⩽ 5

16 International Journal of Combinatorics

This result implies that 119887(119866) ⩽ Δ(119866) + 4 by Corollary 15for any graph 119866 with cr(119866) ⩽ 5 Cao et al established thefollowing relation in terms of maximum planar subgraphs

Lemma 99 (Cao et al [52]) Let 119866 be a connected graph withcrossing number cr(119866) For any maximum planar subgraph119867of 119866

sum

119890isin119864(119867)

119863119866(119890) ⩾ 2 minus

2cr (119866)120575 (119866)

(36)

Using Carlson-Develin method and combiningLemma 98 with Lemma 99 Cao et al proved the followingresults

Theorem 100 (Cao et al [52]) For any connected graph 119866119887(119866) ⩽ Δ(119866) + 2 if 119866 satisfies one of the following conditions

(a) cr(119866) ⩽ 3(b) cr(119866) = 4 and 119866 is not 4-regular(c) cr(119866) = 5 and 119866 contains no vertices of degree 4

Theorem101 (Cao et al [52]) Let119866 be a connected graphwithΔ(119866) = 5 and cr(119866) ⩽ 4 If no triangles contain two vertices ofdegree 5 then 119887(119866) ⩽ 6 = Δ(119866) + 1

Theorem 102 (Cao et al [52]) Let 119866 be a connected graphwith Δ(119866) ⩾ 6 and cr(119866) ⩽ 3 If Δ(119866) ⩾ 7 or if Δ(119866) = 6120575(119866) = 3 and every edge 119890 = 119909119910 with 119889

119866(119909) = 5 and 119889

119866(119910) = 6

is contained in atmost one triangle then 119887(119866) ⩽ min8 Δ(119866)+1

Using Carlson-Develin method it can be proved that119887(119866) ⩽ Δ(119866) + 2 if cr(119866) ⩽ 3 (see the first conclusion inTheorem 100) and not yet proved that 119887(119866) ⩽ 8 if cr(119866) ⩽3 although it has been proved by using other method (seeCorollary 90)

By using Carlson-Develin method Samodivkin [25]obtained some results on the bondage number for graphswithsome given properties

Kim [53] showed that 119887(119866) ⩽ Δ(119866) + 2 for a connectedgraph 119866 with genus 1 Δ(119866) ⩽ 5 and having a toroidalembedding of which at least one region is not 4-sidedRecently Gagarin and Zverovich [54] further have extendedCarlson-Develin ideas to establish a nice upper bound forarbitrary graphs that can be embedded on orientable ornonorientable topological surface

Theorem 103 (Gagarin and Zverovich [54] 2012) Let 119866 bea graph embeddable on an orientable surface of genus ℎ and anonorientable surface of genus 119896 Then 119887(119866) ⩽ minΔ(119866)+ℎ+2 Δ(119866) + 119896 + 1

This result generalizes the corresponding upper boundsin Theorems 74 and 96 for any orientable or nonori-entable topological surface By investigating the proof ofTheorem 103 Huang [55] found that the issue of the ori-entability can be avoided by using the Euler characteristic120594(= 120592(119866) minus 120576(119866) + 120601(119866)) instead of the genera ℎ and 119896 the

relations are 120594 = 2minus2ℎ and 120594 = 2minus119896 To obtain the best resultfromTheorem 103 onewants ℎ and 119896 as small as possible thisis equivalent to having 120594 as large as possible

According toTheorem 103 if119866 is planar (ℎ = 0 120594 = 2) orcan be embedded on the real projective plane (119896 = 1 120594 = 1)then 119887(119866) ⩽ Δ(119866) + 2 In all other cases Huang [55] had thefollowing improvement for Theorem 103 the proof is basedon the technique developed byCarlson-Develin andGagarin-Zverovich and includes some elementary calculus as a newingredient mainly the intermediate-value theorem and themean-value theorem

Theorem 104 (Huang [55] 2012) Let 119866 be a graph embed-dable on a surface whose Euler characteristic 120594 is as large aspossible If 120594 ⩽ 0 then 119887(119866) ⩽ Δ(119866)+lfloor119903rfloor where 119903 is the largestreal root of the following cubic equation in 119911

1199113

+ 21199112

+ (6120594 minus 7) 119911 + 18120594 minus 24 = 0 (37)

In addition if 120594 decreases then 119903 increases

The following result is asymptotically equivalent toTheorem 104

Theorem 105 (Huang [55] 2012) Let 119866 be a graph embed-dable on a surface whose Euler characteristic 120594 is as large aspossible If 120594 ⩽ 0 then 119887(119866) lt Δ(119866) + radic12 minus 6120594 + 12 orequivalently 119887(119866) ⩽ Δ(119866) + lceilradic12 minus 6120594 minus 12rceil

Also Gagarin andZverovich [54] indicated that the upperbound in Theorem 103 can be improved for larger values ofthe genera ℎ and 119896 by adjusting the proofs and stated thefollowing general conjecture

Conjecture 106 (Gagarin and Zverovich [54] 2012) For aconnected graph G of orientable genus ℎ and nonorientablegenus 119896

119887 (119866) ⩽ min 119888ℎ 119888

1015840

119896 Δ (119866) + 119900 (ℎ) Δ (119866) + 119900 (119896) (38)

where 119888ℎand 1198881015840

119896are constants depending respectively on the

orientable and nonorientable genera of 119866

In the recent paper Gagarin and Zverovich [56] providedconstant upper bounds for the bondage number of graphs ontopological surfaces which can be used as the first estimationfor the constants 119888

ℎand 1198881015840

119896of Conjecture 106

Theorem 107 (Gagarin and Zverovich [56] 2013) Let 119866 bea connected graph of order 119899 If 119866 can be embedded on thesurface of its orientable or nonorientable genus of the Eulercharacteristic 120594 then

(a) 120594 ⩾ 1 implies 119887(119866) ⩽ 10(b) 120594 ⩽ 0 and 119899 gt minus12120594 imply 119887(119866) ⩽ 11(c) 120594 ⩽ minus1 and 119899 ⩽ minus12120594 imply 119887(119866) ⩽ 11 +

3120594(radic17 minus 8120594 minus 3)(120594 minus 1) = 11 + 119874(radicminus120594)

Theorem 79 shows that a connected planar triangle-freegraph 119866 has 119887(119866) ⩽ 6 The following result generalizes to itall the other topological surfaces

International Journal of Combinatorics 17

Theorem 108 (Gagarin and Zverovich [56] 2013) Let 119866 be aconnected triangle-free graph of order 119899 If 119866 can be embeddedon the surface of its orientable or nonorientable genus of theEuler characteristic 120594 then

(a) 120594 ⩾ 1 implies 119887(119866) ⩽ 6(b) 120594 ⩽ 0 and 119899 gt minus8120594 imply 119887(119866) ⩽ 7(c) 120594 ⩽ minus1 and 119899 ⩽ minus8120594 imply 119887(119866) ⩽ 7minus4120594(1+radic2 minus 120594)

In [56] the authors also explicitly improved upperbounds in Theorem 103 and gave tight lower bounds forthe number of vertices of graphs 2-cell embeddable ontopological surfaces of a given genusThe interested reader isreferred to the original paper for details The following resultis interesting although its proof is easy

Theorem 109 (Samodivkin [57]) Let119866 be a graph with order119899 and girth 119892 If 119866 is embeddable on a surface whose Eulercharacteristic 120594 is as large as possible then

119887 (119866) ⩽ 3 +8

119892 minus 2minus

4120594119892

119899 (119892 minus 2) (39)

If 119866 is a planar graph with girth 119892 ⩾ 4 + 119894 for any 119894 isin0 1 2 then Theorem 109 leads to 119887(119866) ⩽ 6 minus 119894 which isthe result inTheorem 79 obtained by Fischermann et al [46]Also inmany cases the bound stated inTheorem 109 is betterthan those given byTheorems 103 and 105

8 Conditional Bondage Numbers

Since the concept of the bondage number is based upon thedomination all sorts of dominations which are variationsof the normal domination by adding restricted conditionsto the normal dominating set can develop a new ldquobondagenumberrdquo as long as a variation of domination number isgiven In this section we survey results on the bondagenumber under some restricted conditions

81 Total Bondage Numbers A dominating set 119878 of a graph119866 without isolated vertices is called total if the subgraphinduced by 119878 contains no isolated vertices The minimumcardinality of a total dominating set is called the totaldomination number of 119866 and denoted by 120574

119879(119866) It is clear

that 120574(119866) ⩽ 120574119879(119866) ⩽ 2120574(119866) for any graph 119866 without isolated

verticesThe total domination in graphs was introduced by Cock-

ayne et al [58] in 1980 Pfaff et al [59 60] in 1983 showedthat the problem determining total domination number forgeneral graphs is NP-complete even for bipartite graphs andchordal graphs Even now total domination in graphs hasbeen extensively studied in the literature In 2009 Henning[61] gave a survey of selected recent results on total domina-tion in graphs

The total bondage number of 119866 denoted by 119887119879(119866) is the

smallest cardinality of a subset 119861 sube 119864(119866) with the propertythat119866minus119861 contains no isolated vertices and 120574

119879(119866minus119861) gt 120574

119879(119866)

From definition 119887119879(119866)may not exist for some graphs for

example119866 = 1198701119899 We put 119887

119879(119866) = infin if 119887

119879(119866) does not exist

It is easy to see that 119887119879(119866) is finite for any connected graph 119866

other than 1198701 119870

2 119870

3 and 119870

1119899 In fact for such a graph 119866

since there is a path of length 3 in 119866 we can find 1198611sube 119864(119866)

such that 119866 minus 1198611is a spanning tree 119879 of 119866 containing a path

of length 3 So 120574119879(119879) = 120574

119879(119866minus119861

1) ⩾ 120574

119879(119866) For the tree119879 we

find 1198612sube 119864(119879) such that 120574

119879(119879 minus 119861

2) gt 120574

119879(119879) ⩾ 120574

119879(119866) Thus

we have 120574119879(119866minus119861

2minus119861

1) gt 120574

119879(119866) and so 119887

119879(119866) ⩽ |119861

1|+|119861

2| In

the following discussion we always assume that 119887119879(119866) exists

when a graph 119866 is mentionedIn 1991 Kulli and Patwari [62] first studied the total

bondage number of a graph and calculated the exact valuesof 119887

119879(119866) for some standard graphs

Theorem 110 (Kulli and Patwari [62] 1991) For 119899 ⩾ 4

119887119879(119862

119899) =

3 119894119891 119899 equiv 2 (mod 4) 2 119900119905ℎ119890119903119908119894119904119890

119887119879(119875

119899) =

2 119894119891 119899 equiv 2 (mod 4) 1 119900119905ℎ119890119903119908119894119904119890

119887119879(119870

119898119899) = 119898 119908119894119905ℎ 2 ⩽ 119898 ⩽ 119899

119887119879(119870

119899) =

4 119891119900119903 119899 = 4

2119899 minus 5 119891119900119903 119899 ⩾ 5

(40)

Recently Hu et al [63] have obtained some results on thetotal bondage number of the Cartesian product119875

119898times119875

119899of two

paths 119875119898and 119875

119899

Theorem 111 (Hu et al [63] 2012) For the Cartesian product119875119898times 119875

119899of two paths 119875

119898and 119875

119899

119887119879(119875

2times 119875

119899) =

1 119894119891 119899 equiv 0 (mod 3) 2 119894119891 119899 equiv 2 (mod 3) 3 119894119891 119899 equiv 1 (mod 3)

119887119879(119875

3times 119875

119899) = 1

119887119879(119875

4times 119875

119899)

= 1 119894119891 119899 equiv 1 (mod 5) = 2 119894119891 119899 equiv 4 (mod 5) ⩽ 3 119894119891 119899 equiv 2 (mod 5) ⩽ 4 119894119891 119899 equiv 0 3 (mod 5)

(41)

Generalized Petersen graphs are an important class ofcommonly used interconnection networks and have beenstudied recently By constructing a family of minimum totaldominating sets Cao et al [64] determined the total bondagenumber of the generalized Petersen graphs

FromTheorem 6 we know that 119887(119879) ⩽ 2 for a nontrivialtree 119879 But given any positive integer 119896 Sridharan et al [65]constructed a tree 119879 for which 119887

119879(119879) = 119896 Let119867

119896be the tree

obtained from a star1198701119896+1

by subdividing 119896 edges twice Thetree 119867

7is shown in Figure 6 It can be easily verified that

119887119879(119867

119896) = 119896 This fact can be stated as the following theorem

Theorem 112 (Sridharan et al [65] 2007) For any positiveinteger 119896 there exists a tree 119879 with 119887

119879(119879) = 119896

18 International Journal of Combinatorics

Figure 61198677

Combining Theorem 1 with Theorem 112 we have whatfollows

Corollary 113 (Sridharan et al [65] 2007) The bondagenumber and the total bondage number are unrelated even fortrees

However Sridharan et al [65] gave an upper bound onthe total bondage number of a tree in terms of its maximumdegree For any tree 119879 of order 119899 if 119879 =119870

1119899minus1 then 119887

119879(119879) =

minΔ(119879) (13)(119899 minus 1) Rad and Raczek [66] improved thisupper bound and gave a constructive characterization of acertain class of trees attaching the upper bound

Theorem 114 (Rad and Raczek [66] 2011) 119887119879(119879) ⩽ Δ(119879) minus 1

for any tree 119879 with maximum degree at least three

In general the decision problem for 119887119879(119866) is NP-complete

for any graph 119866 We state the decision problem for the totalbondage as follows

Problem 3 Consider the decision problemTotal Bondage ProblemInstance a graph 119866 and a positive integer 119887Question is 119887

119879(119866) ⩽ 119887

Theorem 115 (Hu and Xu [16] 2012) The total bondageproblem is NP-complete

Consequently in view of its computational hardnessit is significative to establish some sharp bounds on thetotal bondage number of a graph in terms of other graphicparameters In 1991 Kulli and Patwari [62] showed that119887119879(119866) ⩽ 2119899 minus 5 for any graph 119866 with order 119899 ⩾ 5 Sridharan

et al [65] improved this result as follows

Theorem 116 (Sridharan et al [65] 2007) For any graph 119866with order 119899 if 119899 ⩾ 5 then

119887119879(119866) ⩽

1

3(119899 minus 1) 119894119891 119866 119888119900119899119905119886119894119899119904 119899119900 119888119910119888119897119890119904

119899 minus 1 119894119891 119892 (119866) ⩾ 5

119899 minus 2 119894119891 119892 (119866) = 4

(42)

Rad and Raczek [66] also established some upperbounds on 119887

119879(119866) for a general graph 119866 In particular they

gave an upper bound on 119887119879(119866) in terms of the girth of

119866

Theorem 117 (Rad and Raczek [66] 2011) 119887119879(119866)⩽4Δ(119866) minus 5

for any graph 119866 with 119892(119866) ⩾ 4

82 Paired Bondage Numbers A dominating set 119878 of 119866 iscalled to be paired if the subgraph induced by 119878 containsa perfect matching The paired domination number of 119866denoted by 120574

119875(119866) is the minimum cardinality of a paired

dominating set of 119866 Clearly 120574119879(119866) ⩽ 120574

119875(119866) for every

connected graph 119866 with order at least two where 120574119879(119866) is

the total domination number of 119866 and 120574119875(119866) ⩽ 2120574(119866) for

any graph119866without isolated vertices Paired domination wasintroduced by Haynes and Slater [67 68] and further studiedin [69ndash72]

The paired bondage number of 119866 with 120575(119866) ⩾ 1 denotedby 119887P(119866) is the minimum cardinality among all sets of edges119861 sube 119864 such that 120575(119866 minus 119861) ⩾ 1 and 120574

119875(119866 minus 119861) gt 120574

119875(119866)

The concept of the paired bondage number was firstproposed by Raczek [73] in 2008The following observationsfollow immediately from the definition of the paired bondagenumber

Observation 2 Let 119866 be a graph with 120575(119866) ⩾ 1

(a) If 119867 is a subgraph of 119866 such that 120575(119867) ⩾ 1 then119887119875(119867) ⩽ 119887

119875(119866)

(b) If 119867 is a subgraph of 119866 such that 119887119875(119867) = 1 and 119896

is the number of edges removed to form119867 then 1 ⩽119887119875(119866) ⩽ 119896 + 1

(c) If 119909119910 isin 119864(119866) such that 119889119866(119909) 119889

119866(119910) gt 1 and 119909119910

belongs to each perfect matching of each minimumpaired dominating set of 119866 then 119887

119875(119866) = 1

Based on these observations 119887119875(119875

119899) ⩽ 2 In fact the

paired bondage number of a path has been determined

Theorem 118 (Raczek [73] 2008) For any positive integer 119896if 119899 ⩾ 2 then

119887119875(119875

119899) =

0 119894119891 119899 = 2 3 119900119903 5

1 119894119891 119899 = 4119896 4119896 + 3 119900119903 4119896 + 6

2 119900119905ℎ119890119903119908119894119904119890

(43)

For a cycle 119862119899of order 119899 ⩾ 3 since 120574

119875(119862

119899) = 120574

119875(119875

119899) from

Theorem 118 the following result holds immediately

Corollary 119 (Raczek [73] 2008) For any positive integer 119896if 119899 ⩾ 3 then

119887119875(119862

119899) =

0 119894119891 119899 = 3 119900119903 5

2 119894119891 119899 = 4119896 4119896 + 3 119900119903 4119896 + 6

3 119900119905ℎ119890119903119908119894119904119890

(44)

A wheel 119882119899 where 119899 ⩾ 4 is a graph with 119899 vertices

formed by connecting a single vertex to all vertices of a cycle119862119899minus1

Of course 120574119875(119882

119899) = 2

International Journal of Combinatorics 19

119896

Figure 7 A tree 119879 with 119887119875(119879) = 119896

Theorem 120 (Raczek [73] 2008) For positive integers 119899 and119898

119887119875(119870

119898119899) = 119898 119891119900119903 1 lt 119898 ⩽ 119899

119887119875(119882

119899) =

4 119894119891 119899 = 4

3 119894119891 119899 = 5

2 119900119905ℎ119890119903119908119894119904119890

(45)

Theorem 16 shows that if 119879 is a nontrivial tree then119887(119879) ⩽ 2 However no similar result exists for pairedbondage For any nonnegative integer 119896 let 119879

119896be a tree

obtained by subdividing all but one edge of a star 1198701119896+1

(asshown in Figure 7) It is easy to see that 119887

119875(119879

119896) = 119896 We

state this fact as the following theorem which is similar toTheorem 112

Theorem121 (Raczek [73] 2008) For any nonnegative integer119896 there exists a tree 119879 with 119887

119875(119879) = 119896

Consider the tree defined in Figure 5 for 119896 = 3 Then119887(119879

3) ⩽ 2 and 119887

119875(119879

3) = 3 On the other hand 119887(119875

4) = 2

and 119887119875(119875

4) = 1 Thus we have what follows which is similar

to Corollary 113

Corollary 122 The bondage number and the paired bondagenumber are unrelated even for trees

A constructive characterization of trees with 119887(119879) = 2is given by Hartnell and Rall in [12] Raczek [73] provideda constructive characterization of trees with 119887

119875(119879) = 0 In

order to state the characterization we define a labeling andthree simple operations on a tree119879 Let 119910 isin 119881(119879) and let ℓ(119910)be the label assigned to 119910

Operation T1 If ℓ(119910) = 119861 add a vertex 119909 and the

edge 119910119909 and let ℓ(119909) = 119860OperationT

2 If ℓ(119910) = 119862 add a path (119909

1 119909

2) and the

edge 1199101199091 and let ℓ(119909

1) = 119861 ℓ(119909

2) = 119860

Operation T3 If ℓ(119910) = 119861 add a path (119909

1 119909

2 119909

3)

and the edge 1199101199091 and let ℓ(119909

1) = 119862 ℓ(119909

2) = 119861 and

ℓ(1199093) = 119860

Let 1198752= (119906 V) with ℓ(119906) = 119860 and ℓ(V) = 119861 Let T be

the class of all trees obtained from the labeled 1198752by a finite

sequence of operationsT1 T

2 T

3

A tree 119879 in Figure 8 belongs to the familyTRaczek [73] obtained the following characterization of all

trees 119879 with 119887119875(119879) = 0

119860

119860

119860

119860

119860

119861 119862

119861 119862 119861

119861119860

Figure 8 A tree 119879 belong to the familyT

Theorem 123 (Raczek [73] 2008) For a tree 119879 119887119875(119879) = 0 if

and only if 119879 is inT

We state the decision problem for the paired bondage asfollows

Problem 4 Consider the decision problem

Paired Bondage ProblemInstance a graph 119866 and a positive integer 119887Question is 119887

119875(119866) ⩽ 119887

Conjecture 124 The paired bondage problem is NP-complete

83 Restrained Bondage Numbers A dominating set 119878 of 119866 iscalled to be restrained if the subgraph induced by 119878 containsno isolated vertices where 119878 = 119881(119866) 119878 The restraineddomination number of 119866 denoted by 120574

119877(119866) is the smallest

cardinality of a restrained dominating set of 119866 It is clear that120574119877(119866) exists and 120574(119866) ⩽ 120574

119877(119866) for any nonempty graph 119866

The concept of restrained domination was introducedby Telle and Proskurowski [74] in 1997 albeit indirectly asa vertex partitioning problem Concerning the study of therestrained domination numbers the reader is referred to [74ndash79]

In 2008 Hattingh and Plummer [80] defined therestrained bondage number 119887R(119866) of a nonempty graph 119866 tobe the minimum cardinality among all subsets 119861 sube 119864(119866) forwhich 120574

119877(119866 minus 119861) gt 120574

119877(119866)

For some simple graphs their restrained dominationnumbers can be easily determined and so restrained bondagenumbers have been also determined For example it is clearthat 120574

119877(119870

119899) = 1 Domke et al [77] showed that 120574

119877(119862

119899) =

119899 minus 2lfloor1198993rfloor for 119899 ⩾ 3 and 120574119877(119875

119899) = 119899 minus 2lfloor(119899 minus 1)3rfloor for 119899 ⩾ 1

Using these results Hattingh and Plummer [80] obtained therestricted bondage numbers for119870

119899 119862

119899 119875

119899 and119866 = 119870

11989911198992119899119905

as follows

Theorem 125 (Hattingh and Plummer [80] 2008) For 119899 ⩾ 3

119887R (119870119899) =

1 119894119891 119899 = 3

lceil119899

2rceil 119900119905ℎ119890119903119908119894119904119890

119887R (119862119899) =

1 119894119891 119899 equiv 0 (mod 3) 2 119900119905ℎ119890119903119908119894119904119890

119887R (119875119899) = 1 119899 ⩾ 4

20 International Journal of Combinatorics

119887R (119866)

=

lceil119898

2rceil 119894119891 119899

119898= 1 119886119899119889 119899

119898+1⩾ 2 (1 ⩽ 119898 lt 119905)

2119905 minus 2 119894119891 1198991= 119899

2= sdot sdot sdot = 119899

119905= 2 (119905 ⩾ 2)

2 119894119891 1198991= 2 119886119899119889 119899

2⩾ 3 (119905 = 2)

119905minus1

sum

119894=1

119899119894minus 1 119900119905ℎ119890119903119908119894119904119890

(46)

Theorem 126 (Hattingh and Plummer [80] 2008) For a tree119879 of order 119899 (⩾ 4) 119879 ≇ 119870

1119899minus1if and only if 119887R(119879) = 1

Theorem 126 shows that the restrained bondage numberof a tree can be computed in constant time However thedecision problem for 119887R(119866) is NP-complete even for bipartitegraphs

Problem 5 Consider the decision problem

Restrained Bondage Problem

Instance a graph 119866 and a positive integer 119887

Question is 119887R(119866) ⩽ 119887

Theorem 127 (Hattingh and Plummer [80] 2008) Therestrained bondage problem is NP-complete even for bipartitegraphs

Consequently in view of its computational hardnessit is significative to establish some sharp bounds on therestrained bondage number of a graph in terms of othergraphic parameters

Theorem 128 (Hattingh and Plummer [80] 2008) For anygraph 119866 with 120575(119866) ⩾ 2

119887R (119866) ⩽ min119909119910isin119864(119866)

119889119866(119909) + 119889

119866(119910) minus 2 (47)

Corollary 129 119887R(119866) ⩽ Δ(119866)+120575(119866)minus2 for any graph119866 with120575(119866) ⩾ 2

Notice that the bounds stated in Theorem 128 andCorollary 129 are sharp Indeed the class of cycles whoseorders are congruent to 1 2 (mod 3) have a restrained bondagenumber achieving these bounds (see Theorem 125)

Theorem 128 is an analogue of Theorem 14 A quitenatural problem is whether or not for restricted bondagenumber there are analogues of Theorem 16 Theorem 18Theorem 29 and so on Indeed we should consider all resultsfor the bondage number whether or not there are analoguesfor restricted bondage number

Theorem 130 (Hattingh and Plummer [80] 2008) For anygraph 119866 with 120574

119877(119866) = 2

119887R (119866) ⩽ Δ (119866) + 1 (48)

We now consider the relation between 119887119877(119866) and 119887(119866)

Theorem 131 (Hattingh and Plummer [80] 2008) If 120574119877(119866) =

120574(119866) for some graph 119866 then 119887R(119866) ⩽ 119887(119866)

Corollary 129 and Theorem 131 provide a possibility toattack Conjecture 73 by characterizing planar graphs with120574119877= 120574 and 119887R = 119887 when 3 ⩽ Δ ⩽ 6However we do not have 119887R(119866) = 119887(119866) for any graph

119866 even if 120574119877(119866) = 120574(119866) Observe that 120574

119877(119870

3) = 120574(119870

3) yet

119887119877(119870

3) = 1 and 119887(119870

3) = 2 We still may not claim that

119887119877(119866) = 119887(119866) even in the case that every 120574-set is a 120574

119877-set

The example 1198703again demonstrates this

Theorem 132 (Hattingh and Plummer [80] 2008) There isan infinite class of graphs inwhich each graph119866 satisfies 119887(119866) lt119887R(119866)

In fact such an infinite class of graphs can be obtained asfollows Let119867 be a connected graph BC(119867) a graph obtainedfrom 119867 by attaching ℓ (⩾ 2) vertices of degree 1 to eachvertex in 119867 and B = 119866 119866 = BC(119867) for some graph 119867such that 120575(119867) ⩾ 2 By Theorem 3 we can easily check that119887(119866) = 1 lt 2 ⩽ min120575(119867) ℓ = 119887R(119866) for any 119866 isin BMoreover there exists a graph 119866 isin B such that 119887R(119866) canbe much larger than 119887(119866) which is stated as the followingtheorem

Theorem 133 (Hattingh and Plummer [80] 2008) For eachpositive integer 119896 there is a graph119866 such that 119896 = 119887R(119866)minus119887(119866)

Combining Theorem 133 with Theorem 132 we have thefollowing corollary

Corollary 134 The bondage number and the restrainedbondage number are unrelated

Very recently Jafari Rad et al [81] have consideredthe total restrained bondage in graphs based on both totaldominating sets and restrained dominating sets and obtainedseveral properties exact values and bounds for the totalrestrained bondage number of a graph

A restrained dominating set 119878 of a graph 119866 without iso-lated vertices is called to be a total restrained dominating setif 119878 is a total dominating set The total restrained dominationnumber of119866 denoted by 120574TR (119866) is theminimum cardinalityof a total restrained dominating set of 119866 For referenceson this domination in graphs see [3 61 82ndash85] The totalrestrained bondage number 119887TR (119866) of a graph 119866 with noisolated vertex is the cardinality of a smallest set of edges 119861 sube119864(119866) for which119866minus119861 has no isolated vertex and 120574TR (119866minus119861) gt120574TR (119866) In the case that there is no such subset 119861 we define119887TR (119866) = infin

Ma et al [84] determined that 120574TR (119875119899) = 119899minus2lfloor(119899minus2)4rfloorfor 119899 ⩾ 3 120574TR (119862119899

) = 119899 minus 2lfloor1198994rfloor for 119899 ⩾ 3 120574TR (1198703) =

3 and 120574TR (119870119899) = 2 for 119899 = 3 and 120574TR (1198701119899

) = 1 + 119899

and 120574TR (119870119898119899) = 2 for 2 ⩽ 119898 ⩽ 119899 According to these

results Jafari Rad et al [81] determined the exact valuesof total restrained bondage numbers for the correspondinggraphs

International Journal of Combinatorics 21

Theorem 135 (Jafari Rad et al [81]) For an integer 119899

119887TR (119875119899) = infin 119894119891 2 ⩽ 119899 ⩽ 5

1 119894119891 119899 ⩾ 5

119887TR (119862119899) =

infin 119894119891 119899 = 3

1 119894119891 3 lt 119899 119899 equiv 0 1 (mod 4)2 119894119891 3 lt 119899 119899 equiv 2 3 (mod 4)

119887TR (119882119899) = 2 119891119900119903 119899 ⩾ 6

119887TR (119870119899) = 119899 minus 1 119891119900119903 119899 ⩾ 4

119887TR (119870119898119899) = 119898 minus 1 119891119900119903 2 ⩽ 119898 ⩽ 119899

(49)

119887TR (119870119899) = 119899minus1 shows the fact that for any integer 119899 ⩾ 3 there

exists a connected graph namely119870119899+1

such that 119887TR (119870119899+1) =

119899 that is the total restrained bondage number is unboundedfrom the above

A vertex is said to be a support vertex if it is adjacent toa vertex of degree one Let A be the family of all connectedgraphs such that119866 belongs toA if and only if every edge of119866is incident to a support vertex or 119866 is a cycle of length three

Proposition 136 (Cyman and Raczek [82] 2006) For aconnected graph119866 of order 119899 120574TR (119866) = 119899 if and only if119866 isin A

Hence if 119866 isin A then the removal of any subset of edgesof119866 cannot increase the total restrained domination numberof 119866 and thus 119887TR (119866) = infin When 119866 notin A a result similar toTheorem 6 is obtained

Theorem 137 (Jafari Rad et al [81]) For any tree 119879 notin A119887TR (119879) ⩽ 2 This bound is sharp

In the samepaper the authors provided a characterizationfor trees 119879 notin A with 119887TR (119879) = 1 or 2 The following result issimilar to Observation 1

Observation 3 (Jafari Rad et al [81]) If 119896 edges can beremoved from a graph 119866 to obtain a graph 119867 without anisolated vertex and with 119887TR (119867) = 119905 then 119887TR (119866) ⩽ 119896 + 119905

ApplyingTheorem 137 andObservation 3 Jafari Rad et alcharacterized all connected graphs 119866 with 119887TR (119866) = infin

Theorem 138 (Jafari Rad et al [81]) For a connected graph119866119887TR (119866) = infin if and only if 119866 isin A

84 Other Conditional Bondage Numbers A subset 119868 sube 119881(119866)is called an independent set if no two vertices in 119868 are adjacentin 119866 The maximum cardinality among all independent setsis called the independence number of 119866 denoted by 120572(119866)

A dominating set 119878 of a graph 119866 is called to be inde-pendent if 119878 is an independent set of 119866 The minimumcardinality among all independent dominating set is calledthe independence domination number of 119866 and denoted by120574119868(119866)

Since an independent dominating set is not only adominating set but also an independent set 120574(119866) ⩽ 120574

119868(119866) ⩽

120572(119866) for any graph 119866It is clear that a maximal independent set is certainly a

dominating set Thus an independent set is maximal if andonly if it is an independent dominating set and so 120574

119868(119866) is the

minimum cardinality among all maximal independent setsof 119866 This graph-theoretical invariant has been well studiedin the literature see for example Haynes et al [3] for earlyresults Goddard and Henning [86] for recent results onindependent domination in graphs

In 2003 Zhang et al [87] defined the independencebondage number 119887

119868(119866) of a nonempty graph 119866 to be the

minimum cardinality among all subsets 119861 sube 119864(119866) for which120574119868(119866 minus 119861) gt 120574

119868(119866) For some ordinary graphs their indepen-

dence domination numbers can be easily determined and soindependence bondage numbers have been also determinedClearly 119887

119868(119870

119899) = 1 if 119899 ⩾ 2

Theorem 139 (Zhang et al [87] 2003) For any integer 119899 ⩾ 4

119887119868(119862

119899) =

1 119894119891 119899 equiv 1 (mod 2) 2 119894119891 119899 equiv 0 (mod 2)

119887119868(119875

119899) =

1 119894119891 119899 equiv 0 (mod 2) 2 119894119891 119899 equiv 1 (mod 2)

119887119868(119870

119898119899) = 119899 119891119900119903 119898 ⩽ 119899

(50)

Apart from the above-mentioned results as far as we findno other results on the independence bondage number in thepresent literature we never knew much of any result on thisparameter for a tree

A subset 119878 of 119881(119866) is called an equitable dominatingset if for every 119910 isin 119881(119866 minus 119878) there exists a vertex 119909 isin 119878such that 119909119910 isin 119864(119866) and |119889

119866(119909) minus 119889

119866(119910)| ⩽ 1 proposed

by Dharmalingam [88] The minimum cardinality of such adominating set is called the equitable domination number andis denoted by 120574

119864(119866) Deepak et al [89] defined the equitable

bondage number 119887119864(119866) of a graph 119866 to be the cardinality of a

smallest set 119861 sube 119864(119866) of edges for which 120574119864(119866 minus 119861) gt 120574

119864(119866)

and determined 119887119864(119866) for some special graphs

Theorem 140 (Deepak et al [89] 2011) For any integer 119899 ⩾ 2

119887119864(119870

119899) = lceil

119899

2rceil

119887119864(119870

119898119899) = 119898 119891119900119903 0 ⩽ 119899 minus 119898 ⩽ 1

119887119864(119862

119899) =

3 119894119891 119899 equiv 1 (mod 3)2 119900119905ℎ119890119903119908119894119904119890

119887119864(119875

119899) =

2 119894119891 119899 equiv 1 (mod 3)1 119900119905ℎ119890119903119908119894119904119890

119887119864(119879) ⩽ 2 119891119900119903 119886119899119910 119899119900119899119905119903119894119907119894119886119897 119905119903119890119890 119879

119887119864(119866) ⩽ 119899 minus 1 119891119900119903 119886119899119910 119888119900119899119899119890119888119905119890119889 119892119903119886119901ℎ 119866 119900119891 119900119903119889119890119903 119899

(51)

22 International Journal of Combinatorics

As far as we know there are no more results on theequitable bondage number of graphs in the present literature

There are other variations of the normal domination byadding restricted conditions to the normal dominating setFor example the connected domination set of a graph119866 pro-posed by Sampathkumar and Walikar [90] is a dominatingset 119878 such that the subgraph induced by 119878 is connected Theproblem of finding a minimum connected domination set ina graph is equivalent to the problemof finding a spanning treewithmaximumnumber of vertices of degree one [91] and hassome important applications in wireless networks [92] thusit has received much research attention However for suchan important domination there is no result on its bondagenumber We believe that the topic is of significance for futureresearch

9 Generalized Bondage Numbers

There are various generalizations of the classical dominationsuch as 119901-domination distance domination fractional dom-ination Roman domination and rainbow domination Everysuch a generalization can lead to a corresponding bondageIn this section we introduce some of them

91 119901-Bondage Numbers In 1985 Fink and Jacobson [93]introduced the concept of 119901-domination Let 119901 be a positiveinteger A subset 119878 of 119881(119866) is a 119901-dominating set of 119866 if|119878 cap 119873

119866(119910)| ⩾ 119901 for every 119910 isin 119878 The 119901-domination number

120574119901(119866) is the minimum cardinality among all 119901-dominating

sets of 119866 Any 119901-dominating set of 119866 with cardinality 120574119901(119866)

is called a 120574119901-set of 119866 Note that a 120574

1-set is a classic minimum

dominating set Notice that every graph has a 119901-dominatingset since the vertex-set119881(119866) is such a setWe also note that a 1-dominating set is a dominating set and so 120574(119866) = 120574

1(119866) The

119901-domination number has received much research attentionsee a state-of-the-art survey article by Chellali et al [94]

It is clear from definition that every 119901-dominating setof a graph certainly contains all vertices of degree at most119901 minus 1 By this simple observation to avoid the trivial caseoccurrence we always assume that Δ(119866) ⩾ 119901 For 119901 ⩾ 2 Luet al [95] gave a constructive characterization of trees withunique minimum 119901-dominating sets

Recently Lu and Xu [96] have introduced the conceptto the 119901-bondage number of 119866 denoted by 119887

119901(119866) as the

minimum cardinality among all sets of edges 119861 sube 119864(119866) suchthat 120574

119901(119866 minus 119861) gt 120574

119901(119866) Clearly 119887

1(119866) = 119887(119866)

Lu and Xu [96] established a lower bound and an upperbound on 119887

119901(119879) for any integer 119901 ⩾ 2 and any tree 119879 with

Δ(119879) ⩾ 119901 and characterized all trees achieving the lowerbound and the upper bound respectively

Theorem 141 (Lu and Xu [96] 2011) For any integer 119901 ⩾ 2and any tree 119879 with Δ(119879) ⩾ 119901

1 ⩽ 119887119901(119879) ⩽ Δ (119879) minus 119901 + 1 (52)

Let 119878 be a given subset of 119881(119866) and for any 119909 isin 119878 let

119873119901(119909 119878 119866) = 119910 isin 119878 cap 119873

119866(119909)

1003816100381610038161003816119873119866(119910) cap 119878

1003816100381610038161003816 = 119901

(53)

Then all trees achieving the lower bound 1 can be character-ized as follows

Theorem 142 (Lu and Xu [96] 2011) Let 119879 be a tree withΔ(119879) ⩾ 119901 ⩾ 2 Then 119887

119901(119879) = 1 if and only if for any 120574

119901-set

119878 of 119879 there exists an edge 119909119910 isin (119878 119878) such that 119910 isin 119873119901(119909 119878 119879)

The notation 119878(119886 119887) denotes a double star obtained byadding an edge between the central vertices of two stars119870

1119886minus1

and1198701119887minus1

And the vertex with degree 119886 (resp 119887) in 119878(119886 119887) iscalled the 119871-central vertex (resp 119877-central vertex) of 119878(119886 119887)

To characterize all trees attaining the upper bound givenin Theorem 141 we define three types of operations on a tree119879 with Δ(119879) = Δ ⩾ 119901 + 1

Type 1 attach a pendant edge to a vertex 119910with 119889119879(119910) ⩽ 119901minus

2 in 119879

Type 2 attach a star 1198701Δminus1

to a vertex 119910 of 119879 by joining itscentral vertex to 119910 where 119910 in a 120574

119901-set of 119879 and

119889119879(119910) ⩽ Δ minus 1

Type 3 attach a double star 119878(119901 Δ minus 1) to a pendant vertex 119910of 119879 by coinciding its 119877-central vertex with 119910 wherethe unique neighbor of 119910 is in a 120574

119901-set of 119879

Let B = 119879 a tree obtained from 1198701Δ

or 119878(119901 Δ) by afinite sequence of operations of Types 1 2 and 3

Theorem 143 (Lu and Xu [96] 2011) A tree with the maxi-mum Δ ⩾ 119901 + 1 has 119901-bondage number Δ minus 119901 + 1 if and onlyif it belongs toB

Theorem 144 (Lu and Xu [97] 2012) Let 119866119898119899= 119875

119898times 119875

119899

Then

1198872(119866

2119899) = 1 119891119900119903 119899 ⩾ 2

1198872(119866

3119899) =

2 119894119891 119899 equiv 1 (mod 3) 1 119900119905ℎ119890119903119908119894119904119890

for 119899 ⩾ 2

1198872(119866

4119899) =

1 119894119891 119899 equiv 3 (mod 4) 2 119900119905ℎ119890119903119908119894119904119890

for 119899 ⩾ 7

(54)

Recently Krzywkowski [98] has proposed the conceptof the nonisolating 2-bondage of a graph The nonisolating2-bondage number of a graph 119866 denoted by 1198871015840

2(119866) is the

minimum cardinality among all sets of edges 119861 sube 119864(119866) suchthat 119866 minus 119861 has no isolated vertices and 120574

2(119866 minus 119861) gt 120574

2(119866) If

for every 119861 sube 119864(119866) either 1205742(119866 minus 119861) = 120574

2(119866) or 119866 minus 119861 has

isolated vertices then define 11988710158402(119866) = 0 and say that 119866 is a 2-

nonisolatedly strongly stable graph Krzywkowski presentedsome basic properties of nonisolating 2-bondage in graphsshowed that for every nonnegative integer 119896 there exists atree 119879 such 1198871015840

2(119879) = 119896 characterized all 2-non-isolatedly

strongly stable trees and determined 11988710158402(119866) for several special

graphs

International Journal of Combinatorics 23

Theorem 145 (Krzywkowski [98] 2012) For any positiveintegers 119899 and119898 with119898 ⩽ 119899

1198871015840

2(119870

119899) =

0 119894119891 119899 = 1 2 3

lfloor2119899

3rfloor 119900119905ℎ119890119903119908119894119904119890

1198871015840

2(119875

119899) =

0 119894119891 119899 = 1 2 3

1 119900119905ℎ119890119903119908119894119904119890

1198871015840

2(119862

119899) =

0 119894119891 119899 = 3

1 119894119891 119899 119894119904 119890V119890119899

2 119894119891 119899 119894119904 119900119889119889

1198871015840

2(119870

119898119899) =

3 119894119891 119898 = 119899 = 3

1 119894119891 119898 = 119899 = 4

2 119900119905ℎ119890119903119908119894119904119890

(55)

92 Distance Bondage Numbers A subset 119878 of vertices of agraph119866 is said to be a distance 119896-dominating set for119866 if everyvertex in 119866 not in 119878 is at distance at most 119896 from some vertexof 119878 The minimum cardinality of all distance 119896-dominatingsets is called the distance 119896-domination number of 119866 anddenoted by 120574

119896(119866) (do not confuse with the above-mentioned

120574119901(119866)) When 119896 = 1 a distance 1-dominating set is a normal

dominating set and so 1205741(119866) = 120574(119866) for any graph 119866 Thus

the distance 119896-domination is a generalization of the classicaldomination

The relation between 120574119896and 120572

119896(the 119896-independence

number defined in Section 22) for a tree obtained by Meirand Moon [14] who proved that 120574

119896(119879) = 120572

2119896(119879) for any tree

119879 (a special case for 119896 = 1 is stated in Proposition 10) Thefurther research results can be found in Henning et al [99]Tian and Xu [100ndash103] and Liu et al [104]

In 1998 Hartnell et al [15] defined the distance 119896-bondagenumber of 119866 denoted by 119887

119896(119866) to be the cardinality of a

smallest subset 119861 of edges of 119866 with the property that 120574119896(119866 minus

119861) gt 120574119896(119866) FromTheorem 6 it is clear that if119879 is a nontrivial

tree then 1 ⩽ 1198871(119879) ⩽ 2 Hartnell et al [15] generalized this

result to any integer 119896 ⩾ 1

Theorem 146 (Hartnell et al [15] 1998) For every nontrivialtree 119879 and positive integer 119896 1 ⩽ 119887

119896(119879) ⩽ 2

Hartnell et al [15] and Topp and Vestergaard [13] alsocharacterized the trees having distance 119896-bondage number 2In particular the class of trees for which 119887

1(119879) = 2 are just

those which have a unique maximum 2-independent set (seeTheorem 11)

Since when 119896 = 1 the distance 1-bondage number 1198871(119866)

is the classical bondage number 119887(119866) Theorem 12 gives theNP-hardness of deciding the distance 119896-bondage number ofgeneral graphs

Theorem 147 Given a nonempty undirected graph 119866 andpositive integers 119896 and 119887 with 119887 ⩽ 120576(119866) determining wetheror not 119887

119896(119866) ⩽ 119887 is NP-hard

For a vertex 119909 in119866 the open 119896-neighborhood119873119896(119909) of 119909 is

defined as119873119896(119909) = 119910 isin 119881(119866) 1 ⩽ 119889

119866(119909 119910) le 119896The closed

119896-neighborhood119873119896[119909] of 119909 in 119866 is defined as119873

119896(119909) cup 119909 Let

Δ119896(119866) = max 1003816100381610038161003816119873119896

(119909)1003816100381610038161003816 for any 119909 isin 119881 (119866) (56)

Clearly Δ1(119866) = Δ(119866) The 119896th power of a graph 119866 is the

graph119866119896 with vertex-set119881(119866119896

) = 119881(119866) and edge-set119864(119866119896

) =

119909119910 1 ⩽ 119889119866(119909 119910) ⩽ 119896 The following lemma holds directly

from the definition of 119866119896

Proposition 148 (Tian and Xu [101]) Δ(119866119896

) = Δ119896(119866) and

120574(119866119896

) = 120574119896(119866) for any graph 119866 and each 119896 ⩾ 1

A graph 119866 is 119896-distance domination-critical or 120574119896-critical

for short if 120574119896(119866 minus 119909) lt 120574

119896(119866) for every vertex 119909 in 119866

proposed by Henning et al [105]

Proposition 149 (Tian and Xu [101]) For each 119896 ⩾ 1 a graph119866 is 120574

119896-critical if and only if 119866119896 is 120574

119896-critical

By the above facts we suggest the following worthwhileproblem

Problem 6 Can we generalize the results on the bondage for119866 to 119866119896 In particular do the following propositions hold

(a) 119887(119866119896

) = 119887119896(119866) for any graph 119866 and each 119896 ⩾ 1

(b) 119887(119866119896

) ⩽ Δ119896(119866) if 119866 is not 120574

119896-critical

Let 119896 and 119901 be positive integers A subset 119878 of 119881(119866) isdefined to be a (119896 119901)-dominating set of 119866 if for any vertex119909 isin 119878 |119873

119896(119909) cap 119878| ⩾ 119901 The (119896 119901)-domination number of

119866 denoted by 120574119896119901(119866) is the minimum cardinality among

all (119896 119901)-dominating sets of 119866 Clearly for a graph 119866 a(1 1)-dominating set is a classical dominating set a (119896 1)-dominating set is a distance 119896-dominating set and a (1 119901)-dominating set is the above-mentioned 119901-dominating setThat is 120574

11(119866) = 120574(119866) 120574

1198961(119866) = 120574

119896(119866) and 120574

1119901(119866) = 120574

119901(119866)

The concept of (119896 119901)-domination in a graph 119866 is a gen-eralized domination which combines distance 119896-dominationand 119901-domination in 119866 So the investigation of (119896 119901)-domination of 119866 is more interesting and has received theattention of many researchers see for example [106ndash109]

More general Li and Xu [110] proposed the concept of the(ℓ 119908)-domination number of a119908-connected graph A subset119878 of 119881(119866) is defined to be an (ℓ 119908)-dominating set of 119866 iffor any vertex 119909 isin 119878 there are 119908 internally disjoint pathsof length at most ℓ from 119909 to some vertex in 119878 The (ℓ 119908)-domination number of119866 denoted by 120574

ℓ119908(119866) is theminimum

cardinality among all (ℓ 119908)-dominating sets of 119866 Clearly1205741198961(119866) = 120574

119896(119866) and 120574

1119901(119866) = 120574

119901(119866)

It is quite natural to propose the concept of bondage num-ber for (119896 119901)-domination or (ℓ 119908)-domination However asfar as we know none has proposed this concept until todayThis is a worthwhile topic for further research

24 International Journal of Combinatorics

93 Fractional Bondage Numbers If 120590 is a function mappingthe vertex-set 119881 into some set of real numbers then for anysubset 119878 sube 119881 let 120590(119878) = sum

119909isin119878120590(119909) Also let |120590| = 120590(119881)

A real-value function 120590 119881 rarr [0 1] is a dominatingfunction of a graph 119866 if for every 119909 isin 119881(119866) 120590(119873

119866[119909]) ⩾ 1

Thus if 119878 is a dominating set of 119866 120590 is a function where

120590 (119909) = 1 if 119909 isin 1198780 otherwise

(57)

then 120590 is a dominating function of119866 The fractional domina-tion number of 119866 denoted by 120574

119865(119866) is defined as follows

120574119865(119866) = min |120590| 120590 is a domination function of 119866

(58)

The 120574119865-bondage number of 119866 denoted by 119887

119865(119866) is

defined as the minimum cardinality of a subset 119861 sube 119864 whoseremoval results in 120574

119865(119866 minus 119861) gt 120574

119865(119866)

Hedetniemi et al [111] are the first to study fractionaldomination although Farber [112] introduced the idea indi-rectly The concept of the fractional domination numberwas proposed by Domke and Laskar [113] in 1997 Thefractional domination numbers for some ordinary graphs aredetermined

Proposition 150 (Domke et al [114] 1998 Domke and Laskar[113] 1997) (a) If 119866 is a 119896-regular graph with order 119899 then120574119865(119866) = 119899119896 + 1(b) 120574

119865(119862

119899) = 1198993 120574

119865(119875

119899) = lceil1198993rceil 120574

119865(119870

119899) = 1

(c) 120574119865(119866) = 1 if and only if Δ(119866) = 119899 minus 1

(d) 120574119865(119879) = 120574(119879) for any tree 119879

(e) 120574119865(119870

119899119898) = (119899(119898 + 1) + 119898(119899 + 1))(119899119898 minus 1) where

max119899119898 gt 1

The assertions (a)ndash(d) are due to Domke et al [114] andthe assertion (e) is due to Domke and Laskar [113] Accordingto these results Domke and Laskar [113] determined the 120574

119865-

bondage numbers for these graphs

Theorem 151 (Domke and Laskar [113] 1997) 119887119865(119870

119899) =

lceil1198992rceil 119887119865(119870

119899119898) = 1 wheremax119899119898 gt 1

119887119865(119862

119899) =

2 119894119891 119899 equiv 0 (mod 3) 1 119900119905ℎ119890119903119908119894119904119890

119887119865(119875

119899) =

2 119894119891 119899 equiv 1 (mod 3) 1 119900119905ℎ119890119903119908119894119904119890

(59)

It is easy to see that for any tree119879 119887119865(119879) ⩽ 2 In fact since

120574119865(119879) = 120574(119879) for any tree 119879 120574

119865(119879

1015840

) = 120574(1198791015840

) for any subgraph1198791015840 of 119879 It follows that

119887119865(119879) = min 119861 sube 119864 (119879) 120574

119865(119879 minus 119861) gt 120574

119865(119879)

= min 119861 sube 119864 (119879) 120574 (119879 minus 119861) gt 120574 (119879)

= 119887 (119879) ⩽ 2

(60)

Except for these no results on 119887119865(119866) have been known as

yet

94 Roman Bondage Numbers A Roman dominating func-tion on a graph 119866 is a labeling 119891 119881 rarr 0 1 2 such thatevery vertex with label 0 has at least one neighbor with label2 The weight of a Roman dominating function 119891 on 119866 is thevalue

119891 (119866) = sum

119906isin119881(119866)

119891 (119906) (61)

The minimum weight of a Roman dominating function ona graph 119866 is called the Roman domination number of 119866denoted by 120574RM (119866)

A Roman dominating function 119891 119881 rarr 0 1 2

can be represented by the ordered partition (1198810 119881

1 119881

2) (or

(119881119891

0 119881

119891

1 119881

119891

2) to refer to119891) of119881 where119881

119894= V isin 119881 | 119891(V) = 119894

In this representation its weight 119891(119866) = |1198811| + 2|119881

2| It is

clear that119881119891

1cup119881

119891

2is a dominating set of 119866 called the Roman

dominating set denoted by 119863119891

R = (1198811 1198812) Since 119881119891

1cup 119881

119891

2is

a dominating set when 119891 is a Roman dominating functionand since placing weight 2 at the vertices of a dominating setyields a Roman dominating function in [115] it is observedthat

120574 (119866) ⩽ 120574RM (119866) ⩽ 2120574 (119866) (62)

A graph 119866 is called to be Roman if 120574RM (119866) = 2120574(119866)The definition of the Roman domination function is

proposed implicitly by Stewart [116] and ReVelle and Rosing[117] Roman dominating numbers have been deeply studiedIn particular Bahremandpour et al [118] showed that theproblem determining the Roman domination number is NP-complete even for bipartite graphs

Let 119866 be a graph with maximum degree at least twoThe Roman bondage number 119887RM (119866) of 119866 is the minimumcardinality of all sets 1198641015840 sube 119864 for which 120574RM (119866 minus 119864

1015840

) gt

120574RM (119866) Since in study of Roman bondage number theassumption Δ(119866) ⩾ 2 is necessary we always assume thatwhen we discuss 119887RM (119866) all graphs involved satisfy Δ(119866) ⩾2 The Roman bondage number 119887RM (119866) is introduced byJafari Rad and Volkmann in [119]

Recently Bahremandpour et al [118] have shown that theproblem determining the Roman bondage number is NP-hard even for bipartite graphs

Problem 7 Consider the decision problem

Roman Bondage Problem

Instance a nonempty bipartite graph119866 and a positiveinteger 119896

Question is 119887RM (119866) ⩽ 119896

Theorem 152 (Bahremandpour et al [118] 2013) TheRomanbondage problem is NP-hard even for bipartite graphs

The exact value of 119887RM(119866) is known only for a few familyof graphs including the complete graphs cycles and paths

International Journal of Combinatorics 25

Theorem 153 (Jafari Rad and Volkmann [119] 2011) For anypositive integers 119899 and119898 with119898 ⩽ 119899

119887RM (119875119899) = 2 119894119891 119899 equiv 2 (mod 3) 1 119900119905ℎ119890119903119908119894119904119890

119887RM (119862119899) =

3 119894119891 119899 = 2 (mod 3) 2 119900119905ℎ119890119903119908119894119904119890

119887RM (119870119899) = lceil

119899

2rceil 119891119900119903 119899 ⩾ 3

119887RM (119870119898119899) =

1 119894119891 119898 = 1 119886119899119889 119899 = 1

4 119894119891 119898 = 119899 = 3

119898 119900119905ℎ119890119903119908119894119904119890

(63)

Lemma 154 (Cockayne et al [115] 2004) If 119866 is a graph oforder 119899 and contains vertices of degree 119899 minus 1 then 120574RM(119866) = 2

Using Lemma 154 the third conclusion in Theorem 153can be generalized to more general case which is similar toLemma 49

Theorem 155 (Jafari Rad and Volkmann [119] 2011) Let119866 bea graph with order 119899 ge 3 and 119905 the number of vertices of degree119899 minus 1 in 119866 If 119905 ⩾ 1 then 119887RM(119866) = lceil1199052rceil

Ebadi and PushpaLatha [120] conjectured that 119887RM(119866) ⩽119899 minus 1 for any graph 119866 of order 119899 ⩾ 3 Dehgardi et al[121] Akbari andQajar [122] independently showed that thisconjecture is true

Theorem 156 119887RM(119866) ⩽ 119899 minus 1 for any connected graph 119866 oforder 119899 ⩾ 3

For a complete 119905-partite graph we have the followingresult

Theorem 157 (Hu and Xu [123] 2011) Let 119866 = 11987011989811198982119898119905

bea complete 119905-partite graph with 119898

1= sdot sdot sdot = 119898

119894lt 119898

119894+1le sdot sdot sdot ⩽

119898119905 119905 ⩾ 2 and 119899 = sum119905

119895=1119898

119895 Then

119887RM (119866) =

lceil119894

2rceil 119894119891 119898

119894= 1 119886119899119889 119899 ⩾ 3

2 119894119891 119898119894= 2 119886119899119889 119894 = 1

119894 119894119891 119898119894= 2 119886119899119889 119894 ⩾ 2

4 119894119891 119898119894= 3 119886119899119889 119894 = 119905 = 2

119899 minus 1 119894119891 119898119894= 3 119886119899119889 119894 = 119905 ⩾ 3

119899 minus 119898119905

119894119891 119898119894⩾ 3 119886119899119889 119898

119905⩾ 4

(64)

Consider a complete 119905-partite graph 119870119905(3) for 119905 ⩾ 3

119887RM(119870119905(3)) = 119899 minus 1 by Theorem 157 where 119899 = 3119905

which shows that this upper bound of 119899 minus 1 on 119887RM(119866) inTheorem 156 is sharp This fact and 120574

119877(119870

119905(3)) = 4 give

a negative answer to the following two problems posed byDehgardi et al [121]Prove or Disprove for any connected graph 119866 of order 119899 ge 3(i) 119887RM(119866) = 119899 minus 1 if and only if 119866 cong 119870

3 (ii) 119887RM(119866) ⩽ 119899 minus

120574119877(119866) + 1Note that a complete 119905-partite graph 119870

119905(3) is (119899 minus 3)-

regular and 119887RM(119870119905(3)) = 119899 minus 1 for 119905 ⩾ 3 where 119899 = 3119905

Hu and Xu further determined that 119887119877(119866) = 119899 minus 2 for any

(119899 minus 3)-regular graph 119866 of order 119899 ⩽ 5 except 119870119905(3)

Theorem 158 (Hu and Xu [123] 2011) For any (119899minus3)-regulargraph 119866 of order 119899 other than119870

119905(3) 119887RM(119866) = 119899 minus 2 for 119899 ⩾ 5

For a tree 119879 with order 119899 ⩾ 3 Ebadi and PushpaLatha[120] and Jafari Rad and Volkmann [119] independentlyobtained an upper bound on 119887RM(119879)

Theorem 159 119887RM(119879) ⩽ 3 for any tree 119879 with order 119899 ⩾ 3

Theorem 160 (Bahremandpour et al [118] 2013) 119887RM(1198752 times119875119899) = 2 for 119899 ⩾ 2

Theorem 161 (Jafari Rad and Volkmann [119] 2011) Let119866 bea graph with order at least three

(a) If (119909 119910 119911) is a path of length 2 in 119866 then 119887RM(119866) ⩽119889119866(119909) + 119889

119866(119910) + 119889

119866(119911) minus 3 minus |119873

119866(119909) cap 119873

119866(119910)|

(b) If 119866 is connected then 119887RM(119866) ⩽ 120582(119866) + 2Δ(119866) minus 3

Theorem 161 (b) implies 119887RM(119866) ⩽ 120575(119866) + 2Δ(119866) minus 3 for aconnected graph 119866 Note that for a planar graph 119866 120575(119866) ⩽ 5moreover 120575(119866) ⩽ 3 if the girth at least 4 and 120575(119866) ⩽ 2 if thegirth at least 6 These two facts show that 119887RM(119866) ⩽ 2Δ(119866) +2 for a connected planar graph 119866 Jafari Rad and Volkmann[124] improved this bound

Theorem 162 (Jafari Rad and Volkmann [124] 2011) For anyconnected planar graph 119866 of order 119899 ⩾ 3 with girth 119892(119866)

119887RM (119866)

⩽

2Δ (119866)

Δ (119866) + 6

Δ (119866) + 5 119894119891 119866 119888119900119899119905119886119894119899119904 119899119900 V119890119903119905119894119888119890119904 119900119891 119889119890119892119903119890119890 119891119894V119890

Δ (119866) + 4 119894119891 119892 (119866) ⩾ 4

Δ (119866) + 3 119894119891 119892 (119866) ⩾ 5

Δ (119866) + 2 119894119891 119892 (119866) ⩾ 6

Δ (119866) + 1 119894119891 119892 (119866) ⩾ 8

(65)

According toTheorem 157 119887RM(119862119899) = 3 = Δ(119862

119899)+1 for a

cycle 119862119899of length 119899 ⩾ 8 with 119899 equiv 2 (mod 3) and therefore

the last upper bound inTheorem 162 is sharp at least for Δ =2

Combining the fact that every planar graph 119866 withminimum degree 5 contains an edge 119909119910 with 119889

119866(119909) = 5

and 119889119866(119910) isin 5 6 with Theorem 161 (a) Akbari et al [125]

obtained the following result

26 International Journal of Combinatorics

middot middot middot

middot middot middot

middot middot middot

middot middot middot

119866

Figure 9 The graph 119866 is constructed from 119866

Theorem 163 (Akbari et al [125] 2012) 119887RM(119866) ⩽ 15 forevery planar graph 119866

It remains open to show whether the bound inTheorem 163 is sharp or not Though finding a planargraph 119866 with 119887RM(119866) = 15 seems to be difficult Akbari etal [125] constructed an infinite family of planar graphs withRoman bondage number equal to 7 by proving the followingresult

Theorem 164 (Akbari et al [125] 2012) Let 119866 be a graph oforder 119899 and 119866 is the graph of order 5119899 obtained from 119866 byattaching the central vertex of a copy of a path 119875

5to each vertex

of119866 (see Figure 9) Then 120574RM(119866) = 4119899 and 119887RM(119866) = 120575(119866)+2

ByTheorem 164 infinitemany planar graphswith Romanbondage number 7 can be obtained by considering any planargraph 119866 with 120575(119866) = 5 (eg the icosahedron graph)

Conjecture 165 (Akbari et al [125] 2012) The Romanbondage number of every planar graph is at most 7

For general bounds the following observation is directlyobtained which is similar to Observation 1

Observation 4 Let 119866 be a graph of order 119899 with maximumdegree at least two Assume that 119867 is a spanning subgraphobtained from 119866 by removing 119896 edges If 120574RM(119867) = 120574RM(119866)then 119887

119877(119867) ⩽ 119887RM(119866) ⩽ 119887RM(119867) + 119896

Theorem 166 (Bahremandpour et al [118] 2013) For anyconnected graph 119866 of order 119899 if 119899 ⩾ 3 and 120574RM(119866) = 120574(119866) + 1then

119887RM (119866) ⩽ min 119887 (119866) 119899Δ (66)

where 119899Δis the number of vertices with maximum degree Δ in

119866

Observation 5 For any graph 119866 if 120574(119866) = 120574RM(119866) then119887RM(119866) ⩽ 119887(119866)

Theorem 167 (Bahremandpour et al [118] 2013) For everyRoman graph 119866

119887RM (119866) ⩾ 119887 (119866) (67)

The bound is sharp for cycles on 119899 vertices where 119899 equiv 0 (mod3)

The strict inequality in Theorem 167 can hold for exam-ple 119887(119862

3119896+2) = 2 lt 3 = 119887RM(1198623119896+2

) byTheorem 157A graph 119866 is called to be vertex Roman domination-

critical if 120574RM(119866 minus 119909) lt 120574RM(119866) for every vertex 119909 in 119866 If119866 has no isolated vertices then 120574RM(119866) ⩽ 2120574(119866) ⩽ 2120573(119866)If 120574RM(119866) = 2120573(119866) then 120574RM(119866) = 2120574(119866) and hence 119866 is aRoman graph In [30] Volkmann gave a lot of graphs with120574(119866) = 120573(119866) The following result is similar to Theorem 57

Theorem 168 (Bahremandpour et al [118] 2013) For anygraph 119866 if 120574RM(119866) = 2120573(119866) then

(a) 119887RM(119866) ⩾ 120575(119866)

(b) 119887RM(119866) ⩾ 120575(119866)+1 if119866 is a vertex Roman domination-critical graph

If 120574RM(119866) = 2 then obviously 119887RM(119866) ⩽ 120575(119866) So weassume 120574RM(119866) ⩾ 3 Dehgardi et al [121] proved 119887RM(119866) ⩽(120574RM(119866) minus 2)Δ(119866) + 1 and posed the following problem if 119866is a connected graph of order 119899 ⩾ 4 with 120574RM(119866) ⩾ 3 then

119887RM (119866) ⩽ (120574RM (119866) minus 2) Δ (119866) (68)

Theorem 161 (a) shows that the inequality (68) holds if120574RM(119866) ⩾ 5 Thus the bound in (68) is of interest only when120574RM(119866) is 3 or 4

Lemma 169 (Bahremandpour et al [118] 2013) For anynonempty graph 119866 of order 119899 ⩾ 3 120574RM(119866) = 3 if and onlyif Δ(119866) = 119899 minus 2

The following result shows that (68) holds for all graphs119866 of order 119899 ⩾ 4 with 120574RM(119866) = 3 or 4 which improvesTheorem 161 (b)

Theorem 170 (Bahremandpour et al [118] 2013) For anyconnected graph 119866 of order 119899 ⩾ 4

119887RM (119866) le Δ (119866) = 119899 minus 2 119894119891 120574RM (119866) = 3

Δ (119866) + 120575 (119866) minus 1 119894119891 120574RM (119866) = 4(69)

with the first equality if and only if 119866 cong 1198624

Recently Akbari and Qajar [122] have proved the follow-ing result

Theorem 171 (Akbari and Qajar [122]) For any connectedgraph 119866 of order 119899 ⩾ 3

119887RM (119866) ⩽ 119899 minus 120574RM (119866) + 5 (70)

International Journal of Combinatorics 27

We conclude this section with the following problems

Problem 8 Characterize all connected graphs 119866 of order 119899 ⩾3 for which 119887RM(119866) = 119899 minus 1

Problem 9 Prove or disprove if 119866 is a connected graph oforder 119899 ⩾ 3 then

119887RM (119866) ⩽ 119899 minus 120574RM (119866) + 3 (71)

Note that 120574119877(119870

119905(3)) = 4 and 119887RM(119870119905

(3)) = 119899minus1 where 119899 =3119905 for 119905 ⩾ 3 The upper bound on 119887RM(119866) given in Problem 9is sharp if Problem 9 has positive answer

95 Rainbow Bondage Numbers For a positive integer 119896 a119896-rainbow dominating function of a graph 119866 is a function 119891from the vertex-set 119881(119866) to the set of all subsets of the set1 2 119896 such that for any vertex 119909 in 119866 with 119891(119909) = 0 thecondition

⋃

119910isin119873119866(119909)

119891 (119910) = 1 2 119896 (72)

is fulfilledThe weight of a 119896-rainbow dominating function 119891on 119866 is the value

119891 (119866) = sum

119909isin119881(119866)

1003816100381610038161003816119891 (119909)1003816100381610038161003816 (73)

The 119896-rainbow domination number of a graph 119866 denoted by120574119903119896(119866) is the minimum weight of a 119896-rainbow dominating

function 119891 of 119866Note that 120574

1199031(119866) is the classical domination number 120574(119866)

The 119896-rainbow domination number is introduced by Bresaret al [126] Rainbow domination of a graph 119866 coincides withordinary domination of the Cartesian product of 119866 with thecomplete graph in particular 120574

119903119896(119866) = 120574(119866 times 119870

119896) for any

positive integer 119896 and any graph 119866 [126] Hartnell and Rall[127] proved that 120574(119879) lt 120574

1199032(119879) for any tree 119879 no graph has

120574 = 120574119903119896for 119896 ⩾ 3 for any 119896 ⩾ 2 and any graph 119866 of order 119899

min 119899 120574 (119866) + 119896 minus 2 ⩽ 120574119903119896(119866) ⩽ 119896120574 (119866) (74)

The introduction of rainbow domination is motivatedby the study of paired domination in Cartesian productsof graphs where certain upper bounds can be expressed interms of rainbow domination that is for any graph 119866 andany graph 119867 if 119881(119867) can be partitioned into 119896120574

119875-sets then

120574119875(119866 times 119867) ⩽ (1119896)120592(119867)120574

119903119896(119866) proved by Bresar et al [126]

Dehgardi et al [128] introduced the concept of 119896-rainbowbondage number of graphs Let 119866 be a graph with maximumdegree at least two The 119896-rainbow bondage number 119887

119903119896(119866)

of 119866 is the minimum cardinality of all sets 119861 sube 119864(119866) forwhich 120574

119903119896(119866 minus 119861) gt 120574

119903119896(119866) Since in the study of 119896-rainbow

bondage number the assumption Δ(119866) ⩾ 2 is necessarywe always assume that when we discuss 119887

119903119896(119866) all graphs

involved satisfy Δ(119866) ⩾ 2The 2-rainbow bondage number of some special families

of graphs has been determined

Theorem 172 (Dehgardi et al [128]) For any integer 119899 ⩾ 3

1198871199032(119875

119899) = 1 119891119900119903 119899 ⩾ 2

1198871199032(119862

119899) =

1 119894119891 119899 equiv 0 (mod 4) 2 119900119905ℎ119890119903119908119894119904119890

1198871199032(119870

119899119899) =

1 119894119891 119899 = 2

3 119894119891 119899 = 3

6 119894119891 119899 = 4

119898 119894119891 119899 ⩾ 5

(75)

Theorem 173 (Dehgardi et al [128]) For any connected graph119866 of order 119899 ⩾ 3 119887

1199032(119866) ⩽ 120582(119866) + 2Δ(119866) minus 3

Theorem 174 (Dehgardi et al [128]) For any tree 119879 of order119899 ⩾ 3 119887

1199032(119879) ⩽ 2

Theorem 175 (Dehgardi et al [128]) If 119866 is a planar graphwithmaximumdegree at least two then 119887

1199032(119866) ⩽ 15Moreover

if the girth 119892(119866) ⩾ 4 then 1198871199032(119866) ⩽ 11 and if 119892(119866) ⩾ 6 then

1198871199032(119866) ⩽ 9

Theorem 176 (Dehgardi et al [128]) For a complete bipartitegraph 119870

119901119902 if 2119896 + 1 ⩽ 119901 ⩽ 119902 then 119887

119903119896(119870

119901119902) = 119901

Theorem177 (Dehgardi et al [128]) For a complete graph119870119899

if 119899 ⩾ 119896 + 1 then 119887119903119896(119870

119899) = lceil119896119899(119896 + 1)rceil

The special case 119896 = 1 of Theorem 177 can be found inTheorem 1 The following is a simple observation similar toObservation 1

Observation 6 (Dehgardi et al [128]) Let 119866 be a graphwith maximum degree at least two 119867 a spanning subgraphobtained from 119866 by removing 119896 edges If 120574

119903119896(119867) = 120574

119903119896(119866)

then 119887119903119896(119867) ⩽ 119887

119903119896(119866) ⩽ 119887

119903119896(119867) + 119896

Theorem 178 (Dehgardi et al [128]) For every graph 119866 with120574119903119896(119866) = 119896120574(119866) 119887

119903119896(119866) ⩾ 119887(119866)

A graph 119866 is said to be vertex 119896-rainbow domination-critical if 120574

119903119896(119866 minus 119909) lt 120574

119903119896(119866) for every vertex 119909 in 119866 It is

clear that if 119866 has no isolated vertices then 120574119903119896(119866) ⩽ 119896120574(119866) ⩽

119896120573(119866) where 120573(119866) is the vertex-covering number of119866 Thusif 120574

119903119896(119866) = 119896120573(119866) then 120574

119903119896(119866) = 119896120574(119866)

Theorem 179 (Dehgardi et al [128]) For any graph 119866 if120574119903119896(119866) = 119896120573(119866) then

(a) 119887119903119896(119866) ⩾ 120575(119866)

(b) 119887119903119896(119866) ⩾ 120575(119866)+1 if119866 is vertex 119896-rainbow domination-

critical

As far as we know there are no more results on the 119896-rainbow bondage number of graphs in the literature

28 International Journal of Combinatorics

10 Results on Digraphs

Although domination has been extensively studied in undi-rected graphs it is natural to think of a dominating setas a one-way relationship between vertices of the graphIndeed among the earliest literatures on this subject vonNeumann and Morgenstern [129] used what is now calleddomination in digraphs to find solution (or kernels whichare independent dominating sets) for cooperative 119899-persongames Most likely the first formulation of domination byBerge [130] in 1958 was given in the context of digraphs andonly some years latter by Ore [131] in 1962 for undirectedgraphs Despite this history examination of domination andits variants in digraphs has been essentially overlooked (see[4] for an overview of the domination literature) Thus thereare few if any such results on domination for digraphs in theliterature

The bondage number and its related topics for undirectedgraph have become one of major areas both in theoreticaland applied researches However until recently Carlsonand Develin [42] Shan and Kang [132] and Huang andXu [133 134] studied the bondage number for digraphsindependently In this section we introduce their resultsfor general digraphs Results for some special digraphs suchas vertex-transitive digraphs are introduced in the nextsection

101 Upper Bounds for General Digraphs Let 119866 = (119881 119864)

be a digraph without loops and parallel edges A digraph 119866is called to be asymmetric if whenever (119909 119910) isin 119864(119866) then(119910 119909) notin 119864(119866) and to be symmetric if (119909 119910) isin 119864(119866) implies(119910 119909) isin 119864(119866) For a vertex 119909 of 119881(119866) the sets of out-neighbors and in-neighbors of 119909 are respectively defined as119873

+

119866(119909) = 119910 isin 119881(119866) (119909 119910) isin 119864(119866) and 119873minus

119866(119909) = 119909 isin

119881(119866) (119909 119910) isin 119864(119866) the out-degree and the in-degree of119909 are respectively defined as 119889+

119866(119909) = |119873

+

119866(119909)| and 119889minus

119866(119909) =

|119873+

119866(119909)| Denote themaximum and theminimum out-degree

(resp in-degree) of 119866 by Δ+(119866) and 120575+(119866) (resp Δminus(119866) and120575minus

(119866))The degree 119889119866(119909) of 119909 is defined as 119889+

119866(119909)+119889

minus

119866(119909) and

the maximum and the minimum degrees of119866 are denoted byΔ(119866) and 120575(119866) respectively that is Δ(119866) = max119889

119866(119909) 119909 isin

119881(119866) and 120575(119866) = min119889119866(119909) 119909 isin 119881(119866) Note that the

definitions here are different from the ones in the textbookon digraphs see for example Xu [1]

A subset 119878 of119881(119866) is called a dominating set if119881(119866) = 119878cup119873

+

119866(119878) where119873+

119866(119878) is the set of out-neighbors of 119878Then just

as for undirected graphs 120574(119866) is the minimum cardinalityof a dominating set and the bondage number 119887(119866) is thesmallest cardinality of a set 119861 of edges such that 120574(119866 minus 119861) gt120574(119866) if such a subset 119861 sube 119864(119866) exists Otherwise we put119887(119866) = infin

Some basic results for undirected graphs stated inSection 3 can be generalized to digraphs For exampleTheorem 18 is generalized by Carlson and Develin [42]Huang andXu [133] and Shan andKang [132] independentlyas follows

Theorem 180 Let 119866 be a digraph and (119909 119910) isin 119864(119866) Then119887(119866) ⩽ 119889

119866(119910) + 119889

minus

119866(119909) minus |119873

minus

119866(119909) cap 119873

minus

119866(119910)|

Corollary 181 For a digraph 119866 119887(119866) ⩽ 120575minus(119866) + Δ(119866)

Since 120575minus

(119866) ⩽ (12)Δ(119866) from Corollary 181Conjecture 53 is valid for digraphs

Corollary 182 For a digraph 119866 119887(119866) ⩽ (32)Δ(119866)

In the case of undirected graphs the bondage number of119866119899= 119870

119899times 119870

119899achieves this bound However it was shown

by Carlson and Develin in [42] that if we take the symmetricdigraph119866

119899 we haveΔ(119866

119899) = 4(119899minus1) 120574(119866

119899) = 119899 and 119887(119866

119899) ⩽

4(119899 minus 1) + 1 = Δ(119866119899) + 1 So this family of digraphs cannot

show that the bound in Corollary 182 is tightCorresponding to Conjecture 51 which is discredited

for undirected graphs and is valid for digraphs the sameconjecture for digraphs can be proposed as follows

Conjecture 183 (Carlson and Develin [42] 2006) 119887(119866) ⩽Δ(119866) + 1 for any digraph 119866

If Conjecture 183 is true then the following results showsthat this upper bound is tight since 119887(119870

119899times119870

119899) = Δ(119870

119899times119870

119899)+1

for a complete digraph 119870119899

In 2007 Shan and Kang [132] gave some tight upperbounds on the bondage numbers for some asymmetricdigraphs For example 119887(119879) ⩽ Δ(119879) for any asymmetricdirected tree 119879 119887(119866) ⩽ Δ(119866) for any asymmetric digraph 119866with order at least 4 and 120574(119866) ⩽ 2 For planar digraphs theyobtained the following results

Theorem 184 (Shan and Kang [132] 2007) Let 119866 be aasymmetric planar digraphThen 119887(119866) ⩽ Δ(119866)+2 and 119887(119866) ⩽Δ(119866) + 1 if Δ(119866) ⩾ 5 and 119889minus

119866(119909) ⩾ 3 for every vertex 119909 with

119889119866(119909) ⩾ 4

102 Results for Some Special Digraphs The exact values andbounds on 119887(119866) for some standard digraphs are determined

Theorem185 (Huang andXu [133] 2006) For a directed cycle119862119899and a directed path 119875

119899

119887 (119862119899) =

3 119894119891 119899 119894119904 119900119889119889

2 119894119891 119899 119894119904 119890V119890119899

119887 (119875119899) =

2 119894119891 119899 119894119904 119900119889119889

1 119894119891 119899 119894119904 119890V119890119899

(76)

For the de Bruijn digraph 119861(119889 119899) and the Kautz digraph119870(119889 119899)

119887 (119861 (119889 119899)) = 119889 119894119891 119899 119894119904 119900119889119889

119889 ⩽ 119887 (119861 (119889 119899)) ⩽ 2119889 119894119891 119899 119894119904 119890V119890119899

119887 (119870 (119889 119899)) = 119889 + 1

(77)

Like undirected graphs we can define the total domi-nation number and the total bondage number On the totalbondage numbers for some special digraphs the knownresults are as follows

International Journal of Combinatorics 29

Theorem 186 (Huang and Xu [134] 2007) For a directedcycle 119862

119899and a directed path 119875

119899 119887

119879(119875

119899) and 119887

119879(119862

119899) all do not

exist For a complete digraph 119870119899

119887119879(119870

119899) = infin 119894119891 119899 = 1 2

119887119879(119870

119899) = 3 119894119891 119899 = 3

119899 ⩽ 119887119879(119870

119899) ⩽ 2119899 minus 3 119894119891 119899 ⩾ 4

(78)

The extended de Bruijn digraph EB(119889 119899 1199021 119902

119901) and

the extended Kautz digraph EK(119889 119899 1199021 119902

119901) are intro-

duced by Shibata and Gonda [135] If 119901 = 1 then theyare the de Bruijn digraph B(119889 119899) and the Kautz digraphK(119889 119899) respectively Huang and Xu [134] determined theirtotal domination numbers In particular their total bondagenumbers for general cases are determined as follows

Theorem 187 (Huang andXu [134] 2007) If 119889 ⩾ 2 and 119902119894⩾ 2

for each 119894 = 1 2 119901 then

119887119879(EB(119889 119899 119902

1 119902

119901)) = 119889

119901

minus 1

119887119879(EK(119889 119899 119902

1 119902

119901)) = 119889

119901

(79)

In particular for the de Bruijn digraph B(119889 119899) and the Kautzdigraph K(119889 119899)

119887119879(B (119889 119899)) = 119889 minus 1 119887

119879(K (119889 119899)) = 119889 (80)

Zhang et al [136] determined the bondage number incomplete 119905-partite digraphs

Theorem 188 (Zhang et al [136] 2009) For a complete 119905-partite digraph 119870

11989911198992119899119905

where 1198991⩽ 119899

2⩽ sdot sdot sdot ⩽ 119899

119905

119887 (11987011989911198992119899119905

)

=

119898 119894119891 119899119898= 1 119899

119898+1⩾ 2 119891119900119903 119904119900119898119890

119898 (1 ⩽ 119898 lt 119905)

4119905 minus 3 119894119891 1198991= 119899

2= sdot sdot sdot = 119899

119905= 2

119905minus1

sum

119894=1

119899119894+ 2 (119905 minus 1) 119900119905ℎ119890119903119908119894119904119890

(81)

Since an undirected graph can be thought of as a sym-metric digraph any result for digraphs has an analogy forundirected graphs in general In view of this point studyingthe bondage number for digraphs is more significant thanfor undirected graphs Thus we should further study thebondage number of digraphs and try to generalize knownresults on the bondage number and related variants for undi-rected graphs to digraphs prove or disprove Conjecture 183In particular determine the exact values of 119887(B(119889 119899)) for aneven 119899 and 119887

119879(119870

119899) for 119899 ⩾ 4

11 Efficient Dominating Sets

A dominating set 119878 of a graph 119866 is called to be efficient if forevery vertex 119909 in119866 |119873

119866[119909]cap119878| = 1 if119866 is a undirected graph

or |119873minus

119866[119909] cap 119878| = 1 if 119866 is a directed graph The concept of an

efficient set was proposed by Bange et al [137] in 1988 In theliterature an efficient set is also called a perfect dominatingset (see Livingston and Stout [138] for an explanation)

From definition if 119878 is an efficient dominating set of agraph 119866 then 119878 is certainly an independent set and everyvertex not in 119878 is adjacent to exactly one vertex in 119878

It is also clear from definition that a dominating set 119878 isefficient if and only if N

119866[119878] = 119873

119866[119909] 119909 isin 119878 for the

undirected graph 119866 or N+

119866[119878] = 119873

+

119866[119909] 119909 isin 119878 for the

digraph119866 is a partition of119881(119866) where the induced subgraphby119873

119866[119909] or119873+

119866[119909] is a star or an out-star with the root 119909

The efficient domination has important applications inmany areas such as error-correcting codes and receivesmuch attention in the late years

The concept of efficient dominating sets is a measureof the efficiency of domination in graphs Unfortunately asshown in [137] not every graph has an efficient dominatingset andmoreover it is anNP-complete problem to determinewhether a given graph has an efficient dominating set Inaddition it was shown by Clark [139] in 1993 that for a widerange of 119901 almost every random undirected graph 119866 isin

G(120592 119901) has no efficient dominating sets This means thatundirected graphs possessing an efficient dominating set arerare However it is easy to show that every undirected graphhas an orientationwith an efficient dominating set (see Bangeet al [140])

In 1993 Barkauskas and Host [141] showed that deter-mining whether an arbitrary oriented graph has an efficientdominating set is NP-complete Even so the existence ofefficient dominating sets for some special classes of graphs hasbeen examined see for example Dejter and Serra [142] andLee [143] for Cayley graph Gu et al [144] formeshes and toriObradovic et al [145] Huang and Xu [36] Reji Kumar andMacGillivray [146] for circulant graphs Harary graphs andtori and Van Wieren et al [147] for cube-connected cycles

In this section we introduce results of the bondagenumber for some graphs with efficient dominating sets dueto Huang and Xu [36]

111 Results for General Graphs In this subsection we intro-duce some results on bondage numbers obtained by applyingefficient dominating sets We first state the two followinglemmas

Lemma 189 For any 119896-regular graph or digraph 119866 of order 119899120574(119866) ⩾ 119899(119896 + 1) with equality if and only if 119866 has an efficientdominating set In addition if 119866 has an efficient dominatingset then every efficient dominating set is certainly a 120574-set andvice versa

Let 119890 be an edge and 119878 a dominating set in 119866 We say that119890 supports 119878 if 119890 isin (119878 119878) where (119878 119878) = (119909 119910) isin 119864(119866) 119909 isin 119878 119910 isin 119878 Denote by 119904(119866) the minimum number of edgeswhich support all 120574-sets in 119866

Lemma 190 For any graph or digraph 119866 119887(119866) ⩾ 119904(119866) withequality if 119866 is regular and has an efficient dominating set

30 International Journal of Combinatorics

A graph 119866 is called to be vertex-transitive if its automor-phism group Aut(119866) acts transitively on its vertex-set 119881(119866)A vertex-transitive graph is regular Applying Lemmas 189and 190 Huang and Xu obtained some results on bondagenumbers for vertex-transitive graphs or digraphs

Theorem 191 (Huang and Xu [36] 2008) For any vertex-transitive graph or digraph 119866 of order 119899

119887 (119866) ⩾

lceil119899

2120574 (119866)rceil 119894119891 119866 119894119904 119906119899119889119894119903119890119888119905119890119889

lceil119899

120574 (119866)rceil 119894119891 119866 119894119904 119889119894119903119890119888119905119890119889

(82)

Theorem192 (Huang andXu [36] 2008) Let119866 be a 119896-regulargraph of order 119899 Then

119887(119866) ⩽ 119896 if119866 is undirected and 119899 ⩾ 120574(119866)(119896+1)minus119896+1119887(119866) ⩽ 119896+1+ℓ if119866 is directed and 119899 ⩾ 120574(119866)(119896+1)minusℓwith 0 ⩽ ℓ ⩽ 119896 minus 1

Next we introduce a better upper bound on 119887(119866) To thisaim we need the following concept which is a generalizationof the concept of the edge-covering of a graph 119866 For 1198811015840

sube

119881(119866) and 1198641015840 sube 119864(119866) we say that 1198641015840covers 1198811015840 and call 1198641015840an edge-covering for 1198811015840 if there exists an edge (119909 119910) isin 1198641015840 forany vertex 119909 isin 1198811015840 For 119910 isin 119881(119866) let 1205731015840[119910] be the minimumcardinality over all edge-coverings for119873minus

119866[119910]

Theorem 193 (Huang and Xu [36] 2008) If 119866 is a 119896-regulargraph with order 120574(119866)(119896 + 1) then 119887(119866) ⩽ 1205731015840[119910] for any 119910 isin119881(119866)

Theupper bound on 119887(119866) given inTheorem 193 is tight inview of 119887(119862

119899) for a cycle or a directed cycle 119862

119899(seeTheorems

1 and 185 resp)It is easy to see that for a 119896-regular graph 119866 lceil(119896 + 1)2rceil ⩽

1205731015840

[119910] ⩽ 119896 when 119866 is undirected and 1205731015840[119910] = 119896 + 1 when 119866 isdirected By this fact and Lemma 189 the following theoremis merely a simple combination of Theorems 191 and 193

Theorem 194 (Huang and Xu [36] 2008) Let 119866 be a vertex-transitive graph of degree 119896 If 119866 has an efficient dominatingset then

lceil119896 + 1

2rceil ⩽ 119887 (119866) ⩽ 119896 119894119891 119866 119894119904 119906119899119889119894119903119890119888119905119890119889

119887 (119866) = 119896 + 1 119894119891 119866 119894119904 119889119894119903119890119888119905119890119889

(83)

Theorem 195 (Huang and Xu [36] 2008) If 119866 is an undi-rected vertex-transitive cubic graph with order 4120574(119866) and girth119892(119866) ⩽ 5 then 119887(119866) = 2

The proof of Theorem 195 leads to a byproduct If 119892 =5 let (119906

1 119906

2 119906

3 119906

4 119906

5) be a cycle and let D

119894be the family

of efficient dominating sets containing 119906119894for each 119894 isin

1 2 3 4 5 ThenD119894capD

119895= 0 for 119894 = 1 2 5 Then 119866 has

at least 5119904 efficient dominating sets where 119904 = 119904(119866) But thereare only 4s (=ns120574) distinct efficient dominating sets in119866This

contradiction implies that an undirected vertex-transitivecubic graph with girth five has no efficient dominating setsBut a similar argument for 119892(119866) = 3 4 or 119892(119866) ⩾ 6 couldnot give any contradiction This is consistent with the resultthat CCC(119899) a vertex-transitive cubic graph with girth 119899 if3 ⩽ 119899 ⩽ 8 or girth 8 if 119899 ⩾ 9 has efficient dominating setsfor all 119899 ⩾ 3 except 119899 = 5 (see Theorem 199 in the followingsubsection)

112 Results for Cayley Graphs In this subsection we useTheorem 194 to determine the exact values or approximativevalues of bondage numbers for some special vertex-transitivegraphs by characterizing the existence of efficient dominatingsets in these graphs

Let Γ be a nontrivial finite group 119878 a nonempty subset ofΓ without the identity element of Γ A digraph 119866 is defined asfollows

119881 (119866) = Γ (119909 119910) isin 119864 (119866) lArrrArr 119909minus1

119910 isin 119878

for any 119909 119910 isin Γ(84)

is called a Cayley digraph of the group Γ with respect to119878 denoted by 119862

Γ(119878) If 119878minus1 = 119904minus1 119904 isin 119878 = 119878 then

119862Γ(119878) is symmetric and is called a Cayley undirected graph

a Cayley graph for short Cayley graphs or digraphs arecertainly vertex-transitive (see Xu [1])

A circulant graph 119866(119899 119878) of order 119899 is a Cayley graph119862119885119899

(119878) where 119885119899= 0 119899 minus 1 is the addition group of

order 119899 and 119878 is a nonempty subset of119885119899without the identity

element and hence is a vertex-transitive digraph of degree|119878| If 119878minus1 = 119878 then119866(119899 119878) is an undirected graph If 119878 = 1 119896where 2 ⩽ 119896 ⩽ 119899 minus 2 we write 119866(119899 1 119896) for 119866(119899 1 119896) or119866(119899 plusmn1 plusmn119896) and call it a double-loop circulant graph

Huang and Xu [36] showed that for directed 119866 =

119866(119899 1 119896) lceil1198993rceil ⩽ 120574(119866) ⩽ lceil1198992rceil and 119866 has an efficientdominating set if and only if 3 | 119899 and 119896 equiv 2 (mod 3) fordirected 119866 = 119866(119899 1 119896) and 119896 = 1198992 lceil1198995rceil ⩽ 120574(119866) ⩽ lceil1198993rceiland 119866 has an efficient dominating set if and only if 5 | 119899 and119896 equiv plusmn2 (mod 5) By Theorem 194 we obtain the bondagenumber of a double loop circulant graph if it has an efficientdominating set

Theorem 196 (Huang and Xu [36] 2008) Let 119866 be a double-loop circulant graph 119866(119899 1 119896) If 119866 is directed with 3 | 119899and 119896 equiv 2 (mod 3) or 119866 is undirected with 5 | 119899 and119896 equiv plusmn2 (mod 5) then 119887(119866) = 3

The 119898 times 119899 torus is the Cartesian product 119862119898times 119862

119899of

two cycles and is a Cayley graph 119862119885119898times119885119899

(119878) where 119878 =

(0 1) (1 0) for directed cycles and 119878 = (0 plusmn1) (plusmn1 0) forundirected cycles and hence is vertex-transitive Gu et al[144] showed that the undirected torus119862

119898times119862

119899has an efficient

dominating set if and only if both 119898 and 119899 are multiplesof 5 Huang and Xu [36] showed that the directed torus119862119898times 119862

119899has an efficient dominating set if and only if both

119898 and 119899 are multiples of 3 Moreover they found a necessarycondition for a dominating set containing the vertex (0 0)in 119862

119898times 119862

119899to be efficient and obtained the following

result

International Journal of Combinatorics 31

Theorem 197 (Huang and Xu [36] 2008) Let 119866 = 119862119898times 119862

119899

If 119866 is undirected and both119898 and 119899 are multiples of 5 or if 119866 isdirected and both119898 and 119899 are multiples of 3 then 119887(119866) = 3

The hypercube 119876119899

is the Cayley graph 119862Γ(119878)

where Γ = 1198852times sdot sdot sdot times 119885

2= (119885

2)119899 and 119878 =

100 sdot sdot sdot 0 010 sdot sdot sdot 0 00 sdot sdot sdot 01 Lee [143] showed that119876119899has an efficient dominating set if and only if 119899 = 2119898 minus 1

for a positive integer119898 ByTheorem 194 the following resultis obtained immediately

Theorem 198 (Huang and Xu [36] 2008) If 119899 = 2119898 minus 1 for apositive integer119898 then 2119898minus1

⩽ 119887(119876119899) ⩽ 2

119898

minus 1

Problem 10 Determine the exact value of 119887(119876119899) for 119899 = 2119898minus1

The 119899-dimensional cube-connected cycle denoted byCCC(119899) is constructed from the 119899-dimensional hypercube119876119899by replacing each vertex 119909 in 119876

119899with an undirected cycle

119862119899of length 119899 and linking the 119894th vertex of the 119862

119899to the 119894th

neighbor of 119909 It has been proved that CCC(119899) is a Cayleygraph and hence is a vertex-transitive graph with degree 3Van Wieren et al [147] proved that CCC(119899) has an efficientdominating set if and only if 119899 = 5 From Theorems 194 and195 the following result can be derived

Theorem 199 (Huang and Xu [36] 2008) Let 119866 = 119862119862119862(119899)be the 119899-dimensional cube-connected cycles with 119899 ⩾ 3 and119899 = 5 Then 120574(119866) = 1198992119899minus2 and 2 ⩽ 119887(119866) ⩽ 3 In addition119887(119862119862119862(3)) = 119887(119862119862119862(4)) = 2

Problem 11 Determine the exact value of 119887(CCC(119899)) for 119899 ⩾5

Acknowledgments

This work was supported by NNSF of China (Grants nos11071233 61272008) The author would like to express hisgratitude to the anonymous referees for their kind sugges-tions and comments on the original paper to ProfessorSheikholeslami and Professor Jafari Rad for sending thereferences [128] and [81] which resulted in this version

References

[1] J-M Xu Theory and Application of Graphs vol 10 of NetworkTheory and Applications Kluwer Academic Dordrecht TheNetherlands 2003

[2] S THedetniemi andR C Laskar ldquoBibliography on dominationin graphs and some basic definitions of domination parame-tersrdquo Discrete Mathematics vol 86 no 1ndash3 pp 257ndash277 1990

[3] TW Haynes S T Hedetniemi and P J Slater Fundamentals ofDomination in Graphs vol 208 ofMonographs and Textbooks inPure and Applied Mathematics Marcel Dekker New York NYUSA 1998

[4] TWHaynes S THedetniemi and P J Slater EdsDominationin Graphs vol 209 of Monographs and Textbooks in Pure andAppliedMathematicsMarcelDekker NewYorkNYUSA 1998

[5] M R Garey andD S JohnsonComputers and Intractability WH Freeman and Company San Francisco Calif USA 1979

[6] H B Walikar and B D Acharya ldquoDomination critical graphsrdquoNational Academy Science Letters vol 2 pp 70ndash72 1979

[7] D Bauer F Harary J Nieminen and C L Suffel ldquoDominationalteration sets in graphsrdquo Discrete Mathematics vol 47 no 2-3pp 153ndash161 1983

[8] J F Fink M S Jacobson L F Kinch and J Roberts ldquoThebondage number of a graphrdquo Discrete Mathematics vol 86 no1ndash3 pp 47ndash57 1990

[9] F-T Hu and J-M Xu ldquoThe bondage number of (119899 minus 3)-regulargraph G of order nrdquo Ars Combinatoria to appear

[10] U Teschner ldquoNew results about the bondage number of agraphrdquoDiscreteMathematics vol 171 no 1ndash3 pp 249ndash259 1997

[11] B L Hartnell and D F Rall ldquoA bound on the size of a graphwith given order and bondage numberrdquo Discrete Mathematicsvol 197-198 pp 409ndash413 1999 16th British CombinatorialConference (London 1997)

[12] B L Hartnell and D F Rall ldquoA characterization of trees inwhich no edge is essential to the domination numberrdquo ArsCombinatoria vol 33 pp 65ndash76 1992

[13] J Topp and P D Vestergaard ldquo120572119896- and 120574

119896-stable graphsrdquo

Discrete Mathematics vol 212 no 1-2 pp 149ndash160 2000[14] A Meir and J W Moon ldquoRelations between packing and

covering numbers of a treerdquo Pacific Journal of Mathematics vol61 no 1 pp 225ndash233 1975

[15] B L Hartnell L K Joslashrgensen P D Vestergaard and CA Whitehead ldquoEdge stability of the k-domination numberof treesrdquo Bulletin of the Institute of Combinatorics and ItsApplications vol 22 pp 31ndash40 1998

[16] F-T Hu and J-M Xu ldquoOn the complexity of the bondage andreinforcement problemsrdquo Journal of Complexity vol 28 no 2pp 192ndash201 2012

[17] B L Hartnell and D F Rall ldquoBounds on the bondage numberof a graphrdquo Discrete Mathematics vol 128 no 1ndash3 pp 173ndash1771994

[18] Y-L Wang ldquoOn the bondage number of a graphrdquo DiscreteMathematics vol 159 no 1ndash3 pp 291ndash294 1996

[19] G H Fricke TW Haynes S M Hedetniemi S T Hedetniemiand R C Laskar ldquoExcellent treesrdquo Bulletin of the Institute ofCombinatorics and Its Applications vol 34 pp 27ndash38 2002

[20] V Samodivkin ldquoThe bondage number of graphs good and badverticesrdquoDiscussiones Mathematicae GraphTheory vol 28 no3 pp 453ndash462 2008

[21] J E Dunbar T W Haynes U Teschner and L VolkmannldquoBondage insensitivity and reinforcementrdquo in Domination inGraphs vol 209 of Monographs and Textbooks in Pure andAppliedMathematics pp 471ndash489 Dekker NewYork NY USA1998

[22] H Liu and L Sun ldquoThe bondage and connectivity of a graphrdquoDiscrete Mathematics vol 263 no 1ndash3 pp 289ndash293 2003

[23] A-H Esfahanian and S L Hakimi ldquoOn computing a con-ditional edge-connectivity of a graphrdquo Information ProcessingLetters vol 27 no 4 pp 195ndash199 1988

[24] R C Brigham P Z Chinn and R D Dutton ldquoVertexdomination-critical graphsrdquoNetworks vol 18 no 3 pp 173ndash1791988

[25] V Samodivkin ldquoDomination with respect to nondegenerateproperties bondage numberrdquoThe Australasian Journal of Com-binatorics vol 45 pp 217ndash226 2009

[26] U Teschner ldquoA new upper bound for the bondage numberof graphs with small domination numberrdquo The AustralasianJournal of Combinatorics vol 12 pp 27ndash35 1995

32 International Journal of Combinatorics

[27] J Fulman D Hanson and G MacGillivray ldquoVertex dom-ination-critical graphsrdquoNetworks vol 25 no 2 pp 41ndash43 1995

[28] U Teschner ldquoA counterexample to a conjecture on the bondagenumber of a graphrdquo Discrete Mathematics vol 122 no 1ndash3 pp393ndash395 1993

[29] U Teschner ldquoThe bondage number of a graph G can be muchgreater than Δ(G)rdquo Ars Combinatoria vol 43 pp 81ndash87 1996

[30] L Volkmann ldquoOn graphs with equal domination and coveringnumbersrdquoDiscrete AppliedMathematics vol 51 no 1-2 pp 211ndash217 1994

[31] L A Sanchis ldquoMaximum number of edges in connected graphswith a given domination numberrdquoDiscreteMathematics vol 87no 1 pp 65ndash72 1991

[32] S Klavzar and N Seifter ldquoDominating Cartesian products ofcyclesrdquoDiscrete AppliedMathematics vol 59 no 2 pp 129ndash1361995

[33] H K Kim ldquoA note on Vizingrsquos conjecture about the bondagenumberrdquo Korean Journal of Mathematics vol 10 no 2 pp 39ndash42 2003

[34] M Y Sohn X Yuan and H S Jeong ldquoThe bondage number of1198623times119862

119899rdquo Journal of the KoreanMathematical Society vol 44 no

6 pp 1213ndash1231 2007[35] L Kang M Y Sohn and H K Kim ldquoBondage number of the

discrete torus 119862119899times 119862

4rdquo Discrete Mathematics vol 303 no 1ndash3

pp 80ndash86 2005[36] J Huang and J-M Xu ldquoThe bondage numbers and efficient

dominations of vertex-transitive graphsrdquoDiscrete Mathematicsvol 308 no 4 pp 571ndash582 2008

[37] C J Xiang X Yuan andM Y Sohn ldquoDomination and bondagenumber of119862

3times119862

119899rdquoArs Combinatoria vol 97 pp 299ndash310 2010

[38] M S Jacobson and L F Kinch ldquoOn the domination numberof products of graphs Irdquo Ars Combinatoria vol 18 pp 33ndash441984

[39] Y-Q Li ldquoA note on the bondage number of a graphrdquo ChineseQuarterly Journal of Mathematics vol 9 no 4 pp 1ndash3 1994

[40] F-T Hu and J-M Xu ldquoBondage number of mesh networksrdquoFrontiers ofMathematics inChina vol 7 no 5 pp 813ndash826 2012

[41] R Frucht and F Harary ldquoOn the corona of two graphsrdquoAequationes Mathematicae vol 4 pp 322ndash325 1970

[42] K Carlson and M Develin ldquoOn the bondage number of planarand directed graphsrdquoDiscreteMathematics vol 306 no 8-9 pp820ndash826 2006

[43] U Teschner ldquoOn the bondage number of block graphsrdquo ArsCombinatoria vol 46 pp 25ndash32 1997

[44] J Huang and J-M Xu ldquoNote on conjectures of bondagenumbers of planar graphsrdquo Applied Mathematical Sciences vol6 no 65ndash68 pp 3277ndash3287 2012

[45] L Kang and J Yuan ldquoBondage number of planar graphsrdquoDiscrete Mathematics vol 222 no 1ndash3 pp 191ndash198 2000

[46] M Fischermann D Rautenbach and L Volkmann ldquoRemarkson the bondage number of planar graphsrdquo Discrete Mathemat-ics vol 260 no 1ndash3 pp 57ndash67 2003

[47] J Hou G Liu and J Wu ldquoBondage number of planar graphswithout small cyclesrdquoUtilitasMathematica vol 84 pp 189ndash1992011

[48] J Huang and J-M Xu ldquoThe bondage numbers of graphs withsmall crossing numbersrdquo Discrete Mathematics vol 307 no 15pp 1881ndash1897 2007

[49] Q Ma S Zhang and J Wang ldquoBondage number of 1-planargraphrdquo Applied Mathematics vol 1 no 2 pp 101ndash103 2010

[50] J Hou and G Liu ldquoA bound on the bondage number of toroidalgraphsrdquoDiscreteMathematics Algorithms and Applications vol4 no 3 Article ID 1250046 5 pages 2012

[51] Y-C Cao J-M Xu and X-R Xu ldquoOn bondage number oftoroidal graphsrdquo Journal of University of Science and Technologyof China vol 39 no 3 pp 225ndash228 2009

[52] Y-C Cao J Huang and J-M Xu ldquoThe bondage number ofgraphs with crossing number less than fourrdquo Ars Combinatoriato appear

[53] H K Kim ldquoOn the bondage numbers of some graphsrdquo KoreanJournal of Mathematics vol 11 no 2 pp 11ndash15 2004

[54] A Gagarin and V Zverovich ldquoUpper bounds for the bondagenumber of graphs on topological surfacesrdquo Discrete Mathemat-ics 2012

[55] J Huang ldquoAn improved upper bound for the bondage numberof graphs on surfacesrdquoDiscrete Mathematics vol 312 no 18 pp2776ndash2781 2012

[56] A Gagarin and V Zverovich ldquoThe bondage number of graphson topological surfaces and Teschnerrsquos conjecturerdquo DiscreteMathematics vol 313 no 6 pp 796ndash808 2013

[57] V Samodivkin ldquoNote on the bondage number of graphs ontopological surfacesrdquo httparxivorgabs12086203

[58] E J Cockayne R M Dawes and S T Hedetniemi ldquoTotaldomination in graphsrdquoNetworks vol 10 no 3 pp 211ndash219 1980

[59] R Laskar J Pfaff S M Hedetniemi and S T HedetniemildquoOn the algorithmic complexity of total dominationrdquo Society forIndustrial and Applied Mathematics vol 5 no 3 pp 420ndash4251984

[60] J Pfaff R C Laskar and S T Hedetniemi ldquoNP-completeness oftotal and connected domination and irredundance for bipartitegraphsrdquo Tech Rep 428 Department of Mathematical SciencesClemson University 1983

[61] M A Henning ldquoA survey of selected recent results on totaldomination in graphsrdquoDiscrete Mathematics vol 309 no 1 pp32ndash63 2009

[62] V R Kulli and D K Patwari ldquoThe total bondage number ofa graphrdquo in Advances in Graph Theory pp 227ndash235 VishwaGulbarga India 1991

[63] F-T Hu Y Lu and J-M Xu ldquoThe total bondage number ofgrid graphsrdquo Discrete Applied Mathematics vol 160 no 16-17pp 2408ndash2418 2012

[64] J Cao M Shi and B Wu ldquoResearch on the total bondagenumber of a spectial networkrdquo in Proceedings of the 3rdInternational Joint Conference on Computational Sciences andOptimization (CSO rsquo10) pp 456ndash458 May 2010

[65] N Sridharan MD Elias and V S A Subramanian ldquoTotalbondage number of a graphrdquo AKCE International Journal ofGraphs and Combinatorics vol 4 no 2 pp 203ndash209 2007

[66] N Jafari Rad and J Raczek ldquoSome progress on total bondage ingraphsrdquo Graphs and Combinatorics 2011

[67] T W Haynes and P J Slater ldquoPaired-domination and thepaired-domatic numberrdquoCongressusNumerantium vol 109 pp65ndash72 1995

[68] T W Haynes and P J Slater ldquoPaired-domination in graphsrdquoNetworks vol 32 no 3 pp 199ndash206 1998

[69] X Hou ldquoA characterization of 2120574 120574119875-treesrdquoDiscrete Mathemat-

ics vol 308 no 15 pp 3420ndash3426 2008[70] K E Proffitt T W Haynes and P J Slater ldquoPaired-domination

in grid graphsrdquo Congressus Numerantium vol 150 pp 161ndash1722001

International Journal of Combinatorics 33

[71] H Qiao L KangM Cardei andD-Z Du ldquoPaired-dominationof treesrdquo Journal of Global Optimization vol 25 no 1 pp 43ndash542003

[72] E Shan L Kang and M A Henning ldquoA characterizationof trees with equal total domination and paired-dominationnumbersrdquo The Australasian Journal of Combinatorics vol 30pp 31ndash39 2004

[73] J Raczek ldquoPaired bondage in treesrdquo Discrete Mathematics vol308 no 23 pp 5570ndash5575 2008

[74] J A Telle and A Proskurowski ldquoAlgorithms for vertex parti-tioning problems on partial k-treesrdquo SIAM Journal on DiscreteMathematics vol 10 no 4 pp 529ndash550 1997

[75] G S Domke J H Hattingh M A Henning and L R MarkusldquoRestrained domination in treesrdquoDiscreteMathematics vol 211no 1ndash3 pp 1ndash9 2000

[76] G S Domke J H Hattingh M A Henning and L R MarkusldquoRestrained domination in graphs with minimum degree twordquoJournal of Combinatorial Mathematics and Combinatorial Com-puting vol 35 pp 239ndash254 2000

[77] G S Domke J H Hattingh S T Hedetniemi R C Laskarand L R Markus ldquoRestrained domination in graphsrdquo DiscreteMathematics vol 203 no 1ndash3 pp 61ndash69 1999

[78] J H Hattingh and E J Joubert ldquoAn upper bound for therestrained domination number of a graph with minimumdegree at least two in terms of order and minimum degreerdquoDiscrete Applied Mathematics vol 157 no 13 pp 2846ndash28582009

[79] M A Henning ldquoGraphs with large restrained dominationnumberrdquo Discrete Mathematics vol 197-198 pp 415ndash429 199916th British Combinatorial Conference (London 1997)

[80] J H Hattingh and A R Plummer ldquoRestrained bondage ingraphsrdquo Discrete Mathematics vol 308 no 23 pp 5446ndash54532008

[81] N Jafari Rad R Hasni J Raczek and L Volkmann ldquoTotalrestrained bondage in graphsrdquoActaMathematica Sinica vol 29no 6 pp 1033ndash1042 2013

[82] J Cyman and J Raczek ldquoOn the total restrained dominationnumber of a graphrdquoThe Australasian Journal of Combinatoricsvol 36 pp 91ndash100 2006

[83] M A Henning ldquoGraphs with large total domination numberrdquoJournal of Graph Theory vol 35 no 1 pp 21ndash45 2000

[84] D-X Ma X-G Chen and L Sun ldquoOn total restraineddomination in graphsrdquoCzechoslovakMathematical Journal vol55 no 1 pp 165ndash173 2005

[85] B Zelinka ldquoRemarks on restrained domination and totalrestrained domination in graphsrdquo Czechoslovak MathematicalJournal vol 55 no 2 pp 393ndash396 2005

[86] W Goddard and M A Henning ldquoIndependent domination ingraphs a survey and recent resultsrdquo Discrete Mathematics vol313 no 7 pp 839ndash854 2013

[87] J H Zhang H L Liu and L Sun ldquoIndependence bondagenumber and reinforcement number of some graphsrdquo Transac-tions of Beijing Institute of Technology vol 23 no 2 pp 140ndash1422003

[88] K D Dharmalingam Studies in Graph Theorymdashequitabledomination and bottleneck domination [PhD thesis] MaduraiKamaraj University Madurai India 2006

[89] GDeepakND Soner andAAlwardi ldquoThe equitable bondagenumber of a graphrdquo Research Journal of Pure Algebra vol 1 no9 pp 209ndash212 2011

[90] E Sampathkumar and H B Walikar ldquoThe connected domina-tion number of a graphrdquo Journal of Mathematical and PhysicalSciences vol 13 no 6 pp 607ndash613 1979

[91] K White M Farber and W Pulleyblank ldquoSteiner trees con-nected domination and strongly chordal graphsrdquoNetworks vol15 no 1 pp 109ndash124 1985

[92] M Cadei M X Cheng X Cheng and D-Z Du ldquoConnecteddomination in Ad Hoc wireless networksrdquo in Proceedings ofthe 6th International Conference on Computer Science andInformatics (CSampI rsquo02) 2002

[93] J F Fink and M S Jacobson ldquon-domination in graphsrdquo inGraph Theory with Applications to Algorithms and ComputerScience Wiley-Interscience Publications pp 283ndash300 WileyNew York NY USA 1985

[94] M Chellali O Favaron A Hansberg and L Volkmann ldquok-domination and k-independence in graphs a surveyrdquo Graphsand Combinatorics vol 28 no 1 pp 1ndash55 2012

[95] Y Lu X Hou J-M Xu andN Li ldquoTrees with uniqueminimump-dominating setsrdquo Utilitas Mathematica vol 86 pp 193ndash2052011

[96] Y Lu and J-M Xu ldquoThe p-bondage number of treesrdquo Graphsand Combinatorics vol 27 no 1 pp 129ndash141 2011

[97] Y Lu and J-M Xu ldquoThe 2-domination and 2-bondage numbersof grid graphs A manuscriptrdquo httparxivorgabs12044514

[98] M Krzywkowski ldquoNon-isolating 2-bondage in graphsrdquo TheJournal of the Mathematical Society of Japan vol 64 no 4 2012

[99] M A Henning O R Oellermann and H C Swart ldquoBoundson distance domination parametersrdquo Journal of CombinatoricsInformation amp System Sciences vol 16 no 1 pp 11ndash18 1991

[100] F Tian and J-M Xu ldquoBounds for distance domination numbersof graphsrdquo Journal of University of Science and Technology ofChina vol 34 no 5 pp 529ndash534 2004

[101] F Tian and J-M Xu ldquoDistance domination-critical graphsrdquoApplied Mathematics Letters vol 21 no 4 pp 416ndash420 2008

[102] F Tian and J-M Xu ldquoA note on distance domination numbersof graphsrdquo The Australasian Journal of Combinatorics vol 43pp 181ndash190 2009

[103] F Tian and J-M Xu ldquoAverage distances and distance domina-tion numbersrdquo Discrete Applied Mathematics vol 157 no 5 pp1113ndash1127 2009

[104] Z-L Liu F Tian and J-M Xu ldquoProbabilistic analysis of upperbounds for 2-connected distance k-dominating sets in graphsrdquoTheoretical Computer Science vol 410 no 38ndash40 pp 3804ndash3813 2009

[105] M A Henning O R Oellermann and H C Swart ldquoDistancedomination critical graphsrdquo Journal of Combinatorial Mathe-matics and Combinatorial Computing vol 44 pp 33ndash45 2003

[106] T J Bean M A Henning and H C Swart ldquoOn the integrityof distance domination in graphsrdquo The Australasian Journal ofCombinatorics vol 10 pp 29ndash43 1994

[107] M Fischermann and L Volkmann ldquoA remark on a conjecturefor the (kp)-domination numberrdquoUtilitasMathematica vol 67pp 223ndash227 2005

[108] T Korneffel D Meierling and L Volkmann ldquoA remark on the(22)-domination numberrdquo Discussiones Mathematicae GraphTheory vol 28 no 2 pp 361ndash366 2008

[109] Y Lu X Hou and J-M Xu ldquoOn the (22)-domination numberof treesrdquo Discussiones Mathematicae Graph Theory vol 30 no2 pp 185ndash199 2010

34 International Journal of Combinatorics

[110] H Li and J-M Xu ldquo(dm)-domination numbers of m-connected graphsrdquo Rapports de Recherche 1130 LRI UniversiteParis-Sud 1997

[111] S M Hedetniemi S T Hedetniemi and T V Wimer ldquoLineartime resource allocation algorithms for treesrdquo Tech Rep URI-014 Department of Mathematical Sciences Clemson Univer-sity 1987

[112] M Farber ldquoDomination independent domination and dualityin strongly chordal graphsrdquo Discrete Applied Mathematics vol7 no 2 pp 115ndash130 1984

[113] G S Domke and R C Laskar ldquoThe bondage and reinforcementnumbers of 120574

119891for some graphsrdquoDiscrete Mathematics vol 167-

168 pp 249ndash259 1997[114] G S Domke S T Hedetniemi and R C Laskar ldquoFractional

packings coverings and irredundance in graphsrdquo vol 66 pp227ndash238 1988

[115] E J Cockayne P A Dreyer Jr S M Hedetniemi andS T Hedetniemi ldquoRoman domination in graphsrdquo DiscreteMathematics vol 278 no 1ndash3 pp 11ndash22 2004

[116] I Stewart ldquoDefend the roman empirerdquo Scientific American vol281 pp 136ndash139 1999

[117] C S ReVelle and K E Rosing ldquoDefendens imperiumromanum a classical problem in military strategyrdquoThe Ameri-can Mathematical Monthly vol 107 no 7 pp 585ndash594 2000

[118] A Bahremandpour F-T Hu S M Sheikholeslami and J-MXu ldquoOn the Roman bondage number of a graphrdquo DiscreteMathematics Algorithms and Applications vol 5 no 1 ArticleID 1350001 15 pages 2013

[119] N Jafari Rad and L Volkmann ldquoRoman bondage in graphsrdquoDiscussiones Mathematicae Graph Theory vol 31 no 4 pp763ndash773 2011

[120] K Ebadi and L PushpaLatha ldquoRoman bondage and Romanreinforcement numbers of a graphrdquo International Journal ofContemporary Mathematical Sciences vol 5 pp 1487ndash14972010

[121] N Dehgardi S M Sheikholeslami and L Volkman ldquoOn theRoman k-bondage number of a graphrdquo AKCE InternationalJournal of Graphs and Combinatorics vol 8 pp 169ndash180 2011

[122] S Akbari and S Qajar ldquoA note on Roman bondage number ofgraphsrdquo Ars Combinatoria to Appear

[123] F-T Hu and J-M Xu ldquoRoman bondage numbers of somegraphsrdquo httparxivorgabs11093933

[124] N Jafari Rad and L Volkmann ldquoOn the Roman bondagenumber of planar graphsrdquo Graphs and Combinatorics vol 27no 4 pp 531ndash538 2011

[125] S Akbari M Khatirinejad and S Qajar ldquoA note on the romanbondage number of planar graphsrdquo Graphs and Combinatorics2012

[126] B Bresar M A Henning and D F Rall ldquoRainbow dominationin graphsrdquo Taiwanese Journal of Mathematics vol 12 no 1 pp213ndash225 2008

[127] B L Hartnell and D F Rall ldquoOn dominating the Cartesianproduct of a graph and 120572krdquo Discussiones Mathematicae GraphTheory vol 24 no 3 pp 389ndash402 2004

[128] N Dehgardi S M Sheikholeslami and L Volkman ldquoThe k-rainbow bondage number of a graphrdquo to appear in DiscreteApplied Mathematics

[129] J von Neumann and O Morgenstern Theory of Games andEconomic Behavior Princeton University Press Princeton NJUSA 1944

[130] C BergeTheorıe des Graphes et ses Applications 1958[131] O Ore Theory of Graphs vol 38 American Mathematical

Society Colloquium Publications 1962[132] E Shan and L Kang ldquoBondage number in oriented graphsrdquoArs

Combinatoria vol 84 pp 319ndash331 2007[133] J Huang and J-M Xu ldquoThe bondage numbers of extended

de Bruijn and Kautz digraphsrdquo Computers amp Mathematics withApplications vol 51 no 6-7 pp 1137ndash1147 2006

[134] J Huang and J-M Xu ldquoThe total domination and total bondagenumbers of extended de Bruijn and Kautz digraphsrdquoComputersamp Mathematics with Applications vol 53 no 8 pp 1206ndash12132007

[135] Y Shibata and Y Gonda ldquoExtension of de Bruijn graph andKautz graphrdquo Computers amp Mathematics with Applications vol30 no 9 pp 51ndash61 1995

[136] X Zhang J Liu and JMeng ldquoThebondage number in completet-partite digraphsrdquo Information Processing Letters vol 109 no17 pp 997ndash1000 2009

[137] D W Bange A E Barkauskas and P J Slater ldquoEfficient dom-inating sets in graphsrdquo in Applications of Discrete Mathematicspp 189ndash199 SIAM Philadelphia Pa USA 1988

[138] M Livingston and Q F Stout ldquoPerfect dominating setsrdquoCongressus Numerantium vol 79 pp 187ndash203 1990

[139] L Clark ldquoPerfect domination in random graphsrdquo Journal ofCombinatorial Mathematics and Combinatorial Computing vol14 pp 173ndash182 1993

[140] D W Bange A E Barkauskas L H Host and L H ClarkldquoEfficient domination of the orientations of a graphrdquo DiscreteMathematics vol 178 no 1ndash3 pp 1ndash14 1998

[141] A E Barkauskas and L H Host ldquoFinding efficient dominatingsets in oriented graphsrdquo Congressus Numerantium vol 98 pp27ndash32 1993

[142] I J Dejter and O Serra ldquoEfficient dominating sets in CayleygraphsrdquoDiscrete AppliedMathematics vol 129 no 2-3 pp 319ndash328 2003

[143] J Lee ldquoIndependent perfect domination sets in Cayley graphsrdquoJournal of Graph Theory vol 37 no 4 pp 213ndash219 2001

[144] WGu X Jia and J Shen ldquoIndependent perfect domination setsin meshes tori and treesrdquo Unpublished

[145] N Obradovic J Peters and G Ruzic ldquoEfficient domination incirculant graphswith two chord lengthsrdquo Information ProcessingLetters vol 102 no 6 pp 253ndash258 2007

[146] K Reji Kumar and G MacGillivray ldquoEfficient domination incirculant graphsrdquo Discrete Mathematics vol 313 no 6 pp 767ndash771 2013

[147] D Van Wieren M Livingston and Q F Stout ldquoPerfectdominating sets on cube-connected cyclesrdquo Congressus Numer-antium vol 97 pp 51ndash70 1993

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

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Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

International Journal of Combinatorics 9

From Theorem 38 and Corollary 40 we have 119887(119866) ⩽2Δ(119866) if 119866 is a 120574-critical graph with 120574(119866) ⩽ 3 The followingresult shows that this bound is not tight

Theorem 41 (Teschner [26] 1995) Let119866 be a 120574-critical graphgraph with 120574(119866) ⩽ 3 Then 119887(119866) ⩽ (32)Δ(119866)

Until now we have not known whether the bound givenin Theorem 41 is tight or not We state two conjectures on120574-critical graphs proposed by Samodivkin [20] The first ofthem is motivated byTheorems 22 and 19

Conjecture 42 (Samodivkin [20] 2008) For every connectednontrivial 120574-critical graph 119866

min119909isin119860+(119866)119910isin119861(119866minus119909)

119889119866(119909) +

1003816100381610038161003816119873119866(119910) cap 119860 (119866 minus 119909)

1003816100381610038161003816 ⩽3

2Δ (119866)

(15)

To state the second conjecture we need the followingresult on 120574-critical graphs

Proposition 43 If119866 is a 120574-critical graph then 120592(119866) ⩽ (Δ(119866)+1)(120574(119866)minus1)+1 moreover if the equality holds then119866 is regular

The upper bound in Proposition 43 is sharp in thesense that equality holds for the infinite class of 120574-criticalgraphs 119866

119898119899defined in the beginning of this subsection In

Proposition 43 the first result is due to Brigham et al [24] in1988 the second is due to Fulman et al [27] in 1995

For a 120574-critical graph 119866 with 120574 = 3 by Proposition 43120592(119866) ⩽ 2Δ(119866) + 3

Theorem 44 (Teschner [26] 1995) If 119866 is a 120574-critical graphwith 120574 = 3 and 120592(119866) = 2Δ(119866) + 3 then 119887(119866) ⩽ Δ + 1

We have not known yet whether the equality inTheorem 44holds or notHowever Samodivkin proposed thefollowing conjecture

Conjecture 45 (Samodivkin [20] 2008) If 119866 is a 120574-criticalgraph with (Δ(119866)+1)(120574minus1)+1 vertices then 119887(119866) = Δ(119866)+1

In general based on Theorem 41 Teschner proposed thefollowing conjecture

Conjecture 46 (Teschner [26] 1995) If119866 is a 120574-critical graphthen 119887(119866) ⩽ (32)Δ(119866) for 120574 ⩾ 4

We conclude this subsection with some remarks Graphswhich are minimal or critical with respect to a given prop-erty or parameter frequently play an important role in theinvestigation of that property or parameter Not only are suchgraphs of considerable interest in their own right but also aknowledge of their structure often aids in the developmentof the general theory In particular when investigating anyfinite structure a great number of results are proven byinduction Consequently it is desirable to learn as much aspossible about those graphs that are critical with respect toa given property or parameter so as to aid and abet suchinvestigations

In this subsection we survey some results on the bondagenumber for 120574-critical graphs Although these results are notvery perfect they provides a feasible method to approach thebondage number from different viewpoints In particular themethods given in Teschner [26] worth further explorationand development

The following proposition is maybe useful for us tofurther investigate the bondage number of a 120574-critical graph

Proposition 47 (Brigham et al [24] 1988) If 119866 has anonisolated vertex 119909 such that the subgraph induced by119873

119866(119909)

is a complete graph then 119866 is not 120574-critical

This simple fact shows that any 120574-critical graph containsno vertices of degree one

35 Bounds Implied by Domination In the preceding sub-section we introduced some upper bounds on the bondagenumbers for 120574-critical graphs by consideration of domina-tions In this subsection we introduce related results for ageneral graph with given domination number

Theorem 48 (Fink et al [8] 1990) For any connected graph119866 of order 119899 (⩾2)

(a) 119887(119866) ⩽ 119899 minus 1

(b) 119887(119866) ⩽ Δ(119866) + 1 if 120574(119866) = 2

(c) 119887(119866) ⩽ 119899 minus 120574(119866) + 1

While the upper bound of 119899minus1 on 119887(119866) is not particularlygood for many classes of graphs (eg trees and cycles) it isan attainable bound For example if 119866 is a complete 119905-partitegraph 119870

119905(2) for 119905 = 1198992 and 119899 ⩽ 4 an even integer then the

three bounds on 119887(119866) in Theorem 48 are sharpTeschner [26] consider a graphwith 120574(119866) = 1 or 120574(119866) = 2

The next result is almost trivial but useful

Lemma 49 Let 119866 be a graph with order 119899 and 120574(119866) = 1 119905 thenumber of vertices of degree 119899 minus 1 Then 119887(119866) = lceil1199052rceil

Since 119905 ⩽ Δ(119866) clearly Lemma 49 yields the followingresult immediately

Theorem 50 (Teschner [26] 1995) 119887(119866) ⩽ (12)Δ+1 for anygraph 119866 with 120574(119866) = 1

For a complete graph1198702119896+1

119887(1198702119896+1) = 119896+1 = (12)Δ+1

which shows that the upper bound given in Theorem 50 canbe attained For a graph119866with 120574(119866) = 2 byTheorem 59 laterthe upper bound given inTheorem 48 (b) can be also attainedby a 2-critical graph (see Proposition 37)

36 Two Conjectures In 1990 when Fink et al [8] introducedthe concept of the bondage number they proposed thefollowing conjecture

Conjecture 51 119887(119866) ⩽ Δ(119866) + 1 for any nonempty graph 119866

10 International Journal of Combinatorics

Although some results partially support Conjecture 51Teschner [28] in 1993 found a counterexample to this con-jecture the cartesian product 119870

3times 119870

3 which shows that

119887(119866) = Δ(119866) + 2If a graph 119866 is a counterexample to Conjecture 51 it

must be a 120574-critical graph by Theorem 13 That is why thevertex domination-critical graphs are of special interest in theliterature

Now we return to Conjecture 51 Hartnell and Rall [17]and Teschner [29] independently proved that 119887(119866) can bemuch greater than Δ(119866) by showing the following result

Theorem 52 (Hartnell and Rall [17] 1994 Teschner [29]1996) For an integer 119899 ⩾ 3 let 119866

119899be the cartesian product

119870119899times 119870

119899 Then 119887(119866

119899) = 3(119899 minus 1) = (32)Δ(119866

119899)

This theorem shows that there exist no upper boundsof the form 119887(119866) ⩽ Δ(119866) + 119888 for any integer 119888 Teschner[26] proved that 119887(119866) ⩽ (32)Δ(119866) for any graph 119866 with120574(119866) ⩽ 2 (see Theorems 50 and 48) and for 120574-critical graphswith 120574(119866) = 3 and proposed the following conjecture

Conjecture 53 (Teschner [26] 1995) 119887(119866) ⩽ (32)Δ(119866) forany graph 119866

We believe that this conjecture is true but so far no muchwork about it has been known

4 Lower Bounds

Since the bondage number is defined as the smallest numberof edges whose removal results in an increase of dominationnumber each constructive method that creates a concretebondage set leads to an upper bound on the bondage numberFor that reason it is hard to find lower bounds Neverthelessthere are still a few lower bounds

41 Bounds Implied by Subgraphs The first lower boundon the bondage number is gotten in terms of its spanningsubgraph

Theorem 54 (Teschner [10] 1997) Let 119867 be a spanningsubgraph of a nonempty graph119866 If 120574(119867) = 120574(119866) then 119887(119867) ⩽119887(119866)

By Theorem 1 119887(119862119899) = 3 if 119899 equiv 1 (mod 3) and 119887(119862

119899) =

2 otherwise 119887(119875119899) = 2 if 119899 equiv 1 (mod 3) and 119887(119875

119899) = 1

otherwise From these results and Theorem 54 we get thefollowing two corollaries

Corollary 55 If119866 is hamiltonianwith order 119899 ⩾ 3 and 120574(119866) =lceil1198993rceil then 119887(119866) ⩾ 2 and in addition 119887(119866) ⩾ 3 if 119899 equiv 1 (mod3)

Corollary 56 If 119866 with order 119899 ⩾ 2 has a hamiltonian pathand 120574(119866) = lceil1198993rceil then 119887(119866) ⩾ 2 if 119899 equiv 1 (mod 3)

42 Other Bounds The vertex covering number 120573(119866) of 119866 isthe minimum number of vertices that are incident with all

Figure 5 A 120574-critical graph with 120573 = 120574 = 4 120575 = 2 and 119887 = 3

edges in 119866 If 119866 has no isolated vertices then 120574(119866) ⩽ 120573(119866)clearly In [30] Volkmann gave a lot of graphs with 120573 = 120574

Theorem 57 (Teschner [10] 1997) Let119866 be a graph If 120574(119866) =120573(119866) then

(a) 119887(119866) ⩾ 120575(119866)

(b) 119887(119866) ⩾ 120575(119866) + 1 if 119866 is a 120574-critical graph

The graph shown in Figure 5 shows that the bound givenin Theorem 57(b) is sharp For the graph 119866 it is a 120574-criticalgraph and 120574(119866) = 120573(119866) = 4 By Theorem 57 we have 119887(119866) ⩾3 On the other hand 119887(119866) ⩽ 3 by Theorem 16 Thus 119887(119866) =3

Proposition 58 (Sanchis [31] 1991) Let 119866 be a graph 119866 oforder 119899 (⩾ 6) and size 120576 If 119866 has no isolated vertices and3 ⩽ 120574(119866) ⩽ 1198992 then 120576 ⩽ ( 119899minus120574(119866)+1

2)

Using the idea in the proof of Theorem 57 every upperbound for 120574(119866) can lead to a lower bound for 119887(119866) Inthis way Teschner [10] obtained another lower bound fromProposition 58

Theorem59 (Teschner [10] 1997) Let119866 be a graph119866 of order119899 (⩾6) and size 120576 If 2 ⩽ 120574(119866) ⩽ 1198992 minus 1 then

(a) 119887(119866) ⩾ min120575(119866) 120576 minus ( 119899minus120574(119866)2)

(b) 119887(119866) ⩾ min120575(119866) + 1 120576 minus ( 119899minus120574(119866)2) if 119866 is a 120574-critical

graph

The lower bound in Theorem 59(b) is sharp for a classof 120574-critical graphs with domination number 2 Let 119866 be thegraph obtained from complete graph119870

2119905(119905 ⩾ 2) by removing

a perfect matching By Proposition 37 119866 is a 120574-critical graphwith 120574(119866) = 2 Then 119887(119866) ⩾ 120575(119866) + 1 = Δ(119866) + 1 byTheorem 59 and 119887(119866) ⩽ Δ(119866) + 1 by Theorem 38 and so119887(119866) = Δ(119866) + 1 = 2119905 minus 1

As far as the author knows there are no more lowerbounds in the present literature In view of applications ofthe bondage number a network is vulnerable if its bondagenumber is small while it is stable if its bondage number islargeTherefore better lower bounds let us learn better aboutthe stability of networks from this point of view Thus it isof great significance to seek more lower bounds for variousclasses of graphs

International Journal of Combinatorics 11

5 Results on Graph Operations

Generally speaking it is quite difficult to determine the exactvalue of the bondage number for a given graph since itstrongly depends on the dominating number of the graphThus determining bondage numbers for some special graphsis interesting if the dominating numbers of those graphs areknown or can be easily determined In this section we willintroduce some known results on bondage numbers for somespecial classes of graphs obtained by some graph operations

51 Cartesian Product Graphs Let 1198661= (119881

1 119864

1) and 119866

2=

(1198812 119864

2) be two graphs The Cartesian product of 119866

1and 119866

2

is an undirected graph denoted by 1198661times 119866

2 where 119881(119866

1times

1198662) = 119881

1times 119881

2 two distinct vertices 119909

11199092and 119910

11199102 where

1199091 119910

1isin 119881(119866

1) and 119909

2 119910

2isin 119881(119866

2) are linked by an edge in

1198661times 119866

2if and only if either 119909

1= 119910

1and 119909

21199102isin 119864(119866

2) or

1199092= 119910

2and 119909

11199101isin 119864(119866

1) The Cartesian product is a very

effective method for constructing a larger graph from severalspecified small graphs

Theorem 60 (Dunbar et al [21] 1998) For 119899 ⩾ 3

119887 (119862119899times 119870

2) =

2 119894119891 119899 equiv 0 119900119903 1 (mod 4) 3 119894119891 119899 equiv 3 (mod 4) 4 119894119891 119899 equiv 2 (mod 4)

(16)

For the Cartesian product 119862119899times 119862

119898of two cycles 119862

119899and

119862119898 where 119899 ⩾ 4 and 119898 ⩾ 3 Klavzar and Seifter [32]

determined 120574(119862119899times 119862

119898) for some 119899 and 119898 For example

120574(119862119899times119862

4) = 119899 for 119899 ⩾ 3 and 120574(119862

119899times119862

3) = 119899minuslfloor1198994rfloor for 119899 ⩾ 4

Kim [33] determined 119887(1198623times 119862

3) = 6 and 119887(119862

4times 119862

4) = 4 For

a general 119899 ⩾ 4 the exact values of the bondage numbers of119862119899times 119862

3and 119862

119899times 119862

4are determined as follows

Theorem 61 (Sohn et al [34] 2007) For 119899 ⩾ 4

119887 (119862119899times 119862

3) =

2 119894119891 119899 equiv 0 (mod 4) 4 119894119891 119899 equiv 1 119900119903 2 (mod 4) 5 119894119891 119899 equiv 3 (mod 4)

(17)

Theorem62 (Kang et al [35] 2005) 119887(119862119899times119862

4) = 4 for 119899 ⩾ 4

For larger 119898 and 119899 Huang and Xu [36] obtained thefollowing result see Theorem 197 for more details

Theorem 63 (Huang and Xu [36] 2008) 119887(1198625119894times119862

5119895) = 3 for

any positive integers 119894 and 119895

Xiang et al [37] determined that for 119899 ⩾ 5

119887 (119862119899times 119862

5)

= 3 if 119899 equiv 0 or 1 (mod 5) = 4 if 119899 equiv 2 or 4 (mod 5) ⩽ 7 if 119899 equiv 3 (mod 5)

(18)

For the Cartesian product 119875119899times119875

119898of two paths 119875

119899and 119875

119898

Jacobson and Kinch [38] determined 120574(119875119899times119875

2) = lceil(119899+1)2rceil

120574(119875119899times 119875

3) = 119899 minus lfloor(119899 minus 1)4rfloor and 120574(119875

119899times 119875

4) = 119899 + 1 if 119899 =

1 2 3 5 6 9 and 119899 otherwiseThe bondage number of119875119899times119875

119898

for 119899 ⩾ 4 and 2 ⩽ 119898 ⩽ 4 is determined as follows (Li [39] alsodetermined 119887(119875

119899times 119875

2))

Theorem 64 (Hu and Xu [40] 2012) For 119899 ⩾ 4

119887 (119875119899times 119875

2) =

1 119894119891 119899 equiv 1 (mod 2)2 119894119891 119899 equiv 0 (mod 2)

119887 (119875119899times 119875

3) =

1 119894119891 119899 equiv 1 119900119903 2 (mod 4)2 119894119891 119899 equiv 0 119900119903 3 (mod 4)

119887 (119875119899times 119875

4) = 1 119891119900119903 119899 ⩾ 14

(19)

From the proof of Theorem 64 we find that if 119875119899=

0 1 119899 minus 1 and 119875119898= 0 1 119898 minus 1 then the removal

of the vertex (0 0) in 119875119899times119875

119898does not change the domination

number If119898 increases the effect of (0 0) for the dominationnumber will be smaller and smaller from the probabilityTherefore we expect it is possible that 120574(119875

119899times 119875

119898minus (0 0)) =

120574(119875119899times 119875

119898) for119898 ⩾ 5 and give the following conjecture

Conjecture 65 119887(119875119899times 119875

119898) ⩽ 2 for119898 ⩾ 5 and 119899 ⩾ 4

52 Block Graphs and Cactus Graphs In this subsection weintroduce some results for corona graphs block graphs andcactus graphs

The corona1198661∘ 119866

2 proposed by Frucht and Harary [41]

is the graph formed from a copy of 1198661and 120592(119866

1) copies of

1198662by joining the 119894th vertex of 119866

1to the 119894th copy of 119866

2 In

particular we are concernedwith the corona119867∘1198701 the graph

formed by adding a new vertex V119894 and a new edge 119906

119894V119894for

every vertex 119906119894in119867

Theorem 66 (Carlson and Develin [42] 2006) 119887(119867 ∘ 1198701) =

120575(119867) + 1

This is a very important result which is often used toconstruct a graph with required bondage number In otherwords this result implies that for any given positive integer 119887there is a graph 119866 such that 119887(119866) = 119887

A block graph is a graph whose blocks are completegraphs Each block in a cactus graph is either a cycle or a119870

2

If each block of a graph is either a complete graph or a cyclethen we call this graph a block-cactus graph

Theorem67 (Teschner [43] 1997) 119887(119866) le Δ(119866) for any blockgraph 119866

Teschner [43] characterized all block graphs with 119887(119866) =Δ(119866) In the same paper Teschner found that 120574-criticalgraphs are instrumental in determining bounds for thebondage number of cactus and block graphs and obtained thefollowing result

Theorem 68 (Teschner [43] 1997) 119887(119866) ⩽ 3 for anynontrivial cactus graph 119866

This bound can be achieved by 119867 ∘ 1198701where 119867 is

a nontrivial cactus graph without vertices of degree one

12 International Journal of Combinatorics

by Theorem 66 In 1998 Dunbar et al [21] proposed thefollowing problem

Problem 2 Characterize all cactus graphs with bondagenumber 3

Some upper bounds for block-cactus graphs were alsoobtained

Theorem 69 (Dunbar et al [21] 1998) Let 119866 be a connectedblock-cactus graph with at least two blocksThen 119887(119866) ⩽ Δ(119866)

Theorem 70 (Dunbar et al [21] 1998) Let 119866 be a connectedblock-cactus graph which is neither a cactus graph nor a blockgraph Then 119887(119866) ⩽ Δ(119866) minus 1

53 Edge-Deletion and Edge-Contraction In order to obtainfurther results of the bondage number we may consider howthe bondage number changes under some operations of agraph 119866 which remain the domination number unchangedor preserve some property of 119866 such as planarity Twosimple operations satisfying these requirements are the edge-deletion and the edge-contraction

Theorem 71 (Huang and Xu [44] 2012) Let 119866 be any graphand 119890 isin 119864(119866) Then 119887(119866 minus 119890) ⩾ 119887(119866) minus 1 Moreover 119887(119866 minus 119890) ⩽119887(119866) if 120574(119866 minus 119890) = 120574(119866)

FromTheorem 1 119887(119862119899minus 119890) = 119887(119875

119899) = 119887(119862

119899) minus 1 for any 119890

in119862119899 which shows that the lower bound on 119887(119866minus119890) is sharp

The upper bound can reach for any graph 119866 with 119887(119866) ⩾ 2Next we consider the effect of the edge-contraction on

the bondage number Given a graph 119866 the contraction of 119866by the edge 119890 = 119909119910 denoted by 119866119909119910 is the graph obtainedfrom 119866 by contracting two vertices 119909 and 119910 to a new vertexand then deleting all multiedges It is easy to observe that forany graph 119866 120574(119866) minus 1 ⩽ 120574(119866119909119910) ⩽ 120574(119866) for any edge 119909119910 of119866

Theorem 72 (Huang and Xu [44] 2012) Let 119866 be any graphand 119909119910 be any edge in119866 If119873

119866(119909)cap119873

119866(119910) = 0 and 120574(119866119909119910) =

120574(119866) then 119887(119866119909119910) ⩾ 119887(119866)

We can use examples 119862119899and 119870

119899to show that the

conditions ofTheorem 72 are necessary and the lower boundinTheorem 72 is tight Clearly 120574(119862

119899) = lceil1198993rceil and 120574(119870

119899) = 1

for any edge 119909119910 119862119899119909119910 = 119862

119899minus1and 119870

119899119909119910 = 119870

119899minus1 By

Theorem 1 if 119899 equiv 1 (mod 3) then 120574(119862119899119909119910) lt 120574(119866) and

119887(119862119899119909119910) = 2 lt 3 = 119887(119862

119899) Thus the result in Theorem 72

is generally invalid without the hypothesis 120574(119866119909119910) = 120574(119866)Furthermore the condition 119873

119866(119909) cap 119873

119866(119910) = 0 cannot be

omitted even if 120574(119866119909119910) = 120574(119866) since for odd 119899 119887(119870119899119909119910) =

lceil(119899 minus 1)2rceil lt lceil1198992rceil = 119887(119870119899) by Theorem 1

On the other hand 119887(119862119899119909119910) = 119887(119862

119899) = 2 if 119899 equiv 0 (mod

3) and 119887(119870119899119909119910) = 119887(119870

119899) if 119899 is even which shows that the

equality in 119887(119866119909119910) ⩾ 119887(119866) may hold Thus the bound inTheorem 72 is tight However provided all the conditions119887(119866119909119910) can be arbitrarily larger than 119887(119866) Given a graph119867let119866 be the graph formed from119867∘119870

1by adding a new vertex

119909 and joining it to an vertex 119910 of degree one in119867 ∘ 1198701 Then

119866119909119910 = 119867 ∘ 1198701 120574(119866) = 120574(119866119909119910) and 119873

119866(119909) cap 119873

119866(119910) = 0

But 119887(119866) = 1 since 120574(119866 minus 119909119910) = 120574(119866119909119910) + 1 and 119887(119866119909119910) =120575(119867) + 1 byTheorem 66The gap between 119887(119866) and 119887(119866119909119910)is 120575(119867)

6 Results on Planar Graphs

From Section 2 we have seen that the bondage numberfor a tree has been completely solved Moreover a lineartime algorithm to compute the bondage number of a tree isdesigned by Hartnell et al [15] It is quite natural to considerthe bondage number for a planar graph In this section westate some results and problems on the bondage number fora planar graph

61 Conjecture on Bondage Number As mentioned inSection 3 the bondage number can be much larger than themaximum degree But for a planar graph 119866 the bondagenumber 119887(119866) cannot exceed Δ(119866) too much It is clear that119887(119866) ⩽ Δ(119866) + 4 by Corollary 15 since 120575(119866) ⩽ 5 for anyplanar graph119866 In 1998Dunbar et al [21] posed the followingconjecture

Conjecture 73 If 119866 is a planar graph then 119887(119866) ⩽ Δ(119866) + 1

Because of its attraction it immediately became the focusof attention when this conjecture is proposed In fact themain aim concerning the research of the bondage number fora planar graph is focused on this conjecture

It has been mentioned in Theorems 1 and 60 that119887(119862

3119896+1) = 3 = Δ + 1 and 119887(119862

4119896+2times 119870

2) = 4 = Δ + 1 It

is known that 119887(1198706minus 119872) = 5 = Δ + 1 where119872 is a perfect

matching of the complete graph1198706These examples show that

if Conjecture 73 is true then the upper bound is sharp for2 ⩽ Δ ⩽ 4

Here we note that it is sufficient to prove this conjecturefor connected planar graphs since the bondage number of adisconnected graph is simply the minimum of the bondagenumbers of its components

The first paper attacking this conjecture is due to Kangand Yuan [45] which confirmed the conjecture for everyconnected planar graph 119866 with Δ ⩾ 7 The proofs are mainlybased onTheorems 16 and 18

Theorem 74 (Kang and Yuan [45] 2000) If 119866 is a connectedplanar graph then 119887(119866) ⩽ min8 Δ(119866) + 2

Obviously in view of Theorem 74 Conjecture 73 is truefor any connected planar graph with Δ ⩾ 7

62 Bounds Implied by Degree Conditions As we have seenfrom Theorem 74 to attack Conjecture 73 we only need toconsider connected planar graphs with maximum Δ ⩽ 6Thus studying the bondage number of planar graphs bydegree conditions is of significanceThe first result on boundsimplied by degree conditions is obtained by Kang and Yuan[45]

International Journal of Combinatorics 13

Theorem 75 (Kang and Yuan [45] 2000) If 119866 is a connectedplanar graph without vertices of degree 5 then 119887(119866) ⩽ 7

Fischermann et al [46] generalized Theorem 75 as fol-lows

Theorem 76 (Fischermann et al [46] 2003) Let 119866 be aconnected planar graph and 119883 the set of vertices of degree 5which have distance at least 3 to vertices of degrees 1 2 and3 If all vertices in 119883 not adjacent with vertices of degree 4are independent and not adjacent to vertices of degree 6 then119887(119866) ⩽ 7

Clearly if119866 has no vertices of degree 5 then119883 = 0 ThusTheorem 76 yields Theorem 75

Use 119899119894to denote the number of vertices of degree 119894 in 119866

for each 119894 = 1 2 Δ(119866) Using Theorem 18 Fischermannet al obtained the following two theorems

Theorem 77 (Fischermann et al [46] 2003) For any con-nected planar graph 119866

(1) 119887(119866) ⩽ 7 if 1198995lt 2119899

2+ 3119899

3+ 2119899

4+ 12

(2) 119887(119866) ⩽ 6 if 119866 contains no vertices of degrees 4 and 5

Theorem 78 (Fischermann et al [46] 2003) For any con-nected planar graph 119866 119887(119866) ⩽ Δ(119866) + 1 if

(1) Δ(119866) = 6 and every edge 119890 = 119909119910 with 119889119866(119909) = 5 and

119889119866(119910) = 6 is contained in at most one triangle

(2) Δ(119866) = 5 and no triangle contains an edge 119890 = 119909119910 with119889119866(119909) = 5 and 4 ⩽ 119889

119866(119910) ⩽ 5

63 Bounds Implied by Girth Conditions The girth 119892(119866) of agraph 119866 is the length of the shortest cycle in 119866 If 119866 has nocycles we define 119892(119866) = infin

Combining Theorem 74 with Theorem 78 we find thatif a planar graph contains no triangles and has maximumdegree Δ ⩾ 5 then Conjecture 73 holds This fact motivatedFischermann et alrsquos [46] attempt to attack Conjecture 73 bygirth restraints

Theorem 79 (Fischermann et al [46] 2003) For any con-nected planar graph 119866

119887 (119866) ⩽

6 119894119891 119892 (119866) ⩾ 4

5 119894119891 119892 (119866) ⩾ 5

4 119894119891 119892 (119866) ⩾ 6

3 119894119891 119892 (119866) ⩾ 8

(20)

The first result in Theorem 79 shows that Conjecture 73is valid for all connected planar graphs with 119892(119866) ⩾ 4 andΔ(119866) ⩾ 5 It is easy to verify that the second result inTheorem 79 implies that Conjecture 73 is valid for all not3-regular graphs of girth 119892(119866) ⩾ 5 which is stated in thefollowing corollary

Corollary 80 For any connected planar graph 119866 if 119866 is not3-regular and 119892(119866) ⩾ 5 then 119887(119866) ⩽ Δ(119866) + 1

The first result in Theorem 79 also implies that if 119866 isa connected planar graph with no cycles of length 3 then119887(119866) ⩽ 6 Hou et al [47] improved this result as follows

Theorem 81 (Hou et al [47] 2011) For any connected planargraph 119866 if 119866 contains no cycles of length 119894 (4 ⩽ 119894 ⩽ 6) then119887(119866) ⩽ 6

Since 119887(1198623119896+1) = 3 for 119896 ⩾ 3 (see Theorem 1) the

last bound in Theorem 79 is tight Whether other bounds inTheorem 79 are tight remains open In 2003 Fischermann etal [46] posed the following conjecture

Conjecture 82 For any connected planar graph 119866 119887(119866) ⩽ 7and

119887 (119866) ⩽ 5 119894119891 119892 (119866) ⩾ 4

4 119894119891 119892 (119866) ⩾ 5(21)

We conclude this subsection with a question on bondagenumbers of planar graphs

Question 1 (Fischermann et al [46] 2003) Is there a planargraph 119866 with 6 ⩽ 119887(119866) ⩽ 8

In 2006 Carlson andDevelin [42] showed that the corona119866 = 119867 ∘ 119870

1for a planar graph 119867 with 120575(119867) = 5 has the

bondage number 119887(119866) = 120575(119867) + 1 = 6 (see Theorem 66)Since the minimum degree of planar graphs is at most 5then 119887(119866) can attach 6 If we take 119867 as the graph of theicosahedron then 119866 = 119867 ∘ 119870

1is such an example The

question for the existence of planar graphs with bondagenumber 7 or 8 remains open

64 Comments on the Conjectures Conjecture 73 is true forall connected planar graphs with minimum degree 120575 ⩽ 2 byTheorem 16 or maximum degree Δ ⩾ 7 by Theorem 74 ornot 120574-critical planar graphs by Theorem 13 Thus to attackConjecture 73 we only need to consider connected criticalplanar graphs with degree-restriction 3 ⩽ 120575 ⩽ Δ ⩽ 6

Recalling and anatomizing the proofs of all results men-tioned in the preceding subsections on the bondage numberfor connected planar graphs we find that the proofs of theseresults strongly depend upon Theorem 16 or Theorem 18 Inother words a basic way used in the proofs is to find twovertices 119909 and 119910 with distance at most two in a consideredplanar graph 119866 such that

119889119866(119909) + 119889

119866(119910) or 119889

119866(119909) + 119889

119866(119910) minus

1003816100381610038161003816119873119866(119909) cap 119873

119866(119910)1003816100381610038161003816

(22)

which bounds 119887(119866) is as small as possible Very recentlyHuang and Xu [44] have considered the following parameter

119861 (119866) = min119909119910isin119881(119866)

119889119909119910 1 ⩽ 119889

119866(119909 119910) ⩽ 2

cup 119889119909119910minus 119899

119909119910 119889 (119909 119910) = 1

(23)

where 119889119909119910= 119889

119866(119909) + 119889

119866(119910) minus 1 and 119899

119909119910= |119873

119866(119909) cap 119873

119866(119910)|

14 International Journal of Combinatorics

It follows fromTheorems 16 and 18 that

119887 (119866) ⩽ 119861 (119866) (24)

The proofs given in Theorems 74 and 79 indeed imply thefollowing stronger results

Theorem 83 If 119866 is a connected planar graph then

119861 (119866) ⩽

min 8 Δ (119866) + 26 119894119891 119892 (119866) ⩾ 4

5 119894119891 119892 (119866) ⩾ 5

4 119894119891 119892 (119866) ⩾ 6

3 119894119891 119892 (119866) ⩾ 8

(25)

Thus using Theorem 16 or Theorem 18 if we can proveConjectures 73 and 82 then we can prove the followingstatement

Statement If 119866 is a connected planar graph then

119861 (119866) ⩽

Δ (119866) + 1

7

5 if 119892 (119866) ⩾ 44 if 119892 (119866) ⩾ 5

(26)

It follows from (24) that Statement implies Conjectures 73and 82 However Huang and Xu [44] gave examples to showthat none of conclusions in Statement is true As a result theystated the following conclusion

Theorem 84 (Huang and Xu [44] 2012) It is not possible toprove Conjectures 73 and 82 if they are right using Theorems16 and 18

Therefore a new method is needed to prove or disprovethese conjectures At the present time one cannot prove theseconjectures and may consider to disprove them If one ofthese conjectures is invalid then there exists a minimumcounterexample 119866 with respect to 120592(119866) + 120576(119866) In order toobtain a minimum counterexample one may consider twosimple operations satisfying these requirements the edge-deletion and the edge-contractionwhich decrease 120592(119866)+120576(119866)and preserve planarity However applying Theorems 71 and72 Huang and Xu [44] presented the following results

Theorem 85 (Huang and Xu [44] 2012) It is not possible toconstruct minimum counterexamples to Conjectures 73 and 82by the operation of an edge-deletion or an edge-contraction

7 Results on Crossing Number Restraints

It is quite natural to generalize the known results on thebondage number for planar graphs to formore general graphsin terms of graph-theoretical parameters In this section weconsider graphs with crossing number restraints

The crossing number cr(119866) of 119866 is the smallest numberof pairwise intersections of its edges when 119866 is drawn in theplane If cr(119866) = 0 then 119866 is a planar graph

71 General Methods Use 119899119894to denote the number of vertices

of degree 119894 in119866 for 119894 = 1 2 Δ(119866) Huang and Xu obtainedthe following results

Theorem 86 (Huang and Xu [48] 2007) For any connectedgraph 119866

119887 (119866) ⩽

6 119894119891 119892 (119866) ⩾ 4 119886119899119889 2cr (119866) lt 121198861

5 119894119891 119892 (119866) ⩾ 5 119886119899119889 6cr (119866) lt 161198862

4 119894119891 119892 (119866) ⩾ 6 119886119899119889 4cr (119866) lt 141198863

3 119894119891 119892 (119866) ⩾ 8 119886119899119889 6cr (119866) lt 161198864

(27)

where 1198861= 119899

1+ 2119899

2+ 2119899

3+sum

Δ

119894=8(119894 minus 7)119899

119894+ 8 119886

2= 3119899

1+ 6119899

2+

51198993+sum

Δ

119894=7(3119894minus18)119899

119894+20 119886

3= 119899

1+2119899

2+sum

Δ

119894=6(2119894minus10)119899

119894+12

and 1198864= sum

Δ

119894=5(3119894 minus 12)119899

119894+ 16

Simplifying the conditions in Theorem 86 we obtain thefollowing corollaries immediately

Corollary 87 (Huang and Xu [48] 2007) For any connectedgraph 119866

119887 (119866) ⩽

6 119894119891 119892 (119866) ⩾ 4 119886119899119889 cr (119866) ⩽ 3

5 119894119891 119892 (119866) ⩾ 5 119886119899119889 cr (119866) ⩽ 4

4 119894119891 119892 (119866) ⩾ 6 119886119899119889 cr (119866) ⩽ 2

3 119894119891 119892 (119866) ⩾ 8 119886119899119889 cr (119866) ⩽ 2

(28)

Corollary 88 (Huang and Xu [48] 2007) For any connectedgraph 119866

(a) 119887(119866) ⩽ 6 if 119866 is not 4-regular cr(119866) = 4 and 119892(119866) ⩾ 4

(b) 119887(119866) ⩽ Δ(119866) + 1 if 119866 is not 3-regular cr(119866) ⩽ 4 and119892(119866) ⩾ 5

(c) 119887(119866) ⩽ 4 if 119866 is not 3-regular cr(119866) = 3 and 119892(119866) ⩾ 6

(d) 119887(119866) ⩽ 3 if cr(119866) = 3 119892(119866) ⩾ 8 and Δ(119866) ⩾ 5

These corollaries generalize some known results for pla-nar graphs For example Corollary 87 contains Theorem 79Corollary 88 (b) contains Corollary 80

Theorem 89 (Huang and Xu [48] 2007) For any connectedgraph 119866 if cr(119866) ⩽ 119899

3(119866) + 119899

4(119866) + 3 then 119887(119866) ⩽ 8

Corollary 90 (Huang and Xu [48] 2007) For any connectedgraph 119866 if cr(119866) ⩽ 3 then 119887(119866) ⩽ 8

Perhaps being unaware of this result in 2010 Ma et al[49] proved that 119887(119866) ⩽ 12 for any graph 119866 with cr(119866) = 1

Theorem91 (Huang and Xu [48] 2007) Let119866 be a connectedgraph and 119868 = 119909 isin 119881(119866) 119889

119866(119909) = 5 119889

119866(119909 119910) ⩾ 3 if

International Journal of Combinatorics 15

119889119866(119910) ⩽ 3 and 119889

119866(119910) = 4 for every 119910 isin 119873

119866(119909) If 119868 is

independent no vertices adjacent to vertices of degree 6 and

cr (119866) lt max5119899

3+ |119868| minus 2119899

4+ 28

117119899

3+ 40

16 (29)

then 119887(119866) ⩽ 7

Corollary 92 (Huang and Xu [48] 2007) Let 119866 be aconnected graph with cr(119866) ⩽ 2 If 119868 = 119909 isin 119881(119866) 119889

119866(119909) =

5 119889119866(119910 119909) ⩾ 3 if 119889

119866(119910) ⩽ 3 and 119889

119866(119910) = 4 for every 119910 isin

119873119866(119909) is independent and no vertices adjacent to vertices of

degree 6 then 119887(119866) ⩽ 7

Theorem93 (Huang andXu [48] 2007) Let119866 be a connectedgraph If 119866 satisfies

(a) 5cr(119866) + 1198995lt 2119899

2+ 3119899

3+ 2119899

4+ 12 or

(b) 7cr(119866) + 21198995lt 3119899

2+ 4119899

4+ 24

then 119887(119866) ⩽ 7

Proposition 94 (Huang and Xu [48] 2007) Let 119866 be aconnected graph with no vertices of degrees four and five Ifcr(119866) ⩽ 2 then 119887(119866) ⩽ 6

Theorem 95 (Huang and Xu [48] 2007) If 119866 is a connectedgraph with cr(119866) ⩽ 4 and not 4-regular when cr(119866) = 4 then119887(119866) ⩽ Δ(119866) + 2

The above results generalize some known results forplanar graphs For example Corollary 90 and Theorem 95contain Theorem 74 the first condition in Theorem 93 con-tains the second condition inTheorem 77

From Corollary 88 and Theorem 95 we suggest the fol-lowing questions

Question 2 Is there a

(a) 4-regular graph 119866 with cr(119866) = 4 such that 119887(119866) ⩾ 7

(b) 3-regular graph 119866 with cr ⩽ 4 and 119892(119866) ⩾ 5 such that119887(119866) = 5

(c) 3-regular graph 119866 with cr = 3 and 119892(119866) = 6 or 7 suchthat 119887(119866) = 5

72 Carlson-Develin Methods In this subsection we intro-duce an elegant method presented by Carlson and Develin[42] to obtain some upper bounds for the bondage numberof a graph

Suppose that 119866 is a connected graph We say that 119866 hasgenus 120588 if 119866 can be embedded in a surface 119878 with 120588 handlessuch that edges are pairwise disjoint except possibly for end-vertices Let119866 be an embedding of119866 in surface 119878 and let120601(119866)denote the number of regions in 119866 The boundary of everyregion contains at least three edges and every edge is on theboundary of atmost two regions (the two regions are identicalwhen 119890 is a cut-edge) For any edge 119890 of 119866 let 1199031

119866(119890) and 1199032

119866(119890)

be the numbers of edges comprising the regions in 119866 whichthe edge 119890 borders It is clear that every 119890 = 119909119910 isin 119864(119866)

1199032

119866(119890) ⩾ 119903

1

119866(119890) ⩾ 4 if 1003816100381610038161003816119873119866

(119909) cap 119873119866(119910)1003816100381610038161003816 = 0

1199032

119866(119890) ⩾ 4 119903

1

119866(119890) ⩾ 3 if 1003816100381610038161003816119873119866

(119909) cap 119873119866(119910)1003816100381610038161003816 = 1

1199032

119866(119890) ⩾ 119903

1

119866(119890) ⩾ 3 if 1003816100381610038161003816119873119866

(119909) cap 119873119866(119910)1003816100381610038161003816 ⩾ 2

(30)

Following Carlson and Develin [42] for any edge 119890 = 119909119910 of119866 we define

119863119866(119890) =

1

119889119866(119909)+

1

119889119866(119910)

+1

1199031

119866(119890)+

1

1199032

119866(119890)minus 1 (31)

By the well-known Eulerrsquos formula

120592 (119866) minus 120576 (119866) + 120601 (119866) = 2 minus 2120588 (119866) (32)

it is easy to see that

sum

119890isin119864(119866)

119863119866(119890) = 120592 (119866) minus 120576 (119866) + 120601 (119866) = 2 minus 2120588 (119866) (33)

If 119866 is a planar graph that is 120588(119866) = 0 then

sum

119890isin119864(119866)

119863119866(119890) = 120592 (119866) minus 120576 (119866) + 120601 (119866) = 2 (34)

Combining these formulas withTheorem 18 Carlson andDevelin [42] gave a simple and intuitive proof ofTheorem 74and obtained the following result

Theorem 96 (Carlson and Develin [42] 2006) Let 119866 be aconnected graph which can be embedded on a torus Then119887(119866) ⩽ Δ(119866) + 3

Recently Hou and Liu [50] improved Theorem 96 asfollows

Theorem 97 (Hou and Liu [50] 2012) Let 119866 be a connectedgraph which can be embedded on a torus Then 119887(119866) ⩽ 9Moreover if Δ(119866) = 6 then 119887(119866) ⩽ 8

By the way Cao et al [51] generalized the result inTheorem 79 to a connected graph 119866 that can be embeddedon a torus that is

119887 (119866) le

6 if 119892 (119866) ⩾ 4 and 119866 is not 4-regular5 if 119892 (119866) ⩾ 54 if 119892 (119866) ⩾ 6 and 119866 is not 3-regular3 if 119892 (119866) ⩾ 8

(35)

Several authors used this method to obtain some resultson the bondage number For example Fischermann et al[46] used this method to prove the second conclusion inTheorem 78 Recently Cao et al [52] have used thismethod todeal with more general graphs with small crossing numbersFirst they found the following property

Lemma 98 120575(119866) ⩽ 5 for any graph 119866 with cr(119866) ⩽ 5

16 International Journal of Combinatorics

This result implies that 119887(119866) ⩽ Δ(119866) + 4 by Corollary 15for any graph 119866 with cr(119866) ⩽ 5 Cao et al established thefollowing relation in terms of maximum planar subgraphs

Lemma 99 (Cao et al [52]) Let 119866 be a connected graph withcrossing number cr(119866) For any maximum planar subgraph119867of 119866

sum

119890isin119864(119867)

119863119866(119890) ⩾ 2 minus

2cr (119866)120575 (119866)

(36)

Using Carlson-Develin method and combiningLemma 98 with Lemma 99 Cao et al proved the followingresults

Theorem 100 (Cao et al [52]) For any connected graph 119866119887(119866) ⩽ Δ(119866) + 2 if 119866 satisfies one of the following conditions

(a) cr(119866) ⩽ 3(b) cr(119866) = 4 and 119866 is not 4-regular(c) cr(119866) = 5 and 119866 contains no vertices of degree 4

Theorem101 (Cao et al [52]) Let119866 be a connected graphwithΔ(119866) = 5 and cr(119866) ⩽ 4 If no triangles contain two vertices ofdegree 5 then 119887(119866) ⩽ 6 = Δ(119866) + 1

Theorem 102 (Cao et al [52]) Let 119866 be a connected graphwith Δ(119866) ⩾ 6 and cr(119866) ⩽ 3 If Δ(119866) ⩾ 7 or if Δ(119866) = 6120575(119866) = 3 and every edge 119890 = 119909119910 with 119889

119866(119909) = 5 and 119889

119866(119910) = 6

is contained in atmost one triangle then 119887(119866) ⩽ min8 Δ(119866)+1

Using Carlson-Develin method it can be proved that119887(119866) ⩽ Δ(119866) + 2 if cr(119866) ⩽ 3 (see the first conclusion inTheorem 100) and not yet proved that 119887(119866) ⩽ 8 if cr(119866) ⩽3 although it has been proved by using other method (seeCorollary 90)

By using Carlson-Develin method Samodivkin [25]obtained some results on the bondage number for graphswithsome given properties

Kim [53] showed that 119887(119866) ⩽ Δ(119866) + 2 for a connectedgraph 119866 with genus 1 Δ(119866) ⩽ 5 and having a toroidalembedding of which at least one region is not 4-sidedRecently Gagarin and Zverovich [54] further have extendedCarlson-Develin ideas to establish a nice upper bound forarbitrary graphs that can be embedded on orientable ornonorientable topological surface

Theorem 103 (Gagarin and Zverovich [54] 2012) Let 119866 bea graph embeddable on an orientable surface of genus ℎ and anonorientable surface of genus 119896 Then 119887(119866) ⩽ minΔ(119866)+ℎ+2 Δ(119866) + 119896 + 1

This result generalizes the corresponding upper boundsin Theorems 74 and 96 for any orientable or nonori-entable topological surface By investigating the proof ofTheorem 103 Huang [55] found that the issue of the ori-entability can be avoided by using the Euler characteristic120594(= 120592(119866) minus 120576(119866) + 120601(119866)) instead of the genera ℎ and 119896 the

relations are 120594 = 2minus2ℎ and 120594 = 2minus119896 To obtain the best resultfromTheorem 103 onewants ℎ and 119896 as small as possible thisis equivalent to having 120594 as large as possible

According toTheorem 103 if119866 is planar (ℎ = 0 120594 = 2) orcan be embedded on the real projective plane (119896 = 1 120594 = 1)then 119887(119866) ⩽ Δ(119866) + 2 In all other cases Huang [55] had thefollowing improvement for Theorem 103 the proof is basedon the technique developed byCarlson-Develin andGagarin-Zverovich and includes some elementary calculus as a newingredient mainly the intermediate-value theorem and themean-value theorem

Theorem 104 (Huang [55] 2012) Let 119866 be a graph embed-dable on a surface whose Euler characteristic 120594 is as large aspossible If 120594 ⩽ 0 then 119887(119866) ⩽ Δ(119866)+lfloor119903rfloor where 119903 is the largestreal root of the following cubic equation in 119911

1199113

+ 21199112

+ (6120594 minus 7) 119911 + 18120594 minus 24 = 0 (37)

In addition if 120594 decreases then 119903 increases

The following result is asymptotically equivalent toTheorem 104

Theorem 105 (Huang [55] 2012) Let 119866 be a graph embed-dable on a surface whose Euler characteristic 120594 is as large aspossible If 120594 ⩽ 0 then 119887(119866) lt Δ(119866) + radic12 minus 6120594 + 12 orequivalently 119887(119866) ⩽ Δ(119866) + lceilradic12 minus 6120594 minus 12rceil

Also Gagarin andZverovich [54] indicated that the upperbound in Theorem 103 can be improved for larger values ofthe genera ℎ and 119896 by adjusting the proofs and stated thefollowing general conjecture

Conjecture 106 (Gagarin and Zverovich [54] 2012) For aconnected graph G of orientable genus ℎ and nonorientablegenus 119896

119887 (119866) ⩽ min 119888ℎ 119888

1015840

119896 Δ (119866) + 119900 (ℎ) Δ (119866) + 119900 (119896) (38)

where 119888ℎand 1198881015840

119896are constants depending respectively on the

orientable and nonorientable genera of 119866

In the recent paper Gagarin and Zverovich [56] providedconstant upper bounds for the bondage number of graphs ontopological surfaces which can be used as the first estimationfor the constants 119888

ℎand 1198881015840

119896of Conjecture 106

Theorem 107 (Gagarin and Zverovich [56] 2013) Let 119866 bea connected graph of order 119899 If 119866 can be embedded on thesurface of its orientable or nonorientable genus of the Eulercharacteristic 120594 then

(a) 120594 ⩾ 1 implies 119887(119866) ⩽ 10(b) 120594 ⩽ 0 and 119899 gt minus12120594 imply 119887(119866) ⩽ 11(c) 120594 ⩽ minus1 and 119899 ⩽ minus12120594 imply 119887(119866) ⩽ 11 +

3120594(radic17 minus 8120594 minus 3)(120594 minus 1) = 11 + 119874(radicminus120594)

Theorem 79 shows that a connected planar triangle-freegraph 119866 has 119887(119866) ⩽ 6 The following result generalizes to itall the other topological surfaces

International Journal of Combinatorics 17

Theorem 108 (Gagarin and Zverovich [56] 2013) Let 119866 be aconnected triangle-free graph of order 119899 If 119866 can be embeddedon the surface of its orientable or nonorientable genus of theEuler characteristic 120594 then

(a) 120594 ⩾ 1 implies 119887(119866) ⩽ 6(b) 120594 ⩽ 0 and 119899 gt minus8120594 imply 119887(119866) ⩽ 7(c) 120594 ⩽ minus1 and 119899 ⩽ minus8120594 imply 119887(119866) ⩽ 7minus4120594(1+radic2 minus 120594)

In [56] the authors also explicitly improved upperbounds in Theorem 103 and gave tight lower bounds forthe number of vertices of graphs 2-cell embeddable ontopological surfaces of a given genusThe interested reader isreferred to the original paper for details The following resultis interesting although its proof is easy

Theorem 109 (Samodivkin [57]) Let119866 be a graph with order119899 and girth 119892 If 119866 is embeddable on a surface whose Eulercharacteristic 120594 is as large as possible then

119887 (119866) ⩽ 3 +8

119892 minus 2minus

4120594119892

119899 (119892 minus 2) (39)

If 119866 is a planar graph with girth 119892 ⩾ 4 + 119894 for any 119894 isin0 1 2 then Theorem 109 leads to 119887(119866) ⩽ 6 minus 119894 which isthe result inTheorem 79 obtained by Fischermann et al [46]Also inmany cases the bound stated inTheorem 109 is betterthan those given byTheorems 103 and 105

8 Conditional Bondage Numbers

Since the concept of the bondage number is based upon thedomination all sorts of dominations which are variationsof the normal domination by adding restricted conditionsto the normal dominating set can develop a new ldquobondagenumberrdquo as long as a variation of domination number isgiven In this section we survey results on the bondagenumber under some restricted conditions

81 Total Bondage Numbers A dominating set 119878 of a graph119866 without isolated vertices is called total if the subgraphinduced by 119878 contains no isolated vertices The minimumcardinality of a total dominating set is called the totaldomination number of 119866 and denoted by 120574

119879(119866) It is clear

that 120574(119866) ⩽ 120574119879(119866) ⩽ 2120574(119866) for any graph 119866 without isolated

verticesThe total domination in graphs was introduced by Cock-

ayne et al [58] in 1980 Pfaff et al [59 60] in 1983 showedthat the problem determining total domination number forgeneral graphs is NP-complete even for bipartite graphs andchordal graphs Even now total domination in graphs hasbeen extensively studied in the literature In 2009 Henning[61] gave a survey of selected recent results on total domina-tion in graphs

The total bondage number of 119866 denoted by 119887119879(119866) is the

smallest cardinality of a subset 119861 sube 119864(119866) with the propertythat119866minus119861 contains no isolated vertices and 120574

119879(119866minus119861) gt 120574

119879(119866)

From definition 119887119879(119866)may not exist for some graphs for

example119866 = 1198701119899 We put 119887

119879(119866) = infin if 119887

119879(119866) does not exist

It is easy to see that 119887119879(119866) is finite for any connected graph 119866

other than 1198701 119870

2 119870

3 and 119870

1119899 In fact for such a graph 119866

since there is a path of length 3 in 119866 we can find 1198611sube 119864(119866)

such that 119866 minus 1198611is a spanning tree 119879 of 119866 containing a path

of length 3 So 120574119879(119879) = 120574

119879(119866minus119861

1) ⩾ 120574

119879(119866) For the tree119879 we

find 1198612sube 119864(119879) such that 120574

119879(119879 minus 119861

2) gt 120574

119879(119879) ⩾ 120574

119879(119866) Thus

we have 120574119879(119866minus119861

2minus119861

1) gt 120574

119879(119866) and so 119887

119879(119866) ⩽ |119861

1|+|119861

2| In

the following discussion we always assume that 119887119879(119866) exists

when a graph 119866 is mentionedIn 1991 Kulli and Patwari [62] first studied the total

bondage number of a graph and calculated the exact valuesof 119887

119879(119866) for some standard graphs

Theorem 110 (Kulli and Patwari [62] 1991) For 119899 ⩾ 4

119887119879(119862

119899) =

3 119894119891 119899 equiv 2 (mod 4) 2 119900119905ℎ119890119903119908119894119904119890

119887119879(119875

119899) =

2 119894119891 119899 equiv 2 (mod 4) 1 119900119905ℎ119890119903119908119894119904119890

119887119879(119870

119898119899) = 119898 119908119894119905ℎ 2 ⩽ 119898 ⩽ 119899

119887119879(119870

119899) =

4 119891119900119903 119899 = 4

2119899 minus 5 119891119900119903 119899 ⩾ 5

(40)

Recently Hu et al [63] have obtained some results on thetotal bondage number of the Cartesian product119875

119898times119875

119899of two

paths 119875119898and 119875

119899

Theorem 111 (Hu et al [63] 2012) For the Cartesian product119875119898times 119875

119899of two paths 119875

119898and 119875

119899

119887119879(119875

2times 119875

119899) =

1 119894119891 119899 equiv 0 (mod 3) 2 119894119891 119899 equiv 2 (mod 3) 3 119894119891 119899 equiv 1 (mod 3)

119887119879(119875

3times 119875

119899) = 1

119887119879(119875

4times 119875

119899)

= 1 119894119891 119899 equiv 1 (mod 5) = 2 119894119891 119899 equiv 4 (mod 5) ⩽ 3 119894119891 119899 equiv 2 (mod 5) ⩽ 4 119894119891 119899 equiv 0 3 (mod 5)

(41)

Generalized Petersen graphs are an important class ofcommonly used interconnection networks and have beenstudied recently By constructing a family of minimum totaldominating sets Cao et al [64] determined the total bondagenumber of the generalized Petersen graphs

FromTheorem 6 we know that 119887(119879) ⩽ 2 for a nontrivialtree 119879 But given any positive integer 119896 Sridharan et al [65]constructed a tree 119879 for which 119887

119879(119879) = 119896 Let119867

119896be the tree

obtained from a star1198701119896+1

by subdividing 119896 edges twice Thetree 119867

7is shown in Figure 6 It can be easily verified that

119887119879(119867

119896) = 119896 This fact can be stated as the following theorem

Theorem 112 (Sridharan et al [65] 2007) For any positiveinteger 119896 there exists a tree 119879 with 119887

119879(119879) = 119896

18 International Journal of Combinatorics

Figure 61198677

Combining Theorem 1 with Theorem 112 we have whatfollows

Corollary 113 (Sridharan et al [65] 2007) The bondagenumber and the total bondage number are unrelated even fortrees

However Sridharan et al [65] gave an upper bound onthe total bondage number of a tree in terms of its maximumdegree For any tree 119879 of order 119899 if 119879 =119870

1119899minus1 then 119887

119879(119879) =

minΔ(119879) (13)(119899 minus 1) Rad and Raczek [66] improved thisupper bound and gave a constructive characterization of acertain class of trees attaching the upper bound

Theorem 114 (Rad and Raczek [66] 2011) 119887119879(119879) ⩽ Δ(119879) minus 1

for any tree 119879 with maximum degree at least three

In general the decision problem for 119887119879(119866) is NP-complete

for any graph 119866 We state the decision problem for the totalbondage as follows

Problem 3 Consider the decision problemTotal Bondage ProblemInstance a graph 119866 and a positive integer 119887Question is 119887

119879(119866) ⩽ 119887

Theorem 115 (Hu and Xu [16] 2012) The total bondageproblem is NP-complete

Consequently in view of its computational hardnessit is significative to establish some sharp bounds on thetotal bondage number of a graph in terms of other graphicparameters In 1991 Kulli and Patwari [62] showed that119887119879(119866) ⩽ 2119899 minus 5 for any graph 119866 with order 119899 ⩾ 5 Sridharan

et al [65] improved this result as follows

Theorem 116 (Sridharan et al [65] 2007) For any graph 119866with order 119899 if 119899 ⩾ 5 then

119887119879(119866) ⩽

1

3(119899 minus 1) 119894119891 119866 119888119900119899119905119886119894119899119904 119899119900 119888119910119888119897119890119904

119899 minus 1 119894119891 119892 (119866) ⩾ 5

119899 minus 2 119894119891 119892 (119866) = 4

(42)

Rad and Raczek [66] also established some upperbounds on 119887

119879(119866) for a general graph 119866 In particular they

gave an upper bound on 119887119879(119866) in terms of the girth of

119866

Theorem 117 (Rad and Raczek [66] 2011) 119887119879(119866)⩽4Δ(119866) minus 5

for any graph 119866 with 119892(119866) ⩾ 4

82 Paired Bondage Numbers A dominating set 119878 of 119866 iscalled to be paired if the subgraph induced by 119878 containsa perfect matching The paired domination number of 119866denoted by 120574

119875(119866) is the minimum cardinality of a paired

dominating set of 119866 Clearly 120574119879(119866) ⩽ 120574

119875(119866) for every

connected graph 119866 with order at least two where 120574119879(119866) is

the total domination number of 119866 and 120574119875(119866) ⩽ 2120574(119866) for

any graph119866without isolated vertices Paired domination wasintroduced by Haynes and Slater [67 68] and further studiedin [69ndash72]

The paired bondage number of 119866 with 120575(119866) ⩾ 1 denotedby 119887P(119866) is the minimum cardinality among all sets of edges119861 sube 119864 such that 120575(119866 minus 119861) ⩾ 1 and 120574

119875(119866 minus 119861) gt 120574

119875(119866)

The concept of the paired bondage number was firstproposed by Raczek [73] in 2008The following observationsfollow immediately from the definition of the paired bondagenumber

Observation 2 Let 119866 be a graph with 120575(119866) ⩾ 1

(a) If 119867 is a subgraph of 119866 such that 120575(119867) ⩾ 1 then119887119875(119867) ⩽ 119887

119875(119866)

(b) If 119867 is a subgraph of 119866 such that 119887119875(119867) = 1 and 119896

is the number of edges removed to form119867 then 1 ⩽119887119875(119866) ⩽ 119896 + 1

(c) If 119909119910 isin 119864(119866) such that 119889119866(119909) 119889

119866(119910) gt 1 and 119909119910

belongs to each perfect matching of each minimumpaired dominating set of 119866 then 119887

119875(119866) = 1

Based on these observations 119887119875(119875

119899) ⩽ 2 In fact the

paired bondage number of a path has been determined

Theorem 118 (Raczek [73] 2008) For any positive integer 119896if 119899 ⩾ 2 then

119887119875(119875

119899) =

0 119894119891 119899 = 2 3 119900119903 5

1 119894119891 119899 = 4119896 4119896 + 3 119900119903 4119896 + 6

2 119900119905ℎ119890119903119908119894119904119890

(43)

For a cycle 119862119899of order 119899 ⩾ 3 since 120574

119875(119862

119899) = 120574

119875(119875

119899) from

Theorem 118 the following result holds immediately

Corollary 119 (Raczek [73] 2008) For any positive integer 119896if 119899 ⩾ 3 then

119887119875(119862

119899) =

0 119894119891 119899 = 3 119900119903 5

2 119894119891 119899 = 4119896 4119896 + 3 119900119903 4119896 + 6

3 119900119905ℎ119890119903119908119894119904119890

(44)

A wheel 119882119899 where 119899 ⩾ 4 is a graph with 119899 vertices

formed by connecting a single vertex to all vertices of a cycle119862119899minus1

Of course 120574119875(119882

119899) = 2

International Journal of Combinatorics 19

119896

Figure 7 A tree 119879 with 119887119875(119879) = 119896

Theorem 120 (Raczek [73] 2008) For positive integers 119899 and119898

119887119875(119870

119898119899) = 119898 119891119900119903 1 lt 119898 ⩽ 119899

119887119875(119882

119899) =

4 119894119891 119899 = 4

3 119894119891 119899 = 5

2 119900119905ℎ119890119903119908119894119904119890

(45)

Theorem 16 shows that if 119879 is a nontrivial tree then119887(119879) ⩽ 2 However no similar result exists for pairedbondage For any nonnegative integer 119896 let 119879

119896be a tree

obtained by subdividing all but one edge of a star 1198701119896+1

(asshown in Figure 7) It is easy to see that 119887

119875(119879

119896) = 119896 We

state this fact as the following theorem which is similar toTheorem 112

Theorem121 (Raczek [73] 2008) For any nonnegative integer119896 there exists a tree 119879 with 119887

119875(119879) = 119896

Consider the tree defined in Figure 5 for 119896 = 3 Then119887(119879

3) ⩽ 2 and 119887

119875(119879

3) = 3 On the other hand 119887(119875

4) = 2

and 119887119875(119875

4) = 1 Thus we have what follows which is similar

to Corollary 113

Corollary 122 The bondage number and the paired bondagenumber are unrelated even for trees

A constructive characterization of trees with 119887(119879) = 2is given by Hartnell and Rall in [12] Raczek [73] provideda constructive characterization of trees with 119887

119875(119879) = 0 In

order to state the characterization we define a labeling andthree simple operations on a tree119879 Let 119910 isin 119881(119879) and let ℓ(119910)be the label assigned to 119910

Operation T1 If ℓ(119910) = 119861 add a vertex 119909 and the

edge 119910119909 and let ℓ(119909) = 119860OperationT

2 If ℓ(119910) = 119862 add a path (119909

1 119909

2) and the

edge 1199101199091 and let ℓ(119909

1) = 119861 ℓ(119909

2) = 119860

Operation T3 If ℓ(119910) = 119861 add a path (119909

1 119909

2 119909

3)

and the edge 1199101199091 and let ℓ(119909

1) = 119862 ℓ(119909

2) = 119861 and

ℓ(1199093) = 119860

Let 1198752= (119906 V) with ℓ(119906) = 119860 and ℓ(V) = 119861 Let T be

the class of all trees obtained from the labeled 1198752by a finite

sequence of operationsT1 T

2 T

3

A tree 119879 in Figure 8 belongs to the familyTRaczek [73] obtained the following characterization of all

trees 119879 with 119887119875(119879) = 0

119860

119860

119860

119860

119860

119861 119862

119861 119862 119861

119861119860

Figure 8 A tree 119879 belong to the familyT

Theorem 123 (Raczek [73] 2008) For a tree 119879 119887119875(119879) = 0 if

and only if 119879 is inT

We state the decision problem for the paired bondage asfollows

Problem 4 Consider the decision problem

Paired Bondage ProblemInstance a graph 119866 and a positive integer 119887Question is 119887

119875(119866) ⩽ 119887

Conjecture 124 The paired bondage problem is NP-complete

83 Restrained Bondage Numbers A dominating set 119878 of 119866 iscalled to be restrained if the subgraph induced by 119878 containsno isolated vertices where 119878 = 119881(119866) 119878 The restraineddomination number of 119866 denoted by 120574

119877(119866) is the smallest

cardinality of a restrained dominating set of 119866 It is clear that120574119877(119866) exists and 120574(119866) ⩽ 120574

119877(119866) for any nonempty graph 119866

The concept of restrained domination was introducedby Telle and Proskurowski [74] in 1997 albeit indirectly asa vertex partitioning problem Concerning the study of therestrained domination numbers the reader is referred to [74ndash79]

In 2008 Hattingh and Plummer [80] defined therestrained bondage number 119887R(119866) of a nonempty graph 119866 tobe the minimum cardinality among all subsets 119861 sube 119864(119866) forwhich 120574

119877(119866 minus 119861) gt 120574

119877(119866)

For some simple graphs their restrained dominationnumbers can be easily determined and so restrained bondagenumbers have been also determined For example it is clearthat 120574

119877(119870

119899) = 1 Domke et al [77] showed that 120574

119877(119862

119899) =

119899 minus 2lfloor1198993rfloor for 119899 ⩾ 3 and 120574119877(119875

119899) = 119899 minus 2lfloor(119899 minus 1)3rfloor for 119899 ⩾ 1

Using these results Hattingh and Plummer [80] obtained therestricted bondage numbers for119870

119899 119862

119899 119875

119899 and119866 = 119870

11989911198992119899119905

as follows

Theorem 125 (Hattingh and Plummer [80] 2008) For 119899 ⩾ 3

119887R (119870119899) =

1 119894119891 119899 = 3

lceil119899

2rceil 119900119905ℎ119890119903119908119894119904119890

119887R (119862119899) =

1 119894119891 119899 equiv 0 (mod 3) 2 119900119905ℎ119890119903119908119894119904119890

119887R (119875119899) = 1 119899 ⩾ 4

20 International Journal of Combinatorics

119887R (119866)

=

lceil119898

2rceil 119894119891 119899

119898= 1 119886119899119889 119899

119898+1⩾ 2 (1 ⩽ 119898 lt 119905)

2119905 minus 2 119894119891 1198991= 119899

2= sdot sdot sdot = 119899

119905= 2 (119905 ⩾ 2)

2 119894119891 1198991= 2 119886119899119889 119899

2⩾ 3 (119905 = 2)

119905minus1

sum

119894=1

119899119894minus 1 119900119905ℎ119890119903119908119894119904119890

(46)

Theorem 126 (Hattingh and Plummer [80] 2008) For a tree119879 of order 119899 (⩾ 4) 119879 ≇ 119870

1119899minus1if and only if 119887R(119879) = 1

Theorem 126 shows that the restrained bondage numberof a tree can be computed in constant time However thedecision problem for 119887R(119866) is NP-complete even for bipartitegraphs

Problem 5 Consider the decision problem

Restrained Bondage Problem

Instance a graph 119866 and a positive integer 119887

Question is 119887R(119866) ⩽ 119887

Theorem 127 (Hattingh and Plummer [80] 2008) Therestrained bondage problem is NP-complete even for bipartitegraphs

Consequently in view of its computational hardnessit is significative to establish some sharp bounds on therestrained bondage number of a graph in terms of othergraphic parameters

Theorem 128 (Hattingh and Plummer [80] 2008) For anygraph 119866 with 120575(119866) ⩾ 2

119887R (119866) ⩽ min119909119910isin119864(119866)

119889119866(119909) + 119889

119866(119910) minus 2 (47)

Corollary 129 119887R(119866) ⩽ Δ(119866)+120575(119866)minus2 for any graph119866 with120575(119866) ⩾ 2

Notice that the bounds stated in Theorem 128 andCorollary 129 are sharp Indeed the class of cycles whoseorders are congruent to 1 2 (mod 3) have a restrained bondagenumber achieving these bounds (see Theorem 125)

Theorem 128 is an analogue of Theorem 14 A quitenatural problem is whether or not for restricted bondagenumber there are analogues of Theorem 16 Theorem 18Theorem 29 and so on Indeed we should consider all resultsfor the bondage number whether or not there are analoguesfor restricted bondage number

Theorem 130 (Hattingh and Plummer [80] 2008) For anygraph 119866 with 120574

119877(119866) = 2

119887R (119866) ⩽ Δ (119866) + 1 (48)

We now consider the relation between 119887119877(119866) and 119887(119866)

Theorem 131 (Hattingh and Plummer [80] 2008) If 120574119877(119866) =

120574(119866) for some graph 119866 then 119887R(119866) ⩽ 119887(119866)

Corollary 129 and Theorem 131 provide a possibility toattack Conjecture 73 by characterizing planar graphs with120574119877= 120574 and 119887R = 119887 when 3 ⩽ Δ ⩽ 6However we do not have 119887R(119866) = 119887(119866) for any graph

119866 even if 120574119877(119866) = 120574(119866) Observe that 120574

119877(119870

3) = 120574(119870

3) yet

119887119877(119870

3) = 1 and 119887(119870

3) = 2 We still may not claim that

119887119877(119866) = 119887(119866) even in the case that every 120574-set is a 120574

119877-set

The example 1198703again demonstrates this

Theorem 132 (Hattingh and Plummer [80] 2008) There isan infinite class of graphs inwhich each graph119866 satisfies 119887(119866) lt119887R(119866)

In fact such an infinite class of graphs can be obtained asfollows Let119867 be a connected graph BC(119867) a graph obtainedfrom 119867 by attaching ℓ (⩾ 2) vertices of degree 1 to eachvertex in 119867 and B = 119866 119866 = BC(119867) for some graph 119867such that 120575(119867) ⩾ 2 By Theorem 3 we can easily check that119887(119866) = 1 lt 2 ⩽ min120575(119867) ℓ = 119887R(119866) for any 119866 isin BMoreover there exists a graph 119866 isin B such that 119887R(119866) canbe much larger than 119887(119866) which is stated as the followingtheorem

Theorem 133 (Hattingh and Plummer [80] 2008) For eachpositive integer 119896 there is a graph119866 such that 119896 = 119887R(119866)minus119887(119866)

Combining Theorem 133 with Theorem 132 we have thefollowing corollary

Corollary 134 The bondage number and the restrainedbondage number are unrelated

Very recently Jafari Rad et al [81] have consideredthe total restrained bondage in graphs based on both totaldominating sets and restrained dominating sets and obtainedseveral properties exact values and bounds for the totalrestrained bondage number of a graph

A restrained dominating set 119878 of a graph 119866 without iso-lated vertices is called to be a total restrained dominating setif 119878 is a total dominating set The total restrained dominationnumber of119866 denoted by 120574TR (119866) is theminimum cardinalityof a total restrained dominating set of 119866 For referenceson this domination in graphs see [3 61 82ndash85] The totalrestrained bondage number 119887TR (119866) of a graph 119866 with noisolated vertex is the cardinality of a smallest set of edges 119861 sube119864(119866) for which119866minus119861 has no isolated vertex and 120574TR (119866minus119861) gt120574TR (119866) In the case that there is no such subset 119861 we define119887TR (119866) = infin

Ma et al [84] determined that 120574TR (119875119899) = 119899minus2lfloor(119899minus2)4rfloorfor 119899 ⩾ 3 120574TR (119862119899

) = 119899 minus 2lfloor1198994rfloor for 119899 ⩾ 3 120574TR (1198703) =

3 and 120574TR (119870119899) = 2 for 119899 = 3 and 120574TR (1198701119899

) = 1 + 119899

and 120574TR (119870119898119899) = 2 for 2 ⩽ 119898 ⩽ 119899 According to these

results Jafari Rad et al [81] determined the exact valuesof total restrained bondage numbers for the correspondinggraphs

International Journal of Combinatorics 21

Theorem 135 (Jafari Rad et al [81]) For an integer 119899

119887TR (119875119899) = infin 119894119891 2 ⩽ 119899 ⩽ 5

1 119894119891 119899 ⩾ 5

119887TR (119862119899) =

infin 119894119891 119899 = 3

1 119894119891 3 lt 119899 119899 equiv 0 1 (mod 4)2 119894119891 3 lt 119899 119899 equiv 2 3 (mod 4)

119887TR (119882119899) = 2 119891119900119903 119899 ⩾ 6

119887TR (119870119899) = 119899 minus 1 119891119900119903 119899 ⩾ 4

119887TR (119870119898119899) = 119898 minus 1 119891119900119903 2 ⩽ 119898 ⩽ 119899

(49)

119887TR (119870119899) = 119899minus1 shows the fact that for any integer 119899 ⩾ 3 there

exists a connected graph namely119870119899+1

such that 119887TR (119870119899+1) =

119899 that is the total restrained bondage number is unboundedfrom the above

A vertex is said to be a support vertex if it is adjacent toa vertex of degree one Let A be the family of all connectedgraphs such that119866 belongs toA if and only if every edge of119866is incident to a support vertex or 119866 is a cycle of length three

Proposition 136 (Cyman and Raczek [82] 2006) For aconnected graph119866 of order 119899 120574TR (119866) = 119899 if and only if119866 isin A

Hence if 119866 isin A then the removal of any subset of edgesof119866 cannot increase the total restrained domination numberof 119866 and thus 119887TR (119866) = infin When 119866 notin A a result similar toTheorem 6 is obtained

Theorem 137 (Jafari Rad et al [81]) For any tree 119879 notin A119887TR (119879) ⩽ 2 This bound is sharp

In the samepaper the authors provided a characterizationfor trees 119879 notin A with 119887TR (119879) = 1 or 2 The following result issimilar to Observation 1

Observation 3 (Jafari Rad et al [81]) If 119896 edges can beremoved from a graph 119866 to obtain a graph 119867 without anisolated vertex and with 119887TR (119867) = 119905 then 119887TR (119866) ⩽ 119896 + 119905

ApplyingTheorem 137 andObservation 3 Jafari Rad et alcharacterized all connected graphs 119866 with 119887TR (119866) = infin

Theorem 138 (Jafari Rad et al [81]) For a connected graph119866119887TR (119866) = infin if and only if 119866 isin A

84 Other Conditional Bondage Numbers A subset 119868 sube 119881(119866)is called an independent set if no two vertices in 119868 are adjacentin 119866 The maximum cardinality among all independent setsis called the independence number of 119866 denoted by 120572(119866)

A dominating set 119878 of a graph 119866 is called to be inde-pendent if 119878 is an independent set of 119866 The minimumcardinality among all independent dominating set is calledthe independence domination number of 119866 and denoted by120574119868(119866)

Since an independent dominating set is not only adominating set but also an independent set 120574(119866) ⩽ 120574

119868(119866) ⩽

120572(119866) for any graph 119866It is clear that a maximal independent set is certainly a

dominating set Thus an independent set is maximal if andonly if it is an independent dominating set and so 120574

119868(119866) is the

minimum cardinality among all maximal independent setsof 119866 This graph-theoretical invariant has been well studiedin the literature see for example Haynes et al [3] for earlyresults Goddard and Henning [86] for recent results onindependent domination in graphs

In 2003 Zhang et al [87] defined the independencebondage number 119887

119868(119866) of a nonempty graph 119866 to be the

minimum cardinality among all subsets 119861 sube 119864(119866) for which120574119868(119866 minus 119861) gt 120574

119868(119866) For some ordinary graphs their indepen-

dence domination numbers can be easily determined and soindependence bondage numbers have been also determinedClearly 119887

119868(119870

119899) = 1 if 119899 ⩾ 2

Theorem 139 (Zhang et al [87] 2003) For any integer 119899 ⩾ 4

119887119868(119862

119899) =

1 119894119891 119899 equiv 1 (mod 2) 2 119894119891 119899 equiv 0 (mod 2)

119887119868(119875

119899) =

1 119894119891 119899 equiv 0 (mod 2) 2 119894119891 119899 equiv 1 (mod 2)

119887119868(119870

119898119899) = 119899 119891119900119903 119898 ⩽ 119899

(50)

Apart from the above-mentioned results as far as we findno other results on the independence bondage number in thepresent literature we never knew much of any result on thisparameter for a tree

A subset 119878 of 119881(119866) is called an equitable dominatingset if for every 119910 isin 119881(119866 minus 119878) there exists a vertex 119909 isin 119878such that 119909119910 isin 119864(119866) and |119889

119866(119909) minus 119889

119866(119910)| ⩽ 1 proposed

by Dharmalingam [88] The minimum cardinality of such adominating set is called the equitable domination number andis denoted by 120574

119864(119866) Deepak et al [89] defined the equitable

bondage number 119887119864(119866) of a graph 119866 to be the cardinality of a

smallest set 119861 sube 119864(119866) of edges for which 120574119864(119866 minus 119861) gt 120574

119864(119866)

and determined 119887119864(119866) for some special graphs

Theorem 140 (Deepak et al [89] 2011) For any integer 119899 ⩾ 2

119887119864(119870

119899) = lceil

119899

2rceil

119887119864(119870

119898119899) = 119898 119891119900119903 0 ⩽ 119899 minus 119898 ⩽ 1

119887119864(119862

119899) =

3 119894119891 119899 equiv 1 (mod 3)2 119900119905ℎ119890119903119908119894119904119890

119887119864(119875

119899) =

2 119894119891 119899 equiv 1 (mod 3)1 119900119905ℎ119890119903119908119894119904119890

119887119864(119879) ⩽ 2 119891119900119903 119886119899119910 119899119900119899119905119903119894119907119894119886119897 119905119903119890119890 119879

119887119864(119866) ⩽ 119899 minus 1 119891119900119903 119886119899119910 119888119900119899119899119890119888119905119890119889 119892119903119886119901ℎ 119866 119900119891 119900119903119889119890119903 119899

(51)

22 International Journal of Combinatorics

As far as we know there are no more results on theequitable bondage number of graphs in the present literature

There are other variations of the normal domination byadding restricted conditions to the normal dominating setFor example the connected domination set of a graph119866 pro-posed by Sampathkumar and Walikar [90] is a dominatingset 119878 such that the subgraph induced by 119878 is connected Theproblem of finding a minimum connected domination set ina graph is equivalent to the problemof finding a spanning treewithmaximumnumber of vertices of degree one [91] and hassome important applications in wireless networks [92] thusit has received much research attention However for suchan important domination there is no result on its bondagenumber We believe that the topic is of significance for futureresearch

9 Generalized Bondage Numbers

There are various generalizations of the classical dominationsuch as 119901-domination distance domination fractional dom-ination Roman domination and rainbow domination Everysuch a generalization can lead to a corresponding bondageIn this section we introduce some of them

91 119901-Bondage Numbers In 1985 Fink and Jacobson [93]introduced the concept of 119901-domination Let 119901 be a positiveinteger A subset 119878 of 119881(119866) is a 119901-dominating set of 119866 if|119878 cap 119873

119866(119910)| ⩾ 119901 for every 119910 isin 119878 The 119901-domination number

120574119901(119866) is the minimum cardinality among all 119901-dominating

sets of 119866 Any 119901-dominating set of 119866 with cardinality 120574119901(119866)

is called a 120574119901-set of 119866 Note that a 120574

1-set is a classic minimum

dominating set Notice that every graph has a 119901-dominatingset since the vertex-set119881(119866) is such a setWe also note that a 1-dominating set is a dominating set and so 120574(119866) = 120574

1(119866) The

119901-domination number has received much research attentionsee a state-of-the-art survey article by Chellali et al [94]

It is clear from definition that every 119901-dominating setof a graph certainly contains all vertices of degree at most119901 minus 1 By this simple observation to avoid the trivial caseoccurrence we always assume that Δ(119866) ⩾ 119901 For 119901 ⩾ 2 Luet al [95] gave a constructive characterization of trees withunique minimum 119901-dominating sets

Recently Lu and Xu [96] have introduced the conceptto the 119901-bondage number of 119866 denoted by 119887

119901(119866) as the

minimum cardinality among all sets of edges 119861 sube 119864(119866) suchthat 120574

119901(119866 minus 119861) gt 120574

119901(119866) Clearly 119887

1(119866) = 119887(119866)

Lu and Xu [96] established a lower bound and an upperbound on 119887

119901(119879) for any integer 119901 ⩾ 2 and any tree 119879 with

Δ(119879) ⩾ 119901 and characterized all trees achieving the lowerbound and the upper bound respectively

Theorem 141 (Lu and Xu [96] 2011) For any integer 119901 ⩾ 2and any tree 119879 with Δ(119879) ⩾ 119901

1 ⩽ 119887119901(119879) ⩽ Δ (119879) minus 119901 + 1 (52)

Let 119878 be a given subset of 119881(119866) and for any 119909 isin 119878 let

119873119901(119909 119878 119866) = 119910 isin 119878 cap 119873

119866(119909)

1003816100381610038161003816119873119866(119910) cap 119878

1003816100381610038161003816 = 119901

(53)

Then all trees achieving the lower bound 1 can be character-ized as follows

Theorem 142 (Lu and Xu [96] 2011) Let 119879 be a tree withΔ(119879) ⩾ 119901 ⩾ 2 Then 119887

119901(119879) = 1 if and only if for any 120574

119901-set

119878 of 119879 there exists an edge 119909119910 isin (119878 119878) such that 119910 isin 119873119901(119909 119878 119879)

The notation 119878(119886 119887) denotes a double star obtained byadding an edge between the central vertices of two stars119870

1119886minus1

and1198701119887minus1

And the vertex with degree 119886 (resp 119887) in 119878(119886 119887) iscalled the 119871-central vertex (resp 119877-central vertex) of 119878(119886 119887)

To characterize all trees attaining the upper bound givenin Theorem 141 we define three types of operations on a tree119879 with Δ(119879) = Δ ⩾ 119901 + 1

Type 1 attach a pendant edge to a vertex 119910with 119889119879(119910) ⩽ 119901minus

2 in 119879

Type 2 attach a star 1198701Δminus1

to a vertex 119910 of 119879 by joining itscentral vertex to 119910 where 119910 in a 120574

119901-set of 119879 and

119889119879(119910) ⩽ Δ minus 1

Type 3 attach a double star 119878(119901 Δ minus 1) to a pendant vertex 119910of 119879 by coinciding its 119877-central vertex with 119910 wherethe unique neighbor of 119910 is in a 120574

119901-set of 119879

Let B = 119879 a tree obtained from 1198701Δ

or 119878(119901 Δ) by afinite sequence of operations of Types 1 2 and 3

Theorem 143 (Lu and Xu [96] 2011) A tree with the maxi-mum Δ ⩾ 119901 + 1 has 119901-bondage number Δ minus 119901 + 1 if and onlyif it belongs toB

Theorem 144 (Lu and Xu [97] 2012) Let 119866119898119899= 119875

119898times 119875

119899

Then

1198872(119866

2119899) = 1 119891119900119903 119899 ⩾ 2

1198872(119866

3119899) =

2 119894119891 119899 equiv 1 (mod 3) 1 119900119905ℎ119890119903119908119894119904119890

for 119899 ⩾ 2

1198872(119866

4119899) =

1 119894119891 119899 equiv 3 (mod 4) 2 119900119905ℎ119890119903119908119894119904119890

for 119899 ⩾ 7

(54)

Recently Krzywkowski [98] has proposed the conceptof the nonisolating 2-bondage of a graph The nonisolating2-bondage number of a graph 119866 denoted by 1198871015840

2(119866) is the

minimum cardinality among all sets of edges 119861 sube 119864(119866) suchthat 119866 minus 119861 has no isolated vertices and 120574

2(119866 minus 119861) gt 120574

2(119866) If

for every 119861 sube 119864(119866) either 1205742(119866 minus 119861) = 120574

2(119866) or 119866 minus 119861 has

isolated vertices then define 11988710158402(119866) = 0 and say that 119866 is a 2-

nonisolatedly strongly stable graph Krzywkowski presentedsome basic properties of nonisolating 2-bondage in graphsshowed that for every nonnegative integer 119896 there exists atree 119879 such 1198871015840

2(119879) = 119896 characterized all 2-non-isolatedly

strongly stable trees and determined 11988710158402(119866) for several special

graphs

International Journal of Combinatorics 23

Theorem 145 (Krzywkowski [98] 2012) For any positiveintegers 119899 and119898 with119898 ⩽ 119899

1198871015840

2(119870

119899) =

0 119894119891 119899 = 1 2 3

lfloor2119899

3rfloor 119900119905ℎ119890119903119908119894119904119890

1198871015840

2(119875

119899) =

0 119894119891 119899 = 1 2 3

1 119900119905ℎ119890119903119908119894119904119890

1198871015840

2(119862

119899) =

0 119894119891 119899 = 3

1 119894119891 119899 119894119904 119890V119890119899

2 119894119891 119899 119894119904 119900119889119889

1198871015840

2(119870

119898119899) =

3 119894119891 119898 = 119899 = 3

1 119894119891 119898 = 119899 = 4

2 119900119905ℎ119890119903119908119894119904119890

(55)

92 Distance Bondage Numbers A subset 119878 of vertices of agraph119866 is said to be a distance 119896-dominating set for119866 if everyvertex in 119866 not in 119878 is at distance at most 119896 from some vertexof 119878 The minimum cardinality of all distance 119896-dominatingsets is called the distance 119896-domination number of 119866 anddenoted by 120574

119896(119866) (do not confuse with the above-mentioned

120574119901(119866)) When 119896 = 1 a distance 1-dominating set is a normal

dominating set and so 1205741(119866) = 120574(119866) for any graph 119866 Thus

the distance 119896-domination is a generalization of the classicaldomination

The relation between 120574119896and 120572

119896(the 119896-independence

number defined in Section 22) for a tree obtained by Meirand Moon [14] who proved that 120574

119896(119879) = 120572

2119896(119879) for any tree

119879 (a special case for 119896 = 1 is stated in Proposition 10) Thefurther research results can be found in Henning et al [99]Tian and Xu [100ndash103] and Liu et al [104]

In 1998 Hartnell et al [15] defined the distance 119896-bondagenumber of 119866 denoted by 119887

119896(119866) to be the cardinality of a

smallest subset 119861 of edges of 119866 with the property that 120574119896(119866 minus

119861) gt 120574119896(119866) FromTheorem 6 it is clear that if119879 is a nontrivial

tree then 1 ⩽ 1198871(119879) ⩽ 2 Hartnell et al [15] generalized this

result to any integer 119896 ⩾ 1

Theorem 146 (Hartnell et al [15] 1998) For every nontrivialtree 119879 and positive integer 119896 1 ⩽ 119887

119896(119879) ⩽ 2

Hartnell et al [15] and Topp and Vestergaard [13] alsocharacterized the trees having distance 119896-bondage number 2In particular the class of trees for which 119887

1(119879) = 2 are just

those which have a unique maximum 2-independent set (seeTheorem 11)

Since when 119896 = 1 the distance 1-bondage number 1198871(119866)

is the classical bondage number 119887(119866) Theorem 12 gives theNP-hardness of deciding the distance 119896-bondage number ofgeneral graphs

Theorem 147 Given a nonempty undirected graph 119866 andpositive integers 119896 and 119887 with 119887 ⩽ 120576(119866) determining wetheror not 119887

119896(119866) ⩽ 119887 is NP-hard

For a vertex 119909 in119866 the open 119896-neighborhood119873119896(119909) of 119909 is

defined as119873119896(119909) = 119910 isin 119881(119866) 1 ⩽ 119889

119866(119909 119910) le 119896The closed

119896-neighborhood119873119896[119909] of 119909 in 119866 is defined as119873

119896(119909) cup 119909 Let

Δ119896(119866) = max 1003816100381610038161003816119873119896

(119909)1003816100381610038161003816 for any 119909 isin 119881 (119866) (56)

Clearly Δ1(119866) = Δ(119866) The 119896th power of a graph 119866 is the

graph119866119896 with vertex-set119881(119866119896

) = 119881(119866) and edge-set119864(119866119896

) =

119909119910 1 ⩽ 119889119866(119909 119910) ⩽ 119896 The following lemma holds directly

from the definition of 119866119896

Proposition 148 (Tian and Xu [101]) Δ(119866119896

) = Δ119896(119866) and

120574(119866119896

) = 120574119896(119866) for any graph 119866 and each 119896 ⩾ 1

A graph 119866 is 119896-distance domination-critical or 120574119896-critical

for short if 120574119896(119866 minus 119909) lt 120574

119896(119866) for every vertex 119909 in 119866

proposed by Henning et al [105]

Proposition 149 (Tian and Xu [101]) For each 119896 ⩾ 1 a graph119866 is 120574

119896-critical if and only if 119866119896 is 120574

119896-critical

By the above facts we suggest the following worthwhileproblem

Problem 6 Can we generalize the results on the bondage for119866 to 119866119896 In particular do the following propositions hold

(a) 119887(119866119896

) = 119887119896(119866) for any graph 119866 and each 119896 ⩾ 1

(b) 119887(119866119896

) ⩽ Δ119896(119866) if 119866 is not 120574

119896-critical

Let 119896 and 119901 be positive integers A subset 119878 of 119881(119866) isdefined to be a (119896 119901)-dominating set of 119866 if for any vertex119909 isin 119878 |119873

119896(119909) cap 119878| ⩾ 119901 The (119896 119901)-domination number of

119866 denoted by 120574119896119901(119866) is the minimum cardinality among

all (119896 119901)-dominating sets of 119866 Clearly for a graph 119866 a(1 1)-dominating set is a classical dominating set a (119896 1)-dominating set is a distance 119896-dominating set and a (1 119901)-dominating set is the above-mentioned 119901-dominating setThat is 120574

11(119866) = 120574(119866) 120574

1198961(119866) = 120574

119896(119866) and 120574

1119901(119866) = 120574

119901(119866)

The concept of (119896 119901)-domination in a graph 119866 is a gen-eralized domination which combines distance 119896-dominationand 119901-domination in 119866 So the investigation of (119896 119901)-domination of 119866 is more interesting and has received theattention of many researchers see for example [106ndash109]

More general Li and Xu [110] proposed the concept of the(ℓ 119908)-domination number of a119908-connected graph A subset119878 of 119881(119866) is defined to be an (ℓ 119908)-dominating set of 119866 iffor any vertex 119909 isin 119878 there are 119908 internally disjoint pathsof length at most ℓ from 119909 to some vertex in 119878 The (ℓ 119908)-domination number of119866 denoted by 120574

ℓ119908(119866) is theminimum

cardinality among all (ℓ 119908)-dominating sets of 119866 Clearly1205741198961(119866) = 120574

119896(119866) and 120574

1119901(119866) = 120574

119901(119866)

It is quite natural to propose the concept of bondage num-ber for (119896 119901)-domination or (ℓ 119908)-domination However asfar as we know none has proposed this concept until todayThis is a worthwhile topic for further research

24 International Journal of Combinatorics

93 Fractional Bondage Numbers If 120590 is a function mappingthe vertex-set 119881 into some set of real numbers then for anysubset 119878 sube 119881 let 120590(119878) = sum

119909isin119878120590(119909) Also let |120590| = 120590(119881)

A real-value function 120590 119881 rarr [0 1] is a dominatingfunction of a graph 119866 if for every 119909 isin 119881(119866) 120590(119873

119866[119909]) ⩾ 1

Thus if 119878 is a dominating set of 119866 120590 is a function where

120590 (119909) = 1 if 119909 isin 1198780 otherwise

(57)

then 120590 is a dominating function of119866 The fractional domina-tion number of 119866 denoted by 120574

119865(119866) is defined as follows

120574119865(119866) = min |120590| 120590 is a domination function of 119866

(58)

The 120574119865-bondage number of 119866 denoted by 119887

119865(119866) is

defined as the minimum cardinality of a subset 119861 sube 119864 whoseremoval results in 120574

119865(119866 minus 119861) gt 120574

119865(119866)

Hedetniemi et al [111] are the first to study fractionaldomination although Farber [112] introduced the idea indi-rectly The concept of the fractional domination numberwas proposed by Domke and Laskar [113] in 1997 Thefractional domination numbers for some ordinary graphs aredetermined

Proposition 150 (Domke et al [114] 1998 Domke and Laskar[113] 1997) (a) If 119866 is a 119896-regular graph with order 119899 then120574119865(119866) = 119899119896 + 1(b) 120574

119865(119862

119899) = 1198993 120574

119865(119875

119899) = lceil1198993rceil 120574

119865(119870

119899) = 1

(c) 120574119865(119866) = 1 if and only if Δ(119866) = 119899 minus 1

(d) 120574119865(119879) = 120574(119879) for any tree 119879

(e) 120574119865(119870

119899119898) = (119899(119898 + 1) + 119898(119899 + 1))(119899119898 minus 1) where

max119899119898 gt 1

The assertions (a)ndash(d) are due to Domke et al [114] andthe assertion (e) is due to Domke and Laskar [113] Accordingto these results Domke and Laskar [113] determined the 120574

119865-

bondage numbers for these graphs

Theorem 151 (Domke and Laskar [113] 1997) 119887119865(119870

119899) =

lceil1198992rceil 119887119865(119870

119899119898) = 1 wheremax119899119898 gt 1

119887119865(119862

119899) =

2 119894119891 119899 equiv 0 (mod 3) 1 119900119905ℎ119890119903119908119894119904119890

119887119865(119875

119899) =

2 119894119891 119899 equiv 1 (mod 3) 1 119900119905ℎ119890119903119908119894119904119890

(59)

It is easy to see that for any tree119879 119887119865(119879) ⩽ 2 In fact since

120574119865(119879) = 120574(119879) for any tree 119879 120574

119865(119879

1015840

) = 120574(1198791015840

) for any subgraph1198791015840 of 119879 It follows that

119887119865(119879) = min 119861 sube 119864 (119879) 120574

119865(119879 minus 119861) gt 120574

119865(119879)

= min 119861 sube 119864 (119879) 120574 (119879 minus 119861) gt 120574 (119879)

= 119887 (119879) ⩽ 2

(60)

Except for these no results on 119887119865(119866) have been known as

yet

94 Roman Bondage Numbers A Roman dominating func-tion on a graph 119866 is a labeling 119891 119881 rarr 0 1 2 such thatevery vertex with label 0 has at least one neighbor with label2 The weight of a Roman dominating function 119891 on 119866 is thevalue

119891 (119866) = sum

119906isin119881(119866)

119891 (119906) (61)

The minimum weight of a Roman dominating function ona graph 119866 is called the Roman domination number of 119866denoted by 120574RM (119866)

A Roman dominating function 119891 119881 rarr 0 1 2

can be represented by the ordered partition (1198810 119881

1 119881

2) (or

(119881119891

0 119881

119891

1 119881

119891

2) to refer to119891) of119881 where119881

119894= V isin 119881 | 119891(V) = 119894

In this representation its weight 119891(119866) = |1198811| + 2|119881

2| It is

clear that119881119891

1cup119881

119891

2is a dominating set of 119866 called the Roman

dominating set denoted by 119863119891

R = (1198811 1198812) Since 119881119891

1cup 119881

119891

2is

a dominating set when 119891 is a Roman dominating functionand since placing weight 2 at the vertices of a dominating setyields a Roman dominating function in [115] it is observedthat

120574 (119866) ⩽ 120574RM (119866) ⩽ 2120574 (119866) (62)

A graph 119866 is called to be Roman if 120574RM (119866) = 2120574(119866)The definition of the Roman domination function is

proposed implicitly by Stewart [116] and ReVelle and Rosing[117] Roman dominating numbers have been deeply studiedIn particular Bahremandpour et al [118] showed that theproblem determining the Roman domination number is NP-complete even for bipartite graphs

Let 119866 be a graph with maximum degree at least twoThe Roman bondage number 119887RM (119866) of 119866 is the minimumcardinality of all sets 1198641015840 sube 119864 for which 120574RM (119866 minus 119864

1015840

) gt

120574RM (119866) Since in study of Roman bondage number theassumption Δ(119866) ⩾ 2 is necessary we always assume thatwhen we discuss 119887RM (119866) all graphs involved satisfy Δ(119866) ⩾2 The Roman bondage number 119887RM (119866) is introduced byJafari Rad and Volkmann in [119]

Recently Bahremandpour et al [118] have shown that theproblem determining the Roman bondage number is NP-hard even for bipartite graphs

Problem 7 Consider the decision problem

Roman Bondage Problem

Instance a nonempty bipartite graph119866 and a positiveinteger 119896

Question is 119887RM (119866) ⩽ 119896

Theorem 152 (Bahremandpour et al [118] 2013) TheRomanbondage problem is NP-hard even for bipartite graphs

The exact value of 119887RM(119866) is known only for a few familyof graphs including the complete graphs cycles and paths

International Journal of Combinatorics 25

Theorem 153 (Jafari Rad and Volkmann [119] 2011) For anypositive integers 119899 and119898 with119898 ⩽ 119899

119887RM (119875119899) = 2 119894119891 119899 equiv 2 (mod 3) 1 119900119905ℎ119890119903119908119894119904119890

119887RM (119862119899) =

3 119894119891 119899 = 2 (mod 3) 2 119900119905ℎ119890119903119908119894119904119890

119887RM (119870119899) = lceil

119899

2rceil 119891119900119903 119899 ⩾ 3

119887RM (119870119898119899) =

1 119894119891 119898 = 1 119886119899119889 119899 = 1

4 119894119891 119898 = 119899 = 3

119898 119900119905ℎ119890119903119908119894119904119890

(63)

Lemma 154 (Cockayne et al [115] 2004) If 119866 is a graph oforder 119899 and contains vertices of degree 119899 minus 1 then 120574RM(119866) = 2

Using Lemma 154 the third conclusion in Theorem 153can be generalized to more general case which is similar toLemma 49

Theorem 155 (Jafari Rad and Volkmann [119] 2011) Let119866 bea graph with order 119899 ge 3 and 119905 the number of vertices of degree119899 minus 1 in 119866 If 119905 ⩾ 1 then 119887RM(119866) = lceil1199052rceil

Ebadi and PushpaLatha [120] conjectured that 119887RM(119866) ⩽119899 minus 1 for any graph 119866 of order 119899 ⩾ 3 Dehgardi et al[121] Akbari andQajar [122] independently showed that thisconjecture is true

Theorem 156 119887RM(119866) ⩽ 119899 minus 1 for any connected graph 119866 oforder 119899 ⩾ 3

For a complete 119905-partite graph we have the followingresult

Theorem 157 (Hu and Xu [123] 2011) Let 119866 = 11987011989811198982119898119905

bea complete 119905-partite graph with 119898

1= sdot sdot sdot = 119898

119894lt 119898

119894+1le sdot sdot sdot ⩽

119898119905 119905 ⩾ 2 and 119899 = sum119905

119895=1119898

119895 Then

119887RM (119866) =

lceil119894

2rceil 119894119891 119898

119894= 1 119886119899119889 119899 ⩾ 3

2 119894119891 119898119894= 2 119886119899119889 119894 = 1

119894 119894119891 119898119894= 2 119886119899119889 119894 ⩾ 2

4 119894119891 119898119894= 3 119886119899119889 119894 = 119905 = 2

119899 minus 1 119894119891 119898119894= 3 119886119899119889 119894 = 119905 ⩾ 3

119899 minus 119898119905

119894119891 119898119894⩾ 3 119886119899119889 119898

119905⩾ 4

(64)

Consider a complete 119905-partite graph 119870119905(3) for 119905 ⩾ 3

119887RM(119870119905(3)) = 119899 minus 1 by Theorem 157 where 119899 = 3119905

which shows that this upper bound of 119899 minus 1 on 119887RM(119866) inTheorem 156 is sharp This fact and 120574

119877(119870

119905(3)) = 4 give

a negative answer to the following two problems posed byDehgardi et al [121]Prove or Disprove for any connected graph 119866 of order 119899 ge 3(i) 119887RM(119866) = 119899 minus 1 if and only if 119866 cong 119870

3 (ii) 119887RM(119866) ⩽ 119899 minus

120574119877(119866) + 1Note that a complete 119905-partite graph 119870

119905(3) is (119899 minus 3)-

regular and 119887RM(119870119905(3)) = 119899 minus 1 for 119905 ⩾ 3 where 119899 = 3119905

Hu and Xu further determined that 119887119877(119866) = 119899 minus 2 for any

(119899 minus 3)-regular graph 119866 of order 119899 ⩽ 5 except 119870119905(3)

Theorem 158 (Hu and Xu [123] 2011) For any (119899minus3)-regulargraph 119866 of order 119899 other than119870

119905(3) 119887RM(119866) = 119899 minus 2 for 119899 ⩾ 5

For a tree 119879 with order 119899 ⩾ 3 Ebadi and PushpaLatha[120] and Jafari Rad and Volkmann [119] independentlyobtained an upper bound on 119887RM(119879)

Theorem 159 119887RM(119879) ⩽ 3 for any tree 119879 with order 119899 ⩾ 3

Theorem 160 (Bahremandpour et al [118] 2013) 119887RM(1198752 times119875119899) = 2 for 119899 ⩾ 2

Theorem 161 (Jafari Rad and Volkmann [119] 2011) Let119866 bea graph with order at least three

(a) If (119909 119910 119911) is a path of length 2 in 119866 then 119887RM(119866) ⩽119889119866(119909) + 119889

119866(119910) + 119889

119866(119911) minus 3 minus |119873

119866(119909) cap 119873

119866(119910)|

(b) If 119866 is connected then 119887RM(119866) ⩽ 120582(119866) + 2Δ(119866) minus 3

Theorem 161 (b) implies 119887RM(119866) ⩽ 120575(119866) + 2Δ(119866) minus 3 for aconnected graph 119866 Note that for a planar graph 119866 120575(119866) ⩽ 5moreover 120575(119866) ⩽ 3 if the girth at least 4 and 120575(119866) ⩽ 2 if thegirth at least 6 These two facts show that 119887RM(119866) ⩽ 2Δ(119866) +2 for a connected planar graph 119866 Jafari Rad and Volkmann[124] improved this bound

Theorem 162 (Jafari Rad and Volkmann [124] 2011) For anyconnected planar graph 119866 of order 119899 ⩾ 3 with girth 119892(119866)

119887RM (119866)

⩽

2Δ (119866)

Δ (119866) + 6

Δ (119866) + 5 119894119891 119866 119888119900119899119905119886119894119899119904 119899119900 V119890119903119905119894119888119890119904 119900119891 119889119890119892119903119890119890 119891119894V119890

Δ (119866) + 4 119894119891 119892 (119866) ⩾ 4

Δ (119866) + 3 119894119891 119892 (119866) ⩾ 5

Δ (119866) + 2 119894119891 119892 (119866) ⩾ 6

Δ (119866) + 1 119894119891 119892 (119866) ⩾ 8

(65)

According toTheorem 157 119887RM(119862119899) = 3 = Δ(119862

119899)+1 for a

cycle 119862119899of length 119899 ⩾ 8 with 119899 equiv 2 (mod 3) and therefore

the last upper bound inTheorem 162 is sharp at least for Δ =2

Combining the fact that every planar graph 119866 withminimum degree 5 contains an edge 119909119910 with 119889

119866(119909) = 5

and 119889119866(119910) isin 5 6 with Theorem 161 (a) Akbari et al [125]

obtained the following result

26 International Journal of Combinatorics

middot middot middot

middot middot middot

middot middot middot

middot middot middot

119866

Figure 9 The graph 119866 is constructed from 119866

Theorem 163 (Akbari et al [125] 2012) 119887RM(119866) ⩽ 15 forevery planar graph 119866

It remains open to show whether the bound inTheorem 163 is sharp or not Though finding a planargraph 119866 with 119887RM(119866) = 15 seems to be difficult Akbari etal [125] constructed an infinite family of planar graphs withRoman bondage number equal to 7 by proving the followingresult

Theorem 164 (Akbari et al [125] 2012) Let 119866 be a graph oforder 119899 and 119866 is the graph of order 5119899 obtained from 119866 byattaching the central vertex of a copy of a path 119875

5to each vertex

of119866 (see Figure 9) Then 120574RM(119866) = 4119899 and 119887RM(119866) = 120575(119866)+2

ByTheorem 164 infinitemany planar graphswith Romanbondage number 7 can be obtained by considering any planargraph 119866 with 120575(119866) = 5 (eg the icosahedron graph)

Conjecture 165 (Akbari et al [125] 2012) The Romanbondage number of every planar graph is at most 7

For general bounds the following observation is directlyobtained which is similar to Observation 1

Observation 4 Let 119866 be a graph of order 119899 with maximumdegree at least two Assume that 119867 is a spanning subgraphobtained from 119866 by removing 119896 edges If 120574RM(119867) = 120574RM(119866)then 119887

119877(119867) ⩽ 119887RM(119866) ⩽ 119887RM(119867) + 119896

Theorem 166 (Bahremandpour et al [118] 2013) For anyconnected graph 119866 of order 119899 if 119899 ⩾ 3 and 120574RM(119866) = 120574(119866) + 1then

119887RM (119866) ⩽ min 119887 (119866) 119899Δ (66)

where 119899Δis the number of vertices with maximum degree Δ in

119866

Observation 5 For any graph 119866 if 120574(119866) = 120574RM(119866) then119887RM(119866) ⩽ 119887(119866)

Theorem 167 (Bahremandpour et al [118] 2013) For everyRoman graph 119866

119887RM (119866) ⩾ 119887 (119866) (67)

The bound is sharp for cycles on 119899 vertices where 119899 equiv 0 (mod3)

The strict inequality in Theorem 167 can hold for exam-ple 119887(119862

3119896+2) = 2 lt 3 = 119887RM(1198623119896+2

) byTheorem 157A graph 119866 is called to be vertex Roman domination-

critical if 120574RM(119866 minus 119909) lt 120574RM(119866) for every vertex 119909 in 119866 If119866 has no isolated vertices then 120574RM(119866) ⩽ 2120574(119866) ⩽ 2120573(119866)If 120574RM(119866) = 2120573(119866) then 120574RM(119866) = 2120574(119866) and hence 119866 is aRoman graph In [30] Volkmann gave a lot of graphs with120574(119866) = 120573(119866) The following result is similar to Theorem 57

Theorem 168 (Bahremandpour et al [118] 2013) For anygraph 119866 if 120574RM(119866) = 2120573(119866) then

(a) 119887RM(119866) ⩾ 120575(119866)

(b) 119887RM(119866) ⩾ 120575(119866)+1 if119866 is a vertex Roman domination-critical graph

If 120574RM(119866) = 2 then obviously 119887RM(119866) ⩽ 120575(119866) So weassume 120574RM(119866) ⩾ 3 Dehgardi et al [121] proved 119887RM(119866) ⩽(120574RM(119866) minus 2)Δ(119866) + 1 and posed the following problem if 119866is a connected graph of order 119899 ⩾ 4 with 120574RM(119866) ⩾ 3 then

119887RM (119866) ⩽ (120574RM (119866) minus 2) Δ (119866) (68)

Theorem 161 (a) shows that the inequality (68) holds if120574RM(119866) ⩾ 5 Thus the bound in (68) is of interest only when120574RM(119866) is 3 or 4

Lemma 169 (Bahremandpour et al [118] 2013) For anynonempty graph 119866 of order 119899 ⩾ 3 120574RM(119866) = 3 if and onlyif Δ(119866) = 119899 minus 2

The following result shows that (68) holds for all graphs119866 of order 119899 ⩾ 4 with 120574RM(119866) = 3 or 4 which improvesTheorem 161 (b)

Theorem 170 (Bahremandpour et al [118] 2013) For anyconnected graph 119866 of order 119899 ⩾ 4

119887RM (119866) le Δ (119866) = 119899 minus 2 119894119891 120574RM (119866) = 3

Δ (119866) + 120575 (119866) minus 1 119894119891 120574RM (119866) = 4(69)

with the first equality if and only if 119866 cong 1198624

Recently Akbari and Qajar [122] have proved the follow-ing result

Theorem 171 (Akbari and Qajar [122]) For any connectedgraph 119866 of order 119899 ⩾ 3

119887RM (119866) ⩽ 119899 minus 120574RM (119866) + 5 (70)

International Journal of Combinatorics 27

We conclude this section with the following problems

Problem 8 Characterize all connected graphs 119866 of order 119899 ⩾3 for which 119887RM(119866) = 119899 minus 1

Problem 9 Prove or disprove if 119866 is a connected graph oforder 119899 ⩾ 3 then

119887RM (119866) ⩽ 119899 minus 120574RM (119866) + 3 (71)

Note that 120574119877(119870

119905(3)) = 4 and 119887RM(119870119905

(3)) = 119899minus1 where 119899 =3119905 for 119905 ⩾ 3 The upper bound on 119887RM(119866) given in Problem 9is sharp if Problem 9 has positive answer

95 Rainbow Bondage Numbers For a positive integer 119896 a119896-rainbow dominating function of a graph 119866 is a function 119891from the vertex-set 119881(119866) to the set of all subsets of the set1 2 119896 such that for any vertex 119909 in 119866 with 119891(119909) = 0 thecondition

⋃

119910isin119873119866(119909)

119891 (119910) = 1 2 119896 (72)

is fulfilledThe weight of a 119896-rainbow dominating function 119891on 119866 is the value

119891 (119866) = sum

119909isin119881(119866)

1003816100381610038161003816119891 (119909)1003816100381610038161003816 (73)

The 119896-rainbow domination number of a graph 119866 denoted by120574119903119896(119866) is the minimum weight of a 119896-rainbow dominating

function 119891 of 119866Note that 120574

1199031(119866) is the classical domination number 120574(119866)

The 119896-rainbow domination number is introduced by Bresaret al [126] Rainbow domination of a graph 119866 coincides withordinary domination of the Cartesian product of 119866 with thecomplete graph in particular 120574

119903119896(119866) = 120574(119866 times 119870

119896) for any

positive integer 119896 and any graph 119866 [126] Hartnell and Rall[127] proved that 120574(119879) lt 120574

1199032(119879) for any tree 119879 no graph has

120574 = 120574119903119896for 119896 ⩾ 3 for any 119896 ⩾ 2 and any graph 119866 of order 119899

min 119899 120574 (119866) + 119896 minus 2 ⩽ 120574119903119896(119866) ⩽ 119896120574 (119866) (74)

The introduction of rainbow domination is motivatedby the study of paired domination in Cartesian productsof graphs where certain upper bounds can be expressed interms of rainbow domination that is for any graph 119866 andany graph 119867 if 119881(119867) can be partitioned into 119896120574

119875-sets then

120574119875(119866 times 119867) ⩽ (1119896)120592(119867)120574

119903119896(119866) proved by Bresar et al [126]

Dehgardi et al [128] introduced the concept of 119896-rainbowbondage number of graphs Let 119866 be a graph with maximumdegree at least two The 119896-rainbow bondage number 119887

119903119896(119866)

of 119866 is the minimum cardinality of all sets 119861 sube 119864(119866) forwhich 120574

119903119896(119866 minus 119861) gt 120574

119903119896(119866) Since in the study of 119896-rainbow

bondage number the assumption Δ(119866) ⩾ 2 is necessarywe always assume that when we discuss 119887

119903119896(119866) all graphs

involved satisfy Δ(119866) ⩾ 2The 2-rainbow bondage number of some special families

of graphs has been determined

Theorem 172 (Dehgardi et al [128]) For any integer 119899 ⩾ 3

1198871199032(119875

119899) = 1 119891119900119903 119899 ⩾ 2

1198871199032(119862

119899) =

1 119894119891 119899 equiv 0 (mod 4) 2 119900119905ℎ119890119903119908119894119904119890

1198871199032(119870

119899119899) =

1 119894119891 119899 = 2

3 119894119891 119899 = 3

6 119894119891 119899 = 4

119898 119894119891 119899 ⩾ 5

(75)

Theorem 173 (Dehgardi et al [128]) For any connected graph119866 of order 119899 ⩾ 3 119887

1199032(119866) ⩽ 120582(119866) + 2Δ(119866) minus 3

Theorem 174 (Dehgardi et al [128]) For any tree 119879 of order119899 ⩾ 3 119887

1199032(119879) ⩽ 2

Theorem 175 (Dehgardi et al [128]) If 119866 is a planar graphwithmaximumdegree at least two then 119887

1199032(119866) ⩽ 15Moreover

if the girth 119892(119866) ⩾ 4 then 1198871199032(119866) ⩽ 11 and if 119892(119866) ⩾ 6 then

1198871199032(119866) ⩽ 9

Theorem 176 (Dehgardi et al [128]) For a complete bipartitegraph 119870

119901119902 if 2119896 + 1 ⩽ 119901 ⩽ 119902 then 119887

119903119896(119870

119901119902) = 119901

Theorem177 (Dehgardi et al [128]) For a complete graph119870119899

if 119899 ⩾ 119896 + 1 then 119887119903119896(119870

119899) = lceil119896119899(119896 + 1)rceil

The special case 119896 = 1 of Theorem 177 can be found inTheorem 1 The following is a simple observation similar toObservation 1

Observation 6 (Dehgardi et al [128]) Let 119866 be a graphwith maximum degree at least two 119867 a spanning subgraphobtained from 119866 by removing 119896 edges If 120574

119903119896(119867) = 120574

119903119896(119866)

then 119887119903119896(119867) ⩽ 119887

119903119896(119866) ⩽ 119887

119903119896(119867) + 119896

Theorem 178 (Dehgardi et al [128]) For every graph 119866 with120574119903119896(119866) = 119896120574(119866) 119887

119903119896(119866) ⩾ 119887(119866)

A graph 119866 is said to be vertex 119896-rainbow domination-critical if 120574

119903119896(119866 minus 119909) lt 120574

119903119896(119866) for every vertex 119909 in 119866 It is

clear that if 119866 has no isolated vertices then 120574119903119896(119866) ⩽ 119896120574(119866) ⩽

119896120573(119866) where 120573(119866) is the vertex-covering number of119866 Thusif 120574

119903119896(119866) = 119896120573(119866) then 120574

119903119896(119866) = 119896120574(119866)

Theorem 179 (Dehgardi et al [128]) For any graph 119866 if120574119903119896(119866) = 119896120573(119866) then

(a) 119887119903119896(119866) ⩾ 120575(119866)

(b) 119887119903119896(119866) ⩾ 120575(119866)+1 if119866 is vertex 119896-rainbow domination-

critical

As far as we know there are no more results on the 119896-rainbow bondage number of graphs in the literature

28 International Journal of Combinatorics

10 Results on Digraphs

Although domination has been extensively studied in undi-rected graphs it is natural to think of a dominating setas a one-way relationship between vertices of the graphIndeed among the earliest literatures on this subject vonNeumann and Morgenstern [129] used what is now calleddomination in digraphs to find solution (or kernels whichare independent dominating sets) for cooperative 119899-persongames Most likely the first formulation of domination byBerge [130] in 1958 was given in the context of digraphs andonly some years latter by Ore [131] in 1962 for undirectedgraphs Despite this history examination of domination andits variants in digraphs has been essentially overlooked (see[4] for an overview of the domination literature) Thus thereare few if any such results on domination for digraphs in theliterature

The bondage number and its related topics for undirectedgraph have become one of major areas both in theoreticaland applied researches However until recently Carlsonand Develin [42] Shan and Kang [132] and Huang andXu [133 134] studied the bondage number for digraphsindependently In this section we introduce their resultsfor general digraphs Results for some special digraphs suchas vertex-transitive digraphs are introduced in the nextsection

101 Upper Bounds for General Digraphs Let 119866 = (119881 119864)

be a digraph without loops and parallel edges A digraph 119866is called to be asymmetric if whenever (119909 119910) isin 119864(119866) then(119910 119909) notin 119864(119866) and to be symmetric if (119909 119910) isin 119864(119866) implies(119910 119909) isin 119864(119866) For a vertex 119909 of 119881(119866) the sets of out-neighbors and in-neighbors of 119909 are respectively defined as119873

+

119866(119909) = 119910 isin 119881(119866) (119909 119910) isin 119864(119866) and 119873minus

119866(119909) = 119909 isin

119881(119866) (119909 119910) isin 119864(119866) the out-degree and the in-degree of119909 are respectively defined as 119889+

119866(119909) = |119873

+

119866(119909)| and 119889minus

119866(119909) =

|119873+

119866(119909)| Denote themaximum and theminimum out-degree

(resp in-degree) of 119866 by Δ+(119866) and 120575+(119866) (resp Δminus(119866) and120575minus

(119866))The degree 119889119866(119909) of 119909 is defined as 119889+

119866(119909)+119889

minus

119866(119909) and

the maximum and the minimum degrees of119866 are denoted byΔ(119866) and 120575(119866) respectively that is Δ(119866) = max119889

119866(119909) 119909 isin

119881(119866) and 120575(119866) = min119889119866(119909) 119909 isin 119881(119866) Note that the

definitions here are different from the ones in the textbookon digraphs see for example Xu [1]

A subset 119878 of119881(119866) is called a dominating set if119881(119866) = 119878cup119873

+

119866(119878) where119873+

119866(119878) is the set of out-neighbors of 119878Then just

as for undirected graphs 120574(119866) is the minimum cardinalityof a dominating set and the bondage number 119887(119866) is thesmallest cardinality of a set 119861 of edges such that 120574(119866 minus 119861) gt120574(119866) if such a subset 119861 sube 119864(119866) exists Otherwise we put119887(119866) = infin

Some basic results for undirected graphs stated inSection 3 can be generalized to digraphs For exampleTheorem 18 is generalized by Carlson and Develin [42]Huang andXu [133] and Shan andKang [132] independentlyas follows

Theorem 180 Let 119866 be a digraph and (119909 119910) isin 119864(119866) Then119887(119866) ⩽ 119889

119866(119910) + 119889

minus

119866(119909) minus |119873

minus

119866(119909) cap 119873

minus

119866(119910)|

Corollary 181 For a digraph 119866 119887(119866) ⩽ 120575minus(119866) + Δ(119866)

Since 120575minus

(119866) ⩽ (12)Δ(119866) from Corollary 181Conjecture 53 is valid for digraphs

Corollary 182 For a digraph 119866 119887(119866) ⩽ (32)Δ(119866)

In the case of undirected graphs the bondage number of119866119899= 119870

119899times 119870

119899achieves this bound However it was shown

by Carlson and Develin in [42] that if we take the symmetricdigraph119866

119899 we haveΔ(119866

119899) = 4(119899minus1) 120574(119866

119899) = 119899 and 119887(119866

119899) ⩽

4(119899 minus 1) + 1 = Δ(119866119899) + 1 So this family of digraphs cannot

show that the bound in Corollary 182 is tightCorresponding to Conjecture 51 which is discredited

for undirected graphs and is valid for digraphs the sameconjecture for digraphs can be proposed as follows

Conjecture 183 (Carlson and Develin [42] 2006) 119887(119866) ⩽Δ(119866) + 1 for any digraph 119866

If Conjecture 183 is true then the following results showsthat this upper bound is tight since 119887(119870

119899times119870

119899) = Δ(119870

119899times119870

119899)+1

for a complete digraph 119870119899

In 2007 Shan and Kang [132] gave some tight upperbounds on the bondage numbers for some asymmetricdigraphs For example 119887(119879) ⩽ Δ(119879) for any asymmetricdirected tree 119879 119887(119866) ⩽ Δ(119866) for any asymmetric digraph 119866with order at least 4 and 120574(119866) ⩽ 2 For planar digraphs theyobtained the following results

Theorem 184 (Shan and Kang [132] 2007) Let 119866 be aasymmetric planar digraphThen 119887(119866) ⩽ Δ(119866)+2 and 119887(119866) ⩽Δ(119866) + 1 if Δ(119866) ⩾ 5 and 119889minus

119866(119909) ⩾ 3 for every vertex 119909 with

119889119866(119909) ⩾ 4

102 Results for Some Special Digraphs The exact values andbounds on 119887(119866) for some standard digraphs are determined

Theorem185 (Huang andXu [133] 2006) For a directed cycle119862119899and a directed path 119875

119899

119887 (119862119899) =

3 119894119891 119899 119894119904 119900119889119889

2 119894119891 119899 119894119904 119890V119890119899

119887 (119875119899) =

2 119894119891 119899 119894119904 119900119889119889

1 119894119891 119899 119894119904 119890V119890119899

(76)

For the de Bruijn digraph 119861(119889 119899) and the Kautz digraph119870(119889 119899)

119887 (119861 (119889 119899)) = 119889 119894119891 119899 119894119904 119900119889119889

119889 ⩽ 119887 (119861 (119889 119899)) ⩽ 2119889 119894119891 119899 119894119904 119890V119890119899

119887 (119870 (119889 119899)) = 119889 + 1

(77)

Like undirected graphs we can define the total domi-nation number and the total bondage number On the totalbondage numbers for some special digraphs the knownresults are as follows

International Journal of Combinatorics 29

Theorem 186 (Huang and Xu [134] 2007) For a directedcycle 119862

119899and a directed path 119875

119899 119887

119879(119875

119899) and 119887

119879(119862

119899) all do not

exist For a complete digraph 119870119899

119887119879(119870

119899) = infin 119894119891 119899 = 1 2

119887119879(119870

119899) = 3 119894119891 119899 = 3

119899 ⩽ 119887119879(119870

119899) ⩽ 2119899 minus 3 119894119891 119899 ⩾ 4

(78)

The extended de Bruijn digraph EB(119889 119899 1199021 119902

119901) and

the extended Kautz digraph EK(119889 119899 1199021 119902

119901) are intro-

duced by Shibata and Gonda [135] If 119901 = 1 then theyare the de Bruijn digraph B(119889 119899) and the Kautz digraphK(119889 119899) respectively Huang and Xu [134] determined theirtotal domination numbers In particular their total bondagenumbers for general cases are determined as follows

Theorem 187 (Huang andXu [134] 2007) If 119889 ⩾ 2 and 119902119894⩾ 2

for each 119894 = 1 2 119901 then

119887119879(EB(119889 119899 119902

1 119902

119901)) = 119889

119901

minus 1

119887119879(EK(119889 119899 119902

1 119902

119901)) = 119889

119901

(79)

In particular for the de Bruijn digraph B(119889 119899) and the Kautzdigraph K(119889 119899)

119887119879(B (119889 119899)) = 119889 minus 1 119887

119879(K (119889 119899)) = 119889 (80)

Zhang et al [136] determined the bondage number incomplete 119905-partite digraphs

Theorem 188 (Zhang et al [136] 2009) For a complete 119905-partite digraph 119870

11989911198992119899119905

where 1198991⩽ 119899

2⩽ sdot sdot sdot ⩽ 119899

119905

119887 (11987011989911198992119899119905

)

=

119898 119894119891 119899119898= 1 119899

119898+1⩾ 2 119891119900119903 119904119900119898119890

119898 (1 ⩽ 119898 lt 119905)

4119905 minus 3 119894119891 1198991= 119899

2= sdot sdot sdot = 119899

119905= 2

119905minus1

sum

119894=1

119899119894+ 2 (119905 minus 1) 119900119905ℎ119890119903119908119894119904119890

(81)

Since an undirected graph can be thought of as a sym-metric digraph any result for digraphs has an analogy forundirected graphs in general In view of this point studyingthe bondage number for digraphs is more significant thanfor undirected graphs Thus we should further study thebondage number of digraphs and try to generalize knownresults on the bondage number and related variants for undi-rected graphs to digraphs prove or disprove Conjecture 183In particular determine the exact values of 119887(B(119889 119899)) for aneven 119899 and 119887

119879(119870

119899) for 119899 ⩾ 4

11 Efficient Dominating Sets

A dominating set 119878 of a graph 119866 is called to be efficient if forevery vertex 119909 in119866 |119873

119866[119909]cap119878| = 1 if119866 is a undirected graph

or |119873minus

119866[119909] cap 119878| = 1 if 119866 is a directed graph The concept of an

efficient set was proposed by Bange et al [137] in 1988 In theliterature an efficient set is also called a perfect dominatingset (see Livingston and Stout [138] for an explanation)

From definition if 119878 is an efficient dominating set of agraph 119866 then 119878 is certainly an independent set and everyvertex not in 119878 is adjacent to exactly one vertex in 119878

It is also clear from definition that a dominating set 119878 isefficient if and only if N

119866[119878] = 119873

119866[119909] 119909 isin 119878 for the

undirected graph 119866 or N+

119866[119878] = 119873

+

119866[119909] 119909 isin 119878 for the

digraph119866 is a partition of119881(119866) where the induced subgraphby119873

119866[119909] or119873+

119866[119909] is a star or an out-star with the root 119909

The efficient domination has important applications inmany areas such as error-correcting codes and receivesmuch attention in the late years

The concept of efficient dominating sets is a measureof the efficiency of domination in graphs Unfortunately asshown in [137] not every graph has an efficient dominatingset andmoreover it is anNP-complete problem to determinewhether a given graph has an efficient dominating set Inaddition it was shown by Clark [139] in 1993 that for a widerange of 119901 almost every random undirected graph 119866 isin

G(120592 119901) has no efficient dominating sets This means thatundirected graphs possessing an efficient dominating set arerare However it is easy to show that every undirected graphhas an orientationwith an efficient dominating set (see Bangeet al [140])

In 1993 Barkauskas and Host [141] showed that deter-mining whether an arbitrary oriented graph has an efficientdominating set is NP-complete Even so the existence ofefficient dominating sets for some special classes of graphs hasbeen examined see for example Dejter and Serra [142] andLee [143] for Cayley graph Gu et al [144] formeshes and toriObradovic et al [145] Huang and Xu [36] Reji Kumar andMacGillivray [146] for circulant graphs Harary graphs andtori and Van Wieren et al [147] for cube-connected cycles

In this section we introduce results of the bondagenumber for some graphs with efficient dominating sets dueto Huang and Xu [36]

111 Results for General Graphs In this subsection we intro-duce some results on bondage numbers obtained by applyingefficient dominating sets We first state the two followinglemmas

Lemma 189 For any 119896-regular graph or digraph 119866 of order 119899120574(119866) ⩾ 119899(119896 + 1) with equality if and only if 119866 has an efficientdominating set In addition if 119866 has an efficient dominatingset then every efficient dominating set is certainly a 120574-set andvice versa

Let 119890 be an edge and 119878 a dominating set in 119866 We say that119890 supports 119878 if 119890 isin (119878 119878) where (119878 119878) = (119909 119910) isin 119864(119866) 119909 isin 119878 119910 isin 119878 Denote by 119904(119866) the minimum number of edgeswhich support all 120574-sets in 119866

Lemma 190 For any graph or digraph 119866 119887(119866) ⩾ 119904(119866) withequality if 119866 is regular and has an efficient dominating set

30 International Journal of Combinatorics

A graph 119866 is called to be vertex-transitive if its automor-phism group Aut(119866) acts transitively on its vertex-set 119881(119866)A vertex-transitive graph is regular Applying Lemmas 189and 190 Huang and Xu obtained some results on bondagenumbers for vertex-transitive graphs or digraphs

Theorem 191 (Huang and Xu [36] 2008) For any vertex-transitive graph or digraph 119866 of order 119899

119887 (119866) ⩾

lceil119899

2120574 (119866)rceil 119894119891 119866 119894119904 119906119899119889119894119903119890119888119905119890119889

lceil119899

120574 (119866)rceil 119894119891 119866 119894119904 119889119894119903119890119888119905119890119889

(82)

Theorem192 (Huang andXu [36] 2008) Let119866 be a 119896-regulargraph of order 119899 Then

119887(119866) ⩽ 119896 if119866 is undirected and 119899 ⩾ 120574(119866)(119896+1)minus119896+1119887(119866) ⩽ 119896+1+ℓ if119866 is directed and 119899 ⩾ 120574(119866)(119896+1)minusℓwith 0 ⩽ ℓ ⩽ 119896 minus 1

Next we introduce a better upper bound on 119887(119866) To thisaim we need the following concept which is a generalizationof the concept of the edge-covering of a graph 119866 For 1198811015840

sube

119881(119866) and 1198641015840 sube 119864(119866) we say that 1198641015840covers 1198811015840 and call 1198641015840an edge-covering for 1198811015840 if there exists an edge (119909 119910) isin 1198641015840 forany vertex 119909 isin 1198811015840 For 119910 isin 119881(119866) let 1205731015840[119910] be the minimumcardinality over all edge-coverings for119873minus

119866[119910]

Theorem 193 (Huang and Xu [36] 2008) If 119866 is a 119896-regulargraph with order 120574(119866)(119896 + 1) then 119887(119866) ⩽ 1205731015840[119910] for any 119910 isin119881(119866)

Theupper bound on 119887(119866) given inTheorem 193 is tight inview of 119887(119862

119899) for a cycle or a directed cycle 119862

119899(seeTheorems

1 and 185 resp)It is easy to see that for a 119896-regular graph 119866 lceil(119896 + 1)2rceil ⩽

1205731015840

[119910] ⩽ 119896 when 119866 is undirected and 1205731015840[119910] = 119896 + 1 when 119866 isdirected By this fact and Lemma 189 the following theoremis merely a simple combination of Theorems 191 and 193

Theorem 194 (Huang and Xu [36] 2008) Let 119866 be a vertex-transitive graph of degree 119896 If 119866 has an efficient dominatingset then

lceil119896 + 1

2rceil ⩽ 119887 (119866) ⩽ 119896 119894119891 119866 119894119904 119906119899119889119894119903119890119888119905119890119889

119887 (119866) = 119896 + 1 119894119891 119866 119894119904 119889119894119903119890119888119905119890119889

(83)

Theorem 195 (Huang and Xu [36] 2008) If 119866 is an undi-rected vertex-transitive cubic graph with order 4120574(119866) and girth119892(119866) ⩽ 5 then 119887(119866) = 2

The proof of Theorem 195 leads to a byproduct If 119892 =5 let (119906

1 119906

2 119906

3 119906

4 119906

5) be a cycle and let D

119894be the family

of efficient dominating sets containing 119906119894for each 119894 isin

1 2 3 4 5 ThenD119894capD

119895= 0 for 119894 = 1 2 5 Then 119866 has

at least 5119904 efficient dominating sets where 119904 = 119904(119866) But thereare only 4s (=ns120574) distinct efficient dominating sets in119866This

contradiction implies that an undirected vertex-transitivecubic graph with girth five has no efficient dominating setsBut a similar argument for 119892(119866) = 3 4 or 119892(119866) ⩾ 6 couldnot give any contradiction This is consistent with the resultthat CCC(119899) a vertex-transitive cubic graph with girth 119899 if3 ⩽ 119899 ⩽ 8 or girth 8 if 119899 ⩾ 9 has efficient dominating setsfor all 119899 ⩾ 3 except 119899 = 5 (see Theorem 199 in the followingsubsection)

112 Results for Cayley Graphs In this subsection we useTheorem 194 to determine the exact values or approximativevalues of bondage numbers for some special vertex-transitivegraphs by characterizing the existence of efficient dominatingsets in these graphs

Let Γ be a nontrivial finite group 119878 a nonempty subset ofΓ without the identity element of Γ A digraph 119866 is defined asfollows

119881 (119866) = Γ (119909 119910) isin 119864 (119866) lArrrArr 119909minus1

119910 isin 119878

for any 119909 119910 isin Γ(84)

is called a Cayley digraph of the group Γ with respect to119878 denoted by 119862

Γ(119878) If 119878minus1 = 119904minus1 119904 isin 119878 = 119878 then

119862Γ(119878) is symmetric and is called a Cayley undirected graph

a Cayley graph for short Cayley graphs or digraphs arecertainly vertex-transitive (see Xu [1])

A circulant graph 119866(119899 119878) of order 119899 is a Cayley graph119862119885119899

(119878) where 119885119899= 0 119899 minus 1 is the addition group of

order 119899 and 119878 is a nonempty subset of119885119899without the identity

element and hence is a vertex-transitive digraph of degree|119878| If 119878minus1 = 119878 then119866(119899 119878) is an undirected graph If 119878 = 1 119896where 2 ⩽ 119896 ⩽ 119899 minus 2 we write 119866(119899 1 119896) for 119866(119899 1 119896) or119866(119899 plusmn1 plusmn119896) and call it a double-loop circulant graph

Huang and Xu [36] showed that for directed 119866 =

119866(119899 1 119896) lceil1198993rceil ⩽ 120574(119866) ⩽ lceil1198992rceil and 119866 has an efficientdominating set if and only if 3 | 119899 and 119896 equiv 2 (mod 3) fordirected 119866 = 119866(119899 1 119896) and 119896 = 1198992 lceil1198995rceil ⩽ 120574(119866) ⩽ lceil1198993rceiland 119866 has an efficient dominating set if and only if 5 | 119899 and119896 equiv plusmn2 (mod 5) By Theorem 194 we obtain the bondagenumber of a double loop circulant graph if it has an efficientdominating set

Theorem 196 (Huang and Xu [36] 2008) Let 119866 be a double-loop circulant graph 119866(119899 1 119896) If 119866 is directed with 3 | 119899and 119896 equiv 2 (mod 3) or 119866 is undirected with 5 | 119899 and119896 equiv plusmn2 (mod 5) then 119887(119866) = 3

The 119898 times 119899 torus is the Cartesian product 119862119898times 119862

119899of

two cycles and is a Cayley graph 119862119885119898times119885119899

(119878) where 119878 =

(0 1) (1 0) for directed cycles and 119878 = (0 plusmn1) (plusmn1 0) forundirected cycles and hence is vertex-transitive Gu et al[144] showed that the undirected torus119862

119898times119862

119899has an efficient

dominating set if and only if both 119898 and 119899 are multiplesof 5 Huang and Xu [36] showed that the directed torus119862119898times 119862

119899has an efficient dominating set if and only if both

119898 and 119899 are multiples of 3 Moreover they found a necessarycondition for a dominating set containing the vertex (0 0)in 119862

119898times 119862

119899to be efficient and obtained the following

result

International Journal of Combinatorics 31

Theorem 197 (Huang and Xu [36] 2008) Let 119866 = 119862119898times 119862

119899

If 119866 is undirected and both119898 and 119899 are multiples of 5 or if 119866 isdirected and both119898 and 119899 are multiples of 3 then 119887(119866) = 3

The hypercube 119876119899

is the Cayley graph 119862Γ(119878)

where Γ = 1198852times sdot sdot sdot times 119885

2= (119885

2)119899 and 119878 =

100 sdot sdot sdot 0 010 sdot sdot sdot 0 00 sdot sdot sdot 01 Lee [143] showed that119876119899has an efficient dominating set if and only if 119899 = 2119898 minus 1

for a positive integer119898 ByTheorem 194 the following resultis obtained immediately

Theorem 198 (Huang and Xu [36] 2008) If 119899 = 2119898 minus 1 for apositive integer119898 then 2119898minus1

⩽ 119887(119876119899) ⩽ 2

119898

minus 1

Problem 10 Determine the exact value of 119887(119876119899) for 119899 = 2119898minus1

The 119899-dimensional cube-connected cycle denoted byCCC(119899) is constructed from the 119899-dimensional hypercube119876119899by replacing each vertex 119909 in 119876

119899with an undirected cycle

119862119899of length 119899 and linking the 119894th vertex of the 119862

119899to the 119894th

neighbor of 119909 It has been proved that CCC(119899) is a Cayleygraph and hence is a vertex-transitive graph with degree 3Van Wieren et al [147] proved that CCC(119899) has an efficientdominating set if and only if 119899 = 5 From Theorems 194 and195 the following result can be derived

Theorem 199 (Huang and Xu [36] 2008) Let 119866 = 119862119862119862(119899)be the 119899-dimensional cube-connected cycles with 119899 ⩾ 3 and119899 = 5 Then 120574(119866) = 1198992119899minus2 and 2 ⩽ 119887(119866) ⩽ 3 In addition119887(119862119862119862(3)) = 119887(119862119862119862(4)) = 2

Problem 11 Determine the exact value of 119887(CCC(119899)) for 119899 ⩾5

Acknowledgments

This work was supported by NNSF of China (Grants nos11071233 61272008) The author would like to express hisgratitude to the anonymous referees for their kind sugges-tions and comments on the original paper to ProfessorSheikholeslami and Professor Jafari Rad for sending thereferences [128] and [81] which resulted in this version

References

[1] J-M Xu Theory and Application of Graphs vol 10 of NetworkTheory and Applications Kluwer Academic Dordrecht TheNetherlands 2003

[2] S THedetniemi andR C Laskar ldquoBibliography on dominationin graphs and some basic definitions of domination parame-tersrdquo Discrete Mathematics vol 86 no 1ndash3 pp 257ndash277 1990

[3] TW Haynes S T Hedetniemi and P J Slater Fundamentals ofDomination in Graphs vol 208 ofMonographs and Textbooks inPure and Applied Mathematics Marcel Dekker New York NYUSA 1998

[4] TWHaynes S THedetniemi and P J Slater EdsDominationin Graphs vol 209 of Monographs and Textbooks in Pure andAppliedMathematicsMarcelDekker NewYorkNYUSA 1998

[5] M R Garey andD S JohnsonComputers and Intractability WH Freeman and Company San Francisco Calif USA 1979

[6] H B Walikar and B D Acharya ldquoDomination critical graphsrdquoNational Academy Science Letters vol 2 pp 70ndash72 1979

[7] D Bauer F Harary J Nieminen and C L Suffel ldquoDominationalteration sets in graphsrdquo Discrete Mathematics vol 47 no 2-3pp 153ndash161 1983

[8] J F Fink M S Jacobson L F Kinch and J Roberts ldquoThebondage number of a graphrdquo Discrete Mathematics vol 86 no1ndash3 pp 47ndash57 1990

[9] F-T Hu and J-M Xu ldquoThe bondage number of (119899 minus 3)-regulargraph G of order nrdquo Ars Combinatoria to appear

[10] U Teschner ldquoNew results about the bondage number of agraphrdquoDiscreteMathematics vol 171 no 1ndash3 pp 249ndash259 1997

[11] B L Hartnell and D F Rall ldquoA bound on the size of a graphwith given order and bondage numberrdquo Discrete Mathematicsvol 197-198 pp 409ndash413 1999 16th British CombinatorialConference (London 1997)

[12] B L Hartnell and D F Rall ldquoA characterization of trees inwhich no edge is essential to the domination numberrdquo ArsCombinatoria vol 33 pp 65ndash76 1992

[13] J Topp and P D Vestergaard ldquo120572119896- and 120574

119896-stable graphsrdquo

Discrete Mathematics vol 212 no 1-2 pp 149ndash160 2000[14] A Meir and J W Moon ldquoRelations between packing and

covering numbers of a treerdquo Pacific Journal of Mathematics vol61 no 1 pp 225ndash233 1975

[15] B L Hartnell L K Joslashrgensen P D Vestergaard and CA Whitehead ldquoEdge stability of the k-domination numberof treesrdquo Bulletin of the Institute of Combinatorics and ItsApplications vol 22 pp 31ndash40 1998

[16] F-T Hu and J-M Xu ldquoOn the complexity of the bondage andreinforcement problemsrdquo Journal of Complexity vol 28 no 2pp 192ndash201 2012

[17] B L Hartnell and D F Rall ldquoBounds on the bondage numberof a graphrdquo Discrete Mathematics vol 128 no 1ndash3 pp 173ndash1771994

[18] Y-L Wang ldquoOn the bondage number of a graphrdquo DiscreteMathematics vol 159 no 1ndash3 pp 291ndash294 1996

[19] G H Fricke TW Haynes S M Hedetniemi S T Hedetniemiand R C Laskar ldquoExcellent treesrdquo Bulletin of the Institute ofCombinatorics and Its Applications vol 34 pp 27ndash38 2002

[20] V Samodivkin ldquoThe bondage number of graphs good and badverticesrdquoDiscussiones Mathematicae GraphTheory vol 28 no3 pp 453ndash462 2008

[21] J E Dunbar T W Haynes U Teschner and L VolkmannldquoBondage insensitivity and reinforcementrdquo in Domination inGraphs vol 209 of Monographs and Textbooks in Pure andAppliedMathematics pp 471ndash489 Dekker NewYork NY USA1998

[22] H Liu and L Sun ldquoThe bondage and connectivity of a graphrdquoDiscrete Mathematics vol 263 no 1ndash3 pp 289ndash293 2003

[23] A-H Esfahanian and S L Hakimi ldquoOn computing a con-ditional edge-connectivity of a graphrdquo Information ProcessingLetters vol 27 no 4 pp 195ndash199 1988

[24] R C Brigham P Z Chinn and R D Dutton ldquoVertexdomination-critical graphsrdquoNetworks vol 18 no 3 pp 173ndash1791988

[25] V Samodivkin ldquoDomination with respect to nondegenerateproperties bondage numberrdquoThe Australasian Journal of Com-binatorics vol 45 pp 217ndash226 2009

[26] U Teschner ldquoA new upper bound for the bondage numberof graphs with small domination numberrdquo The AustralasianJournal of Combinatorics vol 12 pp 27ndash35 1995

32 International Journal of Combinatorics

[27] J Fulman D Hanson and G MacGillivray ldquoVertex dom-ination-critical graphsrdquoNetworks vol 25 no 2 pp 41ndash43 1995

[28] U Teschner ldquoA counterexample to a conjecture on the bondagenumber of a graphrdquo Discrete Mathematics vol 122 no 1ndash3 pp393ndash395 1993

[29] U Teschner ldquoThe bondage number of a graph G can be muchgreater than Δ(G)rdquo Ars Combinatoria vol 43 pp 81ndash87 1996

[30] L Volkmann ldquoOn graphs with equal domination and coveringnumbersrdquoDiscrete AppliedMathematics vol 51 no 1-2 pp 211ndash217 1994

[31] L A Sanchis ldquoMaximum number of edges in connected graphswith a given domination numberrdquoDiscreteMathematics vol 87no 1 pp 65ndash72 1991

[32] S Klavzar and N Seifter ldquoDominating Cartesian products ofcyclesrdquoDiscrete AppliedMathematics vol 59 no 2 pp 129ndash1361995

[33] H K Kim ldquoA note on Vizingrsquos conjecture about the bondagenumberrdquo Korean Journal of Mathematics vol 10 no 2 pp 39ndash42 2003

[34] M Y Sohn X Yuan and H S Jeong ldquoThe bondage number of1198623times119862

119899rdquo Journal of the KoreanMathematical Society vol 44 no

6 pp 1213ndash1231 2007[35] L Kang M Y Sohn and H K Kim ldquoBondage number of the

discrete torus 119862119899times 119862

4rdquo Discrete Mathematics vol 303 no 1ndash3

pp 80ndash86 2005[36] J Huang and J-M Xu ldquoThe bondage numbers and efficient

dominations of vertex-transitive graphsrdquoDiscrete Mathematicsvol 308 no 4 pp 571ndash582 2008

[37] C J Xiang X Yuan andM Y Sohn ldquoDomination and bondagenumber of119862

3times119862

119899rdquoArs Combinatoria vol 97 pp 299ndash310 2010

[38] M S Jacobson and L F Kinch ldquoOn the domination numberof products of graphs Irdquo Ars Combinatoria vol 18 pp 33ndash441984

[39] Y-Q Li ldquoA note on the bondage number of a graphrdquo ChineseQuarterly Journal of Mathematics vol 9 no 4 pp 1ndash3 1994

[40] F-T Hu and J-M Xu ldquoBondage number of mesh networksrdquoFrontiers ofMathematics inChina vol 7 no 5 pp 813ndash826 2012

[41] R Frucht and F Harary ldquoOn the corona of two graphsrdquoAequationes Mathematicae vol 4 pp 322ndash325 1970

[42] K Carlson and M Develin ldquoOn the bondage number of planarand directed graphsrdquoDiscreteMathematics vol 306 no 8-9 pp820ndash826 2006

[43] U Teschner ldquoOn the bondage number of block graphsrdquo ArsCombinatoria vol 46 pp 25ndash32 1997

[44] J Huang and J-M Xu ldquoNote on conjectures of bondagenumbers of planar graphsrdquo Applied Mathematical Sciences vol6 no 65ndash68 pp 3277ndash3287 2012

[45] L Kang and J Yuan ldquoBondage number of planar graphsrdquoDiscrete Mathematics vol 222 no 1ndash3 pp 191ndash198 2000

[46] M Fischermann D Rautenbach and L Volkmann ldquoRemarkson the bondage number of planar graphsrdquo Discrete Mathemat-ics vol 260 no 1ndash3 pp 57ndash67 2003

[47] J Hou G Liu and J Wu ldquoBondage number of planar graphswithout small cyclesrdquoUtilitasMathematica vol 84 pp 189ndash1992011

[48] J Huang and J-M Xu ldquoThe bondage numbers of graphs withsmall crossing numbersrdquo Discrete Mathematics vol 307 no 15pp 1881ndash1897 2007

[49] Q Ma S Zhang and J Wang ldquoBondage number of 1-planargraphrdquo Applied Mathematics vol 1 no 2 pp 101ndash103 2010

[50] J Hou and G Liu ldquoA bound on the bondage number of toroidalgraphsrdquoDiscreteMathematics Algorithms and Applications vol4 no 3 Article ID 1250046 5 pages 2012

[51] Y-C Cao J-M Xu and X-R Xu ldquoOn bondage number oftoroidal graphsrdquo Journal of University of Science and Technologyof China vol 39 no 3 pp 225ndash228 2009

[52] Y-C Cao J Huang and J-M Xu ldquoThe bondage number ofgraphs with crossing number less than fourrdquo Ars Combinatoriato appear

[53] H K Kim ldquoOn the bondage numbers of some graphsrdquo KoreanJournal of Mathematics vol 11 no 2 pp 11ndash15 2004

[54] A Gagarin and V Zverovich ldquoUpper bounds for the bondagenumber of graphs on topological surfacesrdquo Discrete Mathemat-ics 2012

[55] J Huang ldquoAn improved upper bound for the bondage numberof graphs on surfacesrdquoDiscrete Mathematics vol 312 no 18 pp2776ndash2781 2012

[56] A Gagarin and V Zverovich ldquoThe bondage number of graphson topological surfaces and Teschnerrsquos conjecturerdquo DiscreteMathematics vol 313 no 6 pp 796ndash808 2013

[57] V Samodivkin ldquoNote on the bondage number of graphs ontopological surfacesrdquo httparxivorgabs12086203

[58] E J Cockayne R M Dawes and S T Hedetniemi ldquoTotaldomination in graphsrdquoNetworks vol 10 no 3 pp 211ndash219 1980

[59] R Laskar J Pfaff S M Hedetniemi and S T HedetniemildquoOn the algorithmic complexity of total dominationrdquo Society forIndustrial and Applied Mathematics vol 5 no 3 pp 420ndash4251984

[60] J Pfaff R C Laskar and S T Hedetniemi ldquoNP-completeness oftotal and connected domination and irredundance for bipartitegraphsrdquo Tech Rep 428 Department of Mathematical SciencesClemson University 1983

[61] M A Henning ldquoA survey of selected recent results on totaldomination in graphsrdquoDiscrete Mathematics vol 309 no 1 pp32ndash63 2009

[62] V R Kulli and D K Patwari ldquoThe total bondage number ofa graphrdquo in Advances in Graph Theory pp 227ndash235 VishwaGulbarga India 1991

[63] F-T Hu Y Lu and J-M Xu ldquoThe total bondage number ofgrid graphsrdquo Discrete Applied Mathematics vol 160 no 16-17pp 2408ndash2418 2012

[64] J Cao M Shi and B Wu ldquoResearch on the total bondagenumber of a spectial networkrdquo in Proceedings of the 3rdInternational Joint Conference on Computational Sciences andOptimization (CSO rsquo10) pp 456ndash458 May 2010

[65] N Sridharan MD Elias and V S A Subramanian ldquoTotalbondage number of a graphrdquo AKCE International Journal ofGraphs and Combinatorics vol 4 no 2 pp 203ndash209 2007

[66] N Jafari Rad and J Raczek ldquoSome progress on total bondage ingraphsrdquo Graphs and Combinatorics 2011

[67] T W Haynes and P J Slater ldquoPaired-domination and thepaired-domatic numberrdquoCongressusNumerantium vol 109 pp65ndash72 1995

[68] T W Haynes and P J Slater ldquoPaired-domination in graphsrdquoNetworks vol 32 no 3 pp 199ndash206 1998

[69] X Hou ldquoA characterization of 2120574 120574119875-treesrdquoDiscrete Mathemat-

ics vol 308 no 15 pp 3420ndash3426 2008[70] K E Proffitt T W Haynes and P J Slater ldquoPaired-domination

in grid graphsrdquo Congressus Numerantium vol 150 pp 161ndash1722001

International Journal of Combinatorics 33

[71] H Qiao L KangM Cardei andD-Z Du ldquoPaired-dominationof treesrdquo Journal of Global Optimization vol 25 no 1 pp 43ndash542003

[72] E Shan L Kang and M A Henning ldquoA characterizationof trees with equal total domination and paired-dominationnumbersrdquo The Australasian Journal of Combinatorics vol 30pp 31ndash39 2004

[73] J Raczek ldquoPaired bondage in treesrdquo Discrete Mathematics vol308 no 23 pp 5570ndash5575 2008

[74] J A Telle and A Proskurowski ldquoAlgorithms for vertex parti-tioning problems on partial k-treesrdquo SIAM Journal on DiscreteMathematics vol 10 no 4 pp 529ndash550 1997

[75] G S Domke J H Hattingh M A Henning and L R MarkusldquoRestrained domination in treesrdquoDiscreteMathematics vol 211no 1ndash3 pp 1ndash9 2000

[76] G S Domke J H Hattingh M A Henning and L R MarkusldquoRestrained domination in graphs with minimum degree twordquoJournal of Combinatorial Mathematics and Combinatorial Com-puting vol 35 pp 239ndash254 2000

[77] G S Domke J H Hattingh S T Hedetniemi R C Laskarand L R Markus ldquoRestrained domination in graphsrdquo DiscreteMathematics vol 203 no 1ndash3 pp 61ndash69 1999

[78] J H Hattingh and E J Joubert ldquoAn upper bound for therestrained domination number of a graph with minimumdegree at least two in terms of order and minimum degreerdquoDiscrete Applied Mathematics vol 157 no 13 pp 2846ndash28582009

[79] M A Henning ldquoGraphs with large restrained dominationnumberrdquo Discrete Mathematics vol 197-198 pp 415ndash429 199916th British Combinatorial Conference (London 1997)

[80] J H Hattingh and A R Plummer ldquoRestrained bondage ingraphsrdquo Discrete Mathematics vol 308 no 23 pp 5446ndash54532008

[81] N Jafari Rad R Hasni J Raczek and L Volkmann ldquoTotalrestrained bondage in graphsrdquoActaMathematica Sinica vol 29no 6 pp 1033ndash1042 2013

[82] J Cyman and J Raczek ldquoOn the total restrained dominationnumber of a graphrdquoThe Australasian Journal of Combinatoricsvol 36 pp 91ndash100 2006

[83] M A Henning ldquoGraphs with large total domination numberrdquoJournal of Graph Theory vol 35 no 1 pp 21ndash45 2000

[84] D-X Ma X-G Chen and L Sun ldquoOn total restraineddomination in graphsrdquoCzechoslovakMathematical Journal vol55 no 1 pp 165ndash173 2005

[85] B Zelinka ldquoRemarks on restrained domination and totalrestrained domination in graphsrdquo Czechoslovak MathematicalJournal vol 55 no 2 pp 393ndash396 2005

[86] W Goddard and M A Henning ldquoIndependent domination ingraphs a survey and recent resultsrdquo Discrete Mathematics vol313 no 7 pp 839ndash854 2013

[87] J H Zhang H L Liu and L Sun ldquoIndependence bondagenumber and reinforcement number of some graphsrdquo Transac-tions of Beijing Institute of Technology vol 23 no 2 pp 140ndash1422003

[88] K D Dharmalingam Studies in Graph Theorymdashequitabledomination and bottleneck domination [PhD thesis] MaduraiKamaraj University Madurai India 2006

[89] GDeepakND Soner andAAlwardi ldquoThe equitable bondagenumber of a graphrdquo Research Journal of Pure Algebra vol 1 no9 pp 209ndash212 2011

[90] E Sampathkumar and H B Walikar ldquoThe connected domina-tion number of a graphrdquo Journal of Mathematical and PhysicalSciences vol 13 no 6 pp 607ndash613 1979

[91] K White M Farber and W Pulleyblank ldquoSteiner trees con-nected domination and strongly chordal graphsrdquoNetworks vol15 no 1 pp 109ndash124 1985

[92] M Cadei M X Cheng X Cheng and D-Z Du ldquoConnecteddomination in Ad Hoc wireless networksrdquo in Proceedings ofthe 6th International Conference on Computer Science andInformatics (CSampI rsquo02) 2002

[93] J F Fink and M S Jacobson ldquon-domination in graphsrdquo inGraph Theory with Applications to Algorithms and ComputerScience Wiley-Interscience Publications pp 283ndash300 WileyNew York NY USA 1985

[94] M Chellali O Favaron A Hansberg and L Volkmann ldquok-domination and k-independence in graphs a surveyrdquo Graphsand Combinatorics vol 28 no 1 pp 1ndash55 2012

[95] Y Lu X Hou J-M Xu andN Li ldquoTrees with uniqueminimump-dominating setsrdquo Utilitas Mathematica vol 86 pp 193ndash2052011

[96] Y Lu and J-M Xu ldquoThe p-bondage number of treesrdquo Graphsand Combinatorics vol 27 no 1 pp 129ndash141 2011

[97] Y Lu and J-M Xu ldquoThe 2-domination and 2-bondage numbersof grid graphs A manuscriptrdquo httparxivorgabs12044514

[98] M Krzywkowski ldquoNon-isolating 2-bondage in graphsrdquo TheJournal of the Mathematical Society of Japan vol 64 no 4 2012

[99] M A Henning O R Oellermann and H C Swart ldquoBoundson distance domination parametersrdquo Journal of CombinatoricsInformation amp System Sciences vol 16 no 1 pp 11ndash18 1991

[100] F Tian and J-M Xu ldquoBounds for distance domination numbersof graphsrdquo Journal of University of Science and Technology ofChina vol 34 no 5 pp 529ndash534 2004

[101] F Tian and J-M Xu ldquoDistance domination-critical graphsrdquoApplied Mathematics Letters vol 21 no 4 pp 416ndash420 2008

[102] F Tian and J-M Xu ldquoA note on distance domination numbersof graphsrdquo The Australasian Journal of Combinatorics vol 43pp 181ndash190 2009

[103] F Tian and J-M Xu ldquoAverage distances and distance domina-tion numbersrdquo Discrete Applied Mathematics vol 157 no 5 pp1113ndash1127 2009

[104] Z-L Liu F Tian and J-M Xu ldquoProbabilistic analysis of upperbounds for 2-connected distance k-dominating sets in graphsrdquoTheoretical Computer Science vol 410 no 38ndash40 pp 3804ndash3813 2009

[105] M A Henning O R Oellermann and H C Swart ldquoDistancedomination critical graphsrdquo Journal of Combinatorial Mathe-matics and Combinatorial Computing vol 44 pp 33ndash45 2003

[106] T J Bean M A Henning and H C Swart ldquoOn the integrityof distance domination in graphsrdquo The Australasian Journal ofCombinatorics vol 10 pp 29ndash43 1994

[107] M Fischermann and L Volkmann ldquoA remark on a conjecturefor the (kp)-domination numberrdquoUtilitasMathematica vol 67pp 223ndash227 2005

[108] T Korneffel D Meierling and L Volkmann ldquoA remark on the(22)-domination numberrdquo Discussiones Mathematicae GraphTheory vol 28 no 2 pp 361ndash366 2008

[109] Y Lu X Hou and J-M Xu ldquoOn the (22)-domination numberof treesrdquo Discussiones Mathematicae Graph Theory vol 30 no2 pp 185ndash199 2010

34 International Journal of Combinatorics

[110] H Li and J-M Xu ldquo(dm)-domination numbers of m-connected graphsrdquo Rapports de Recherche 1130 LRI UniversiteParis-Sud 1997

[111] S M Hedetniemi S T Hedetniemi and T V Wimer ldquoLineartime resource allocation algorithms for treesrdquo Tech Rep URI-014 Department of Mathematical Sciences Clemson Univer-sity 1987

[112] M Farber ldquoDomination independent domination and dualityin strongly chordal graphsrdquo Discrete Applied Mathematics vol7 no 2 pp 115ndash130 1984

[113] G S Domke and R C Laskar ldquoThe bondage and reinforcementnumbers of 120574

119891for some graphsrdquoDiscrete Mathematics vol 167-

168 pp 249ndash259 1997[114] G S Domke S T Hedetniemi and R C Laskar ldquoFractional

packings coverings and irredundance in graphsrdquo vol 66 pp227ndash238 1988

[115] E J Cockayne P A Dreyer Jr S M Hedetniemi andS T Hedetniemi ldquoRoman domination in graphsrdquo DiscreteMathematics vol 278 no 1ndash3 pp 11ndash22 2004

[116] I Stewart ldquoDefend the roman empirerdquo Scientific American vol281 pp 136ndash139 1999

[117] C S ReVelle and K E Rosing ldquoDefendens imperiumromanum a classical problem in military strategyrdquoThe Ameri-can Mathematical Monthly vol 107 no 7 pp 585ndash594 2000

[118] A Bahremandpour F-T Hu S M Sheikholeslami and J-MXu ldquoOn the Roman bondage number of a graphrdquo DiscreteMathematics Algorithms and Applications vol 5 no 1 ArticleID 1350001 15 pages 2013

[119] N Jafari Rad and L Volkmann ldquoRoman bondage in graphsrdquoDiscussiones Mathematicae Graph Theory vol 31 no 4 pp763ndash773 2011

[120] K Ebadi and L PushpaLatha ldquoRoman bondage and Romanreinforcement numbers of a graphrdquo International Journal ofContemporary Mathematical Sciences vol 5 pp 1487ndash14972010

[121] N Dehgardi S M Sheikholeslami and L Volkman ldquoOn theRoman k-bondage number of a graphrdquo AKCE InternationalJournal of Graphs and Combinatorics vol 8 pp 169ndash180 2011

[122] S Akbari and S Qajar ldquoA note on Roman bondage number ofgraphsrdquo Ars Combinatoria to Appear

[123] F-T Hu and J-M Xu ldquoRoman bondage numbers of somegraphsrdquo httparxivorgabs11093933

[124] N Jafari Rad and L Volkmann ldquoOn the Roman bondagenumber of planar graphsrdquo Graphs and Combinatorics vol 27no 4 pp 531ndash538 2011

[125] S Akbari M Khatirinejad and S Qajar ldquoA note on the romanbondage number of planar graphsrdquo Graphs and Combinatorics2012

[126] B Bresar M A Henning and D F Rall ldquoRainbow dominationin graphsrdquo Taiwanese Journal of Mathematics vol 12 no 1 pp213ndash225 2008

[127] B L Hartnell and D F Rall ldquoOn dominating the Cartesianproduct of a graph and 120572krdquo Discussiones Mathematicae GraphTheory vol 24 no 3 pp 389ndash402 2004

[128] N Dehgardi S M Sheikholeslami and L Volkman ldquoThe k-rainbow bondage number of a graphrdquo to appear in DiscreteApplied Mathematics

[129] J von Neumann and O Morgenstern Theory of Games andEconomic Behavior Princeton University Press Princeton NJUSA 1944

[130] C BergeTheorıe des Graphes et ses Applications 1958[131] O Ore Theory of Graphs vol 38 American Mathematical

Society Colloquium Publications 1962[132] E Shan and L Kang ldquoBondage number in oriented graphsrdquoArs

Combinatoria vol 84 pp 319ndash331 2007[133] J Huang and J-M Xu ldquoThe bondage numbers of extended

de Bruijn and Kautz digraphsrdquo Computers amp Mathematics withApplications vol 51 no 6-7 pp 1137ndash1147 2006

[134] J Huang and J-M Xu ldquoThe total domination and total bondagenumbers of extended de Bruijn and Kautz digraphsrdquoComputersamp Mathematics with Applications vol 53 no 8 pp 1206ndash12132007

[135] Y Shibata and Y Gonda ldquoExtension of de Bruijn graph andKautz graphrdquo Computers amp Mathematics with Applications vol30 no 9 pp 51ndash61 1995

[136] X Zhang J Liu and JMeng ldquoThebondage number in completet-partite digraphsrdquo Information Processing Letters vol 109 no17 pp 997ndash1000 2009

[137] D W Bange A E Barkauskas and P J Slater ldquoEfficient dom-inating sets in graphsrdquo in Applications of Discrete Mathematicspp 189ndash199 SIAM Philadelphia Pa USA 1988

[138] M Livingston and Q F Stout ldquoPerfect dominating setsrdquoCongressus Numerantium vol 79 pp 187ndash203 1990

[139] L Clark ldquoPerfect domination in random graphsrdquo Journal ofCombinatorial Mathematics and Combinatorial Computing vol14 pp 173ndash182 1993

[140] D W Bange A E Barkauskas L H Host and L H ClarkldquoEfficient domination of the orientations of a graphrdquo DiscreteMathematics vol 178 no 1ndash3 pp 1ndash14 1998

[141] A E Barkauskas and L H Host ldquoFinding efficient dominatingsets in oriented graphsrdquo Congressus Numerantium vol 98 pp27ndash32 1993

[142] I J Dejter and O Serra ldquoEfficient dominating sets in CayleygraphsrdquoDiscrete AppliedMathematics vol 129 no 2-3 pp 319ndash328 2003

[143] J Lee ldquoIndependent perfect domination sets in Cayley graphsrdquoJournal of Graph Theory vol 37 no 4 pp 213ndash219 2001

[144] WGu X Jia and J Shen ldquoIndependent perfect domination setsin meshes tori and treesrdquo Unpublished

[145] N Obradovic J Peters and G Ruzic ldquoEfficient domination incirculant graphswith two chord lengthsrdquo Information ProcessingLetters vol 102 no 6 pp 253ndash258 2007

[146] K Reji Kumar and G MacGillivray ldquoEfficient domination incirculant graphsrdquo Discrete Mathematics vol 313 no 6 pp 767ndash771 2013

[147] D Van Wieren M Livingston and Q F Stout ldquoPerfectdominating sets on cube-connected cyclesrdquo Congressus Numer-antium vol 97 pp 51ndash70 1993

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 International Journal of Combinatorics

Although some results partially support Conjecture 51Teschner [28] in 1993 found a counterexample to this con-jecture the cartesian product 119870

3times 119870

3 which shows that

119887(119866) = Δ(119866) + 2If a graph 119866 is a counterexample to Conjecture 51 it

must be a 120574-critical graph by Theorem 13 That is why thevertex domination-critical graphs are of special interest in theliterature

Now we return to Conjecture 51 Hartnell and Rall [17]and Teschner [29] independently proved that 119887(119866) can bemuch greater than Δ(119866) by showing the following result

Theorem 52 (Hartnell and Rall [17] 1994 Teschner [29]1996) For an integer 119899 ⩾ 3 let 119866

119899be the cartesian product

119870119899times 119870

119899 Then 119887(119866

119899) = 3(119899 minus 1) = (32)Δ(119866

119899)

This theorem shows that there exist no upper boundsof the form 119887(119866) ⩽ Δ(119866) + 119888 for any integer 119888 Teschner[26] proved that 119887(119866) ⩽ (32)Δ(119866) for any graph 119866 with120574(119866) ⩽ 2 (see Theorems 50 and 48) and for 120574-critical graphswith 120574(119866) = 3 and proposed the following conjecture

Conjecture 53 (Teschner [26] 1995) 119887(119866) ⩽ (32)Δ(119866) forany graph 119866

We believe that this conjecture is true but so far no muchwork about it has been known

4 Lower Bounds

Since the bondage number is defined as the smallest numberof edges whose removal results in an increase of dominationnumber each constructive method that creates a concretebondage set leads to an upper bound on the bondage numberFor that reason it is hard to find lower bounds Neverthelessthere are still a few lower bounds

41 Bounds Implied by Subgraphs The first lower boundon the bondage number is gotten in terms of its spanningsubgraph

Theorem 54 (Teschner [10] 1997) Let 119867 be a spanningsubgraph of a nonempty graph119866 If 120574(119867) = 120574(119866) then 119887(119867) ⩽119887(119866)

By Theorem 1 119887(119862119899) = 3 if 119899 equiv 1 (mod 3) and 119887(119862

119899) =

2 otherwise 119887(119875119899) = 2 if 119899 equiv 1 (mod 3) and 119887(119875

119899) = 1

otherwise From these results and Theorem 54 we get thefollowing two corollaries

Corollary 55 If119866 is hamiltonianwith order 119899 ⩾ 3 and 120574(119866) =lceil1198993rceil then 119887(119866) ⩾ 2 and in addition 119887(119866) ⩾ 3 if 119899 equiv 1 (mod3)

Corollary 56 If 119866 with order 119899 ⩾ 2 has a hamiltonian pathand 120574(119866) = lceil1198993rceil then 119887(119866) ⩾ 2 if 119899 equiv 1 (mod 3)

42 Other Bounds The vertex covering number 120573(119866) of 119866 isthe minimum number of vertices that are incident with all

Figure 5 A 120574-critical graph with 120573 = 120574 = 4 120575 = 2 and 119887 = 3

edges in 119866 If 119866 has no isolated vertices then 120574(119866) ⩽ 120573(119866)clearly In [30] Volkmann gave a lot of graphs with 120573 = 120574

Theorem 57 (Teschner [10] 1997) Let119866 be a graph If 120574(119866) =120573(119866) then

(a) 119887(119866) ⩾ 120575(119866)

(b) 119887(119866) ⩾ 120575(119866) + 1 if 119866 is a 120574-critical graph

The graph shown in Figure 5 shows that the bound givenin Theorem 57(b) is sharp For the graph 119866 it is a 120574-criticalgraph and 120574(119866) = 120573(119866) = 4 By Theorem 57 we have 119887(119866) ⩾3 On the other hand 119887(119866) ⩽ 3 by Theorem 16 Thus 119887(119866) =3

Proposition 58 (Sanchis [31] 1991) Let 119866 be a graph 119866 oforder 119899 (⩾ 6) and size 120576 If 119866 has no isolated vertices and3 ⩽ 120574(119866) ⩽ 1198992 then 120576 ⩽ ( 119899minus120574(119866)+1

2)

Using the idea in the proof of Theorem 57 every upperbound for 120574(119866) can lead to a lower bound for 119887(119866) Inthis way Teschner [10] obtained another lower bound fromProposition 58

Theorem59 (Teschner [10] 1997) Let119866 be a graph119866 of order119899 (⩾6) and size 120576 If 2 ⩽ 120574(119866) ⩽ 1198992 minus 1 then

(a) 119887(119866) ⩾ min120575(119866) 120576 minus ( 119899minus120574(119866)2)

(b) 119887(119866) ⩾ min120575(119866) + 1 120576 minus ( 119899minus120574(119866)2) if 119866 is a 120574-critical

graph

The lower bound in Theorem 59(b) is sharp for a classof 120574-critical graphs with domination number 2 Let 119866 be thegraph obtained from complete graph119870

2119905(119905 ⩾ 2) by removing

a perfect matching By Proposition 37 119866 is a 120574-critical graphwith 120574(119866) = 2 Then 119887(119866) ⩾ 120575(119866) + 1 = Δ(119866) + 1 byTheorem 59 and 119887(119866) ⩽ Δ(119866) + 1 by Theorem 38 and so119887(119866) = Δ(119866) + 1 = 2119905 minus 1

As far as the author knows there are no more lowerbounds in the present literature In view of applications ofthe bondage number a network is vulnerable if its bondagenumber is small while it is stable if its bondage number islargeTherefore better lower bounds let us learn better aboutthe stability of networks from this point of view Thus it isof great significance to seek more lower bounds for variousclasses of graphs

International Journal of Combinatorics 11

5 Results on Graph Operations

Generally speaking it is quite difficult to determine the exactvalue of the bondage number for a given graph since itstrongly depends on the dominating number of the graphThus determining bondage numbers for some special graphsis interesting if the dominating numbers of those graphs areknown or can be easily determined In this section we willintroduce some known results on bondage numbers for somespecial classes of graphs obtained by some graph operations

51 Cartesian Product Graphs Let 1198661= (119881

1 119864

1) and 119866

2=

(1198812 119864

2) be two graphs The Cartesian product of 119866

1and 119866

2

is an undirected graph denoted by 1198661times 119866

2 where 119881(119866

1times

1198662) = 119881

1times 119881

2 two distinct vertices 119909

11199092and 119910

11199102 where

1199091 119910

1isin 119881(119866

1) and 119909

2 119910

2isin 119881(119866

2) are linked by an edge in

1198661times 119866

2if and only if either 119909

1= 119910

1and 119909

21199102isin 119864(119866

2) or

1199092= 119910

2and 119909

11199101isin 119864(119866

1) The Cartesian product is a very

effective method for constructing a larger graph from severalspecified small graphs

Theorem 60 (Dunbar et al [21] 1998) For 119899 ⩾ 3

119887 (119862119899times 119870

2) =

2 119894119891 119899 equiv 0 119900119903 1 (mod 4) 3 119894119891 119899 equiv 3 (mod 4) 4 119894119891 119899 equiv 2 (mod 4)

(16)

For the Cartesian product 119862119899times 119862

119898of two cycles 119862

119899and

119862119898 where 119899 ⩾ 4 and 119898 ⩾ 3 Klavzar and Seifter [32]

determined 120574(119862119899times 119862

119898) for some 119899 and 119898 For example

120574(119862119899times119862

4) = 119899 for 119899 ⩾ 3 and 120574(119862

119899times119862

3) = 119899minuslfloor1198994rfloor for 119899 ⩾ 4

Kim [33] determined 119887(1198623times 119862

3) = 6 and 119887(119862

4times 119862

4) = 4 For

a general 119899 ⩾ 4 the exact values of the bondage numbers of119862119899times 119862

3and 119862

119899times 119862

4are determined as follows

Theorem 61 (Sohn et al [34] 2007) For 119899 ⩾ 4

119887 (119862119899times 119862

3) =

2 119894119891 119899 equiv 0 (mod 4) 4 119894119891 119899 equiv 1 119900119903 2 (mod 4) 5 119894119891 119899 equiv 3 (mod 4)

(17)

Theorem62 (Kang et al [35] 2005) 119887(119862119899times119862

4) = 4 for 119899 ⩾ 4

For larger 119898 and 119899 Huang and Xu [36] obtained thefollowing result see Theorem 197 for more details

Theorem 63 (Huang and Xu [36] 2008) 119887(1198625119894times119862

5119895) = 3 for

any positive integers 119894 and 119895

Xiang et al [37] determined that for 119899 ⩾ 5

119887 (119862119899times 119862

5)

= 3 if 119899 equiv 0 or 1 (mod 5) = 4 if 119899 equiv 2 or 4 (mod 5) ⩽ 7 if 119899 equiv 3 (mod 5)

(18)

For the Cartesian product 119875119899times119875

119898of two paths 119875

119899and 119875

119898

Jacobson and Kinch [38] determined 120574(119875119899times119875

2) = lceil(119899+1)2rceil

120574(119875119899times 119875

3) = 119899 minus lfloor(119899 minus 1)4rfloor and 120574(119875

119899times 119875

4) = 119899 + 1 if 119899 =

1 2 3 5 6 9 and 119899 otherwiseThe bondage number of119875119899times119875

119898

for 119899 ⩾ 4 and 2 ⩽ 119898 ⩽ 4 is determined as follows (Li [39] alsodetermined 119887(119875

119899times 119875

2))

Theorem 64 (Hu and Xu [40] 2012) For 119899 ⩾ 4

119887 (119875119899times 119875

2) =

1 119894119891 119899 equiv 1 (mod 2)2 119894119891 119899 equiv 0 (mod 2)

119887 (119875119899times 119875

3) =

1 119894119891 119899 equiv 1 119900119903 2 (mod 4)2 119894119891 119899 equiv 0 119900119903 3 (mod 4)

119887 (119875119899times 119875

4) = 1 119891119900119903 119899 ⩾ 14

(19)

From the proof of Theorem 64 we find that if 119875119899=

0 1 119899 minus 1 and 119875119898= 0 1 119898 minus 1 then the removal

of the vertex (0 0) in 119875119899times119875

119898does not change the domination

number If119898 increases the effect of (0 0) for the dominationnumber will be smaller and smaller from the probabilityTherefore we expect it is possible that 120574(119875

119899times 119875

119898minus (0 0)) =

120574(119875119899times 119875

119898) for119898 ⩾ 5 and give the following conjecture

Conjecture 65 119887(119875119899times 119875

119898) ⩽ 2 for119898 ⩾ 5 and 119899 ⩾ 4

52 Block Graphs and Cactus Graphs In this subsection weintroduce some results for corona graphs block graphs andcactus graphs

The corona1198661∘ 119866

2 proposed by Frucht and Harary [41]

is the graph formed from a copy of 1198661and 120592(119866

1) copies of

1198662by joining the 119894th vertex of 119866

1to the 119894th copy of 119866

2 In

particular we are concernedwith the corona119867∘1198701 the graph

formed by adding a new vertex V119894 and a new edge 119906

119894V119894for

every vertex 119906119894in119867

Theorem 66 (Carlson and Develin [42] 2006) 119887(119867 ∘ 1198701) =

120575(119867) + 1

This is a very important result which is often used toconstruct a graph with required bondage number In otherwords this result implies that for any given positive integer 119887there is a graph 119866 such that 119887(119866) = 119887

A block graph is a graph whose blocks are completegraphs Each block in a cactus graph is either a cycle or a119870

2

If each block of a graph is either a complete graph or a cyclethen we call this graph a block-cactus graph

Theorem67 (Teschner [43] 1997) 119887(119866) le Δ(119866) for any blockgraph 119866

Teschner [43] characterized all block graphs with 119887(119866) =Δ(119866) In the same paper Teschner found that 120574-criticalgraphs are instrumental in determining bounds for thebondage number of cactus and block graphs and obtained thefollowing result

Theorem 68 (Teschner [43] 1997) 119887(119866) ⩽ 3 for anynontrivial cactus graph 119866

This bound can be achieved by 119867 ∘ 1198701where 119867 is

a nontrivial cactus graph without vertices of degree one

12 International Journal of Combinatorics

by Theorem 66 In 1998 Dunbar et al [21] proposed thefollowing problem

Problem 2 Characterize all cactus graphs with bondagenumber 3

Some upper bounds for block-cactus graphs were alsoobtained

Theorem 69 (Dunbar et al [21] 1998) Let 119866 be a connectedblock-cactus graph with at least two blocksThen 119887(119866) ⩽ Δ(119866)

Theorem 70 (Dunbar et al [21] 1998) Let 119866 be a connectedblock-cactus graph which is neither a cactus graph nor a blockgraph Then 119887(119866) ⩽ Δ(119866) minus 1

53 Edge-Deletion and Edge-Contraction In order to obtainfurther results of the bondage number we may consider howthe bondage number changes under some operations of agraph 119866 which remain the domination number unchangedor preserve some property of 119866 such as planarity Twosimple operations satisfying these requirements are the edge-deletion and the edge-contraction

Theorem 71 (Huang and Xu [44] 2012) Let 119866 be any graphand 119890 isin 119864(119866) Then 119887(119866 minus 119890) ⩾ 119887(119866) minus 1 Moreover 119887(119866 minus 119890) ⩽119887(119866) if 120574(119866 minus 119890) = 120574(119866)

FromTheorem 1 119887(119862119899minus 119890) = 119887(119875

119899) = 119887(119862

119899) minus 1 for any 119890

in119862119899 which shows that the lower bound on 119887(119866minus119890) is sharp

The upper bound can reach for any graph 119866 with 119887(119866) ⩾ 2Next we consider the effect of the edge-contraction on

the bondage number Given a graph 119866 the contraction of 119866by the edge 119890 = 119909119910 denoted by 119866119909119910 is the graph obtainedfrom 119866 by contracting two vertices 119909 and 119910 to a new vertexand then deleting all multiedges It is easy to observe that forany graph 119866 120574(119866) minus 1 ⩽ 120574(119866119909119910) ⩽ 120574(119866) for any edge 119909119910 of119866

Theorem 72 (Huang and Xu [44] 2012) Let 119866 be any graphand 119909119910 be any edge in119866 If119873

119866(119909)cap119873

119866(119910) = 0 and 120574(119866119909119910) =

120574(119866) then 119887(119866119909119910) ⩾ 119887(119866)

We can use examples 119862119899and 119870

119899to show that the

conditions ofTheorem 72 are necessary and the lower boundinTheorem 72 is tight Clearly 120574(119862

119899) = lceil1198993rceil and 120574(119870

119899) = 1

for any edge 119909119910 119862119899119909119910 = 119862

119899minus1and 119870

119899119909119910 = 119870

119899minus1 By

Theorem 1 if 119899 equiv 1 (mod 3) then 120574(119862119899119909119910) lt 120574(119866) and

119887(119862119899119909119910) = 2 lt 3 = 119887(119862

119899) Thus the result in Theorem 72

is generally invalid without the hypothesis 120574(119866119909119910) = 120574(119866)Furthermore the condition 119873

119866(119909) cap 119873

119866(119910) = 0 cannot be

omitted even if 120574(119866119909119910) = 120574(119866) since for odd 119899 119887(119870119899119909119910) =

lceil(119899 minus 1)2rceil lt lceil1198992rceil = 119887(119870119899) by Theorem 1

On the other hand 119887(119862119899119909119910) = 119887(119862

119899) = 2 if 119899 equiv 0 (mod

3) and 119887(119870119899119909119910) = 119887(119870

119899) if 119899 is even which shows that the

equality in 119887(119866119909119910) ⩾ 119887(119866) may hold Thus the bound inTheorem 72 is tight However provided all the conditions119887(119866119909119910) can be arbitrarily larger than 119887(119866) Given a graph119867let119866 be the graph formed from119867∘119870

1by adding a new vertex

119909 and joining it to an vertex 119910 of degree one in119867 ∘ 1198701 Then

119866119909119910 = 119867 ∘ 1198701 120574(119866) = 120574(119866119909119910) and 119873

119866(119909) cap 119873

119866(119910) = 0

But 119887(119866) = 1 since 120574(119866 minus 119909119910) = 120574(119866119909119910) + 1 and 119887(119866119909119910) =120575(119867) + 1 byTheorem 66The gap between 119887(119866) and 119887(119866119909119910)is 120575(119867)

6 Results on Planar Graphs

From Section 2 we have seen that the bondage numberfor a tree has been completely solved Moreover a lineartime algorithm to compute the bondage number of a tree isdesigned by Hartnell et al [15] It is quite natural to considerthe bondage number for a planar graph In this section westate some results and problems on the bondage number fora planar graph

61 Conjecture on Bondage Number As mentioned inSection 3 the bondage number can be much larger than themaximum degree But for a planar graph 119866 the bondagenumber 119887(119866) cannot exceed Δ(119866) too much It is clear that119887(119866) ⩽ Δ(119866) + 4 by Corollary 15 since 120575(119866) ⩽ 5 for anyplanar graph119866 In 1998Dunbar et al [21] posed the followingconjecture

Conjecture 73 If 119866 is a planar graph then 119887(119866) ⩽ Δ(119866) + 1

Because of its attraction it immediately became the focusof attention when this conjecture is proposed In fact themain aim concerning the research of the bondage number fora planar graph is focused on this conjecture

It has been mentioned in Theorems 1 and 60 that119887(119862

3119896+1) = 3 = Δ + 1 and 119887(119862

4119896+2times 119870

2) = 4 = Δ + 1 It

is known that 119887(1198706minus 119872) = 5 = Δ + 1 where119872 is a perfect

matching of the complete graph1198706These examples show that

if Conjecture 73 is true then the upper bound is sharp for2 ⩽ Δ ⩽ 4

Here we note that it is sufficient to prove this conjecturefor connected planar graphs since the bondage number of adisconnected graph is simply the minimum of the bondagenumbers of its components

The first paper attacking this conjecture is due to Kangand Yuan [45] which confirmed the conjecture for everyconnected planar graph 119866 with Δ ⩾ 7 The proofs are mainlybased onTheorems 16 and 18

Theorem 74 (Kang and Yuan [45] 2000) If 119866 is a connectedplanar graph then 119887(119866) ⩽ min8 Δ(119866) + 2

Obviously in view of Theorem 74 Conjecture 73 is truefor any connected planar graph with Δ ⩾ 7

62 Bounds Implied by Degree Conditions As we have seenfrom Theorem 74 to attack Conjecture 73 we only need toconsider connected planar graphs with maximum Δ ⩽ 6Thus studying the bondage number of planar graphs bydegree conditions is of significanceThe first result on boundsimplied by degree conditions is obtained by Kang and Yuan[45]

International Journal of Combinatorics 13

Theorem 75 (Kang and Yuan [45] 2000) If 119866 is a connectedplanar graph without vertices of degree 5 then 119887(119866) ⩽ 7

Fischermann et al [46] generalized Theorem 75 as fol-lows

Theorem 76 (Fischermann et al [46] 2003) Let 119866 be aconnected planar graph and 119883 the set of vertices of degree 5which have distance at least 3 to vertices of degrees 1 2 and3 If all vertices in 119883 not adjacent with vertices of degree 4are independent and not adjacent to vertices of degree 6 then119887(119866) ⩽ 7

Clearly if119866 has no vertices of degree 5 then119883 = 0 ThusTheorem 76 yields Theorem 75

Use 119899119894to denote the number of vertices of degree 119894 in 119866

for each 119894 = 1 2 Δ(119866) Using Theorem 18 Fischermannet al obtained the following two theorems

Theorem 77 (Fischermann et al [46] 2003) For any con-nected planar graph 119866

(1) 119887(119866) ⩽ 7 if 1198995lt 2119899

2+ 3119899

3+ 2119899

4+ 12

(2) 119887(119866) ⩽ 6 if 119866 contains no vertices of degrees 4 and 5

Theorem 78 (Fischermann et al [46] 2003) For any con-nected planar graph 119866 119887(119866) ⩽ Δ(119866) + 1 if

(1) Δ(119866) = 6 and every edge 119890 = 119909119910 with 119889119866(119909) = 5 and

119889119866(119910) = 6 is contained in at most one triangle

(2) Δ(119866) = 5 and no triangle contains an edge 119890 = 119909119910 with119889119866(119909) = 5 and 4 ⩽ 119889

119866(119910) ⩽ 5

63 Bounds Implied by Girth Conditions The girth 119892(119866) of agraph 119866 is the length of the shortest cycle in 119866 If 119866 has nocycles we define 119892(119866) = infin

Combining Theorem 74 with Theorem 78 we find thatif a planar graph contains no triangles and has maximumdegree Δ ⩾ 5 then Conjecture 73 holds This fact motivatedFischermann et alrsquos [46] attempt to attack Conjecture 73 bygirth restraints

Theorem 79 (Fischermann et al [46] 2003) For any con-nected planar graph 119866

119887 (119866) ⩽

6 119894119891 119892 (119866) ⩾ 4

5 119894119891 119892 (119866) ⩾ 5

4 119894119891 119892 (119866) ⩾ 6

3 119894119891 119892 (119866) ⩾ 8

(20)

The first result in Theorem 79 shows that Conjecture 73is valid for all connected planar graphs with 119892(119866) ⩾ 4 andΔ(119866) ⩾ 5 It is easy to verify that the second result inTheorem 79 implies that Conjecture 73 is valid for all not3-regular graphs of girth 119892(119866) ⩾ 5 which is stated in thefollowing corollary

Corollary 80 For any connected planar graph 119866 if 119866 is not3-regular and 119892(119866) ⩾ 5 then 119887(119866) ⩽ Δ(119866) + 1

The first result in Theorem 79 also implies that if 119866 isa connected planar graph with no cycles of length 3 then119887(119866) ⩽ 6 Hou et al [47] improved this result as follows

Theorem 81 (Hou et al [47] 2011) For any connected planargraph 119866 if 119866 contains no cycles of length 119894 (4 ⩽ 119894 ⩽ 6) then119887(119866) ⩽ 6

Since 119887(1198623119896+1) = 3 for 119896 ⩾ 3 (see Theorem 1) the

last bound in Theorem 79 is tight Whether other bounds inTheorem 79 are tight remains open In 2003 Fischermann etal [46] posed the following conjecture

Conjecture 82 For any connected planar graph 119866 119887(119866) ⩽ 7and

119887 (119866) ⩽ 5 119894119891 119892 (119866) ⩾ 4

4 119894119891 119892 (119866) ⩾ 5(21)

We conclude this subsection with a question on bondagenumbers of planar graphs

Question 1 (Fischermann et al [46] 2003) Is there a planargraph 119866 with 6 ⩽ 119887(119866) ⩽ 8

In 2006 Carlson andDevelin [42] showed that the corona119866 = 119867 ∘ 119870

1for a planar graph 119867 with 120575(119867) = 5 has the

bondage number 119887(119866) = 120575(119867) + 1 = 6 (see Theorem 66)Since the minimum degree of planar graphs is at most 5then 119887(119866) can attach 6 If we take 119867 as the graph of theicosahedron then 119866 = 119867 ∘ 119870

1is such an example The

question for the existence of planar graphs with bondagenumber 7 or 8 remains open

64 Comments on the Conjectures Conjecture 73 is true forall connected planar graphs with minimum degree 120575 ⩽ 2 byTheorem 16 or maximum degree Δ ⩾ 7 by Theorem 74 ornot 120574-critical planar graphs by Theorem 13 Thus to attackConjecture 73 we only need to consider connected criticalplanar graphs with degree-restriction 3 ⩽ 120575 ⩽ Δ ⩽ 6

Recalling and anatomizing the proofs of all results men-tioned in the preceding subsections on the bondage numberfor connected planar graphs we find that the proofs of theseresults strongly depend upon Theorem 16 or Theorem 18 Inother words a basic way used in the proofs is to find twovertices 119909 and 119910 with distance at most two in a consideredplanar graph 119866 such that

119889119866(119909) + 119889

119866(119910) or 119889

119866(119909) + 119889

119866(119910) minus

1003816100381610038161003816119873119866(119909) cap 119873

119866(119910)1003816100381610038161003816

(22)

which bounds 119887(119866) is as small as possible Very recentlyHuang and Xu [44] have considered the following parameter

119861 (119866) = min119909119910isin119881(119866)

119889119909119910 1 ⩽ 119889

119866(119909 119910) ⩽ 2

cup 119889119909119910minus 119899

119909119910 119889 (119909 119910) = 1

(23)

where 119889119909119910= 119889

119866(119909) + 119889

119866(119910) minus 1 and 119899

119909119910= |119873

119866(119909) cap 119873

119866(119910)|

14 International Journal of Combinatorics

It follows fromTheorems 16 and 18 that

119887 (119866) ⩽ 119861 (119866) (24)

The proofs given in Theorems 74 and 79 indeed imply thefollowing stronger results

Theorem 83 If 119866 is a connected planar graph then

119861 (119866) ⩽

min 8 Δ (119866) + 26 119894119891 119892 (119866) ⩾ 4

5 119894119891 119892 (119866) ⩾ 5

4 119894119891 119892 (119866) ⩾ 6

3 119894119891 119892 (119866) ⩾ 8

(25)

Thus using Theorem 16 or Theorem 18 if we can proveConjectures 73 and 82 then we can prove the followingstatement

Statement If 119866 is a connected planar graph then

119861 (119866) ⩽

Δ (119866) + 1

7

5 if 119892 (119866) ⩾ 44 if 119892 (119866) ⩾ 5

(26)

It follows from (24) that Statement implies Conjectures 73and 82 However Huang and Xu [44] gave examples to showthat none of conclusions in Statement is true As a result theystated the following conclusion

Theorem 84 (Huang and Xu [44] 2012) It is not possible toprove Conjectures 73 and 82 if they are right using Theorems16 and 18

Therefore a new method is needed to prove or disprovethese conjectures At the present time one cannot prove theseconjectures and may consider to disprove them If one ofthese conjectures is invalid then there exists a minimumcounterexample 119866 with respect to 120592(119866) + 120576(119866) In order toobtain a minimum counterexample one may consider twosimple operations satisfying these requirements the edge-deletion and the edge-contractionwhich decrease 120592(119866)+120576(119866)and preserve planarity However applying Theorems 71 and72 Huang and Xu [44] presented the following results

Theorem 85 (Huang and Xu [44] 2012) It is not possible toconstruct minimum counterexamples to Conjectures 73 and 82by the operation of an edge-deletion or an edge-contraction

7 Results on Crossing Number Restraints

It is quite natural to generalize the known results on thebondage number for planar graphs to formore general graphsin terms of graph-theoretical parameters In this section weconsider graphs with crossing number restraints

The crossing number cr(119866) of 119866 is the smallest numberof pairwise intersections of its edges when 119866 is drawn in theplane If cr(119866) = 0 then 119866 is a planar graph

71 General Methods Use 119899119894to denote the number of vertices

of degree 119894 in119866 for 119894 = 1 2 Δ(119866) Huang and Xu obtainedthe following results

Theorem 86 (Huang and Xu [48] 2007) For any connectedgraph 119866

119887 (119866) ⩽

6 119894119891 119892 (119866) ⩾ 4 119886119899119889 2cr (119866) lt 121198861

5 119894119891 119892 (119866) ⩾ 5 119886119899119889 6cr (119866) lt 161198862

4 119894119891 119892 (119866) ⩾ 6 119886119899119889 4cr (119866) lt 141198863

3 119894119891 119892 (119866) ⩾ 8 119886119899119889 6cr (119866) lt 161198864

(27)

where 1198861= 119899

1+ 2119899

2+ 2119899

3+sum

Δ

119894=8(119894 minus 7)119899

119894+ 8 119886

2= 3119899

1+ 6119899

2+

51198993+sum

Δ

119894=7(3119894minus18)119899

119894+20 119886

3= 119899

1+2119899

2+sum

Δ

119894=6(2119894minus10)119899

119894+12

and 1198864= sum

Δ

119894=5(3119894 minus 12)119899

119894+ 16

Simplifying the conditions in Theorem 86 we obtain thefollowing corollaries immediately

Corollary 87 (Huang and Xu [48] 2007) For any connectedgraph 119866

119887 (119866) ⩽

6 119894119891 119892 (119866) ⩾ 4 119886119899119889 cr (119866) ⩽ 3

5 119894119891 119892 (119866) ⩾ 5 119886119899119889 cr (119866) ⩽ 4

4 119894119891 119892 (119866) ⩾ 6 119886119899119889 cr (119866) ⩽ 2

3 119894119891 119892 (119866) ⩾ 8 119886119899119889 cr (119866) ⩽ 2

(28)

Corollary 88 (Huang and Xu [48] 2007) For any connectedgraph 119866

(a) 119887(119866) ⩽ 6 if 119866 is not 4-regular cr(119866) = 4 and 119892(119866) ⩾ 4

(b) 119887(119866) ⩽ Δ(119866) + 1 if 119866 is not 3-regular cr(119866) ⩽ 4 and119892(119866) ⩾ 5

(c) 119887(119866) ⩽ 4 if 119866 is not 3-regular cr(119866) = 3 and 119892(119866) ⩾ 6

(d) 119887(119866) ⩽ 3 if cr(119866) = 3 119892(119866) ⩾ 8 and Δ(119866) ⩾ 5

These corollaries generalize some known results for pla-nar graphs For example Corollary 87 contains Theorem 79Corollary 88 (b) contains Corollary 80

Theorem 89 (Huang and Xu [48] 2007) For any connectedgraph 119866 if cr(119866) ⩽ 119899

3(119866) + 119899

4(119866) + 3 then 119887(119866) ⩽ 8

Corollary 90 (Huang and Xu [48] 2007) For any connectedgraph 119866 if cr(119866) ⩽ 3 then 119887(119866) ⩽ 8

Perhaps being unaware of this result in 2010 Ma et al[49] proved that 119887(119866) ⩽ 12 for any graph 119866 with cr(119866) = 1

Theorem91 (Huang and Xu [48] 2007) Let119866 be a connectedgraph and 119868 = 119909 isin 119881(119866) 119889

119866(119909) = 5 119889

119866(119909 119910) ⩾ 3 if

International Journal of Combinatorics 15

119889119866(119910) ⩽ 3 and 119889

119866(119910) = 4 for every 119910 isin 119873

119866(119909) If 119868 is

independent no vertices adjacent to vertices of degree 6 and

cr (119866) lt max5119899

3+ |119868| minus 2119899

4+ 28

117119899

3+ 40

16 (29)

then 119887(119866) ⩽ 7

Corollary 92 (Huang and Xu [48] 2007) Let 119866 be aconnected graph with cr(119866) ⩽ 2 If 119868 = 119909 isin 119881(119866) 119889

119866(119909) =

5 119889119866(119910 119909) ⩾ 3 if 119889

119866(119910) ⩽ 3 and 119889

119866(119910) = 4 for every 119910 isin

119873119866(119909) is independent and no vertices adjacent to vertices of

degree 6 then 119887(119866) ⩽ 7

Theorem93 (Huang andXu [48] 2007) Let119866 be a connectedgraph If 119866 satisfies

(a) 5cr(119866) + 1198995lt 2119899

2+ 3119899

3+ 2119899

4+ 12 or

(b) 7cr(119866) + 21198995lt 3119899

2+ 4119899

4+ 24

then 119887(119866) ⩽ 7

Proposition 94 (Huang and Xu [48] 2007) Let 119866 be aconnected graph with no vertices of degrees four and five Ifcr(119866) ⩽ 2 then 119887(119866) ⩽ 6

Theorem 95 (Huang and Xu [48] 2007) If 119866 is a connectedgraph with cr(119866) ⩽ 4 and not 4-regular when cr(119866) = 4 then119887(119866) ⩽ Δ(119866) + 2

The above results generalize some known results forplanar graphs For example Corollary 90 and Theorem 95contain Theorem 74 the first condition in Theorem 93 con-tains the second condition inTheorem 77

From Corollary 88 and Theorem 95 we suggest the fol-lowing questions

Question 2 Is there a

(a) 4-regular graph 119866 with cr(119866) = 4 such that 119887(119866) ⩾ 7

(b) 3-regular graph 119866 with cr ⩽ 4 and 119892(119866) ⩾ 5 such that119887(119866) = 5

(c) 3-regular graph 119866 with cr = 3 and 119892(119866) = 6 or 7 suchthat 119887(119866) = 5

72 Carlson-Develin Methods In this subsection we intro-duce an elegant method presented by Carlson and Develin[42] to obtain some upper bounds for the bondage numberof a graph

Suppose that 119866 is a connected graph We say that 119866 hasgenus 120588 if 119866 can be embedded in a surface 119878 with 120588 handlessuch that edges are pairwise disjoint except possibly for end-vertices Let119866 be an embedding of119866 in surface 119878 and let120601(119866)denote the number of regions in 119866 The boundary of everyregion contains at least three edges and every edge is on theboundary of atmost two regions (the two regions are identicalwhen 119890 is a cut-edge) For any edge 119890 of 119866 let 1199031

119866(119890) and 1199032

119866(119890)

be the numbers of edges comprising the regions in 119866 whichthe edge 119890 borders It is clear that every 119890 = 119909119910 isin 119864(119866)

1199032

119866(119890) ⩾ 119903

1

119866(119890) ⩾ 4 if 1003816100381610038161003816119873119866

(119909) cap 119873119866(119910)1003816100381610038161003816 = 0

1199032

119866(119890) ⩾ 4 119903

1

119866(119890) ⩾ 3 if 1003816100381610038161003816119873119866

(119909) cap 119873119866(119910)1003816100381610038161003816 = 1

1199032

119866(119890) ⩾ 119903

1

119866(119890) ⩾ 3 if 1003816100381610038161003816119873119866

(119909) cap 119873119866(119910)1003816100381610038161003816 ⩾ 2

(30)

Following Carlson and Develin [42] for any edge 119890 = 119909119910 of119866 we define

119863119866(119890) =

1

119889119866(119909)+

1

119889119866(119910)

+1

1199031

119866(119890)+

1

1199032

119866(119890)minus 1 (31)

By the well-known Eulerrsquos formula

120592 (119866) minus 120576 (119866) + 120601 (119866) = 2 minus 2120588 (119866) (32)

it is easy to see that

sum

119890isin119864(119866)

119863119866(119890) = 120592 (119866) minus 120576 (119866) + 120601 (119866) = 2 minus 2120588 (119866) (33)

If 119866 is a planar graph that is 120588(119866) = 0 then

sum

119890isin119864(119866)

119863119866(119890) = 120592 (119866) minus 120576 (119866) + 120601 (119866) = 2 (34)

Combining these formulas withTheorem 18 Carlson andDevelin [42] gave a simple and intuitive proof ofTheorem 74and obtained the following result

Theorem 96 (Carlson and Develin [42] 2006) Let 119866 be aconnected graph which can be embedded on a torus Then119887(119866) ⩽ Δ(119866) + 3

Recently Hou and Liu [50] improved Theorem 96 asfollows

Theorem 97 (Hou and Liu [50] 2012) Let 119866 be a connectedgraph which can be embedded on a torus Then 119887(119866) ⩽ 9Moreover if Δ(119866) = 6 then 119887(119866) ⩽ 8

By the way Cao et al [51] generalized the result inTheorem 79 to a connected graph 119866 that can be embeddedon a torus that is

119887 (119866) le

6 if 119892 (119866) ⩾ 4 and 119866 is not 4-regular5 if 119892 (119866) ⩾ 54 if 119892 (119866) ⩾ 6 and 119866 is not 3-regular3 if 119892 (119866) ⩾ 8

(35)

Several authors used this method to obtain some resultson the bondage number For example Fischermann et al[46] used this method to prove the second conclusion inTheorem 78 Recently Cao et al [52] have used thismethod todeal with more general graphs with small crossing numbersFirst they found the following property

Lemma 98 120575(119866) ⩽ 5 for any graph 119866 with cr(119866) ⩽ 5

16 International Journal of Combinatorics

This result implies that 119887(119866) ⩽ Δ(119866) + 4 by Corollary 15for any graph 119866 with cr(119866) ⩽ 5 Cao et al established thefollowing relation in terms of maximum planar subgraphs

Lemma 99 (Cao et al [52]) Let 119866 be a connected graph withcrossing number cr(119866) For any maximum planar subgraph119867of 119866

sum

119890isin119864(119867)

119863119866(119890) ⩾ 2 minus

2cr (119866)120575 (119866)

(36)

Using Carlson-Develin method and combiningLemma 98 with Lemma 99 Cao et al proved the followingresults

Theorem 100 (Cao et al [52]) For any connected graph 119866119887(119866) ⩽ Δ(119866) + 2 if 119866 satisfies one of the following conditions

(a) cr(119866) ⩽ 3(b) cr(119866) = 4 and 119866 is not 4-regular(c) cr(119866) = 5 and 119866 contains no vertices of degree 4

Theorem101 (Cao et al [52]) Let119866 be a connected graphwithΔ(119866) = 5 and cr(119866) ⩽ 4 If no triangles contain two vertices ofdegree 5 then 119887(119866) ⩽ 6 = Δ(119866) + 1

Theorem 102 (Cao et al [52]) Let 119866 be a connected graphwith Δ(119866) ⩾ 6 and cr(119866) ⩽ 3 If Δ(119866) ⩾ 7 or if Δ(119866) = 6120575(119866) = 3 and every edge 119890 = 119909119910 with 119889

119866(119909) = 5 and 119889

119866(119910) = 6

is contained in atmost one triangle then 119887(119866) ⩽ min8 Δ(119866)+1

Using Carlson-Develin method it can be proved that119887(119866) ⩽ Δ(119866) + 2 if cr(119866) ⩽ 3 (see the first conclusion inTheorem 100) and not yet proved that 119887(119866) ⩽ 8 if cr(119866) ⩽3 although it has been proved by using other method (seeCorollary 90)

By using Carlson-Develin method Samodivkin [25]obtained some results on the bondage number for graphswithsome given properties

Kim [53] showed that 119887(119866) ⩽ Δ(119866) + 2 for a connectedgraph 119866 with genus 1 Δ(119866) ⩽ 5 and having a toroidalembedding of which at least one region is not 4-sidedRecently Gagarin and Zverovich [54] further have extendedCarlson-Develin ideas to establish a nice upper bound forarbitrary graphs that can be embedded on orientable ornonorientable topological surface

Theorem 103 (Gagarin and Zverovich [54] 2012) Let 119866 bea graph embeddable on an orientable surface of genus ℎ and anonorientable surface of genus 119896 Then 119887(119866) ⩽ minΔ(119866)+ℎ+2 Δ(119866) + 119896 + 1

This result generalizes the corresponding upper boundsin Theorems 74 and 96 for any orientable or nonori-entable topological surface By investigating the proof ofTheorem 103 Huang [55] found that the issue of the ori-entability can be avoided by using the Euler characteristic120594(= 120592(119866) minus 120576(119866) + 120601(119866)) instead of the genera ℎ and 119896 the

relations are 120594 = 2minus2ℎ and 120594 = 2minus119896 To obtain the best resultfromTheorem 103 onewants ℎ and 119896 as small as possible thisis equivalent to having 120594 as large as possible

According toTheorem 103 if119866 is planar (ℎ = 0 120594 = 2) orcan be embedded on the real projective plane (119896 = 1 120594 = 1)then 119887(119866) ⩽ Δ(119866) + 2 In all other cases Huang [55] had thefollowing improvement for Theorem 103 the proof is basedon the technique developed byCarlson-Develin andGagarin-Zverovich and includes some elementary calculus as a newingredient mainly the intermediate-value theorem and themean-value theorem

Theorem 104 (Huang [55] 2012) Let 119866 be a graph embed-dable on a surface whose Euler characteristic 120594 is as large aspossible If 120594 ⩽ 0 then 119887(119866) ⩽ Δ(119866)+lfloor119903rfloor where 119903 is the largestreal root of the following cubic equation in 119911

1199113

+ 21199112

+ (6120594 minus 7) 119911 + 18120594 minus 24 = 0 (37)

In addition if 120594 decreases then 119903 increases

The following result is asymptotically equivalent toTheorem 104

Theorem 105 (Huang [55] 2012) Let 119866 be a graph embed-dable on a surface whose Euler characteristic 120594 is as large aspossible If 120594 ⩽ 0 then 119887(119866) lt Δ(119866) + radic12 minus 6120594 + 12 orequivalently 119887(119866) ⩽ Δ(119866) + lceilradic12 minus 6120594 minus 12rceil

Also Gagarin andZverovich [54] indicated that the upperbound in Theorem 103 can be improved for larger values ofthe genera ℎ and 119896 by adjusting the proofs and stated thefollowing general conjecture

Conjecture 106 (Gagarin and Zverovich [54] 2012) For aconnected graph G of orientable genus ℎ and nonorientablegenus 119896

119887 (119866) ⩽ min 119888ℎ 119888

1015840

119896 Δ (119866) + 119900 (ℎ) Δ (119866) + 119900 (119896) (38)

where 119888ℎand 1198881015840

119896are constants depending respectively on the

orientable and nonorientable genera of 119866

In the recent paper Gagarin and Zverovich [56] providedconstant upper bounds for the bondage number of graphs ontopological surfaces which can be used as the first estimationfor the constants 119888

ℎand 1198881015840

119896of Conjecture 106

Theorem 107 (Gagarin and Zverovich [56] 2013) Let 119866 bea connected graph of order 119899 If 119866 can be embedded on thesurface of its orientable or nonorientable genus of the Eulercharacteristic 120594 then

(a) 120594 ⩾ 1 implies 119887(119866) ⩽ 10(b) 120594 ⩽ 0 and 119899 gt minus12120594 imply 119887(119866) ⩽ 11(c) 120594 ⩽ minus1 and 119899 ⩽ minus12120594 imply 119887(119866) ⩽ 11 +

3120594(radic17 minus 8120594 minus 3)(120594 minus 1) = 11 + 119874(radicminus120594)

Theorem 79 shows that a connected planar triangle-freegraph 119866 has 119887(119866) ⩽ 6 The following result generalizes to itall the other topological surfaces

International Journal of Combinatorics 17

Theorem 108 (Gagarin and Zverovich [56] 2013) Let 119866 be aconnected triangle-free graph of order 119899 If 119866 can be embeddedon the surface of its orientable or nonorientable genus of theEuler characteristic 120594 then

(a) 120594 ⩾ 1 implies 119887(119866) ⩽ 6(b) 120594 ⩽ 0 and 119899 gt minus8120594 imply 119887(119866) ⩽ 7(c) 120594 ⩽ minus1 and 119899 ⩽ minus8120594 imply 119887(119866) ⩽ 7minus4120594(1+radic2 minus 120594)

In [56] the authors also explicitly improved upperbounds in Theorem 103 and gave tight lower bounds forthe number of vertices of graphs 2-cell embeddable ontopological surfaces of a given genusThe interested reader isreferred to the original paper for details The following resultis interesting although its proof is easy

Theorem 109 (Samodivkin [57]) Let119866 be a graph with order119899 and girth 119892 If 119866 is embeddable on a surface whose Eulercharacteristic 120594 is as large as possible then

119887 (119866) ⩽ 3 +8

119892 minus 2minus

4120594119892

119899 (119892 minus 2) (39)

If 119866 is a planar graph with girth 119892 ⩾ 4 + 119894 for any 119894 isin0 1 2 then Theorem 109 leads to 119887(119866) ⩽ 6 minus 119894 which isthe result inTheorem 79 obtained by Fischermann et al [46]Also inmany cases the bound stated inTheorem 109 is betterthan those given byTheorems 103 and 105

8 Conditional Bondage Numbers

Since the concept of the bondage number is based upon thedomination all sorts of dominations which are variationsof the normal domination by adding restricted conditionsto the normal dominating set can develop a new ldquobondagenumberrdquo as long as a variation of domination number isgiven In this section we survey results on the bondagenumber under some restricted conditions

81 Total Bondage Numbers A dominating set 119878 of a graph119866 without isolated vertices is called total if the subgraphinduced by 119878 contains no isolated vertices The minimumcardinality of a total dominating set is called the totaldomination number of 119866 and denoted by 120574

119879(119866) It is clear

that 120574(119866) ⩽ 120574119879(119866) ⩽ 2120574(119866) for any graph 119866 without isolated

verticesThe total domination in graphs was introduced by Cock-

ayne et al [58] in 1980 Pfaff et al [59 60] in 1983 showedthat the problem determining total domination number forgeneral graphs is NP-complete even for bipartite graphs andchordal graphs Even now total domination in graphs hasbeen extensively studied in the literature In 2009 Henning[61] gave a survey of selected recent results on total domina-tion in graphs

The total bondage number of 119866 denoted by 119887119879(119866) is the

smallest cardinality of a subset 119861 sube 119864(119866) with the propertythat119866minus119861 contains no isolated vertices and 120574

119879(119866minus119861) gt 120574

119879(119866)

From definition 119887119879(119866)may not exist for some graphs for

example119866 = 1198701119899 We put 119887

119879(119866) = infin if 119887

119879(119866) does not exist

It is easy to see that 119887119879(119866) is finite for any connected graph 119866

other than 1198701 119870

2 119870

3 and 119870

1119899 In fact for such a graph 119866

since there is a path of length 3 in 119866 we can find 1198611sube 119864(119866)

such that 119866 minus 1198611is a spanning tree 119879 of 119866 containing a path

of length 3 So 120574119879(119879) = 120574

119879(119866minus119861

1) ⩾ 120574

119879(119866) For the tree119879 we

find 1198612sube 119864(119879) such that 120574

119879(119879 minus 119861

2) gt 120574

119879(119879) ⩾ 120574

119879(119866) Thus

we have 120574119879(119866minus119861

2minus119861

1) gt 120574

119879(119866) and so 119887

119879(119866) ⩽ |119861

1|+|119861

2| In

the following discussion we always assume that 119887119879(119866) exists

when a graph 119866 is mentionedIn 1991 Kulli and Patwari [62] first studied the total

bondage number of a graph and calculated the exact valuesof 119887

119879(119866) for some standard graphs

Theorem 110 (Kulli and Patwari [62] 1991) For 119899 ⩾ 4

119887119879(119862

119899) =

3 119894119891 119899 equiv 2 (mod 4) 2 119900119905ℎ119890119903119908119894119904119890

119887119879(119875

119899) =

2 119894119891 119899 equiv 2 (mod 4) 1 119900119905ℎ119890119903119908119894119904119890

119887119879(119870

119898119899) = 119898 119908119894119905ℎ 2 ⩽ 119898 ⩽ 119899

119887119879(119870

119899) =

4 119891119900119903 119899 = 4

2119899 minus 5 119891119900119903 119899 ⩾ 5

(40)

Recently Hu et al [63] have obtained some results on thetotal bondage number of the Cartesian product119875

119898times119875

119899of two

paths 119875119898and 119875

119899

Theorem 111 (Hu et al [63] 2012) For the Cartesian product119875119898times 119875

119899of two paths 119875

119898and 119875

119899

119887119879(119875

2times 119875

119899) =

1 119894119891 119899 equiv 0 (mod 3) 2 119894119891 119899 equiv 2 (mod 3) 3 119894119891 119899 equiv 1 (mod 3)

119887119879(119875

3times 119875

119899) = 1

119887119879(119875

4times 119875

119899)

= 1 119894119891 119899 equiv 1 (mod 5) = 2 119894119891 119899 equiv 4 (mod 5) ⩽ 3 119894119891 119899 equiv 2 (mod 5) ⩽ 4 119894119891 119899 equiv 0 3 (mod 5)

(41)

Generalized Petersen graphs are an important class ofcommonly used interconnection networks and have beenstudied recently By constructing a family of minimum totaldominating sets Cao et al [64] determined the total bondagenumber of the generalized Petersen graphs

FromTheorem 6 we know that 119887(119879) ⩽ 2 for a nontrivialtree 119879 But given any positive integer 119896 Sridharan et al [65]constructed a tree 119879 for which 119887

119879(119879) = 119896 Let119867

119896be the tree

obtained from a star1198701119896+1

by subdividing 119896 edges twice Thetree 119867

7is shown in Figure 6 It can be easily verified that

119887119879(119867

119896) = 119896 This fact can be stated as the following theorem

Theorem 112 (Sridharan et al [65] 2007) For any positiveinteger 119896 there exists a tree 119879 with 119887

119879(119879) = 119896

18 International Journal of Combinatorics

Figure 61198677

Combining Theorem 1 with Theorem 112 we have whatfollows

Corollary 113 (Sridharan et al [65] 2007) The bondagenumber and the total bondage number are unrelated even fortrees

However Sridharan et al [65] gave an upper bound onthe total bondage number of a tree in terms of its maximumdegree For any tree 119879 of order 119899 if 119879 =119870

1119899minus1 then 119887

119879(119879) =

minΔ(119879) (13)(119899 minus 1) Rad and Raczek [66] improved thisupper bound and gave a constructive characterization of acertain class of trees attaching the upper bound

Theorem 114 (Rad and Raczek [66] 2011) 119887119879(119879) ⩽ Δ(119879) minus 1

for any tree 119879 with maximum degree at least three

In general the decision problem for 119887119879(119866) is NP-complete

for any graph 119866 We state the decision problem for the totalbondage as follows

Problem 3 Consider the decision problemTotal Bondage ProblemInstance a graph 119866 and a positive integer 119887Question is 119887

119879(119866) ⩽ 119887

Theorem 115 (Hu and Xu [16] 2012) The total bondageproblem is NP-complete

Consequently in view of its computational hardnessit is significative to establish some sharp bounds on thetotal bondage number of a graph in terms of other graphicparameters In 1991 Kulli and Patwari [62] showed that119887119879(119866) ⩽ 2119899 minus 5 for any graph 119866 with order 119899 ⩾ 5 Sridharan

et al [65] improved this result as follows

Theorem 116 (Sridharan et al [65] 2007) For any graph 119866with order 119899 if 119899 ⩾ 5 then

119887119879(119866) ⩽

1

3(119899 minus 1) 119894119891 119866 119888119900119899119905119886119894119899119904 119899119900 119888119910119888119897119890119904

119899 minus 1 119894119891 119892 (119866) ⩾ 5

119899 minus 2 119894119891 119892 (119866) = 4

(42)

Rad and Raczek [66] also established some upperbounds on 119887

119879(119866) for a general graph 119866 In particular they

gave an upper bound on 119887119879(119866) in terms of the girth of

119866

Theorem 117 (Rad and Raczek [66] 2011) 119887119879(119866)⩽4Δ(119866) minus 5

for any graph 119866 with 119892(119866) ⩾ 4

82 Paired Bondage Numbers A dominating set 119878 of 119866 iscalled to be paired if the subgraph induced by 119878 containsa perfect matching The paired domination number of 119866denoted by 120574

119875(119866) is the minimum cardinality of a paired

dominating set of 119866 Clearly 120574119879(119866) ⩽ 120574

119875(119866) for every

connected graph 119866 with order at least two where 120574119879(119866) is

the total domination number of 119866 and 120574119875(119866) ⩽ 2120574(119866) for

any graph119866without isolated vertices Paired domination wasintroduced by Haynes and Slater [67 68] and further studiedin [69ndash72]

The paired bondage number of 119866 with 120575(119866) ⩾ 1 denotedby 119887P(119866) is the minimum cardinality among all sets of edges119861 sube 119864 such that 120575(119866 minus 119861) ⩾ 1 and 120574

119875(119866 minus 119861) gt 120574

119875(119866)

The concept of the paired bondage number was firstproposed by Raczek [73] in 2008The following observationsfollow immediately from the definition of the paired bondagenumber

Observation 2 Let 119866 be a graph with 120575(119866) ⩾ 1

(a) If 119867 is a subgraph of 119866 such that 120575(119867) ⩾ 1 then119887119875(119867) ⩽ 119887

119875(119866)

(b) If 119867 is a subgraph of 119866 such that 119887119875(119867) = 1 and 119896

is the number of edges removed to form119867 then 1 ⩽119887119875(119866) ⩽ 119896 + 1

(c) If 119909119910 isin 119864(119866) such that 119889119866(119909) 119889

119866(119910) gt 1 and 119909119910

belongs to each perfect matching of each minimumpaired dominating set of 119866 then 119887

119875(119866) = 1

Based on these observations 119887119875(119875

119899) ⩽ 2 In fact the

paired bondage number of a path has been determined

Theorem 118 (Raczek [73] 2008) For any positive integer 119896if 119899 ⩾ 2 then

119887119875(119875

119899) =

0 119894119891 119899 = 2 3 119900119903 5

1 119894119891 119899 = 4119896 4119896 + 3 119900119903 4119896 + 6

2 119900119905ℎ119890119903119908119894119904119890

(43)

For a cycle 119862119899of order 119899 ⩾ 3 since 120574

119875(119862

119899) = 120574

119875(119875

119899) from

Theorem 118 the following result holds immediately

Corollary 119 (Raczek [73] 2008) For any positive integer 119896if 119899 ⩾ 3 then

119887119875(119862

119899) =

0 119894119891 119899 = 3 119900119903 5

2 119894119891 119899 = 4119896 4119896 + 3 119900119903 4119896 + 6

3 119900119905ℎ119890119903119908119894119904119890

(44)

A wheel 119882119899 where 119899 ⩾ 4 is a graph with 119899 vertices

formed by connecting a single vertex to all vertices of a cycle119862119899minus1

Of course 120574119875(119882

119899) = 2

International Journal of Combinatorics 19

119896

Figure 7 A tree 119879 with 119887119875(119879) = 119896

Theorem 120 (Raczek [73] 2008) For positive integers 119899 and119898

119887119875(119870

119898119899) = 119898 119891119900119903 1 lt 119898 ⩽ 119899

119887119875(119882

119899) =

4 119894119891 119899 = 4

3 119894119891 119899 = 5

2 119900119905ℎ119890119903119908119894119904119890

(45)

Theorem 16 shows that if 119879 is a nontrivial tree then119887(119879) ⩽ 2 However no similar result exists for pairedbondage For any nonnegative integer 119896 let 119879

119896be a tree

obtained by subdividing all but one edge of a star 1198701119896+1

(asshown in Figure 7) It is easy to see that 119887

119875(119879

119896) = 119896 We

state this fact as the following theorem which is similar toTheorem 112

Theorem121 (Raczek [73] 2008) For any nonnegative integer119896 there exists a tree 119879 with 119887

119875(119879) = 119896

Consider the tree defined in Figure 5 for 119896 = 3 Then119887(119879

3) ⩽ 2 and 119887

119875(119879

3) = 3 On the other hand 119887(119875

4) = 2

and 119887119875(119875

4) = 1 Thus we have what follows which is similar

to Corollary 113

Corollary 122 The bondage number and the paired bondagenumber are unrelated even for trees

A constructive characterization of trees with 119887(119879) = 2is given by Hartnell and Rall in [12] Raczek [73] provideda constructive characterization of trees with 119887

119875(119879) = 0 In

order to state the characterization we define a labeling andthree simple operations on a tree119879 Let 119910 isin 119881(119879) and let ℓ(119910)be the label assigned to 119910

Operation T1 If ℓ(119910) = 119861 add a vertex 119909 and the

edge 119910119909 and let ℓ(119909) = 119860OperationT

2 If ℓ(119910) = 119862 add a path (119909

1 119909

2) and the

edge 1199101199091 and let ℓ(119909

1) = 119861 ℓ(119909

2) = 119860

Operation T3 If ℓ(119910) = 119861 add a path (119909

1 119909

2 119909

3)

and the edge 1199101199091 and let ℓ(119909

1) = 119862 ℓ(119909

2) = 119861 and

ℓ(1199093) = 119860

Let 1198752= (119906 V) with ℓ(119906) = 119860 and ℓ(V) = 119861 Let T be

the class of all trees obtained from the labeled 1198752by a finite

sequence of operationsT1 T

2 T

3

A tree 119879 in Figure 8 belongs to the familyTRaczek [73] obtained the following characterization of all

trees 119879 with 119887119875(119879) = 0

119860

119860

119860

119860

119860

119861 119862

119861 119862 119861

119861119860

Figure 8 A tree 119879 belong to the familyT

Theorem 123 (Raczek [73] 2008) For a tree 119879 119887119875(119879) = 0 if

and only if 119879 is inT

We state the decision problem for the paired bondage asfollows

Problem 4 Consider the decision problem

Paired Bondage ProblemInstance a graph 119866 and a positive integer 119887Question is 119887

119875(119866) ⩽ 119887

Conjecture 124 The paired bondage problem is NP-complete

83 Restrained Bondage Numbers A dominating set 119878 of 119866 iscalled to be restrained if the subgraph induced by 119878 containsno isolated vertices where 119878 = 119881(119866) 119878 The restraineddomination number of 119866 denoted by 120574

119877(119866) is the smallest

cardinality of a restrained dominating set of 119866 It is clear that120574119877(119866) exists and 120574(119866) ⩽ 120574

119877(119866) for any nonempty graph 119866

The concept of restrained domination was introducedby Telle and Proskurowski [74] in 1997 albeit indirectly asa vertex partitioning problem Concerning the study of therestrained domination numbers the reader is referred to [74ndash79]

In 2008 Hattingh and Plummer [80] defined therestrained bondage number 119887R(119866) of a nonempty graph 119866 tobe the minimum cardinality among all subsets 119861 sube 119864(119866) forwhich 120574

119877(119866 minus 119861) gt 120574

119877(119866)

For some simple graphs their restrained dominationnumbers can be easily determined and so restrained bondagenumbers have been also determined For example it is clearthat 120574

119877(119870

119899) = 1 Domke et al [77] showed that 120574

119877(119862

119899) =

119899 minus 2lfloor1198993rfloor for 119899 ⩾ 3 and 120574119877(119875

119899) = 119899 minus 2lfloor(119899 minus 1)3rfloor for 119899 ⩾ 1

Using these results Hattingh and Plummer [80] obtained therestricted bondage numbers for119870

119899 119862

119899 119875

119899 and119866 = 119870

11989911198992119899119905

as follows

Theorem 125 (Hattingh and Plummer [80] 2008) For 119899 ⩾ 3

119887R (119870119899) =

1 119894119891 119899 = 3

lceil119899

2rceil 119900119905ℎ119890119903119908119894119904119890

119887R (119862119899) =

1 119894119891 119899 equiv 0 (mod 3) 2 119900119905ℎ119890119903119908119894119904119890

119887R (119875119899) = 1 119899 ⩾ 4

20 International Journal of Combinatorics

119887R (119866)

=

lceil119898

2rceil 119894119891 119899

119898= 1 119886119899119889 119899

119898+1⩾ 2 (1 ⩽ 119898 lt 119905)

2119905 minus 2 119894119891 1198991= 119899

2= sdot sdot sdot = 119899

119905= 2 (119905 ⩾ 2)

2 119894119891 1198991= 2 119886119899119889 119899

2⩾ 3 (119905 = 2)

119905minus1

sum

119894=1

119899119894minus 1 119900119905ℎ119890119903119908119894119904119890

(46)

Theorem 126 (Hattingh and Plummer [80] 2008) For a tree119879 of order 119899 (⩾ 4) 119879 ≇ 119870

1119899minus1if and only if 119887R(119879) = 1

Theorem 126 shows that the restrained bondage numberof a tree can be computed in constant time However thedecision problem for 119887R(119866) is NP-complete even for bipartitegraphs

Problem 5 Consider the decision problem

Restrained Bondage Problem

Instance a graph 119866 and a positive integer 119887

Question is 119887R(119866) ⩽ 119887

Theorem 127 (Hattingh and Plummer [80] 2008) Therestrained bondage problem is NP-complete even for bipartitegraphs

Consequently in view of its computational hardnessit is significative to establish some sharp bounds on therestrained bondage number of a graph in terms of othergraphic parameters

Theorem 128 (Hattingh and Plummer [80] 2008) For anygraph 119866 with 120575(119866) ⩾ 2

119887R (119866) ⩽ min119909119910isin119864(119866)

119889119866(119909) + 119889

119866(119910) minus 2 (47)

Corollary 129 119887R(119866) ⩽ Δ(119866)+120575(119866)minus2 for any graph119866 with120575(119866) ⩾ 2

Notice that the bounds stated in Theorem 128 andCorollary 129 are sharp Indeed the class of cycles whoseorders are congruent to 1 2 (mod 3) have a restrained bondagenumber achieving these bounds (see Theorem 125)

Theorem 128 is an analogue of Theorem 14 A quitenatural problem is whether or not for restricted bondagenumber there are analogues of Theorem 16 Theorem 18Theorem 29 and so on Indeed we should consider all resultsfor the bondage number whether or not there are analoguesfor restricted bondage number

Theorem 130 (Hattingh and Plummer [80] 2008) For anygraph 119866 with 120574

119877(119866) = 2

119887R (119866) ⩽ Δ (119866) + 1 (48)

We now consider the relation between 119887119877(119866) and 119887(119866)

Theorem 131 (Hattingh and Plummer [80] 2008) If 120574119877(119866) =

120574(119866) for some graph 119866 then 119887R(119866) ⩽ 119887(119866)

Corollary 129 and Theorem 131 provide a possibility toattack Conjecture 73 by characterizing planar graphs with120574119877= 120574 and 119887R = 119887 when 3 ⩽ Δ ⩽ 6However we do not have 119887R(119866) = 119887(119866) for any graph

119866 even if 120574119877(119866) = 120574(119866) Observe that 120574

119877(119870

3) = 120574(119870

3) yet

119887119877(119870

3) = 1 and 119887(119870

3) = 2 We still may not claim that

119887119877(119866) = 119887(119866) even in the case that every 120574-set is a 120574

119877-set

The example 1198703again demonstrates this

Theorem 132 (Hattingh and Plummer [80] 2008) There isan infinite class of graphs inwhich each graph119866 satisfies 119887(119866) lt119887R(119866)

In fact such an infinite class of graphs can be obtained asfollows Let119867 be a connected graph BC(119867) a graph obtainedfrom 119867 by attaching ℓ (⩾ 2) vertices of degree 1 to eachvertex in 119867 and B = 119866 119866 = BC(119867) for some graph 119867such that 120575(119867) ⩾ 2 By Theorem 3 we can easily check that119887(119866) = 1 lt 2 ⩽ min120575(119867) ℓ = 119887R(119866) for any 119866 isin BMoreover there exists a graph 119866 isin B such that 119887R(119866) canbe much larger than 119887(119866) which is stated as the followingtheorem

Theorem 133 (Hattingh and Plummer [80] 2008) For eachpositive integer 119896 there is a graph119866 such that 119896 = 119887R(119866)minus119887(119866)

Combining Theorem 133 with Theorem 132 we have thefollowing corollary

Corollary 134 The bondage number and the restrainedbondage number are unrelated

Very recently Jafari Rad et al [81] have consideredthe total restrained bondage in graphs based on both totaldominating sets and restrained dominating sets and obtainedseveral properties exact values and bounds for the totalrestrained bondage number of a graph

A restrained dominating set 119878 of a graph 119866 without iso-lated vertices is called to be a total restrained dominating setif 119878 is a total dominating set The total restrained dominationnumber of119866 denoted by 120574TR (119866) is theminimum cardinalityof a total restrained dominating set of 119866 For referenceson this domination in graphs see [3 61 82ndash85] The totalrestrained bondage number 119887TR (119866) of a graph 119866 with noisolated vertex is the cardinality of a smallest set of edges 119861 sube119864(119866) for which119866minus119861 has no isolated vertex and 120574TR (119866minus119861) gt120574TR (119866) In the case that there is no such subset 119861 we define119887TR (119866) = infin

Ma et al [84] determined that 120574TR (119875119899) = 119899minus2lfloor(119899minus2)4rfloorfor 119899 ⩾ 3 120574TR (119862119899

) = 119899 minus 2lfloor1198994rfloor for 119899 ⩾ 3 120574TR (1198703) =

3 and 120574TR (119870119899) = 2 for 119899 = 3 and 120574TR (1198701119899

) = 1 + 119899

and 120574TR (119870119898119899) = 2 for 2 ⩽ 119898 ⩽ 119899 According to these

results Jafari Rad et al [81] determined the exact valuesof total restrained bondage numbers for the correspondinggraphs

International Journal of Combinatorics 21

Theorem 135 (Jafari Rad et al [81]) For an integer 119899

119887TR (119875119899) = infin 119894119891 2 ⩽ 119899 ⩽ 5

1 119894119891 119899 ⩾ 5

119887TR (119862119899) =

infin 119894119891 119899 = 3

1 119894119891 3 lt 119899 119899 equiv 0 1 (mod 4)2 119894119891 3 lt 119899 119899 equiv 2 3 (mod 4)

119887TR (119882119899) = 2 119891119900119903 119899 ⩾ 6

119887TR (119870119899) = 119899 minus 1 119891119900119903 119899 ⩾ 4

119887TR (119870119898119899) = 119898 minus 1 119891119900119903 2 ⩽ 119898 ⩽ 119899

(49)

119887TR (119870119899) = 119899minus1 shows the fact that for any integer 119899 ⩾ 3 there

exists a connected graph namely119870119899+1

such that 119887TR (119870119899+1) =

119899 that is the total restrained bondage number is unboundedfrom the above

A vertex is said to be a support vertex if it is adjacent toa vertex of degree one Let A be the family of all connectedgraphs such that119866 belongs toA if and only if every edge of119866is incident to a support vertex or 119866 is a cycle of length three

Proposition 136 (Cyman and Raczek [82] 2006) For aconnected graph119866 of order 119899 120574TR (119866) = 119899 if and only if119866 isin A

Hence if 119866 isin A then the removal of any subset of edgesof119866 cannot increase the total restrained domination numberof 119866 and thus 119887TR (119866) = infin When 119866 notin A a result similar toTheorem 6 is obtained

Theorem 137 (Jafari Rad et al [81]) For any tree 119879 notin A119887TR (119879) ⩽ 2 This bound is sharp

In the samepaper the authors provided a characterizationfor trees 119879 notin A with 119887TR (119879) = 1 or 2 The following result issimilar to Observation 1

Observation 3 (Jafari Rad et al [81]) If 119896 edges can beremoved from a graph 119866 to obtain a graph 119867 without anisolated vertex and with 119887TR (119867) = 119905 then 119887TR (119866) ⩽ 119896 + 119905

ApplyingTheorem 137 andObservation 3 Jafari Rad et alcharacterized all connected graphs 119866 with 119887TR (119866) = infin

Theorem 138 (Jafari Rad et al [81]) For a connected graph119866119887TR (119866) = infin if and only if 119866 isin A

84 Other Conditional Bondage Numbers A subset 119868 sube 119881(119866)is called an independent set if no two vertices in 119868 are adjacentin 119866 The maximum cardinality among all independent setsis called the independence number of 119866 denoted by 120572(119866)

A dominating set 119878 of a graph 119866 is called to be inde-pendent if 119878 is an independent set of 119866 The minimumcardinality among all independent dominating set is calledthe independence domination number of 119866 and denoted by120574119868(119866)

Since an independent dominating set is not only adominating set but also an independent set 120574(119866) ⩽ 120574

119868(119866) ⩽

120572(119866) for any graph 119866It is clear that a maximal independent set is certainly a

dominating set Thus an independent set is maximal if andonly if it is an independent dominating set and so 120574

119868(119866) is the

minimum cardinality among all maximal independent setsof 119866 This graph-theoretical invariant has been well studiedin the literature see for example Haynes et al [3] for earlyresults Goddard and Henning [86] for recent results onindependent domination in graphs

In 2003 Zhang et al [87] defined the independencebondage number 119887

119868(119866) of a nonempty graph 119866 to be the

minimum cardinality among all subsets 119861 sube 119864(119866) for which120574119868(119866 minus 119861) gt 120574

119868(119866) For some ordinary graphs their indepen-

dence domination numbers can be easily determined and soindependence bondage numbers have been also determinedClearly 119887

119868(119870

119899) = 1 if 119899 ⩾ 2

Theorem 139 (Zhang et al [87] 2003) For any integer 119899 ⩾ 4

119887119868(119862

119899) =

1 119894119891 119899 equiv 1 (mod 2) 2 119894119891 119899 equiv 0 (mod 2)

119887119868(119875

119899) =

1 119894119891 119899 equiv 0 (mod 2) 2 119894119891 119899 equiv 1 (mod 2)

119887119868(119870

119898119899) = 119899 119891119900119903 119898 ⩽ 119899

(50)

Apart from the above-mentioned results as far as we findno other results on the independence bondage number in thepresent literature we never knew much of any result on thisparameter for a tree

A subset 119878 of 119881(119866) is called an equitable dominatingset if for every 119910 isin 119881(119866 minus 119878) there exists a vertex 119909 isin 119878such that 119909119910 isin 119864(119866) and |119889

119866(119909) minus 119889

119866(119910)| ⩽ 1 proposed

by Dharmalingam [88] The minimum cardinality of such adominating set is called the equitable domination number andis denoted by 120574

119864(119866) Deepak et al [89] defined the equitable

bondage number 119887119864(119866) of a graph 119866 to be the cardinality of a

smallest set 119861 sube 119864(119866) of edges for which 120574119864(119866 minus 119861) gt 120574

119864(119866)

and determined 119887119864(119866) for some special graphs

Theorem 140 (Deepak et al [89] 2011) For any integer 119899 ⩾ 2

119887119864(119870

119899) = lceil

119899

2rceil

119887119864(119870

119898119899) = 119898 119891119900119903 0 ⩽ 119899 minus 119898 ⩽ 1

119887119864(119862

119899) =

3 119894119891 119899 equiv 1 (mod 3)2 119900119905ℎ119890119903119908119894119904119890

119887119864(119875

119899) =

2 119894119891 119899 equiv 1 (mod 3)1 119900119905ℎ119890119903119908119894119904119890

119887119864(119879) ⩽ 2 119891119900119903 119886119899119910 119899119900119899119905119903119894119907119894119886119897 119905119903119890119890 119879

119887119864(119866) ⩽ 119899 minus 1 119891119900119903 119886119899119910 119888119900119899119899119890119888119905119890119889 119892119903119886119901ℎ 119866 119900119891 119900119903119889119890119903 119899

(51)

22 International Journal of Combinatorics

As far as we know there are no more results on theequitable bondage number of graphs in the present literature

There are other variations of the normal domination byadding restricted conditions to the normal dominating setFor example the connected domination set of a graph119866 pro-posed by Sampathkumar and Walikar [90] is a dominatingset 119878 such that the subgraph induced by 119878 is connected Theproblem of finding a minimum connected domination set ina graph is equivalent to the problemof finding a spanning treewithmaximumnumber of vertices of degree one [91] and hassome important applications in wireless networks [92] thusit has received much research attention However for suchan important domination there is no result on its bondagenumber We believe that the topic is of significance for futureresearch

9 Generalized Bondage Numbers

There are various generalizations of the classical dominationsuch as 119901-domination distance domination fractional dom-ination Roman domination and rainbow domination Everysuch a generalization can lead to a corresponding bondageIn this section we introduce some of them

91 119901-Bondage Numbers In 1985 Fink and Jacobson [93]introduced the concept of 119901-domination Let 119901 be a positiveinteger A subset 119878 of 119881(119866) is a 119901-dominating set of 119866 if|119878 cap 119873

119866(119910)| ⩾ 119901 for every 119910 isin 119878 The 119901-domination number

120574119901(119866) is the minimum cardinality among all 119901-dominating

sets of 119866 Any 119901-dominating set of 119866 with cardinality 120574119901(119866)

is called a 120574119901-set of 119866 Note that a 120574

1-set is a classic minimum

dominating set Notice that every graph has a 119901-dominatingset since the vertex-set119881(119866) is such a setWe also note that a 1-dominating set is a dominating set and so 120574(119866) = 120574

1(119866) The

119901-domination number has received much research attentionsee a state-of-the-art survey article by Chellali et al [94]

It is clear from definition that every 119901-dominating setof a graph certainly contains all vertices of degree at most119901 minus 1 By this simple observation to avoid the trivial caseoccurrence we always assume that Δ(119866) ⩾ 119901 For 119901 ⩾ 2 Luet al [95] gave a constructive characterization of trees withunique minimum 119901-dominating sets

Recently Lu and Xu [96] have introduced the conceptto the 119901-bondage number of 119866 denoted by 119887

119901(119866) as the

minimum cardinality among all sets of edges 119861 sube 119864(119866) suchthat 120574

119901(119866 minus 119861) gt 120574

119901(119866) Clearly 119887

1(119866) = 119887(119866)

Lu and Xu [96] established a lower bound and an upperbound on 119887

119901(119879) for any integer 119901 ⩾ 2 and any tree 119879 with

Δ(119879) ⩾ 119901 and characterized all trees achieving the lowerbound and the upper bound respectively

Theorem 141 (Lu and Xu [96] 2011) For any integer 119901 ⩾ 2and any tree 119879 with Δ(119879) ⩾ 119901

1 ⩽ 119887119901(119879) ⩽ Δ (119879) minus 119901 + 1 (52)

Let 119878 be a given subset of 119881(119866) and for any 119909 isin 119878 let

119873119901(119909 119878 119866) = 119910 isin 119878 cap 119873

119866(119909)

1003816100381610038161003816119873119866(119910) cap 119878

1003816100381610038161003816 = 119901

(53)

Then all trees achieving the lower bound 1 can be character-ized as follows

Theorem 142 (Lu and Xu [96] 2011) Let 119879 be a tree withΔ(119879) ⩾ 119901 ⩾ 2 Then 119887

119901(119879) = 1 if and only if for any 120574

119901-set

119878 of 119879 there exists an edge 119909119910 isin (119878 119878) such that 119910 isin 119873119901(119909 119878 119879)

The notation 119878(119886 119887) denotes a double star obtained byadding an edge between the central vertices of two stars119870

1119886minus1

and1198701119887minus1

And the vertex with degree 119886 (resp 119887) in 119878(119886 119887) iscalled the 119871-central vertex (resp 119877-central vertex) of 119878(119886 119887)

To characterize all trees attaining the upper bound givenin Theorem 141 we define three types of operations on a tree119879 with Δ(119879) = Δ ⩾ 119901 + 1

Type 1 attach a pendant edge to a vertex 119910with 119889119879(119910) ⩽ 119901minus

2 in 119879

Type 2 attach a star 1198701Δminus1

to a vertex 119910 of 119879 by joining itscentral vertex to 119910 where 119910 in a 120574

119901-set of 119879 and

119889119879(119910) ⩽ Δ minus 1

Type 3 attach a double star 119878(119901 Δ minus 1) to a pendant vertex 119910of 119879 by coinciding its 119877-central vertex with 119910 wherethe unique neighbor of 119910 is in a 120574

119901-set of 119879

Let B = 119879 a tree obtained from 1198701Δ

or 119878(119901 Δ) by afinite sequence of operations of Types 1 2 and 3

Theorem 143 (Lu and Xu [96] 2011) A tree with the maxi-mum Δ ⩾ 119901 + 1 has 119901-bondage number Δ minus 119901 + 1 if and onlyif it belongs toB

Theorem 144 (Lu and Xu [97] 2012) Let 119866119898119899= 119875

119898times 119875

119899

Then

1198872(119866

2119899) = 1 119891119900119903 119899 ⩾ 2

1198872(119866

3119899) =

2 119894119891 119899 equiv 1 (mod 3) 1 119900119905ℎ119890119903119908119894119904119890

for 119899 ⩾ 2

1198872(119866

4119899) =

1 119894119891 119899 equiv 3 (mod 4) 2 119900119905ℎ119890119903119908119894119904119890

for 119899 ⩾ 7

(54)

Recently Krzywkowski [98] has proposed the conceptof the nonisolating 2-bondage of a graph The nonisolating2-bondage number of a graph 119866 denoted by 1198871015840

2(119866) is the

minimum cardinality among all sets of edges 119861 sube 119864(119866) suchthat 119866 minus 119861 has no isolated vertices and 120574

2(119866 minus 119861) gt 120574

2(119866) If

for every 119861 sube 119864(119866) either 1205742(119866 minus 119861) = 120574

2(119866) or 119866 minus 119861 has

isolated vertices then define 11988710158402(119866) = 0 and say that 119866 is a 2-

nonisolatedly strongly stable graph Krzywkowski presentedsome basic properties of nonisolating 2-bondage in graphsshowed that for every nonnegative integer 119896 there exists atree 119879 such 1198871015840

2(119879) = 119896 characterized all 2-non-isolatedly

strongly stable trees and determined 11988710158402(119866) for several special

graphs

International Journal of Combinatorics 23

Theorem 145 (Krzywkowski [98] 2012) For any positiveintegers 119899 and119898 with119898 ⩽ 119899

1198871015840

2(119870

119899) =

0 119894119891 119899 = 1 2 3

lfloor2119899

3rfloor 119900119905ℎ119890119903119908119894119904119890

1198871015840

2(119875

119899) =

0 119894119891 119899 = 1 2 3

1 119900119905ℎ119890119903119908119894119904119890

1198871015840

2(119862

119899) =

0 119894119891 119899 = 3

1 119894119891 119899 119894119904 119890V119890119899

2 119894119891 119899 119894119904 119900119889119889

1198871015840

2(119870

119898119899) =

3 119894119891 119898 = 119899 = 3

1 119894119891 119898 = 119899 = 4

2 119900119905ℎ119890119903119908119894119904119890

(55)

92 Distance Bondage Numbers A subset 119878 of vertices of agraph119866 is said to be a distance 119896-dominating set for119866 if everyvertex in 119866 not in 119878 is at distance at most 119896 from some vertexof 119878 The minimum cardinality of all distance 119896-dominatingsets is called the distance 119896-domination number of 119866 anddenoted by 120574

119896(119866) (do not confuse with the above-mentioned

120574119901(119866)) When 119896 = 1 a distance 1-dominating set is a normal

dominating set and so 1205741(119866) = 120574(119866) for any graph 119866 Thus

the distance 119896-domination is a generalization of the classicaldomination

The relation between 120574119896and 120572

119896(the 119896-independence

number defined in Section 22) for a tree obtained by Meirand Moon [14] who proved that 120574

119896(119879) = 120572

2119896(119879) for any tree

119879 (a special case for 119896 = 1 is stated in Proposition 10) Thefurther research results can be found in Henning et al [99]Tian and Xu [100ndash103] and Liu et al [104]

In 1998 Hartnell et al [15] defined the distance 119896-bondagenumber of 119866 denoted by 119887

119896(119866) to be the cardinality of a

smallest subset 119861 of edges of 119866 with the property that 120574119896(119866 minus

119861) gt 120574119896(119866) FromTheorem 6 it is clear that if119879 is a nontrivial

tree then 1 ⩽ 1198871(119879) ⩽ 2 Hartnell et al [15] generalized this

result to any integer 119896 ⩾ 1

Theorem 146 (Hartnell et al [15] 1998) For every nontrivialtree 119879 and positive integer 119896 1 ⩽ 119887

119896(119879) ⩽ 2

Hartnell et al [15] and Topp and Vestergaard [13] alsocharacterized the trees having distance 119896-bondage number 2In particular the class of trees for which 119887

1(119879) = 2 are just

those which have a unique maximum 2-independent set (seeTheorem 11)

Since when 119896 = 1 the distance 1-bondage number 1198871(119866)

is the classical bondage number 119887(119866) Theorem 12 gives theNP-hardness of deciding the distance 119896-bondage number ofgeneral graphs

Theorem 147 Given a nonempty undirected graph 119866 andpositive integers 119896 and 119887 with 119887 ⩽ 120576(119866) determining wetheror not 119887

119896(119866) ⩽ 119887 is NP-hard

For a vertex 119909 in119866 the open 119896-neighborhood119873119896(119909) of 119909 is

defined as119873119896(119909) = 119910 isin 119881(119866) 1 ⩽ 119889

119866(119909 119910) le 119896The closed

119896-neighborhood119873119896[119909] of 119909 in 119866 is defined as119873

119896(119909) cup 119909 Let

Δ119896(119866) = max 1003816100381610038161003816119873119896

(119909)1003816100381610038161003816 for any 119909 isin 119881 (119866) (56)

Clearly Δ1(119866) = Δ(119866) The 119896th power of a graph 119866 is the

graph119866119896 with vertex-set119881(119866119896

) = 119881(119866) and edge-set119864(119866119896

) =

119909119910 1 ⩽ 119889119866(119909 119910) ⩽ 119896 The following lemma holds directly

from the definition of 119866119896

Proposition 148 (Tian and Xu [101]) Δ(119866119896

) = Δ119896(119866) and

120574(119866119896

) = 120574119896(119866) for any graph 119866 and each 119896 ⩾ 1

A graph 119866 is 119896-distance domination-critical or 120574119896-critical

for short if 120574119896(119866 minus 119909) lt 120574

119896(119866) for every vertex 119909 in 119866

proposed by Henning et al [105]

Proposition 149 (Tian and Xu [101]) For each 119896 ⩾ 1 a graph119866 is 120574

119896-critical if and only if 119866119896 is 120574

119896-critical

By the above facts we suggest the following worthwhileproblem

Problem 6 Can we generalize the results on the bondage for119866 to 119866119896 In particular do the following propositions hold

(a) 119887(119866119896

) = 119887119896(119866) for any graph 119866 and each 119896 ⩾ 1

(b) 119887(119866119896

) ⩽ Δ119896(119866) if 119866 is not 120574

119896-critical

Let 119896 and 119901 be positive integers A subset 119878 of 119881(119866) isdefined to be a (119896 119901)-dominating set of 119866 if for any vertex119909 isin 119878 |119873

119896(119909) cap 119878| ⩾ 119901 The (119896 119901)-domination number of

119866 denoted by 120574119896119901(119866) is the minimum cardinality among

all (119896 119901)-dominating sets of 119866 Clearly for a graph 119866 a(1 1)-dominating set is a classical dominating set a (119896 1)-dominating set is a distance 119896-dominating set and a (1 119901)-dominating set is the above-mentioned 119901-dominating setThat is 120574

11(119866) = 120574(119866) 120574

1198961(119866) = 120574

119896(119866) and 120574

1119901(119866) = 120574

119901(119866)

The concept of (119896 119901)-domination in a graph 119866 is a gen-eralized domination which combines distance 119896-dominationand 119901-domination in 119866 So the investigation of (119896 119901)-domination of 119866 is more interesting and has received theattention of many researchers see for example [106ndash109]

More general Li and Xu [110] proposed the concept of the(ℓ 119908)-domination number of a119908-connected graph A subset119878 of 119881(119866) is defined to be an (ℓ 119908)-dominating set of 119866 iffor any vertex 119909 isin 119878 there are 119908 internally disjoint pathsof length at most ℓ from 119909 to some vertex in 119878 The (ℓ 119908)-domination number of119866 denoted by 120574

ℓ119908(119866) is theminimum

cardinality among all (ℓ 119908)-dominating sets of 119866 Clearly1205741198961(119866) = 120574

119896(119866) and 120574

1119901(119866) = 120574

119901(119866)

It is quite natural to propose the concept of bondage num-ber for (119896 119901)-domination or (ℓ 119908)-domination However asfar as we know none has proposed this concept until todayThis is a worthwhile topic for further research

24 International Journal of Combinatorics

93 Fractional Bondage Numbers If 120590 is a function mappingthe vertex-set 119881 into some set of real numbers then for anysubset 119878 sube 119881 let 120590(119878) = sum

119909isin119878120590(119909) Also let |120590| = 120590(119881)

A real-value function 120590 119881 rarr [0 1] is a dominatingfunction of a graph 119866 if for every 119909 isin 119881(119866) 120590(119873

119866[119909]) ⩾ 1

Thus if 119878 is a dominating set of 119866 120590 is a function where

120590 (119909) = 1 if 119909 isin 1198780 otherwise

(57)

then 120590 is a dominating function of119866 The fractional domina-tion number of 119866 denoted by 120574

119865(119866) is defined as follows

120574119865(119866) = min |120590| 120590 is a domination function of 119866

(58)

The 120574119865-bondage number of 119866 denoted by 119887

119865(119866) is

defined as the minimum cardinality of a subset 119861 sube 119864 whoseremoval results in 120574

119865(119866 minus 119861) gt 120574

119865(119866)

Hedetniemi et al [111] are the first to study fractionaldomination although Farber [112] introduced the idea indi-rectly The concept of the fractional domination numberwas proposed by Domke and Laskar [113] in 1997 Thefractional domination numbers for some ordinary graphs aredetermined

Proposition 150 (Domke et al [114] 1998 Domke and Laskar[113] 1997) (a) If 119866 is a 119896-regular graph with order 119899 then120574119865(119866) = 119899119896 + 1(b) 120574

119865(119862

119899) = 1198993 120574

119865(119875

119899) = lceil1198993rceil 120574

119865(119870

119899) = 1

(c) 120574119865(119866) = 1 if and only if Δ(119866) = 119899 minus 1

(d) 120574119865(119879) = 120574(119879) for any tree 119879

(e) 120574119865(119870

119899119898) = (119899(119898 + 1) + 119898(119899 + 1))(119899119898 minus 1) where

max119899119898 gt 1

The assertions (a)ndash(d) are due to Domke et al [114] andthe assertion (e) is due to Domke and Laskar [113] Accordingto these results Domke and Laskar [113] determined the 120574

119865-

bondage numbers for these graphs

Theorem 151 (Domke and Laskar [113] 1997) 119887119865(119870

119899) =

lceil1198992rceil 119887119865(119870

119899119898) = 1 wheremax119899119898 gt 1

119887119865(119862

119899) =

2 119894119891 119899 equiv 0 (mod 3) 1 119900119905ℎ119890119903119908119894119904119890

119887119865(119875

119899) =

2 119894119891 119899 equiv 1 (mod 3) 1 119900119905ℎ119890119903119908119894119904119890

(59)

It is easy to see that for any tree119879 119887119865(119879) ⩽ 2 In fact since

120574119865(119879) = 120574(119879) for any tree 119879 120574

119865(119879

1015840

) = 120574(1198791015840

) for any subgraph1198791015840 of 119879 It follows that

119887119865(119879) = min 119861 sube 119864 (119879) 120574

119865(119879 minus 119861) gt 120574

119865(119879)

= min 119861 sube 119864 (119879) 120574 (119879 minus 119861) gt 120574 (119879)

= 119887 (119879) ⩽ 2

(60)

Except for these no results on 119887119865(119866) have been known as

yet

94 Roman Bondage Numbers A Roman dominating func-tion on a graph 119866 is a labeling 119891 119881 rarr 0 1 2 such thatevery vertex with label 0 has at least one neighbor with label2 The weight of a Roman dominating function 119891 on 119866 is thevalue

119891 (119866) = sum

119906isin119881(119866)

119891 (119906) (61)

The minimum weight of a Roman dominating function ona graph 119866 is called the Roman domination number of 119866denoted by 120574RM (119866)

A Roman dominating function 119891 119881 rarr 0 1 2

can be represented by the ordered partition (1198810 119881

1 119881

2) (or

(119881119891

0 119881

119891

1 119881

119891

2) to refer to119891) of119881 where119881

119894= V isin 119881 | 119891(V) = 119894

In this representation its weight 119891(119866) = |1198811| + 2|119881

2| It is

clear that119881119891

1cup119881

119891

2is a dominating set of 119866 called the Roman

dominating set denoted by 119863119891

R = (1198811 1198812) Since 119881119891

1cup 119881

119891

2is

a dominating set when 119891 is a Roman dominating functionand since placing weight 2 at the vertices of a dominating setyields a Roman dominating function in [115] it is observedthat

120574 (119866) ⩽ 120574RM (119866) ⩽ 2120574 (119866) (62)

A graph 119866 is called to be Roman if 120574RM (119866) = 2120574(119866)The definition of the Roman domination function is

proposed implicitly by Stewart [116] and ReVelle and Rosing[117] Roman dominating numbers have been deeply studiedIn particular Bahremandpour et al [118] showed that theproblem determining the Roman domination number is NP-complete even for bipartite graphs

Let 119866 be a graph with maximum degree at least twoThe Roman bondage number 119887RM (119866) of 119866 is the minimumcardinality of all sets 1198641015840 sube 119864 for which 120574RM (119866 minus 119864

1015840

) gt

120574RM (119866) Since in study of Roman bondage number theassumption Δ(119866) ⩾ 2 is necessary we always assume thatwhen we discuss 119887RM (119866) all graphs involved satisfy Δ(119866) ⩾2 The Roman bondage number 119887RM (119866) is introduced byJafari Rad and Volkmann in [119]

Recently Bahremandpour et al [118] have shown that theproblem determining the Roman bondage number is NP-hard even for bipartite graphs

Problem 7 Consider the decision problem

Roman Bondage Problem

Instance a nonempty bipartite graph119866 and a positiveinteger 119896

Question is 119887RM (119866) ⩽ 119896

Theorem 152 (Bahremandpour et al [118] 2013) TheRomanbondage problem is NP-hard even for bipartite graphs

The exact value of 119887RM(119866) is known only for a few familyof graphs including the complete graphs cycles and paths

International Journal of Combinatorics 25

Theorem 153 (Jafari Rad and Volkmann [119] 2011) For anypositive integers 119899 and119898 with119898 ⩽ 119899

119887RM (119875119899) = 2 119894119891 119899 equiv 2 (mod 3) 1 119900119905ℎ119890119903119908119894119904119890

119887RM (119862119899) =

3 119894119891 119899 = 2 (mod 3) 2 119900119905ℎ119890119903119908119894119904119890

119887RM (119870119899) = lceil

119899

2rceil 119891119900119903 119899 ⩾ 3

119887RM (119870119898119899) =

1 119894119891 119898 = 1 119886119899119889 119899 = 1

4 119894119891 119898 = 119899 = 3

119898 119900119905ℎ119890119903119908119894119904119890

(63)

Lemma 154 (Cockayne et al [115] 2004) If 119866 is a graph oforder 119899 and contains vertices of degree 119899 minus 1 then 120574RM(119866) = 2

Using Lemma 154 the third conclusion in Theorem 153can be generalized to more general case which is similar toLemma 49

Theorem 155 (Jafari Rad and Volkmann [119] 2011) Let119866 bea graph with order 119899 ge 3 and 119905 the number of vertices of degree119899 minus 1 in 119866 If 119905 ⩾ 1 then 119887RM(119866) = lceil1199052rceil

Ebadi and PushpaLatha [120] conjectured that 119887RM(119866) ⩽119899 minus 1 for any graph 119866 of order 119899 ⩾ 3 Dehgardi et al[121] Akbari andQajar [122] independently showed that thisconjecture is true

Theorem 156 119887RM(119866) ⩽ 119899 minus 1 for any connected graph 119866 oforder 119899 ⩾ 3

For a complete 119905-partite graph we have the followingresult

Theorem 157 (Hu and Xu [123] 2011) Let 119866 = 11987011989811198982119898119905

bea complete 119905-partite graph with 119898

1= sdot sdot sdot = 119898

119894lt 119898

119894+1le sdot sdot sdot ⩽

119898119905 119905 ⩾ 2 and 119899 = sum119905

119895=1119898

119895 Then

119887RM (119866) =

lceil119894

2rceil 119894119891 119898

119894= 1 119886119899119889 119899 ⩾ 3

2 119894119891 119898119894= 2 119886119899119889 119894 = 1

119894 119894119891 119898119894= 2 119886119899119889 119894 ⩾ 2

4 119894119891 119898119894= 3 119886119899119889 119894 = 119905 = 2

119899 minus 1 119894119891 119898119894= 3 119886119899119889 119894 = 119905 ⩾ 3

119899 minus 119898119905

119894119891 119898119894⩾ 3 119886119899119889 119898

119905⩾ 4

(64)

Consider a complete 119905-partite graph 119870119905(3) for 119905 ⩾ 3

119887RM(119870119905(3)) = 119899 minus 1 by Theorem 157 where 119899 = 3119905

which shows that this upper bound of 119899 minus 1 on 119887RM(119866) inTheorem 156 is sharp This fact and 120574

119877(119870

119905(3)) = 4 give

a negative answer to the following two problems posed byDehgardi et al [121]Prove or Disprove for any connected graph 119866 of order 119899 ge 3(i) 119887RM(119866) = 119899 minus 1 if and only if 119866 cong 119870

3 (ii) 119887RM(119866) ⩽ 119899 minus

120574119877(119866) + 1Note that a complete 119905-partite graph 119870

119905(3) is (119899 minus 3)-

regular and 119887RM(119870119905(3)) = 119899 minus 1 for 119905 ⩾ 3 where 119899 = 3119905

Hu and Xu further determined that 119887119877(119866) = 119899 minus 2 for any

(119899 minus 3)-regular graph 119866 of order 119899 ⩽ 5 except 119870119905(3)

Theorem 158 (Hu and Xu [123] 2011) For any (119899minus3)-regulargraph 119866 of order 119899 other than119870

119905(3) 119887RM(119866) = 119899 minus 2 for 119899 ⩾ 5

For a tree 119879 with order 119899 ⩾ 3 Ebadi and PushpaLatha[120] and Jafari Rad and Volkmann [119] independentlyobtained an upper bound on 119887RM(119879)

Theorem 159 119887RM(119879) ⩽ 3 for any tree 119879 with order 119899 ⩾ 3

Theorem 160 (Bahremandpour et al [118] 2013) 119887RM(1198752 times119875119899) = 2 for 119899 ⩾ 2

Theorem 161 (Jafari Rad and Volkmann [119] 2011) Let119866 bea graph with order at least three

(a) If (119909 119910 119911) is a path of length 2 in 119866 then 119887RM(119866) ⩽119889119866(119909) + 119889

119866(119910) + 119889

119866(119911) minus 3 minus |119873

119866(119909) cap 119873

119866(119910)|

(b) If 119866 is connected then 119887RM(119866) ⩽ 120582(119866) + 2Δ(119866) minus 3

Theorem 161 (b) implies 119887RM(119866) ⩽ 120575(119866) + 2Δ(119866) minus 3 for aconnected graph 119866 Note that for a planar graph 119866 120575(119866) ⩽ 5moreover 120575(119866) ⩽ 3 if the girth at least 4 and 120575(119866) ⩽ 2 if thegirth at least 6 These two facts show that 119887RM(119866) ⩽ 2Δ(119866) +2 for a connected planar graph 119866 Jafari Rad and Volkmann[124] improved this bound

Theorem 162 (Jafari Rad and Volkmann [124] 2011) For anyconnected planar graph 119866 of order 119899 ⩾ 3 with girth 119892(119866)

119887RM (119866)

⩽

2Δ (119866)

Δ (119866) + 6

Δ (119866) + 5 119894119891 119866 119888119900119899119905119886119894119899119904 119899119900 V119890119903119905119894119888119890119904 119900119891 119889119890119892119903119890119890 119891119894V119890

Δ (119866) + 4 119894119891 119892 (119866) ⩾ 4

Δ (119866) + 3 119894119891 119892 (119866) ⩾ 5

Δ (119866) + 2 119894119891 119892 (119866) ⩾ 6

Δ (119866) + 1 119894119891 119892 (119866) ⩾ 8

(65)

According toTheorem 157 119887RM(119862119899) = 3 = Δ(119862

119899)+1 for a

cycle 119862119899of length 119899 ⩾ 8 with 119899 equiv 2 (mod 3) and therefore

the last upper bound inTheorem 162 is sharp at least for Δ =2

Combining the fact that every planar graph 119866 withminimum degree 5 contains an edge 119909119910 with 119889

119866(119909) = 5

and 119889119866(119910) isin 5 6 with Theorem 161 (a) Akbari et al [125]

obtained the following result

26 International Journal of Combinatorics

middot middot middot

middot middot middot

middot middot middot

middot middot middot

119866

Figure 9 The graph 119866 is constructed from 119866

Theorem 163 (Akbari et al [125] 2012) 119887RM(119866) ⩽ 15 forevery planar graph 119866

It remains open to show whether the bound inTheorem 163 is sharp or not Though finding a planargraph 119866 with 119887RM(119866) = 15 seems to be difficult Akbari etal [125] constructed an infinite family of planar graphs withRoman bondage number equal to 7 by proving the followingresult

Theorem 164 (Akbari et al [125] 2012) Let 119866 be a graph oforder 119899 and 119866 is the graph of order 5119899 obtained from 119866 byattaching the central vertex of a copy of a path 119875

5to each vertex

of119866 (see Figure 9) Then 120574RM(119866) = 4119899 and 119887RM(119866) = 120575(119866)+2

ByTheorem 164 infinitemany planar graphswith Romanbondage number 7 can be obtained by considering any planargraph 119866 with 120575(119866) = 5 (eg the icosahedron graph)

Conjecture 165 (Akbari et al [125] 2012) The Romanbondage number of every planar graph is at most 7

For general bounds the following observation is directlyobtained which is similar to Observation 1

Observation 4 Let 119866 be a graph of order 119899 with maximumdegree at least two Assume that 119867 is a spanning subgraphobtained from 119866 by removing 119896 edges If 120574RM(119867) = 120574RM(119866)then 119887

119877(119867) ⩽ 119887RM(119866) ⩽ 119887RM(119867) + 119896

Theorem 166 (Bahremandpour et al [118] 2013) For anyconnected graph 119866 of order 119899 if 119899 ⩾ 3 and 120574RM(119866) = 120574(119866) + 1then

119887RM (119866) ⩽ min 119887 (119866) 119899Δ (66)

where 119899Δis the number of vertices with maximum degree Δ in

119866

Observation 5 For any graph 119866 if 120574(119866) = 120574RM(119866) then119887RM(119866) ⩽ 119887(119866)

Theorem 167 (Bahremandpour et al [118] 2013) For everyRoman graph 119866

119887RM (119866) ⩾ 119887 (119866) (67)

The bound is sharp for cycles on 119899 vertices where 119899 equiv 0 (mod3)

The strict inequality in Theorem 167 can hold for exam-ple 119887(119862

3119896+2) = 2 lt 3 = 119887RM(1198623119896+2

) byTheorem 157A graph 119866 is called to be vertex Roman domination-

critical if 120574RM(119866 minus 119909) lt 120574RM(119866) for every vertex 119909 in 119866 If119866 has no isolated vertices then 120574RM(119866) ⩽ 2120574(119866) ⩽ 2120573(119866)If 120574RM(119866) = 2120573(119866) then 120574RM(119866) = 2120574(119866) and hence 119866 is aRoman graph In [30] Volkmann gave a lot of graphs with120574(119866) = 120573(119866) The following result is similar to Theorem 57

Theorem 168 (Bahremandpour et al [118] 2013) For anygraph 119866 if 120574RM(119866) = 2120573(119866) then

(a) 119887RM(119866) ⩾ 120575(119866)

(b) 119887RM(119866) ⩾ 120575(119866)+1 if119866 is a vertex Roman domination-critical graph

If 120574RM(119866) = 2 then obviously 119887RM(119866) ⩽ 120575(119866) So weassume 120574RM(119866) ⩾ 3 Dehgardi et al [121] proved 119887RM(119866) ⩽(120574RM(119866) minus 2)Δ(119866) + 1 and posed the following problem if 119866is a connected graph of order 119899 ⩾ 4 with 120574RM(119866) ⩾ 3 then

119887RM (119866) ⩽ (120574RM (119866) minus 2) Δ (119866) (68)

Theorem 161 (a) shows that the inequality (68) holds if120574RM(119866) ⩾ 5 Thus the bound in (68) is of interest only when120574RM(119866) is 3 or 4

Lemma 169 (Bahremandpour et al [118] 2013) For anynonempty graph 119866 of order 119899 ⩾ 3 120574RM(119866) = 3 if and onlyif Δ(119866) = 119899 minus 2

The following result shows that (68) holds for all graphs119866 of order 119899 ⩾ 4 with 120574RM(119866) = 3 or 4 which improvesTheorem 161 (b)

Theorem 170 (Bahremandpour et al [118] 2013) For anyconnected graph 119866 of order 119899 ⩾ 4

119887RM (119866) le Δ (119866) = 119899 minus 2 119894119891 120574RM (119866) = 3

Δ (119866) + 120575 (119866) minus 1 119894119891 120574RM (119866) = 4(69)

with the first equality if and only if 119866 cong 1198624

Recently Akbari and Qajar [122] have proved the follow-ing result

Theorem 171 (Akbari and Qajar [122]) For any connectedgraph 119866 of order 119899 ⩾ 3

119887RM (119866) ⩽ 119899 minus 120574RM (119866) + 5 (70)

International Journal of Combinatorics 27

We conclude this section with the following problems

Problem 8 Characterize all connected graphs 119866 of order 119899 ⩾3 for which 119887RM(119866) = 119899 minus 1

Problem 9 Prove or disprove if 119866 is a connected graph oforder 119899 ⩾ 3 then

119887RM (119866) ⩽ 119899 minus 120574RM (119866) + 3 (71)

Note that 120574119877(119870

119905(3)) = 4 and 119887RM(119870119905

(3)) = 119899minus1 where 119899 =3119905 for 119905 ⩾ 3 The upper bound on 119887RM(119866) given in Problem 9is sharp if Problem 9 has positive answer

95 Rainbow Bondage Numbers For a positive integer 119896 a119896-rainbow dominating function of a graph 119866 is a function 119891from the vertex-set 119881(119866) to the set of all subsets of the set1 2 119896 such that for any vertex 119909 in 119866 with 119891(119909) = 0 thecondition

⋃

119910isin119873119866(119909)

119891 (119910) = 1 2 119896 (72)

is fulfilledThe weight of a 119896-rainbow dominating function 119891on 119866 is the value

119891 (119866) = sum

119909isin119881(119866)

1003816100381610038161003816119891 (119909)1003816100381610038161003816 (73)

The 119896-rainbow domination number of a graph 119866 denoted by120574119903119896(119866) is the minimum weight of a 119896-rainbow dominating

function 119891 of 119866Note that 120574

1199031(119866) is the classical domination number 120574(119866)

The 119896-rainbow domination number is introduced by Bresaret al [126] Rainbow domination of a graph 119866 coincides withordinary domination of the Cartesian product of 119866 with thecomplete graph in particular 120574

119903119896(119866) = 120574(119866 times 119870

119896) for any

positive integer 119896 and any graph 119866 [126] Hartnell and Rall[127] proved that 120574(119879) lt 120574

1199032(119879) for any tree 119879 no graph has

120574 = 120574119903119896for 119896 ⩾ 3 for any 119896 ⩾ 2 and any graph 119866 of order 119899

min 119899 120574 (119866) + 119896 minus 2 ⩽ 120574119903119896(119866) ⩽ 119896120574 (119866) (74)

The introduction of rainbow domination is motivatedby the study of paired domination in Cartesian productsof graphs where certain upper bounds can be expressed interms of rainbow domination that is for any graph 119866 andany graph 119867 if 119881(119867) can be partitioned into 119896120574

119875-sets then

120574119875(119866 times 119867) ⩽ (1119896)120592(119867)120574

119903119896(119866) proved by Bresar et al [126]

Dehgardi et al [128] introduced the concept of 119896-rainbowbondage number of graphs Let 119866 be a graph with maximumdegree at least two The 119896-rainbow bondage number 119887

119903119896(119866)

of 119866 is the minimum cardinality of all sets 119861 sube 119864(119866) forwhich 120574

119903119896(119866 minus 119861) gt 120574

119903119896(119866) Since in the study of 119896-rainbow

bondage number the assumption Δ(119866) ⩾ 2 is necessarywe always assume that when we discuss 119887

119903119896(119866) all graphs

involved satisfy Δ(119866) ⩾ 2The 2-rainbow bondage number of some special families

of graphs has been determined

Theorem 172 (Dehgardi et al [128]) For any integer 119899 ⩾ 3

1198871199032(119875

119899) = 1 119891119900119903 119899 ⩾ 2

1198871199032(119862

119899) =

1 119894119891 119899 equiv 0 (mod 4) 2 119900119905ℎ119890119903119908119894119904119890

1198871199032(119870

119899119899) =

1 119894119891 119899 = 2

3 119894119891 119899 = 3

6 119894119891 119899 = 4

119898 119894119891 119899 ⩾ 5

(75)

Theorem 173 (Dehgardi et al [128]) For any connected graph119866 of order 119899 ⩾ 3 119887

1199032(119866) ⩽ 120582(119866) + 2Δ(119866) minus 3

Theorem 174 (Dehgardi et al [128]) For any tree 119879 of order119899 ⩾ 3 119887

1199032(119879) ⩽ 2

Theorem 175 (Dehgardi et al [128]) If 119866 is a planar graphwithmaximumdegree at least two then 119887

1199032(119866) ⩽ 15Moreover

if the girth 119892(119866) ⩾ 4 then 1198871199032(119866) ⩽ 11 and if 119892(119866) ⩾ 6 then

1198871199032(119866) ⩽ 9

Theorem 176 (Dehgardi et al [128]) For a complete bipartitegraph 119870

119901119902 if 2119896 + 1 ⩽ 119901 ⩽ 119902 then 119887

119903119896(119870

119901119902) = 119901

Theorem177 (Dehgardi et al [128]) For a complete graph119870119899

if 119899 ⩾ 119896 + 1 then 119887119903119896(119870

119899) = lceil119896119899(119896 + 1)rceil

The special case 119896 = 1 of Theorem 177 can be found inTheorem 1 The following is a simple observation similar toObservation 1

Observation 6 (Dehgardi et al [128]) Let 119866 be a graphwith maximum degree at least two 119867 a spanning subgraphobtained from 119866 by removing 119896 edges If 120574

119903119896(119867) = 120574

119903119896(119866)

then 119887119903119896(119867) ⩽ 119887

119903119896(119866) ⩽ 119887

119903119896(119867) + 119896

Theorem 178 (Dehgardi et al [128]) For every graph 119866 with120574119903119896(119866) = 119896120574(119866) 119887

119903119896(119866) ⩾ 119887(119866)

A graph 119866 is said to be vertex 119896-rainbow domination-critical if 120574

119903119896(119866 minus 119909) lt 120574

119903119896(119866) for every vertex 119909 in 119866 It is

clear that if 119866 has no isolated vertices then 120574119903119896(119866) ⩽ 119896120574(119866) ⩽

119896120573(119866) where 120573(119866) is the vertex-covering number of119866 Thusif 120574

119903119896(119866) = 119896120573(119866) then 120574

119903119896(119866) = 119896120574(119866)

Theorem 179 (Dehgardi et al [128]) For any graph 119866 if120574119903119896(119866) = 119896120573(119866) then

(a) 119887119903119896(119866) ⩾ 120575(119866)

(b) 119887119903119896(119866) ⩾ 120575(119866)+1 if119866 is vertex 119896-rainbow domination-

critical

As far as we know there are no more results on the 119896-rainbow bondage number of graphs in the literature

28 International Journal of Combinatorics

10 Results on Digraphs

Although domination has been extensively studied in undi-rected graphs it is natural to think of a dominating setas a one-way relationship between vertices of the graphIndeed among the earliest literatures on this subject vonNeumann and Morgenstern [129] used what is now calleddomination in digraphs to find solution (or kernels whichare independent dominating sets) for cooperative 119899-persongames Most likely the first formulation of domination byBerge [130] in 1958 was given in the context of digraphs andonly some years latter by Ore [131] in 1962 for undirectedgraphs Despite this history examination of domination andits variants in digraphs has been essentially overlooked (see[4] for an overview of the domination literature) Thus thereare few if any such results on domination for digraphs in theliterature

The bondage number and its related topics for undirectedgraph have become one of major areas both in theoreticaland applied researches However until recently Carlsonand Develin [42] Shan and Kang [132] and Huang andXu [133 134] studied the bondage number for digraphsindependently In this section we introduce their resultsfor general digraphs Results for some special digraphs suchas vertex-transitive digraphs are introduced in the nextsection

101 Upper Bounds for General Digraphs Let 119866 = (119881 119864)

be a digraph without loops and parallel edges A digraph 119866is called to be asymmetric if whenever (119909 119910) isin 119864(119866) then(119910 119909) notin 119864(119866) and to be symmetric if (119909 119910) isin 119864(119866) implies(119910 119909) isin 119864(119866) For a vertex 119909 of 119881(119866) the sets of out-neighbors and in-neighbors of 119909 are respectively defined as119873

+

119866(119909) = 119910 isin 119881(119866) (119909 119910) isin 119864(119866) and 119873minus

119866(119909) = 119909 isin

119881(119866) (119909 119910) isin 119864(119866) the out-degree and the in-degree of119909 are respectively defined as 119889+

119866(119909) = |119873

+

119866(119909)| and 119889minus

119866(119909) =

|119873+

119866(119909)| Denote themaximum and theminimum out-degree

(resp in-degree) of 119866 by Δ+(119866) and 120575+(119866) (resp Δminus(119866) and120575minus

(119866))The degree 119889119866(119909) of 119909 is defined as 119889+

119866(119909)+119889

minus

119866(119909) and

the maximum and the minimum degrees of119866 are denoted byΔ(119866) and 120575(119866) respectively that is Δ(119866) = max119889

119866(119909) 119909 isin

119881(119866) and 120575(119866) = min119889119866(119909) 119909 isin 119881(119866) Note that the

definitions here are different from the ones in the textbookon digraphs see for example Xu [1]

A subset 119878 of119881(119866) is called a dominating set if119881(119866) = 119878cup119873

+

119866(119878) where119873+

119866(119878) is the set of out-neighbors of 119878Then just

as for undirected graphs 120574(119866) is the minimum cardinalityof a dominating set and the bondage number 119887(119866) is thesmallest cardinality of a set 119861 of edges such that 120574(119866 minus 119861) gt120574(119866) if such a subset 119861 sube 119864(119866) exists Otherwise we put119887(119866) = infin

Some basic results for undirected graphs stated inSection 3 can be generalized to digraphs For exampleTheorem 18 is generalized by Carlson and Develin [42]Huang andXu [133] and Shan andKang [132] independentlyas follows

Theorem 180 Let 119866 be a digraph and (119909 119910) isin 119864(119866) Then119887(119866) ⩽ 119889

119866(119910) + 119889

minus

119866(119909) minus |119873

minus

119866(119909) cap 119873

minus

119866(119910)|

Corollary 181 For a digraph 119866 119887(119866) ⩽ 120575minus(119866) + Δ(119866)

Since 120575minus

(119866) ⩽ (12)Δ(119866) from Corollary 181Conjecture 53 is valid for digraphs

Corollary 182 For a digraph 119866 119887(119866) ⩽ (32)Δ(119866)

In the case of undirected graphs the bondage number of119866119899= 119870

119899times 119870

119899achieves this bound However it was shown

by Carlson and Develin in [42] that if we take the symmetricdigraph119866

119899 we haveΔ(119866

119899) = 4(119899minus1) 120574(119866

119899) = 119899 and 119887(119866

119899) ⩽

4(119899 minus 1) + 1 = Δ(119866119899) + 1 So this family of digraphs cannot

show that the bound in Corollary 182 is tightCorresponding to Conjecture 51 which is discredited

for undirected graphs and is valid for digraphs the sameconjecture for digraphs can be proposed as follows

Conjecture 183 (Carlson and Develin [42] 2006) 119887(119866) ⩽Δ(119866) + 1 for any digraph 119866

If Conjecture 183 is true then the following results showsthat this upper bound is tight since 119887(119870

119899times119870

119899) = Δ(119870

119899times119870

119899)+1

for a complete digraph 119870119899

In 2007 Shan and Kang [132] gave some tight upperbounds on the bondage numbers for some asymmetricdigraphs For example 119887(119879) ⩽ Δ(119879) for any asymmetricdirected tree 119879 119887(119866) ⩽ Δ(119866) for any asymmetric digraph 119866with order at least 4 and 120574(119866) ⩽ 2 For planar digraphs theyobtained the following results

Theorem 184 (Shan and Kang [132] 2007) Let 119866 be aasymmetric planar digraphThen 119887(119866) ⩽ Δ(119866)+2 and 119887(119866) ⩽Δ(119866) + 1 if Δ(119866) ⩾ 5 and 119889minus

119866(119909) ⩾ 3 for every vertex 119909 with

119889119866(119909) ⩾ 4

102 Results for Some Special Digraphs The exact values andbounds on 119887(119866) for some standard digraphs are determined

Theorem185 (Huang andXu [133] 2006) For a directed cycle119862119899and a directed path 119875

119899

119887 (119862119899) =

3 119894119891 119899 119894119904 119900119889119889

2 119894119891 119899 119894119904 119890V119890119899

119887 (119875119899) =

2 119894119891 119899 119894119904 119900119889119889

1 119894119891 119899 119894119904 119890V119890119899

(76)

For the de Bruijn digraph 119861(119889 119899) and the Kautz digraph119870(119889 119899)

119887 (119861 (119889 119899)) = 119889 119894119891 119899 119894119904 119900119889119889

119889 ⩽ 119887 (119861 (119889 119899)) ⩽ 2119889 119894119891 119899 119894119904 119890V119890119899

119887 (119870 (119889 119899)) = 119889 + 1

(77)

Like undirected graphs we can define the total domi-nation number and the total bondage number On the totalbondage numbers for some special digraphs the knownresults are as follows

International Journal of Combinatorics 29

Theorem 186 (Huang and Xu [134] 2007) For a directedcycle 119862

119899and a directed path 119875

119899 119887

119879(119875

119899) and 119887

119879(119862

119899) all do not

exist For a complete digraph 119870119899

119887119879(119870

119899) = infin 119894119891 119899 = 1 2

119887119879(119870

119899) = 3 119894119891 119899 = 3

119899 ⩽ 119887119879(119870

119899) ⩽ 2119899 minus 3 119894119891 119899 ⩾ 4

(78)

The extended de Bruijn digraph EB(119889 119899 1199021 119902

119901) and

the extended Kautz digraph EK(119889 119899 1199021 119902

119901) are intro-

duced by Shibata and Gonda [135] If 119901 = 1 then theyare the de Bruijn digraph B(119889 119899) and the Kautz digraphK(119889 119899) respectively Huang and Xu [134] determined theirtotal domination numbers In particular their total bondagenumbers for general cases are determined as follows

Theorem 187 (Huang andXu [134] 2007) If 119889 ⩾ 2 and 119902119894⩾ 2

for each 119894 = 1 2 119901 then

119887119879(EB(119889 119899 119902

1 119902

119901)) = 119889

119901

minus 1

119887119879(EK(119889 119899 119902

1 119902

119901)) = 119889

119901

(79)

In particular for the de Bruijn digraph B(119889 119899) and the Kautzdigraph K(119889 119899)

119887119879(B (119889 119899)) = 119889 minus 1 119887

119879(K (119889 119899)) = 119889 (80)

Zhang et al [136] determined the bondage number incomplete 119905-partite digraphs

Theorem 188 (Zhang et al [136] 2009) For a complete 119905-partite digraph 119870

11989911198992119899119905

where 1198991⩽ 119899

2⩽ sdot sdot sdot ⩽ 119899

119905

119887 (11987011989911198992119899119905

)

=

119898 119894119891 119899119898= 1 119899

119898+1⩾ 2 119891119900119903 119904119900119898119890

119898 (1 ⩽ 119898 lt 119905)

4119905 minus 3 119894119891 1198991= 119899

2= sdot sdot sdot = 119899

119905= 2

119905minus1

sum

119894=1

119899119894+ 2 (119905 minus 1) 119900119905ℎ119890119903119908119894119904119890

(81)

Since an undirected graph can be thought of as a sym-metric digraph any result for digraphs has an analogy forundirected graphs in general In view of this point studyingthe bondage number for digraphs is more significant thanfor undirected graphs Thus we should further study thebondage number of digraphs and try to generalize knownresults on the bondage number and related variants for undi-rected graphs to digraphs prove or disprove Conjecture 183In particular determine the exact values of 119887(B(119889 119899)) for aneven 119899 and 119887

119879(119870

119899) for 119899 ⩾ 4

11 Efficient Dominating Sets

A dominating set 119878 of a graph 119866 is called to be efficient if forevery vertex 119909 in119866 |119873

119866[119909]cap119878| = 1 if119866 is a undirected graph

or |119873minus

119866[119909] cap 119878| = 1 if 119866 is a directed graph The concept of an

efficient set was proposed by Bange et al [137] in 1988 In theliterature an efficient set is also called a perfect dominatingset (see Livingston and Stout [138] for an explanation)

From definition if 119878 is an efficient dominating set of agraph 119866 then 119878 is certainly an independent set and everyvertex not in 119878 is adjacent to exactly one vertex in 119878

It is also clear from definition that a dominating set 119878 isefficient if and only if N

119866[119878] = 119873

119866[119909] 119909 isin 119878 for the

undirected graph 119866 or N+

119866[119878] = 119873

+

119866[119909] 119909 isin 119878 for the

digraph119866 is a partition of119881(119866) where the induced subgraphby119873

119866[119909] or119873+

119866[119909] is a star or an out-star with the root 119909

The efficient domination has important applications inmany areas such as error-correcting codes and receivesmuch attention in the late years

The concept of efficient dominating sets is a measureof the efficiency of domination in graphs Unfortunately asshown in [137] not every graph has an efficient dominatingset andmoreover it is anNP-complete problem to determinewhether a given graph has an efficient dominating set Inaddition it was shown by Clark [139] in 1993 that for a widerange of 119901 almost every random undirected graph 119866 isin

G(120592 119901) has no efficient dominating sets This means thatundirected graphs possessing an efficient dominating set arerare However it is easy to show that every undirected graphhas an orientationwith an efficient dominating set (see Bangeet al [140])

In 1993 Barkauskas and Host [141] showed that deter-mining whether an arbitrary oriented graph has an efficientdominating set is NP-complete Even so the existence ofefficient dominating sets for some special classes of graphs hasbeen examined see for example Dejter and Serra [142] andLee [143] for Cayley graph Gu et al [144] formeshes and toriObradovic et al [145] Huang and Xu [36] Reji Kumar andMacGillivray [146] for circulant graphs Harary graphs andtori and Van Wieren et al [147] for cube-connected cycles

In this section we introduce results of the bondagenumber for some graphs with efficient dominating sets dueto Huang and Xu [36]

111 Results for General Graphs In this subsection we intro-duce some results on bondage numbers obtained by applyingefficient dominating sets We first state the two followinglemmas

Lemma 189 For any 119896-regular graph or digraph 119866 of order 119899120574(119866) ⩾ 119899(119896 + 1) with equality if and only if 119866 has an efficientdominating set In addition if 119866 has an efficient dominatingset then every efficient dominating set is certainly a 120574-set andvice versa

Let 119890 be an edge and 119878 a dominating set in 119866 We say that119890 supports 119878 if 119890 isin (119878 119878) where (119878 119878) = (119909 119910) isin 119864(119866) 119909 isin 119878 119910 isin 119878 Denote by 119904(119866) the minimum number of edgeswhich support all 120574-sets in 119866

Lemma 190 For any graph or digraph 119866 119887(119866) ⩾ 119904(119866) withequality if 119866 is regular and has an efficient dominating set

30 International Journal of Combinatorics

A graph 119866 is called to be vertex-transitive if its automor-phism group Aut(119866) acts transitively on its vertex-set 119881(119866)A vertex-transitive graph is regular Applying Lemmas 189and 190 Huang and Xu obtained some results on bondagenumbers for vertex-transitive graphs or digraphs

Theorem 191 (Huang and Xu [36] 2008) For any vertex-transitive graph or digraph 119866 of order 119899

119887 (119866) ⩾

lceil119899

2120574 (119866)rceil 119894119891 119866 119894119904 119906119899119889119894119903119890119888119905119890119889

lceil119899

120574 (119866)rceil 119894119891 119866 119894119904 119889119894119903119890119888119905119890119889

(82)

Theorem192 (Huang andXu [36] 2008) Let119866 be a 119896-regulargraph of order 119899 Then

119887(119866) ⩽ 119896 if119866 is undirected and 119899 ⩾ 120574(119866)(119896+1)minus119896+1119887(119866) ⩽ 119896+1+ℓ if119866 is directed and 119899 ⩾ 120574(119866)(119896+1)minusℓwith 0 ⩽ ℓ ⩽ 119896 minus 1

Next we introduce a better upper bound on 119887(119866) To thisaim we need the following concept which is a generalizationof the concept of the edge-covering of a graph 119866 For 1198811015840

sube

119881(119866) and 1198641015840 sube 119864(119866) we say that 1198641015840covers 1198811015840 and call 1198641015840an edge-covering for 1198811015840 if there exists an edge (119909 119910) isin 1198641015840 forany vertex 119909 isin 1198811015840 For 119910 isin 119881(119866) let 1205731015840[119910] be the minimumcardinality over all edge-coverings for119873minus

119866[119910]

Theorem 193 (Huang and Xu [36] 2008) If 119866 is a 119896-regulargraph with order 120574(119866)(119896 + 1) then 119887(119866) ⩽ 1205731015840[119910] for any 119910 isin119881(119866)

Theupper bound on 119887(119866) given inTheorem 193 is tight inview of 119887(119862

119899) for a cycle or a directed cycle 119862

119899(seeTheorems

1 and 185 resp)It is easy to see that for a 119896-regular graph 119866 lceil(119896 + 1)2rceil ⩽

1205731015840

[119910] ⩽ 119896 when 119866 is undirected and 1205731015840[119910] = 119896 + 1 when 119866 isdirected By this fact and Lemma 189 the following theoremis merely a simple combination of Theorems 191 and 193

Theorem 194 (Huang and Xu [36] 2008) Let 119866 be a vertex-transitive graph of degree 119896 If 119866 has an efficient dominatingset then

lceil119896 + 1

2rceil ⩽ 119887 (119866) ⩽ 119896 119894119891 119866 119894119904 119906119899119889119894119903119890119888119905119890119889

119887 (119866) = 119896 + 1 119894119891 119866 119894119904 119889119894119903119890119888119905119890119889

(83)

Theorem 195 (Huang and Xu [36] 2008) If 119866 is an undi-rected vertex-transitive cubic graph with order 4120574(119866) and girth119892(119866) ⩽ 5 then 119887(119866) = 2

The proof of Theorem 195 leads to a byproduct If 119892 =5 let (119906

1 119906

2 119906

3 119906

4 119906

5) be a cycle and let D

119894be the family

of efficient dominating sets containing 119906119894for each 119894 isin

1 2 3 4 5 ThenD119894capD

119895= 0 for 119894 = 1 2 5 Then 119866 has

at least 5119904 efficient dominating sets where 119904 = 119904(119866) But thereare only 4s (=ns120574) distinct efficient dominating sets in119866This

contradiction implies that an undirected vertex-transitivecubic graph with girth five has no efficient dominating setsBut a similar argument for 119892(119866) = 3 4 or 119892(119866) ⩾ 6 couldnot give any contradiction This is consistent with the resultthat CCC(119899) a vertex-transitive cubic graph with girth 119899 if3 ⩽ 119899 ⩽ 8 or girth 8 if 119899 ⩾ 9 has efficient dominating setsfor all 119899 ⩾ 3 except 119899 = 5 (see Theorem 199 in the followingsubsection)

112 Results for Cayley Graphs In this subsection we useTheorem 194 to determine the exact values or approximativevalues of bondage numbers for some special vertex-transitivegraphs by characterizing the existence of efficient dominatingsets in these graphs

Let Γ be a nontrivial finite group 119878 a nonempty subset ofΓ without the identity element of Γ A digraph 119866 is defined asfollows

119881 (119866) = Γ (119909 119910) isin 119864 (119866) lArrrArr 119909minus1

119910 isin 119878

for any 119909 119910 isin Γ(84)

is called a Cayley digraph of the group Γ with respect to119878 denoted by 119862

Γ(119878) If 119878minus1 = 119904minus1 119904 isin 119878 = 119878 then

119862Γ(119878) is symmetric and is called a Cayley undirected graph

a Cayley graph for short Cayley graphs or digraphs arecertainly vertex-transitive (see Xu [1])

A circulant graph 119866(119899 119878) of order 119899 is a Cayley graph119862119885119899

(119878) where 119885119899= 0 119899 minus 1 is the addition group of

order 119899 and 119878 is a nonempty subset of119885119899without the identity

element and hence is a vertex-transitive digraph of degree|119878| If 119878minus1 = 119878 then119866(119899 119878) is an undirected graph If 119878 = 1 119896where 2 ⩽ 119896 ⩽ 119899 minus 2 we write 119866(119899 1 119896) for 119866(119899 1 119896) or119866(119899 plusmn1 plusmn119896) and call it a double-loop circulant graph

Huang and Xu [36] showed that for directed 119866 =

119866(119899 1 119896) lceil1198993rceil ⩽ 120574(119866) ⩽ lceil1198992rceil and 119866 has an efficientdominating set if and only if 3 | 119899 and 119896 equiv 2 (mod 3) fordirected 119866 = 119866(119899 1 119896) and 119896 = 1198992 lceil1198995rceil ⩽ 120574(119866) ⩽ lceil1198993rceiland 119866 has an efficient dominating set if and only if 5 | 119899 and119896 equiv plusmn2 (mod 5) By Theorem 194 we obtain the bondagenumber of a double loop circulant graph if it has an efficientdominating set

Theorem 196 (Huang and Xu [36] 2008) Let 119866 be a double-loop circulant graph 119866(119899 1 119896) If 119866 is directed with 3 | 119899and 119896 equiv 2 (mod 3) or 119866 is undirected with 5 | 119899 and119896 equiv plusmn2 (mod 5) then 119887(119866) = 3

The 119898 times 119899 torus is the Cartesian product 119862119898times 119862

119899of

two cycles and is a Cayley graph 119862119885119898times119885119899

(119878) where 119878 =

(0 1) (1 0) for directed cycles and 119878 = (0 plusmn1) (plusmn1 0) forundirected cycles and hence is vertex-transitive Gu et al[144] showed that the undirected torus119862

119898times119862

119899has an efficient

dominating set if and only if both 119898 and 119899 are multiplesof 5 Huang and Xu [36] showed that the directed torus119862119898times 119862

119899has an efficient dominating set if and only if both

119898 and 119899 are multiples of 3 Moreover they found a necessarycondition for a dominating set containing the vertex (0 0)in 119862

119898times 119862

119899to be efficient and obtained the following

result

International Journal of Combinatorics 31

Theorem 197 (Huang and Xu [36] 2008) Let 119866 = 119862119898times 119862

119899

If 119866 is undirected and both119898 and 119899 are multiples of 5 or if 119866 isdirected and both119898 and 119899 are multiples of 3 then 119887(119866) = 3

The hypercube 119876119899

is the Cayley graph 119862Γ(119878)

where Γ = 1198852times sdot sdot sdot times 119885

2= (119885

2)119899 and 119878 =

100 sdot sdot sdot 0 010 sdot sdot sdot 0 00 sdot sdot sdot 01 Lee [143] showed that119876119899has an efficient dominating set if and only if 119899 = 2119898 minus 1

for a positive integer119898 ByTheorem 194 the following resultis obtained immediately

Theorem 198 (Huang and Xu [36] 2008) If 119899 = 2119898 minus 1 for apositive integer119898 then 2119898minus1

⩽ 119887(119876119899) ⩽ 2

119898

minus 1

Problem 10 Determine the exact value of 119887(119876119899) for 119899 = 2119898minus1

The 119899-dimensional cube-connected cycle denoted byCCC(119899) is constructed from the 119899-dimensional hypercube119876119899by replacing each vertex 119909 in 119876

119899with an undirected cycle

119862119899of length 119899 and linking the 119894th vertex of the 119862

119899to the 119894th

neighbor of 119909 It has been proved that CCC(119899) is a Cayleygraph and hence is a vertex-transitive graph with degree 3Van Wieren et al [147] proved that CCC(119899) has an efficientdominating set if and only if 119899 = 5 From Theorems 194 and195 the following result can be derived

Theorem 199 (Huang and Xu [36] 2008) Let 119866 = 119862119862119862(119899)be the 119899-dimensional cube-connected cycles with 119899 ⩾ 3 and119899 = 5 Then 120574(119866) = 1198992119899minus2 and 2 ⩽ 119887(119866) ⩽ 3 In addition119887(119862119862119862(3)) = 119887(119862119862119862(4)) = 2

Problem 11 Determine the exact value of 119887(CCC(119899)) for 119899 ⩾5

Acknowledgments

This work was supported by NNSF of China (Grants nos11071233 61272008) The author would like to express hisgratitude to the anonymous referees for their kind sugges-tions and comments on the original paper to ProfessorSheikholeslami and Professor Jafari Rad for sending thereferences [128] and [81] which resulted in this version

References

[1] J-M Xu Theory and Application of Graphs vol 10 of NetworkTheory and Applications Kluwer Academic Dordrecht TheNetherlands 2003

[2] S THedetniemi andR C Laskar ldquoBibliography on dominationin graphs and some basic definitions of domination parame-tersrdquo Discrete Mathematics vol 86 no 1ndash3 pp 257ndash277 1990

[3] TW Haynes S T Hedetniemi and P J Slater Fundamentals ofDomination in Graphs vol 208 ofMonographs and Textbooks inPure and Applied Mathematics Marcel Dekker New York NYUSA 1998

[4] TWHaynes S THedetniemi and P J Slater EdsDominationin Graphs vol 209 of Monographs and Textbooks in Pure andAppliedMathematicsMarcelDekker NewYorkNYUSA 1998

[5] M R Garey andD S JohnsonComputers and Intractability WH Freeman and Company San Francisco Calif USA 1979

[6] H B Walikar and B D Acharya ldquoDomination critical graphsrdquoNational Academy Science Letters vol 2 pp 70ndash72 1979

[7] D Bauer F Harary J Nieminen and C L Suffel ldquoDominationalteration sets in graphsrdquo Discrete Mathematics vol 47 no 2-3pp 153ndash161 1983

[8] J F Fink M S Jacobson L F Kinch and J Roberts ldquoThebondage number of a graphrdquo Discrete Mathematics vol 86 no1ndash3 pp 47ndash57 1990

[9] F-T Hu and J-M Xu ldquoThe bondage number of (119899 minus 3)-regulargraph G of order nrdquo Ars Combinatoria to appear

[10] U Teschner ldquoNew results about the bondage number of agraphrdquoDiscreteMathematics vol 171 no 1ndash3 pp 249ndash259 1997

[11] B L Hartnell and D F Rall ldquoA bound on the size of a graphwith given order and bondage numberrdquo Discrete Mathematicsvol 197-198 pp 409ndash413 1999 16th British CombinatorialConference (London 1997)

[12] B L Hartnell and D F Rall ldquoA characterization of trees inwhich no edge is essential to the domination numberrdquo ArsCombinatoria vol 33 pp 65ndash76 1992

[13] J Topp and P D Vestergaard ldquo120572119896- and 120574

119896-stable graphsrdquo

Discrete Mathematics vol 212 no 1-2 pp 149ndash160 2000[14] A Meir and J W Moon ldquoRelations between packing and

covering numbers of a treerdquo Pacific Journal of Mathematics vol61 no 1 pp 225ndash233 1975

[15] B L Hartnell L K Joslashrgensen P D Vestergaard and CA Whitehead ldquoEdge stability of the k-domination numberof treesrdquo Bulletin of the Institute of Combinatorics and ItsApplications vol 22 pp 31ndash40 1998

[16] F-T Hu and J-M Xu ldquoOn the complexity of the bondage andreinforcement problemsrdquo Journal of Complexity vol 28 no 2pp 192ndash201 2012

[17] B L Hartnell and D F Rall ldquoBounds on the bondage numberof a graphrdquo Discrete Mathematics vol 128 no 1ndash3 pp 173ndash1771994

[18] Y-L Wang ldquoOn the bondage number of a graphrdquo DiscreteMathematics vol 159 no 1ndash3 pp 291ndash294 1996

[19] G H Fricke TW Haynes S M Hedetniemi S T Hedetniemiand R C Laskar ldquoExcellent treesrdquo Bulletin of the Institute ofCombinatorics and Its Applications vol 34 pp 27ndash38 2002

[20] V Samodivkin ldquoThe bondage number of graphs good and badverticesrdquoDiscussiones Mathematicae GraphTheory vol 28 no3 pp 453ndash462 2008

[21] J E Dunbar T W Haynes U Teschner and L VolkmannldquoBondage insensitivity and reinforcementrdquo in Domination inGraphs vol 209 of Monographs and Textbooks in Pure andAppliedMathematics pp 471ndash489 Dekker NewYork NY USA1998

[22] H Liu and L Sun ldquoThe bondage and connectivity of a graphrdquoDiscrete Mathematics vol 263 no 1ndash3 pp 289ndash293 2003

[23] A-H Esfahanian and S L Hakimi ldquoOn computing a con-ditional edge-connectivity of a graphrdquo Information ProcessingLetters vol 27 no 4 pp 195ndash199 1988

[24] R C Brigham P Z Chinn and R D Dutton ldquoVertexdomination-critical graphsrdquoNetworks vol 18 no 3 pp 173ndash1791988

[25] V Samodivkin ldquoDomination with respect to nondegenerateproperties bondage numberrdquoThe Australasian Journal of Com-binatorics vol 45 pp 217ndash226 2009

[26] U Teschner ldquoA new upper bound for the bondage numberof graphs with small domination numberrdquo The AustralasianJournal of Combinatorics vol 12 pp 27ndash35 1995

32 International Journal of Combinatorics

[27] J Fulman D Hanson and G MacGillivray ldquoVertex dom-ination-critical graphsrdquoNetworks vol 25 no 2 pp 41ndash43 1995

[28] U Teschner ldquoA counterexample to a conjecture on the bondagenumber of a graphrdquo Discrete Mathematics vol 122 no 1ndash3 pp393ndash395 1993

[29] U Teschner ldquoThe bondage number of a graph G can be muchgreater than Δ(G)rdquo Ars Combinatoria vol 43 pp 81ndash87 1996

[30] L Volkmann ldquoOn graphs with equal domination and coveringnumbersrdquoDiscrete AppliedMathematics vol 51 no 1-2 pp 211ndash217 1994

[31] L A Sanchis ldquoMaximum number of edges in connected graphswith a given domination numberrdquoDiscreteMathematics vol 87no 1 pp 65ndash72 1991

[32] S Klavzar and N Seifter ldquoDominating Cartesian products ofcyclesrdquoDiscrete AppliedMathematics vol 59 no 2 pp 129ndash1361995

[33] H K Kim ldquoA note on Vizingrsquos conjecture about the bondagenumberrdquo Korean Journal of Mathematics vol 10 no 2 pp 39ndash42 2003

[34] M Y Sohn X Yuan and H S Jeong ldquoThe bondage number of1198623times119862

119899rdquo Journal of the KoreanMathematical Society vol 44 no

6 pp 1213ndash1231 2007[35] L Kang M Y Sohn and H K Kim ldquoBondage number of the

discrete torus 119862119899times 119862

4rdquo Discrete Mathematics vol 303 no 1ndash3

pp 80ndash86 2005[36] J Huang and J-M Xu ldquoThe bondage numbers and efficient

dominations of vertex-transitive graphsrdquoDiscrete Mathematicsvol 308 no 4 pp 571ndash582 2008

[37] C J Xiang X Yuan andM Y Sohn ldquoDomination and bondagenumber of119862

3times119862

119899rdquoArs Combinatoria vol 97 pp 299ndash310 2010

[38] M S Jacobson and L F Kinch ldquoOn the domination numberof products of graphs Irdquo Ars Combinatoria vol 18 pp 33ndash441984

[39] Y-Q Li ldquoA note on the bondage number of a graphrdquo ChineseQuarterly Journal of Mathematics vol 9 no 4 pp 1ndash3 1994

[40] F-T Hu and J-M Xu ldquoBondage number of mesh networksrdquoFrontiers ofMathematics inChina vol 7 no 5 pp 813ndash826 2012

[41] R Frucht and F Harary ldquoOn the corona of two graphsrdquoAequationes Mathematicae vol 4 pp 322ndash325 1970

[42] K Carlson and M Develin ldquoOn the bondage number of planarand directed graphsrdquoDiscreteMathematics vol 306 no 8-9 pp820ndash826 2006

[43] U Teschner ldquoOn the bondage number of block graphsrdquo ArsCombinatoria vol 46 pp 25ndash32 1997

[44] J Huang and J-M Xu ldquoNote on conjectures of bondagenumbers of planar graphsrdquo Applied Mathematical Sciences vol6 no 65ndash68 pp 3277ndash3287 2012

[45] L Kang and J Yuan ldquoBondage number of planar graphsrdquoDiscrete Mathematics vol 222 no 1ndash3 pp 191ndash198 2000

[46] M Fischermann D Rautenbach and L Volkmann ldquoRemarkson the bondage number of planar graphsrdquo Discrete Mathemat-ics vol 260 no 1ndash3 pp 57ndash67 2003

[47] J Hou G Liu and J Wu ldquoBondage number of planar graphswithout small cyclesrdquoUtilitasMathematica vol 84 pp 189ndash1992011

[48] J Huang and J-M Xu ldquoThe bondage numbers of graphs withsmall crossing numbersrdquo Discrete Mathematics vol 307 no 15pp 1881ndash1897 2007

[49] Q Ma S Zhang and J Wang ldquoBondage number of 1-planargraphrdquo Applied Mathematics vol 1 no 2 pp 101ndash103 2010

[50] J Hou and G Liu ldquoA bound on the bondage number of toroidalgraphsrdquoDiscreteMathematics Algorithms and Applications vol4 no 3 Article ID 1250046 5 pages 2012

[51] Y-C Cao J-M Xu and X-R Xu ldquoOn bondage number oftoroidal graphsrdquo Journal of University of Science and Technologyof China vol 39 no 3 pp 225ndash228 2009

[52] Y-C Cao J Huang and J-M Xu ldquoThe bondage number ofgraphs with crossing number less than fourrdquo Ars Combinatoriato appear

[53] H K Kim ldquoOn the bondage numbers of some graphsrdquo KoreanJournal of Mathematics vol 11 no 2 pp 11ndash15 2004

[54] A Gagarin and V Zverovich ldquoUpper bounds for the bondagenumber of graphs on topological surfacesrdquo Discrete Mathemat-ics 2012

[55] J Huang ldquoAn improved upper bound for the bondage numberof graphs on surfacesrdquoDiscrete Mathematics vol 312 no 18 pp2776ndash2781 2012

[56] A Gagarin and V Zverovich ldquoThe bondage number of graphson topological surfaces and Teschnerrsquos conjecturerdquo DiscreteMathematics vol 313 no 6 pp 796ndash808 2013

[57] V Samodivkin ldquoNote on the bondage number of graphs ontopological surfacesrdquo httparxivorgabs12086203

[58] E J Cockayne R M Dawes and S T Hedetniemi ldquoTotaldomination in graphsrdquoNetworks vol 10 no 3 pp 211ndash219 1980

[59] R Laskar J Pfaff S M Hedetniemi and S T HedetniemildquoOn the algorithmic complexity of total dominationrdquo Society forIndustrial and Applied Mathematics vol 5 no 3 pp 420ndash4251984

[60] J Pfaff R C Laskar and S T Hedetniemi ldquoNP-completeness oftotal and connected domination and irredundance for bipartitegraphsrdquo Tech Rep 428 Department of Mathematical SciencesClemson University 1983

[61] M A Henning ldquoA survey of selected recent results on totaldomination in graphsrdquoDiscrete Mathematics vol 309 no 1 pp32ndash63 2009

[62] V R Kulli and D K Patwari ldquoThe total bondage number ofa graphrdquo in Advances in Graph Theory pp 227ndash235 VishwaGulbarga India 1991

[63] F-T Hu Y Lu and J-M Xu ldquoThe total bondage number ofgrid graphsrdquo Discrete Applied Mathematics vol 160 no 16-17pp 2408ndash2418 2012

[64] J Cao M Shi and B Wu ldquoResearch on the total bondagenumber of a spectial networkrdquo in Proceedings of the 3rdInternational Joint Conference on Computational Sciences andOptimization (CSO rsquo10) pp 456ndash458 May 2010

[65] N Sridharan MD Elias and V S A Subramanian ldquoTotalbondage number of a graphrdquo AKCE International Journal ofGraphs and Combinatorics vol 4 no 2 pp 203ndash209 2007

[66] N Jafari Rad and J Raczek ldquoSome progress on total bondage ingraphsrdquo Graphs and Combinatorics 2011

[67] T W Haynes and P J Slater ldquoPaired-domination and thepaired-domatic numberrdquoCongressusNumerantium vol 109 pp65ndash72 1995

[68] T W Haynes and P J Slater ldquoPaired-domination in graphsrdquoNetworks vol 32 no 3 pp 199ndash206 1998

[69] X Hou ldquoA characterization of 2120574 120574119875-treesrdquoDiscrete Mathemat-

ics vol 308 no 15 pp 3420ndash3426 2008[70] K E Proffitt T W Haynes and P J Slater ldquoPaired-domination

in grid graphsrdquo Congressus Numerantium vol 150 pp 161ndash1722001

International Journal of Combinatorics 33

[71] H Qiao L KangM Cardei andD-Z Du ldquoPaired-dominationof treesrdquo Journal of Global Optimization vol 25 no 1 pp 43ndash542003

[72] E Shan L Kang and M A Henning ldquoA characterizationof trees with equal total domination and paired-dominationnumbersrdquo The Australasian Journal of Combinatorics vol 30pp 31ndash39 2004

[73] J Raczek ldquoPaired bondage in treesrdquo Discrete Mathematics vol308 no 23 pp 5570ndash5575 2008

[74] J A Telle and A Proskurowski ldquoAlgorithms for vertex parti-tioning problems on partial k-treesrdquo SIAM Journal on DiscreteMathematics vol 10 no 4 pp 529ndash550 1997

[75] G S Domke J H Hattingh M A Henning and L R MarkusldquoRestrained domination in treesrdquoDiscreteMathematics vol 211no 1ndash3 pp 1ndash9 2000

[76] G S Domke J H Hattingh M A Henning and L R MarkusldquoRestrained domination in graphs with minimum degree twordquoJournal of Combinatorial Mathematics and Combinatorial Com-puting vol 35 pp 239ndash254 2000

[77] G S Domke J H Hattingh S T Hedetniemi R C Laskarand L R Markus ldquoRestrained domination in graphsrdquo DiscreteMathematics vol 203 no 1ndash3 pp 61ndash69 1999

[78] J H Hattingh and E J Joubert ldquoAn upper bound for therestrained domination number of a graph with minimumdegree at least two in terms of order and minimum degreerdquoDiscrete Applied Mathematics vol 157 no 13 pp 2846ndash28582009

[79] M A Henning ldquoGraphs with large restrained dominationnumberrdquo Discrete Mathematics vol 197-198 pp 415ndash429 199916th British Combinatorial Conference (London 1997)

[80] J H Hattingh and A R Plummer ldquoRestrained bondage ingraphsrdquo Discrete Mathematics vol 308 no 23 pp 5446ndash54532008

[81] N Jafari Rad R Hasni J Raczek and L Volkmann ldquoTotalrestrained bondage in graphsrdquoActaMathematica Sinica vol 29no 6 pp 1033ndash1042 2013

[82] J Cyman and J Raczek ldquoOn the total restrained dominationnumber of a graphrdquoThe Australasian Journal of Combinatoricsvol 36 pp 91ndash100 2006

[83] M A Henning ldquoGraphs with large total domination numberrdquoJournal of Graph Theory vol 35 no 1 pp 21ndash45 2000

[84] D-X Ma X-G Chen and L Sun ldquoOn total restraineddomination in graphsrdquoCzechoslovakMathematical Journal vol55 no 1 pp 165ndash173 2005

[85] B Zelinka ldquoRemarks on restrained domination and totalrestrained domination in graphsrdquo Czechoslovak MathematicalJournal vol 55 no 2 pp 393ndash396 2005

[86] W Goddard and M A Henning ldquoIndependent domination ingraphs a survey and recent resultsrdquo Discrete Mathematics vol313 no 7 pp 839ndash854 2013

[87] J H Zhang H L Liu and L Sun ldquoIndependence bondagenumber and reinforcement number of some graphsrdquo Transac-tions of Beijing Institute of Technology vol 23 no 2 pp 140ndash1422003

[88] K D Dharmalingam Studies in Graph Theorymdashequitabledomination and bottleneck domination [PhD thesis] MaduraiKamaraj University Madurai India 2006

[89] GDeepakND Soner andAAlwardi ldquoThe equitable bondagenumber of a graphrdquo Research Journal of Pure Algebra vol 1 no9 pp 209ndash212 2011

[90] E Sampathkumar and H B Walikar ldquoThe connected domina-tion number of a graphrdquo Journal of Mathematical and PhysicalSciences vol 13 no 6 pp 607ndash613 1979

[91] K White M Farber and W Pulleyblank ldquoSteiner trees con-nected domination and strongly chordal graphsrdquoNetworks vol15 no 1 pp 109ndash124 1985

[92] M Cadei M X Cheng X Cheng and D-Z Du ldquoConnecteddomination in Ad Hoc wireless networksrdquo in Proceedings ofthe 6th International Conference on Computer Science andInformatics (CSampI rsquo02) 2002

[93] J F Fink and M S Jacobson ldquon-domination in graphsrdquo inGraph Theory with Applications to Algorithms and ComputerScience Wiley-Interscience Publications pp 283ndash300 WileyNew York NY USA 1985

[94] M Chellali O Favaron A Hansberg and L Volkmann ldquok-domination and k-independence in graphs a surveyrdquo Graphsand Combinatorics vol 28 no 1 pp 1ndash55 2012

[95] Y Lu X Hou J-M Xu andN Li ldquoTrees with uniqueminimump-dominating setsrdquo Utilitas Mathematica vol 86 pp 193ndash2052011

[96] Y Lu and J-M Xu ldquoThe p-bondage number of treesrdquo Graphsand Combinatorics vol 27 no 1 pp 129ndash141 2011

[97] Y Lu and J-M Xu ldquoThe 2-domination and 2-bondage numbersof grid graphs A manuscriptrdquo httparxivorgabs12044514

[98] M Krzywkowski ldquoNon-isolating 2-bondage in graphsrdquo TheJournal of the Mathematical Society of Japan vol 64 no 4 2012

[99] M A Henning O R Oellermann and H C Swart ldquoBoundson distance domination parametersrdquo Journal of CombinatoricsInformation amp System Sciences vol 16 no 1 pp 11ndash18 1991

[100] F Tian and J-M Xu ldquoBounds for distance domination numbersof graphsrdquo Journal of University of Science and Technology ofChina vol 34 no 5 pp 529ndash534 2004

[101] F Tian and J-M Xu ldquoDistance domination-critical graphsrdquoApplied Mathematics Letters vol 21 no 4 pp 416ndash420 2008

[102] F Tian and J-M Xu ldquoA note on distance domination numbersof graphsrdquo The Australasian Journal of Combinatorics vol 43pp 181ndash190 2009

[103] F Tian and J-M Xu ldquoAverage distances and distance domina-tion numbersrdquo Discrete Applied Mathematics vol 157 no 5 pp1113ndash1127 2009

[104] Z-L Liu F Tian and J-M Xu ldquoProbabilistic analysis of upperbounds for 2-connected distance k-dominating sets in graphsrdquoTheoretical Computer Science vol 410 no 38ndash40 pp 3804ndash3813 2009

[105] M A Henning O R Oellermann and H C Swart ldquoDistancedomination critical graphsrdquo Journal of Combinatorial Mathe-matics and Combinatorial Computing vol 44 pp 33ndash45 2003

[106] T J Bean M A Henning and H C Swart ldquoOn the integrityof distance domination in graphsrdquo The Australasian Journal ofCombinatorics vol 10 pp 29ndash43 1994

[107] M Fischermann and L Volkmann ldquoA remark on a conjecturefor the (kp)-domination numberrdquoUtilitasMathematica vol 67pp 223ndash227 2005

[108] T Korneffel D Meierling and L Volkmann ldquoA remark on the(22)-domination numberrdquo Discussiones Mathematicae GraphTheory vol 28 no 2 pp 361ndash366 2008

[109] Y Lu X Hou and J-M Xu ldquoOn the (22)-domination numberof treesrdquo Discussiones Mathematicae Graph Theory vol 30 no2 pp 185ndash199 2010

34 International Journal of Combinatorics

[110] H Li and J-M Xu ldquo(dm)-domination numbers of m-connected graphsrdquo Rapports de Recherche 1130 LRI UniversiteParis-Sud 1997

[111] S M Hedetniemi S T Hedetniemi and T V Wimer ldquoLineartime resource allocation algorithms for treesrdquo Tech Rep URI-014 Department of Mathematical Sciences Clemson Univer-sity 1987

[112] M Farber ldquoDomination independent domination and dualityin strongly chordal graphsrdquo Discrete Applied Mathematics vol7 no 2 pp 115ndash130 1984

[113] G S Domke and R C Laskar ldquoThe bondage and reinforcementnumbers of 120574

119891for some graphsrdquoDiscrete Mathematics vol 167-

168 pp 249ndash259 1997[114] G S Domke S T Hedetniemi and R C Laskar ldquoFractional

packings coverings and irredundance in graphsrdquo vol 66 pp227ndash238 1988

[115] E J Cockayne P A Dreyer Jr S M Hedetniemi andS T Hedetniemi ldquoRoman domination in graphsrdquo DiscreteMathematics vol 278 no 1ndash3 pp 11ndash22 2004

[116] I Stewart ldquoDefend the roman empirerdquo Scientific American vol281 pp 136ndash139 1999

[117] C S ReVelle and K E Rosing ldquoDefendens imperiumromanum a classical problem in military strategyrdquoThe Ameri-can Mathematical Monthly vol 107 no 7 pp 585ndash594 2000

[118] A Bahremandpour F-T Hu S M Sheikholeslami and J-MXu ldquoOn the Roman bondage number of a graphrdquo DiscreteMathematics Algorithms and Applications vol 5 no 1 ArticleID 1350001 15 pages 2013

[119] N Jafari Rad and L Volkmann ldquoRoman bondage in graphsrdquoDiscussiones Mathematicae Graph Theory vol 31 no 4 pp763ndash773 2011

[120] K Ebadi and L PushpaLatha ldquoRoman bondage and Romanreinforcement numbers of a graphrdquo International Journal ofContemporary Mathematical Sciences vol 5 pp 1487ndash14972010

[121] N Dehgardi S M Sheikholeslami and L Volkman ldquoOn theRoman k-bondage number of a graphrdquo AKCE InternationalJournal of Graphs and Combinatorics vol 8 pp 169ndash180 2011

[122] S Akbari and S Qajar ldquoA note on Roman bondage number ofgraphsrdquo Ars Combinatoria to Appear

[123] F-T Hu and J-M Xu ldquoRoman bondage numbers of somegraphsrdquo httparxivorgabs11093933

[124] N Jafari Rad and L Volkmann ldquoOn the Roman bondagenumber of planar graphsrdquo Graphs and Combinatorics vol 27no 4 pp 531ndash538 2011

[125] S Akbari M Khatirinejad and S Qajar ldquoA note on the romanbondage number of planar graphsrdquo Graphs and Combinatorics2012

[126] B Bresar M A Henning and D F Rall ldquoRainbow dominationin graphsrdquo Taiwanese Journal of Mathematics vol 12 no 1 pp213ndash225 2008

[127] B L Hartnell and D F Rall ldquoOn dominating the Cartesianproduct of a graph and 120572krdquo Discussiones Mathematicae GraphTheory vol 24 no 3 pp 389ndash402 2004

[128] N Dehgardi S M Sheikholeslami and L Volkman ldquoThe k-rainbow bondage number of a graphrdquo to appear in DiscreteApplied Mathematics

[129] J von Neumann and O Morgenstern Theory of Games andEconomic Behavior Princeton University Press Princeton NJUSA 1944

[130] C BergeTheorıe des Graphes et ses Applications 1958[131] O Ore Theory of Graphs vol 38 American Mathematical

Society Colloquium Publications 1962[132] E Shan and L Kang ldquoBondage number in oriented graphsrdquoArs

Combinatoria vol 84 pp 319ndash331 2007[133] J Huang and J-M Xu ldquoThe bondage numbers of extended

de Bruijn and Kautz digraphsrdquo Computers amp Mathematics withApplications vol 51 no 6-7 pp 1137ndash1147 2006

[134] J Huang and J-M Xu ldquoThe total domination and total bondagenumbers of extended de Bruijn and Kautz digraphsrdquoComputersamp Mathematics with Applications vol 53 no 8 pp 1206ndash12132007

[135] Y Shibata and Y Gonda ldquoExtension of de Bruijn graph andKautz graphrdquo Computers amp Mathematics with Applications vol30 no 9 pp 51ndash61 1995

[136] X Zhang J Liu and JMeng ldquoThebondage number in completet-partite digraphsrdquo Information Processing Letters vol 109 no17 pp 997ndash1000 2009

[137] D W Bange A E Barkauskas and P J Slater ldquoEfficient dom-inating sets in graphsrdquo in Applications of Discrete Mathematicspp 189ndash199 SIAM Philadelphia Pa USA 1988

[138] M Livingston and Q F Stout ldquoPerfect dominating setsrdquoCongressus Numerantium vol 79 pp 187ndash203 1990

[139] L Clark ldquoPerfect domination in random graphsrdquo Journal ofCombinatorial Mathematics and Combinatorial Computing vol14 pp 173ndash182 1993

[140] D W Bange A E Barkauskas L H Host and L H ClarkldquoEfficient domination of the orientations of a graphrdquo DiscreteMathematics vol 178 no 1ndash3 pp 1ndash14 1998

[141] A E Barkauskas and L H Host ldquoFinding efficient dominatingsets in oriented graphsrdquo Congressus Numerantium vol 98 pp27ndash32 1993

[142] I J Dejter and O Serra ldquoEfficient dominating sets in CayleygraphsrdquoDiscrete AppliedMathematics vol 129 no 2-3 pp 319ndash328 2003

[143] J Lee ldquoIndependent perfect domination sets in Cayley graphsrdquoJournal of Graph Theory vol 37 no 4 pp 213ndash219 2001

[144] WGu X Jia and J Shen ldquoIndependent perfect domination setsin meshes tori and treesrdquo Unpublished

[145] N Obradovic J Peters and G Ruzic ldquoEfficient domination incirculant graphswith two chord lengthsrdquo Information ProcessingLetters vol 102 no 6 pp 253ndash258 2007

[146] K Reji Kumar and G MacGillivray ldquoEfficient domination incirculant graphsrdquo Discrete Mathematics vol 313 no 6 pp 767ndash771 2013

[147] D Van Wieren M Livingston and Q F Stout ldquoPerfectdominating sets on cube-connected cyclesrdquo Congressus Numer-antium vol 97 pp 51ndash70 1993

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

International Journal of Combinatorics 11

5 Results on Graph Operations

Generally speaking it is quite difficult to determine the exactvalue of the bondage number for a given graph since itstrongly depends on the dominating number of the graphThus determining bondage numbers for some special graphsis interesting if the dominating numbers of those graphs areknown or can be easily determined In this section we willintroduce some known results on bondage numbers for somespecial classes of graphs obtained by some graph operations

51 Cartesian Product Graphs Let 1198661= (119881

1 119864

1) and 119866

2=

(1198812 119864

2) be two graphs The Cartesian product of 119866

1and 119866

2

is an undirected graph denoted by 1198661times 119866

2 where 119881(119866

1times

1198662) = 119881

1times 119881

2 two distinct vertices 119909

11199092and 119910

11199102 where

1199091 119910

1isin 119881(119866

1) and 119909

2 119910

2isin 119881(119866

2) are linked by an edge in

1198661times 119866

2if and only if either 119909

1= 119910

1and 119909

21199102isin 119864(119866

2) or

1199092= 119910

2and 119909

11199101isin 119864(119866

1) The Cartesian product is a very

effective method for constructing a larger graph from severalspecified small graphs

Theorem 60 (Dunbar et al [21] 1998) For 119899 ⩾ 3

119887 (119862119899times 119870

2) =

2 119894119891 119899 equiv 0 119900119903 1 (mod 4) 3 119894119891 119899 equiv 3 (mod 4) 4 119894119891 119899 equiv 2 (mod 4)

(16)

For the Cartesian product 119862119899times 119862

119898of two cycles 119862

119899and

119862119898 where 119899 ⩾ 4 and 119898 ⩾ 3 Klavzar and Seifter [32]

determined 120574(119862119899times 119862

119898) for some 119899 and 119898 For example

120574(119862119899times119862

4) = 119899 for 119899 ⩾ 3 and 120574(119862

119899times119862

3) = 119899minuslfloor1198994rfloor for 119899 ⩾ 4

Kim [33] determined 119887(1198623times 119862

3) = 6 and 119887(119862

4times 119862

4) = 4 For

a general 119899 ⩾ 4 the exact values of the bondage numbers of119862119899times 119862

3and 119862

119899times 119862

4are determined as follows

Theorem 61 (Sohn et al [34] 2007) For 119899 ⩾ 4

119887 (119862119899times 119862

3) =

2 119894119891 119899 equiv 0 (mod 4) 4 119894119891 119899 equiv 1 119900119903 2 (mod 4) 5 119894119891 119899 equiv 3 (mod 4)

(17)

Theorem62 (Kang et al [35] 2005) 119887(119862119899times119862

4) = 4 for 119899 ⩾ 4

For larger 119898 and 119899 Huang and Xu [36] obtained thefollowing result see Theorem 197 for more details

Theorem 63 (Huang and Xu [36] 2008) 119887(1198625119894times119862

5119895) = 3 for

any positive integers 119894 and 119895

Xiang et al [37] determined that for 119899 ⩾ 5

119887 (119862119899times 119862

5)

= 3 if 119899 equiv 0 or 1 (mod 5) = 4 if 119899 equiv 2 or 4 (mod 5) ⩽ 7 if 119899 equiv 3 (mod 5)

(18)

For the Cartesian product 119875119899times119875

119898of two paths 119875

119899and 119875

119898

Jacobson and Kinch [38] determined 120574(119875119899times119875

2) = lceil(119899+1)2rceil

120574(119875119899times 119875

3) = 119899 minus lfloor(119899 minus 1)4rfloor and 120574(119875

119899times 119875

4) = 119899 + 1 if 119899 =

1 2 3 5 6 9 and 119899 otherwiseThe bondage number of119875119899times119875

119898

for 119899 ⩾ 4 and 2 ⩽ 119898 ⩽ 4 is determined as follows (Li [39] alsodetermined 119887(119875

119899times 119875

2))

Theorem 64 (Hu and Xu [40] 2012) For 119899 ⩾ 4

119887 (119875119899times 119875

2) =

1 119894119891 119899 equiv 1 (mod 2)2 119894119891 119899 equiv 0 (mod 2)

119887 (119875119899times 119875

3) =

1 119894119891 119899 equiv 1 119900119903 2 (mod 4)2 119894119891 119899 equiv 0 119900119903 3 (mod 4)

119887 (119875119899times 119875

4) = 1 119891119900119903 119899 ⩾ 14

(19)

From the proof of Theorem 64 we find that if 119875119899=

0 1 119899 minus 1 and 119875119898= 0 1 119898 minus 1 then the removal

of the vertex (0 0) in 119875119899times119875

119898does not change the domination

number If119898 increases the effect of (0 0) for the dominationnumber will be smaller and smaller from the probabilityTherefore we expect it is possible that 120574(119875

119899times 119875

119898minus (0 0)) =

120574(119875119899times 119875

119898) for119898 ⩾ 5 and give the following conjecture

Conjecture 65 119887(119875119899times 119875

119898) ⩽ 2 for119898 ⩾ 5 and 119899 ⩾ 4

52 Block Graphs and Cactus Graphs In this subsection weintroduce some results for corona graphs block graphs andcactus graphs

The corona1198661∘ 119866

2 proposed by Frucht and Harary [41]

is the graph formed from a copy of 1198661and 120592(119866

1) copies of

1198662by joining the 119894th vertex of 119866

1to the 119894th copy of 119866

2 In

particular we are concernedwith the corona119867∘1198701 the graph

formed by adding a new vertex V119894 and a new edge 119906

119894V119894for

every vertex 119906119894in119867

Theorem 66 (Carlson and Develin [42] 2006) 119887(119867 ∘ 1198701) =

120575(119867) + 1

This is a very important result which is often used toconstruct a graph with required bondage number In otherwords this result implies that for any given positive integer 119887there is a graph 119866 such that 119887(119866) = 119887

A block graph is a graph whose blocks are completegraphs Each block in a cactus graph is either a cycle or a119870

2

If each block of a graph is either a complete graph or a cyclethen we call this graph a block-cactus graph

Theorem67 (Teschner [43] 1997) 119887(119866) le Δ(119866) for any blockgraph 119866

Teschner [43] characterized all block graphs with 119887(119866) =Δ(119866) In the same paper Teschner found that 120574-criticalgraphs are instrumental in determining bounds for thebondage number of cactus and block graphs and obtained thefollowing result

Theorem 68 (Teschner [43] 1997) 119887(119866) ⩽ 3 for anynontrivial cactus graph 119866

This bound can be achieved by 119867 ∘ 1198701where 119867 is

a nontrivial cactus graph without vertices of degree one

12 International Journal of Combinatorics

by Theorem 66 In 1998 Dunbar et al [21] proposed thefollowing problem

Problem 2 Characterize all cactus graphs with bondagenumber 3

Some upper bounds for block-cactus graphs were alsoobtained

Theorem 69 (Dunbar et al [21] 1998) Let 119866 be a connectedblock-cactus graph with at least two blocksThen 119887(119866) ⩽ Δ(119866)

Theorem 70 (Dunbar et al [21] 1998) Let 119866 be a connectedblock-cactus graph which is neither a cactus graph nor a blockgraph Then 119887(119866) ⩽ Δ(119866) minus 1

53 Edge-Deletion and Edge-Contraction In order to obtainfurther results of the bondage number we may consider howthe bondage number changes under some operations of agraph 119866 which remain the domination number unchangedor preserve some property of 119866 such as planarity Twosimple operations satisfying these requirements are the edge-deletion and the edge-contraction

Theorem 71 (Huang and Xu [44] 2012) Let 119866 be any graphand 119890 isin 119864(119866) Then 119887(119866 minus 119890) ⩾ 119887(119866) minus 1 Moreover 119887(119866 minus 119890) ⩽119887(119866) if 120574(119866 minus 119890) = 120574(119866)

FromTheorem 1 119887(119862119899minus 119890) = 119887(119875

119899) = 119887(119862

119899) minus 1 for any 119890

in119862119899 which shows that the lower bound on 119887(119866minus119890) is sharp

The upper bound can reach for any graph 119866 with 119887(119866) ⩾ 2Next we consider the effect of the edge-contraction on

the bondage number Given a graph 119866 the contraction of 119866by the edge 119890 = 119909119910 denoted by 119866119909119910 is the graph obtainedfrom 119866 by contracting two vertices 119909 and 119910 to a new vertexand then deleting all multiedges It is easy to observe that forany graph 119866 120574(119866) minus 1 ⩽ 120574(119866119909119910) ⩽ 120574(119866) for any edge 119909119910 of119866

Theorem 72 (Huang and Xu [44] 2012) Let 119866 be any graphand 119909119910 be any edge in119866 If119873

119866(119909)cap119873

119866(119910) = 0 and 120574(119866119909119910) =

120574(119866) then 119887(119866119909119910) ⩾ 119887(119866)

We can use examples 119862119899and 119870

119899to show that the

conditions ofTheorem 72 are necessary and the lower boundinTheorem 72 is tight Clearly 120574(119862

119899) = lceil1198993rceil and 120574(119870

119899) = 1

for any edge 119909119910 119862119899119909119910 = 119862

119899minus1and 119870

119899119909119910 = 119870

119899minus1 By

Theorem 1 if 119899 equiv 1 (mod 3) then 120574(119862119899119909119910) lt 120574(119866) and

119887(119862119899119909119910) = 2 lt 3 = 119887(119862

119899) Thus the result in Theorem 72

is generally invalid without the hypothesis 120574(119866119909119910) = 120574(119866)Furthermore the condition 119873

119866(119909) cap 119873

119866(119910) = 0 cannot be

omitted even if 120574(119866119909119910) = 120574(119866) since for odd 119899 119887(119870119899119909119910) =

lceil(119899 minus 1)2rceil lt lceil1198992rceil = 119887(119870119899) by Theorem 1

On the other hand 119887(119862119899119909119910) = 119887(119862

119899) = 2 if 119899 equiv 0 (mod

3) and 119887(119870119899119909119910) = 119887(119870

119899) if 119899 is even which shows that the

equality in 119887(119866119909119910) ⩾ 119887(119866) may hold Thus the bound inTheorem 72 is tight However provided all the conditions119887(119866119909119910) can be arbitrarily larger than 119887(119866) Given a graph119867let119866 be the graph formed from119867∘119870

1by adding a new vertex

119909 and joining it to an vertex 119910 of degree one in119867 ∘ 1198701 Then

119866119909119910 = 119867 ∘ 1198701 120574(119866) = 120574(119866119909119910) and 119873

119866(119909) cap 119873

119866(119910) = 0

But 119887(119866) = 1 since 120574(119866 minus 119909119910) = 120574(119866119909119910) + 1 and 119887(119866119909119910) =120575(119867) + 1 byTheorem 66The gap between 119887(119866) and 119887(119866119909119910)is 120575(119867)

6 Results on Planar Graphs

From Section 2 we have seen that the bondage numberfor a tree has been completely solved Moreover a lineartime algorithm to compute the bondage number of a tree isdesigned by Hartnell et al [15] It is quite natural to considerthe bondage number for a planar graph In this section westate some results and problems on the bondage number fora planar graph

61 Conjecture on Bondage Number As mentioned inSection 3 the bondage number can be much larger than themaximum degree But for a planar graph 119866 the bondagenumber 119887(119866) cannot exceed Δ(119866) too much It is clear that119887(119866) ⩽ Δ(119866) + 4 by Corollary 15 since 120575(119866) ⩽ 5 for anyplanar graph119866 In 1998Dunbar et al [21] posed the followingconjecture

Conjecture 73 If 119866 is a planar graph then 119887(119866) ⩽ Δ(119866) + 1

Because of its attraction it immediately became the focusof attention when this conjecture is proposed In fact themain aim concerning the research of the bondage number fora planar graph is focused on this conjecture

It has been mentioned in Theorems 1 and 60 that119887(119862

3119896+1) = 3 = Δ + 1 and 119887(119862

4119896+2times 119870

2) = 4 = Δ + 1 It

is known that 119887(1198706minus 119872) = 5 = Δ + 1 where119872 is a perfect

matching of the complete graph1198706These examples show that

if Conjecture 73 is true then the upper bound is sharp for2 ⩽ Δ ⩽ 4

Here we note that it is sufficient to prove this conjecturefor connected planar graphs since the bondage number of adisconnected graph is simply the minimum of the bondagenumbers of its components

The first paper attacking this conjecture is due to Kangand Yuan [45] which confirmed the conjecture for everyconnected planar graph 119866 with Δ ⩾ 7 The proofs are mainlybased onTheorems 16 and 18

Theorem 74 (Kang and Yuan [45] 2000) If 119866 is a connectedplanar graph then 119887(119866) ⩽ min8 Δ(119866) + 2

Obviously in view of Theorem 74 Conjecture 73 is truefor any connected planar graph with Δ ⩾ 7

62 Bounds Implied by Degree Conditions As we have seenfrom Theorem 74 to attack Conjecture 73 we only need toconsider connected planar graphs with maximum Δ ⩽ 6Thus studying the bondage number of planar graphs bydegree conditions is of significanceThe first result on boundsimplied by degree conditions is obtained by Kang and Yuan[45]

International Journal of Combinatorics 13

Theorem 75 (Kang and Yuan [45] 2000) If 119866 is a connectedplanar graph without vertices of degree 5 then 119887(119866) ⩽ 7

Fischermann et al [46] generalized Theorem 75 as fol-lows

Theorem 76 (Fischermann et al [46] 2003) Let 119866 be aconnected planar graph and 119883 the set of vertices of degree 5which have distance at least 3 to vertices of degrees 1 2 and3 If all vertices in 119883 not adjacent with vertices of degree 4are independent and not adjacent to vertices of degree 6 then119887(119866) ⩽ 7

Clearly if119866 has no vertices of degree 5 then119883 = 0 ThusTheorem 76 yields Theorem 75

Use 119899119894to denote the number of vertices of degree 119894 in 119866

for each 119894 = 1 2 Δ(119866) Using Theorem 18 Fischermannet al obtained the following two theorems

Theorem 77 (Fischermann et al [46] 2003) For any con-nected planar graph 119866

(1) 119887(119866) ⩽ 7 if 1198995lt 2119899

2+ 3119899

3+ 2119899

4+ 12

(2) 119887(119866) ⩽ 6 if 119866 contains no vertices of degrees 4 and 5

Theorem 78 (Fischermann et al [46] 2003) For any con-nected planar graph 119866 119887(119866) ⩽ Δ(119866) + 1 if

(1) Δ(119866) = 6 and every edge 119890 = 119909119910 with 119889119866(119909) = 5 and

119889119866(119910) = 6 is contained in at most one triangle

(2) Δ(119866) = 5 and no triangle contains an edge 119890 = 119909119910 with119889119866(119909) = 5 and 4 ⩽ 119889

119866(119910) ⩽ 5

63 Bounds Implied by Girth Conditions The girth 119892(119866) of agraph 119866 is the length of the shortest cycle in 119866 If 119866 has nocycles we define 119892(119866) = infin

Combining Theorem 74 with Theorem 78 we find thatif a planar graph contains no triangles and has maximumdegree Δ ⩾ 5 then Conjecture 73 holds This fact motivatedFischermann et alrsquos [46] attempt to attack Conjecture 73 bygirth restraints

Theorem 79 (Fischermann et al [46] 2003) For any con-nected planar graph 119866

119887 (119866) ⩽

6 119894119891 119892 (119866) ⩾ 4

5 119894119891 119892 (119866) ⩾ 5

4 119894119891 119892 (119866) ⩾ 6

3 119894119891 119892 (119866) ⩾ 8

(20)

The first result in Theorem 79 shows that Conjecture 73is valid for all connected planar graphs with 119892(119866) ⩾ 4 andΔ(119866) ⩾ 5 It is easy to verify that the second result inTheorem 79 implies that Conjecture 73 is valid for all not3-regular graphs of girth 119892(119866) ⩾ 5 which is stated in thefollowing corollary

Corollary 80 For any connected planar graph 119866 if 119866 is not3-regular and 119892(119866) ⩾ 5 then 119887(119866) ⩽ Δ(119866) + 1

The first result in Theorem 79 also implies that if 119866 isa connected planar graph with no cycles of length 3 then119887(119866) ⩽ 6 Hou et al [47] improved this result as follows

Theorem 81 (Hou et al [47] 2011) For any connected planargraph 119866 if 119866 contains no cycles of length 119894 (4 ⩽ 119894 ⩽ 6) then119887(119866) ⩽ 6

Since 119887(1198623119896+1) = 3 for 119896 ⩾ 3 (see Theorem 1) the

last bound in Theorem 79 is tight Whether other bounds inTheorem 79 are tight remains open In 2003 Fischermann etal [46] posed the following conjecture

Conjecture 82 For any connected planar graph 119866 119887(119866) ⩽ 7and

119887 (119866) ⩽ 5 119894119891 119892 (119866) ⩾ 4

4 119894119891 119892 (119866) ⩾ 5(21)

We conclude this subsection with a question on bondagenumbers of planar graphs

Question 1 (Fischermann et al [46] 2003) Is there a planargraph 119866 with 6 ⩽ 119887(119866) ⩽ 8

In 2006 Carlson andDevelin [42] showed that the corona119866 = 119867 ∘ 119870

1for a planar graph 119867 with 120575(119867) = 5 has the

bondage number 119887(119866) = 120575(119867) + 1 = 6 (see Theorem 66)Since the minimum degree of planar graphs is at most 5then 119887(119866) can attach 6 If we take 119867 as the graph of theicosahedron then 119866 = 119867 ∘ 119870

1is such an example The

question for the existence of planar graphs with bondagenumber 7 or 8 remains open

64 Comments on the Conjectures Conjecture 73 is true forall connected planar graphs with minimum degree 120575 ⩽ 2 byTheorem 16 or maximum degree Δ ⩾ 7 by Theorem 74 ornot 120574-critical planar graphs by Theorem 13 Thus to attackConjecture 73 we only need to consider connected criticalplanar graphs with degree-restriction 3 ⩽ 120575 ⩽ Δ ⩽ 6

Recalling and anatomizing the proofs of all results men-tioned in the preceding subsections on the bondage numberfor connected planar graphs we find that the proofs of theseresults strongly depend upon Theorem 16 or Theorem 18 Inother words a basic way used in the proofs is to find twovertices 119909 and 119910 with distance at most two in a consideredplanar graph 119866 such that

119889119866(119909) + 119889

119866(119910) or 119889

119866(119909) + 119889

119866(119910) minus

1003816100381610038161003816119873119866(119909) cap 119873

119866(119910)1003816100381610038161003816

(22)

which bounds 119887(119866) is as small as possible Very recentlyHuang and Xu [44] have considered the following parameter

119861 (119866) = min119909119910isin119881(119866)

119889119909119910 1 ⩽ 119889

119866(119909 119910) ⩽ 2

cup 119889119909119910minus 119899

119909119910 119889 (119909 119910) = 1

(23)

where 119889119909119910= 119889

119866(119909) + 119889

119866(119910) minus 1 and 119899

119909119910= |119873

119866(119909) cap 119873

119866(119910)|

14 International Journal of Combinatorics

It follows fromTheorems 16 and 18 that

119887 (119866) ⩽ 119861 (119866) (24)

The proofs given in Theorems 74 and 79 indeed imply thefollowing stronger results

Theorem 83 If 119866 is a connected planar graph then

119861 (119866) ⩽

min 8 Δ (119866) + 26 119894119891 119892 (119866) ⩾ 4

5 119894119891 119892 (119866) ⩾ 5

4 119894119891 119892 (119866) ⩾ 6

3 119894119891 119892 (119866) ⩾ 8

(25)

Thus using Theorem 16 or Theorem 18 if we can proveConjectures 73 and 82 then we can prove the followingstatement

Statement If 119866 is a connected planar graph then

119861 (119866) ⩽

Δ (119866) + 1

7

5 if 119892 (119866) ⩾ 44 if 119892 (119866) ⩾ 5

(26)

It follows from (24) that Statement implies Conjectures 73and 82 However Huang and Xu [44] gave examples to showthat none of conclusions in Statement is true As a result theystated the following conclusion

Theorem 84 (Huang and Xu [44] 2012) It is not possible toprove Conjectures 73 and 82 if they are right using Theorems16 and 18

Therefore a new method is needed to prove or disprovethese conjectures At the present time one cannot prove theseconjectures and may consider to disprove them If one ofthese conjectures is invalid then there exists a minimumcounterexample 119866 with respect to 120592(119866) + 120576(119866) In order toobtain a minimum counterexample one may consider twosimple operations satisfying these requirements the edge-deletion and the edge-contractionwhich decrease 120592(119866)+120576(119866)and preserve planarity However applying Theorems 71 and72 Huang and Xu [44] presented the following results

Theorem 85 (Huang and Xu [44] 2012) It is not possible toconstruct minimum counterexamples to Conjectures 73 and 82by the operation of an edge-deletion or an edge-contraction

7 Results on Crossing Number Restraints

It is quite natural to generalize the known results on thebondage number for planar graphs to formore general graphsin terms of graph-theoretical parameters In this section weconsider graphs with crossing number restraints

The crossing number cr(119866) of 119866 is the smallest numberof pairwise intersections of its edges when 119866 is drawn in theplane If cr(119866) = 0 then 119866 is a planar graph

71 General Methods Use 119899119894to denote the number of vertices

of degree 119894 in119866 for 119894 = 1 2 Δ(119866) Huang and Xu obtainedthe following results

Theorem 86 (Huang and Xu [48] 2007) For any connectedgraph 119866

119887 (119866) ⩽

6 119894119891 119892 (119866) ⩾ 4 119886119899119889 2cr (119866) lt 121198861

5 119894119891 119892 (119866) ⩾ 5 119886119899119889 6cr (119866) lt 161198862

4 119894119891 119892 (119866) ⩾ 6 119886119899119889 4cr (119866) lt 141198863

3 119894119891 119892 (119866) ⩾ 8 119886119899119889 6cr (119866) lt 161198864

(27)

where 1198861= 119899

1+ 2119899

2+ 2119899

3+sum

Δ

119894=8(119894 minus 7)119899

119894+ 8 119886

2= 3119899

1+ 6119899

2+

51198993+sum

Δ

119894=7(3119894minus18)119899

119894+20 119886

3= 119899

1+2119899

2+sum

Δ

119894=6(2119894minus10)119899

119894+12

and 1198864= sum

Δ

119894=5(3119894 minus 12)119899

119894+ 16

Simplifying the conditions in Theorem 86 we obtain thefollowing corollaries immediately

Corollary 87 (Huang and Xu [48] 2007) For any connectedgraph 119866

119887 (119866) ⩽

6 119894119891 119892 (119866) ⩾ 4 119886119899119889 cr (119866) ⩽ 3

5 119894119891 119892 (119866) ⩾ 5 119886119899119889 cr (119866) ⩽ 4

4 119894119891 119892 (119866) ⩾ 6 119886119899119889 cr (119866) ⩽ 2

3 119894119891 119892 (119866) ⩾ 8 119886119899119889 cr (119866) ⩽ 2

(28)

Corollary 88 (Huang and Xu [48] 2007) For any connectedgraph 119866

(a) 119887(119866) ⩽ 6 if 119866 is not 4-regular cr(119866) = 4 and 119892(119866) ⩾ 4

(b) 119887(119866) ⩽ Δ(119866) + 1 if 119866 is not 3-regular cr(119866) ⩽ 4 and119892(119866) ⩾ 5

(c) 119887(119866) ⩽ 4 if 119866 is not 3-regular cr(119866) = 3 and 119892(119866) ⩾ 6

(d) 119887(119866) ⩽ 3 if cr(119866) = 3 119892(119866) ⩾ 8 and Δ(119866) ⩾ 5

These corollaries generalize some known results for pla-nar graphs For example Corollary 87 contains Theorem 79Corollary 88 (b) contains Corollary 80

Theorem 89 (Huang and Xu [48] 2007) For any connectedgraph 119866 if cr(119866) ⩽ 119899

3(119866) + 119899

4(119866) + 3 then 119887(119866) ⩽ 8

Corollary 90 (Huang and Xu [48] 2007) For any connectedgraph 119866 if cr(119866) ⩽ 3 then 119887(119866) ⩽ 8

Perhaps being unaware of this result in 2010 Ma et al[49] proved that 119887(119866) ⩽ 12 for any graph 119866 with cr(119866) = 1

Theorem91 (Huang and Xu [48] 2007) Let119866 be a connectedgraph and 119868 = 119909 isin 119881(119866) 119889

119866(119909) = 5 119889

119866(119909 119910) ⩾ 3 if

International Journal of Combinatorics 15

119889119866(119910) ⩽ 3 and 119889

119866(119910) = 4 for every 119910 isin 119873

119866(119909) If 119868 is

independent no vertices adjacent to vertices of degree 6 and

cr (119866) lt max5119899

3+ |119868| minus 2119899

4+ 28

117119899

3+ 40

16 (29)

then 119887(119866) ⩽ 7

Corollary 92 (Huang and Xu [48] 2007) Let 119866 be aconnected graph with cr(119866) ⩽ 2 If 119868 = 119909 isin 119881(119866) 119889

119866(119909) =

5 119889119866(119910 119909) ⩾ 3 if 119889

119866(119910) ⩽ 3 and 119889

119866(119910) = 4 for every 119910 isin

119873119866(119909) is independent and no vertices adjacent to vertices of

degree 6 then 119887(119866) ⩽ 7

Theorem93 (Huang andXu [48] 2007) Let119866 be a connectedgraph If 119866 satisfies

(a) 5cr(119866) + 1198995lt 2119899

2+ 3119899

3+ 2119899

4+ 12 or

(b) 7cr(119866) + 21198995lt 3119899

2+ 4119899

4+ 24

then 119887(119866) ⩽ 7

Proposition 94 (Huang and Xu [48] 2007) Let 119866 be aconnected graph with no vertices of degrees four and five Ifcr(119866) ⩽ 2 then 119887(119866) ⩽ 6

Theorem 95 (Huang and Xu [48] 2007) If 119866 is a connectedgraph with cr(119866) ⩽ 4 and not 4-regular when cr(119866) = 4 then119887(119866) ⩽ Δ(119866) + 2

The above results generalize some known results forplanar graphs For example Corollary 90 and Theorem 95contain Theorem 74 the first condition in Theorem 93 con-tains the second condition inTheorem 77

From Corollary 88 and Theorem 95 we suggest the fol-lowing questions

Question 2 Is there a

(a) 4-regular graph 119866 with cr(119866) = 4 such that 119887(119866) ⩾ 7

(b) 3-regular graph 119866 with cr ⩽ 4 and 119892(119866) ⩾ 5 such that119887(119866) = 5

(c) 3-regular graph 119866 with cr = 3 and 119892(119866) = 6 or 7 suchthat 119887(119866) = 5

72 Carlson-Develin Methods In this subsection we intro-duce an elegant method presented by Carlson and Develin[42] to obtain some upper bounds for the bondage numberof a graph

Suppose that 119866 is a connected graph We say that 119866 hasgenus 120588 if 119866 can be embedded in a surface 119878 with 120588 handlessuch that edges are pairwise disjoint except possibly for end-vertices Let119866 be an embedding of119866 in surface 119878 and let120601(119866)denote the number of regions in 119866 The boundary of everyregion contains at least three edges and every edge is on theboundary of atmost two regions (the two regions are identicalwhen 119890 is a cut-edge) For any edge 119890 of 119866 let 1199031

119866(119890) and 1199032

119866(119890)

be the numbers of edges comprising the regions in 119866 whichthe edge 119890 borders It is clear that every 119890 = 119909119910 isin 119864(119866)

1199032

119866(119890) ⩾ 119903

1

119866(119890) ⩾ 4 if 1003816100381610038161003816119873119866

(119909) cap 119873119866(119910)1003816100381610038161003816 = 0

1199032

119866(119890) ⩾ 4 119903

1

119866(119890) ⩾ 3 if 1003816100381610038161003816119873119866

(119909) cap 119873119866(119910)1003816100381610038161003816 = 1

1199032

119866(119890) ⩾ 119903

1

119866(119890) ⩾ 3 if 1003816100381610038161003816119873119866

(119909) cap 119873119866(119910)1003816100381610038161003816 ⩾ 2

(30)

Following Carlson and Develin [42] for any edge 119890 = 119909119910 of119866 we define

119863119866(119890) =

1

119889119866(119909)+

1

119889119866(119910)

+1

1199031

119866(119890)+

1

1199032

119866(119890)minus 1 (31)

By the well-known Eulerrsquos formula

120592 (119866) minus 120576 (119866) + 120601 (119866) = 2 minus 2120588 (119866) (32)

it is easy to see that

sum

119890isin119864(119866)

119863119866(119890) = 120592 (119866) minus 120576 (119866) + 120601 (119866) = 2 minus 2120588 (119866) (33)

If 119866 is a planar graph that is 120588(119866) = 0 then

sum

119890isin119864(119866)

119863119866(119890) = 120592 (119866) minus 120576 (119866) + 120601 (119866) = 2 (34)

Combining these formulas withTheorem 18 Carlson andDevelin [42] gave a simple and intuitive proof ofTheorem 74and obtained the following result

Theorem 96 (Carlson and Develin [42] 2006) Let 119866 be aconnected graph which can be embedded on a torus Then119887(119866) ⩽ Δ(119866) + 3

Recently Hou and Liu [50] improved Theorem 96 asfollows

Theorem 97 (Hou and Liu [50] 2012) Let 119866 be a connectedgraph which can be embedded on a torus Then 119887(119866) ⩽ 9Moreover if Δ(119866) = 6 then 119887(119866) ⩽ 8

By the way Cao et al [51] generalized the result inTheorem 79 to a connected graph 119866 that can be embeddedon a torus that is

119887 (119866) le

6 if 119892 (119866) ⩾ 4 and 119866 is not 4-regular5 if 119892 (119866) ⩾ 54 if 119892 (119866) ⩾ 6 and 119866 is not 3-regular3 if 119892 (119866) ⩾ 8

(35)

Several authors used this method to obtain some resultson the bondage number For example Fischermann et al[46] used this method to prove the second conclusion inTheorem 78 Recently Cao et al [52] have used thismethod todeal with more general graphs with small crossing numbersFirst they found the following property

Lemma 98 120575(119866) ⩽ 5 for any graph 119866 with cr(119866) ⩽ 5

16 International Journal of Combinatorics

This result implies that 119887(119866) ⩽ Δ(119866) + 4 by Corollary 15for any graph 119866 with cr(119866) ⩽ 5 Cao et al established thefollowing relation in terms of maximum planar subgraphs

Lemma 99 (Cao et al [52]) Let 119866 be a connected graph withcrossing number cr(119866) For any maximum planar subgraph119867of 119866

sum

119890isin119864(119867)

119863119866(119890) ⩾ 2 minus

2cr (119866)120575 (119866)

(36)

Using Carlson-Develin method and combiningLemma 98 with Lemma 99 Cao et al proved the followingresults

Theorem 100 (Cao et al [52]) For any connected graph 119866119887(119866) ⩽ Δ(119866) + 2 if 119866 satisfies one of the following conditions

(a) cr(119866) ⩽ 3(b) cr(119866) = 4 and 119866 is not 4-regular(c) cr(119866) = 5 and 119866 contains no vertices of degree 4

Theorem101 (Cao et al [52]) Let119866 be a connected graphwithΔ(119866) = 5 and cr(119866) ⩽ 4 If no triangles contain two vertices ofdegree 5 then 119887(119866) ⩽ 6 = Δ(119866) + 1

Theorem 102 (Cao et al [52]) Let 119866 be a connected graphwith Δ(119866) ⩾ 6 and cr(119866) ⩽ 3 If Δ(119866) ⩾ 7 or if Δ(119866) = 6120575(119866) = 3 and every edge 119890 = 119909119910 with 119889

119866(119909) = 5 and 119889

119866(119910) = 6

is contained in atmost one triangle then 119887(119866) ⩽ min8 Δ(119866)+1

Using Carlson-Develin method it can be proved that119887(119866) ⩽ Δ(119866) + 2 if cr(119866) ⩽ 3 (see the first conclusion inTheorem 100) and not yet proved that 119887(119866) ⩽ 8 if cr(119866) ⩽3 although it has been proved by using other method (seeCorollary 90)

By using Carlson-Develin method Samodivkin [25]obtained some results on the bondage number for graphswithsome given properties

Kim [53] showed that 119887(119866) ⩽ Δ(119866) + 2 for a connectedgraph 119866 with genus 1 Δ(119866) ⩽ 5 and having a toroidalembedding of which at least one region is not 4-sidedRecently Gagarin and Zverovich [54] further have extendedCarlson-Develin ideas to establish a nice upper bound forarbitrary graphs that can be embedded on orientable ornonorientable topological surface

Theorem 103 (Gagarin and Zverovich [54] 2012) Let 119866 bea graph embeddable on an orientable surface of genus ℎ and anonorientable surface of genus 119896 Then 119887(119866) ⩽ minΔ(119866)+ℎ+2 Δ(119866) + 119896 + 1

This result generalizes the corresponding upper boundsin Theorems 74 and 96 for any orientable or nonori-entable topological surface By investigating the proof ofTheorem 103 Huang [55] found that the issue of the ori-entability can be avoided by using the Euler characteristic120594(= 120592(119866) minus 120576(119866) + 120601(119866)) instead of the genera ℎ and 119896 the

relations are 120594 = 2minus2ℎ and 120594 = 2minus119896 To obtain the best resultfromTheorem 103 onewants ℎ and 119896 as small as possible thisis equivalent to having 120594 as large as possible

According toTheorem 103 if119866 is planar (ℎ = 0 120594 = 2) orcan be embedded on the real projective plane (119896 = 1 120594 = 1)then 119887(119866) ⩽ Δ(119866) + 2 In all other cases Huang [55] had thefollowing improvement for Theorem 103 the proof is basedon the technique developed byCarlson-Develin andGagarin-Zverovich and includes some elementary calculus as a newingredient mainly the intermediate-value theorem and themean-value theorem

Theorem 104 (Huang [55] 2012) Let 119866 be a graph embed-dable on a surface whose Euler characteristic 120594 is as large aspossible If 120594 ⩽ 0 then 119887(119866) ⩽ Δ(119866)+lfloor119903rfloor where 119903 is the largestreal root of the following cubic equation in 119911

1199113

+ 21199112

+ (6120594 minus 7) 119911 + 18120594 minus 24 = 0 (37)

In addition if 120594 decreases then 119903 increases

The following result is asymptotically equivalent toTheorem 104

Theorem 105 (Huang [55] 2012) Let 119866 be a graph embed-dable on a surface whose Euler characteristic 120594 is as large aspossible If 120594 ⩽ 0 then 119887(119866) lt Δ(119866) + radic12 minus 6120594 + 12 orequivalently 119887(119866) ⩽ Δ(119866) + lceilradic12 minus 6120594 minus 12rceil

Also Gagarin andZverovich [54] indicated that the upperbound in Theorem 103 can be improved for larger values ofthe genera ℎ and 119896 by adjusting the proofs and stated thefollowing general conjecture

Conjecture 106 (Gagarin and Zverovich [54] 2012) For aconnected graph G of orientable genus ℎ and nonorientablegenus 119896

119887 (119866) ⩽ min 119888ℎ 119888

1015840

119896 Δ (119866) + 119900 (ℎ) Δ (119866) + 119900 (119896) (38)

where 119888ℎand 1198881015840

119896are constants depending respectively on the

orientable and nonorientable genera of 119866

In the recent paper Gagarin and Zverovich [56] providedconstant upper bounds for the bondage number of graphs ontopological surfaces which can be used as the first estimationfor the constants 119888

ℎand 1198881015840

119896of Conjecture 106

Theorem 107 (Gagarin and Zverovich [56] 2013) Let 119866 bea connected graph of order 119899 If 119866 can be embedded on thesurface of its orientable or nonorientable genus of the Eulercharacteristic 120594 then

(a) 120594 ⩾ 1 implies 119887(119866) ⩽ 10(b) 120594 ⩽ 0 and 119899 gt minus12120594 imply 119887(119866) ⩽ 11(c) 120594 ⩽ minus1 and 119899 ⩽ minus12120594 imply 119887(119866) ⩽ 11 +

3120594(radic17 minus 8120594 minus 3)(120594 minus 1) = 11 + 119874(radicminus120594)

Theorem 79 shows that a connected planar triangle-freegraph 119866 has 119887(119866) ⩽ 6 The following result generalizes to itall the other topological surfaces

International Journal of Combinatorics 17

Theorem 108 (Gagarin and Zverovich [56] 2013) Let 119866 be aconnected triangle-free graph of order 119899 If 119866 can be embeddedon the surface of its orientable or nonorientable genus of theEuler characteristic 120594 then

(a) 120594 ⩾ 1 implies 119887(119866) ⩽ 6(b) 120594 ⩽ 0 and 119899 gt minus8120594 imply 119887(119866) ⩽ 7(c) 120594 ⩽ minus1 and 119899 ⩽ minus8120594 imply 119887(119866) ⩽ 7minus4120594(1+radic2 minus 120594)

In [56] the authors also explicitly improved upperbounds in Theorem 103 and gave tight lower bounds forthe number of vertices of graphs 2-cell embeddable ontopological surfaces of a given genusThe interested reader isreferred to the original paper for details The following resultis interesting although its proof is easy

Theorem 109 (Samodivkin [57]) Let119866 be a graph with order119899 and girth 119892 If 119866 is embeddable on a surface whose Eulercharacteristic 120594 is as large as possible then

119887 (119866) ⩽ 3 +8

119892 minus 2minus

4120594119892

119899 (119892 minus 2) (39)

If 119866 is a planar graph with girth 119892 ⩾ 4 + 119894 for any 119894 isin0 1 2 then Theorem 109 leads to 119887(119866) ⩽ 6 minus 119894 which isthe result inTheorem 79 obtained by Fischermann et al [46]Also inmany cases the bound stated inTheorem 109 is betterthan those given byTheorems 103 and 105

8 Conditional Bondage Numbers

Since the concept of the bondage number is based upon thedomination all sorts of dominations which are variationsof the normal domination by adding restricted conditionsto the normal dominating set can develop a new ldquobondagenumberrdquo as long as a variation of domination number isgiven In this section we survey results on the bondagenumber under some restricted conditions

81 Total Bondage Numbers A dominating set 119878 of a graph119866 without isolated vertices is called total if the subgraphinduced by 119878 contains no isolated vertices The minimumcardinality of a total dominating set is called the totaldomination number of 119866 and denoted by 120574

119879(119866) It is clear

that 120574(119866) ⩽ 120574119879(119866) ⩽ 2120574(119866) for any graph 119866 without isolated

verticesThe total domination in graphs was introduced by Cock-

ayne et al [58] in 1980 Pfaff et al [59 60] in 1983 showedthat the problem determining total domination number forgeneral graphs is NP-complete even for bipartite graphs andchordal graphs Even now total domination in graphs hasbeen extensively studied in the literature In 2009 Henning[61] gave a survey of selected recent results on total domina-tion in graphs

The total bondage number of 119866 denoted by 119887119879(119866) is the

smallest cardinality of a subset 119861 sube 119864(119866) with the propertythat119866minus119861 contains no isolated vertices and 120574

119879(119866minus119861) gt 120574

119879(119866)

From definition 119887119879(119866)may not exist for some graphs for

example119866 = 1198701119899 We put 119887

119879(119866) = infin if 119887

119879(119866) does not exist

It is easy to see that 119887119879(119866) is finite for any connected graph 119866

other than 1198701 119870

2 119870

3 and 119870

1119899 In fact for such a graph 119866

since there is a path of length 3 in 119866 we can find 1198611sube 119864(119866)

such that 119866 minus 1198611is a spanning tree 119879 of 119866 containing a path

of length 3 So 120574119879(119879) = 120574

119879(119866minus119861

1) ⩾ 120574

119879(119866) For the tree119879 we

find 1198612sube 119864(119879) such that 120574

119879(119879 minus 119861

2) gt 120574

119879(119879) ⩾ 120574

119879(119866) Thus

we have 120574119879(119866minus119861

2minus119861

1) gt 120574

119879(119866) and so 119887

119879(119866) ⩽ |119861

1|+|119861

2| In

the following discussion we always assume that 119887119879(119866) exists

when a graph 119866 is mentionedIn 1991 Kulli and Patwari [62] first studied the total

bondage number of a graph and calculated the exact valuesof 119887

119879(119866) for some standard graphs

Theorem 110 (Kulli and Patwari [62] 1991) For 119899 ⩾ 4

119887119879(119862

119899) =

3 119894119891 119899 equiv 2 (mod 4) 2 119900119905ℎ119890119903119908119894119904119890

119887119879(119875

119899) =

2 119894119891 119899 equiv 2 (mod 4) 1 119900119905ℎ119890119903119908119894119904119890

119887119879(119870

119898119899) = 119898 119908119894119905ℎ 2 ⩽ 119898 ⩽ 119899

119887119879(119870

119899) =

4 119891119900119903 119899 = 4

2119899 minus 5 119891119900119903 119899 ⩾ 5

(40)

Recently Hu et al [63] have obtained some results on thetotal bondage number of the Cartesian product119875

119898times119875

119899of two

paths 119875119898and 119875

119899

Theorem 111 (Hu et al [63] 2012) For the Cartesian product119875119898times 119875

119899of two paths 119875

119898and 119875

119899

119887119879(119875

2times 119875

119899) =

1 119894119891 119899 equiv 0 (mod 3) 2 119894119891 119899 equiv 2 (mod 3) 3 119894119891 119899 equiv 1 (mod 3)

119887119879(119875

3times 119875

119899) = 1

119887119879(119875

4times 119875

119899)

= 1 119894119891 119899 equiv 1 (mod 5) = 2 119894119891 119899 equiv 4 (mod 5) ⩽ 3 119894119891 119899 equiv 2 (mod 5) ⩽ 4 119894119891 119899 equiv 0 3 (mod 5)

(41)

Generalized Petersen graphs are an important class ofcommonly used interconnection networks and have beenstudied recently By constructing a family of minimum totaldominating sets Cao et al [64] determined the total bondagenumber of the generalized Petersen graphs

FromTheorem 6 we know that 119887(119879) ⩽ 2 for a nontrivialtree 119879 But given any positive integer 119896 Sridharan et al [65]constructed a tree 119879 for which 119887

119879(119879) = 119896 Let119867

119896be the tree

obtained from a star1198701119896+1

by subdividing 119896 edges twice Thetree 119867

7is shown in Figure 6 It can be easily verified that

119887119879(119867

119896) = 119896 This fact can be stated as the following theorem

Theorem 112 (Sridharan et al [65] 2007) For any positiveinteger 119896 there exists a tree 119879 with 119887

119879(119879) = 119896

18 International Journal of Combinatorics

Figure 61198677

Combining Theorem 1 with Theorem 112 we have whatfollows

Corollary 113 (Sridharan et al [65] 2007) The bondagenumber and the total bondage number are unrelated even fortrees

However Sridharan et al [65] gave an upper bound onthe total bondage number of a tree in terms of its maximumdegree For any tree 119879 of order 119899 if 119879 =119870

1119899minus1 then 119887

119879(119879) =

minΔ(119879) (13)(119899 minus 1) Rad and Raczek [66] improved thisupper bound and gave a constructive characterization of acertain class of trees attaching the upper bound

Theorem 114 (Rad and Raczek [66] 2011) 119887119879(119879) ⩽ Δ(119879) minus 1

for any tree 119879 with maximum degree at least three

In general the decision problem for 119887119879(119866) is NP-complete

for any graph 119866 We state the decision problem for the totalbondage as follows

Problem 3 Consider the decision problemTotal Bondage ProblemInstance a graph 119866 and a positive integer 119887Question is 119887

119879(119866) ⩽ 119887

Theorem 115 (Hu and Xu [16] 2012) The total bondageproblem is NP-complete

Consequently in view of its computational hardnessit is significative to establish some sharp bounds on thetotal bondage number of a graph in terms of other graphicparameters In 1991 Kulli and Patwari [62] showed that119887119879(119866) ⩽ 2119899 minus 5 for any graph 119866 with order 119899 ⩾ 5 Sridharan

et al [65] improved this result as follows

Theorem 116 (Sridharan et al [65] 2007) For any graph 119866with order 119899 if 119899 ⩾ 5 then

119887119879(119866) ⩽

1

3(119899 minus 1) 119894119891 119866 119888119900119899119905119886119894119899119904 119899119900 119888119910119888119897119890119904

119899 minus 1 119894119891 119892 (119866) ⩾ 5

119899 minus 2 119894119891 119892 (119866) = 4

(42)

Rad and Raczek [66] also established some upperbounds on 119887

119879(119866) for a general graph 119866 In particular they

gave an upper bound on 119887119879(119866) in terms of the girth of

119866

Theorem 117 (Rad and Raczek [66] 2011) 119887119879(119866)⩽4Δ(119866) minus 5

for any graph 119866 with 119892(119866) ⩾ 4

82 Paired Bondage Numbers A dominating set 119878 of 119866 iscalled to be paired if the subgraph induced by 119878 containsa perfect matching The paired domination number of 119866denoted by 120574

119875(119866) is the minimum cardinality of a paired

dominating set of 119866 Clearly 120574119879(119866) ⩽ 120574

119875(119866) for every

connected graph 119866 with order at least two where 120574119879(119866) is

the total domination number of 119866 and 120574119875(119866) ⩽ 2120574(119866) for

any graph119866without isolated vertices Paired domination wasintroduced by Haynes and Slater [67 68] and further studiedin [69ndash72]

The paired bondage number of 119866 with 120575(119866) ⩾ 1 denotedby 119887P(119866) is the minimum cardinality among all sets of edges119861 sube 119864 such that 120575(119866 minus 119861) ⩾ 1 and 120574

119875(119866 minus 119861) gt 120574

119875(119866)

The concept of the paired bondage number was firstproposed by Raczek [73] in 2008The following observationsfollow immediately from the definition of the paired bondagenumber

Observation 2 Let 119866 be a graph with 120575(119866) ⩾ 1

(a) If 119867 is a subgraph of 119866 such that 120575(119867) ⩾ 1 then119887119875(119867) ⩽ 119887

119875(119866)

(b) If 119867 is a subgraph of 119866 such that 119887119875(119867) = 1 and 119896

is the number of edges removed to form119867 then 1 ⩽119887119875(119866) ⩽ 119896 + 1

(c) If 119909119910 isin 119864(119866) such that 119889119866(119909) 119889

119866(119910) gt 1 and 119909119910

belongs to each perfect matching of each minimumpaired dominating set of 119866 then 119887

119875(119866) = 1

Based on these observations 119887119875(119875

119899) ⩽ 2 In fact the

paired bondage number of a path has been determined

Theorem 118 (Raczek [73] 2008) For any positive integer 119896if 119899 ⩾ 2 then

119887119875(119875

119899) =

0 119894119891 119899 = 2 3 119900119903 5

1 119894119891 119899 = 4119896 4119896 + 3 119900119903 4119896 + 6

2 119900119905ℎ119890119903119908119894119904119890

(43)

For a cycle 119862119899of order 119899 ⩾ 3 since 120574

119875(119862

119899) = 120574

119875(119875

119899) from

Theorem 118 the following result holds immediately

Corollary 119 (Raczek [73] 2008) For any positive integer 119896if 119899 ⩾ 3 then

119887119875(119862

119899) =

0 119894119891 119899 = 3 119900119903 5

2 119894119891 119899 = 4119896 4119896 + 3 119900119903 4119896 + 6

3 119900119905ℎ119890119903119908119894119904119890

(44)

A wheel 119882119899 where 119899 ⩾ 4 is a graph with 119899 vertices

formed by connecting a single vertex to all vertices of a cycle119862119899minus1

Of course 120574119875(119882

119899) = 2

International Journal of Combinatorics 19

119896

Figure 7 A tree 119879 with 119887119875(119879) = 119896

Theorem 120 (Raczek [73] 2008) For positive integers 119899 and119898

119887119875(119870

119898119899) = 119898 119891119900119903 1 lt 119898 ⩽ 119899

119887119875(119882

119899) =

4 119894119891 119899 = 4

3 119894119891 119899 = 5

2 119900119905ℎ119890119903119908119894119904119890

(45)

Theorem 16 shows that if 119879 is a nontrivial tree then119887(119879) ⩽ 2 However no similar result exists for pairedbondage For any nonnegative integer 119896 let 119879

119896be a tree

obtained by subdividing all but one edge of a star 1198701119896+1

(asshown in Figure 7) It is easy to see that 119887

119875(119879

119896) = 119896 We

state this fact as the following theorem which is similar toTheorem 112

Theorem121 (Raczek [73] 2008) For any nonnegative integer119896 there exists a tree 119879 with 119887

119875(119879) = 119896

Consider the tree defined in Figure 5 for 119896 = 3 Then119887(119879

3) ⩽ 2 and 119887

119875(119879

3) = 3 On the other hand 119887(119875

4) = 2

and 119887119875(119875

4) = 1 Thus we have what follows which is similar

to Corollary 113

Corollary 122 The bondage number and the paired bondagenumber are unrelated even for trees

A constructive characterization of trees with 119887(119879) = 2is given by Hartnell and Rall in [12] Raczek [73] provideda constructive characterization of trees with 119887

119875(119879) = 0 In

order to state the characterization we define a labeling andthree simple operations on a tree119879 Let 119910 isin 119881(119879) and let ℓ(119910)be the label assigned to 119910

Operation T1 If ℓ(119910) = 119861 add a vertex 119909 and the

edge 119910119909 and let ℓ(119909) = 119860OperationT

2 If ℓ(119910) = 119862 add a path (119909

1 119909

2) and the

edge 1199101199091 and let ℓ(119909

1) = 119861 ℓ(119909

2) = 119860

Operation T3 If ℓ(119910) = 119861 add a path (119909

1 119909

2 119909

3)

and the edge 1199101199091 and let ℓ(119909

1) = 119862 ℓ(119909

2) = 119861 and

ℓ(1199093) = 119860

Let 1198752= (119906 V) with ℓ(119906) = 119860 and ℓ(V) = 119861 Let T be

the class of all trees obtained from the labeled 1198752by a finite

sequence of operationsT1 T

2 T

3

A tree 119879 in Figure 8 belongs to the familyTRaczek [73] obtained the following characterization of all

trees 119879 with 119887119875(119879) = 0

119860

119860

119860

119860

119860

119861 119862

119861 119862 119861

119861119860

Figure 8 A tree 119879 belong to the familyT

Theorem 123 (Raczek [73] 2008) For a tree 119879 119887119875(119879) = 0 if

and only if 119879 is inT

We state the decision problem for the paired bondage asfollows

Problem 4 Consider the decision problem

Paired Bondage ProblemInstance a graph 119866 and a positive integer 119887Question is 119887

119875(119866) ⩽ 119887

Conjecture 124 The paired bondage problem is NP-complete

83 Restrained Bondage Numbers A dominating set 119878 of 119866 iscalled to be restrained if the subgraph induced by 119878 containsno isolated vertices where 119878 = 119881(119866) 119878 The restraineddomination number of 119866 denoted by 120574

119877(119866) is the smallest

cardinality of a restrained dominating set of 119866 It is clear that120574119877(119866) exists and 120574(119866) ⩽ 120574

119877(119866) for any nonempty graph 119866

The concept of restrained domination was introducedby Telle and Proskurowski [74] in 1997 albeit indirectly asa vertex partitioning problem Concerning the study of therestrained domination numbers the reader is referred to [74ndash79]

In 2008 Hattingh and Plummer [80] defined therestrained bondage number 119887R(119866) of a nonempty graph 119866 tobe the minimum cardinality among all subsets 119861 sube 119864(119866) forwhich 120574

119877(119866 minus 119861) gt 120574

119877(119866)

For some simple graphs their restrained dominationnumbers can be easily determined and so restrained bondagenumbers have been also determined For example it is clearthat 120574

119877(119870

119899) = 1 Domke et al [77] showed that 120574

119877(119862

119899) =

119899 minus 2lfloor1198993rfloor for 119899 ⩾ 3 and 120574119877(119875

119899) = 119899 minus 2lfloor(119899 minus 1)3rfloor for 119899 ⩾ 1

Using these results Hattingh and Plummer [80] obtained therestricted bondage numbers for119870

119899 119862

119899 119875

119899 and119866 = 119870

11989911198992119899119905

as follows

Theorem 125 (Hattingh and Plummer [80] 2008) For 119899 ⩾ 3

119887R (119870119899) =

1 119894119891 119899 = 3

lceil119899

2rceil 119900119905ℎ119890119903119908119894119904119890

119887R (119862119899) =

1 119894119891 119899 equiv 0 (mod 3) 2 119900119905ℎ119890119903119908119894119904119890

119887R (119875119899) = 1 119899 ⩾ 4

20 International Journal of Combinatorics

119887R (119866)

=

lceil119898

2rceil 119894119891 119899

119898= 1 119886119899119889 119899

119898+1⩾ 2 (1 ⩽ 119898 lt 119905)

2119905 minus 2 119894119891 1198991= 119899

2= sdot sdot sdot = 119899

119905= 2 (119905 ⩾ 2)

2 119894119891 1198991= 2 119886119899119889 119899

2⩾ 3 (119905 = 2)

119905minus1

sum

119894=1

119899119894minus 1 119900119905ℎ119890119903119908119894119904119890

(46)

Theorem 126 (Hattingh and Plummer [80] 2008) For a tree119879 of order 119899 (⩾ 4) 119879 ≇ 119870

1119899minus1if and only if 119887R(119879) = 1

Theorem 126 shows that the restrained bondage numberof a tree can be computed in constant time However thedecision problem for 119887R(119866) is NP-complete even for bipartitegraphs

Problem 5 Consider the decision problem

Restrained Bondage Problem

Instance a graph 119866 and a positive integer 119887

Question is 119887R(119866) ⩽ 119887

Theorem 127 (Hattingh and Plummer [80] 2008) Therestrained bondage problem is NP-complete even for bipartitegraphs

Consequently in view of its computational hardnessit is significative to establish some sharp bounds on therestrained bondage number of a graph in terms of othergraphic parameters

Theorem 128 (Hattingh and Plummer [80] 2008) For anygraph 119866 with 120575(119866) ⩾ 2

119887R (119866) ⩽ min119909119910isin119864(119866)

119889119866(119909) + 119889

119866(119910) minus 2 (47)

Corollary 129 119887R(119866) ⩽ Δ(119866)+120575(119866)minus2 for any graph119866 with120575(119866) ⩾ 2

Notice that the bounds stated in Theorem 128 andCorollary 129 are sharp Indeed the class of cycles whoseorders are congruent to 1 2 (mod 3) have a restrained bondagenumber achieving these bounds (see Theorem 125)

Theorem 128 is an analogue of Theorem 14 A quitenatural problem is whether or not for restricted bondagenumber there are analogues of Theorem 16 Theorem 18Theorem 29 and so on Indeed we should consider all resultsfor the bondage number whether or not there are analoguesfor restricted bondage number

Theorem 130 (Hattingh and Plummer [80] 2008) For anygraph 119866 with 120574

119877(119866) = 2

119887R (119866) ⩽ Δ (119866) + 1 (48)

We now consider the relation between 119887119877(119866) and 119887(119866)

Theorem 131 (Hattingh and Plummer [80] 2008) If 120574119877(119866) =

120574(119866) for some graph 119866 then 119887R(119866) ⩽ 119887(119866)

Corollary 129 and Theorem 131 provide a possibility toattack Conjecture 73 by characterizing planar graphs with120574119877= 120574 and 119887R = 119887 when 3 ⩽ Δ ⩽ 6However we do not have 119887R(119866) = 119887(119866) for any graph

119866 even if 120574119877(119866) = 120574(119866) Observe that 120574

119877(119870

3) = 120574(119870

3) yet

119887119877(119870

3) = 1 and 119887(119870

3) = 2 We still may not claim that

119887119877(119866) = 119887(119866) even in the case that every 120574-set is a 120574

119877-set

The example 1198703again demonstrates this

Theorem 132 (Hattingh and Plummer [80] 2008) There isan infinite class of graphs inwhich each graph119866 satisfies 119887(119866) lt119887R(119866)

In fact such an infinite class of graphs can be obtained asfollows Let119867 be a connected graph BC(119867) a graph obtainedfrom 119867 by attaching ℓ (⩾ 2) vertices of degree 1 to eachvertex in 119867 and B = 119866 119866 = BC(119867) for some graph 119867such that 120575(119867) ⩾ 2 By Theorem 3 we can easily check that119887(119866) = 1 lt 2 ⩽ min120575(119867) ℓ = 119887R(119866) for any 119866 isin BMoreover there exists a graph 119866 isin B such that 119887R(119866) canbe much larger than 119887(119866) which is stated as the followingtheorem

Theorem 133 (Hattingh and Plummer [80] 2008) For eachpositive integer 119896 there is a graph119866 such that 119896 = 119887R(119866)minus119887(119866)

Combining Theorem 133 with Theorem 132 we have thefollowing corollary

Corollary 134 The bondage number and the restrainedbondage number are unrelated

Very recently Jafari Rad et al [81] have consideredthe total restrained bondage in graphs based on both totaldominating sets and restrained dominating sets and obtainedseveral properties exact values and bounds for the totalrestrained bondage number of a graph

A restrained dominating set 119878 of a graph 119866 without iso-lated vertices is called to be a total restrained dominating setif 119878 is a total dominating set The total restrained dominationnumber of119866 denoted by 120574TR (119866) is theminimum cardinalityof a total restrained dominating set of 119866 For referenceson this domination in graphs see [3 61 82ndash85] The totalrestrained bondage number 119887TR (119866) of a graph 119866 with noisolated vertex is the cardinality of a smallest set of edges 119861 sube119864(119866) for which119866minus119861 has no isolated vertex and 120574TR (119866minus119861) gt120574TR (119866) In the case that there is no such subset 119861 we define119887TR (119866) = infin

Ma et al [84] determined that 120574TR (119875119899) = 119899minus2lfloor(119899minus2)4rfloorfor 119899 ⩾ 3 120574TR (119862119899

) = 119899 minus 2lfloor1198994rfloor for 119899 ⩾ 3 120574TR (1198703) =

3 and 120574TR (119870119899) = 2 for 119899 = 3 and 120574TR (1198701119899

) = 1 + 119899

and 120574TR (119870119898119899) = 2 for 2 ⩽ 119898 ⩽ 119899 According to these

results Jafari Rad et al [81] determined the exact valuesof total restrained bondage numbers for the correspondinggraphs

International Journal of Combinatorics 21

Theorem 135 (Jafari Rad et al [81]) For an integer 119899

119887TR (119875119899) = infin 119894119891 2 ⩽ 119899 ⩽ 5

1 119894119891 119899 ⩾ 5

119887TR (119862119899) =

infin 119894119891 119899 = 3

1 119894119891 3 lt 119899 119899 equiv 0 1 (mod 4)2 119894119891 3 lt 119899 119899 equiv 2 3 (mod 4)

119887TR (119882119899) = 2 119891119900119903 119899 ⩾ 6

119887TR (119870119899) = 119899 minus 1 119891119900119903 119899 ⩾ 4

119887TR (119870119898119899) = 119898 minus 1 119891119900119903 2 ⩽ 119898 ⩽ 119899

(49)

119887TR (119870119899) = 119899minus1 shows the fact that for any integer 119899 ⩾ 3 there

exists a connected graph namely119870119899+1

such that 119887TR (119870119899+1) =

119899 that is the total restrained bondage number is unboundedfrom the above

A vertex is said to be a support vertex if it is adjacent toa vertex of degree one Let A be the family of all connectedgraphs such that119866 belongs toA if and only if every edge of119866is incident to a support vertex or 119866 is a cycle of length three

Proposition 136 (Cyman and Raczek [82] 2006) For aconnected graph119866 of order 119899 120574TR (119866) = 119899 if and only if119866 isin A

Hence if 119866 isin A then the removal of any subset of edgesof119866 cannot increase the total restrained domination numberof 119866 and thus 119887TR (119866) = infin When 119866 notin A a result similar toTheorem 6 is obtained

Theorem 137 (Jafari Rad et al [81]) For any tree 119879 notin A119887TR (119879) ⩽ 2 This bound is sharp

In the samepaper the authors provided a characterizationfor trees 119879 notin A with 119887TR (119879) = 1 or 2 The following result issimilar to Observation 1

Observation 3 (Jafari Rad et al [81]) If 119896 edges can beremoved from a graph 119866 to obtain a graph 119867 without anisolated vertex and with 119887TR (119867) = 119905 then 119887TR (119866) ⩽ 119896 + 119905

ApplyingTheorem 137 andObservation 3 Jafari Rad et alcharacterized all connected graphs 119866 with 119887TR (119866) = infin

Theorem 138 (Jafari Rad et al [81]) For a connected graph119866119887TR (119866) = infin if and only if 119866 isin A

84 Other Conditional Bondage Numbers A subset 119868 sube 119881(119866)is called an independent set if no two vertices in 119868 are adjacentin 119866 The maximum cardinality among all independent setsis called the independence number of 119866 denoted by 120572(119866)

A dominating set 119878 of a graph 119866 is called to be inde-pendent if 119878 is an independent set of 119866 The minimumcardinality among all independent dominating set is calledthe independence domination number of 119866 and denoted by120574119868(119866)

Since an independent dominating set is not only adominating set but also an independent set 120574(119866) ⩽ 120574

119868(119866) ⩽

120572(119866) for any graph 119866It is clear that a maximal independent set is certainly a

dominating set Thus an independent set is maximal if andonly if it is an independent dominating set and so 120574

119868(119866) is the

minimum cardinality among all maximal independent setsof 119866 This graph-theoretical invariant has been well studiedin the literature see for example Haynes et al [3] for earlyresults Goddard and Henning [86] for recent results onindependent domination in graphs

In 2003 Zhang et al [87] defined the independencebondage number 119887

119868(119866) of a nonempty graph 119866 to be the

minimum cardinality among all subsets 119861 sube 119864(119866) for which120574119868(119866 minus 119861) gt 120574

119868(119866) For some ordinary graphs their indepen-

dence domination numbers can be easily determined and soindependence bondage numbers have been also determinedClearly 119887

119868(119870

119899) = 1 if 119899 ⩾ 2

Theorem 139 (Zhang et al [87] 2003) For any integer 119899 ⩾ 4

119887119868(119862

119899) =

1 119894119891 119899 equiv 1 (mod 2) 2 119894119891 119899 equiv 0 (mod 2)

119887119868(119875

119899) =

1 119894119891 119899 equiv 0 (mod 2) 2 119894119891 119899 equiv 1 (mod 2)

119887119868(119870

119898119899) = 119899 119891119900119903 119898 ⩽ 119899

(50)

Apart from the above-mentioned results as far as we findno other results on the independence bondage number in thepresent literature we never knew much of any result on thisparameter for a tree

A subset 119878 of 119881(119866) is called an equitable dominatingset if for every 119910 isin 119881(119866 minus 119878) there exists a vertex 119909 isin 119878such that 119909119910 isin 119864(119866) and |119889

119866(119909) minus 119889

119866(119910)| ⩽ 1 proposed

by Dharmalingam [88] The minimum cardinality of such adominating set is called the equitable domination number andis denoted by 120574

119864(119866) Deepak et al [89] defined the equitable

bondage number 119887119864(119866) of a graph 119866 to be the cardinality of a

smallest set 119861 sube 119864(119866) of edges for which 120574119864(119866 minus 119861) gt 120574

119864(119866)

and determined 119887119864(119866) for some special graphs

Theorem 140 (Deepak et al [89] 2011) For any integer 119899 ⩾ 2

119887119864(119870

119899) = lceil

119899

2rceil

119887119864(119870

119898119899) = 119898 119891119900119903 0 ⩽ 119899 minus 119898 ⩽ 1

119887119864(119862

119899) =

3 119894119891 119899 equiv 1 (mod 3)2 119900119905ℎ119890119903119908119894119904119890

119887119864(119875

119899) =

2 119894119891 119899 equiv 1 (mod 3)1 119900119905ℎ119890119903119908119894119904119890

119887119864(119879) ⩽ 2 119891119900119903 119886119899119910 119899119900119899119905119903119894119907119894119886119897 119905119903119890119890 119879

119887119864(119866) ⩽ 119899 minus 1 119891119900119903 119886119899119910 119888119900119899119899119890119888119905119890119889 119892119903119886119901ℎ 119866 119900119891 119900119903119889119890119903 119899

(51)

22 International Journal of Combinatorics

As far as we know there are no more results on theequitable bondage number of graphs in the present literature

There are other variations of the normal domination byadding restricted conditions to the normal dominating setFor example the connected domination set of a graph119866 pro-posed by Sampathkumar and Walikar [90] is a dominatingset 119878 such that the subgraph induced by 119878 is connected Theproblem of finding a minimum connected domination set ina graph is equivalent to the problemof finding a spanning treewithmaximumnumber of vertices of degree one [91] and hassome important applications in wireless networks [92] thusit has received much research attention However for suchan important domination there is no result on its bondagenumber We believe that the topic is of significance for futureresearch

9 Generalized Bondage Numbers

There are various generalizations of the classical dominationsuch as 119901-domination distance domination fractional dom-ination Roman domination and rainbow domination Everysuch a generalization can lead to a corresponding bondageIn this section we introduce some of them

91 119901-Bondage Numbers In 1985 Fink and Jacobson [93]introduced the concept of 119901-domination Let 119901 be a positiveinteger A subset 119878 of 119881(119866) is a 119901-dominating set of 119866 if|119878 cap 119873

119866(119910)| ⩾ 119901 for every 119910 isin 119878 The 119901-domination number

120574119901(119866) is the minimum cardinality among all 119901-dominating

sets of 119866 Any 119901-dominating set of 119866 with cardinality 120574119901(119866)

is called a 120574119901-set of 119866 Note that a 120574

1-set is a classic minimum

dominating set Notice that every graph has a 119901-dominatingset since the vertex-set119881(119866) is such a setWe also note that a 1-dominating set is a dominating set and so 120574(119866) = 120574

1(119866) The

119901-domination number has received much research attentionsee a state-of-the-art survey article by Chellali et al [94]

It is clear from definition that every 119901-dominating setof a graph certainly contains all vertices of degree at most119901 minus 1 By this simple observation to avoid the trivial caseoccurrence we always assume that Δ(119866) ⩾ 119901 For 119901 ⩾ 2 Luet al [95] gave a constructive characterization of trees withunique minimum 119901-dominating sets

Recently Lu and Xu [96] have introduced the conceptto the 119901-bondage number of 119866 denoted by 119887

119901(119866) as the

minimum cardinality among all sets of edges 119861 sube 119864(119866) suchthat 120574

119901(119866 minus 119861) gt 120574

119901(119866) Clearly 119887

1(119866) = 119887(119866)

Lu and Xu [96] established a lower bound and an upperbound on 119887

119901(119879) for any integer 119901 ⩾ 2 and any tree 119879 with

Δ(119879) ⩾ 119901 and characterized all trees achieving the lowerbound and the upper bound respectively

Theorem 141 (Lu and Xu [96] 2011) For any integer 119901 ⩾ 2and any tree 119879 with Δ(119879) ⩾ 119901

1 ⩽ 119887119901(119879) ⩽ Δ (119879) minus 119901 + 1 (52)

Let 119878 be a given subset of 119881(119866) and for any 119909 isin 119878 let

119873119901(119909 119878 119866) = 119910 isin 119878 cap 119873

119866(119909)

1003816100381610038161003816119873119866(119910) cap 119878

1003816100381610038161003816 = 119901

(53)

Then all trees achieving the lower bound 1 can be character-ized as follows

Theorem 142 (Lu and Xu [96] 2011) Let 119879 be a tree withΔ(119879) ⩾ 119901 ⩾ 2 Then 119887

119901(119879) = 1 if and only if for any 120574

119901-set

119878 of 119879 there exists an edge 119909119910 isin (119878 119878) such that 119910 isin 119873119901(119909 119878 119879)

The notation 119878(119886 119887) denotes a double star obtained byadding an edge between the central vertices of two stars119870

1119886minus1

and1198701119887minus1

And the vertex with degree 119886 (resp 119887) in 119878(119886 119887) iscalled the 119871-central vertex (resp 119877-central vertex) of 119878(119886 119887)

To characterize all trees attaining the upper bound givenin Theorem 141 we define three types of operations on a tree119879 with Δ(119879) = Δ ⩾ 119901 + 1

Type 1 attach a pendant edge to a vertex 119910with 119889119879(119910) ⩽ 119901minus

2 in 119879

Type 2 attach a star 1198701Δminus1

to a vertex 119910 of 119879 by joining itscentral vertex to 119910 where 119910 in a 120574

119901-set of 119879 and

119889119879(119910) ⩽ Δ minus 1

Type 3 attach a double star 119878(119901 Δ minus 1) to a pendant vertex 119910of 119879 by coinciding its 119877-central vertex with 119910 wherethe unique neighbor of 119910 is in a 120574

119901-set of 119879

Let B = 119879 a tree obtained from 1198701Δ

or 119878(119901 Δ) by afinite sequence of operations of Types 1 2 and 3

Theorem 143 (Lu and Xu [96] 2011) A tree with the maxi-mum Δ ⩾ 119901 + 1 has 119901-bondage number Δ minus 119901 + 1 if and onlyif it belongs toB

Theorem 144 (Lu and Xu [97] 2012) Let 119866119898119899= 119875

119898times 119875

119899

Then

1198872(119866

2119899) = 1 119891119900119903 119899 ⩾ 2

1198872(119866

3119899) =

2 119894119891 119899 equiv 1 (mod 3) 1 119900119905ℎ119890119903119908119894119904119890

for 119899 ⩾ 2

1198872(119866

4119899) =

1 119894119891 119899 equiv 3 (mod 4) 2 119900119905ℎ119890119903119908119894119904119890

for 119899 ⩾ 7

(54)

Recently Krzywkowski [98] has proposed the conceptof the nonisolating 2-bondage of a graph The nonisolating2-bondage number of a graph 119866 denoted by 1198871015840

2(119866) is the

minimum cardinality among all sets of edges 119861 sube 119864(119866) suchthat 119866 minus 119861 has no isolated vertices and 120574

2(119866 minus 119861) gt 120574

2(119866) If

for every 119861 sube 119864(119866) either 1205742(119866 minus 119861) = 120574

2(119866) or 119866 minus 119861 has

isolated vertices then define 11988710158402(119866) = 0 and say that 119866 is a 2-

nonisolatedly strongly stable graph Krzywkowski presentedsome basic properties of nonisolating 2-bondage in graphsshowed that for every nonnegative integer 119896 there exists atree 119879 such 1198871015840

2(119879) = 119896 characterized all 2-non-isolatedly

strongly stable trees and determined 11988710158402(119866) for several special

graphs

International Journal of Combinatorics 23

Theorem 145 (Krzywkowski [98] 2012) For any positiveintegers 119899 and119898 with119898 ⩽ 119899

1198871015840

2(119870

119899) =

0 119894119891 119899 = 1 2 3

lfloor2119899

3rfloor 119900119905ℎ119890119903119908119894119904119890

1198871015840

2(119875

119899) =

0 119894119891 119899 = 1 2 3

1 119900119905ℎ119890119903119908119894119904119890

1198871015840

2(119862

119899) =

0 119894119891 119899 = 3

1 119894119891 119899 119894119904 119890V119890119899

2 119894119891 119899 119894119904 119900119889119889

1198871015840

2(119870

119898119899) =

3 119894119891 119898 = 119899 = 3

1 119894119891 119898 = 119899 = 4

2 119900119905ℎ119890119903119908119894119904119890

(55)

92 Distance Bondage Numbers A subset 119878 of vertices of agraph119866 is said to be a distance 119896-dominating set for119866 if everyvertex in 119866 not in 119878 is at distance at most 119896 from some vertexof 119878 The minimum cardinality of all distance 119896-dominatingsets is called the distance 119896-domination number of 119866 anddenoted by 120574

119896(119866) (do not confuse with the above-mentioned

120574119901(119866)) When 119896 = 1 a distance 1-dominating set is a normal

dominating set and so 1205741(119866) = 120574(119866) for any graph 119866 Thus

the distance 119896-domination is a generalization of the classicaldomination

The relation between 120574119896and 120572

119896(the 119896-independence

number defined in Section 22) for a tree obtained by Meirand Moon [14] who proved that 120574

119896(119879) = 120572

2119896(119879) for any tree

119879 (a special case for 119896 = 1 is stated in Proposition 10) Thefurther research results can be found in Henning et al [99]Tian and Xu [100ndash103] and Liu et al [104]

In 1998 Hartnell et al [15] defined the distance 119896-bondagenumber of 119866 denoted by 119887

119896(119866) to be the cardinality of a

smallest subset 119861 of edges of 119866 with the property that 120574119896(119866 minus

119861) gt 120574119896(119866) FromTheorem 6 it is clear that if119879 is a nontrivial

tree then 1 ⩽ 1198871(119879) ⩽ 2 Hartnell et al [15] generalized this

result to any integer 119896 ⩾ 1

Theorem 146 (Hartnell et al [15] 1998) For every nontrivialtree 119879 and positive integer 119896 1 ⩽ 119887

119896(119879) ⩽ 2

Hartnell et al [15] and Topp and Vestergaard [13] alsocharacterized the trees having distance 119896-bondage number 2In particular the class of trees for which 119887

1(119879) = 2 are just

those which have a unique maximum 2-independent set (seeTheorem 11)

Since when 119896 = 1 the distance 1-bondage number 1198871(119866)

is the classical bondage number 119887(119866) Theorem 12 gives theNP-hardness of deciding the distance 119896-bondage number ofgeneral graphs

Theorem 147 Given a nonempty undirected graph 119866 andpositive integers 119896 and 119887 with 119887 ⩽ 120576(119866) determining wetheror not 119887

119896(119866) ⩽ 119887 is NP-hard

For a vertex 119909 in119866 the open 119896-neighborhood119873119896(119909) of 119909 is

defined as119873119896(119909) = 119910 isin 119881(119866) 1 ⩽ 119889

119866(119909 119910) le 119896The closed

119896-neighborhood119873119896[119909] of 119909 in 119866 is defined as119873

119896(119909) cup 119909 Let

Δ119896(119866) = max 1003816100381610038161003816119873119896

(119909)1003816100381610038161003816 for any 119909 isin 119881 (119866) (56)

Clearly Δ1(119866) = Δ(119866) The 119896th power of a graph 119866 is the

graph119866119896 with vertex-set119881(119866119896

) = 119881(119866) and edge-set119864(119866119896

) =

119909119910 1 ⩽ 119889119866(119909 119910) ⩽ 119896 The following lemma holds directly

from the definition of 119866119896

Proposition 148 (Tian and Xu [101]) Δ(119866119896

) = Δ119896(119866) and

120574(119866119896

) = 120574119896(119866) for any graph 119866 and each 119896 ⩾ 1

A graph 119866 is 119896-distance domination-critical or 120574119896-critical

for short if 120574119896(119866 minus 119909) lt 120574

119896(119866) for every vertex 119909 in 119866

proposed by Henning et al [105]

Proposition 149 (Tian and Xu [101]) For each 119896 ⩾ 1 a graph119866 is 120574

119896-critical if and only if 119866119896 is 120574

119896-critical

By the above facts we suggest the following worthwhileproblem

Problem 6 Can we generalize the results on the bondage for119866 to 119866119896 In particular do the following propositions hold

(a) 119887(119866119896

) = 119887119896(119866) for any graph 119866 and each 119896 ⩾ 1

(b) 119887(119866119896

) ⩽ Δ119896(119866) if 119866 is not 120574

119896-critical

Let 119896 and 119901 be positive integers A subset 119878 of 119881(119866) isdefined to be a (119896 119901)-dominating set of 119866 if for any vertex119909 isin 119878 |119873

119896(119909) cap 119878| ⩾ 119901 The (119896 119901)-domination number of

119866 denoted by 120574119896119901(119866) is the minimum cardinality among

all (119896 119901)-dominating sets of 119866 Clearly for a graph 119866 a(1 1)-dominating set is a classical dominating set a (119896 1)-dominating set is a distance 119896-dominating set and a (1 119901)-dominating set is the above-mentioned 119901-dominating setThat is 120574

11(119866) = 120574(119866) 120574

1198961(119866) = 120574

119896(119866) and 120574

1119901(119866) = 120574

119901(119866)

The concept of (119896 119901)-domination in a graph 119866 is a gen-eralized domination which combines distance 119896-dominationand 119901-domination in 119866 So the investigation of (119896 119901)-domination of 119866 is more interesting and has received theattention of many researchers see for example [106ndash109]

More general Li and Xu [110] proposed the concept of the(ℓ 119908)-domination number of a119908-connected graph A subset119878 of 119881(119866) is defined to be an (ℓ 119908)-dominating set of 119866 iffor any vertex 119909 isin 119878 there are 119908 internally disjoint pathsof length at most ℓ from 119909 to some vertex in 119878 The (ℓ 119908)-domination number of119866 denoted by 120574

ℓ119908(119866) is theminimum

cardinality among all (ℓ 119908)-dominating sets of 119866 Clearly1205741198961(119866) = 120574

119896(119866) and 120574

1119901(119866) = 120574

119901(119866)

It is quite natural to propose the concept of bondage num-ber for (119896 119901)-domination or (ℓ 119908)-domination However asfar as we know none has proposed this concept until todayThis is a worthwhile topic for further research

24 International Journal of Combinatorics

93 Fractional Bondage Numbers If 120590 is a function mappingthe vertex-set 119881 into some set of real numbers then for anysubset 119878 sube 119881 let 120590(119878) = sum

119909isin119878120590(119909) Also let |120590| = 120590(119881)

A real-value function 120590 119881 rarr [0 1] is a dominatingfunction of a graph 119866 if for every 119909 isin 119881(119866) 120590(119873

119866[119909]) ⩾ 1

Thus if 119878 is a dominating set of 119866 120590 is a function where

120590 (119909) = 1 if 119909 isin 1198780 otherwise

(57)

then 120590 is a dominating function of119866 The fractional domina-tion number of 119866 denoted by 120574

119865(119866) is defined as follows

120574119865(119866) = min |120590| 120590 is a domination function of 119866

(58)

The 120574119865-bondage number of 119866 denoted by 119887

119865(119866) is

defined as the minimum cardinality of a subset 119861 sube 119864 whoseremoval results in 120574

119865(119866 minus 119861) gt 120574

119865(119866)

Hedetniemi et al [111] are the first to study fractionaldomination although Farber [112] introduced the idea indi-rectly The concept of the fractional domination numberwas proposed by Domke and Laskar [113] in 1997 Thefractional domination numbers for some ordinary graphs aredetermined

Proposition 150 (Domke et al [114] 1998 Domke and Laskar[113] 1997) (a) If 119866 is a 119896-regular graph with order 119899 then120574119865(119866) = 119899119896 + 1(b) 120574

119865(119862

119899) = 1198993 120574

119865(119875

119899) = lceil1198993rceil 120574

119865(119870

119899) = 1

(c) 120574119865(119866) = 1 if and only if Δ(119866) = 119899 minus 1

(d) 120574119865(119879) = 120574(119879) for any tree 119879

(e) 120574119865(119870

119899119898) = (119899(119898 + 1) + 119898(119899 + 1))(119899119898 minus 1) where

max119899119898 gt 1

The assertions (a)ndash(d) are due to Domke et al [114] andthe assertion (e) is due to Domke and Laskar [113] Accordingto these results Domke and Laskar [113] determined the 120574

119865-

bondage numbers for these graphs

Theorem 151 (Domke and Laskar [113] 1997) 119887119865(119870

119899) =

lceil1198992rceil 119887119865(119870

119899119898) = 1 wheremax119899119898 gt 1

119887119865(119862

119899) =

2 119894119891 119899 equiv 0 (mod 3) 1 119900119905ℎ119890119903119908119894119904119890

119887119865(119875

119899) =

2 119894119891 119899 equiv 1 (mod 3) 1 119900119905ℎ119890119903119908119894119904119890

(59)

It is easy to see that for any tree119879 119887119865(119879) ⩽ 2 In fact since

120574119865(119879) = 120574(119879) for any tree 119879 120574

119865(119879

1015840

) = 120574(1198791015840

) for any subgraph1198791015840 of 119879 It follows that

119887119865(119879) = min 119861 sube 119864 (119879) 120574

119865(119879 minus 119861) gt 120574

119865(119879)

= min 119861 sube 119864 (119879) 120574 (119879 minus 119861) gt 120574 (119879)

= 119887 (119879) ⩽ 2

(60)

Except for these no results on 119887119865(119866) have been known as

yet

94 Roman Bondage Numbers A Roman dominating func-tion on a graph 119866 is a labeling 119891 119881 rarr 0 1 2 such thatevery vertex with label 0 has at least one neighbor with label2 The weight of a Roman dominating function 119891 on 119866 is thevalue

119891 (119866) = sum

119906isin119881(119866)

119891 (119906) (61)

The minimum weight of a Roman dominating function ona graph 119866 is called the Roman domination number of 119866denoted by 120574RM (119866)

A Roman dominating function 119891 119881 rarr 0 1 2

can be represented by the ordered partition (1198810 119881

1 119881

2) (or

(119881119891

0 119881

119891

1 119881

119891

2) to refer to119891) of119881 where119881

119894= V isin 119881 | 119891(V) = 119894

In this representation its weight 119891(119866) = |1198811| + 2|119881

2| It is

clear that119881119891

1cup119881

119891

2is a dominating set of 119866 called the Roman

dominating set denoted by 119863119891

R = (1198811 1198812) Since 119881119891

1cup 119881

119891

2is

a dominating set when 119891 is a Roman dominating functionand since placing weight 2 at the vertices of a dominating setyields a Roman dominating function in [115] it is observedthat

120574 (119866) ⩽ 120574RM (119866) ⩽ 2120574 (119866) (62)

A graph 119866 is called to be Roman if 120574RM (119866) = 2120574(119866)The definition of the Roman domination function is

proposed implicitly by Stewart [116] and ReVelle and Rosing[117] Roman dominating numbers have been deeply studiedIn particular Bahremandpour et al [118] showed that theproblem determining the Roman domination number is NP-complete even for bipartite graphs

Let 119866 be a graph with maximum degree at least twoThe Roman bondage number 119887RM (119866) of 119866 is the minimumcardinality of all sets 1198641015840 sube 119864 for which 120574RM (119866 minus 119864

1015840

) gt

120574RM (119866) Since in study of Roman bondage number theassumption Δ(119866) ⩾ 2 is necessary we always assume thatwhen we discuss 119887RM (119866) all graphs involved satisfy Δ(119866) ⩾2 The Roman bondage number 119887RM (119866) is introduced byJafari Rad and Volkmann in [119]

Recently Bahremandpour et al [118] have shown that theproblem determining the Roman bondage number is NP-hard even for bipartite graphs

Problem 7 Consider the decision problem

Roman Bondage Problem

Instance a nonempty bipartite graph119866 and a positiveinteger 119896

Question is 119887RM (119866) ⩽ 119896

Theorem 152 (Bahremandpour et al [118] 2013) TheRomanbondage problem is NP-hard even for bipartite graphs

The exact value of 119887RM(119866) is known only for a few familyof graphs including the complete graphs cycles and paths

International Journal of Combinatorics 25

Theorem 153 (Jafari Rad and Volkmann [119] 2011) For anypositive integers 119899 and119898 with119898 ⩽ 119899

119887RM (119875119899) = 2 119894119891 119899 equiv 2 (mod 3) 1 119900119905ℎ119890119903119908119894119904119890

119887RM (119862119899) =

3 119894119891 119899 = 2 (mod 3) 2 119900119905ℎ119890119903119908119894119904119890

119887RM (119870119899) = lceil

119899

2rceil 119891119900119903 119899 ⩾ 3

119887RM (119870119898119899) =

1 119894119891 119898 = 1 119886119899119889 119899 = 1

4 119894119891 119898 = 119899 = 3

119898 119900119905ℎ119890119903119908119894119904119890

(63)

Lemma 154 (Cockayne et al [115] 2004) If 119866 is a graph oforder 119899 and contains vertices of degree 119899 minus 1 then 120574RM(119866) = 2

Using Lemma 154 the third conclusion in Theorem 153can be generalized to more general case which is similar toLemma 49

Theorem 155 (Jafari Rad and Volkmann [119] 2011) Let119866 bea graph with order 119899 ge 3 and 119905 the number of vertices of degree119899 minus 1 in 119866 If 119905 ⩾ 1 then 119887RM(119866) = lceil1199052rceil

Ebadi and PushpaLatha [120] conjectured that 119887RM(119866) ⩽119899 minus 1 for any graph 119866 of order 119899 ⩾ 3 Dehgardi et al[121] Akbari andQajar [122] independently showed that thisconjecture is true

Theorem 156 119887RM(119866) ⩽ 119899 minus 1 for any connected graph 119866 oforder 119899 ⩾ 3

For a complete 119905-partite graph we have the followingresult

Theorem 157 (Hu and Xu [123] 2011) Let 119866 = 11987011989811198982119898119905

bea complete 119905-partite graph with 119898

1= sdot sdot sdot = 119898

119894lt 119898

119894+1le sdot sdot sdot ⩽

119898119905 119905 ⩾ 2 and 119899 = sum119905

119895=1119898

119895 Then

119887RM (119866) =

lceil119894

2rceil 119894119891 119898

119894= 1 119886119899119889 119899 ⩾ 3

2 119894119891 119898119894= 2 119886119899119889 119894 = 1

119894 119894119891 119898119894= 2 119886119899119889 119894 ⩾ 2

4 119894119891 119898119894= 3 119886119899119889 119894 = 119905 = 2

119899 minus 1 119894119891 119898119894= 3 119886119899119889 119894 = 119905 ⩾ 3

119899 minus 119898119905

119894119891 119898119894⩾ 3 119886119899119889 119898

119905⩾ 4

(64)

Consider a complete 119905-partite graph 119870119905(3) for 119905 ⩾ 3

119887RM(119870119905(3)) = 119899 minus 1 by Theorem 157 where 119899 = 3119905

which shows that this upper bound of 119899 minus 1 on 119887RM(119866) inTheorem 156 is sharp This fact and 120574

119877(119870

119905(3)) = 4 give

a negative answer to the following two problems posed byDehgardi et al [121]Prove or Disprove for any connected graph 119866 of order 119899 ge 3(i) 119887RM(119866) = 119899 minus 1 if and only if 119866 cong 119870

3 (ii) 119887RM(119866) ⩽ 119899 minus

120574119877(119866) + 1Note that a complete 119905-partite graph 119870

119905(3) is (119899 minus 3)-

regular and 119887RM(119870119905(3)) = 119899 minus 1 for 119905 ⩾ 3 where 119899 = 3119905

Hu and Xu further determined that 119887119877(119866) = 119899 minus 2 for any

(119899 minus 3)-regular graph 119866 of order 119899 ⩽ 5 except 119870119905(3)

Theorem 158 (Hu and Xu [123] 2011) For any (119899minus3)-regulargraph 119866 of order 119899 other than119870

119905(3) 119887RM(119866) = 119899 minus 2 for 119899 ⩾ 5

For a tree 119879 with order 119899 ⩾ 3 Ebadi and PushpaLatha[120] and Jafari Rad and Volkmann [119] independentlyobtained an upper bound on 119887RM(119879)

Theorem 159 119887RM(119879) ⩽ 3 for any tree 119879 with order 119899 ⩾ 3

Theorem 160 (Bahremandpour et al [118] 2013) 119887RM(1198752 times119875119899) = 2 for 119899 ⩾ 2

Theorem 161 (Jafari Rad and Volkmann [119] 2011) Let119866 bea graph with order at least three

(a) If (119909 119910 119911) is a path of length 2 in 119866 then 119887RM(119866) ⩽119889119866(119909) + 119889

119866(119910) + 119889

119866(119911) minus 3 minus |119873

119866(119909) cap 119873

119866(119910)|

(b) If 119866 is connected then 119887RM(119866) ⩽ 120582(119866) + 2Δ(119866) minus 3

Theorem 161 (b) implies 119887RM(119866) ⩽ 120575(119866) + 2Δ(119866) minus 3 for aconnected graph 119866 Note that for a planar graph 119866 120575(119866) ⩽ 5moreover 120575(119866) ⩽ 3 if the girth at least 4 and 120575(119866) ⩽ 2 if thegirth at least 6 These two facts show that 119887RM(119866) ⩽ 2Δ(119866) +2 for a connected planar graph 119866 Jafari Rad and Volkmann[124] improved this bound

Theorem 162 (Jafari Rad and Volkmann [124] 2011) For anyconnected planar graph 119866 of order 119899 ⩾ 3 with girth 119892(119866)

119887RM (119866)

⩽

2Δ (119866)

Δ (119866) + 6

Δ (119866) + 5 119894119891 119866 119888119900119899119905119886119894119899119904 119899119900 V119890119903119905119894119888119890119904 119900119891 119889119890119892119903119890119890 119891119894V119890

Δ (119866) + 4 119894119891 119892 (119866) ⩾ 4

Δ (119866) + 3 119894119891 119892 (119866) ⩾ 5

Δ (119866) + 2 119894119891 119892 (119866) ⩾ 6

Δ (119866) + 1 119894119891 119892 (119866) ⩾ 8

(65)

According toTheorem 157 119887RM(119862119899) = 3 = Δ(119862

119899)+1 for a

cycle 119862119899of length 119899 ⩾ 8 with 119899 equiv 2 (mod 3) and therefore

the last upper bound inTheorem 162 is sharp at least for Δ =2

Combining the fact that every planar graph 119866 withminimum degree 5 contains an edge 119909119910 with 119889

119866(119909) = 5

and 119889119866(119910) isin 5 6 with Theorem 161 (a) Akbari et al [125]

obtained the following result

26 International Journal of Combinatorics

middot middot middot

middot middot middot

middot middot middot

middot middot middot

119866

Figure 9 The graph 119866 is constructed from 119866

Theorem 163 (Akbari et al [125] 2012) 119887RM(119866) ⩽ 15 forevery planar graph 119866

It remains open to show whether the bound inTheorem 163 is sharp or not Though finding a planargraph 119866 with 119887RM(119866) = 15 seems to be difficult Akbari etal [125] constructed an infinite family of planar graphs withRoman bondage number equal to 7 by proving the followingresult

Theorem 164 (Akbari et al [125] 2012) Let 119866 be a graph oforder 119899 and 119866 is the graph of order 5119899 obtained from 119866 byattaching the central vertex of a copy of a path 119875

5to each vertex

of119866 (see Figure 9) Then 120574RM(119866) = 4119899 and 119887RM(119866) = 120575(119866)+2

ByTheorem 164 infinitemany planar graphswith Romanbondage number 7 can be obtained by considering any planargraph 119866 with 120575(119866) = 5 (eg the icosahedron graph)

Conjecture 165 (Akbari et al [125] 2012) The Romanbondage number of every planar graph is at most 7

For general bounds the following observation is directlyobtained which is similar to Observation 1

Observation 4 Let 119866 be a graph of order 119899 with maximumdegree at least two Assume that 119867 is a spanning subgraphobtained from 119866 by removing 119896 edges If 120574RM(119867) = 120574RM(119866)then 119887

119877(119867) ⩽ 119887RM(119866) ⩽ 119887RM(119867) + 119896

Theorem 166 (Bahremandpour et al [118] 2013) For anyconnected graph 119866 of order 119899 if 119899 ⩾ 3 and 120574RM(119866) = 120574(119866) + 1then

119887RM (119866) ⩽ min 119887 (119866) 119899Δ (66)

where 119899Δis the number of vertices with maximum degree Δ in

119866

Observation 5 For any graph 119866 if 120574(119866) = 120574RM(119866) then119887RM(119866) ⩽ 119887(119866)

Theorem 167 (Bahremandpour et al [118] 2013) For everyRoman graph 119866

119887RM (119866) ⩾ 119887 (119866) (67)

The bound is sharp for cycles on 119899 vertices where 119899 equiv 0 (mod3)

The strict inequality in Theorem 167 can hold for exam-ple 119887(119862

3119896+2) = 2 lt 3 = 119887RM(1198623119896+2

) byTheorem 157A graph 119866 is called to be vertex Roman domination-

critical if 120574RM(119866 minus 119909) lt 120574RM(119866) for every vertex 119909 in 119866 If119866 has no isolated vertices then 120574RM(119866) ⩽ 2120574(119866) ⩽ 2120573(119866)If 120574RM(119866) = 2120573(119866) then 120574RM(119866) = 2120574(119866) and hence 119866 is aRoman graph In [30] Volkmann gave a lot of graphs with120574(119866) = 120573(119866) The following result is similar to Theorem 57

Theorem 168 (Bahremandpour et al [118] 2013) For anygraph 119866 if 120574RM(119866) = 2120573(119866) then

(a) 119887RM(119866) ⩾ 120575(119866)

(b) 119887RM(119866) ⩾ 120575(119866)+1 if119866 is a vertex Roman domination-critical graph

If 120574RM(119866) = 2 then obviously 119887RM(119866) ⩽ 120575(119866) So weassume 120574RM(119866) ⩾ 3 Dehgardi et al [121] proved 119887RM(119866) ⩽(120574RM(119866) minus 2)Δ(119866) + 1 and posed the following problem if 119866is a connected graph of order 119899 ⩾ 4 with 120574RM(119866) ⩾ 3 then

119887RM (119866) ⩽ (120574RM (119866) minus 2) Δ (119866) (68)

Theorem 161 (a) shows that the inequality (68) holds if120574RM(119866) ⩾ 5 Thus the bound in (68) is of interest only when120574RM(119866) is 3 or 4

Lemma 169 (Bahremandpour et al [118] 2013) For anynonempty graph 119866 of order 119899 ⩾ 3 120574RM(119866) = 3 if and onlyif Δ(119866) = 119899 minus 2

The following result shows that (68) holds for all graphs119866 of order 119899 ⩾ 4 with 120574RM(119866) = 3 or 4 which improvesTheorem 161 (b)

Theorem 170 (Bahremandpour et al [118] 2013) For anyconnected graph 119866 of order 119899 ⩾ 4

119887RM (119866) le Δ (119866) = 119899 minus 2 119894119891 120574RM (119866) = 3

Δ (119866) + 120575 (119866) minus 1 119894119891 120574RM (119866) = 4(69)

with the first equality if and only if 119866 cong 1198624

Recently Akbari and Qajar [122] have proved the follow-ing result

Theorem 171 (Akbari and Qajar [122]) For any connectedgraph 119866 of order 119899 ⩾ 3

119887RM (119866) ⩽ 119899 minus 120574RM (119866) + 5 (70)

International Journal of Combinatorics 27

We conclude this section with the following problems

Problem 8 Characterize all connected graphs 119866 of order 119899 ⩾3 for which 119887RM(119866) = 119899 minus 1

Problem 9 Prove or disprove if 119866 is a connected graph oforder 119899 ⩾ 3 then

119887RM (119866) ⩽ 119899 minus 120574RM (119866) + 3 (71)

Note that 120574119877(119870

119905(3)) = 4 and 119887RM(119870119905

(3)) = 119899minus1 where 119899 =3119905 for 119905 ⩾ 3 The upper bound on 119887RM(119866) given in Problem 9is sharp if Problem 9 has positive answer

95 Rainbow Bondage Numbers For a positive integer 119896 a119896-rainbow dominating function of a graph 119866 is a function 119891from the vertex-set 119881(119866) to the set of all subsets of the set1 2 119896 such that for any vertex 119909 in 119866 with 119891(119909) = 0 thecondition

⋃

119910isin119873119866(119909)

119891 (119910) = 1 2 119896 (72)

is fulfilledThe weight of a 119896-rainbow dominating function 119891on 119866 is the value

119891 (119866) = sum

119909isin119881(119866)

1003816100381610038161003816119891 (119909)1003816100381610038161003816 (73)

The 119896-rainbow domination number of a graph 119866 denoted by120574119903119896(119866) is the minimum weight of a 119896-rainbow dominating

function 119891 of 119866Note that 120574

1199031(119866) is the classical domination number 120574(119866)

The 119896-rainbow domination number is introduced by Bresaret al [126] Rainbow domination of a graph 119866 coincides withordinary domination of the Cartesian product of 119866 with thecomplete graph in particular 120574

119903119896(119866) = 120574(119866 times 119870

119896) for any

positive integer 119896 and any graph 119866 [126] Hartnell and Rall[127] proved that 120574(119879) lt 120574

1199032(119879) for any tree 119879 no graph has

120574 = 120574119903119896for 119896 ⩾ 3 for any 119896 ⩾ 2 and any graph 119866 of order 119899

min 119899 120574 (119866) + 119896 minus 2 ⩽ 120574119903119896(119866) ⩽ 119896120574 (119866) (74)

The introduction of rainbow domination is motivatedby the study of paired domination in Cartesian productsof graphs where certain upper bounds can be expressed interms of rainbow domination that is for any graph 119866 andany graph 119867 if 119881(119867) can be partitioned into 119896120574

119875-sets then

120574119875(119866 times 119867) ⩽ (1119896)120592(119867)120574

119903119896(119866) proved by Bresar et al [126]

Dehgardi et al [128] introduced the concept of 119896-rainbowbondage number of graphs Let 119866 be a graph with maximumdegree at least two The 119896-rainbow bondage number 119887

119903119896(119866)

of 119866 is the minimum cardinality of all sets 119861 sube 119864(119866) forwhich 120574

119903119896(119866 minus 119861) gt 120574

119903119896(119866) Since in the study of 119896-rainbow

bondage number the assumption Δ(119866) ⩾ 2 is necessarywe always assume that when we discuss 119887

119903119896(119866) all graphs

involved satisfy Δ(119866) ⩾ 2The 2-rainbow bondage number of some special families

of graphs has been determined

Theorem 172 (Dehgardi et al [128]) For any integer 119899 ⩾ 3

1198871199032(119875

119899) = 1 119891119900119903 119899 ⩾ 2

1198871199032(119862

119899) =

1 119894119891 119899 equiv 0 (mod 4) 2 119900119905ℎ119890119903119908119894119904119890

1198871199032(119870

119899119899) =

1 119894119891 119899 = 2

3 119894119891 119899 = 3

6 119894119891 119899 = 4

119898 119894119891 119899 ⩾ 5

(75)

Theorem 173 (Dehgardi et al [128]) For any connected graph119866 of order 119899 ⩾ 3 119887

1199032(119866) ⩽ 120582(119866) + 2Δ(119866) minus 3

Theorem 174 (Dehgardi et al [128]) For any tree 119879 of order119899 ⩾ 3 119887

1199032(119879) ⩽ 2

Theorem 175 (Dehgardi et al [128]) If 119866 is a planar graphwithmaximumdegree at least two then 119887

1199032(119866) ⩽ 15Moreover

if the girth 119892(119866) ⩾ 4 then 1198871199032(119866) ⩽ 11 and if 119892(119866) ⩾ 6 then

1198871199032(119866) ⩽ 9

Theorem 176 (Dehgardi et al [128]) For a complete bipartitegraph 119870

119901119902 if 2119896 + 1 ⩽ 119901 ⩽ 119902 then 119887

119903119896(119870

119901119902) = 119901

Theorem177 (Dehgardi et al [128]) For a complete graph119870119899

if 119899 ⩾ 119896 + 1 then 119887119903119896(119870

119899) = lceil119896119899(119896 + 1)rceil

The special case 119896 = 1 of Theorem 177 can be found inTheorem 1 The following is a simple observation similar toObservation 1

Observation 6 (Dehgardi et al [128]) Let 119866 be a graphwith maximum degree at least two 119867 a spanning subgraphobtained from 119866 by removing 119896 edges If 120574

119903119896(119867) = 120574

119903119896(119866)

then 119887119903119896(119867) ⩽ 119887

119903119896(119866) ⩽ 119887

119903119896(119867) + 119896

Theorem 178 (Dehgardi et al [128]) For every graph 119866 with120574119903119896(119866) = 119896120574(119866) 119887

119903119896(119866) ⩾ 119887(119866)

A graph 119866 is said to be vertex 119896-rainbow domination-critical if 120574

119903119896(119866 minus 119909) lt 120574

119903119896(119866) for every vertex 119909 in 119866 It is

clear that if 119866 has no isolated vertices then 120574119903119896(119866) ⩽ 119896120574(119866) ⩽

119896120573(119866) where 120573(119866) is the vertex-covering number of119866 Thusif 120574

119903119896(119866) = 119896120573(119866) then 120574

119903119896(119866) = 119896120574(119866)

Theorem 179 (Dehgardi et al [128]) For any graph 119866 if120574119903119896(119866) = 119896120573(119866) then

(a) 119887119903119896(119866) ⩾ 120575(119866)

(b) 119887119903119896(119866) ⩾ 120575(119866)+1 if119866 is vertex 119896-rainbow domination-

critical

As far as we know there are no more results on the 119896-rainbow bondage number of graphs in the literature

28 International Journal of Combinatorics

10 Results on Digraphs

Although domination has been extensively studied in undi-rected graphs it is natural to think of a dominating setas a one-way relationship between vertices of the graphIndeed among the earliest literatures on this subject vonNeumann and Morgenstern [129] used what is now calleddomination in digraphs to find solution (or kernels whichare independent dominating sets) for cooperative 119899-persongames Most likely the first formulation of domination byBerge [130] in 1958 was given in the context of digraphs andonly some years latter by Ore [131] in 1962 for undirectedgraphs Despite this history examination of domination andits variants in digraphs has been essentially overlooked (see[4] for an overview of the domination literature) Thus thereare few if any such results on domination for digraphs in theliterature

The bondage number and its related topics for undirectedgraph have become one of major areas both in theoreticaland applied researches However until recently Carlsonand Develin [42] Shan and Kang [132] and Huang andXu [133 134] studied the bondage number for digraphsindependently In this section we introduce their resultsfor general digraphs Results for some special digraphs suchas vertex-transitive digraphs are introduced in the nextsection

101 Upper Bounds for General Digraphs Let 119866 = (119881 119864)

be a digraph without loops and parallel edges A digraph 119866is called to be asymmetric if whenever (119909 119910) isin 119864(119866) then(119910 119909) notin 119864(119866) and to be symmetric if (119909 119910) isin 119864(119866) implies(119910 119909) isin 119864(119866) For a vertex 119909 of 119881(119866) the sets of out-neighbors and in-neighbors of 119909 are respectively defined as119873

+

119866(119909) = 119910 isin 119881(119866) (119909 119910) isin 119864(119866) and 119873minus

119866(119909) = 119909 isin

119881(119866) (119909 119910) isin 119864(119866) the out-degree and the in-degree of119909 are respectively defined as 119889+

119866(119909) = |119873

+

119866(119909)| and 119889minus

119866(119909) =

|119873+

119866(119909)| Denote themaximum and theminimum out-degree

(resp in-degree) of 119866 by Δ+(119866) and 120575+(119866) (resp Δminus(119866) and120575minus

(119866))The degree 119889119866(119909) of 119909 is defined as 119889+

119866(119909)+119889

minus

119866(119909) and

the maximum and the minimum degrees of119866 are denoted byΔ(119866) and 120575(119866) respectively that is Δ(119866) = max119889

119866(119909) 119909 isin

119881(119866) and 120575(119866) = min119889119866(119909) 119909 isin 119881(119866) Note that the

definitions here are different from the ones in the textbookon digraphs see for example Xu [1]

A subset 119878 of119881(119866) is called a dominating set if119881(119866) = 119878cup119873

+

119866(119878) where119873+

119866(119878) is the set of out-neighbors of 119878Then just

as for undirected graphs 120574(119866) is the minimum cardinalityof a dominating set and the bondage number 119887(119866) is thesmallest cardinality of a set 119861 of edges such that 120574(119866 minus 119861) gt120574(119866) if such a subset 119861 sube 119864(119866) exists Otherwise we put119887(119866) = infin

Some basic results for undirected graphs stated inSection 3 can be generalized to digraphs For exampleTheorem 18 is generalized by Carlson and Develin [42]Huang andXu [133] and Shan andKang [132] independentlyas follows

Theorem 180 Let 119866 be a digraph and (119909 119910) isin 119864(119866) Then119887(119866) ⩽ 119889

119866(119910) + 119889

minus

119866(119909) minus |119873

minus

119866(119909) cap 119873

minus

119866(119910)|

Corollary 181 For a digraph 119866 119887(119866) ⩽ 120575minus(119866) + Δ(119866)

Since 120575minus

(119866) ⩽ (12)Δ(119866) from Corollary 181Conjecture 53 is valid for digraphs

Corollary 182 For a digraph 119866 119887(119866) ⩽ (32)Δ(119866)

In the case of undirected graphs the bondage number of119866119899= 119870

119899times 119870

119899achieves this bound However it was shown

by Carlson and Develin in [42] that if we take the symmetricdigraph119866

119899 we haveΔ(119866

119899) = 4(119899minus1) 120574(119866

119899) = 119899 and 119887(119866

119899) ⩽

4(119899 minus 1) + 1 = Δ(119866119899) + 1 So this family of digraphs cannot

show that the bound in Corollary 182 is tightCorresponding to Conjecture 51 which is discredited

for undirected graphs and is valid for digraphs the sameconjecture for digraphs can be proposed as follows

Conjecture 183 (Carlson and Develin [42] 2006) 119887(119866) ⩽Δ(119866) + 1 for any digraph 119866

If Conjecture 183 is true then the following results showsthat this upper bound is tight since 119887(119870

119899times119870

119899) = Δ(119870

119899times119870

119899)+1

for a complete digraph 119870119899

In 2007 Shan and Kang [132] gave some tight upperbounds on the bondage numbers for some asymmetricdigraphs For example 119887(119879) ⩽ Δ(119879) for any asymmetricdirected tree 119879 119887(119866) ⩽ Δ(119866) for any asymmetric digraph 119866with order at least 4 and 120574(119866) ⩽ 2 For planar digraphs theyobtained the following results

Theorem 184 (Shan and Kang [132] 2007) Let 119866 be aasymmetric planar digraphThen 119887(119866) ⩽ Δ(119866)+2 and 119887(119866) ⩽Δ(119866) + 1 if Δ(119866) ⩾ 5 and 119889minus

119866(119909) ⩾ 3 for every vertex 119909 with

119889119866(119909) ⩾ 4

102 Results for Some Special Digraphs The exact values andbounds on 119887(119866) for some standard digraphs are determined

Theorem185 (Huang andXu [133] 2006) For a directed cycle119862119899and a directed path 119875

119899

119887 (119862119899) =

3 119894119891 119899 119894119904 119900119889119889

2 119894119891 119899 119894119904 119890V119890119899

119887 (119875119899) =

2 119894119891 119899 119894119904 119900119889119889

1 119894119891 119899 119894119904 119890V119890119899

(76)

For the de Bruijn digraph 119861(119889 119899) and the Kautz digraph119870(119889 119899)

119887 (119861 (119889 119899)) = 119889 119894119891 119899 119894119904 119900119889119889

119889 ⩽ 119887 (119861 (119889 119899)) ⩽ 2119889 119894119891 119899 119894119904 119890V119890119899

119887 (119870 (119889 119899)) = 119889 + 1

(77)

Like undirected graphs we can define the total domi-nation number and the total bondage number On the totalbondage numbers for some special digraphs the knownresults are as follows

International Journal of Combinatorics 29

Theorem 186 (Huang and Xu [134] 2007) For a directedcycle 119862

119899and a directed path 119875

119899 119887

119879(119875

119899) and 119887

119879(119862

119899) all do not

exist For a complete digraph 119870119899

119887119879(119870

119899) = infin 119894119891 119899 = 1 2

119887119879(119870

119899) = 3 119894119891 119899 = 3

119899 ⩽ 119887119879(119870

119899) ⩽ 2119899 minus 3 119894119891 119899 ⩾ 4

(78)

The extended de Bruijn digraph EB(119889 119899 1199021 119902

119901) and

the extended Kautz digraph EK(119889 119899 1199021 119902

119901) are intro-

duced by Shibata and Gonda [135] If 119901 = 1 then theyare the de Bruijn digraph B(119889 119899) and the Kautz digraphK(119889 119899) respectively Huang and Xu [134] determined theirtotal domination numbers In particular their total bondagenumbers for general cases are determined as follows

Theorem 187 (Huang andXu [134] 2007) If 119889 ⩾ 2 and 119902119894⩾ 2

for each 119894 = 1 2 119901 then

119887119879(EB(119889 119899 119902

1 119902

119901)) = 119889

119901

minus 1

119887119879(EK(119889 119899 119902

1 119902

119901)) = 119889

119901

(79)

In particular for the de Bruijn digraph B(119889 119899) and the Kautzdigraph K(119889 119899)

119887119879(B (119889 119899)) = 119889 minus 1 119887

119879(K (119889 119899)) = 119889 (80)

Zhang et al [136] determined the bondage number incomplete 119905-partite digraphs

Theorem 188 (Zhang et al [136] 2009) For a complete 119905-partite digraph 119870

11989911198992119899119905

where 1198991⩽ 119899

2⩽ sdot sdot sdot ⩽ 119899

119905

119887 (11987011989911198992119899119905

)

=

119898 119894119891 119899119898= 1 119899

119898+1⩾ 2 119891119900119903 119904119900119898119890

119898 (1 ⩽ 119898 lt 119905)

4119905 minus 3 119894119891 1198991= 119899

2= sdot sdot sdot = 119899

119905= 2

119905minus1

sum

119894=1

119899119894+ 2 (119905 minus 1) 119900119905ℎ119890119903119908119894119904119890

(81)

Since an undirected graph can be thought of as a sym-metric digraph any result for digraphs has an analogy forundirected graphs in general In view of this point studyingthe bondage number for digraphs is more significant thanfor undirected graphs Thus we should further study thebondage number of digraphs and try to generalize knownresults on the bondage number and related variants for undi-rected graphs to digraphs prove or disprove Conjecture 183In particular determine the exact values of 119887(B(119889 119899)) for aneven 119899 and 119887

119879(119870

119899) for 119899 ⩾ 4

11 Efficient Dominating Sets

A dominating set 119878 of a graph 119866 is called to be efficient if forevery vertex 119909 in119866 |119873

119866[119909]cap119878| = 1 if119866 is a undirected graph

or |119873minus

119866[119909] cap 119878| = 1 if 119866 is a directed graph The concept of an

efficient set was proposed by Bange et al [137] in 1988 In theliterature an efficient set is also called a perfect dominatingset (see Livingston and Stout [138] for an explanation)

From definition if 119878 is an efficient dominating set of agraph 119866 then 119878 is certainly an independent set and everyvertex not in 119878 is adjacent to exactly one vertex in 119878

It is also clear from definition that a dominating set 119878 isefficient if and only if N

119866[119878] = 119873

119866[119909] 119909 isin 119878 for the

undirected graph 119866 or N+

119866[119878] = 119873

+

119866[119909] 119909 isin 119878 for the

digraph119866 is a partition of119881(119866) where the induced subgraphby119873

119866[119909] or119873+

119866[119909] is a star or an out-star with the root 119909

The efficient domination has important applications inmany areas such as error-correcting codes and receivesmuch attention in the late years

The concept of efficient dominating sets is a measureof the efficiency of domination in graphs Unfortunately asshown in [137] not every graph has an efficient dominatingset andmoreover it is anNP-complete problem to determinewhether a given graph has an efficient dominating set Inaddition it was shown by Clark [139] in 1993 that for a widerange of 119901 almost every random undirected graph 119866 isin

G(120592 119901) has no efficient dominating sets This means thatundirected graphs possessing an efficient dominating set arerare However it is easy to show that every undirected graphhas an orientationwith an efficient dominating set (see Bangeet al [140])

In 1993 Barkauskas and Host [141] showed that deter-mining whether an arbitrary oriented graph has an efficientdominating set is NP-complete Even so the existence ofefficient dominating sets for some special classes of graphs hasbeen examined see for example Dejter and Serra [142] andLee [143] for Cayley graph Gu et al [144] formeshes and toriObradovic et al [145] Huang and Xu [36] Reji Kumar andMacGillivray [146] for circulant graphs Harary graphs andtori and Van Wieren et al [147] for cube-connected cycles

In this section we introduce results of the bondagenumber for some graphs with efficient dominating sets dueto Huang and Xu [36]

111 Results for General Graphs In this subsection we intro-duce some results on bondage numbers obtained by applyingefficient dominating sets We first state the two followinglemmas

Lemma 189 For any 119896-regular graph or digraph 119866 of order 119899120574(119866) ⩾ 119899(119896 + 1) with equality if and only if 119866 has an efficientdominating set In addition if 119866 has an efficient dominatingset then every efficient dominating set is certainly a 120574-set andvice versa

Let 119890 be an edge and 119878 a dominating set in 119866 We say that119890 supports 119878 if 119890 isin (119878 119878) where (119878 119878) = (119909 119910) isin 119864(119866) 119909 isin 119878 119910 isin 119878 Denote by 119904(119866) the minimum number of edgeswhich support all 120574-sets in 119866

Lemma 190 For any graph or digraph 119866 119887(119866) ⩾ 119904(119866) withequality if 119866 is regular and has an efficient dominating set

30 International Journal of Combinatorics

A graph 119866 is called to be vertex-transitive if its automor-phism group Aut(119866) acts transitively on its vertex-set 119881(119866)A vertex-transitive graph is regular Applying Lemmas 189and 190 Huang and Xu obtained some results on bondagenumbers for vertex-transitive graphs or digraphs

Theorem 191 (Huang and Xu [36] 2008) For any vertex-transitive graph or digraph 119866 of order 119899

119887 (119866) ⩾

lceil119899

2120574 (119866)rceil 119894119891 119866 119894119904 119906119899119889119894119903119890119888119905119890119889

lceil119899

120574 (119866)rceil 119894119891 119866 119894119904 119889119894119903119890119888119905119890119889

(82)

Theorem192 (Huang andXu [36] 2008) Let119866 be a 119896-regulargraph of order 119899 Then

119887(119866) ⩽ 119896 if119866 is undirected and 119899 ⩾ 120574(119866)(119896+1)minus119896+1119887(119866) ⩽ 119896+1+ℓ if119866 is directed and 119899 ⩾ 120574(119866)(119896+1)minusℓwith 0 ⩽ ℓ ⩽ 119896 minus 1

Next we introduce a better upper bound on 119887(119866) To thisaim we need the following concept which is a generalizationof the concept of the edge-covering of a graph 119866 For 1198811015840

sube

119881(119866) and 1198641015840 sube 119864(119866) we say that 1198641015840covers 1198811015840 and call 1198641015840an edge-covering for 1198811015840 if there exists an edge (119909 119910) isin 1198641015840 forany vertex 119909 isin 1198811015840 For 119910 isin 119881(119866) let 1205731015840[119910] be the minimumcardinality over all edge-coverings for119873minus

119866[119910]

Theorem 193 (Huang and Xu [36] 2008) If 119866 is a 119896-regulargraph with order 120574(119866)(119896 + 1) then 119887(119866) ⩽ 1205731015840[119910] for any 119910 isin119881(119866)

Theupper bound on 119887(119866) given inTheorem 193 is tight inview of 119887(119862

119899) for a cycle or a directed cycle 119862

119899(seeTheorems

1 and 185 resp)It is easy to see that for a 119896-regular graph 119866 lceil(119896 + 1)2rceil ⩽

1205731015840

[119910] ⩽ 119896 when 119866 is undirected and 1205731015840[119910] = 119896 + 1 when 119866 isdirected By this fact and Lemma 189 the following theoremis merely a simple combination of Theorems 191 and 193

Theorem 194 (Huang and Xu [36] 2008) Let 119866 be a vertex-transitive graph of degree 119896 If 119866 has an efficient dominatingset then

lceil119896 + 1

2rceil ⩽ 119887 (119866) ⩽ 119896 119894119891 119866 119894119904 119906119899119889119894119903119890119888119905119890119889

119887 (119866) = 119896 + 1 119894119891 119866 119894119904 119889119894119903119890119888119905119890119889

(83)

Theorem 195 (Huang and Xu [36] 2008) If 119866 is an undi-rected vertex-transitive cubic graph with order 4120574(119866) and girth119892(119866) ⩽ 5 then 119887(119866) = 2

The proof of Theorem 195 leads to a byproduct If 119892 =5 let (119906

1 119906

2 119906

3 119906

4 119906

5) be a cycle and let D

119894be the family

of efficient dominating sets containing 119906119894for each 119894 isin

1 2 3 4 5 ThenD119894capD

119895= 0 for 119894 = 1 2 5 Then 119866 has

at least 5119904 efficient dominating sets where 119904 = 119904(119866) But thereare only 4s (=ns120574) distinct efficient dominating sets in119866This

contradiction implies that an undirected vertex-transitivecubic graph with girth five has no efficient dominating setsBut a similar argument for 119892(119866) = 3 4 or 119892(119866) ⩾ 6 couldnot give any contradiction This is consistent with the resultthat CCC(119899) a vertex-transitive cubic graph with girth 119899 if3 ⩽ 119899 ⩽ 8 or girth 8 if 119899 ⩾ 9 has efficient dominating setsfor all 119899 ⩾ 3 except 119899 = 5 (see Theorem 199 in the followingsubsection)

112 Results for Cayley Graphs In this subsection we useTheorem 194 to determine the exact values or approximativevalues of bondage numbers for some special vertex-transitivegraphs by characterizing the existence of efficient dominatingsets in these graphs

Let Γ be a nontrivial finite group 119878 a nonempty subset ofΓ without the identity element of Γ A digraph 119866 is defined asfollows

119881 (119866) = Γ (119909 119910) isin 119864 (119866) lArrrArr 119909minus1

119910 isin 119878

for any 119909 119910 isin Γ(84)

is called a Cayley digraph of the group Γ with respect to119878 denoted by 119862

Γ(119878) If 119878minus1 = 119904minus1 119904 isin 119878 = 119878 then

119862Γ(119878) is symmetric and is called a Cayley undirected graph

a Cayley graph for short Cayley graphs or digraphs arecertainly vertex-transitive (see Xu [1])

A circulant graph 119866(119899 119878) of order 119899 is a Cayley graph119862119885119899

(119878) where 119885119899= 0 119899 minus 1 is the addition group of

order 119899 and 119878 is a nonempty subset of119885119899without the identity

element and hence is a vertex-transitive digraph of degree|119878| If 119878minus1 = 119878 then119866(119899 119878) is an undirected graph If 119878 = 1 119896where 2 ⩽ 119896 ⩽ 119899 minus 2 we write 119866(119899 1 119896) for 119866(119899 1 119896) or119866(119899 plusmn1 plusmn119896) and call it a double-loop circulant graph

Huang and Xu [36] showed that for directed 119866 =

119866(119899 1 119896) lceil1198993rceil ⩽ 120574(119866) ⩽ lceil1198992rceil and 119866 has an efficientdominating set if and only if 3 | 119899 and 119896 equiv 2 (mod 3) fordirected 119866 = 119866(119899 1 119896) and 119896 = 1198992 lceil1198995rceil ⩽ 120574(119866) ⩽ lceil1198993rceiland 119866 has an efficient dominating set if and only if 5 | 119899 and119896 equiv plusmn2 (mod 5) By Theorem 194 we obtain the bondagenumber of a double loop circulant graph if it has an efficientdominating set

Theorem 196 (Huang and Xu [36] 2008) Let 119866 be a double-loop circulant graph 119866(119899 1 119896) If 119866 is directed with 3 | 119899and 119896 equiv 2 (mod 3) or 119866 is undirected with 5 | 119899 and119896 equiv plusmn2 (mod 5) then 119887(119866) = 3

The 119898 times 119899 torus is the Cartesian product 119862119898times 119862

119899of

two cycles and is a Cayley graph 119862119885119898times119885119899

(119878) where 119878 =

(0 1) (1 0) for directed cycles and 119878 = (0 plusmn1) (plusmn1 0) forundirected cycles and hence is vertex-transitive Gu et al[144] showed that the undirected torus119862

119898times119862

119899has an efficient

dominating set if and only if both 119898 and 119899 are multiplesof 5 Huang and Xu [36] showed that the directed torus119862119898times 119862

119899has an efficient dominating set if and only if both

119898 and 119899 are multiples of 3 Moreover they found a necessarycondition for a dominating set containing the vertex (0 0)in 119862

119898times 119862

119899to be efficient and obtained the following

result

International Journal of Combinatorics 31

Theorem 197 (Huang and Xu [36] 2008) Let 119866 = 119862119898times 119862

119899

If 119866 is undirected and both119898 and 119899 are multiples of 5 or if 119866 isdirected and both119898 and 119899 are multiples of 3 then 119887(119866) = 3

The hypercube 119876119899

is the Cayley graph 119862Γ(119878)

where Γ = 1198852times sdot sdot sdot times 119885

2= (119885

2)119899 and 119878 =

100 sdot sdot sdot 0 010 sdot sdot sdot 0 00 sdot sdot sdot 01 Lee [143] showed that119876119899has an efficient dominating set if and only if 119899 = 2119898 minus 1

for a positive integer119898 ByTheorem 194 the following resultis obtained immediately

Theorem 198 (Huang and Xu [36] 2008) If 119899 = 2119898 minus 1 for apositive integer119898 then 2119898minus1

⩽ 119887(119876119899) ⩽ 2

119898

minus 1

Problem 10 Determine the exact value of 119887(119876119899) for 119899 = 2119898minus1

The 119899-dimensional cube-connected cycle denoted byCCC(119899) is constructed from the 119899-dimensional hypercube119876119899by replacing each vertex 119909 in 119876

119899with an undirected cycle

119862119899of length 119899 and linking the 119894th vertex of the 119862

119899to the 119894th

neighbor of 119909 It has been proved that CCC(119899) is a Cayleygraph and hence is a vertex-transitive graph with degree 3Van Wieren et al [147] proved that CCC(119899) has an efficientdominating set if and only if 119899 = 5 From Theorems 194 and195 the following result can be derived

Theorem 199 (Huang and Xu [36] 2008) Let 119866 = 119862119862119862(119899)be the 119899-dimensional cube-connected cycles with 119899 ⩾ 3 and119899 = 5 Then 120574(119866) = 1198992119899minus2 and 2 ⩽ 119887(119866) ⩽ 3 In addition119887(119862119862119862(3)) = 119887(119862119862119862(4)) = 2

Problem 11 Determine the exact value of 119887(CCC(119899)) for 119899 ⩾5

Acknowledgments

This work was supported by NNSF of China (Grants nos11071233 61272008) The author would like to express hisgratitude to the anonymous referees for their kind sugges-tions and comments on the original paper to ProfessorSheikholeslami and Professor Jafari Rad for sending thereferences [128] and [81] which resulted in this version

References

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[2] S THedetniemi andR C Laskar ldquoBibliography on dominationin graphs and some basic definitions of domination parame-tersrdquo Discrete Mathematics vol 86 no 1ndash3 pp 257ndash277 1990

[3] TW Haynes S T Hedetniemi and P J Slater Fundamentals ofDomination in Graphs vol 208 ofMonographs and Textbooks inPure and Applied Mathematics Marcel Dekker New York NYUSA 1998

[4] TWHaynes S THedetniemi and P J Slater EdsDominationin Graphs vol 209 of Monographs and Textbooks in Pure andAppliedMathematicsMarcelDekker NewYorkNYUSA 1998

[5] M R Garey andD S JohnsonComputers and Intractability WH Freeman and Company San Francisco Calif USA 1979

[6] H B Walikar and B D Acharya ldquoDomination critical graphsrdquoNational Academy Science Letters vol 2 pp 70ndash72 1979

[7] D Bauer F Harary J Nieminen and C L Suffel ldquoDominationalteration sets in graphsrdquo Discrete Mathematics vol 47 no 2-3pp 153ndash161 1983

[8] J F Fink M S Jacobson L F Kinch and J Roberts ldquoThebondage number of a graphrdquo Discrete Mathematics vol 86 no1ndash3 pp 47ndash57 1990

[9] F-T Hu and J-M Xu ldquoThe bondage number of (119899 minus 3)-regulargraph G of order nrdquo Ars Combinatoria to appear

[10] U Teschner ldquoNew results about the bondage number of agraphrdquoDiscreteMathematics vol 171 no 1ndash3 pp 249ndash259 1997

[11] B L Hartnell and D F Rall ldquoA bound on the size of a graphwith given order and bondage numberrdquo Discrete Mathematicsvol 197-198 pp 409ndash413 1999 16th British CombinatorialConference (London 1997)

[12] B L Hartnell and D F Rall ldquoA characterization of trees inwhich no edge is essential to the domination numberrdquo ArsCombinatoria vol 33 pp 65ndash76 1992

[13] J Topp and P D Vestergaard ldquo120572119896- and 120574

119896-stable graphsrdquo

Discrete Mathematics vol 212 no 1-2 pp 149ndash160 2000[14] A Meir and J W Moon ldquoRelations between packing and

covering numbers of a treerdquo Pacific Journal of Mathematics vol61 no 1 pp 225ndash233 1975

[15] B L Hartnell L K Joslashrgensen P D Vestergaard and CA Whitehead ldquoEdge stability of the k-domination numberof treesrdquo Bulletin of the Institute of Combinatorics and ItsApplications vol 22 pp 31ndash40 1998

[16] F-T Hu and J-M Xu ldquoOn the complexity of the bondage andreinforcement problemsrdquo Journal of Complexity vol 28 no 2pp 192ndash201 2012

[17] B L Hartnell and D F Rall ldquoBounds on the bondage numberof a graphrdquo Discrete Mathematics vol 128 no 1ndash3 pp 173ndash1771994

[18] Y-L Wang ldquoOn the bondage number of a graphrdquo DiscreteMathematics vol 159 no 1ndash3 pp 291ndash294 1996

[19] G H Fricke TW Haynes S M Hedetniemi S T Hedetniemiand R C Laskar ldquoExcellent treesrdquo Bulletin of the Institute ofCombinatorics and Its Applications vol 34 pp 27ndash38 2002

[20] V Samodivkin ldquoThe bondage number of graphs good and badverticesrdquoDiscussiones Mathematicae GraphTheory vol 28 no3 pp 453ndash462 2008

[21] J E Dunbar T W Haynes U Teschner and L VolkmannldquoBondage insensitivity and reinforcementrdquo in Domination inGraphs vol 209 of Monographs and Textbooks in Pure andAppliedMathematics pp 471ndash489 Dekker NewYork NY USA1998

[22] H Liu and L Sun ldquoThe bondage and connectivity of a graphrdquoDiscrete Mathematics vol 263 no 1ndash3 pp 289ndash293 2003

[23] A-H Esfahanian and S L Hakimi ldquoOn computing a con-ditional edge-connectivity of a graphrdquo Information ProcessingLetters vol 27 no 4 pp 195ndash199 1988

[24] R C Brigham P Z Chinn and R D Dutton ldquoVertexdomination-critical graphsrdquoNetworks vol 18 no 3 pp 173ndash1791988

[25] V Samodivkin ldquoDomination with respect to nondegenerateproperties bondage numberrdquoThe Australasian Journal of Com-binatorics vol 45 pp 217ndash226 2009

[26] U Teschner ldquoA new upper bound for the bondage numberof graphs with small domination numberrdquo The AustralasianJournal of Combinatorics vol 12 pp 27ndash35 1995

32 International Journal of Combinatorics

[27] J Fulman D Hanson and G MacGillivray ldquoVertex dom-ination-critical graphsrdquoNetworks vol 25 no 2 pp 41ndash43 1995

[28] U Teschner ldquoA counterexample to a conjecture on the bondagenumber of a graphrdquo Discrete Mathematics vol 122 no 1ndash3 pp393ndash395 1993

[29] U Teschner ldquoThe bondage number of a graph G can be muchgreater than Δ(G)rdquo Ars Combinatoria vol 43 pp 81ndash87 1996

[30] L Volkmann ldquoOn graphs with equal domination and coveringnumbersrdquoDiscrete AppliedMathematics vol 51 no 1-2 pp 211ndash217 1994

[31] L A Sanchis ldquoMaximum number of edges in connected graphswith a given domination numberrdquoDiscreteMathematics vol 87no 1 pp 65ndash72 1991

[32] S Klavzar and N Seifter ldquoDominating Cartesian products ofcyclesrdquoDiscrete AppliedMathematics vol 59 no 2 pp 129ndash1361995

[33] H K Kim ldquoA note on Vizingrsquos conjecture about the bondagenumberrdquo Korean Journal of Mathematics vol 10 no 2 pp 39ndash42 2003

[34] M Y Sohn X Yuan and H S Jeong ldquoThe bondage number of1198623times119862

119899rdquo Journal of the KoreanMathematical Society vol 44 no

6 pp 1213ndash1231 2007[35] L Kang M Y Sohn and H K Kim ldquoBondage number of the

discrete torus 119862119899times 119862

4rdquo Discrete Mathematics vol 303 no 1ndash3

pp 80ndash86 2005[36] J Huang and J-M Xu ldquoThe bondage numbers and efficient

dominations of vertex-transitive graphsrdquoDiscrete Mathematicsvol 308 no 4 pp 571ndash582 2008

[37] C J Xiang X Yuan andM Y Sohn ldquoDomination and bondagenumber of119862

3times119862

119899rdquoArs Combinatoria vol 97 pp 299ndash310 2010

[38] M S Jacobson and L F Kinch ldquoOn the domination numberof products of graphs Irdquo Ars Combinatoria vol 18 pp 33ndash441984

[39] Y-Q Li ldquoA note on the bondage number of a graphrdquo ChineseQuarterly Journal of Mathematics vol 9 no 4 pp 1ndash3 1994

[40] F-T Hu and J-M Xu ldquoBondage number of mesh networksrdquoFrontiers ofMathematics inChina vol 7 no 5 pp 813ndash826 2012

[41] R Frucht and F Harary ldquoOn the corona of two graphsrdquoAequationes Mathematicae vol 4 pp 322ndash325 1970

[42] K Carlson and M Develin ldquoOn the bondage number of planarand directed graphsrdquoDiscreteMathematics vol 306 no 8-9 pp820ndash826 2006

[43] U Teschner ldquoOn the bondage number of block graphsrdquo ArsCombinatoria vol 46 pp 25ndash32 1997

[44] J Huang and J-M Xu ldquoNote on conjectures of bondagenumbers of planar graphsrdquo Applied Mathematical Sciences vol6 no 65ndash68 pp 3277ndash3287 2012

[45] L Kang and J Yuan ldquoBondage number of planar graphsrdquoDiscrete Mathematics vol 222 no 1ndash3 pp 191ndash198 2000

[46] M Fischermann D Rautenbach and L Volkmann ldquoRemarkson the bondage number of planar graphsrdquo Discrete Mathemat-ics vol 260 no 1ndash3 pp 57ndash67 2003

[47] J Hou G Liu and J Wu ldquoBondage number of planar graphswithout small cyclesrdquoUtilitasMathematica vol 84 pp 189ndash1992011

[48] J Huang and J-M Xu ldquoThe bondage numbers of graphs withsmall crossing numbersrdquo Discrete Mathematics vol 307 no 15pp 1881ndash1897 2007

[49] Q Ma S Zhang and J Wang ldquoBondage number of 1-planargraphrdquo Applied Mathematics vol 1 no 2 pp 101ndash103 2010

[50] J Hou and G Liu ldquoA bound on the bondage number of toroidalgraphsrdquoDiscreteMathematics Algorithms and Applications vol4 no 3 Article ID 1250046 5 pages 2012

[51] Y-C Cao J-M Xu and X-R Xu ldquoOn bondage number oftoroidal graphsrdquo Journal of University of Science and Technologyof China vol 39 no 3 pp 225ndash228 2009

[52] Y-C Cao J Huang and J-M Xu ldquoThe bondage number ofgraphs with crossing number less than fourrdquo Ars Combinatoriato appear

[53] H K Kim ldquoOn the bondage numbers of some graphsrdquo KoreanJournal of Mathematics vol 11 no 2 pp 11ndash15 2004

[54] A Gagarin and V Zverovich ldquoUpper bounds for the bondagenumber of graphs on topological surfacesrdquo Discrete Mathemat-ics 2012

[55] J Huang ldquoAn improved upper bound for the bondage numberof graphs on surfacesrdquoDiscrete Mathematics vol 312 no 18 pp2776ndash2781 2012

[56] A Gagarin and V Zverovich ldquoThe bondage number of graphson topological surfaces and Teschnerrsquos conjecturerdquo DiscreteMathematics vol 313 no 6 pp 796ndash808 2013

[57] V Samodivkin ldquoNote on the bondage number of graphs ontopological surfacesrdquo httparxivorgabs12086203

[58] E J Cockayne R M Dawes and S T Hedetniemi ldquoTotaldomination in graphsrdquoNetworks vol 10 no 3 pp 211ndash219 1980

[59] R Laskar J Pfaff S M Hedetniemi and S T HedetniemildquoOn the algorithmic complexity of total dominationrdquo Society forIndustrial and Applied Mathematics vol 5 no 3 pp 420ndash4251984

[60] J Pfaff R C Laskar and S T Hedetniemi ldquoNP-completeness oftotal and connected domination and irredundance for bipartitegraphsrdquo Tech Rep 428 Department of Mathematical SciencesClemson University 1983

[61] M A Henning ldquoA survey of selected recent results on totaldomination in graphsrdquoDiscrete Mathematics vol 309 no 1 pp32ndash63 2009

[62] V R Kulli and D K Patwari ldquoThe total bondage number ofa graphrdquo in Advances in Graph Theory pp 227ndash235 VishwaGulbarga India 1991

[63] F-T Hu Y Lu and J-M Xu ldquoThe total bondage number ofgrid graphsrdquo Discrete Applied Mathematics vol 160 no 16-17pp 2408ndash2418 2012

[64] J Cao M Shi and B Wu ldquoResearch on the total bondagenumber of a spectial networkrdquo in Proceedings of the 3rdInternational Joint Conference on Computational Sciences andOptimization (CSO rsquo10) pp 456ndash458 May 2010

[65] N Sridharan MD Elias and V S A Subramanian ldquoTotalbondage number of a graphrdquo AKCE International Journal ofGraphs and Combinatorics vol 4 no 2 pp 203ndash209 2007

[66] N Jafari Rad and J Raczek ldquoSome progress on total bondage ingraphsrdquo Graphs and Combinatorics 2011

[67] T W Haynes and P J Slater ldquoPaired-domination and thepaired-domatic numberrdquoCongressusNumerantium vol 109 pp65ndash72 1995

[68] T W Haynes and P J Slater ldquoPaired-domination in graphsrdquoNetworks vol 32 no 3 pp 199ndash206 1998

[69] X Hou ldquoA characterization of 2120574 120574119875-treesrdquoDiscrete Mathemat-

ics vol 308 no 15 pp 3420ndash3426 2008[70] K E Proffitt T W Haynes and P J Slater ldquoPaired-domination

in grid graphsrdquo Congressus Numerantium vol 150 pp 161ndash1722001

International Journal of Combinatorics 33

[71] H Qiao L KangM Cardei andD-Z Du ldquoPaired-dominationof treesrdquo Journal of Global Optimization vol 25 no 1 pp 43ndash542003

[72] E Shan L Kang and M A Henning ldquoA characterizationof trees with equal total domination and paired-dominationnumbersrdquo The Australasian Journal of Combinatorics vol 30pp 31ndash39 2004

[73] J Raczek ldquoPaired bondage in treesrdquo Discrete Mathematics vol308 no 23 pp 5570ndash5575 2008

[74] J A Telle and A Proskurowski ldquoAlgorithms for vertex parti-tioning problems on partial k-treesrdquo SIAM Journal on DiscreteMathematics vol 10 no 4 pp 529ndash550 1997

[75] G S Domke J H Hattingh M A Henning and L R MarkusldquoRestrained domination in treesrdquoDiscreteMathematics vol 211no 1ndash3 pp 1ndash9 2000

[76] G S Domke J H Hattingh M A Henning and L R MarkusldquoRestrained domination in graphs with minimum degree twordquoJournal of Combinatorial Mathematics and Combinatorial Com-puting vol 35 pp 239ndash254 2000

[77] G S Domke J H Hattingh S T Hedetniemi R C Laskarand L R Markus ldquoRestrained domination in graphsrdquo DiscreteMathematics vol 203 no 1ndash3 pp 61ndash69 1999

[78] J H Hattingh and E J Joubert ldquoAn upper bound for therestrained domination number of a graph with minimumdegree at least two in terms of order and minimum degreerdquoDiscrete Applied Mathematics vol 157 no 13 pp 2846ndash28582009

[79] M A Henning ldquoGraphs with large restrained dominationnumberrdquo Discrete Mathematics vol 197-198 pp 415ndash429 199916th British Combinatorial Conference (London 1997)

[80] J H Hattingh and A R Plummer ldquoRestrained bondage ingraphsrdquo Discrete Mathematics vol 308 no 23 pp 5446ndash54532008

[81] N Jafari Rad R Hasni J Raczek and L Volkmann ldquoTotalrestrained bondage in graphsrdquoActaMathematica Sinica vol 29no 6 pp 1033ndash1042 2013

[82] J Cyman and J Raczek ldquoOn the total restrained dominationnumber of a graphrdquoThe Australasian Journal of Combinatoricsvol 36 pp 91ndash100 2006

[83] M A Henning ldquoGraphs with large total domination numberrdquoJournal of Graph Theory vol 35 no 1 pp 21ndash45 2000

[84] D-X Ma X-G Chen and L Sun ldquoOn total restraineddomination in graphsrdquoCzechoslovakMathematical Journal vol55 no 1 pp 165ndash173 2005

[85] B Zelinka ldquoRemarks on restrained domination and totalrestrained domination in graphsrdquo Czechoslovak MathematicalJournal vol 55 no 2 pp 393ndash396 2005

[86] W Goddard and M A Henning ldquoIndependent domination ingraphs a survey and recent resultsrdquo Discrete Mathematics vol313 no 7 pp 839ndash854 2013

[87] J H Zhang H L Liu and L Sun ldquoIndependence bondagenumber and reinforcement number of some graphsrdquo Transac-tions of Beijing Institute of Technology vol 23 no 2 pp 140ndash1422003

[88] K D Dharmalingam Studies in Graph Theorymdashequitabledomination and bottleneck domination [PhD thesis] MaduraiKamaraj University Madurai India 2006

[89] GDeepakND Soner andAAlwardi ldquoThe equitable bondagenumber of a graphrdquo Research Journal of Pure Algebra vol 1 no9 pp 209ndash212 2011

[90] E Sampathkumar and H B Walikar ldquoThe connected domina-tion number of a graphrdquo Journal of Mathematical and PhysicalSciences vol 13 no 6 pp 607ndash613 1979

[91] K White M Farber and W Pulleyblank ldquoSteiner trees con-nected domination and strongly chordal graphsrdquoNetworks vol15 no 1 pp 109ndash124 1985

[92] M Cadei M X Cheng X Cheng and D-Z Du ldquoConnecteddomination in Ad Hoc wireless networksrdquo in Proceedings ofthe 6th International Conference on Computer Science andInformatics (CSampI rsquo02) 2002

[93] J F Fink and M S Jacobson ldquon-domination in graphsrdquo inGraph Theory with Applications to Algorithms and ComputerScience Wiley-Interscience Publications pp 283ndash300 WileyNew York NY USA 1985

[94] M Chellali O Favaron A Hansberg and L Volkmann ldquok-domination and k-independence in graphs a surveyrdquo Graphsand Combinatorics vol 28 no 1 pp 1ndash55 2012

[95] Y Lu X Hou J-M Xu andN Li ldquoTrees with uniqueminimump-dominating setsrdquo Utilitas Mathematica vol 86 pp 193ndash2052011

[96] Y Lu and J-M Xu ldquoThe p-bondage number of treesrdquo Graphsand Combinatorics vol 27 no 1 pp 129ndash141 2011

[97] Y Lu and J-M Xu ldquoThe 2-domination and 2-bondage numbersof grid graphs A manuscriptrdquo httparxivorgabs12044514

[98] M Krzywkowski ldquoNon-isolating 2-bondage in graphsrdquo TheJournal of the Mathematical Society of Japan vol 64 no 4 2012

[99] M A Henning O R Oellermann and H C Swart ldquoBoundson distance domination parametersrdquo Journal of CombinatoricsInformation amp System Sciences vol 16 no 1 pp 11ndash18 1991

[100] F Tian and J-M Xu ldquoBounds for distance domination numbersof graphsrdquo Journal of University of Science and Technology ofChina vol 34 no 5 pp 529ndash534 2004

[101] F Tian and J-M Xu ldquoDistance domination-critical graphsrdquoApplied Mathematics Letters vol 21 no 4 pp 416ndash420 2008

[102] F Tian and J-M Xu ldquoA note on distance domination numbersof graphsrdquo The Australasian Journal of Combinatorics vol 43pp 181ndash190 2009

[103] F Tian and J-M Xu ldquoAverage distances and distance domina-tion numbersrdquo Discrete Applied Mathematics vol 157 no 5 pp1113ndash1127 2009

[104] Z-L Liu F Tian and J-M Xu ldquoProbabilistic analysis of upperbounds for 2-connected distance k-dominating sets in graphsrdquoTheoretical Computer Science vol 410 no 38ndash40 pp 3804ndash3813 2009

[105] M A Henning O R Oellermann and H C Swart ldquoDistancedomination critical graphsrdquo Journal of Combinatorial Mathe-matics and Combinatorial Computing vol 44 pp 33ndash45 2003

[106] T J Bean M A Henning and H C Swart ldquoOn the integrityof distance domination in graphsrdquo The Australasian Journal ofCombinatorics vol 10 pp 29ndash43 1994

[107] M Fischermann and L Volkmann ldquoA remark on a conjecturefor the (kp)-domination numberrdquoUtilitasMathematica vol 67pp 223ndash227 2005

[108] T Korneffel D Meierling and L Volkmann ldquoA remark on the(22)-domination numberrdquo Discussiones Mathematicae GraphTheory vol 28 no 2 pp 361ndash366 2008

[109] Y Lu X Hou and J-M Xu ldquoOn the (22)-domination numberof treesrdquo Discussiones Mathematicae Graph Theory vol 30 no2 pp 185ndash199 2010

34 International Journal of Combinatorics

[110] H Li and J-M Xu ldquo(dm)-domination numbers of m-connected graphsrdquo Rapports de Recherche 1130 LRI UniversiteParis-Sud 1997

[111] S M Hedetniemi S T Hedetniemi and T V Wimer ldquoLineartime resource allocation algorithms for treesrdquo Tech Rep URI-014 Department of Mathematical Sciences Clemson Univer-sity 1987

[112] M Farber ldquoDomination independent domination and dualityin strongly chordal graphsrdquo Discrete Applied Mathematics vol7 no 2 pp 115ndash130 1984

[113] G S Domke and R C Laskar ldquoThe bondage and reinforcementnumbers of 120574

119891for some graphsrdquoDiscrete Mathematics vol 167-

168 pp 249ndash259 1997[114] G S Domke S T Hedetniemi and R C Laskar ldquoFractional

packings coverings and irredundance in graphsrdquo vol 66 pp227ndash238 1988

[115] E J Cockayne P A Dreyer Jr S M Hedetniemi andS T Hedetniemi ldquoRoman domination in graphsrdquo DiscreteMathematics vol 278 no 1ndash3 pp 11ndash22 2004

[116] I Stewart ldquoDefend the roman empirerdquo Scientific American vol281 pp 136ndash139 1999

[117] C S ReVelle and K E Rosing ldquoDefendens imperiumromanum a classical problem in military strategyrdquoThe Ameri-can Mathematical Monthly vol 107 no 7 pp 585ndash594 2000

[118] A Bahremandpour F-T Hu S M Sheikholeslami and J-MXu ldquoOn the Roman bondage number of a graphrdquo DiscreteMathematics Algorithms and Applications vol 5 no 1 ArticleID 1350001 15 pages 2013

[119] N Jafari Rad and L Volkmann ldquoRoman bondage in graphsrdquoDiscussiones Mathematicae Graph Theory vol 31 no 4 pp763ndash773 2011

[120] K Ebadi and L PushpaLatha ldquoRoman bondage and Romanreinforcement numbers of a graphrdquo International Journal ofContemporary Mathematical Sciences vol 5 pp 1487ndash14972010

[121] N Dehgardi S M Sheikholeslami and L Volkman ldquoOn theRoman k-bondage number of a graphrdquo AKCE InternationalJournal of Graphs and Combinatorics vol 8 pp 169ndash180 2011

[122] S Akbari and S Qajar ldquoA note on Roman bondage number ofgraphsrdquo Ars Combinatoria to Appear

[123] F-T Hu and J-M Xu ldquoRoman bondage numbers of somegraphsrdquo httparxivorgabs11093933

[124] N Jafari Rad and L Volkmann ldquoOn the Roman bondagenumber of planar graphsrdquo Graphs and Combinatorics vol 27no 4 pp 531ndash538 2011

[125] S Akbari M Khatirinejad and S Qajar ldquoA note on the romanbondage number of planar graphsrdquo Graphs and Combinatorics2012

[126] B Bresar M A Henning and D F Rall ldquoRainbow dominationin graphsrdquo Taiwanese Journal of Mathematics vol 12 no 1 pp213ndash225 2008

[127] B L Hartnell and D F Rall ldquoOn dominating the Cartesianproduct of a graph and 120572krdquo Discussiones Mathematicae GraphTheory vol 24 no 3 pp 389ndash402 2004

[128] N Dehgardi S M Sheikholeslami and L Volkman ldquoThe k-rainbow bondage number of a graphrdquo to appear in DiscreteApplied Mathematics

[129] J von Neumann and O Morgenstern Theory of Games andEconomic Behavior Princeton University Press Princeton NJUSA 1944

[130] C BergeTheorıe des Graphes et ses Applications 1958[131] O Ore Theory of Graphs vol 38 American Mathematical

Society Colloquium Publications 1962[132] E Shan and L Kang ldquoBondage number in oriented graphsrdquoArs

Combinatoria vol 84 pp 319ndash331 2007[133] J Huang and J-M Xu ldquoThe bondage numbers of extended

de Bruijn and Kautz digraphsrdquo Computers amp Mathematics withApplications vol 51 no 6-7 pp 1137ndash1147 2006

[134] J Huang and J-M Xu ldquoThe total domination and total bondagenumbers of extended de Bruijn and Kautz digraphsrdquoComputersamp Mathematics with Applications vol 53 no 8 pp 1206ndash12132007

[135] Y Shibata and Y Gonda ldquoExtension of de Bruijn graph andKautz graphrdquo Computers amp Mathematics with Applications vol30 no 9 pp 51ndash61 1995

[136] X Zhang J Liu and JMeng ldquoThebondage number in completet-partite digraphsrdquo Information Processing Letters vol 109 no17 pp 997ndash1000 2009

[137] D W Bange A E Barkauskas and P J Slater ldquoEfficient dom-inating sets in graphsrdquo in Applications of Discrete Mathematicspp 189ndash199 SIAM Philadelphia Pa USA 1988

[138] M Livingston and Q F Stout ldquoPerfect dominating setsrdquoCongressus Numerantium vol 79 pp 187ndash203 1990

[139] L Clark ldquoPerfect domination in random graphsrdquo Journal ofCombinatorial Mathematics and Combinatorial Computing vol14 pp 173ndash182 1993

[140] D W Bange A E Barkauskas L H Host and L H ClarkldquoEfficient domination of the orientations of a graphrdquo DiscreteMathematics vol 178 no 1ndash3 pp 1ndash14 1998

[141] A E Barkauskas and L H Host ldquoFinding efficient dominatingsets in oriented graphsrdquo Congressus Numerantium vol 98 pp27ndash32 1993

[142] I J Dejter and O Serra ldquoEfficient dominating sets in CayleygraphsrdquoDiscrete AppliedMathematics vol 129 no 2-3 pp 319ndash328 2003

[143] J Lee ldquoIndependent perfect domination sets in Cayley graphsrdquoJournal of Graph Theory vol 37 no 4 pp 213ndash219 2001

[144] WGu X Jia and J Shen ldquoIndependent perfect domination setsin meshes tori and treesrdquo Unpublished

[145] N Obradovic J Peters and G Ruzic ldquoEfficient domination incirculant graphswith two chord lengthsrdquo Information ProcessingLetters vol 102 no 6 pp 253ndash258 2007

[146] K Reji Kumar and G MacGillivray ldquoEfficient domination incirculant graphsrdquo Discrete Mathematics vol 313 no 6 pp 767ndash771 2013

[147] D Van Wieren M Livingston and Q F Stout ldquoPerfectdominating sets on cube-connected cyclesrdquo Congressus Numer-antium vol 97 pp 51ndash70 1993

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

12 International Journal of Combinatorics

by Theorem 66 In 1998 Dunbar et al [21] proposed thefollowing problem

Problem 2 Characterize all cactus graphs with bondagenumber 3

Some upper bounds for block-cactus graphs were alsoobtained

Theorem 69 (Dunbar et al [21] 1998) Let 119866 be a connectedblock-cactus graph with at least two blocksThen 119887(119866) ⩽ Δ(119866)

Theorem 70 (Dunbar et al [21] 1998) Let 119866 be a connectedblock-cactus graph which is neither a cactus graph nor a blockgraph Then 119887(119866) ⩽ Δ(119866) minus 1

53 Edge-Deletion and Edge-Contraction In order to obtainfurther results of the bondage number we may consider howthe bondage number changes under some operations of agraph 119866 which remain the domination number unchangedor preserve some property of 119866 such as planarity Twosimple operations satisfying these requirements are the edge-deletion and the edge-contraction

Theorem 71 (Huang and Xu [44] 2012) Let 119866 be any graphand 119890 isin 119864(119866) Then 119887(119866 minus 119890) ⩾ 119887(119866) minus 1 Moreover 119887(119866 minus 119890) ⩽119887(119866) if 120574(119866 minus 119890) = 120574(119866)

FromTheorem 1 119887(119862119899minus 119890) = 119887(119875

119899) = 119887(119862

119899) minus 1 for any 119890

in119862119899 which shows that the lower bound on 119887(119866minus119890) is sharp

The upper bound can reach for any graph 119866 with 119887(119866) ⩾ 2Next we consider the effect of the edge-contraction on

the bondage number Given a graph 119866 the contraction of 119866by the edge 119890 = 119909119910 denoted by 119866119909119910 is the graph obtainedfrom 119866 by contracting two vertices 119909 and 119910 to a new vertexand then deleting all multiedges It is easy to observe that forany graph 119866 120574(119866) minus 1 ⩽ 120574(119866119909119910) ⩽ 120574(119866) for any edge 119909119910 of119866

Theorem 72 (Huang and Xu [44] 2012) Let 119866 be any graphand 119909119910 be any edge in119866 If119873

119866(119909)cap119873

119866(119910) = 0 and 120574(119866119909119910) =

120574(119866) then 119887(119866119909119910) ⩾ 119887(119866)

We can use examples 119862119899and 119870

119899to show that the

conditions ofTheorem 72 are necessary and the lower boundinTheorem 72 is tight Clearly 120574(119862

119899) = lceil1198993rceil and 120574(119870

119899) = 1

for any edge 119909119910 119862119899119909119910 = 119862

119899minus1and 119870

119899119909119910 = 119870

119899minus1 By

Theorem 1 if 119899 equiv 1 (mod 3) then 120574(119862119899119909119910) lt 120574(119866) and

119887(119862119899119909119910) = 2 lt 3 = 119887(119862

119899) Thus the result in Theorem 72

is generally invalid without the hypothesis 120574(119866119909119910) = 120574(119866)Furthermore the condition 119873

119866(119909) cap 119873

119866(119910) = 0 cannot be

omitted even if 120574(119866119909119910) = 120574(119866) since for odd 119899 119887(119870119899119909119910) =

lceil(119899 minus 1)2rceil lt lceil1198992rceil = 119887(119870119899) by Theorem 1

On the other hand 119887(119862119899119909119910) = 119887(119862

119899) = 2 if 119899 equiv 0 (mod

3) and 119887(119870119899119909119910) = 119887(119870

119899) if 119899 is even which shows that the

equality in 119887(119866119909119910) ⩾ 119887(119866) may hold Thus the bound inTheorem 72 is tight However provided all the conditions119887(119866119909119910) can be arbitrarily larger than 119887(119866) Given a graph119867let119866 be the graph formed from119867∘119870

1by adding a new vertex

119909 and joining it to an vertex 119910 of degree one in119867 ∘ 1198701 Then

119866119909119910 = 119867 ∘ 1198701 120574(119866) = 120574(119866119909119910) and 119873

119866(119909) cap 119873

119866(119910) = 0

But 119887(119866) = 1 since 120574(119866 minus 119909119910) = 120574(119866119909119910) + 1 and 119887(119866119909119910) =120575(119867) + 1 byTheorem 66The gap between 119887(119866) and 119887(119866119909119910)is 120575(119867)

6 Results on Planar Graphs

From Section 2 we have seen that the bondage numberfor a tree has been completely solved Moreover a lineartime algorithm to compute the bondage number of a tree isdesigned by Hartnell et al [15] It is quite natural to considerthe bondage number for a planar graph In this section westate some results and problems on the bondage number fora planar graph

61 Conjecture on Bondage Number As mentioned inSection 3 the bondage number can be much larger than themaximum degree But for a planar graph 119866 the bondagenumber 119887(119866) cannot exceed Δ(119866) too much It is clear that119887(119866) ⩽ Δ(119866) + 4 by Corollary 15 since 120575(119866) ⩽ 5 for anyplanar graph119866 In 1998Dunbar et al [21] posed the followingconjecture

Conjecture 73 If 119866 is a planar graph then 119887(119866) ⩽ Δ(119866) + 1

Because of its attraction it immediately became the focusof attention when this conjecture is proposed In fact themain aim concerning the research of the bondage number fora planar graph is focused on this conjecture

It has been mentioned in Theorems 1 and 60 that119887(119862

3119896+1) = 3 = Δ + 1 and 119887(119862

4119896+2times 119870

2) = 4 = Δ + 1 It

is known that 119887(1198706minus 119872) = 5 = Δ + 1 where119872 is a perfect

matching of the complete graph1198706These examples show that

if Conjecture 73 is true then the upper bound is sharp for2 ⩽ Δ ⩽ 4

Here we note that it is sufficient to prove this conjecturefor connected planar graphs since the bondage number of adisconnected graph is simply the minimum of the bondagenumbers of its components

The first paper attacking this conjecture is due to Kangand Yuan [45] which confirmed the conjecture for everyconnected planar graph 119866 with Δ ⩾ 7 The proofs are mainlybased onTheorems 16 and 18

Theorem 74 (Kang and Yuan [45] 2000) If 119866 is a connectedplanar graph then 119887(119866) ⩽ min8 Δ(119866) + 2

Obviously in view of Theorem 74 Conjecture 73 is truefor any connected planar graph with Δ ⩾ 7

62 Bounds Implied by Degree Conditions As we have seenfrom Theorem 74 to attack Conjecture 73 we only need toconsider connected planar graphs with maximum Δ ⩽ 6Thus studying the bondage number of planar graphs bydegree conditions is of significanceThe first result on boundsimplied by degree conditions is obtained by Kang and Yuan[45]

International Journal of Combinatorics 13

Theorem 75 (Kang and Yuan [45] 2000) If 119866 is a connectedplanar graph without vertices of degree 5 then 119887(119866) ⩽ 7

Fischermann et al [46] generalized Theorem 75 as fol-lows

Theorem 76 (Fischermann et al [46] 2003) Let 119866 be aconnected planar graph and 119883 the set of vertices of degree 5which have distance at least 3 to vertices of degrees 1 2 and3 If all vertices in 119883 not adjacent with vertices of degree 4are independent and not adjacent to vertices of degree 6 then119887(119866) ⩽ 7

Clearly if119866 has no vertices of degree 5 then119883 = 0 ThusTheorem 76 yields Theorem 75

Use 119899119894to denote the number of vertices of degree 119894 in 119866

for each 119894 = 1 2 Δ(119866) Using Theorem 18 Fischermannet al obtained the following two theorems

Theorem 77 (Fischermann et al [46] 2003) For any con-nected planar graph 119866

(1) 119887(119866) ⩽ 7 if 1198995lt 2119899

2+ 3119899

3+ 2119899

4+ 12

(2) 119887(119866) ⩽ 6 if 119866 contains no vertices of degrees 4 and 5

Theorem 78 (Fischermann et al [46] 2003) For any con-nected planar graph 119866 119887(119866) ⩽ Δ(119866) + 1 if

(1) Δ(119866) = 6 and every edge 119890 = 119909119910 with 119889119866(119909) = 5 and

119889119866(119910) = 6 is contained in at most one triangle

(2) Δ(119866) = 5 and no triangle contains an edge 119890 = 119909119910 with119889119866(119909) = 5 and 4 ⩽ 119889

119866(119910) ⩽ 5

63 Bounds Implied by Girth Conditions The girth 119892(119866) of agraph 119866 is the length of the shortest cycle in 119866 If 119866 has nocycles we define 119892(119866) = infin

Combining Theorem 74 with Theorem 78 we find thatif a planar graph contains no triangles and has maximumdegree Δ ⩾ 5 then Conjecture 73 holds This fact motivatedFischermann et alrsquos [46] attempt to attack Conjecture 73 bygirth restraints

Theorem 79 (Fischermann et al [46] 2003) For any con-nected planar graph 119866

119887 (119866) ⩽

6 119894119891 119892 (119866) ⩾ 4

5 119894119891 119892 (119866) ⩾ 5

4 119894119891 119892 (119866) ⩾ 6

3 119894119891 119892 (119866) ⩾ 8

(20)

The first result in Theorem 79 shows that Conjecture 73is valid for all connected planar graphs with 119892(119866) ⩾ 4 andΔ(119866) ⩾ 5 It is easy to verify that the second result inTheorem 79 implies that Conjecture 73 is valid for all not3-regular graphs of girth 119892(119866) ⩾ 5 which is stated in thefollowing corollary

Corollary 80 For any connected planar graph 119866 if 119866 is not3-regular and 119892(119866) ⩾ 5 then 119887(119866) ⩽ Δ(119866) + 1

The first result in Theorem 79 also implies that if 119866 isa connected planar graph with no cycles of length 3 then119887(119866) ⩽ 6 Hou et al [47] improved this result as follows

Theorem 81 (Hou et al [47] 2011) For any connected planargraph 119866 if 119866 contains no cycles of length 119894 (4 ⩽ 119894 ⩽ 6) then119887(119866) ⩽ 6

Since 119887(1198623119896+1) = 3 for 119896 ⩾ 3 (see Theorem 1) the

last bound in Theorem 79 is tight Whether other bounds inTheorem 79 are tight remains open In 2003 Fischermann etal [46] posed the following conjecture

Conjecture 82 For any connected planar graph 119866 119887(119866) ⩽ 7and

119887 (119866) ⩽ 5 119894119891 119892 (119866) ⩾ 4

4 119894119891 119892 (119866) ⩾ 5(21)

We conclude this subsection with a question on bondagenumbers of planar graphs

Question 1 (Fischermann et al [46] 2003) Is there a planargraph 119866 with 6 ⩽ 119887(119866) ⩽ 8

In 2006 Carlson andDevelin [42] showed that the corona119866 = 119867 ∘ 119870

1for a planar graph 119867 with 120575(119867) = 5 has the

bondage number 119887(119866) = 120575(119867) + 1 = 6 (see Theorem 66)Since the minimum degree of planar graphs is at most 5then 119887(119866) can attach 6 If we take 119867 as the graph of theicosahedron then 119866 = 119867 ∘ 119870

1is such an example The

question for the existence of planar graphs with bondagenumber 7 or 8 remains open

64 Comments on the Conjectures Conjecture 73 is true forall connected planar graphs with minimum degree 120575 ⩽ 2 byTheorem 16 or maximum degree Δ ⩾ 7 by Theorem 74 ornot 120574-critical planar graphs by Theorem 13 Thus to attackConjecture 73 we only need to consider connected criticalplanar graphs with degree-restriction 3 ⩽ 120575 ⩽ Δ ⩽ 6

Recalling and anatomizing the proofs of all results men-tioned in the preceding subsections on the bondage numberfor connected planar graphs we find that the proofs of theseresults strongly depend upon Theorem 16 or Theorem 18 Inother words a basic way used in the proofs is to find twovertices 119909 and 119910 with distance at most two in a consideredplanar graph 119866 such that

119889119866(119909) + 119889

119866(119910) or 119889

119866(119909) + 119889

119866(119910) minus

1003816100381610038161003816119873119866(119909) cap 119873

119866(119910)1003816100381610038161003816

(22)

which bounds 119887(119866) is as small as possible Very recentlyHuang and Xu [44] have considered the following parameter

119861 (119866) = min119909119910isin119881(119866)

119889119909119910 1 ⩽ 119889

119866(119909 119910) ⩽ 2

cup 119889119909119910minus 119899

119909119910 119889 (119909 119910) = 1

(23)

where 119889119909119910= 119889

119866(119909) + 119889

119866(119910) minus 1 and 119899

119909119910= |119873

119866(119909) cap 119873

119866(119910)|

14 International Journal of Combinatorics

It follows fromTheorems 16 and 18 that

119887 (119866) ⩽ 119861 (119866) (24)

The proofs given in Theorems 74 and 79 indeed imply thefollowing stronger results

Theorem 83 If 119866 is a connected planar graph then

119861 (119866) ⩽

min 8 Δ (119866) + 26 119894119891 119892 (119866) ⩾ 4

5 119894119891 119892 (119866) ⩾ 5

4 119894119891 119892 (119866) ⩾ 6

3 119894119891 119892 (119866) ⩾ 8

(25)

Thus using Theorem 16 or Theorem 18 if we can proveConjectures 73 and 82 then we can prove the followingstatement

Statement If 119866 is a connected planar graph then

119861 (119866) ⩽

Δ (119866) + 1

7

5 if 119892 (119866) ⩾ 44 if 119892 (119866) ⩾ 5

(26)

It follows from (24) that Statement implies Conjectures 73and 82 However Huang and Xu [44] gave examples to showthat none of conclusions in Statement is true As a result theystated the following conclusion

Theorem 84 (Huang and Xu [44] 2012) It is not possible toprove Conjectures 73 and 82 if they are right using Theorems16 and 18

Therefore a new method is needed to prove or disprovethese conjectures At the present time one cannot prove theseconjectures and may consider to disprove them If one ofthese conjectures is invalid then there exists a minimumcounterexample 119866 with respect to 120592(119866) + 120576(119866) In order toobtain a minimum counterexample one may consider twosimple operations satisfying these requirements the edge-deletion and the edge-contractionwhich decrease 120592(119866)+120576(119866)and preserve planarity However applying Theorems 71 and72 Huang and Xu [44] presented the following results

Theorem 85 (Huang and Xu [44] 2012) It is not possible toconstruct minimum counterexamples to Conjectures 73 and 82by the operation of an edge-deletion or an edge-contraction

7 Results on Crossing Number Restraints

It is quite natural to generalize the known results on thebondage number for planar graphs to formore general graphsin terms of graph-theoretical parameters In this section weconsider graphs with crossing number restraints

The crossing number cr(119866) of 119866 is the smallest numberof pairwise intersections of its edges when 119866 is drawn in theplane If cr(119866) = 0 then 119866 is a planar graph

71 General Methods Use 119899119894to denote the number of vertices

of degree 119894 in119866 for 119894 = 1 2 Δ(119866) Huang and Xu obtainedthe following results

Theorem 86 (Huang and Xu [48] 2007) For any connectedgraph 119866

119887 (119866) ⩽

6 119894119891 119892 (119866) ⩾ 4 119886119899119889 2cr (119866) lt 121198861

5 119894119891 119892 (119866) ⩾ 5 119886119899119889 6cr (119866) lt 161198862

4 119894119891 119892 (119866) ⩾ 6 119886119899119889 4cr (119866) lt 141198863

3 119894119891 119892 (119866) ⩾ 8 119886119899119889 6cr (119866) lt 161198864

(27)

where 1198861= 119899

1+ 2119899

2+ 2119899

3+sum

Δ

119894=8(119894 minus 7)119899

119894+ 8 119886

2= 3119899

1+ 6119899

2+

51198993+sum

Δ

119894=7(3119894minus18)119899

119894+20 119886

3= 119899

1+2119899

2+sum

Δ

119894=6(2119894minus10)119899

119894+12

and 1198864= sum

Δ

119894=5(3119894 minus 12)119899

119894+ 16

Simplifying the conditions in Theorem 86 we obtain thefollowing corollaries immediately

Corollary 87 (Huang and Xu [48] 2007) For any connectedgraph 119866

119887 (119866) ⩽

6 119894119891 119892 (119866) ⩾ 4 119886119899119889 cr (119866) ⩽ 3

5 119894119891 119892 (119866) ⩾ 5 119886119899119889 cr (119866) ⩽ 4

4 119894119891 119892 (119866) ⩾ 6 119886119899119889 cr (119866) ⩽ 2

3 119894119891 119892 (119866) ⩾ 8 119886119899119889 cr (119866) ⩽ 2

(28)

Corollary 88 (Huang and Xu [48] 2007) For any connectedgraph 119866

(a) 119887(119866) ⩽ 6 if 119866 is not 4-regular cr(119866) = 4 and 119892(119866) ⩾ 4

(b) 119887(119866) ⩽ Δ(119866) + 1 if 119866 is not 3-regular cr(119866) ⩽ 4 and119892(119866) ⩾ 5

(c) 119887(119866) ⩽ 4 if 119866 is not 3-regular cr(119866) = 3 and 119892(119866) ⩾ 6

(d) 119887(119866) ⩽ 3 if cr(119866) = 3 119892(119866) ⩾ 8 and Δ(119866) ⩾ 5

These corollaries generalize some known results for pla-nar graphs For example Corollary 87 contains Theorem 79Corollary 88 (b) contains Corollary 80

Theorem 89 (Huang and Xu [48] 2007) For any connectedgraph 119866 if cr(119866) ⩽ 119899

3(119866) + 119899

4(119866) + 3 then 119887(119866) ⩽ 8

Corollary 90 (Huang and Xu [48] 2007) For any connectedgraph 119866 if cr(119866) ⩽ 3 then 119887(119866) ⩽ 8

Perhaps being unaware of this result in 2010 Ma et al[49] proved that 119887(119866) ⩽ 12 for any graph 119866 with cr(119866) = 1

Theorem91 (Huang and Xu [48] 2007) Let119866 be a connectedgraph and 119868 = 119909 isin 119881(119866) 119889

119866(119909) = 5 119889

119866(119909 119910) ⩾ 3 if

International Journal of Combinatorics 15

119889119866(119910) ⩽ 3 and 119889

119866(119910) = 4 for every 119910 isin 119873

119866(119909) If 119868 is

independent no vertices adjacent to vertices of degree 6 and

cr (119866) lt max5119899

3+ |119868| minus 2119899

4+ 28

117119899

3+ 40

16 (29)

then 119887(119866) ⩽ 7

Corollary 92 (Huang and Xu [48] 2007) Let 119866 be aconnected graph with cr(119866) ⩽ 2 If 119868 = 119909 isin 119881(119866) 119889

119866(119909) =

5 119889119866(119910 119909) ⩾ 3 if 119889

119866(119910) ⩽ 3 and 119889

119866(119910) = 4 for every 119910 isin

119873119866(119909) is independent and no vertices adjacent to vertices of

degree 6 then 119887(119866) ⩽ 7

Theorem93 (Huang andXu [48] 2007) Let119866 be a connectedgraph If 119866 satisfies

(a) 5cr(119866) + 1198995lt 2119899

2+ 3119899

3+ 2119899

4+ 12 or

(b) 7cr(119866) + 21198995lt 3119899

2+ 4119899

4+ 24

then 119887(119866) ⩽ 7

Proposition 94 (Huang and Xu [48] 2007) Let 119866 be aconnected graph with no vertices of degrees four and five Ifcr(119866) ⩽ 2 then 119887(119866) ⩽ 6

Theorem 95 (Huang and Xu [48] 2007) If 119866 is a connectedgraph with cr(119866) ⩽ 4 and not 4-regular when cr(119866) = 4 then119887(119866) ⩽ Δ(119866) + 2

The above results generalize some known results forplanar graphs For example Corollary 90 and Theorem 95contain Theorem 74 the first condition in Theorem 93 con-tains the second condition inTheorem 77

From Corollary 88 and Theorem 95 we suggest the fol-lowing questions

Question 2 Is there a

(a) 4-regular graph 119866 with cr(119866) = 4 such that 119887(119866) ⩾ 7

(b) 3-regular graph 119866 with cr ⩽ 4 and 119892(119866) ⩾ 5 such that119887(119866) = 5

(c) 3-regular graph 119866 with cr = 3 and 119892(119866) = 6 or 7 suchthat 119887(119866) = 5

72 Carlson-Develin Methods In this subsection we intro-duce an elegant method presented by Carlson and Develin[42] to obtain some upper bounds for the bondage numberof a graph

Suppose that 119866 is a connected graph We say that 119866 hasgenus 120588 if 119866 can be embedded in a surface 119878 with 120588 handlessuch that edges are pairwise disjoint except possibly for end-vertices Let119866 be an embedding of119866 in surface 119878 and let120601(119866)denote the number of regions in 119866 The boundary of everyregion contains at least three edges and every edge is on theboundary of atmost two regions (the two regions are identicalwhen 119890 is a cut-edge) For any edge 119890 of 119866 let 1199031

119866(119890) and 1199032

119866(119890)

be the numbers of edges comprising the regions in 119866 whichthe edge 119890 borders It is clear that every 119890 = 119909119910 isin 119864(119866)

1199032

119866(119890) ⩾ 119903

1

119866(119890) ⩾ 4 if 1003816100381610038161003816119873119866

(119909) cap 119873119866(119910)1003816100381610038161003816 = 0

1199032

119866(119890) ⩾ 4 119903

1

119866(119890) ⩾ 3 if 1003816100381610038161003816119873119866

(119909) cap 119873119866(119910)1003816100381610038161003816 = 1

1199032

119866(119890) ⩾ 119903

1

119866(119890) ⩾ 3 if 1003816100381610038161003816119873119866

(119909) cap 119873119866(119910)1003816100381610038161003816 ⩾ 2

(30)

Following Carlson and Develin [42] for any edge 119890 = 119909119910 of119866 we define

119863119866(119890) =

1

119889119866(119909)+

1

119889119866(119910)

+1

1199031

119866(119890)+

1

1199032

119866(119890)minus 1 (31)

By the well-known Eulerrsquos formula

120592 (119866) minus 120576 (119866) + 120601 (119866) = 2 minus 2120588 (119866) (32)

it is easy to see that

sum

119890isin119864(119866)

119863119866(119890) = 120592 (119866) minus 120576 (119866) + 120601 (119866) = 2 minus 2120588 (119866) (33)

If 119866 is a planar graph that is 120588(119866) = 0 then

sum

119890isin119864(119866)

119863119866(119890) = 120592 (119866) minus 120576 (119866) + 120601 (119866) = 2 (34)

Combining these formulas withTheorem 18 Carlson andDevelin [42] gave a simple and intuitive proof ofTheorem 74and obtained the following result

Theorem 96 (Carlson and Develin [42] 2006) Let 119866 be aconnected graph which can be embedded on a torus Then119887(119866) ⩽ Δ(119866) + 3

Recently Hou and Liu [50] improved Theorem 96 asfollows

Theorem 97 (Hou and Liu [50] 2012) Let 119866 be a connectedgraph which can be embedded on a torus Then 119887(119866) ⩽ 9Moreover if Δ(119866) = 6 then 119887(119866) ⩽ 8

By the way Cao et al [51] generalized the result inTheorem 79 to a connected graph 119866 that can be embeddedon a torus that is

119887 (119866) le

6 if 119892 (119866) ⩾ 4 and 119866 is not 4-regular5 if 119892 (119866) ⩾ 54 if 119892 (119866) ⩾ 6 and 119866 is not 3-regular3 if 119892 (119866) ⩾ 8

(35)

Several authors used this method to obtain some resultson the bondage number For example Fischermann et al[46] used this method to prove the second conclusion inTheorem 78 Recently Cao et al [52] have used thismethod todeal with more general graphs with small crossing numbersFirst they found the following property

Lemma 98 120575(119866) ⩽ 5 for any graph 119866 with cr(119866) ⩽ 5

16 International Journal of Combinatorics

This result implies that 119887(119866) ⩽ Δ(119866) + 4 by Corollary 15for any graph 119866 with cr(119866) ⩽ 5 Cao et al established thefollowing relation in terms of maximum planar subgraphs

Lemma 99 (Cao et al [52]) Let 119866 be a connected graph withcrossing number cr(119866) For any maximum planar subgraph119867of 119866

sum

119890isin119864(119867)

119863119866(119890) ⩾ 2 minus

2cr (119866)120575 (119866)

(36)

Using Carlson-Develin method and combiningLemma 98 with Lemma 99 Cao et al proved the followingresults

Theorem 100 (Cao et al [52]) For any connected graph 119866119887(119866) ⩽ Δ(119866) + 2 if 119866 satisfies one of the following conditions

(a) cr(119866) ⩽ 3(b) cr(119866) = 4 and 119866 is not 4-regular(c) cr(119866) = 5 and 119866 contains no vertices of degree 4

Theorem101 (Cao et al [52]) Let119866 be a connected graphwithΔ(119866) = 5 and cr(119866) ⩽ 4 If no triangles contain two vertices ofdegree 5 then 119887(119866) ⩽ 6 = Δ(119866) + 1

Theorem 102 (Cao et al [52]) Let 119866 be a connected graphwith Δ(119866) ⩾ 6 and cr(119866) ⩽ 3 If Δ(119866) ⩾ 7 or if Δ(119866) = 6120575(119866) = 3 and every edge 119890 = 119909119910 with 119889

119866(119909) = 5 and 119889

119866(119910) = 6

is contained in atmost one triangle then 119887(119866) ⩽ min8 Δ(119866)+1

Using Carlson-Develin method it can be proved that119887(119866) ⩽ Δ(119866) + 2 if cr(119866) ⩽ 3 (see the first conclusion inTheorem 100) and not yet proved that 119887(119866) ⩽ 8 if cr(119866) ⩽3 although it has been proved by using other method (seeCorollary 90)

By using Carlson-Develin method Samodivkin [25]obtained some results on the bondage number for graphswithsome given properties

Kim [53] showed that 119887(119866) ⩽ Δ(119866) + 2 for a connectedgraph 119866 with genus 1 Δ(119866) ⩽ 5 and having a toroidalembedding of which at least one region is not 4-sidedRecently Gagarin and Zverovich [54] further have extendedCarlson-Develin ideas to establish a nice upper bound forarbitrary graphs that can be embedded on orientable ornonorientable topological surface

Theorem 103 (Gagarin and Zverovich [54] 2012) Let 119866 bea graph embeddable on an orientable surface of genus ℎ and anonorientable surface of genus 119896 Then 119887(119866) ⩽ minΔ(119866)+ℎ+2 Δ(119866) + 119896 + 1

This result generalizes the corresponding upper boundsin Theorems 74 and 96 for any orientable or nonori-entable topological surface By investigating the proof ofTheorem 103 Huang [55] found that the issue of the ori-entability can be avoided by using the Euler characteristic120594(= 120592(119866) minus 120576(119866) + 120601(119866)) instead of the genera ℎ and 119896 the

relations are 120594 = 2minus2ℎ and 120594 = 2minus119896 To obtain the best resultfromTheorem 103 onewants ℎ and 119896 as small as possible thisis equivalent to having 120594 as large as possible

According toTheorem 103 if119866 is planar (ℎ = 0 120594 = 2) orcan be embedded on the real projective plane (119896 = 1 120594 = 1)then 119887(119866) ⩽ Δ(119866) + 2 In all other cases Huang [55] had thefollowing improvement for Theorem 103 the proof is basedon the technique developed byCarlson-Develin andGagarin-Zverovich and includes some elementary calculus as a newingredient mainly the intermediate-value theorem and themean-value theorem

Theorem 104 (Huang [55] 2012) Let 119866 be a graph embed-dable on a surface whose Euler characteristic 120594 is as large aspossible If 120594 ⩽ 0 then 119887(119866) ⩽ Δ(119866)+lfloor119903rfloor where 119903 is the largestreal root of the following cubic equation in 119911

1199113

+ 21199112

+ (6120594 minus 7) 119911 + 18120594 minus 24 = 0 (37)

In addition if 120594 decreases then 119903 increases

The following result is asymptotically equivalent toTheorem 104

Theorem 105 (Huang [55] 2012) Let 119866 be a graph embed-dable on a surface whose Euler characteristic 120594 is as large aspossible If 120594 ⩽ 0 then 119887(119866) lt Δ(119866) + radic12 minus 6120594 + 12 orequivalently 119887(119866) ⩽ Δ(119866) + lceilradic12 minus 6120594 minus 12rceil

Also Gagarin andZverovich [54] indicated that the upperbound in Theorem 103 can be improved for larger values ofthe genera ℎ and 119896 by adjusting the proofs and stated thefollowing general conjecture

Conjecture 106 (Gagarin and Zverovich [54] 2012) For aconnected graph G of orientable genus ℎ and nonorientablegenus 119896

119887 (119866) ⩽ min 119888ℎ 119888

1015840

119896 Δ (119866) + 119900 (ℎ) Δ (119866) + 119900 (119896) (38)

where 119888ℎand 1198881015840

119896are constants depending respectively on the

orientable and nonorientable genera of 119866

In the recent paper Gagarin and Zverovich [56] providedconstant upper bounds for the bondage number of graphs ontopological surfaces which can be used as the first estimationfor the constants 119888

ℎand 1198881015840

119896of Conjecture 106

Theorem 107 (Gagarin and Zverovich [56] 2013) Let 119866 bea connected graph of order 119899 If 119866 can be embedded on thesurface of its orientable or nonorientable genus of the Eulercharacteristic 120594 then

(a) 120594 ⩾ 1 implies 119887(119866) ⩽ 10(b) 120594 ⩽ 0 and 119899 gt minus12120594 imply 119887(119866) ⩽ 11(c) 120594 ⩽ minus1 and 119899 ⩽ minus12120594 imply 119887(119866) ⩽ 11 +

3120594(radic17 minus 8120594 minus 3)(120594 minus 1) = 11 + 119874(radicminus120594)

Theorem 79 shows that a connected planar triangle-freegraph 119866 has 119887(119866) ⩽ 6 The following result generalizes to itall the other topological surfaces

International Journal of Combinatorics 17

Theorem 108 (Gagarin and Zverovich [56] 2013) Let 119866 be aconnected triangle-free graph of order 119899 If 119866 can be embeddedon the surface of its orientable or nonorientable genus of theEuler characteristic 120594 then

(a) 120594 ⩾ 1 implies 119887(119866) ⩽ 6(b) 120594 ⩽ 0 and 119899 gt minus8120594 imply 119887(119866) ⩽ 7(c) 120594 ⩽ minus1 and 119899 ⩽ minus8120594 imply 119887(119866) ⩽ 7minus4120594(1+radic2 minus 120594)

In [56] the authors also explicitly improved upperbounds in Theorem 103 and gave tight lower bounds forthe number of vertices of graphs 2-cell embeddable ontopological surfaces of a given genusThe interested reader isreferred to the original paper for details The following resultis interesting although its proof is easy

Theorem 109 (Samodivkin [57]) Let119866 be a graph with order119899 and girth 119892 If 119866 is embeddable on a surface whose Eulercharacteristic 120594 is as large as possible then

119887 (119866) ⩽ 3 +8

119892 minus 2minus

4120594119892

119899 (119892 minus 2) (39)

If 119866 is a planar graph with girth 119892 ⩾ 4 + 119894 for any 119894 isin0 1 2 then Theorem 109 leads to 119887(119866) ⩽ 6 minus 119894 which isthe result inTheorem 79 obtained by Fischermann et al [46]Also inmany cases the bound stated inTheorem 109 is betterthan those given byTheorems 103 and 105

8 Conditional Bondage Numbers

Since the concept of the bondage number is based upon thedomination all sorts of dominations which are variationsof the normal domination by adding restricted conditionsto the normal dominating set can develop a new ldquobondagenumberrdquo as long as a variation of domination number isgiven In this section we survey results on the bondagenumber under some restricted conditions

81 Total Bondage Numbers A dominating set 119878 of a graph119866 without isolated vertices is called total if the subgraphinduced by 119878 contains no isolated vertices The minimumcardinality of a total dominating set is called the totaldomination number of 119866 and denoted by 120574

119879(119866) It is clear

that 120574(119866) ⩽ 120574119879(119866) ⩽ 2120574(119866) for any graph 119866 without isolated

verticesThe total domination in graphs was introduced by Cock-

ayne et al [58] in 1980 Pfaff et al [59 60] in 1983 showedthat the problem determining total domination number forgeneral graphs is NP-complete even for bipartite graphs andchordal graphs Even now total domination in graphs hasbeen extensively studied in the literature In 2009 Henning[61] gave a survey of selected recent results on total domina-tion in graphs

The total bondage number of 119866 denoted by 119887119879(119866) is the

smallest cardinality of a subset 119861 sube 119864(119866) with the propertythat119866minus119861 contains no isolated vertices and 120574

119879(119866minus119861) gt 120574

119879(119866)

From definition 119887119879(119866)may not exist for some graphs for

example119866 = 1198701119899 We put 119887

119879(119866) = infin if 119887

119879(119866) does not exist

It is easy to see that 119887119879(119866) is finite for any connected graph 119866

other than 1198701 119870

2 119870

3 and 119870

1119899 In fact for such a graph 119866

since there is a path of length 3 in 119866 we can find 1198611sube 119864(119866)

such that 119866 minus 1198611is a spanning tree 119879 of 119866 containing a path

of length 3 So 120574119879(119879) = 120574

119879(119866minus119861

1) ⩾ 120574

119879(119866) For the tree119879 we

find 1198612sube 119864(119879) such that 120574

119879(119879 minus 119861

2) gt 120574

119879(119879) ⩾ 120574

119879(119866) Thus

we have 120574119879(119866minus119861

2minus119861

1) gt 120574

119879(119866) and so 119887

119879(119866) ⩽ |119861

1|+|119861

2| In

the following discussion we always assume that 119887119879(119866) exists

when a graph 119866 is mentionedIn 1991 Kulli and Patwari [62] first studied the total

bondage number of a graph and calculated the exact valuesof 119887

119879(119866) for some standard graphs

Theorem 110 (Kulli and Patwari [62] 1991) For 119899 ⩾ 4

119887119879(119862

119899) =

3 119894119891 119899 equiv 2 (mod 4) 2 119900119905ℎ119890119903119908119894119904119890

119887119879(119875

119899) =

2 119894119891 119899 equiv 2 (mod 4) 1 119900119905ℎ119890119903119908119894119904119890

119887119879(119870

119898119899) = 119898 119908119894119905ℎ 2 ⩽ 119898 ⩽ 119899

119887119879(119870

119899) =

4 119891119900119903 119899 = 4

2119899 minus 5 119891119900119903 119899 ⩾ 5

(40)

Recently Hu et al [63] have obtained some results on thetotal bondage number of the Cartesian product119875

119898times119875

119899of two

paths 119875119898and 119875

119899

Theorem 111 (Hu et al [63] 2012) For the Cartesian product119875119898times 119875

119899of two paths 119875

119898and 119875

119899

119887119879(119875

2times 119875

119899) =

1 119894119891 119899 equiv 0 (mod 3) 2 119894119891 119899 equiv 2 (mod 3) 3 119894119891 119899 equiv 1 (mod 3)

119887119879(119875

3times 119875

119899) = 1

119887119879(119875

4times 119875

119899)

= 1 119894119891 119899 equiv 1 (mod 5) = 2 119894119891 119899 equiv 4 (mod 5) ⩽ 3 119894119891 119899 equiv 2 (mod 5) ⩽ 4 119894119891 119899 equiv 0 3 (mod 5)

(41)

Generalized Petersen graphs are an important class ofcommonly used interconnection networks and have beenstudied recently By constructing a family of minimum totaldominating sets Cao et al [64] determined the total bondagenumber of the generalized Petersen graphs

FromTheorem 6 we know that 119887(119879) ⩽ 2 for a nontrivialtree 119879 But given any positive integer 119896 Sridharan et al [65]constructed a tree 119879 for which 119887

119879(119879) = 119896 Let119867

119896be the tree

obtained from a star1198701119896+1

by subdividing 119896 edges twice Thetree 119867

7is shown in Figure 6 It can be easily verified that

119887119879(119867

119896) = 119896 This fact can be stated as the following theorem

Theorem 112 (Sridharan et al [65] 2007) For any positiveinteger 119896 there exists a tree 119879 with 119887

119879(119879) = 119896

18 International Journal of Combinatorics

Figure 61198677

Combining Theorem 1 with Theorem 112 we have whatfollows

Corollary 113 (Sridharan et al [65] 2007) The bondagenumber and the total bondage number are unrelated even fortrees

However Sridharan et al [65] gave an upper bound onthe total bondage number of a tree in terms of its maximumdegree For any tree 119879 of order 119899 if 119879 =119870

1119899minus1 then 119887

119879(119879) =

minΔ(119879) (13)(119899 minus 1) Rad and Raczek [66] improved thisupper bound and gave a constructive characterization of acertain class of trees attaching the upper bound

Theorem 114 (Rad and Raczek [66] 2011) 119887119879(119879) ⩽ Δ(119879) minus 1

for any tree 119879 with maximum degree at least three

In general the decision problem for 119887119879(119866) is NP-complete

for any graph 119866 We state the decision problem for the totalbondage as follows

Problem 3 Consider the decision problemTotal Bondage ProblemInstance a graph 119866 and a positive integer 119887Question is 119887

119879(119866) ⩽ 119887

Theorem 115 (Hu and Xu [16] 2012) The total bondageproblem is NP-complete

Consequently in view of its computational hardnessit is significative to establish some sharp bounds on thetotal bondage number of a graph in terms of other graphicparameters In 1991 Kulli and Patwari [62] showed that119887119879(119866) ⩽ 2119899 minus 5 for any graph 119866 with order 119899 ⩾ 5 Sridharan

et al [65] improved this result as follows

Theorem 116 (Sridharan et al [65] 2007) For any graph 119866with order 119899 if 119899 ⩾ 5 then

119887119879(119866) ⩽

1

3(119899 minus 1) 119894119891 119866 119888119900119899119905119886119894119899119904 119899119900 119888119910119888119897119890119904

119899 minus 1 119894119891 119892 (119866) ⩾ 5

119899 minus 2 119894119891 119892 (119866) = 4

(42)

Rad and Raczek [66] also established some upperbounds on 119887

119879(119866) for a general graph 119866 In particular they

gave an upper bound on 119887119879(119866) in terms of the girth of

119866

Theorem 117 (Rad and Raczek [66] 2011) 119887119879(119866)⩽4Δ(119866) minus 5

for any graph 119866 with 119892(119866) ⩾ 4

82 Paired Bondage Numbers A dominating set 119878 of 119866 iscalled to be paired if the subgraph induced by 119878 containsa perfect matching The paired domination number of 119866denoted by 120574

119875(119866) is the minimum cardinality of a paired

dominating set of 119866 Clearly 120574119879(119866) ⩽ 120574

119875(119866) for every

connected graph 119866 with order at least two where 120574119879(119866) is

the total domination number of 119866 and 120574119875(119866) ⩽ 2120574(119866) for

any graph119866without isolated vertices Paired domination wasintroduced by Haynes and Slater [67 68] and further studiedin [69ndash72]

The paired bondage number of 119866 with 120575(119866) ⩾ 1 denotedby 119887P(119866) is the minimum cardinality among all sets of edges119861 sube 119864 such that 120575(119866 minus 119861) ⩾ 1 and 120574

119875(119866 minus 119861) gt 120574

119875(119866)

The concept of the paired bondage number was firstproposed by Raczek [73] in 2008The following observationsfollow immediately from the definition of the paired bondagenumber

Observation 2 Let 119866 be a graph with 120575(119866) ⩾ 1

(a) If 119867 is a subgraph of 119866 such that 120575(119867) ⩾ 1 then119887119875(119867) ⩽ 119887

119875(119866)

(b) If 119867 is a subgraph of 119866 such that 119887119875(119867) = 1 and 119896

is the number of edges removed to form119867 then 1 ⩽119887119875(119866) ⩽ 119896 + 1

(c) If 119909119910 isin 119864(119866) such that 119889119866(119909) 119889

119866(119910) gt 1 and 119909119910

belongs to each perfect matching of each minimumpaired dominating set of 119866 then 119887

119875(119866) = 1

Based on these observations 119887119875(119875

119899) ⩽ 2 In fact the

paired bondage number of a path has been determined

Theorem 118 (Raczek [73] 2008) For any positive integer 119896if 119899 ⩾ 2 then

119887119875(119875

119899) =

0 119894119891 119899 = 2 3 119900119903 5

1 119894119891 119899 = 4119896 4119896 + 3 119900119903 4119896 + 6

2 119900119905ℎ119890119903119908119894119904119890

(43)

For a cycle 119862119899of order 119899 ⩾ 3 since 120574

119875(119862

119899) = 120574

119875(119875

119899) from

Theorem 118 the following result holds immediately

Corollary 119 (Raczek [73] 2008) For any positive integer 119896if 119899 ⩾ 3 then

119887119875(119862

119899) =

0 119894119891 119899 = 3 119900119903 5

2 119894119891 119899 = 4119896 4119896 + 3 119900119903 4119896 + 6

3 119900119905ℎ119890119903119908119894119904119890

(44)

A wheel 119882119899 where 119899 ⩾ 4 is a graph with 119899 vertices

formed by connecting a single vertex to all vertices of a cycle119862119899minus1

Of course 120574119875(119882

119899) = 2

International Journal of Combinatorics 19

119896

Figure 7 A tree 119879 with 119887119875(119879) = 119896

Theorem 120 (Raczek [73] 2008) For positive integers 119899 and119898

119887119875(119870

119898119899) = 119898 119891119900119903 1 lt 119898 ⩽ 119899

119887119875(119882

119899) =

4 119894119891 119899 = 4

3 119894119891 119899 = 5

2 119900119905ℎ119890119903119908119894119904119890

(45)

Theorem 16 shows that if 119879 is a nontrivial tree then119887(119879) ⩽ 2 However no similar result exists for pairedbondage For any nonnegative integer 119896 let 119879

119896be a tree

obtained by subdividing all but one edge of a star 1198701119896+1

(asshown in Figure 7) It is easy to see that 119887

119875(119879

119896) = 119896 We

state this fact as the following theorem which is similar toTheorem 112

Theorem121 (Raczek [73] 2008) For any nonnegative integer119896 there exists a tree 119879 with 119887

119875(119879) = 119896

Consider the tree defined in Figure 5 for 119896 = 3 Then119887(119879

3) ⩽ 2 and 119887

119875(119879

3) = 3 On the other hand 119887(119875

4) = 2

and 119887119875(119875

4) = 1 Thus we have what follows which is similar

to Corollary 113

Corollary 122 The bondage number and the paired bondagenumber are unrelated even for trees

A constructive characterization of trees with 119887(119879) = 2is given by Hartnell and Rall in [12] Raczek [73] provideda constructive characterization of trees with 119887

119875(119879) = 0 In

order to state the characterization we define a labeling andthree simple operations on a tree119879 Let 119910 isin 119881(119879) and let ℓ(119910)be the label assigned to 119910

Operation T1 If ℓ(119910) = 119861 add a vertex 119909 and the

edge 119910119909 and let ℓ(119909) = 119860OperationT

2 If ℓ(119910) = 119862 add a path (119909

1 119909

2) and the

edge 1199101199091 and let ℓ(119909

1) = 119861 ℓ(119909

2) = 119860

Operation T3 If ℓ(119910) = 119861 add a path (119909

1 119909

2 119909

3)

and the edge 1199101199091 and let ℓ(119909

1) = 119862 ℓ(119909

2) = 119861 and

ℓ(1199093) = 119860

Let 1198752= (119906 V) with ℓ(119906) = 119860 and ℓ(V) = 119861 Let T be

the class of all trees obtained from the labeled 1198752by a finite

sequence of operationsT1 T

2 T

3

A tree 119879 in Figure 8 belongs to the familyTRaczek [73] obtained the following characterization of all

trees 119879 with 119887119875(119879) = 0

119860

119860

119860

119860

119860

119861 119862

119861 119862 119861

119861119860

Figure 8 A tree 119879 belong to the familyT

Theorem 123 (Raczek [73] 2008) For a tree 119879 119887119875(119879) = 0 if

and only if 119879 is inT

We state the decision problem for the paired bondage asfollows

Problem 4 Consider the decision problem

Paired Bondage ProblemInstance a graph 119866 and a positive integer 119887Question is 119887

119875(119866) ⩽ 119887

Conjecture 124 The paired bondage problem is NP-complete

83 Restrained Bondage Numbers A dominating set 119878 of 119866 iscalled to be restrained if the subgraph induced by 119878 containsno isolated vertices where 119878 = 119881(119866) 119878 The restraineddomination number of 119866 denoted by 120574

119877(119866) is the smallest

cardinality of a restrained dominating set of 119866 It is clear that120574119877(119866) exists and 120574(119866) ⩽ 120574

119877(119866) for any nonempty graph 119866

The concept of restrained domination was introducedby Telle and Proskurowski [74] in 1997 albeit indirectly asa vertex partitioning problem Concerning the study of therestrained domination numbers the reader is referred to [74ndash79]

In 2008 Hattingh and Plummer [80] defined therestrained bondage number 119887R(119866) of a nonempty graph 119866 tobe the minimum cardinality among all subsets 119861 sube 119864(119866) forwhich 120574

119877(119866 minus 119861) gt 120574

119877(119866)

For some simple graphs their restrained dominationnumbers can be easily determined and so restrained bondagenumbers have been also determined For example it is clearthat 120574

119877(119870

119899) = 1 Domke et al [77] showed that 120574

119877(119862

119899) =

119899 minus 2lfloor1198993rfloor for 119899 ⩾ 3 and 120574119877(119875

119899) = 119899 minus 2lfloor(119899 minus 1)3rfloor for 119899 ⩾ 1

Using these results Hattingh and Plummer [80] obtained therestricted bondage numbers for119870

119899 119862

119899 119875

119899 and119866 = 119870

11989911198992119899119905

as follows

Theorem 125 (Hattingh and Plummer [80] 2008) For 119899 ⩾ 3

119887R (119870119899) =

1 119894119891 119899 = 3

lceil119899

2rceil 119900119905ℎ119890119903119908119894119904119890

119887R (119862119899) =

1 119894119891 119899 equiv 0 (mod 3) 2 119900119905ℎ119890119903119908119894119904119890

119887R (119875119899) = 1 119899 ⩾ 4

20 International Journal of Combinatorics

119887R (119866)

=

lceil119898

2rceil 119894119891 119899

119898= 1 119886119899119889 119899

119898+1⩾ 2 (1 ⩽ 119898 lt 119905)

2119905 minus 2 119894119891 1198991= 119899

2= sdot sdot sdot = 119899

119905= 2 (119905 ⩾ 2)

2 119894119891 1198991= 2 119886119899119889 119899

2⩾ 3 (119905 = 2)

119905minus1

sum

119894=1

119899119894minus 1 119900119905ℎ119890119903119908119894119904119890

(46)

Theorem 126 (Hattingh and Plummer [80] 2008) For a tree119879 of order 119899 (⩾ 4) 119879 ≇ 119870

1119899minus1if and only if 119887R(119879) = 1

Theorem 126 shows that the restrained bondage numberof a tree can be computed in constant time However thedecision problem for 119887R(119866) is NP-complete even for bipartitegraphs

Problem 5 Consider the decision problem

Restrained Bondage Problem

Instance a graph 119866 and a positive integer 119887

Question is 119887R(119866) ⩽ 119887

Theorem 127 (Hattingh and Plummer [80] 2008) Therestrained bondage problem is NP-complete even for bipartitegraphs

Consequently in view of its computational hardnessit is significative to establish some sharp bounds on therestrained bondage number of a graph in terms of othergraphic parameters

Theorem 128 (Hattingh and Plummer [80] 2008) For anygraph 119866 with 120575(119866) ⩾ 2

119887R (119866) ⩽ min119909119910isin119864(119866)

119889119866(119909) + 119889

119866(119910) minus 2 (47)

Corollary 129 119887R(119866) ⩽ Δ(119866)+120575(119866)minus2 for any graph119866 with120575(119866) ⩾ 2

Notice that the bounds stated in Theorem 128 andCorollary 129 are sharp Indeed the class of cycles whoseorders are congruent to 1 2 (mod 3) have a restrained bondagenumber achieving these bounds (see Theorem 125)

Theorem 128 is an analogue of Theorem 14 A quitenatural problem is whether or not for restricted bondagenumber there are analogues of Theorem 16 Theorem 18Theorem 29 and so on Indeed we should consider all resultsfor the bondage number whether or not there are analoguesfor restricted bondage number

Theorem 130 (Hattingh and Plummer [80] 2008) For anygraph 119866 with 120574

119877(119866) = 2

119887R (119866) ⩽ Δ (119866) + 1 (48)

We now consider the relation between 119887119877(119866) and 119887(119866)

Theorem 131 (Hattingh and Plummer [80] 2008) If 120574119877(119866) =

120574(119866) for some graph 119866 then 119887R(119866) ⩽ 119887(119866)

Corollary 129 and Theorem 131 provide a possibility toattack Conjecture 73 by characterizing planar graphs with120574119877= 120574 and 119887R = 119887 when 3 ⩽ Δ ⩽ 6However we do not have 119887R(119866) = 119887(119866) for any graph

119866 even if 120574119877(119866) = 120574(119866) Observe that 120574

119877(119870

3) = 120574(119870

3) yet

119887119877(119870

3) = 1 and 119887(119870

3) = 2 We still may not claim that

119887119877(119866) = 119887(119866) even in the case that every 120574-set is a 120574

119877-set

The example 1198703again demonstrates this

Theorem 132 (Hattingh and Plummer [80] 2008) There isan infinite class of graphs inwhich each graph119866 satisfies 119887(119866) lt119887R(119866)

In fact such an infinite class of graphs can be obtained asfollows Let119867 be a connected graph BC(119867) a graph obtainedfrom 119867 by attaching ℓ (⩾ 2) vertices of degree 1 to eachvertex in 119867 and B = 119866 119866 = BC(119867) for some graph 119867such that 120575(119867) ⩾ 2 By Theorem 3 we can easily check that119887(119866) = 1 lt 2 ⩽ min120575(119867) ℓ = 119887R(119866) for any 119866 isin BMoreover there exists a graph 119866 isin B such that 119887R(119866) canbe much larger than 119887(119866) which is stated as the followingtheorem

Theorem 133 (Hattingh and Plummer [80] 2008) For eachpositive integer 119896 there is a graph119866 such that 119896 = 119887R(119866)minus119887(119866)

Combining Theorem 133 with Theorem 132 we have thefollowing corollary

Corollary 134 The bondage number and the restrainedbondage number are unrelated

Very recently Jafari Rad et al [81] have consideredthe total restrained bondage in graphs based on both totaldominating sets and restrained dominating sets and obtainedseveral properties exact values and bounds for the totalrestrained bondage number of a graph

A restrained dominating set 119878 of a graph 119866 without iso-lated vertices is called to be a total restrained dominating setif 119878 is a total dominating set The total restrained dominationnumber of119866 denoted by 120574TR (119866) is theminimum cardinalityof a total restrained dominating set of 119866 For referenceson this domination in graphs see [3 61 82ndash85] The totalrestrained bondage number 119887TR (119866) of a graph 119866 with noisolated vertex is the cardinality of a smallest set of edges 119861 sube119864(119866) for which119866minus119861 has no isolated vertex and 120574TR (119866minus119861) gt120574TR (119866) In the case that there is no such subset 119861 we define119887TR (119866) = infin

Ma et al [84] determined that 120574TR (119875119899) = 119899minus2lfloor(119899minus2)4rfloorfor 119899 ⩾ 3 120574TR (119862119899

) = 119899 minus 2lfloor1198994rfloor for 119899 ⩾ 3 120574TR (1198703) =

3 and 120574TR (119870119899) = 2 for 119899 = 3 and 120574TR (1198701119899

) = 1 + 119899

and 120574TR (119870119898119899) = 2 for 2 ⩽ 119898 ⩽ 119899 According to these

results Jafari Rad et al [81] determined the exact valuesof total restrained bondage numbers for the correspondinggraphs

International Journal of Combinatorics 21

Theorem 135 (Jafari Rad et al [81]) For an integer 119899

119887TR (119875119899) = infin 119894119891 2 ⩽ 119899 ⩽ 5

1 119894119891 119899 ⩾ 5

119887TR (119862119899) =

infin 119894119891 119899 = 3

1 119894119891 3 lt 119899 119899 equiv 0 1 (mod 4)2 119894119891 3 lt 119899 119899 equiv 2 3 (mod 4)

119887TR (119882119899) = 2 119891119900119903 119899 ⩾ 6

119887TR (119870119899) = 119899 minus 1 119891119900119903 119899 ⩾ 4

119887TR (119870119898119899) = 119898 minus 1 119891119900119903 2 ⩽ 119898 ⩽ 119899

(49)

119887TR (119870119899) = 119899minus1 shows the fact that for any integer 119899 ⩾ 3 there

exists a connected graph namely119870119899+1

such that 119887TR (119870119899+1) =

119899 that is the total restrained bondage number is unboundedfrom the above

A vertex is said to be a support vertex if it is adjacent toa vertex of degree one Let A be the family of all connectedgraphs such that119866 belongs toA if and only if every edge of119866is incident to a support vertex or 119866 is a cycle of length three

Proposition 136 (Cyman and Raczek [82] 2006) For aconnected graph119866 of order 119899 120574TR (119866) = 119899 if and only if119866 isin A

Hence if 119866 isin A then the removal of any subset of edgesof119866 cannot increase the total restrained domination numberof 119866 and thus 119887TR (119866) = infin When 119866 notin A a result similar toTheorem 6 is obtained

Theorem 137 (Jafari Rad et al [81]) For any tree 119879 notin A119887TR (119879) ⩽ 2 This bound is sharp

In the samepaper the authors provided a characterizationfor trees 119879 notin A with 119887TR (119879) = 1 or 2 The following result issimilar to Observation 1

Observation 3 (Jafari Rad et al [81]) If 119896 edges can beremoved from a graph 119866 to obtain a graph 119867 without anisolated vertex and with 119887TR (119867) = 119905 then 119887TR (119866) ⩽ 119896 + 119905

ApplyingTheorem 137 andObservation 3 Jafari Rad et alcharacterized all connected graphs 119866 with 119887TR (119866) = infin

Theorem 138 (Jafari Rad et al [81]) For a connected graph119866119887TR (119866) = infin if and only if 119866 isin A

84 Other Conditional Bondage Numbers A subset 119868 sube 119881(119866)is called an independent set if no two vertices in 119868 are adjacentin 119866 The maximum cardinality among all independent setsis called the independence number of 119866 denoted by 120572(119866)

A dominating set 119878 of a graph 119866 is called to be inde-pendent if 119878 is an independent set of 119866 The minimumcardinality among all independent dominating set is calledthe independence domination number of 119866 and denoted by120574119868(119866)

Since an independent dominating set is not only adominating set but also an independent set 120574(119866) ⩽ 120574

119868(119866) ⩽

120572(119866) for any graph 119866It is clear that a maximal independent set is certainly a

dominating set Thus an independent set is maximal if andonly if it is an independent dominating set and so 120574

119868(119866) is the

minimum cardinality among all maximal independent setsof 119866 This graph-theoretical invariant has been well studiedin the literature see for example Haynes et al [3] for earlyresults Goddard and Henning [86] for recent results onindependent domination in graphs

In 2003 Zhang et al [87] defined the independencebondage number 119887

119868(119866) of a nonempty graph 119866 to be the

minimum cardinality among all subsets 119861 sube 119864(119866) for which120574119868(119866 minus 119861) gt 120574

119868(119866) For some ordinary graphs their indepen-

dence domination numbers can be easily determined and soindependence bondage numbers have been also determinedClearly 119887

119868(119870

119899) = 1 if 119899 ⩾ 2

Theorem 139 (Zhang et al [87] 2003) For any integer 119899 ⩾ 4

119887119868(119862

119899) =

1 119894119891 119899 equiv 1 (mod 2) 2 119894119891 119899 equiv 0 (mod 2)

119887119868(119875

119899) =

1 119894119891 119899 equiv 0 (mod 2) 2 119894119891 119899 equiv 1 (mod 2)

119887119868(119870

119898119899) = 119899 119891119900119903 119898 ⩽ 119899

(50)

Apart from the above-mentioned results as far as we findno other results on the independence bondage number in thepresent literature we never knew much of any result on thisparameter for a tree

A subset 119878 of 119881(119866) is called an equitable dominatingset if for every 119910 isin 119881(119866 minus 119878) there exists a vertex 119909 isin 119878such that 119909119910 isin 119864(119866) and |119889

119866(119909) minus 119889

119866(119910)| ⩽ 1 proposed

by Dharmalingam [88] The minimum cardinality of such adominating set is called the equitable domination number andis denoted by 120574

119864(119866) Deepak et al [89] defined the equitable

bondage number 119887119864(119866) of a graph 119866 to be the cardinality of a

smallest set 119861 sube 119864(119866) of edges for which 120574119864(119866 minus 119861) gt 120574

119864(119866)

and determined 119887119864(119866) for some special graphs

Theorem 140 (Deepak et al [89] 2011) For any integer 119899 ⩾ 2

119887119864(119870

119899) = lceil

119899

2rceil

119887119864(119870

119898119899) = 119898 119891119900119903 0 ⩽ 119899 minus 119898 ⩽ 1

119887119864(119862

119899) =

3 119894119891 119899 equiv 1 (mod 3)2 119900119905ℎ119890119903119908119894119904119890

119887119864(119875

119899) =

2 119894119891 119899 equiv 1 (mod 3)1 119900119905ℎ119890119903119908119894119904119890

119887119864(119879) ⩽ 2 119891119900119903 119886119899119910 119899119900119899119905119903119894119907119894119886119897 119905119903119890119890 119879

119887119864(119866) ⩽ 119899 minus 1 119891119900119903 119886119899119910 119888119900119899119899119890119888119905119890119889 119892119903119886119901ℎ 119866 119900119891 119900119903119889119890119903 119899

(51)

22 International Journal of Combinatorics

As far as we know there are no more results on theequitable bondage number of graphs in the present literature

There are other variations of the normal domination byadding restricted conditions to the normal dominating setFor example the connected domination set of a graph119866 pro-posed by Sampathkumar and Walikar [90] is a dominatingset 119878 such that the subgraph induced by 119878 is connected Theproblem of finding a minimum connected domination set ina graph is equivalent to the problemof finding a spanning treewithmaximumnumber of vertices of degree one [91] and hassome important applications in wireless networks [92] thusit has received much research attention However for suchan important domination there is no result on its bondagenumber We believe that the topic is of significance for futureresearch

9 Generalized Bondage Numbers

There are various generalizations of the classical dominationsuch as 119901-domination distance domination fractional dom-ination Roman domination and rainbow domination Everysuch a generalization can lead to a corresponding bondageIn this section we introduce some of them

91 119901-Bondage Numbers In 1985 Fink and Jacobson [93]introduced the concept of 119901-domination Let 119901 be a positiveinteger A subset 119878 of 119881(119866) is a 119901-dominating set of 119866 if|119878 cap 119873

119866(119910)| ⩾ 119901 for every 119910 isin 119878 The 119901-domination number

120574119901(119866) is the minimum cardinality among all 119901-dominating

sets of 119866 Any 119901-dominating set of 119866 with cardinality 120574119901(119866)

is called a 120574119901-set of 119866 Note that a 120574

1-set is a classic minimum

dominating set Notice that every graph has a 119901-dominatingset since the vertex-set119881(119866) is such a setWe also note that a 1-dominating set is a dominating set and so 120574(119866) = 120574

1(119866) The

119901-domination number has received much research attentionsee a state-of-the-art survey article by Chellali et al [94]

It is clear from definition that every 119901-dominating setof a graph certainly contains all vertices of degree at most119901 minus 1 By this simple observation to avoid the trivial caseoccurrence we always assume that Δ(119866) ⩾ 119901 For 119901 ⩾ 2 Luet al [95] gave a constructive characterization of trees withunique minimum 119901-dominating sets

Recently Lu and Xu [96] have introduced the conceptto the 119901-bondage number of 119866 denoted by 119887

119901(119866) as the

minimum cardinality among all sets of edges 119861 sube 119864(119866) suchthat 120574

119901(119866 minus 119861) gt 120574

119901(119866) Clearly 119887

1(119866) = 119887(119866)

Lu and Xu [96] established a lower bound and an upperbound on 119887

119901(119879) for any integer 119901 ⩾ 2 and any tree 119879 with

Δ(119879) ⩾ 119901 and characterized all trees achieving the lowerbound and the upper bound respectively

Theorem 141 (Lu and Xu [96] 2011) For any integer 119901 ⩾ 2and any tree 119879 with Δ(119879) ⩾ 119901

1 ⩽ 119887119901(119879) ⩽ Δ (119879) minus 119901 + 1 (52)

Let 119878 be a given subset of 119881(119866) and for any 119909 isin 119878 let

119873119901(119909 119878 119866) = 119910 isin 119878 cap 119873

119866(119909)

1003816100381610038161003816119873119866(119910) cap 119878

1003816100381610038161003816 = 119901

(53)

Then all trees achieving the lower bound 1 can be character-ized as follows

Theorem 142 (Lu and Xu [96] 2011) Let 119879 be a tree withΔ(119879) ⩾ 119901 ⩾ 2 Then 119887

119901(119879) = 1 if and only if for any 120574

119901-set

119878 of 119879 there exists an edge 119909119910 isin (119878 119878) such that 119910 isin 119873119901(119909 119878 119879)

The notation 119878(119886 119887) denotes a double star obtained byadding an edge between the central vertices of two stars119870

1119886minus1

and1198701119887minus1

And the vertex with degree 119886 (resp 119887) in 119878(119886 119887) iscalled the 119871-central vertex (resp 119877-central vertex) of 119878(119886 119887)

To characterize all trees attaining the upper bound givenin Theorem 141 we define three types of operations on a tree119879 with Δ(119879) = Δ ⩾ 119901 + 1

Type 1 attach a pendant edge to a vertex 119910with 119889119879(119910) ⩽ 119901minus

2 in 119879

Type 2 attach a star 1198701Δminus1

to a vertex 119910 of 119879 by joining itscentral vertex to 119910 where 119910 in a 120574

119901-set of 119879 and

119889119879(119910) ⩽ Δ minus 1

Type 3 attach a double star 119878(119901 Δ minus 1) to a pendant vertex 119910of 119879 by coinciding its 119877-central vertex with 119910 wherethe unique neighbor of 119910 is in a 120574

119901-set of 119879

Let B = 119879 a tree obtained from 1198701Δ

or 119878(119901 Δ) by afinite sequence of operations of Types 1 2 and 3

Theorem 143 (Lu and Xu [96] 2011) A tree with the maxi-mum Δ ⩾ 119901 + 1 has 119901-bondage number Δ minus 119901 + 1 if and onlyif it belongs toB

Theorem 144 (Lu and Xu [97] 2012) Let 119866119898119899= 119875

119898times 119875

119899

Then

1198872(119866

2119899) = 1 119891119900119903 119899 ⩾ 2

1198872(119866

3119899) =

2 119894119891 119899 equiv 1 (mod 3) 1 119900119905ℎ119890119903119908119894119904119890

for 119899 ⩾ 2

1198872(119866

4119899) =

1 119894119891 119899 equiv 3 (mod 4) 2 119900119905ℎ119890119903119908119894119904119890

for 119899 ⩾ 7

(54)

Recently Krzywkowski [98] has proposed the conceptof the nonisolating 2-bondage of a graph The nonisolating2-bondage number of a graph 119866 denoted by 1198871015840

2(119866) is the

minimum cardinality among all sets of edges 119861 sube 119864(119866) suchthat 119866 minus 119861 has no isolated vertices and 120574

2(119866 minus 119861) gt 120574

2(119866) If

for every 119861 sube 119864(119866) either 1205742(119866 minus 119861) = 120574

2(119866) or 119866 minus 119861 has

isolated vertices then define 11988710158402(119866) = 0 and say that 119866 is a 2-

nonisolatedly strongly stable graph Krzywkowski presentedsome basic properties of nonisolating 2-bondage in graphsshowed that for every nonnegative integer 119896 there exists atree 119879 such 1198871015840

2(119879) = 119896 characterized all 2-non-isolatedly

strongly stable trees and determined 11988710158402(119866) for several special

graphs

International Journal of Combinatorics 23

Theorem 145 (Krzywkowski [98] 2012) For any positiveintegers 119899 and119898 with119898 ⩽ 119899

1198871015840

2(119870

119899) =

0 119894119891 119899 = 1 2 3

lfloor2119899

3rfloor 119900119905ℎ119890119903119908119894119904119890

1198871015840

2(119875

119899) =

0 119894119891 119899 = 1 2 3

1 119900119905ℎ119890119903119908119894119904119890

1198871015840

2(119862

119899) =

0 119894119891 119899 = 3

1 119894119891 119899 119894119904 119890V119890119899

2 119894119891 119899 119894119904 119900119889119889

1198871015840

2(119870

119898119899) =

3 119894119891 119898 = 119899 = 3

1 119894119891 119898 = 119899 = 4

2 119900119905ℎ119890119903119908119894119904119890

(55)

92 Distance Bondage Numbers A subset 119878 of vertices of agraph119866 is said to be a distance 119896-dominating set for119866 if everyvertex in 119866 not in 119878 is at distance at most 119896 from some vertexof 119878 The minimum cardinality of all distance 119896-dominatingsets is called the distance 119896-domination number of 119866 anddenoted by 120574

119896(119866) (do not confuse with the above-mentioned

120574119901(119866)) When 119896 = 1 a distance 1-dominating set is a normal

dominating set and so 1205741(119866) = 120574(119866) for any graph 119866 Thus

the distance 119896-domination is a generalization of the classicaldomination

The relation between 120574119896and 120572

119896(the 119896-independence

number defined in Section 22) for a tree obtained by Meirand Moon [14] who proved that 120574

119896(119879) = 120572

2119896(119879) for any tree

119879 (a special case for 119896 = 1 is stated in Proposition 10) Thefurther research results can be found in Henning et al [99]Tian and Xu [100ndash103] and Liu et al [104]

In 1998 Hartnell et al [15] defined the distance 119896-bondagenumber of 119866 denoted by 119887

119896(119866) to be the cardinality of a

smallest subset 119861 of edges of 119866 with the property that 120574119896(119866 minus

119861) gt 120574119896(119866) FromTheorem 6 it is clear that if119879 is a nontrivial

tree then 1 ⩽ 1198871(119879) ⩽ 2 Hartnell et al [15] generalized this

result to any integer 119896 ⩾ 1

Theorem 146 (Hartnell et al [15] 1998) For every nontrivialtree 119879 and positive integer 119896 1 ⩽ 119887

119896(119879) ⩽ 2

Hartnell et al [15] and Topp and Vestergaard [13] alsocharacterized the trees having distance 119896-bondage number 2In particular the class of trees for which 119887

1(119879) = 2 are just

those which have a unique maximum 2-independent set (seeTheorem 11)

Since when 119896 = 1 the distance 1-bondage number 1198871(119866)

is the classical bondage number 119887(119866) Theorem 12 gives theNP-hardness of deciding the distance 119896-bondage number ofgeneral graphs

Theorem 147 Given a nonempty undirected graph 119866 andpositive integers 119896 and 119887 with 119887 ⩽ 120576(119866) determining wetheror not 119887

119896(119866) ⩽ 119887 is NP-hard

For a vertex 119909 in119866 the open 119896-neighborhood119873119896(119909) of 119909 is

defined as119873119896(119909) = 119910 isin 119881(119866) 1 ⩽ 119889

119866(119909 119910) le 119896The closed

119896-neighborhood119873119896[119909] of 119909 in 119866 is defined as119873

119896(119909) cup 119909 Let

Δ119896(119866) = max 1003816100381610038161003816119873119896

(119909)1003816100381610038161003816 for any 119909 isin 119881 (119866) (56)

Clearly Δ1(119866) = Δ(119866) The 119896th power of a graph 119866 is the

graph119866119896 with vertex-set119881(119866119896

) = 119881(119866) and edge-set119864(119866119896

) =

119909119910 1 ⩽ 119889119866(119909 119910) ⩽ 119896 The following lemma holds directly

from the definition of 119866119896

Proposition 148 (Tian and Xu [101]) Δ(119866119896

) = Δ119896(119866) and

120574(119866119896

) = 120574119896(119866) for any graph 119866 and each 119896 ⩾ 1

A graph 119866 is 119896-distance domination-critical or 120574119896-critical

for short if 120574119896(119866 minus 119909) lt 120574

119896(119866) for every vertex 119909 in 119866

proposed by Henning et al [105]

Proposition 149 (Tian and Xu [101]) For each 119896 ⩾ 1 a graph119866 is 120574

119896-critical if and only if 119866119896 is 120574

119896-critical

By the above facts we suggest the following worthwhileproblem

Problem 6 Can we generalize the results on the bondage for119866 to 119866119896 In particular do the following propositions hold

(a) 119887(119866119896

) = 119887119896(119866) for any graph 119866 and each 119896 ⩾ 1

(b) 119887(119866119896

) ⩽ Δ119896(119866) if 119866 is not 120574

119896-critical

Let 119896 and 119901 be positive integers A subset 119878 of 119881(119866) isdefined to be a (119896 119901)-dominating set of 119866 if for any vertex119909 isin 119878 |119873

119896(119909) cap 119878| ⩾ 119901 The (119896 119901)-domination number of

119866 denoted by 120574119896119901(119866) is the minimum cardinality among

all (119896 119901)-dominating sets of 119866 Clearly for a graph 119866 a(1 1)-dominating set is a classical dominating set a (119896 1)-dominating set is a distance 119896-dominating set and a (1 119901)-dominating set is the above-mentioned 119901-dominating setThat is 120574

11(119866) = 120574(119866) 120574

1198961(119866) = 120574

119896(119866) and 120574

1119901(119866) = 120574

119901(119866)

The concept of (119896 119901)-domination in a graph 119866 is a gen-eralized domination which combines distance 119896-dominationand 119901-domination in 119866 So the investigation of (119896 119901)-domination of 119866 is more interesting and has received theattention of many researchers see for example [106ndash109]

More general Li and Xu [110] proposed the concept of the(ℓ 119908)-domination number of a119908-connected graph A subset119878 of 119881(119866) is defined to be an (ℓ 119908)-dominating set of 119866 iffor any vertex 119909 isin 119878 there are 119908 internally disjoint pathsof length at most ℓ from 119909 to some vertex in 119878 The (ℓ 119908)-domination number of119866 denoted by 120574

ℓ119908(119866) is theminimum

cardinality among all (ℓ 119908)-dominating sets of 119866 Clearly1205741198961(119866) = 120574

119896(119866) and 120574

1119901(119866) = 120574

119901(119866)

It is quite natural to propose the concept of bondage num-ber for (119896 119901)-domination or (ℓ 119908)-domination However asfar as we know none has proposed this concept until todayThis is a worthwhile topic for further research

24 International Journal of Combinatorics

93 Fractional Bondage Numbers If 120590 is a function mappingthe vertex-set 119881 into some set of real numbers then for anysubset 119878 sube 119881 let 120590(119878) = sum

119909isin119878120590(119909) Also let |120590| = 120590(119881)

A real-value function 120590 119881 rarr [0 1] is a dominatingfunction of a graph 119866 if for every 119909 isin 119881(119866) 120590(119873

119866[119909]) ⩾ 1

Thus if 119878 is a dominating set of 119866 120590 is a function where

120590 (119909) = 1 if 119909 isin 1198780 otherwise

(57)

then 120590 is a dominating function of119866 The fractional domina-tion number of 119866 denoted by 120574

119865(119866) is defined as follows

120574119865(119866) = min |120590| 120590 is a domination function of 119866

(58)

The 120574119865-bondage number of 119866 denoted by 119887

119865(119866) is

defined as the minimum cardinality of a subset 119861 sube 119864 whoseremoval results in 120574

119865(119866 minus 119861) gt 120574

119865(119866)

Hedetniemi et al [111] are the first to study fractionaldomination although Farber [112] introduced the idea indi-rectly The concept of the fractional domination numberwas proposed by Domke and Laskar [113] in 1997 Thefractional domination numbers for some ordinary graphs aredetermined

Proposition 150 (Domke et al [114] 1998 Domke and Laskar[113] 1997) (a) If 119866 is a 119896-regular graph with order 119899 then120574119865(119866) = 119899119896 + 1(b) 120574

119865(119862

119899) = 1198993 120574

119865(119875

119899) = lceil1198993rceil 120574

119865(119870

119899) = 1

(c) 120574119865(119866) = 1 if and only if Δ(119866) = 119899 minus 1

(d) 120574119865(119879) = 120574(119879) for any tree 119879

(e) 120574119865(119870

119899119898) = (119899(119898 + 1) + 119898(119899 + 1))(119899119898 minus 1) where

max119899119898 gt 1

The assertions (a)ndash(d) are due to Domke et al [114] andthe assertion (e) is due to Domke and Laskar [113] Accordingto these results Domke and Laskar [113] determined the 120574

119865-

bondage numbers for these graphs

Theorem 151 (Domke and Laskar [113] 1997) 119887119865(119870

119899) =

lceil1198992rceil 119887119865(119870

119899119898) = 1 wheremax119899119898 gt 1

119887119865(119862

119899) =

2 119894119891 119899 equiv 0 (mod 3) 1 119900119905ℎ119890119903119908119894119904119890

119887119865(119875

119899) =

2 119894119891 119899 equiv 1 (mod 3) 1 119900119905ℎ119890119903119908119894119904119890

(59)

It is easy to see that for any tree119879 119887119865(119879) ⩽ 2 In fact since

120574119865(119879) = 120574(119879) for any tree 119879 120574

119865(119879

1015840

) = 120574(1198791015840

) for any subgraph1198791015840 of 119879 It follows that

119887119865(119879) = min 119861 sube 119864 (119879) 120574

119865(119879 minus 119861) gt 120574

119865(119879)

= min 119861 sube 119864 (119879) 120574 (119879 minus 119861) gt 120574 (119879)

= 119887 (119879) ⩽ 2

(60)

Except for these no results on 119887119865(119866) have been known as

yet

94 Roman Bondage Numbers A Roman dominating func-tion on a graph 119866 is a labeling 119891 119881 rarr 0 1 2 such thatevery vertex with label 0 has at least one neighbor with label2 The weight of a Roman dominating function 119891 on 119866 is thevalue

119891 (119866) = sum

119906isin119881(119866)

119891 (119906) (61)

The minimum weight of a Roman dominating function ona graph 119866 is called the Roman domination number of 119866denoted by 120574RM (119866)

A Roman dominating function 119891 119881 rarr 0 1 2

can be represented by the ordered partition (1198810 119881

1 119881

2) (or

(119881119891

0 119881

119891

1 119881

119891

2) to refer to119891) of119881 where119881

119894= V isin 119881 | 119891(V) = 119894

In this representation its weight 119891(119866) = |1198811| + 2|119881

2| It is

clear that119881119891

1cup119881

119891

2is a dominating set of 119866 called the Roman

dominating set denoted by 119863119891

R = (1198811 1198812) Since 119881119891

1cup 119881

119891

2is

a dominating set when 119891 is a Roman dominating functionand since placing weight 2 at the vertices of a dominating setyields a Roman dominating function in [115] it is observedthat

120574 (119866) ⩽ 120574RM (119866) ⩽ 2120574 (119866) (62)

A graph 119866 is called to be Roman if 120574RM (119866) = 2120574(119866)The definition of the Roman domination function is

proposed implicitly by Stewart [116] and ReVelle and Rosing[117] Roman dominating numbers have been deeply studiedIn particular Bahremandpour et al [118] showed that theproblem determining the Roman domination number is NP-complete even for bipartite graphs

Let 119866 be a graph with maximum degree at least twoThe Roman bondage number 119887RM (119866) of 119866 is the minimumcardinality of all sets 1198641015840 sube 119864 for which 120574RM (119866 minus 119864

1015840

) gt

120574RM (119866) Since in study of Roman bondage number theassumption Δ(119866) ⩾ 2 is necessary we always assume thatwhen we discuss 119887RM (119866) all graphs involved satisfy Δ(119866) ⩾2 The Roman bondage number 119887RM (119866) is introduced byJafari Rad and Volkmann in [119]

Recently Bahremandpour et al [118] have shown that theproblem determining the Roman bondage number is NP-hard even for bipartite graphs

Problem 7 Consider the decision problem

Roman Bondage Problem

Instance a nonempty bipartite graph119866 and a positiveinteger 119896

Question is 119887RM (119866) ⩽ 119896

Theorem 152 (Bahremandpour et al [118] 2013) TheRomanbondage problem is NP-hard even for bipartite graphs

The exact value of 119887RM(119866) is known only for a few familyof graphs including the complete graphs cycles and paths

International Journal of Combinatorics 25

Theorem 153 (Jafari Rad and Volkmann [119] 2011) For anypositive integers 119899 and119898 with119898 ⩽ 119899

119887RM (119875119899) = 2 119894119891 119899 equiv 2 (mod 3) 1 119900119905ℎ119890119903119908119894119904119890

119887RM (119862119899) =

3 119894119891 119899 = 2 (mod 3) 2 119900119905ℎ119890119903119908119894119904119890

119887RM (119870119899) = lceil

119899

2rceil 119891119900119903 119899 ⩾ 3

119887RM (119870119898119899) =

1 119894119891 119898 = 1 119886119899119889 119899 = 1

4 119894119891 119898 = 119899 = 3

119898 119900119905ℎ119890119903119908119894119904119890

(63)

Lemma 154 (Cockayne et al [115] 2004) If 119866 is a graph oforder 119899 and contains vertices of degree 119899 minus 1 then 120574RM(119866) = 2

Using Lemma 154 the third conclusion in Theorem 153can be generalized to more general case which is similar toLemma 49

Theorem 155 (Jafari Rad and Volkmann [119] 2011) Let119866 bea graph with order 119899 ge 3 and 119905 the number of vertices of degree119899 minus 1 in 119866 If 119905 ⩾ 1 then 119887RM(119866) = lceil1199052rceil

Ebadi and PushpaLatha [120] conjectured that 119887RM(119866) ⩽119899 minus 1 for any graph 119866 of order 119899 ⩾ 3 Dehgardi et al[121] Akbari andQajar [122] independently showed that thisconjecture is true

Theorem 156 119887RM(119866) ⩽ 119899 minus 1 for any connected graph 119866 oforder 119899 ⩾ 3

For a complete 119905-partite graph we have the followingresult

Theorem 157 (Hu and Xu [123] 2011) Let 119866 = 11987011989811198982119898119905

bea complete 119905-partite graph with 119898

1= sdot sdot sdot = 119898

119894lt 119898

119894+1le sdot sdot sdot ⩽

119898119905 119905 ⩾ 2 and 119899 = sum119905

119895=1119898

119895 Then

119887RM (119866) =

lceil119894

2rceil 119894119891 119898

119894= 1 119886119899119889 119899 ⩾ 3

2 119894119891 119898119894= 2 119886119899119889 119894 = 1

119894 119894119891 119898119894= 2 119886119899119889 119894 ⩾ 2

4 119894119891 119898119894= 3 119886119899119889 119894 = 119905 = 2

119899 minus 1 119894119891 119898119894= 3 119886119899119889 119894 = 119905 ⩾ 3

119899 minus 119898119905

119894119891 119898119894⩾ 3 119886119899119889 119898

119905⩾ 4

(64)

Consider a complete 119905-partite graph 119870119905(3) for 119905 ⩾ 3

119887RM(119870119905(3)) = 119899 minus 1 by Theorem 157 where 119899 = 3119905

which shows that this upper bound of 119899 minus 1 on 119887RM(119866) inTheorem 156 is sharp This fact and 120574

119877(119870

119905(3)) = 4 give

a negative answer to the following two problems posed byDehgardi et al [121]Prove or Disprove for any connected graph 119866 of order 119899 ge 3(i) 119887RM(119866) = 119899 minus 1 if and only if 119866 cong 119870

3 (ii) 119887RM(119866) ⩽ 119899 minus

120574119877(119866) + 1Note that a complete 119905-partite graph 119870

119905(3) is (119899 minus 3)-

regular and 119887RM(119870119905(3)) = 119899 minus 1 for 119905 ⩾ 3 where 119899 = 3119905

Hu and Xu further determined that 119887119877(119866) = 119899 minus 2 for any

(119899 minus 3)-regular graph 119866 of order 119899 ⩽ 5 except 119870119905(3)

Theorem 158 (Hu and Xu [123] 2011) For any (119899minus3)-regulargraph 119866 of order 119899 other than119870

119905(3) 119887RM(119866) = 119899 minus 2 for 119899 ⩾ 5

For a tree 119879 with order 119899 ⩾ 3 Ebadi and PushpaLatha[120] and Jafari Rad and Volkmann [119] independentlyobtained an upper bound on 119887RM(119879)

Theorem 159 119887RM(119879) ⩽ 3 for any tree 119879 with order 119899 ⩾ 3

Theorem 160 (Bahremandpour et al [118] 2013) 119887RM(1198752 times119875119899) = 2 for 119899 ⩾ 2

Theorem 161 (Jafari Rad and Volkmann [119] 2011) Let119866 bea graph with order at least three

(a) If (119909 119910 119911) is a path of length 2 in 119866 then 119887RM(119866) ⩽119889119866(119909) + 119889

119866(119910) + 119889

119866(119911) minus 3 minus |119873

119866(119909) cap 119873

119866(119910)|

(b) If 119866 is connected then 119887RM(119866) ⩽ 120582(119866) + 2Δ(119866) minus 3

Theorem 161 (b) implies 119887RM(119866) ⩽ 120575(119866) + 2Δ(119866) minus 3 for aconnected graph 119866 Note that for a planar graph 119866 120575(119866) ⩽ 5moreover 120575(119866) ⩽ 3 if the girth at least 4 and 120575(119866) ⩽ 2 if thegirth at least 6 These two facts show that 119887RM(119866) ⩽ 2Δ(119866) +2 for a connected planar graph 119866 Jafari Rad and Volkmann[124] improved this bound

Theorem 162 (Jafari Rad and Volkmann [124] 2011) For anyconnected planar graph 119866 of order 119899 ⩾ 3 with girth 119892(119866)

119887RM (119866)

⩽

2Δ (119866)

Δ (119866) + 6

Δ (119866) + 5 119894119891 119866 119888119900119899119905119886119894119899119904 119899119900 V119890119903119905119894119888119890119904 119900119891 119889119890119892119903119890119890 119891119894V119890

Δ (119866) + 4 119894119891 119892 (119866) ⩾ 4

Δ (119866) + 3 119894119891 119892 (119866) ⩾ 5

Δ (119866) + 2 119894119891 119892 (119866) ⩾ 6

Δ (119866) + 1 119894119891 119892 (119866) ⩾ 8

(65)

According toTheorem 157 119887RM(119862119899) = 3 = Δ(119862

119899)+1 for a

cycle 119862119899of length 119899 ⩾ 8 with 119899 equiv 2 (mod 3) and therefore

the last upper bound inTheorem 162 is sharp at least for Δ =2

Combining the fact that every planar graph 119866 withminimum degree 5 contains an edge 119909119910 with 119889

119866(119909) = 5

and 119889119866(119910) isin 5 6 with Theorem 161 (a) Akbari et al [125]

obtained the following result

26 International Journal of Combinatorics

middot middot middot

middot middot middot

middot middot middot

middot middot middot

119866

Figure 9 The graph 119866 is constructed from 119866

Theorem 163 (Akbari et al [125] 2012) 119887RM(119866) ⩽ 15 forevery planar graph 119866

It remains open to show whether the bound inTheorem 163 is sharp or not Though finding a planargraph 119866 with 119887RM(119866) = 15 seems to be difficult Akbari etal [125] constructed an infinite family of planar graphs withRoman bondage number equal to 7 by proving the followingresult

Theorem 164 (Akbari et al [125] 2012) Let 119866 be a graph oforder 119899 and 119866 is the graph of order 5119899 obtained from 119866 byattaching the central vertex of a copy of a path 119875

5to each vertex

of119866 (see Figure 9) Then 120574RM(119866) = 4119899 and 119887RM(119866) = 120575(119866)+2

ByTheorem 164 infinitemany planar graphswith Romanbondage number 7 can be obtained by considering any planargraph 119866 with 120575(119866) = 5 (eg the icosahedron graph)

Conjecture 165 (Akbari et al [125] 2012) The Romanbondage number of every planar graph is at most 7

For general bounds the following observation is directlyobtained which is similar to Observation 1

Observation 4 Let 119866 be a graph of order 119899 with maximumdegree at least two Assume that 119867 is a spanning subgraphobtained from 119866 by removing 119896 edges If 120574RM(119867) = 120574RM(119866)then 119887

119877(119867) ⩽ 119887RM(119866) ⩽ 119887RM(119867) + 119896

Theorem 166 (Bahremandpour et al [118] 2013) For anyconnected graph 119866 of order 119899 if 119899 ⩾ 3 and 120574RM(119866) = 120574(119866) + 1then

119887RM (119866) ⩽ min 119887 (119866) 119899Δ (66)

where 119899Δis the number of vertices with maximum degree Δ in

119866

Observation 5 For any graph 119866 if 120574(119866) = 120574RM(119866) then119887RM(119866) ⩽ 119887(119866)

Theorem 167 (Bahremandpour et al [118] 2013) For everyRoman graph 119866

119887RM (119866) ⩾ 119887 (119866) (67)

The bound is sharp for cycles on 119899 vertices where 119899 equiv 0 (mod3)

The strict inequality in Theorem 167 can hold for exam-ple 119887(119862

3119896+2) = 2 lt 3 = 119887RM(1198623119896+2

) byTheorem 157A graph 119866 is called to be vertex Roman domination-

critical if 120574RM(119866 minus 119909) lt 120574RM(119866) for every vertex 119909 in 119866 If119866 has no isolated vertices then 120574RM(119866) ⩽ 2120574(119866) ⩽ 2120573(119866)If 120574RM(119866) = 2120573(119866) then 120574RM(119866) = 2120574(119866) and hence 119866 is aRoman graph In [30] Volkmann gave a lot of graphs with120574(119866) = 120573(119866) The following result is similar to Theorem 57

Theorem 168 (Bahremandpour et al [118] 2013) For anygraph 119866 if 120574RM(119866) = 2120573(119866) then

(a) 119887RM(119866) ⩾ 120575(119866)

(b) 119887RM(119866) ⩾ 120575(119866)+1 if119866 is a vertex Roman domination-critical graph

If 120574RM(119866) = 2 then obviously 119887RM(119866) ⩽ 120575(119866) So weassume 120574RM(119866) ⩾ 3 Dehgardi et al [121] proved 119887RM(119866) ⩽(120574RM(119866) minus 2)Δ(119866) + 1 and posed the following problem if 119866is a connected graph of order 119899 ⩾ 4 with 120574RM(119866) ⩾ 3 then

119887RM (119866) ⩽ (120574RM (119866) minus 2) Δ (119866) (68)

Theorem 161 (a) shows that the inequality (68) holds if120574RM(119866) ⩾ 5 Thus the bound in (68) is of interest only when120574RM(119866) is 3 or 4

Lemma 169 (Bahremandpour et al [118] 2013) For anynonempty graph 119866 of order 119899 ⩾ 3 120574RM(119866) = 3 if and onlyif Δ(119866) = 119899 minus 2

The following result shows that (68) holds for all graphs119866 of order 119899 ⩾ 4 with 120574RM(119866) = 3 or 4 which improvesTheorem 161 (b)

Theorem 170 (Bahremandpour et al [118] 2013) For anyconnected graph 119866 of order 119899 ⩾ 4

119887RM (119866) le Δ (119866) = 119899 minus 2 119894119891 120574RM (119866) = 3

Δ (119866) + 120575 (119866) minus 1 119894119891 120574RM (119866) = 4(69)

with the first equality if and only if 119866 cong 1198624

Recently Akbari and Qajar [122] have proved the follow-ing result

Theorem 171 (Akbari and Qajar [122]) For any connectedgraph 119866 of order 119899 ⩾ 3

119887RM (119866) ⩽ 119899 minus 120574RM (119866) + 5 (70)

International Journal of Combinatorics 27

We conclude this section with the following problems

Problem 8 Characterize all connected graphs 119866 of order 119899 ⩾3 for which 119887RM(119866) = 119899 minus 1

Problem 9 Prove or disprove if 119866 is a connected graph oforder 119899 ⩾ 3 then

119887RM (119866) ⩽ 119899 minus 120574RM (119866) + 3 (71)

Note that 120574119877(119870

119905(3)) = 4 and 119887RM(119870119905

(3)) = 119899minus1 where 119899 =3119905 for 119905 ⩾ 3 The upper bound on 119887RM(119866) given in Problem 9is sharp if Problem 9 has positive answer

95 Rainbow Bondage Numbers For a positive integer 119896 a119896-rainbow dominating function of a graph 119866 is a function 119891from the vertex-set 119881(119866) to the set of all subsets of the set1 2 119896 such that for any vertex 119909 in 119866 with 119891(119909) = 0 thecondition

⋃

119910isin119873119866(119909)

119891 (119910) = 1 2 119896 (72)

is fulfilledThe weight of a 119896-rainbow dominating function 119891on 119866 is the value

119891 (119866) = sum

119909isin119881(119866)

1003816100381610038161003816119891 (119909)1003816100381610038161003816 (73)

The 119896-rainbow domination number of a graph 119866 denoted by120574119903119896(119866) is the minimum weight of a 119896-rainbow dominating

function 119891 of 119866Note that 120574

1199031(119866) is the classical domination number 120574(119866)

The 119896-rainbow domination number is introduced by Bresaret al [126] Rainbow domination of a graph 119866 coincides withordinary domination of the Cartesian product of 119866 with thecomplete graph in particular 120574

119903119896(119866) = 120574(119866 times 119870

119896) for any

positive integer 119896 and any graph 119866 [126] Hartnell and Rall[127] proved that 120574(119879) lt 120574

1199032(119879) for any tree 119879 no graph has

120574 = 120574119903119896for 119896 ⩾ 3 for any 119896 ⩾ 2 and any graph 119866 of order 119899

min 119899 120574 (119866) + 119896 minus 2 ⩽ 120574119903119896(119866) ⩽ 119896120574 (119866) (74)

The introduction of rainbow domination is motivatedby the study of paired domination in Cartesian productsof graphs where certain upper bounds can be expressed interms of rainbow domination that is for any graph 119866 andany graph 119867 if 119881(119867) can be partitioned into 119896120574

119875-sets then

120574119875(119866 times 119867) ⩽ (1119896)120592(119867)120574

119903119896(119866) proved by Bresar et al [126]

Dehgardi et al [128] introduced the concept of 119896-rainbowbondage number of graphs Let 119866 be a graph with maximumdegree at least two The 119896-rainbow bondage number 119887

119903119896(119866)

of 119866 is the minimum cardinality of all sets 119861 sube 119864(119866) forwhich 120574

119903119896(119866 minus 119861) gt 120574

119903119896(119866) Since in the study of 119896-rainbow

bondage number the assumption Δ(119866) ⩾ 2 is necessarywe always assume that when we discuss 119887

119903119896(119866) all graphs

involved satisfy Δ(119866) ⩾ 2The 2-rainbow bondage number of some special families

of graphs has been determined

Theorem 172 (Dehgardi et al [128]) For any integer 119899 ⩾ 3

1198871199032(119875

119899) = 1 119891119900119903 119899 ⩾ 2

1198871199032(119862

119899) =

1 119894119891 119899 equiv 0 (mod 4) 2 119900119905ℎ119890119903119908119894119904119890

1198871199032(119870

119899119899) =

1 119894119891 119899 = 2

3 119894119891 119899 = 3

6 119894119891 119899 = 4

119898 119894119891 119899 ⩾ 5

(75)

Theorem 173 (Dehgardi et al [128]) For any connected graph119866 of order 119899 ⩾ 3 119887

1199032(119866) ⩽ 120582(119866) + 2Δ(119866) minus 3

Theorem 174 (Dehgardi et al [128]) For any tree 119879 of order119899 ⩾ 3 119887

1199032(119879) ⩽ 2

Theorem 175 (Dehgardi et al [128]) If 119866 is a planar graphwithmaximumdegree at least two then 119887

1199032(119866) ⩽ 15Moreover

if the girth 119892(119866) ⩾ 4 then 1198871199032(119866) ⩽ 11 and if 119892(119866) ⩾ 6 then

1198871199032(119866) ⩽ 9

Theorem 176 (Dehgardi et al [128]) For a complete bipartitegraph 119870

119901119902 if 2119896 + 1 ⩽ 119901 ⩽ 119902 then 119887

119903119896(119870

119901119902) = 119901

Theorem177 (Dehgardi et al [128]) For a complete graph119870119899

if 119899 ⩾ 119896 + 1 then 119887119903119896(119870

119899) = lceil119896119899(119896 + 1)rceil

The special case 119896 = 1 of Theorem 177 can be found inTheorem 1 The following is a simple observation similar toObservation 1

Observation 6 (Dehgardi et al [128]) Let 119866 be a graphwith maximum degree at least two 119867 a spanning subgraphobtained from 119866 by removing 119896 edges If 120574

119903119896(119867) = 120574

119903119896(119866)

then 119887119903119896(119867) ⩽ 119887

119903119896(119866) ⩽ 119887

119903119896(119867) + 119896

Theorem 178 (Dehgardi et al [128]) For every graph 119866 with120574119903119896(119866) = 119896120574(119866) 119887

119903119896(119866) ⩾ 119887(119866)

A graph 119866 is said to be vertex 119896-rainbow domination-critical if 120574

119903119896(119866 minus 119909) lt 120574

119903119896(119866) for every vertex 119909 in 119866 It is

clear that if 119866 has no isolated vertices then 120574119903119896(119866) ⩽ 119896120574(119866) ⩽

119896120573(119866) where 120573(119866) is the vertex-covering number of119866 Thusif 120574

119903119896(119866) = 119896120573(119866) then 120574

119903119896(119866) = 119896120574(119866)

Theorem 179 (Dehgardi et al [128]) For any graph 119866 if120574119903119896(119866) = 119896120573(119866) then

(a) 119887119903119896(119866) ⩾ 120575(119866)

(b) 119887119903119896(119866) ⩾ 120575(119866)+1 if119866 is vertex 119896-rainbow domination-

critical

As far as we know there are no more results on the 119896-rainbow bondage number of graphs in the literature

28 International Journal of Combinatorics

10 Results on Digraphs

Although domination has been extensively studied in undi-rected graphs it is natural to think of a dominating setas a one-way relationship between vertices of the graphIndeed among the earliest literatures on this subject vonNeumann and Morgenstern [129] used what is now calleddomination in digraphs to find solution (or kernels whichare independent dominating sets) for cooperative 119899-persongames Most likely the first formulation of domination byBerge [130] in 1958 was given in the context of digraphs andonly some years latter by Ore [131] in 1962 for undirectedgraphs Despite this history examination of domination andits variants in digraphs has been essentially overlooked (see[4] for an overview of the domination literature) Thus thereare few if any such results on domination for digraphs in theliterature

The bondage number and its related topics for undirectedgraph have become one of major areas both in theoreticaland applied researches However until recently Carlsonand Develin [42] Shan and Kang [132] and Huang andXu [133 134] studied the bondage number for digraphsindependently In this section we introduce their resultsfor general digraphs Results for some special digraphs suchas vertex-transitive digraphs are introduced in the nextsection

101 Upper Bounds for General Digraphs Let 119866 = (119881 119864)

be a digraph without loops and parallel edges A digraph 119866is called to be asymmetric if whenever (119909 119910) isin 119864(119866) then(119910 119909) notin 119864(119866) and to be symmetric if (119909 119910) isin 119864(119866) implies(119910 119909) isin 119864(119866) For a vertex 119909 of 119881(119866) the sets of out-neighbors and in-neighbors of 119909 are respectively defined as119873

+

119866(119909) = 119910 isin 119881(119866) (119909 119910) isin 119864(119866) and 119873minus

119866(119909) = 119909 isin

119881(119866) (119909 119910) isin 119864(119866) the out-degree and the in-degree of119909 are respectively defined as 119889+

119866(119909) = |119873

+

119866(119909)| and 119889minus

119866(119909) =

|119873+

119866(119909)| Denote themaximum and theminimum out-degree

(resp in-degree) of 119866 by Δ+(119866) and 120575+(119866) (resp Δminus(119866) and120575minus

(119866))The degree 119889119866(119909) of 119909 is defined as 119889+

119866(119909)+119889

minus

119866(119909) and

the maximum and the minimum degrees of119866 are denoted byΔ(119866) and 120575(119866) respectively that is Δ(119866) = max119889

119866(119909) 119909 isin

119881(119866) and 120575(119866) = min119889119866(119909) 119909 isin 119881(119866) Note that the

definitions here are different from the ones in the textbookon digraphs see for example Xu [1]

A subset 119878 of119881(119866) is called a dominating set if119881(119866) = 119878cup119873

+

119866(119878) where119873+

119866(119878) is the set of out-neighbors of 119878Then just

as for undirected graphs 120574(119866) is the minimum cardinalityof a dominating set and the bondage number 119887(119866) is thesmallest cardinality of a set 119861 of edges such that 120574(119866 minus 119861) gt120574(119866) if such a subset 119861 sube 119864(119866) exists Otherwise we put119887(119866) = infin

Some basic results for undirected graphs stated inSection 3 can be generalized to digraphs For exampleTheorem 18 is generalized by Carlson and Develin [42]Huang andXu [133] and Shan andKang [132] independentlyas follows

Theorem 180 Let 119866 be a digraph and (119909 119910) isin 119864(119866) Then119887(119866) ⩽ 119889

119866(119910) + 119889

minus

119866(119909) minus |119873

minus

119866(119909) cap 119873

minus

119866(119910)|

Corollary 181 For a digraph 119866 119887(119866) ⩽ 120575minus(119866) + Δ(119866)

Since 120575minus

(119866) ⩽ (12)Δ(119866) from Corollary 181Conjecture 53 is valid for digraphs

Corollary 182 For a digraph 119866 119887(119866) ⩽ (32)Δ(119866)

In the case of undirected graphs the bondage number of119866119899= 119870

119899times 119870

119899achieves this bound However it was shown

by Carlson and Develin in [42] that if we take the symmetricdigraph119866

119899 we haveΔ(119866

119899) = 4(119899minus1) 120574(119866

119899) = 119899 and 119887(119866

119899) ⩽

4(119899 minus 1) + 1 = Δ(119866119899) + 1 So this family of digraphs cannot

show that the bound in Corollary 182 is tightCorresponding to Conjecture 51 which is discredited

for undirected graphs and is valid for digraphs the sameconjecture for digraphs can be proposed as follows

Conjecture 183 (Carlson and Develin [42] 2006) 119887(119866) ⩽Δ(119866) + 1 for any digraph 119866

If Conjecture 183 is true then the following results showsthat this upper bound is tight since 119887(119870

119899times119870

119899) = Δ(119870

119899times119870

119899)+1

for a complete digraph 119870119899

In 2007 Shan and Kang [132] gave some tight upperbounds on the bondage numbers for some asymmetricdigraphs For example 119887(119879) ⩽ Δ(119879) for any asymmetricdirected tree 119879 119887(119866) ⩽ Δ(119866) for any asymmetric digraph 119866with order at least 4 and 120574(119866) ⩽ 2 For planar digraphs theyobtained the following results

Theorem 184 (Shan and Kang [132] 2007) Let 119866 be aasymmetric planar digraphThen 119887(119866) ⩽ Δ(119866)+2 and 119887(119866) ⩽Δ(119866) + 1 if Δ(119866) ⩾ 5 and 119889minus

119866(119909) ⩾ 3 for every vertex 119909 with

119889119866(119909) ⩾ 4

102 Results for Some Special Digraphs The exact values andbounds on 119887(119866) for some standard digraphs are determined

Theorem185 (Huang andXu [133] 2006) For a directed cycle119862119899and a directed path 119875

119899

119887 (119862119899) =

3 119894119891 119899 119894119904 119900119889119889

2 119894119891 119899 119894119904 119890V119890119899

119887 (119875119899) =

2 119894119891 119899 119894119904 119900119889119889

1 119894119891 119899 119894119904 119890V119890119899

(76)

For the de Bruijn digraph 119861(119889 119899) and the Kautz digraph119870(119889 119899)

119887 (119861 (119889 119899)) = 119889 119894119891 119899 119894119904 119900119889119889

119889 ⩽ 119887 (119861 (119889 119899)) ⩽ 2119889 119894119891 119899 119894119904 119890V119890119899

119887 (119870 (119889 119899)) = 119889 + 1

(77)

Like undirected graphs we can define the total domi-nation number and the total bondage number On the totalbondage numbers for some special digraphs the knownresults are as follows

International Journal of Combinatorics 29

Theorem 186 (Huang and Xu [134] 2007) For a directedcycle 119862

119899and a directed path 119875

119899 119887

119879(119875

119899) and 119887

119879(119862

119899) all do not

exist For a complete digraph 119870119899

119887119879(119870

119899) = infin 119894119891 119899 = 1 2

119887119879(119870

119899) = 3 119894119891 119899 = 3

119899 ⩽ 119887119879(119870

119899) ⩽ 2119899 minus 3 119894119891 119899 ⩾ 4

(78)

The extended de Bruijn digraph EB(119889 119899 1199021 119902

119901) and

the extended Kautz digraph EK(119889 119899 1199021 119902

119901) are intro-

duced by Shibata and Gonda [135] If 119901 = 1 then theyare the de Bruijn digraph B(119889 119899) and the Kautz digraphK(119889 119899) respectively Huang and Xu [134] determined theirtotal domination numbers In particular their total bondagenumbers for general cases are determined as follows

Theorem 187 (Huang andXu [134] 2007) If 119889 ⩾ 2 and 119902119894⩾ 2

for each 119894 = 1 2 119901 then

119887119879(EB(119889 119899 119902

1 119902

119901)) = 119889

119901

minus 1

119887119879(EK(119889 119899 119902

1 119902

119901)) = 119889

119901

(79)

In particular for the de Bruijn digraph B(119889 119899) and the Kautzdigraph K(119889 119899)

119887119879(B (119889 119899)) = 119889 minus 1 119887

119879(K (119889 119899)) = 119889 (80)

Zhang et al [136] determined the bondage number incomplete 119905-partite digraphs

Theorem 188 (Zhang et al [136] 2009) For a complete 119905-partite digraph 119870

11989911198992119899119905

where 1198991⩽ 119899

2⩽ sdot sdot sdot ⩽ 119899

119905

119887 (11987011989911198992119899119905

)

=

119898 119894119891 119899119898= 1 119899

119898+1⩾ 2 119891119900119903 119904119900119898119890

119898 (1 ⩽ 119898 lt 119905)

4119905 minus 3 119894119891 1198991= 119899

2= sdot sdot sdot = 119899

119905= 2

119905minus1

sum

119894=1

119899119894+ 2 (119905 minus 1) 119900119905ℎ119890119903119908119894119904119890

(81)

Since an undirected graph can be thought of as a sym-metric digraph any result for digraphs has an analogy forundirected graphs in general In view of this point studyingthe bondage number for digraphs is more significant thanfor undirected graphs Thus we should further study thebondage number of digraphs and try to generalize knownresults on the bondage number and related variants for undi-rected graphs to digraphs prove or disprove Conjecture 183In particular determine the exact values of 119887(B(119889 119899)) for aneven 119899 and 119887

119879(119870

119899) for 119899 ⩾ 4

11 Efficient Dominating Sets

A dominating set 119878 of a graph 119866 is called to be efficient if forevery vertex 119909 in119866 |119873

119866[119909]cap119878| = 1 if119866 is a undirected graph

or |119873minus

119866[119909] cap 119878| = 1 if 119866 is a directed graph The concept of an

efficient set was proposed by Bange et al [137] in 1988 In theliterature an efficient set is also called a perfect dominatingset (see Livingston and Stout [138] for an explanation)

From definition if 119878 is an efficient dominating set of agraph 119866 then 119878 is certainly an independent set and everyvertex not in 119878 is adjacent to exactly one vertex in 119878

It is also clear from definition that a dominating set 119878 isefficient if and only if N

119866[119878] = 119873

119866[119909] 119909 isin 119878 for the

undirected graph 119866 or N+

119866[119878] = 119873

+

119866[119909] 119909 isin 119878 for the

digraph119866 is a partition of119881(119866) where the induced subgraphby119873

119866[119909] or119873+

119866[119909] is a star or an out-star with the root 119909

The efficient domination has important applications inmany areas such as error-correcting codes and receivesmuch attention in the late years

The concept of efficient dominating sets is a measureof the efficiency of domination in graphs Unfortunately asshown in [137] not every graph has an efficient dominatingset andmoreover it is anNP-complete problem to determinewhether a given graph has an efficient dominating set Inaddition it was shown by Clark [139] in 1993 that for a widerange of 119901 almost every random undirected graph 119866 isin

G(120592 119901) has no efficient dominating sets This means thatundirected graphs possessing an efficient dominating set arerare However it is easy to show that every undirected graphhas an orientationwith an efficient dominating set (see Bangeet al [140])

In 1993 Barkauskas and Host [141] showed that deter-mining whether an arbitrary oriented graph has an efficientdominating set is NP-complete Even so the existence ofefficient dominating sets for some special classes of graphs hasbeen examined see for example Dejter and Serra [142] andLee [143] for Cayley graph Gu et al [144] formeshes and toriObradovic et al [145] Huang and Xu [36] Reji Kumar andMacGillivray [146] for circulant graphs Harary graphs andtori and Van Wieren et al [147] for cube-connected cycles

In this section we introduce results of the bondagenumber for some graphs with efficient dominating sets dueto Huang and Xu [36]

111 Results for General Graphs In this subsection we intro-duce some results on bondage numbers obtained by applyingefficient dominating sets We first state the two followinglemmas

Lemma 189 For any 119896-regular graph or digraph 119866 of order 119899120574(119866) ⩾ 119899(119896 + 1) with equality if and only if 119866 has an efficientdominating set In addition if 119866 has an efficient dominatingset then every efficient dominating set is certainly a 120574-set andvice versa

Let 119890 be an edge and 119878 a dominating set in 119866 We say that119890 supports 119878 if 119890 isin (119878 119878) where (119878 119878) = (119909 119910) isin 119864(119866) 119909 isin 119878 119910 isin 119878 Denote by 119904(119866) the minimum number of edgeswhich support all 120574-sets in 119866

Lemma 190 For any graph or digraph 119866 119887(119866) ⩾ 119904(119866) withequality if 119866 is regular and has an efficient dominating set

30 International Journal of Combinatorics

A graph 119866 is called to be vertex-transitive if its automor-phism group Aut(119866) acts transitively on its vertex-set 119881(119866)A vertex-transitive graph is regular Applying Lemmas 189and 190 Huang and Xu obtained some results on bondagenumbers for vertex-transitive graphs or digraphs

Theorem 191 (Huang and Xu [36] 2008) For any vertex-transitive graph or digraph 119866 of order 119899

119887 (119866) ⩾

lceil119899

2120574 (119866)rceil 119894119891 119866 119894119904 119906119899119889119894119903119890119888119905119890119889

lceil119899

120574 (119866)rceil 119894119891 119866 119894119904 119889119894119903119890119888119905119890119889

(82)

Theorem192 (Huang andXu [36] 2008) Let119866 be a 119896-regulargraph of order 119899 Then

119887(119866) ⩽ 119896 if119866 is undirected and 119899 ⩾ 120574(119866)(119896+1)minus119896+1119887(119866) ⩽ 119896+1+ℓ if119866 is directed and 119899 ⩾ 120574(119866)(119896+1)minusℓwith 0 ⩽ ℓ ⩽ 119896 minus 1

Next we introduce a better upper bound on 119887(119866) To thisaim we need the following concept which is a generalizationof the concept of the edge-covering of a graph 119866 For 1198811015840

sube

119881(119866) and 1198641015840 sube 119864(119866) we say that 1198641015840covers 1198811015840 and call 1198641015840an edge-covering for 1198811015840 if there exists an edge (119909 119910) isin 1198641015840 forany vertex 119909 isin 1198811015840 For 119910 isin 119881(119866) let 1205731015840[119910] be the minimumcardinality over all edge-coverings for119873minus

119866[119910]

Theorem 193 (Huang and Xu [36] 2008) If 119866 is a 119896-regulargraph with order 120574(119866)(119896 + 1) then 119887(119866) ⩽ 1205731015840[119910] for any 119910 isin119881(119866)

Theupper bound on 119887(119866) given inTheorem 193 is tight inview of 119887(119862

119899) for a cycle or a directed cycle 119862

119899(seeTheorems

1 and 185 resp)It is easy to see that for a 119896-regular graph 119866 lceil(119896 + 1)2rceil ⩽

1205731015840

[119910] ⩽ 119896 when 119866 is undirected and 1205731015840[119910] = 119896 + 1 when 119866 isdirected By this fact and Lemma 189 the following theoremis merely a simple combination of Theorems 191 and 193

Theorem 194 (Huang and Xu [36] 2008) Let 119866 be a vertex-transitive graph of degree 119896 If 119866 has an efficient dominatingset then

lceil119896 + 1

2rceil ⩽ 119887 (119866) ⩽ 119896 119894119891 119866 119894119904 119906119899119889119894119903119890119888119905119890119889

119887 (119866) = 119896 + 1 119894119891 119866 119894119904 119889119894119903119890119888119905119890119889

(83)

Theorem 195 (Huang and Xu [36] 2008) If 119866 is an undi-rected vertex-transitive cubic graph with order 4120574(119866) and girth119892(119866) ⩽ 5 then 119887(119866) = 2

The proof of Theorem 195 leads to a byproduct If 119892 =5 let (119906

1 119906

2 119906

3 119906

4 119906

5) be a cycle and let D

119894be the family

of efficient dominating sets containing 119906119894for each 119894 isin

1 2 3 4 5 ThenD119894capD

119895= 0 for 119894 = 1 2 5 Then 119866 has

at least 5119904 efficient dominating sets where 119904 = 119904(119866) But thereare only 4s (=ns120574) distinct efficient dominating sets in119866This

contradiction implies that an undirected vertex-transitivecubic graph with girth five has no efficient dominating setsBut a similar argument for 119892(119866) = 3 4 or 119892(119866) ⩾ 6 couldnot give any contradiction This is consistent with the resultthat CCC(119899) a vertex-transitive cubic graph with girth 119899 if3 ⩽ 119899 ⩽ 8 or girth 8 if 119899 ⩾ 9 has efficient dominating setsfor all 119899 ⩾ 3 except 119899 = 5 (see Theorem 199 in the followingsubsection)

112 Results for Cayley Graphs In this subsection we useTheorem 194 to determine the exact values or approximativevalues of bondage numbers for some special vertex-transitivegraphs by characterizing the existence of efficient dominatingsets in these graphs

Let Γ be a nontrivial finite group 119878 a nonempty subset ofΓ without the identity element of Γ A digraph 119866 is defined asfollows

119881 (119866) = Γ (119909 119910) isin 119864 (119866) lArrrArr 119909minus1

119910 isin 119878

for any 119909 119910 isin Γ(84)

is called a Cayley digraph of the group Γ with respect to119878 denoted by 119862

Γ(119878) If 119878minus1 = 119904minus1 119904 isin 119878 = 119878 then

119862Γ(119878) is symmetric and is called a Cayley undirected graph

a Cayley graph for short Cayley graphs or digraphs arecertainly vertex-transitive (see Xu [1])

A circulant graph 119866(119899 119878) of order 119899 is a Cayley graph119862119885119899

(119878) where 119885119899= 0 119899 minus 1 is the addition group of

order 119899 and 119878 is a nonempty subset of119885119899without the identity

element and hence is a vertex-transitive digraph of degree|119878| If 119878minus1 = 119878 then119866(119899 119878) is an undirected graph If 119878 = 1 119896where 2 ⩽ 119896 ⩽ 119899 minus 2 we write 119866(119899 1 119896) for 119866(119899 1 119896) or119866(119899 plusmn1 plusmn119896) and call it a double-loop circulant graph

Huang and Xu [36] showed that for directed 119866 =

119866(119899 1 119896) lceil1198993rceil ⩽ 120574(119866) ⩽ lceil1198992rceil and 119866 has an efficientdominating set if and only if 3 | 119899 and 119896 equiv 2 (mod 3) fordirected 119866 = 119866(119899 1 119896) and 119896 = 1198992 lceil1198995rceil ⩽ 120574(119866) ⩽ lceil1198993rceiland 119866 has an efficient dominating set if and only if 5 | 119899 and119896 equiv plusmn2 (mod 5) By Theorem 194 we obtain the bondagenumber of a double loop circulant graph if it has an efficientdominating set

Theorem 196 (Huang and Xu [36] 2008) Let 119866 be a double-loop circulant graph 119866(119899 1 119896) If 119866 is directed with 3 | 119899and 119896 equiv 2 (mod 3) or 119866 is undirected with 5 | 119899 and119896 equiv plusmn2 (mod 5) then 119887(119866) = 3

The 119898 times 119899 torus is the Cartesian product 119862119898times 119862

119899of

two cycles and is a Cayley graph 119862119885119898times119885119899

(119878) where 119878 =

(0 1) (1 0) for directed cycles and 119878 = (0 plusmn1) (plusmn1 0) forundirected cycles and hence is vertex-transitive Gu et al[144] showed that the undirected torus119862

119898times119862

119899has an efficient

dominating set if and only if both 119898 and 119899 are multiplesof 5 Huang and Xu [36] showed that the directed torus119862119898times 119862

119899has an efficient dominating set if and only if both

119898 and 119899 are multiples of 3 Moreover they found a necessarycondition for a dominating set containing the vertex (0 0)in 119862

119898times 119862

119899to be efficient and obtained the following

result

International Journal of Combinatorics 31

Theorem 197 (Huang and Xu [36] 2008) Let 119866 = 119862119898times 119862

119899

If 119866 is undirected and both119898 and 119899 are multiples of 5 or if 119866 isdirected and both119898 and 119899 are multiples of 3 then 119887(119866) = 3

The hypercube 119876119899

is the Cayley graph 119862Γ(119878)

where Γ = 1198852times sdot sdot sdot times 119885

2= (119885

2)119899 and 119878 =

100 sdot sdot sdot 0 010 sdot sdot sdot 0 00 sdot sdot sdot 01 Lee [143] showed that119876119899has an efficient dominating set if and only if 119899 = 2119898 minus 1

for a positive integer119898 ByTheorem 194 the following resultis obtained immediately

Theorem 198 (Huang and Xu [36] 2008) If 119899 = 2119898 minus 1 for apositive integer119898 then 2119898minus1

⩽ 119887(119876119899) ⩽ 2

119898

minus 1

Problem 10 Determine the exact value of 119887(119876119899) for 119899 = 2119898minus1

The 119899-dimensional cube-connected cycle denoted byCCC(119899) is constructed from the 119899-dimensional hypercube119876119899by replacing each vertex 119909 in 119876

119899with an undirected cycle

119862119899of length 119899 and linking the 119894th vertex of the 119862

119899to the 119894th

neighbor of 119909 It has been proved that CCC(119899) is a Cayleygraph and hence is a vertex-transitive graph with degree 3Van Wieren et al [147] proved that CCC(119899) has an efficientdominating set if and only if 119899 = 5 From Theorems 194 and195 the following result can be derived

Theorem 199 (Huang and Xu [36] 2008) Let 119866 = 119862119862119862(119899)be the 119899-dimensional cube-connected cycles with 119899 ⩾ 3 and119899 = 5 Then 120574(119866) = 1198992119899minus2 and 2 ⩽ 119887(119866) ⩽ 3 In addition119887(119862119862119862(3)) = 119887(119862119862119862(4)) = 2

Problem 11 Determine the exact value of 119887(CCC(119899)) for 119899 ⩾5

Acknowledgments

This work was supported by NNSF of China (Grants nos11071233 61272008) The author would like to express hisgratitude to the anonymous referees for their kind sugges-tions and comments on the original paper to ProfessorSheikholeslami and Professor Jafari Rad for sending thereferences [128] and [81] which resulted in this version

References

[1] J-M Xu Theory and Application of Graphs vol 10 of NetworkTheory and Applications Kluwer Academic Dordrecht TheNetherlands 2003

[2] S THedetniemi andR C Laskar ldquoBibliography on dominationin graphs and some basic definitions of domination parame-tersrdquo Discrete Mathematics vol 86 no 1ndash3 pp 257ndash277 1990

[3] TW Haynes S T Hedetniemi and P J Slater Fundamentals ofDomination in Graphs vol 208 ofMonographs and Textbooks inPure and Applied Mathematics Marcel Dekker New York NYUSA 1998

[4] TWHaynes S THedetniemi and P J Slater EdsDominationin Graphs vol 209 of Monographs and Textbooks in Pure andAppliedMathematicsMarcelDekker NewYorkNYUSA 1998

[5] M R Garey andD S JohnsonComputers and Intractability WH Freeman and Company San Francisco Calif USA 1979

[6] H B Walikar and B D Acharya ldquoDomination critical graphsrdquoNational Academy Science Letters vol 2 pp 70ndash72 1979

[7] D Bauer F Harary J Nieminen and C L Suffel ldquoDominationalteration sets in graphsrdquo Discrete Mathematics vol 47 no 2-3pp 153ndash161 1983

[8] J F Fink M S Jacobson L F Kinch and J Roberts ldquoThebondage number of a graphrdquo Discrete Mathematics vol 86 no1ndash3 pp 47ndash57 1990

[9] F-T Hu and J-M Xu ldquoThe bondage number of (119899 minus 3)-regulargraph G of order nrdquo Ars Combinatoria to appear

[10] U Teschner ldquoNew results about the bondage number of agraphrdquoDiscreteMathematics vol 171 no 1ndash3 pp 249ndash259 1997

[11] B L Hartnell and D F Rall ldquoA bound on the size of a graphwith given order and bondage numberrdquo Discrete Mathematicsvol 197-198 pp 409ndash413 1999 16th British CombinatorialConference (London 1997)

[12] B L Hartnell and D F Rall ldquoA characterization of trees inwhich no edge is essential to the domination numberrdquo ArsCombinatoria vol 33 pp 65ndash76 1992

[13] J Topp and P D Vestergaard ldquo120572119896- and 120574

119896-stable graphsrdquo

Discrete Mathematics vol 212 no 1-2 pp 149ndash160 2000[14] A Meir and J W Moon ldquoRelations between packing and

covering numbers of a treerdquo Pacific Journal of Mathematics vol61 no 1 pp 225ndash233 1975

[15] B L Hartnell L K Joslashrgensen P D Vestergaard and CA Whitehead ldquoEdge stability of the k-domination numberof treesrdquo Bulletin of the Institute of Combinatorics and ItsApplications vol 22 pp 31ndash40 1998

[16] F-T Hu and J-M Xu ldquoOn the complexity of the bondage andreinforcement problemsrdquo Journal of Complexity vol 28 no 2pp 192ndash201 2012

[17] B L Hartnell and D F Rall ldquoBounds on the bondage numberof a graphrdquo Discrete Mathematics vol 128 no 1ndash3 pp 173ndash1771994

[18] Y-L Wang ldquoOn the bondage number of a graphrdquo DiscreteMathematics vol 159 no 1ndash3 pp 291ndash294 1996

[19] G H Fricke TW Haynes S M Hedetniemi S T Hedetniemiand R C Laskar ldquoExcellent treesrdquo Bulletin of the Institute ofCombinatorics and Its Applications vol 34 pp 27ndash38 2002

[20] V Samodivkin ldquoThe bondage number of graphs good and badverticesrdquoDiscussiones Mathematicae GraphTheory vol 28 no3 pp 453ndash462 2008

[21] J E Dunbar T W Haynes U Teschner and L VolkmannldquoBondage insensitivity and reinforcementrdquo in Domination inGraphs vol 209 of Monographs and Textbooks in Pure andAppliedMathematics pp 471ndash489 Dekker NewYork NY USA1998

[22] H Liu and L Sun ldquoThe bondage and connectivity of a graphrdquoDiscrete Mathematics vol 263 no 1ndash3 pp 289ndash293 2003

[23] A-H Esfahanian and S L Hakimi ldquoOn computing a con-ditional edge-connectivity of a graphrdquo Information ProcessingLetters vol 27 no 4 pp 195ndash199 1988

[24] R C Brigham P Z Chinn and R D Dutton ldquoVertexdomination-critical graphsrdquoNetworks vol 18 no 3 pp 173ndash1791988

[25] V Samodivkin ldquoDomination with respect to nondegenerateproperties bondage numberrdquoThe Australasian Journal of Com-binatorics vol 45 pp 217ndash226 2009

[26] U Teschner ldquoA new upper bound for the bondage numberof graphs with small domination numberrdquo The AustralasianJournal of Combinatorics vol 12 pp 27ndash35 1995

32 International Journal of Combinatorics

[27] J Fulman D Hanson and G MacGillivray ldquoVertex dom-ination-critical graphsrdquoNetworks vol 25 no 2 pp 41ndash43 1995

[28] U Teschner ldquoA counterexample to a conjecture on the bondagenumber of a graphrdquo Discrete Mathematics vol 122 no 1ndash3 pp393ndash395 1993

[29] U Teschner ldquoThe bondage number of a graph G can be muchgreater than Δ(G)rdquo Ars Combinatoria vol 43 pp 81ndash87 1996

[30] L Volkmann ldquoOn graphs with equal domination and coveringnumbersrdquoDiscrete AppliedMathematics vol 51 no 1-2 pp 211ndash217 1994

[31] L A Sanchis ldquoMaximum number of edges in connected graphswith a given domination numberrdquoDiscreteMathematics vol 87no 1 pp 65ndash72 1991

[32] S Klavzar and N Seifter ldquoDominating Cartesian products ofcyclesrdquoDiscrete AppliedMathematics vol 59 no 2 pp 129ndash1361995

[33] H K Kim ldquoA note on Vizingrsquos conjecture about the bondagenumberrdquo Korean Journal of Mathematics vol 10 no 2 pp 39ndash42 2003

[34] M Y Sohn X Yuan and H S Jeong ldquoThe bondage number of1198623times119862

119899rdquo Journal of the KoreanMathematical Society vol 44 no

6 pp 1213ndash1231 2007[35] L Kang M Y Sohn and H K Kim ldquoBondage number of the

discrete torus 119862119899times 119862

4rdquo Discrete Mathematics vol 303 no 1ndash3

pp 80ndash86 2005[36] J Huang and J-M Xu ldquoThe bondage numbers and efficient

dominations of vertex-transitive graphsrdquoDiscrete Mathematicsvol 308 no 4 pp 571ndash582 2008

[37] C J Xiang X Yuan andM Y Sohn ldquoDomination and bondagenumber of119862

3times119862

119899rdquoArs Combinatoria vol 97 pp 299ndash310 2010

[38] M S Jacobson and L F Kinch ldquoOn the domination numberof products of graphs Irdquo Ars Combinatoria vol 18 pp 33ndash441984

[39] Y-Q Li ldquoA note on the bondage number of a graphrdquo ChineseQuarterly Journal of Mathematics vol 9 no 4 pp 1ndash3 1994

[40] F-T Hu and J-M Xu ldquoBondage number of mesh networksrdquoFrontiers ofMathematics inChina vol 7 no 5 pp 813ndash826 2012

[41] R Frucht and F Harary ldquoOn the corona of two graphsrdquoAequationes Mathematicae vol 4 pp 322ndash325 1970

[42] K Carlson and M Develin ldquoOn the bondage number of planarand directed graphsrdquoDiscreteMathematics vol 306 no 8-9 pp820ndash826 2006

[43] U Teschner ldquoOn the bondage number of block graphsrdquo ArsCombinatoria vol 46 pp 25ndash32 1997

[44] J Huang and J-M Xu ldquoNote on conjectures of bondagenumbers of planar graphsrdquo Applied Mathematical Sciences vol6 no 65ndash68 pp 3277ndash3287 2012

[45] L Kang and J Yuan ldquoBondage number of planar graphsrdquoDiscrete Mathematics vol 222 no 1ndash3 pp 191ndash198 2000

[46] M Fischermann D Rautenbach and L Volkmann ldquoRemarkson the bondage number of planar graphsrdquo Discrete Mathemat-ics vol 260 no 1ndash3 pp 57ndash67 2003

[47] J Hou G Liu and J Wu ldquoBondage number of planar graphswithout small cyclesrdquoUtilitasMathematica vol 84 pp 189ndash1992011

[48] J Huang and J-M Xu ldquoThe bondage numbers of graphs withsmall crossing numbersrdquo Discrete Mathematics vol 307 no 15pp 1881ndash1897 2007

[49] Q Ma S Zhang and J Wang ldquoBondage number of 1-planargraphrdquo Applied Mathematics vol 1 no 2 pp 101ndash103 2010

[50] J Hou and G Liu ldquoA bound on the bondage number of toroidalgraphsrdquoDiscreteMathematics Algorithms and Applications vol4 no 3 Article ID 1250046 5 pages 2012

[51] Y-C Cao J-M Xu and X-R Xu ldquoOn bondage number oftoroidal graphsrdquo Journal of University of Science and Technologyof China vol 39 no 3 pp 225ndash228 2009

[52] Y-C Cao J Huang and J-M Xu ldquoThe bondage number ofgraphs with crossing number less than fourrdquo Ars Combinatoriato appear

[53] H K Kim ldquoOn the bondage numbers of some graphsrdquo KoreanJournal of Mathematics vol 11 no 2 pp 11ndash15 2004

[54] A Gagarin and V Zverovich ldquoUpper bounds for the bondagenumber of graphs on topological surfacesrdquo Discrete Mathemat-ics 2012

[55] J Huang ldquoAn improved upper bound for the bondage numberof graphs on surfacesrdquoDiscrete Mathematics vol 312 no 18 pp2776ndash2781 2012

[56] A Gagarin and V Zverovich ldquoThe bondage number of graphson topological surfaces and Teschnerrsquos conjecturerdquo DiscreteMathematics vol 313 no 6 pp 796ndash808 2013

[57] V Samodivkin ldquoNote on the bondage number of graphs ontopological surfacesrdquo httparxivorgabs12086203

[58] E J Cockayne R M Dawes and S T Hedetniemi ldquoTotaldomination in graphsrdquoNetworks vol 10 no 3 pp 211ndash219 1980

[59] R Laskar J Pfaff S M Hedetniemi and S T HedetniemildquoOn the algorithmic complexity of total dominationrdquo Society forIndustrial and Applied Mathematics vol 5 no 3 pp 420ndash4251984

[60] J Pfaff R C Laskar and S T Hedetniemi ldquoNP-completeness oftotal and connected domination and irredundance for bipartitegraphsrdquo Tech Rep 428 Department of Mathematical SciencesClemson University 1983

[61] M A Henning ldquoA survey of selected recent results on totaldomination in graphsrdquoDiscrete Mathematics vol 309 no 1 pp32ndash63 2009

[62] V R Kulli and D K Patwari ldquoThe total bondage number ofa graphrdquo in Advances in Graph Theory pp 227ndash235 VishwaGulbarga India 1991

[63] F-T Hu Y Lu and J-M Xu ldquoThe total bondage number ofgrid graphsrdquo Discrete Applied Mathematics vol 160 no 16-17pp 2408ndash2418 2012

[64] J Cao M Shi and B Wu ldquoResearch on the total bondagenumber of a spectial networkrdquo in Proceedings of the 3rdInternational Joint Conference on Computational Sciences andOptimization (CSO rsquo10) pp 456ndash458 May 2010

[65] N Sridharan MD Elias and V S A Subramanian ldquoTotalbondage number of a graphrdquo AKCE International Journal ofGraphs and Combinatorics vol 4 no 2 pp 203ndash209 2007

[66] N Jafari Rad and J Raczek ldquoSome progress on total bondage ingraphsrdquo Graphs and Combinatorics 2011

[67] T W Haynes and P J Slater ldquoPaired-domination and thepaired-domatic numberrdquoCongressusNumerantium vol 109 pp65ndash72 1995

[68] T W Haynes and P J Slater ldquoPaired-domination in graphsrdquoNetworks vol 32 no 3 pp 199ndash206 1998

[69] X Hou ldquoA characterization of 2120574 120574119875-treesrdquoDiscrete Mathemat-

ics vol 308 no 15 pp 3420ndash3426 2008[70] K E Proffitt T W Haynes and P J Slater ldquoPaired-domination

in grid graphsrdquo Congressus Numerantium vol 150 pp 161ndash1722001

International Journal of Combinatorics 33

[71] H Qiao L KangM Cardei andD-Z Du ldquoPaired-dominationof treesrdquo Journal of Global Optimization vol 25 no 1 pp 43ndash542003

[72] E Shan L Kang and M A Henning ldquoA characterizationof trees with equal total domination and paired-dominationnumbersrdquo The Australasian Journal of Combinatorics vol 30pp 31ndash39 2004

[73] J Raczek ldquoPaired bondage in treesrdquo Discrete Mathematics vol308 no 23 pp 5570ndash5575 2008

[74] J A Telle and A Proskurowski ldquoAlgorithms for vertex parti-tioning problems on partial k-treesrdquo SIAM Journal on DiscreteMathematics vol 10 no 4 pp 529ndash550 1997

[75] G S Domke J H Hattingh M A Henning and L R MarkusldquoRestrained domination in treesrdquoDiscreteMathematics vol 211no 1ndash3 pp 1ndash9 2000

[76] G S Domke J H Hattingh M A Henning and L R MarkusldquoRestrained domination in graphs with minimum degree twordquoJournal of Combinatorial Mathematics and Combinatorial Com-puting vol 35 pp 239ndash254 2000

[77] G S Domke J H Hattingh S T Hedetniemi R C Laskarand L R Markus ldquoRestrained domination in graphsrdquo DiscreteMathematics vol 203 no 1ndash3 pp 61ndash69 1999

[78] J H Hattingh and E J Joubert ldquoAn upper bound for therestrained domination number of a graph with minimumdegree at least two in terms of order and minimum degreerdquoDiscrete Applied Mathematics vol 157 no 13 pp 2846ndash28582009

[79] M A Henning ldquoGraphs with large restrained dominationnumberrdquo Discrete Mathematics vol 197-198 pp 415ndash429 199916th British Combinatorial Conference (London 1997)

[80] J H Hattingh and A R Plummer ldquoRestrained bondage ingraphsrdquo Discrete Mathematics vol 308 no 23 pp 5446ndash54532008

[81] N Jafari Rad R Hasni J Raczek and L Volkmann ldquoTotalrestrained bondage in graphsrdquoActaMathematica Sinica vol 29no 6 pp 1033ndash1042 2013

[82] J Cyman and J Raczek ldquoOn the total restrained dominationnumber of a graphrdquoThe Australasian Journal of Combinatoricsvol 36 pp 91ndash100 2006

[83] M A Henning ldquoGraphs with large total domination numberrdquoJournal of Graph Theory vol 35 no 1 pp 21ndash45 2000

[84] D-X Ma X-G Chen and L Sun ldquoOn total restraineddomination in graphsrdquoCzechoslovakMathematical Journal vol55 no 1 pp 165ndash173 2005

[85] B Zelinka ldquoRemarks on restrained domination and totalrestrained domination in graphsrdquo Czechoslovak MathematicalJournal vol 55 no 2 pp 393ndash396 2005

[86] W Goddard and M A Henning ldquoIndependent domination ingraphs a survey and recent resultsrdquo Discrete Mathematics vol313 no 7 pp 839ndash854 2013

[87] J H Zhang H L Liu and L Sun ldquoIndependence bondagenumber and reinforcement number of some graphsrdquo Transac-tions of Beijing Institute of Technology vol 23 no 2 pp 140ndash1422003

[88] K D Dharmalingam Studies in Graph Theorymdashequitabledomination and bottleneck domination [PhD thesis] MaduraiKamaraj University Madurai India 2006

[89] GDeepakND Soner andAAlwardi ldquoThe equitable bondagenumber of a graphrdquo Research Journal of Pure Algebra vol 1 no9 pp 209ndash212 2011

[90] E Sampathkumar and H B Walikar ldquoThe connected domina-tion number of a graphrdquo Journal of Mathematical and PhysicalSciences vol 13 no 6 pp 607ndash613 1979

[91] K White M Farber and W Pulleyblank ldquoSteiner trees con-nected domination and strongly chordal graphsrdquoNetworks vol15 no 1 pp 109ndash124 1985

[92] M Cadei M X Cheng X Cheng and D-Z Du ldquoConnecteddomination in Ad Hoc wireless networksrdquo in Proceedings ofthe 6th International Conference on Computer Science andInformatics (CSampI rsquo02) 2002

[93] J F Fink and M S Jacobson ldquon-domination in graphsrdquo inGraph Theory with Applications to Algorithms and ComputerScience Wiley-Interscience Publications pp 283ndash300 WileyNew York NY USA 1985

[94] M Chellali O Favaron A Hansberg and L Volkmann ldquok-domination and k-independence in graphs a surveyrdquo Graphsand Combinatorics vol 28 no 1 pp 1ndash55 2012

[95] Y Lu X Hou J-M Xu andN Li ldquoTrees with uniqueminimump-dominating setsrdquo Utilitas Mathematica vol 86 pp 193ndash2052011

[96] Y Lu and J-M Xu ldquoThe p-bondage number of treesrdquo Graphsand Combinatorics vol 27 no 1 pp 129ndash141 2011

[97] Y Lu and J-M Xu ldquoThe 2-domination and 2-bondage numbersof grid graphs A manuscriptrdquo httparxivorgabs12044514

[98] M Krzywkowski ldquoNon-isolating 2-bondage in graphsrdquo TheJournal of the Mathematical Society of Japan vol 64 no 4 2012

[99] M A Henning O R Oellermann and H C Swart ldquoBoundson distance domination parametersrdquo Journal of CombinatoricsInformation amp System Sciences vol 16 no 1 pp 11ndash18 1991

[100] F Tian and J-M Xu ldquoBounds for distance domination numbersof graphsrdquo Journal of University of Science and Technology ofChina vol 34 no 5 pp 529ndash534 2004

[101] F Tian and J-M Xu ldquoDistance domination-critical graphsrdquoApplied Mathematics Letters vol 21 no 4 pp 416ndash420 2008

[102] F Tian and J-M Xu ldquoA note on distance domination numbersof graphsrdquo The Australasian Journal of Combinatorics vol 43pp 181ndash190 2009

[103] F Tian and J-M Xu ldquoAverage distances and distance domina-tion numbersrdquo Discrete Applied Mathematics vol 157 no 5 pp1113ndash1127 2009

[104] Z-L Liu F Tian and J-M Xu ldquoProbabilistic analysis of upperbounds for 2-connected distance k-dominating sets in graphsrdquoTheoretical Computer Science vol 410 no 38ndash40 pp 3804ndash3813 2009

[105] M A Henning O R Oellermann and H C Swart ldquoDistancedomination critical graphsrdquo Journal of Combinatorial Mathe-matics and Combinatorial Computing vol 44 pp 33ndash45 2003

[106] T J Bean M A Henning and H C Swart ldquoOn the integrityof distance domination in graphsrdquo The Australasian Journal ofCombinatorics vol 10 pp 29ndash43 1994

[107] M Fischermann and L Volkmann ldquoA remark on a conjecturefor the (kp)-domination numberrdquoUtilitasMathematica vol 67pp 223ndash227 2005

[108] T Korneffel D Meierling and L Volkmann ldquoA remark on the(22)-domination numberrdquo Discussiones Mathematicae GraphTheory vol 28 no 2 pp 361ndash366 2008

[109] Y Lu X Hou and J-M Xu ldquoOn the (22)-domination numberof treesrdquo Discussiones Mathematicae Graph Theory vol 30 no2 pp 185ndash199 2010

34 International Journal of Combinatorics

[110] H Li and J-M Xu ldquo(dm)-domination numbers of m-connected graphsrdquo Rapports de Recherche 1130 LRI UniversiteParis-Sud 1997

[111] S M Hedetniemi S T Hedetniemi and T V Wimer ldquoLineartime resource allocation algorithms for treesrdquo Tech Rep URI-014 Department of Mathematical Sciences Clemson Univer-sity 1987

[112] M Farber ldquoDomination independent domination and dualityin strongly chordal graphsrdquo Discrete Applied Mathematics vol7 no 2 pp 115ndash130 1984

[113] G S Domke and R C Laskar ldquoThe bondage and reinforcementnumbers of 120574

119891for some graphsrdquoDiscrete Mathematics vol 167-

168 pp 249ndash259 1997[114] G S Domke S T Hedetniemi and R C Laskar ldquoFractional

packings coverings and irredundance in graphsrdquo vol 66 pp227ndash238 1988

[115] E J Cockayne P A Dreyer Jr S M Hedetniemi andS T Hedetniemi ldquoRoman domination in graphsrdquo DiscreteMathematics vol 278 no 1ndash3 pp 11ndash22 2004

[116] I Stewart ldquoDefend the roman empirerdquo Scientific American vol281 pp 136ndash139 1999

[117] C S ReVelle and K E Rosing ldquoDefendens imperiumromanum a classical problem in military strategyrdquoThe Ameri-can Mathematical Monthly vol 107 no 7 pp 585ndash594 2000

[118] A Bahremandpour F-T Hu S M Sheikholeslami and J-MXu ldquoOn the Roman bondage number of a graphrdquo DiscreteMathematics Algorithms and Applications vol 5 no 1 ArticleID 1350001 15 pages 2013

[119] N Jafari Rad and L Volkmann ldquoRoman bondage in graphsrdquoDiscussiones Mathematicae Graph Theory vol 31 no 4 pp763ndash773 2011

[120] K Ebadi and L PushpaLatha ldquoRoman bondage and Romanreinforcement numbers of a graphrdquo International Journal ofContemporary Mathematical Sciences vol 5 pp 1487ndash14972010

[121] N Dehgardi S M Sheikholeslami and L Volkman ldquoOn theRoman k-bondage number of a graphrdquo AKCE InternationalJournal of Graphs and Combinatorics vol 8 pp 169ndash180 2011

[122] S Akbari and S Qajar ldquoA note on Roman bondage number ofgraphsrdquo Ars Combinatoria to Appear

[123] F-T Hu and J-M Xu ldquoRoman bondage numbers of somegraphsrdquo httparxivorgabs11093933

[124] N Jafari Rad and L Volkmann ldquoOn the Roman bondagenumber of planar graphsrdquo Graphs and Combinatorics vol 27no 4 pp 531ndash538 2011

[125] S Akbari M Khatirinejad and S Qajar ldquoA note on the romanbondage number of planar graphsrdquo Graphs and Combinatorics2012

[126] B Bresar M A Henning and D F Rall ldquoRainbow dominationin graphsrdquo Taiwanese Journal of Mathematics vol 12 no 1 pp213ndash225 2008

[127] B L Hartnell and D F Rall ldquoOn dominating the Cartesianproduct of a graph and 120572krdquo Discussiones Mathematicae GraphTheory vol 24 no 3 pp 389ndash402 2004

[128] N Dehgardi S M Sheikholeslami and L Volkman ldquoThe k-rainbow bondage number of a graphrdquo to appear in DiscreteApplied Mathematics

[129] J von Neumann and O Morgenstern Theory of Games andEconomic Behavior Princeton University Press Princeton NJUSA 1944

[130] C BergeTheorıe des Graphes et ses Applications 1958[131] O Ore Theory of Graphs vol 38 American Mathematical

Society Colloquium Publications 1962[132] E Shan and L Kang ldquoBondage number in oriented graphsrdquoArs

Combinatoria vol 84 pp 319ndash331 2007[133] J Huang and J-M Xu ldquoThe bondage numbers of extended

de Bruijn and Kautz digraphsrdquo Computers amp Mathematics withApplications vol 51 no 6-7 pp 1137ndash1147 2006

[134] J Huang and J-M Xu ldquoThe total domination and total bondagenumbers of extended de Bruijn and Kautz digraphsrdquoComputersamp Mathematics with Applications vol 53 no 8 pp 1206ndash12132007

[135] Y Shibata and Y Gonda ldquoExtension of de Bruijn graph andKautz graphrdquo Computers amp Mathematics with Applications vol30 no 9 pp 51ndash61 1995

[136] X Zhang J Liu and JMeng ldquoThebondage number in completet-partite digraphsrdquo Information Processing Letters vol 109 no17 pp 997ndash1000 2009

[137] D W Bange A E Barkauskas and P J Slater ldquoEfficient dom-inating sets in graphsrdquo in Applications of Discrete Mathematicspp 189ndash199 SIAM Philadelphia Pa USA 1988

[138] M Livingston and Q F Stout ldquoPerfect dominating setsrdquoCongressus Numerantium vol 79 pp 187ndash203 1990

[139] L Clark ldquoPerfect domination in random graphsrdquo Journal ofCombinatorial Mathematics and Combinatorial Computing vol14 pp 173ndash182 1993

[140] D W Bange A E Barkauskas L H Host and L H ClarkldquoEfficient domination of the orientations of a graphrdquo DiscreteMathematics vol 178 no 1ndash3 pp 1ndash14 1998

[141] A E Barkauskas and L H Host ldquoFinding efficient dominatingsets in oriented graphsrdquo Congressus Numerantium vol 98 pp27ndash32 1993

[142] I J Dejter and O Serra ldquoEfficient dominating sets in CayleygraphsrdquoDiscrete AppliedMathematics vol 129 no 2-3 pp 319ndash328 2003

[143] J Lee ldquoIndependent perfect domination sets in Cayley graphsrdquoJournal of Graph Theory vol 37 no 4 pp 213ndash219 2001

[144] WGu X Jia and J Shen ldquoIndependent perfect domination setsin meshes tori and treesrdquo Unpublished

[145] N Obradovic J Peters and G Ruzic ldquoEfficient domination incirculant graphswith two chord lengthsrdquo Information ProcessingLetters vol 102 no 6 pp 253ndash258 2007

[146] K Reji Kumar and G MacGillivray ldquoEfficient domination incirculant graphsrdquo Discrete Mathematics vol 313 no 6 pp 767ndash771 2013

[147] D Van Wieren M Livingston and Q F Stout ldquoPerfectdominating sets on cube-connected cyclesrdquo Congressus Numer-antium vol 97 pp 51ndash70 1993

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

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Journal of

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Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

International Journal of Combinatorics 13

Theorem 75 (Kang and Yuan [45] 2000) If 119866 is a connectedplanar graph without vertices of degree 5 then 119887(119866) ⩽ 7

Fischermann et al [46] generalized Theorem 75 as fol-lows

Theorem 76 (Fischermann et al [46] 2003) Let 119866 be aconnected planar graph and 119883 the set of vertices of degree 5which have distance at least 3 to vertices of degrees 1 2 and3 If all vertices in 119883 not adjacent with vertices of degree 4are independent and not adjacent to vertices of degree 6 then119887(119866) ⩽ 7

Clearly if119866 has no vertices of degree 5 then119883 = 0 ThusTheorem 76 yields Theorem 75

Use 119899119894to denote the number of vertices of degree 119894 in 119866

for each 119894 = 1 2 Δ(119866) Using Theorem 18 Fischermannet al obtained the following two theorems

Theorem 77 (Fischermann et al [46] 2003) For any con-nected planar graph 119866

(1) 119887(119866) ⩽ 7 if 1198995lt 2119899

2+ 3119899

3+ 2119899

4+ 12

(2) 119887(119866) ⩽ 6 if 119866 contains no vertices of degrees 4 and 5

Theorem 78 (Fischermann et al [46] 2003) For any con-nected planar graph 119866 119887(119866) ⩽ Δ(119866) + 1 if

(1) Δ(119866) = 6 and every edge 119890 = 119909119910 with 119889119866(119909) = 5 and

119889119866(119910) = 6 is contained in at most one triangle

(2) Δ(119866) = 5 and no triangle contains an edge 119890 = 119909119910 with119889119866(119909) = 5 and 4 ⩽ 119889

119866(119910) ⩽ 5

63 Bounds Implied by Girth Conditions The girth 119892(119866) of agraph 119866 is the length of the shortest cycle in 119866 If 119866 has nocycles we define 119892(119866) = infin

Combining Theorem 74 with Theorem 78 we find thatif a planar graph contains no triangles and has maximumdegree Δ ⩾ 5 then Conjecture 73 holds This fact motivatedFischermann et alrsquos [46] attempt to attack Conjecture 73 bygirth restraints

Theorem 79 (Fischermann et al [46] 2003) For any con-nected planar graph 119866

119887 (119866) ⩽

6 119894119891 119892 (119866) ⩾ 4

5 119894119891 119892 (119866) ⩾ 5

4 119894119891 119892 (119866) ⩾ 6

3 119894119891 119892 (119866) ⩾ 8

(20)

The first result in Theorem 79 shows that Conjecture 73is valid for all connected planar graphs with 119892(119866) ⩾ 4 andΔ(119866) ⩾ 5 It is easy to verify that the second result inTheorem 79 implies that Conjecture 73 is valid for all not3-regular graphs of girth 119892(119866) ⩾ 5 which is stated in thefollowing corollary

Corollary 80 For any connected planar graph 119866 if 119866 is not3-regular and 119892(119866) ⩾ 5 then 119887(119866) ⩽ Δ(119866) + 1

The first result in Theorem 79 also implies that if 119866 isa connected planar graph with no cycles of length 3 then119887(119866) ⩽ 6 Hou et al [47] improved this result as follows

Theorem 81 (Hou et al [47] 2011) For any connected planargraph 119866 if 119866 contains no cycles of length 119894 (4 ⩽ 119894 ⩽ 6) then119887(119866) ⩽ 6

Since 119887(1198623119896+1) = 3 for 119896 ⩾ 3 (see Theorem 1) the

last bound in Theorem 79 is tight Whether other bounds inTheorem 79 are tight remains open In 2003 Fischermann etal [46] posed the following conjecture

Conjecture 82 For any connected planar graph 119866 119887(119866) ⩽ 7and

119887 (119866) ⩽ 5 119894119891 119892 (119866) ⩾ 4

4 119894119891 119892 (119866) ⩾ 5(21)

We conclude this subsection with a question on bondagenumbers of planar graphs

Question 1 (Fischermann et al [46] 2003) Is there a planargraph 119866 with 6 ⩽ 119887(119866) ⩽ 8

In 2006 Carlson andDevelin [42] showed that the corona119866 = 119867 ∘ 119870

1for a planar graph 119867 with 120575(119867) = 5 has the

bondage number 119887(119866) = 120575(119867) + 1 = 6 (see Theorem 66)Since the minimum degree of planar graphs is at most 5then 119887(119866) can attach 6 If we take 119867 as the graph of theicosahedron then 119866 = 119867 ∘ 119870

1is such an example The

question for the existence of planar graphs with bondagenumber 7 or 8 remains open

64 Comments on the Conjectures Conjecture 73 is true forall connected planar graphs with minimum degree 120575 ⩽ 2 byTheorem 16 or maximum degree Δ ⩾ 7 by Theorem 74 ornot 120574-critical planar graphs by Theorem 13 Thus to attackConjecture 73 we only need to consider connected criticalplanar graphs with degree-restriction 3 ⩽ 120575 ⩽ Δ ⩽ 6

Recalling and anatomizing the proofs of all results men-tioned in the preceding subsections on the bondage numberfor connected planar graphs we find that the proofs of theseresults strongly depend upon Theorem 16 or Theorem 18 Inother words a basic way used in the proofs is to find twovertices 119909 and 119910 with distance at most two in a consideredplanar graph 119866 such that

119889119866(119909) + 119889

119866(119910) or 119889

119866(119909) + 119889

119866(119910) minus

1003816100381610038161003816119873119866(119909) cap 119873

119866(119910)1003816100381610038161003816

(22)

which bounds 119887(119866) is as small as possible Very recentlyHuang and Xu [44] have considered the following parameter

119861 (119866) = min119909119910isin119881(119866)

119889119909119910 1 ⩽ 119889

119866(119909 119910) ⩽ 2

cup 119889119909119910minus 119899

119909119910 119889 (119909 119910) = 1

(23)

where 119889119909119910= 119889

119866(119909) + 119889

119866(119910) minus 1 and 119899

119909119910= |119873

119866(119909) cap 119873

119866(119910)|

14 International Journal of Combinatorics

It follows fromTheorems 16 and 18 that

119887 (119866) ⩽ 119861 (119866) (24)

The proofs given in Theorems 74 and 79 indeed imply thefollowing stronger results

Theorem 83 If 119866 is a connected planar graph then

119861 (119866) ⩽

min 8 Δ (119866) + 26 119894119891 119892 (119866) ⩾ 4

5 119894119891 119892 (119866) ⩾ 5

4 119894119891 119892 (119866) ⩾ 6

3 119894119891 119892 (119866) ⩾ 8

(25)

Thus using Theorem 16 or Theorem 18 if we can proveConjectures 73 and 82 then we can prove the followingstatement

Statement If 119866 is a connected planar graph then

119861 (119866) ⩽

Δ (119866) + 1

7

5 if 119892 (119866) ⩾ 44 if 119892 (119866) ⩾ 5

(26)

It follows from (24) that Statement implies Conjectures 73and 82 However Huang and Xu [44] gave examples to showthat none of conclusions in Statement is true As a result theystated the following conclusion

Theorem 84 (Huang and Xu [44] 2012) It is not possible toprove Conjectures 73 and 82 if they are right using Theorems16 and 18

Therefore a new method is needed to prove or disprovethese conjectures At the present time one cannot prove theseconjectures and may consider to disprove them If one ofthese conjectures is invalid then there exists a minimumcounterexample 119866 with respect to 120592(119866) + 120576(119866) In order toobtain a minimum counterexample one may consider twosimple operations satisfying these requirements the edge-deletion and the edge-contractionwhich decrease 120592(119866)+120576(119866)and preserve planarity However applying Theorems 71 and72 Huang and Xu [44] presented the following results

Theorem 85 (Huang and Xu [44] 2012) It is not possible toconstruct minimum counterexamples to Conjectures 73 and 82by the operation of an edge-deletion or an edge-contraction

7 Results on Crossing Number Restraints

It is quite natural to generalize the known results on thebondage number for planar graphs to formore general graphsin terms of graph-theoretical parameters In this section weconsider graphs with crossing number restraints

The crossing number cr(119866) of 119866 is the smallest numberof pairwise intersections of its edges when 119866 is drawn in theplane If cr(119866) = 0 then 119866 is a planar graph

71 General Methods Use 119899119894to denote the number of vertices

of degree 119894 in119866 for 119894 = 1 2 Δ(119866) Huang and Xu obtainedthe following results

Theorem 86 (Huang and Xu [48] 2007) For any connectedgraph 119866

119887 (119866) ⩽

6 119894119891 119892 (119866) ⩾ 4 119886119899119889 2cr (119866) lt 121198861

5 119894119891 119892 (119866) ⩾ 5 119886119899119889 6cr (119866) lt 161198862

4 119894119891 119892 (119866) ⩾ 6 119886119899119889 4cr (119866) lt 141198863

3 119894119891 119892 (119866) ⩾ 8 119886119899119889 6cr (119866) lt 161198864

(27)

where 1198861= 119899

1+ 2119899

2+ 2119899

3+sum

Δ

119894=8(119894 minus 7)119899

119894+ 8 119886

2= 3119899

1+ 6119899

2+

51198993+sum

Δ

119894=7(3119894minus18)119899

119894+20 119886

3= 119899

1+2119899

2+sum

Δ

119894=6(2119894minus10)119899

119894+12

and 1198864= sum

Δ

119894=5(3119894 minus 12)119899

119894+ 16

Simplifying the conditions in Theorem 86 we obtain thefollowing corollaries immediately

Corollary 87 (Huang and Xu [48] 2007) For any connectedgraph 119866

119887 (119866) ⩽

6 119894119891 119892 (119866) ⩾ 4 119886119899119889 cr (119866) ⩽ 3

5 119894119891 119892 (119866) ⩾ 5 119886119899119889 cr (119866) ⩽ 4

4 119894119891 119892 (119866) ⩾ 6 119886119899119889 cr (119866) ⩽ 2

3 119894119891 119892 (119866) ⩾ 8 119886119899119889 cr (119866) ⩽ 2

(28)

Corollary 88 (Huang and Xu [48] 2007) For any connectedgraph 119866

(a) 119887(119866) ⩽ 6 if 119866 is not 4-regular cr(119866) = 4 and 119892(119866) ⩾ 4

(b) 119887(119866) ⩽ Δ(119866) + 1 if 119866 is not 3-regular cr(119866) ⩽ 4 and119892(119866) ⩾ 5

(c) 119887(119866) ⩽ 4 if 119866 is not 3-regular cr(119866) = 3 and 119892(119866) ⩾ 6

(d) 119887(119866) ⩽ 3 if cr(119866) = 3 119892(119866) ⩾ 8 and Δ(119866) ⩾ 5

These corollaries generalize some known results for pla-nar graphs For example Corollary 87 contains Theorem 79Corollary 88 (b) contains Corollary 80

Theorem 89 (Huang and Xu [48] 2007) For any connectedgraph 119866 if cr(119866) ⩽ 119899

3(119866) + 119899

4(119866) + 3 then 119887(119866) ⩽ 8

Corollary 90 (Huang and Xu [48] 2007) For any connectedgraph 119866 if cr(119866) ⩽ 3 then 119887(119866) ⩽ 8

Perhaps being unaware of this result in 2010 Ma et al[49] proved that 119887(119866) ⩽ 12 for any graph 119866 with cr(119866) = 1

Theorem91 (Huang and Xu [48] 2007) Let119866 be a connectedgraph and 119868 = 119909 isin 119881(119866) 119889

119866(119909) = 5 119889

119866(119909 119910) ⩾ 3 if

International Journal of Combinatorics 15

119889119866(119910) ⩽ 3 and 119889

119866(119910) = 4 for every 119910 isin 119873

119866(119909) If 119868 is

independent no vertices adjacent to vertices of degree 6 and

cr (119866) lt max5119899

3+ |119868| minus 2119899

4+ 28

117119899

3+ 40

16 (29)

then 119887(119866) ⩽ 7

Corollary 92 (Huang and Xu [48] 2007) Let 119866 be aconnected graph with cr(119866) ⩽ 2 If 119868 = 119909 isin 119881(119866) 119889

119866(119909) =

5 119889119866(119910 119909) ⩾ 3 if 119889

119866(119910) ⩽ 3 and 119889

119866(119910) = 4 for every 119910 isin

119873119866(119909) is independent and no vertices adjacent to vertices of

degree 6 then 119887(119866) ⩽ 7

Theorem93 (Huang andXu [48] 2007) Let119866 be a connectedgraph If 119866 satisfies

(a) 5cr(119866) + 1198995lt 2119899

2+ 3119899

3+ 2119899

4+ 12 or

(b) 7cr(119866) + 21198995lt 3119899

2+ 4119899

4+ 24

then 119887(119866) ⩽ 7

Proposition 94 (Huang and Xu [48] 2007) Let 119866 be aconnected graph with no vertices of degrees four and five Ifcr(119866) ⩽ 2 then 119887(119866) ⩽ 6

Theorem 95 (Huang and Xu [48] 2007) If 119866 is a connectedgraph with cr(119866) ⩽ 4 and not 4-regular when cr(119866) = 4 then119887(119866) ⩽ Δ(119866) + 2

The above results generalize some known results forplanar graphs For example Corollary 90 and Theorem 95contain Theorem 74 the first condition in Theorem 93 con-tains the second condition inTheorem 77

From Corollary 88 and Theorem 95 we suggest the fol-lowing questions

Question 2 Is there a

(a) 4-regular graph 119866 with cr(119866) = 4 such that 119887(119866) ⩾ 7

(b) 3-regular graph 119866 with cr ⩽ 4 and 119892(119866) ⩾ 5 such that119887(119866) = 5

(c) 3-regular graph 119866 with cr = 3 and 119892(119866) = 6 or 7 suchthat 119887(119866) = 5

72 Carlson-Develin Methods In this subsection we intro-duce an elegant method presented by Carlson and Develin[42] to obtain some upper bounds for the bondage numberof a graph

Suppose that 119866 is a connected graph We say that 119866 hasgenus 120588 if 119866 can be embedded in a surface 119878 with 120588 handlessuch that edges are pairwise disjoint except possibly for end-vertices Let119866 be an embedding of119866 in surface 119878 and let120601(119866)denote the number of regions in 119866 The boundary of everyregion contains at least three edges and every edge is on theboundary of atmost two regions (the two regions are identicalwhen 119890 is a cut-edge) For any edge 119890 of 119866 let 1199031

119866(119890) and 1199032

119866(119890)

be the numbers of edges comprising the regions in 119866 whichthe edge 119890 borders It is clear that every 119890 = 119909119910 isin 119864(119866)

1199032

119866(119890) ⩾ 119903

1

119866(119890) ⩾ 4 if 1003816100381610038161003816119873119866

(119909) cap 119873119866(119910)1003816100381610038161003816 = 0

1199032

119866(119890) ⩾ 4 119903

1

119866(119890) ⩾ 3 if 1003816100381610038161003816119873119866

(119909) cap 119873119866(119910)1003816100381610038161003816 = 1

1199032

119866(119890) ⩾ 119903

1

119866(119890) ⩾ 3 if 1003816100381610038161003816119873119866

(119909) cap 119873119866(119910)1003816100381610038161003816 ⩾ 2

(30)

Following Carlson and Develin [42] for any edge 119890 = 119909119910 of119866 we define

119863119866(119890) =

1

119889119866(119909)+

1

119889119866(119910)

+1

1199031

119866(119890)+

1

1199032

119866(119890)minus 1 (31)

By the well-known Eulerrsquos formula

120592 (119866) minus 120576 (119866) + 120601 (119866) = 2 minus 2120588 (119866) (32)

it is easy to see that

sum

119890isin119864(119866)

119863119866(119890) = 120592 (119866) minus 120576 (119866) + 120601 (119866) = 2 minus 2120588 (119866) (33)

If 119866 is a planar graph that is 120588(119866) = 0 then

sum

119890isin119864(119866)

119863119866(119890) = 120592 (119866) minus 120576 (119866) + 120601 (119866) = 2 (34)

Combining these formulas withTheorem 18 Carlson andDevelin [42] gave a simple and intuitive proof ofTheorem 74and obtained the following result

Theorem 96 (Carlson and Develin [42] 2006) Let 119866 be aconnected graph which can be embedded on a torus Then119887(119866) ⩽ Δ(119866) + 3

Recently Hou and Liu [50] improved Theorem 96 asfollows

Theorem 97 (Hou and Liu [50] 2012) Let 119866 be a connectedgraph which can be embedded on a torus Then 119887(119866) ⩽ 9Moreover if Δ(119866) = 6 then 119887(119866) ⩽ 8

By the way Cao et al [51] generalized the result inTheorem 79 to a connected graph 119866 that can be embeddedon a torus that is

119887 (119866) le

6 if 119892 (119866) ⩾ 4 and 119866 is not 4-regular5 if 119892 (119866) ⩾ 54 if 119892 (119866) ⩾ 6 and 119866 is not 3-regular3 if 119892 (119866) ⩾ 8

(35)

Several authors used this method to obtain some resultson the bondage number For example Fischermann et al[46] used this method to prove the second conclusion inTheorem 78 Recently Cao et al [52] have used thismethod todeal with more general graphs with small crossing numbersFirst they found the following property

Lemma 98 120575(119866) ⩽ 5 for any graph 119866 with cr(119866) ⩽ 5

16 International Journal of Combinatorics

This result implies that 119887(119866) ⩽ Δ(119866) + 4 by Corollary 15for any graph 119866 with cr(119866) ⩽ 5 Cao et al established thefollowing relation in terms of maximum planar subgraphs

Lemma 99 (Cao et al [52]) Let 119866 be a connected graph withcrossing number cr(119866) For any maximum planar subgraph119867of 119866

sum

119890isin119864(119867)

119863119866(119890) ⩾ 2 minus

2cr (119866)120575 (119866)

(36)

Using Carlson-Develin method and combiningLemma 98 with Lemma 99 Cao et al proved the followingresults

Theorem 100 (Cao et al [52]) For any connected graph 119866119887(119866) ⩽ Δ(119866) + 2 if 119866 satisfies one of the following conditions

(a) cr(119866) ⩽ 3(b) cr(119866) = 4 and 119866 is not 4-regular(c) cr(119866) = 5 and 119866 contains no vertices of degree 4

Theorem101 (Cao et al [52]) Let119866 be a connected graphwithΔ(119866) = 5 and cr(119866) ⩽ 4 If no triangles contain two vertices ofdegree 5 then 119887(119866) ⩽ 6 = Δ(119866) + 1

Theorem 102 (Cao et al [52]) Let 119866 be a connected graphwith Δ(119866) ⩾ 6 and cr(119866) ⩽ 3 If Δ(119866) ⩾ 7 or if Δ(119866) = 6120575(119866) = 3 and every edge 119890 = 119909119910 with 119889

119866(119909) = 5 and 119889

119866(119910) = 6

is contained in atmost one triangle then 119887(119866) ⩽ min8 Δ(119866)+1

Using Carlson-Develin method it can be proved that119887(119866) ⩽ Δ(119866) + 2 if cr(119866) ⩽ 3 (see the first conclusion inTheorem 100) and not yet proved that 119887(119866) ⩽ 8 if cr(119866) ⩽3 although it has been proved by using other method (seeCorollary 90)

By using Carlson-Develin method Samodivkin [25]obtained some results on the bondage number for graphswithsome given properties

Kim [53] showed that 119887(119866) ⩽ Δ(119866) + 2 for a connectedgraph 119866 with genus 1 Δ(119866) ⩽ 5 and having a toroidalembedding of which at least one region is not 4-sidedRecently Gagarin and Zverovich [54] further have extendedCarlson-Develin ideas to establish a nice upper bound forarbitrary graphs that can be embedded on orientable ornonorientable topological surface

Theorem 103 (Gagarin and Zverovich [54] 2012) Let 119866 bea graph embeddable on an orientable surface of genus ℎ and anonorientable surface of genus 119896 Then 119887(119866) ⩽ minΔ(119866)+ℎ+2 Δ(119866) + 119896 + 1

This result generalizes the corresponding upper boundsin Theorems 74 and 96 for any orientable or nonori-entable topological surface By investigating the proof ofTheorem 103 Huang [55] found that the issue of the ori-entability can be avoided by using the Euler characteristic120594(= 120592(119866) minus 120576(119866) + 120601(119866)) instead of the genera ℎ and 119896 the

relations are 120594 = 2minus2ℎ and 120594 = 2minus119896 To obtain the best resultfromTheorem 103 onewants ℎ and 119896 as small as possible thisis equivalent to having 120594 as large as possible

According toTheorem 103 if119866 is planar (ℎ = 0 120594 = 2) orcan be embedded on the real projective plane (119896 = 1 120594 = 1)then 119887(119866) ⩽ Δ(119866) + 2 In all other cases Huang [55] had thefollowing improvement for Theorem 103 the proof is basedon the technique developed byCarlson-Develin andGagarin-Zverovich and includes some elementary calculus as a newingredient mainly the intermediate-value theorem and themean-value theorem

Theorem 104 (Huang [55] 2012) Let 119866 be a graph embed-dable on a surface whose Euler characteristic 120594 is as large aspossible If 120594 ⩽ 0 then 119887(119866) ⩽ Δ(119866)+lfloor119903rfloor where 119903 is the largestreal root of the following cubic equation in 119911

1199113

+ 21199112

+ (6120594 minus 7) 119911 + 18120594 minus 24 = 0 (37)

In addition if 120594 decreases then 119903 increases

The following result is asymptotically equivalent toTheorem 104

Theorem 105 (Huang [55] 2012) Let 119866 be a graph embed-dable on a surface whose Euler characteristic 120594 is as large aspossible If 120594 ⩽ 0 then 119887(119866) lt Δ(119866) + radic12 minus 6120594 + 12 orequivalently 119887(119866) ⩽ Δ(119866) + lceilradic12 minus 6120594 minus 12rceil

Also Gagarin andZverovich [54] indicated that the upperbound in Theorem 103 can be improved for larger values ofthe genera ℎ and 119896 by adjusting the proofs and stated thefollowing general conjecture

Conjecture 106 (Gagarin and Zverovich [54] 2012) For aconnected graph G of orientable genus ℎ and nonorientablegenus 119896

119887 (119866) ⩽ min 119888ℎ 119888

1015840

119896 Δ (119866) + 119900 (ℎ) Δ (119866) + 119900 (119896) (38)

where 119888ℎand 1198881015840

119896are constants depending respectively on the

orientable and nonorientable genera of 119866

In the recent paper Gagarin and Zverovich [56] providedconstant upper bounds for the bondage number of graphs ontopological surfaces which can be used as the first estimationfor the constants 119888

ℎand 1198881015840

119896of Conjecture 106

Theorem 107 (Gagarin and Zverovich [56] 2013) Let 119866 bea connected graph of order 119899 If 119866 can be embedded on thesurface of its orientable or nonorientable genus of the Eulercharacteristic 120594 then

(a) 120594 ⩾ 1 implies 119887(119866) ⩽ 10(b) 120594 ⩽ 0 and 119899 gt minus12120594 imply 119887(119866) ⩽ 11(c) 120594 ⩽ minus1 and 119899 ⩽ minus12120594 imply 119887(119866) ⩽ 11 +

3120594(radic17 minus 8120594 minus 3)(120594 minus 1) = 11 + 119874(radicminus120594)

Theorem 79 shows that a connected planar triangle-freegraph 119866 has 119887(119866) ⩽ 6 The following result generalizes to itall the other topological surfaces

International Journal of Combinatorics 17

Theorem 108 (Gagarin and Zverovich [56] 2013) Let 119866 be aconnected triangle-free graph of order 119899 If 119866 can be embeddedon the surface of its orientable or nonorientable genus of theEuler characteristic 120594 then

(a) 120594 ⩾ 1 implies 119887(119866) ⩽ 6(b) 120594 ⩽ 0 and 119899 gt minus8120594 imply 119887(119866) ⩽ 7(c) 120594 ⩽ minus1 and 119899 ⩽ minus8120594 imply 119887(119866) ⩽ 7minus4120594(1+radic2 minus 120594)

In [56] the authors also explicitly improved upperbounds in Theorem 103 and gave tight lower bounds forthe number of vertices of graphs 2-cell embeddable ontopological surfaces of a given genusThe interested reader isreferred to the original paper for details The following resultis interesting although its proof is easy

Theorem 109 (Samodivkin [57]) Let119866 be a graph with order119899 and girth 119892 If 119866 is embeddable on a surface whose Eulercharacteristic 120594 is as large as possible then

119887 (119866) ⩽ 3 +8

119892 minus 2minus

4120594119892

119899 (119892 minus 2) (39)

If 119866 is a planar graph with girth 119892 ⩾ 4 + 119894 for any 119894 isin0 1 2 then Theorem 109 leads to 119887(119866) ⩽ 6 minus 119894 which isthe result inTheorem 79 obtained by Fischermann et al [46]Also inmany cases the bound stated inTheorem 109 is betterthan those given byTheorems 103 and 105

8 Conditional Bondage Numbers

Since the concept of the bondage number is based upon thedomination all sorts of dominations which are variationsof the normal domination by adding restricted conditionsto the normal dominating set can develop a new ldquobondagenumberrdquo as long as a variation of domination number isgiven In this section we survey results on the bondagenumber under some restricted conditions

81 Total Bondage Numbers A dominating set 119878 of a graph119866 without isolated vertices is called total if the subgraphinduced by 119878 contains no isolated vertices The minimumcardinality of a total dominating set is called the totaldomination number of 119866 and denoted by 120574

119879(119866) It is clear

that 120574(119866) ⩽ 120574119879(119866) ⩽ 2120574(119866) for any graph 119866 without isolated

verticesThe total domination in graphs was introduced by Cock-

ayne et al [58] in 1980 Pfaff et al [59 60] in 1983 showedthat the problem determining total domination number forgeneral graphs is NP-complete even for bipartite graphs andchordal graphs Even now total domination in graphs hasbeen extensively studied in the literature In 2009 Henning[61] gave a survey of selected recent results on total domina-tion in graphs

The total bondage number of 119866 denoted by 119887119879(119866) is the

smallest cardinality of a subset 119861 sube 119864(119866) with the propertythat119866minus119861 contains no isolated vertices and 120574

119879(119866minus119861) gt 120574

119879(119866)

From definition 119887119879(119866)may not exist for some graphs for

example119866 = 1198701119899 We put 119887

119879(119866) = infin if 119887

119879(119866) does not exist

It is easy to see that 119887119879(119866) is finite for any connected graph 119866

other than 1198701 119870

2 119870

3 and 119870

1119899 In fact for such a graph 119866

since there is a path of length 3 in 119866 we can find 1198611sube 119864(119866)

such that 119866 minus 1198611is a spanning tree 119879 of 119866 containing a path

of length 3 So 120574119879(119879) = 120574

119879(119866minus119861

1) ⩾ 120574

119879(119866) For the tree119879 we

find 1198612sube 119864(119879) such that 120574

119879(119879 minus 119861

2) gt 120574

119879(119879) ⩾ 120574

119879(119866) Thus

we have 120574119879(119866minus119861

2minus119861

1) gt 120574

119879(119866) and so 119887

119879(119866) ⩽ |119861

1|+|119861

2| In

the following discussion we always assume that 119887119879(119866) exists

when a graph 119866 is mentionedIn 1991 Kulli and Patwari [62] first studied the total

bondage number of a graph and calculated the exact valuesof 119887

119879(119866) for some standard graphs

Theorem 110 (Kulli and Patwari [62] 1991) For 119899 ⩾ 4

119887119879(119862

119899) =

3 119894119891 119899 equiv 2 (mod 4) 2 119900119905ℎ119890119903119908119894119904119890

119887119879(119875

119899) =

2 119894119891 119899 equiv 2 (mod 4) 1 119900119905ℎ119890119903119908119894119904119890

119887119879(119870

119898119899) = 119898 119908119894119905ℎ 2 ⩽ 119898 ⩽ 119899

119887119879(119870

119899) =

4 119891119900119903 119899 = 4

2119899 minus 5 119891119900119903 119899 ⩾ 5

(40)

Recently Hu et al [63] have obtained some results on thetotal bondage number of the Cartesian product119875

119898times119875

119899of two

paths 119875119898and 119875

119899

Theorem 111 (Hu et al [63] 2012) For the Cartesian product119875119898times 119875

119899of two paths 119875

119898and 119875

119899

119887119879(119875

2times 119875

119899) =

1 119894119891 119899 equiv 0 (mod 3) 2 119894119891 119899 equiv 2 (mod 3) 3 119894119891 119899 equiv 1 (mod 3)

119887119879(119875

3times 119875

119899) = 1

119887119879(119875

4times 119875

119899)

= 1 119894119891 119899 equiv 1 (mod 5) = 2 119894119891 119899 equiv 4 (mod 5) ⩽ 3 119894119891 119899 equiv 2 (mod 5) ⩽ 4 119894119891 119899 equiv 0 3 (mod 5)

(41)

Generalized Petersen graphs are an important class ofcommonly used interconnection networks and have beenstudied recently By constructing a family of minimum totaldominating sets Cao et al [64] determined the total bondagenumber of the generalized Petersen graphs

FromTheorem 6 we know that 119887(119879) ⩽ 2 for a nontrivialtree 119879 But given any positive integer 119896 Sridharan et al [65]constructed a tree 119879 for which 119887

119879(119879) = 119896 Let119867

119896be the tree

obtained from a star1198701119896+1

by subdividing 119896 edges twice Thetree 119867

7is shown in Figure 6 It can be easily verified that

119887119879(119867

119896) = 119896 This fact can be stated as the following theorem

Theorem 112 (Sridharan et al [65] 2007) For any positiveinteger 119896 there exists a tree 119879 with 119887

119879(119879) = 119896

18 International Journal of Combinatorics

Figure 61198677

Combining Theorem 1 with Theorem 112 we have whatfollows

Corollary 113 (Sridharan et al [65] 2007) The bondagenumber and the total bondage number are unrelated even fortrees

However Sridharan et al [65] gave an upper bound onthe total bondage number of a tree in terms of its maximumdegree For any tree 119879 of order 119899 if 119879 =119870

1119899minus1 then 119887

119879(119879) =

minΔ(119879) (13)(119899 minus 1) Rad and Raczek [66] improved thisupper bound and gave a constructive characterization of acertain class of trees attaching the upper bound

Theorem 114 (Rad and Raczek [66] 2011) 119887119879(119879) ⩽ Δ(119879) minus 1

for any tree 119879 with maximum degree at least three

In general the decision problem for 119887119879(119866) is NP-complete

for any graph 119866 We state the decision problem for the totalbondage as follows

Problem 3 Consider the decision problemTotal Bondage ProblemInstance a graph 119866 and a positive integer 119887Question is 119887

119879(119866) ⩽ 119887

Theorem 115 (Hu and Xu [16] 2012) The total bondageproblem is NP-complete

Consequently in view of its computational hardnessit is significative to establish some sharp bounds on thetotal bondage number of a graph in terms of other graphicparameters In 1991 Kulli and Patwari [62] showed that119887119879(119866) ⩽ 2119899 minus 5 for any graph 119866 with order 119899 ⩾ 5 Sridharan

et al [65] improved this result as follows

Theorem 116 (Sridharan et al [65] 2007) For any graph 119866with order 119899 if 119899 ⩾ 5 then

119887119879(119866) ⩽

1

3(119899 minus 1) 119894119891 119866 119888119900119899119905119886119894119899119904 119899119900 119888119910119888119897119890119904

119899 minus 1 119894119891 119892 (119866) ⩾ 5

119899 minus 2 119894119891 119892 (119866) = 4

(42)

Rad and Raczek [66] also established some upperbounds on 119887

119879(119866) for a general graph 119866 In particular they

gave an upper bound on 119887119879(119866) in terms of the girth of

119866

Theorem 117 (Rad and Raczek [66] 2011) 119887119879(119866)⩽4Δ(119866) minus 5

for any graph 119866 with 119892(119866) ⩾ 4

82 Paired Bondage Numbers A dominating set 119878 of 119866 iscalled to be paired if the subgraph induced by 119878 containsa perfect matching The paired domination number of 119866denoted by 120574

119875(119866) is the minimum cardinality of a paired

dominating set of 119866 Clearly 120574119879(119866) ⩽ 120574

119875(119866) for every

connected graph 119866 with order at least two where 120574119879(119866) is

the total domination number of 119866 and 120574119875(119866) ⩽ 2120574(119866) for

any graph119866without isolated vertices Paired domination wasintroduced by Haynes and Slater [67 68] and further studiedin [69ndash72]

The paired bondage number of 119866 with 120575(119866) ⩾ 1 denotedby 119887P(119866) is the minimum cardinality among all sets of edges119861 sube 119864 such that 120575(119866 minus 119861) ⩾ 1 and 120574

119875(119866 minus 119861) gt 120574

119875(119866)

The concept of the paired bondage number was firstproposed by Raczek [73] in 2008The following observationsfollow immediately from the definition of the paired bondagenumber

Observation 2 Let 119866 be a graph with 120575(119866) ⩾ 1

(a) If 119867 is a subgraph of 119866 such that 120575(119867) ⩾ 1 then119887119875(119867) ⩽ 119887

119875(119866)

(b) If 119867 is a subgraph of 119866 such that 119887119875(119867) = 1 and 119896

is the number of edges removed to form119867 then 1 ⩽119887119875(119866) ⩽ 119896 + 1

(c) If 119909119910 isin 119864(119866) such that 119889119866(119909) 119889

119866(119910) gt 1 and 119909119910

belongs to each perfect matching of each minimumpaired dominating set of 119866 then 119887

119875(119866) = 1

Based on these observations 119887119875(119875

119899) ⩽ 2 In fact the

paired bondage number of a path has been determined

Theorem 118 (Raczek [73] 2008) For any positive integer 119896if 119899 ⩾ 2 then

119887119875(119875

119899) =

0 119894119891 119899 = 2 3 119900119903 5

1 119894119891 119899 = 4119896 4119896 + 3 119900119903 4119896 + 6

2 119900119905ℎ119890119903119908119894119904119890

(43)

For a cycle 119862119899of order 119899 ⩾ 3 since 120574

119875(119862

119899) = 120574

119875(119875

119899) from

Theorem 118 the following result holds immediately

Corollary 119 (Raczek [73] 2008) For any positive integer 119896if 119899 ⩾ 3 then

119887119875(119862

119899) =

0 119894119891 119899 = 3 119900119903 5

2 119894119891 119899 = 4119896 4119896 + 3 119900119903 4119896 + 6

3 119900119905ℎ119890119903119908119894119904119890

(44)

A wheel 119882119899 where 119899 ⩾ 4 is a graph with 119899 vertices

formed by connecting a single vertex to all vertices of a cycle119862119899minus1

Of course 120574119875(119882

119899) = 2

International Journal of Combinatorics 19

119896

Figure 7 A tree 119879 with 119887119875(119879) = 119896

Theorem 120 (Raczek [73] 2008) For positive integers 119899 and119898

119887119875(119870

119898119899) = 119898 119891119900119903 1 lt 119898 ⩽ 119899

119887119875(119882

119899) =

4 119894119891 119899 = 4

3 119894119891 119899 = 5

2 119900119905ℎ119890119903119908119894119904119890

(45)

Theorem 16 shows that if 119879 is a nontrivial tree then119887(119879) ⩽ 2 However no similar result exists for pairedbondage For any nonnegative integer 119896 let 119879

119896be a tree

obtained by subdividing all but one edge of a star 1198701119896+1

(asshown in Figure 7) It is easy to see that 119887

119875(119879

119896) = 119896 We

state this fact as the following theorem which is similar toTheorem 112

Theorem121 (Raczek [73] 2008) For any nonnegative integer119896 there exists a tree 119879 with 119887

119875(119879) = 119896

Consider the tree defined in Figure 5 for 119896 = 3 Then119887(119879

3) ⩽ 2 and 119887

119875(119879

3) = 3 On the other hand 119887(119875

4) = 2

and 119887119875(119875

4) = 1 Thus we have what follows which is similar

to Corollary 113

Corollary 122 The bondage number and the paired bondagenumber are unrelated even for trees

A constructive characterization of trees with 119887(119879) = 2is given by Hartnell and Rall in [12] Raczek [73] provideda constructive characterization of trees with 119887

119875(119879) = 0 In

order to state the characterization we define a labeling andthree simple operations on a tree119879 Let 119910 isin 119881(119879) and let ℓ(119910)be the label assigned to 119910

Operation T1 If ℓ(119910) = 119861 add a vertex 119909 and the

edge 119910119909 and let ℓ(119909) = 119860OperationT

2 If ℓ(119910) = 119862 add a path (119909

1 119909

2) and the

edge 1199101199091 and let ℓ(119909

1) = 119861 ℓ(119909

2) = 119860

Operation T3 If ℓ(119910) = 119861 add a path (119909

1 119909

2 119909

3)

and the edge 1199101199091 and let ℓ(119909

1) = 119862 ℓ(119909

2) = 119861 and

ℓ(1199093) = 119860

Let 1198752= (119906 V) with ℓ(119906) = 119860 and ℓ(V) = 119861 Let T be

the class of all trees obtained from the labeled 1198752by a finite

sequence of operationsT1 T

2 T

3

A tree 119879 in Figure 8 belongs to the familyTRaczek [73] obtained the following characterization of all

trees 119879 with 119887119875(119879) = 0

119860

119860

119860

119860

119860

119861 119862

119861 119862 119861

119861119860

Figure 8 A tree 119879 belong to the familyT

Theorem 123 (Raczek [73] 2008) For a tree 119879 119887119875(119879) = 0 if

and only if 119879 is inT

We state the decision problem for the paired bondage asfollows

Problem 4 Consider the decision problem

Paired Bondage ProblemInstance a graph 119866 and a positive integer 119887Question is 119887

119875(119866) ⩽ 119887

Conjecture 124 The paired bondage problem is NP-complete

83 Restrained Bondage Numbers A dominating set 119878 of 119866 iscalled to be restrained if the subgraph induced by 119878 containsno isolated vertices where 119878 = 119881(119866) 119878 The restraineddomination number of 119866 denoted by 120574

119877(119866) is the smallest

cardinality of a restrained dominating set of 119866 It is clear that120574119877(119866) exists and 120574(119866) ⩽ 120574

119877(119866) for any nonempty graph 119866

The concept of restrained domination was introducedby Telle and Proskurowski [74] in 1997 albeit indirectly asa vertex partitioning problem Concerning the study of therestrained domination numbers the reader is referred to [74ndash79]

In 2008 Hattingh and Plummer [80] defined therestrained bondage number 119887R(119866) of a nonempty graph 119866 tobe the minimum cardinality among all subsets 119861 sube 119864(119866) forwhich 120574

119877(119866 minus 119861) gt 120574

119877(119866)

For some simple graphs their restrained dominationnumbers can be easily determined and so restrained bondagenumbers have been also determined For example it is clearthat 120574

119877(119870

119899) = 1 Domke et al [77] showed that 120574

119877(119862

119899) =

119899 minus 2lfloor1198993rfloor for 119899 ⩾ 3 and 120574119877(119875

119899) = 119899 minus 2lfloor(119899 minus 1)3rfloor for 119899 ⩾ 1

Using these results Hattingh and Plummer [80] obtained therestricted bondage numbers for119870

119899 119862

119899 119875

119899 and119866 = 119870

11989911198992119899119905

as follows

Theorem 125 (Hattingh and Plummer [80] 2008) For 119899 ⩾ 3

119887R (119870119899) =

1 119894119891 119899 = 3

lceil119899

2rceil 119900119905ℎ119890119903119908119894119904119890

119887R (119862119899) =

1 119894119891 119899 equiv 0 (mod 3) 2 119900119905ℎ119890119903119908119894119904119890

119887R (119875119899) = 1 119899 ⩾ 4

20 International Journal of Combinatorics

119887R (119866)

=

lceil119898

2rceil 119894119891 119899

119898= 1 119886119899119889 119899

119898+1⩾ 2 (1 ⩽ 119898 lt 119905)

2119905 minus 2 119894119891 1198991= 119899

2= sdot sdot sdot = 119899

119905= 2 (119905 ⩾ 2)

2 119894119891 1198991= 2 119886119899119889 119899

2⩾ 3 (119905 = 2)

119905minus1

sum

119894=1

119899119894minus 1 119900119905ℎ119890119903119908119894119904119890

(46)

Theorem 126 (Hattingh and Plummer [80] 2008) For a tree119879 of order 119899 (⩾ 4) 119879 ≇ 119870

1119899minus1if and only if 119887R(119879) = 1

Theorem 126 shows that the restrained bondage numberof a tree can be computed in constant time However thedecision problem for 119887R(119866) is NP-complete even for bipartitegraphs

Problem 5 Consider the decision problem

Restrained Bondage Problem

Instance a graph 119866 and a positive integer 119887

Question is 119887R(119866) ⩽ 119887

Theorem 127 (Hattingh and Plummer [80] 2008) Therestrained bondage problem is NP-complete even for bipartitegraphs

Consequently in view of its computational hardnessit is significative to establish some sharp bounds on therestrained bondage number of a graph in terms of othergraphic parameters

Theorem 128 (Hattingh and Plummer [80] 2008) For anygraph 119866 with 120575(119866) ⩾ 2

119887R (119866) ⩽ min119909119910isin119864(119866)

119889119866(119909) + 119889

119866(119910) minus 2 (47)

Corollary 129 119887R(119866) ⩽ Δ(119866)+120575(119866)minus2 for any graph119866 with120575(119866) ⩾ 2

Notice that the bounds stated in Theorem 128 andCorollary 129 are sharp Indeed the class of cycles whoseorders are congruent to 1 2 (mod 3) have a restrained bondagenumber achieving these bounds (see Theorem 125)

Theorem 128 is an analogue of Theorem 14 A quitenatural problem is whether or not for restricted bondagenumber there are analogues of Theorem 16 Theorem 18Theorem 29 and so on Indeed we should consider all resultsfor the bondage number whether or not there are analoguesfor restricted bondage number

Theorem 130 (Hattingh and Plummer [80] 2008) For anygraph 119866 with 120574

119877(119866) = 2

119887R (119866) ⩽ Δ (119866) + 1 (48)

We now consider the relation between 119887119877(119866) and 119887(119866)

Theorem 131 (Hattingh and Plummer [80] 2008) If 120574119877(119866) =

120574(119866) for some graph 119866 then 119887R(119866) ⩽ 119887(119866)

Corollary 129 and Theorem 131 provide a possibility toattack Conjecture 73 by characterizing planar graphs with120574119877= 120574 and 119887R = 119887 when 3 ⩽ Δ ⩽ 6However we do not have 119887R(119866) = 119887(119866) for any graph

119866 even if 120574119877(119866) = 120574(119866) Observe that 120574

119877(119870

3) = 120574(119870

3) yet

119887119877(119870

3) = 1 and 119887(119870

3) = 2 We still may not claim that

119887119877(119866) = 119887(119866) even in the case that every 120574-set is a 120574

119877-set

The example 1198703again demonstrates this

Theorem 132 (Hattingh and Plummer [80] 2008) There isan infinite class of graphs inwhich each graph119866 satisfies 119887(119866) lt119887R(119866)

In fact such an infinite class of graphs can be obtained asfollows Let119867 be a connected graph BC(119867) a graph obtainedfrom 119867 by attaching ℓ (⩾ 2) vertices of degree 1 to eachvertex in 119867 and B = 119866 119866 = BC(119867) for some graph 119867such that 120575(119867) ⩾ 2 By Theorem 3 we can easily check that119887(119866) = 1 lt 2 ⩽ min120575(119867) ℓ = 119887R(119866) for any 119866 isin BMoreover there exists a graph 119866 isin B such that 119887R(119866) canbe much larger than 119887(119866) which is stated as the followingtheorem

Theorem 133 (Hattingh and Plummer [80] 2008) For eachpositive integer 119896 there is a graph119866 such that 119896 = 119887R(119866)minus119887(119866)

Combining Theorem 133 with Theorem 132 we have thefollowing corollary

Corollary 134 The bondage number and the restrainedbondage number are unrelated

Very recently Jafari Rad et al [81] have consideredthe total restrained bondage in graphs based on both totaldominating sets and restrained dominating sets and obtainedseveral properties exact values and bounds for the totalrestrained bondage number of a graph

A restrained dominating set 119878 of a graph 119866 without iso-lated vertices is called to be a total restrained dominating setif 119878 is a total dominating set The total restrained dominationnumber of119866 denoted by 120574TR (119866) is theminimum cardinalityof a total restrained dominating set of 119866 For referenceson this domination in graphs see [3 61 82ndash85] The totalrestrained bondage number 119887TR (119866) of a graph 119866 with noisolated vertex is the cardinality of a smallest set of edges 119861 sube119864(119866) for which119866minus119861 has no isolated vertex and 120574TR (119866minus119861) gt120574TR (119866) In the case that there is no such subset 119861 we define119887TR (119866) = infin

Ma et al [84] determined that 120574TR (119875119899) = 119899minus2lfloor(119899minus2)4rfloorfor 119899 ⩾ 3 120574TR (119862119899

) = 119899 minus 2lfloor1198994rfloor for 119899 ⩾ 3 120574TR (1198703) =

3 and 120574TR (119870119899) = 2 for 119899 = 3 and 120574TR (1198701119899

) = 1 + 119899

and 120574TR (119870119898119899) = 2 for 2 ⩽ 119898 ⩽ 119899 According to these

results Jafari Rad et al [81] determined the exact valuesof total restrained bondage numbers for the correspondinggraphs

International Journal of Combinatorics 21

Theorem 135 (Jafari Rad et al [81]) For an integer 119899

119887TR (119875119899) = infin 119894119891 2 ⩽ 119899 ⩽ 5

1 119894119891 119899 ⩾ 5

119887TR (119862119899) =

infin 119894119891 119899 = 3

1 119894119891 3 lt 119899 119899 equiv 0 1 (mod 4)2 119894119891 3 lt 119899 119899 equiv 2 3 (mod 4)

119887TR (119882119899) = 2 119891119900119903 119899 ⩾ 6

119887TR (119870119899) = 119899 minus 1 119891119900119903 119899 ⩾ 4

119887TR (119870119898119899) = 119898 minus 1 119891119900119903 2 ⩽ 119898 ⩽ 119899

(49)

119887TR (119870119899) = 119899minus1 shows the fact that for any integer 119899 ⩾ 3 there

exists a connected graph namely119870119899+1

such that 119887TR (119870119899+1) =

119899 that is the total restrained bondage number is unboundedfrom the above

A vertex is said to be a support vertex if it is adjacent toa vertex of degree one Let A be the family of all connectedgraphs such that119866 belongs toA if and only if every edge of119866is incident to a support vertex or 119866 is a cycle of length three

Proposition 136 (Cyman and Raczek [82] 2006) For aconnected graph119866 of order 119899 120574TR (119866) = 119899 if and only if119866 isin A

Hence if 119866 isin A then the removal of any subset of edgesof119866 cannot increase the total restrained domination numberof 119866 and thus 119887TR (119866) = infin When 119866 notin A a result similar toTheorem 6 is obtained

Theorem 137 (Jafari Rad et al [81]) For any tree 119879 notin A119887TR (119879) ⩽ 2 This bound is sharp

In the samepaper the authors provided a characterizationfor trees 119879 notin A with 119887TR (119879) = 1 or 2 The following result issimilar to Observation 1

Observation 3 (Jafari Rad et al [81]) If 119896 edges can beremoved from a graph 119866 to obtain a graph 119867 without anisolated vertex and with 119887TR (119867) = 119905 then 119887TR (119866) ⩽ 119896 + 119905

ApplyingTheorem 137 andObservation 3 Jafari Rad et alcharacterized all connected graphs 119866 with 119887TR (119866) = infin

Theorem 138 (Jafari Rad et al [81]) For a connected graph119866119887TR (119866) = infin if and only if 119866 isin A

84 Other Conditional Bondage Numbers A subset 119868 sube 119881(119866)is called an independent set if no two vertices in 119868 are adjacentin 119866 The maximum cardinality among all independent setsis called the independence number of 119866 denoted by 120572(119866)

A dominating set 119878 of a graph 119866 is called to be inde-pendent if 119878 is an independent set of 119866 The minimumcardinality among all independent dominating set is calledthe independence domination number of 119866 and denoted by120574119868(119866)

Since an independent dominating set is not only adominating set but also an independent set 120574(119866) ⩽ 120574

119868(119866) ⩽

120572(119866) for any graph 119866It is clear that a maximal independent set is certainly a

dominating set Thus an independent set is maximal if andonly if it is an independent dominating set and so 120574

119868(119866) is the

minimum cardinality among all maximal independent setsof 119866 This graph-theoretical invariant has been well studiedin the literature see for example Haynes et al [3] for earlyresults Goddard and Henning [86] for recent results onindependent domination in graphs

In 2003 Zhang et al [87] defined the independencebondage number 119887

119868(119866) of a nonempty graph 119866 to be the

minimum cardinality among all subsets 119861 sube 119864(119866) for which120574119868(119866 minus 119861) gt 120574

119868(119866) For some ordinary graphs their indepen-

dence domination numbers can be easily determined and soindependence bondage numbers have been also determinedClearly 119887

119868(119870

119899) = 1 if 119899 ⩾ 2

Theorem 139 (Zhang et al [87] 2003) For any integer 119899 ⩾ 4

119887119868(119862

119899) =

1 119894119891 119899 equiv 1 (mod 2) 2 119894119891 119899 equiv 0 (mod 2)

119887119868(119875

119899) =

1 119894119891 119899 equiv 0 (mod 2) 2 119894119891 119899 equiv 1 (mod 2)

119887119868(119870

119898119899) = 119899 119891119900119903 119898 ⩽ 119899

(50)

Apart from the above-mentioned results as far as we findno other results on the independence bondage number in thepresent literature we never knew much of any result on thisparameter for a tree

A subset 119878 of 119881(119866) is called an equitable dominatingset if for every 119910 isin 119881(119866 minus 119878) there exists a vertex 119909 isin 119878such that 119909119910 isin 119864(119866) and |119889

119866(119909) minus 119889

119866(119910)| ⩽ 1 proposed

by Dharmalingam [88] The minimum cardinality of such adominating set is called the equitable domination number andis denoted by 120574

119864(119866) Deepak et al [89] defined the equitable

bondage number 119887119864(119866) of a graph 119866 to be the cardinality of a

smallest set 119861 sube 119864(119866) of edges for which 120574119864(119866 minus 119861) gt 120574

119864(119866)

and determined 119887119864(119866) for some special graphs

Theorem 140 (Deepak et al [89] 2011) For any integer 119899 ⩾ 2

119887119864(119870

119899) = lceil

119899

2rceil

119887119864(119870

119898119899) = 119898 119891119900119903 0 ⩽ 119899 minus 119898 ⩽ 1

119887119864(119862

119899) =

3 119894119891 119899 equiv 1 (mod 3)2 119900119905ℎ119890119903119908119894119904119890

119887119864(119875

119899) =

2 119894119891 119899 equiv 1 (mod 3)1 119900119905ℎ119890119903119908119894119904119890

119887119864(119879) ⩽ 2 119891119900119903 119886119899119910 119899119900119899119905119903119894119907119894119886119897 119905119903119890119890 119879

119887119864(119866) ⩽ 119899 minus 1 119891119900119903 119886119899119910 119888119900119899119899119890119888119905119890119889 119892119903119886119901ℎ 119866 119900119891 119900119903119889119890119903 119899

(51)

22 International Journal of Combinatorics

As far as we know there are no more results on theequitable bondage number of graphs in the present literature

There are other variations of the normal domination byadding restricted conditions to the normal dominating setFor example the connected domination set of a graph119866 pro-posed by Sampathkumar and Walikar [90] is a dominatingset 119878 such that the subgraph induced by 119878 is connected Theproblem of finding a minimum connected domination set ina graph is equivalent to the problemof finding a spanning treewithmaximumnumber of vertices of degree one [91] and hassome important applications in wireless networks [92] thusit has received much research attention However for suchan important domination there is no result on its bondagenumber We believe that the topic is of significance for futureresearch

9 Generalized Bondage Numbers

There are various generalizations of the classical dominationsuch as 119901-domination distance domination fractional dom-ination Roman domination and rainbow domination Everysuch a generalization can lead to a corresponding bondageIn this section we introduce some of them

91 119901-Bondage Numbers In 1985 Fink and Jacobson [93]introduced the concept of 119901-domination Let 119901 be a positiveinteger A subset 119878 of 119881(119866) is a 119901-dominating set of 119866 if|119878 cap 119873

119866(119910)| ⩾ 119901 for every 119910 isin 119878 The 119901-domination number

120574119901(119866) is the minimum cardinality among all 119901-dominating

sets of 119866 Any 119901-dominating set of 119866 with cardinality 120574119901(119866)

is called a 120574119901-set of 119866 Note that a 120574

1-set is a classic minimum

dominating set Notice that every graph has a 119901-dominatingset since the vertex-set119881(119866) is such a setWe also note that a 1-dominating set is a dominating set and so 120574(119866) = 120574

1(119866) The

119901-domination number has received much research attentionsee a state-of-the-art survey article by Chellali et al [94]

It is clear from definition that every 119901-dominating setof a graph certainly contains all vertices of degree at most119901 minus 1 By this simple observation to avoid the trivial caseoccurrence we always assume that Δ(119866) ⩾ 119901 For 119901 ⩾ 2 Luet al [95] gave a constructive characterization of trees withunique minimum 119901-dominating sets

Recently Lu and Xu [96] have introduced the conceptto the 119901-bondage number of 119866 denoted by 119887

119901(119866) as the

minimum cardinality among all sets of edges 119861 sube 119864(119866) suchthat 120574

119901(119866 minus 119861) gt 120574

119901(119866) Clearly 119887

1(119866) = 119887(119866)

Lu and Xu [96] established a lower bound and an upperbound on 119887

119901(119879) for any integer 119901 ⩾ 2 and any tree 119879 with

Δ(119879) ⩾ 119901 and characterized all trees achieving the lowerbound and the upper bound respectively

Theorem 141 (Lu and Xu [96] 2011) For any integer 119901 ⩾ 2and any tree 119879 with Δ(119879) ⩾ 119901

1 ⩽ 119887119901(119879) ⩽ Δ (119879) minus 119901 + 1 (52)

Let 119878 be a given subset of 119881(119866) and for any 119909 isin 119878 let

119873119901(119909 119878 119866) = 119910 isin 119878 cap 119873

119866(119909)

1003816100381610038161003816119873119866(119910) cap 119878

1003816100381610038161003816 = 119901

(53)

Then all trees achieving the lower bound 1 can be character-ized as follows

Theorem 142 (Lu and Xu [96] 2011) Let 119879 be a tree withΔ(119879) ⩾ 119901 ⩾ 2 Then 119887

119901(119879) = 1 if and only if for any 120574

119901-set

119878 of 119879 there exists an edge 119909119910 isin (119878 119878) such that 119910 isin 119873119901(119909 119878 119879)

The notation 119878(119886 119887) denotes a double star obtained byadding an edge between the central vertices of two stars119870

1119886minus1

and1198701119887minus1

And the vertex with degree 119886 (resp 119887) in 119878(119886 119887) iscalled the 119871-central vertex (resp 119877-central vertex) of 119878(119886 119887)

To characterize all trees attaining the upper bound givenin Theorem 141 we define three types of operations on a tree119879 with Δ(119879) = Δ ⩾ 119901 + 1

Type 1 attach a pendant edge to a vertex 119910with 119889119879(119910) ⩽ 119901minus

2 in 119879

Type 2 attach a star 1198701Δminus1

to a vertex 119910 of 119879 by joining itscentral vertex to 119910 where 119910 in a 120574

119901-set of 119879 and

119889119879(119910) ⩽ Δ minus 1

Type 3 attach a double star 119878(119901 Δ minus 1) to a pendant vertex 119910of 119879 by coinciding its 119877-central vertex with 119910 wherethe unique neighbor of 119910 is in a 120574

119901-set of 119879

Let B = 119879 a tree obtained from 1198701Δ

or 119878(119901 Δ) by afinite sequence of operations of Types 1 2 and 3

Theorem 143 (Lu and Xu [96] 2011) A tree with the maxi-mum Δ ⩾ 119901 + 1 has 119901-bondage number Δ minus 119901 + 1 if and onlyif it belongs toB

Theorem 144 (Lu and Xu [97] 2012) Let 119866119898119899= 119875

119898times 119875

119899

Then

1198872(119866

2119899) = 1 119891119900119903 119899 ⩾ 2

1198872(119866

3119899) =

2 119894119891 119899 equiv 1 (mod 3) 1 119900119905ℎ119890119903119908119894119904119890

for 119899 ⩾ 2

1198872(119866

4119899) =

1 119894119891 119899 equiv 3 (mod 4) 2 119900119905ℎ119890119903119908119894119904119890

for 119899 ⩾ 7

(54)

Recently Krzywkowski [98] has proposed the conceptof the nonisolating 2-bondage of a graph The nonisolating2-bondage number of a graph 119866 denoted by 1198871015840

2(119866) is the

minimum cardinality among all sets of edges 119861 sube 119864(119866) suchthat 119866 minus 119861 has no isolated vertices and 120574

2(119866 minus 119861) gt 120574

2(119866) If

for every 119861 sube 119864(119866) either 1205742(119866 minus 119861) = 120574

2(119866) or 119866 minus 119861 has

isolated vertices then define 11988710158402(119866) = 0 and say that 119866 is a 2-

nonisolatedly strongly stable graph Krzywkowski presentedsome basic properties of nonisolating 2-bondage in graphsshowed that for every nonnegative integer 119896 there exists atree 119879 such 1198871015840

2(119879) = 119896 characterized all 2-non-isolatedly

strongly stable trees and determined 11988710158402(119866) for several special

graphs

International Journal of Combinatorics 23

Theorem 145 (Krzywkowski [98] 2012) For any positiveintegers 119899 and119898 with119898 ⩽ 119899

1198871015840

2(119870

119899) =

0 119894119891 119899 = 1 2 3

lfloor2119899

3rfloor 119900119905ℎ119890119903119908119894119904119890

1198871015840

2(119875

119899) =

0 119894119891 119899 = 1 2 3

1 119900119905ℎ119890119903119908119894119904119890

1198871015840

2(119862

119899) =

0 119894119891 119899 = 3

1 119894119891 119899 119894119904 119890V119890119899

2 119894119891 119899 119894119904 119900119889119889

1198871015840

2(119870

119898119899) =

3 119894119891 119898 = 119899 = 3

1 119894119891 119898 = 119899 = 4

2 119900119905ℎ119890119903119908119894119904119890

(55)

92 Distance Bondage Numbers A subset 119878 of vertices of agraph119866 is said to be a distance 119896-dominating set for119866 if everyvertex in 119866 not in 119878 is at distance at most 119896 from some vertexof 119878 The minimum cardinality of all distance 119896-dominatingsets is called the distance 119896-domination number of 119866 anddenoted by 120574

119896(119866) (do not confuse with the above-mentioned

120574119901(119866)) When 119896 = 1 a distance 1-dominating set is a normal

dominating set and so 1205741(119866) = 120574(119866) for any graph 119866 Thus

the distance 119896-domination is a generalization of the classicaldomination

The relation between 120574119896and 120572

119896(the 119896-independence

number defined in Section 22) for a tree obtained by Meirand Moon [14] who proved that 120574

119896(119879) = 120572

2119896(119879) for any tree

119879 (a special case for 119896 = 1 is stated in Proposition 10) Thefurther research results can be found in Henning et al [99]Tian and Xu [100ndash103] and Liu et al [104]

In 1998 Hartnell et al [15] defined the distance 119896-bondagenumber of 119866 denoted by 119887

119896(119866) to be the cardinality of a

smallest subset 119861 of edges of 119866 with the property that 120574119896(119866 minus

119861) gt 120574119896(119866) FromTheorem 6 it is clear that if119879 is a nontrivial

tree then 1 ⩽ 1198871(119879) ⩽ 2 Hartnell et al [15] generalized this

result to any integer 119896 ⩾ 1

Theorem 146 (Hartnell et al [15] 1998) For every nontrivialtree 119879 and positive integer 119896 1 ⩽ 119887

119896(119879) ⩽ 2

Hartnell et al [15] and Topp and Vestergaard [13] alsocharacterized the trees having distance 119896-bondage number 2In particular the class of trees for which 119887

1(119879) = 2 are just

those which have a unique maximum 2-independent set (seeTheorem 11)

Since when 119896 = 1 the distance 1-bondage number 1198871(119866)

is the classical bondage number 119887(119866) Theorem 12 gives theNP-hardness of deciding the distance 119896-bondage number ofgeneral graphs

Theorem 147 Given a nonempty undirected graph 119866 andpositive integers 119896 and 119887 with 119887 ⩽ 120576(119866) determining wetheror not 119887

119896(119866) ⩽ 119887 is NP-hard

For a vertex 119909 in119866 the open 119896-neighborhood119873119896(119909) of 119909 is

defined as119873119896(119909) = 119910 isin 119881(119866) 1 ⩽ 119889

119866(119909 119910) le 119896The closed

119896-neighborhood119873119896[119909] of 119909 in 119866 is defined as119873

119896(119909) cup 119909 Let

Δ119896(119866) = max 1003816100381610038161003816119873119896

(119909)1003816100381610038161003816 for any 119909 isin 119881 (119866) (56)

Clearly Δ1(119866) = Δ(119866) The 119896th power of a graph 119866 is the

graph119866119896 with vertex-set119881(119866119896

) = 119881(119866) and edge-set119864(119866119896

) =

119909119910 1 ⩽ 119889119866(119909 119910) ⩽ 119896 The following lemma holds directly

from the definition of 119866119896

Proposition 148 (Tian and Xu [101]) Δ(119866119896

) = Δ119896(119866) and

120574(119866119896

) = 120574119896(119866) for any graph 119866 and each 119896 ⩾ 1

A graph 119866 is 119896-distance domination-critical or 120574119896-critical

for short if 120574119896(119866 minus 119909) lt 120574

119896(119866) for every vertex 119909 in 119866

proposed by Henning et al [105]

Proposition 149 (Tian and Xu [101]) For each 119896 ⩾ 1 a graph119866 is 120574

119896-critical if and only if 119866119896 is 120574

119896-critical

By the above facts we suggest the following worthwhileproblem

Problem 6 Can we generalize the results on the bondage for119866 to 119866119896 In particular do the following propositions hold

(a) 119887(119866119896

) = 119887119896(119866) for any graph 119866 and each 119896 ⩾ 1

(b) 119887(119866119896

) ⩽ Δ119896(119866) if 119866 is not 120574

119896-critical

Let 119896 and 119901 be positive integers A subset 119878 of 119881(119866) isdefined to be a (119896 119901)-dominating set of 119866 if for any vertex119909 isin 119878 |119873

119896(119909) cap 119878| ⩾ 119901 The (119896 119901)-domination number of

119866 denoted by 120574119896119901(119866) is the minimum cardinality among

all (119896 119901)-dominating sets of 119866 Clearly for a graph 119866 a(1 1)-dominating set is a classical dominating set a (119896 1)-dominating set is a distance 119896-dominating set and a (1 119901)-dominating set is the above-mentioned 119901-dominating setThat is 120574

11(119866) = 120574(119866) 120574

1198961(119866) = 120574

119896(119866) and 120574

1119901(119866) = 120574

119901(119866)

The concept of (119896 119901)-domination in a graph 119866 is a gen-eralized domination which combines distance 119896-dominationand 119901-domination in 119866 So the investigation of (119896 119901)-domination of 119866 is more interesting and has received theattention of many researchers see for example [106ndash109]

More general Li and Xu [110] proposed the concept of the(ℓ 119908)-domination number of a119908-connected graph A subset119878 of 119881(119866) is defined to be an (ℓ 119908)-dominating set of 119866 iffor any vertex 119909 isin 119878 there are 119908 internally disjoint pathsof length at most ℓ from 119909 to some vertex in 119878 The (ℓ 119908)-domination number of119866 denoted by 120574

ℓ119908(119866) is theminimum

cardinality among all (ℓ 119908)-dominating sets of 119866 Clearly1205741198961(119866) = 120574

119896(119866) and 120574

1119901(119866) = 120574

119901(119866)

It is quite natural to propose the concept of bondage num-ber for (119896 119901)-domination or (ℓ 119908)-domination However asfar as we know none has proposed this concept until todayThis is a worthwhile topic for further research

24 International Journal of Combinatorics

93 Fractional Bondage Numbers If 120590 is a function mappingthe vertex-set 119881 into some set of real numbers then for anysubset 119878 sube 119881 let 120590(119878) = sum

119909isin119878120590(119909) Also let |120590| = 120590(119881)

A real-value function 120590 119881 rarr [0 1] is a dominatingfunction of a graph 119866 if for every 119909 isin 119881(119866) 120590(119873

119866[119909]) ⩾ 1

Thus if 119878 is a dominating set of 119866 120590 is a function where

120590 (119909) = 1 if 119909 isin 1198780 otherwise

(57)

then 120590 is a dominating function of119866 The fractional domina-tion number of 119866 denoted by 120574

119865(119866) is defined as follows

120574119865(119866) = min |120590| 120590 is a domination function of 119866

(58)

The 120574119865-bondage number of 119866 denoted by 119887

119865(119866) is

defined as the minimum cardinality of a subset 119861 sube 119864 whoseremoval results in 120574

119865(119866 minus 119861) gt 120574

119865(119866)

Hedetniemi et al [111] are the first to study fractionaldomination although Farber [112] introduced the idea indi-rectly The concept of the fractional domination numberwas proposed by Domke and Laskar [113] in 1997 Thefractional domination numbers for some ordinary graphs aredetermined

Proposition 150 (Domke et al [114] 1998 Domke and Laskar[113] 1997) (a) If 119866 is a 119896-regular graph with order 119899 then120574119865(119866) = 119899119896 + 1(b) 120574

119865(119862

119899) = 1198993 120574

119865(119875

119899) = lceil1198993rceil 120574

119865(119870

119899) = 1

(c) 120574119865(119866) = 1 if and only if Δ(119866) = 119899 minus 1

(d) 120574119865(119879) = 120574(119879) for any tree 119879

(e) 120574119865(119870

119899119898) = (119899(119898 + 1) + 119898(119899 + 1))(119899119898 minus 1) where

max119899119898 gt 1

The assertions (a)ndash(d) are due to Domke et al [114] andthe assertion (e) is due to Domke and Laskar [113] Accordingto these results Domke and Laskar [113] determined the 120574

119865-

bondage numbers for these graphs

Theorem 151 (Domke and Laskar [113] 1997) 119887119865(119870

119899) =

lceil1198992rceil 119887119865(119870

119899119898) = 1 wheremax119899119898 gt 1

119887119865(119862

119899) =

2 119894119891 119899 equiv 0 (mod 3) 1 119900119905ℎ119890119903119908119894119904119890

119887119865(119875

119899) =

2 119894119891 119899 equiv 1 (mod 3) 1 119900119905ℎ119890119903119908119894119904119890

(59)

It is easy to see that for any tree119879 119887119865(119879) ⩽ 2 In fact since

120574119865(119879) = 120574(119879) for any tree 119879 120574

119865(119879

1015840

) = 120574(1198791015840

) for any subgraph1198791015840 of 119879 It follows that

119887119865(119879) = min 119861 sube 119864 (119879) 120574

119865(119879 minus 119861) gt 120574

119865(119879)

= min 119861 sube 119864 (119879) 120574 (119879 minus 119861) gt 120574 (119879)

= 119887 (119879) ⩽ 2

(60)

Except for these no results on 119887119865(119866) have been known as

yet

94 Roman Bondage Numbers A Roman dominating func-tion on a graph 119866 is a labeling 119891 119881 rarr 0 1 2 such thatevery vertex with label 0 has at least one neighbor with label2 The weight of a Roman dominating function 119891 on 119866 is thevalue

119891 (119866) = sum

119906isin119881(119866)

119891 (119906) (61)

The minimum weight of a Roman dominating function ona graph 119866 is called the Roman domination number of 119866denoted by 120574RM (119866)

A Roman dominating function 119891 119881 rarr 0 1 2

can be represented by the ordered partition (1198810 119881

1 119881

2) (or

(119881119891

0 119881

119891

1 119881

119891

2) to refer to119891) of119881 where119881

119894= V isin 119881 | 119891(V) = 119894

In this representation its weight 119891(119866) = |1198811| + 2|119881

2| It is

clear that119881119891

1cup119881

119891

2is a dominating set of 119866 called the Roman

dominating set denoted by 119863119891

R = (1198811 1198812) Since 119881119891

1cup 119881

119891

2is

a dominating set when 119891 is a Roman dominating functionand since placing weight 2 at the vertices of a dominating setyields a Roman dominating function in [115] it is observedthat

120574 (119866) ⩽ 120574RM (119866) ⩽ 2120574 (119866) (62)

A graph 119866 is called to be Roman if 120574RM (119866) = 2120574(119866)The definition of the Roman domination function is

proposed implicitly by Stewart [116] and ReVelle and Rosing[117] Roman dominating numbers have been deeply studiedIn particular Bahremandpour et al [118] showed that theproblem determining the Roman domination number is NP-complete even for bipartite graphs

Let 119866 be a graph with maximum degree at least twoThe Roman bondage number 119887RM (119866) of 119866 is the minimumcardinality of all sets 1198641015840 sube 119864 for which 120574RM (119866 minus 119864

1015840

) gt

120574RM (119866) Since in study of Roman bondage number theassumption Δ(119866) ⩾ 2 is necessary we always assume thatwhen we discuss 119887RM (119866) all graphs involved satisfy Δ(119866) ⩾2 The Roman bondage number 119887RM (119866) is introduced byJafari Rad and Volkmann in [119]

Recently Bahremandpour et al [118] have shown that theproblem determining the Roman bondage number is NP-hard even for bipartite graphs

Problem 7 Consider the decision problem

Roman Bondage Problem

Instance a nonempty bipartite graph119866 and a positiveinteger 119896

Question is 119887RM (119866) ⩽ 119896

Theorem 152 (Bahremandpour et al [118] 2013) TheRomanbondage problem is NP-hard even for bipartite graphs

The exact value of 119887RM(119866) is known only for a few familyof graphs including the complete graphs cycles and paths

International Journal of Combinatorics 25

Theorem 153 (Jafari Rad and Volkmann [119] 2011) For anypositive integers 119899 and119898 with119898 ⩽ 119899

119887RM (119875119899) = 2 119894119891 119899 equiv 2 (mod 3) 1 119900119905ℎ119890119903119908119894119904119890

119887RM (119862119899) =

3 119894119891 119899 = 2 (mod 3) 2 119900119905ℎ119890119903119908119894119904119890

119887RM (119870119899) = lceil

119899

2rceil 119891119900119903 119899 ⩾ 3

119887RM (119870119898119899) =

1 119894119891 119898 = 1 119886119899119889 119899 = 1

4 119894119891 119898 = 119899 = 3

119898 119900119905ℎ119890119903119908119894119904119890

(63)

Lemma 154 (Cockayne et al [115] 2004) If 119866 is a graph oforder 119899 and contains vertices of degree 119899 minus 1 then 120574RM(119866) = 2

Using Lemma 154 the third conclusion in Theorem 153can be generalized to more general case which is similar toLemma 49

Theorem 155 (Jafari Rad and Volkmann [119] 2011) Let119866 bea graph with order 119899 ge 3 and 119905 the number of vertices of degree119899 minus 1 in 119866 If 119905 ⩾ 1 then 119887RM(119866) = lceil1199052rceil

Ebadi and PushpaLatha [120] conjectured that 119887RM(119866) ⩽119899 minus 1 for any graph 119866 of order 119899 ⩾ 3 Dehgardi et al[121] Akbari andQajar [122] independently showed that thisconjecture is true

Theorem 156 119887RM(119866) ⩽ 119899 minus 1 for any connected graph 119866 oforder 119899 ⩾ 3

For a complete 119905-partite graph we have the followingresult

Theorem 157 (Hu and Xu [123] 2011) Let 119866 = 11987011989811198982119898119905

bea complete 119905-partite graph with 119898

1= sdot sdot sdot = 119898

119894lt 119898

119894+1le sdot sdot sdot ⩽

119898119905 119905 ⩾ 2 and 119899 = sum119905

119895=1119898

119895 Then

119887RM (119866) =

lceil119894

2rceil 119894119891 119898

119894= 1 119886119899119889 119899 ⩾ 3

2 119894119891 119898119894= 2 119886119899119889 119894 = 1

119894 119894119891 119898119894= 2 119886119899119889 119894 ⩾ 2

4 119894119891 119898119894= 3 119886119899119889 119894 = 119905 = 2

119899 minus 1 119894119891 119898119894= 3 119886119899119889 119894 = 119905 ⩾ 3

119899 minus 119898119905

119894119891 119898119894⩾ 3 119886119899119889 119898

119905⩾ 4

(64)

Consider a complete 119905-partite graph 119870119905(3) for 119905 ⩾ 3

119887RM(119870119905(3)) = 119899 minus 1 by Theorem 157 where 119899 = 3119905

which shows that this upper bound of 119899 minus 1 on 119887RM(119866) inTheorem 156 is sharp This fact and 120574

119877(119870

119905(3)) = 4 give

a negative answer to the following two problems posed byDehgardi et al [121]Prove or Disprove for any connected graph 119866 of order 119899 ge 3(i) 119887RM(119866) = 119899 minus 1 if and only if 119866 cong 119870

3 (ii) 119887RM(119866) ⩽ 119899 minus

120574119877(119866) + 1Note that a complete 119905-partite graph 119870

119905(3) is (119899 minus 3)-

regular and 119887RM(119870119905(3)) = 119899 minus 1 for 119905 ⩾ 3 where 119899 = 3119905

Hu and Xu further determined that 119887119877(119866) = 119899 minus 2 for any

(119899 minus 3)-regular graph 119866 of order 119899 ⩽ 5 except 119870119905(3)

Theorem 158 (Hu and Xu [123] 2011) For any (119899minus3)-regulargraph 119866 of order 119899 other than119870

119905(3) 119887RM(119866) = 119899 minus 2 for 119899 ⩾ 5

For a tree 119879 with order 119899 ⩾ 3 Ebadi and PushpaLatha[120] and Jafari Rad and Volkmann [119] independentlyobtained an upper bound on 119887RM(119879)

Theorem 159 119887RM(119879) ⩽ 3 for any tree 119879 with order 119899 ⩾ 3

Theorem 160 (Bahremandpour et al [118] 2013) 119887RM(1198752 times119875119899) = 2 for 119899 ⩾ 2

Theorem 161 (Jafari Rad and Volkmann [119] 2011) Let119866 bea graph with order at least three

(a) If (119909 119910 119911) is a path of length 2 in 119866 then 119887RM(119866) ⩽119889119866(119909) + 119889

119866(119910) + 119889

119866(119911) minus 3 minus |119873

119866(119909) cap 119873

119866(119910)|

(b) If 119866 is connected then 119887RM(119866) ⩽ 120582(119866) + 2Δ(119866) minus 3

Theorem 161 (b) implies 119887RM(119866) ⩽ 120575(119866) + 2Δ(119866) minus 3 for aconnected graph 119866 Note that for a planar graph 119866 120575(119866) ⩽ 5moreover 120575(119866) ⩽ 3 if the girth at least 4 and 120575(119866) ⩽ 2 if thegirth at least 6 These two facts show that 119887RM(119866) ⩽ 2Δ(119866) +2 for a connected planar graph 119866 Jafari Rad and Volkmann[124] improved this bound

Theorem 162 (Jafari Rad and Volkmann [124] 2011) For anyconnected planar graph 119866 of order 119899 ⩾ 3 with girth 119892(119866)

119887RM (119866)

⩽

2Δ (119866)

Δ (119866) + 6

Δ (119866) + 5 119894119891 119866 119888119900119899119905119886119894119899119904 119899119900 V119890119903119905119894119888119890119904 119900119891 119889119890119892119903119890119890 119891119894V119890

Δ (119866) + 4 119894119891 119892 (119866) ⩾ 4

Δ (119866) + 3 119894119891 119892 (119866) ⩾ 5

Δ (119866) + 2 119894119891 119892 (119866) ⩾ 6

Δ (119866) + 1 119894119891 119892 (119866) ⩾ 8

(65)

According toTheorem 157 119887RM(119862119899) = 3 = Δ(119862

119899)+1 for a

cycle 119862119899of length 119899 ⩾ 8 with 119899 equiv 2 (mod 3) and therefore

the last upper bound inTheorem 162 is sharp at least for Δ =2

Combining the fact that every planar graph 119866 withminimum degree 5 contains an edge 119909119910 with 119889

119866(119909) = 5

and 119889119866(119910) isin 5 6 with Theorem 161 (a) Akbari et al [125]

obtained the following result

26 International Journal of Combinatorics

middot middot middot

middot middot middot

middot middot middot

middot middot middot

119866

Figure 9 The graph 119866 is constructed from 119866

Theorem 163 (Akbari et al [125] 2012) 119887RM(119866) ⩽ 15 forevery planar graph 119866

It remains open to show whether the bound inTheorem 163 is sharp or not Though finding a planargraph 119866 with 119887RM(119866) = 15 seems to be difficult Akbari etal [125] constructed an infinite family of planar graphs withRoman bondage number equal to 7 by proving the followingresult

Theorem 164 (Akbari et al [125] 2012) Let 119866 be a graph oforder 119899 and 119866 is the graph of order 5119899 obtained from 119866 byattaching the central vertex of a copy of a path 119875

5to each vertex

of119866 (see Figure 9) Then 120574RM(119866) = 4119899 and 119887RM(119866) = 120575(119866)+2

ByTheorem 164 infinitemany planar graphswith Romanbondage number 7 can be obtained by considering any planargraph 119866 with 120575(119866) = 5 (eg the icosahedron graph)

Conjecture 165 (Akbari et al [125] 2012) The Romanbondage number of every planar graph is at most 7

For general bounds the following observation is directlyobtained which is similar to Observation 1

Observation 4 Let 119866 be a graph of order 119899 with maximumdegree at least two Assume that 119867 is a spanning subgraphobtained from 119866 by removing 119896 edges If 120574RM(119867) = 120574RM(119866)then 119887

119877(119867) ⩽ 119887RM(119866) ⩽ 119887RM(119867) + 119896

Theorem 166 (Bahremandpour et al [118] 2013) For anyconnected graph 119866 of order 119899 if 119899 ⩾ 3 and 120574RM(119866) = 120574(119866) + 1then

119887RM (119866) ⩽ min 119887 (119866) 119899Δ (66)

where 119899Δis the number of vertices with maximum degree Δ in

119866

Observation 5 For any graph 119866 if 120574(119866) = 120574RM(119866) then119887RM(119866) ⩽ 119887(119866)

Theorem 167 (Bahremandpour et al [118] 2013) For everyRoman graph 119866

119887RM (119866) ⩾ 119887 (119866) (67)

The bound is sharp for cycles on 119899 vertices where 119899 equiv 0 (mod3)

The strict inequality in Theorem 167 can hold for exam-ple 119887(119862

3119896+2) = 2 lt 3 = 119887RM(1198623119896+2

) byTheorem 157A graph 119866 is called to be vertex Roman domination-

critical if 120574RM(119866 minus 119909) lt 120574RM(119866) for every vertex 119909 in 119866 If119866 has no isolated vertices then 120574RM(119866) ⩽ 2120574(119866) ⩽ 2120573(119866)If 120574RM(119866) = 2120573(119866) then 120574RM(119866) = 2120574(119866) and hence 119866 is aRoman graph In [30] Volkmann gave a lot of graphs with120574(119866) = 120573(119866) The following result is similar to Theorem 57

Theorem 168 (Bahremandpour et al [118] 2013) For anygraph 119866 if 120574RM(119866) = 2120573(119866) then

(a) 119887RM(119866) ⩾ 120575(119866)

(b) 119887RM(119866) ⩾ 120575(119866)+1 if119866 is a vertex Roman domination-critical graph

If 120574RM(119866) = 2 then obviously 119887RM(119866) ⩽ 120575(119866) So weassume 120574RM(119866) ⩾ 3 Dehgardi et al [121] proved 119887RM(119866) ⩽(120574RM(119866) minus 2)Δ(119866) + 1 and posed the following problem if 119866is a connected graph of order 119899 ⩾ 4 with 120574RM(119866) ⩾ 3 then

119887RM (119866) ⩽ (120574RM (119866) minus 2) Δ (119866) (68)

Theorem 161 (a) shows that the inequality (68) holds if120574RM(119866) ⩾ 5 Thus the bound in (68) is of interest only when120574RM(119866) is 3 or 4

Lemma 169 (Bahremandpour et al [118] 2013) For anynonempty graph 119866 of order 119899 ⩾ 3 120574RM(119866) = 3 if and onlyif Δ(119866) = 119899 minus 2

The following result shows that (68) holds for all graphs119866 of order 119899 ⩾ 4 with 120574RM(119866) = 3 or 4 which improvesTheorem 161 (b)

Theorem 170 (Bahremandpour et al [118] 2013) For anyconnected graph 119866 of order 119899 ⩾ 4

119887RM (119866) le Δ (119866) = 119899 minus 2 119894119891 120574RM (119866) = 3

Δ (119866) + 120575 (119866) minus 1 119894119891 120574RM (119866) = 4(69)

with the first equality if and only if 119866 cong 1198624

Recently Akbari and Qajar [122] have proved the follow-ing result

Theorem 171 (Akbari and Qajar [122]) For any connectedgraph 119866 of order 119899 ⩾ 3

119887RM (119866) ⩽ 119899 minus 120574RM (119866) + 5 (70)

International Journal of Combinatorics 27

We conclude this section with the following problems

Problem 8 Characterize all connected graphs 119866 of order 119899 ⩾3 for which 119887RM(119866) = 119899 minus 1

Problem 9 Prove or disprove if 119866 is a connected graph oforder 119899 ⩾ 3 then

119887RM (119866) ⩽ 119899 minus 120574RM (119866) + 3 (71)

Note that 120574119877(119870

119905(3)) = 4 and 119887RM(119870119905

(3)) = 119899minus1 where 119899 =3119905 for 119905 ⩾ 3 The upper bound on 119887RM(119866) given in Problem 9is sharp if Problem 9 has positive answer

95 Rainbow Bondage Numbers For a positive integer 119896 a119896-rainbow dominating function of a graph 119866 is a function 119891from the vertex-set 119881(119866) to the set of all subsets of the set1 2 119896 such that for any vertex 119909 in 119866 with 119891(119909) = 0 thecondition

⋃

119910isin119873119866(119909)

119891 (119910) = 1 2 119896 (72)

is fulfilledThe weight of a 119896-rainbow dominating function 119891on 119866 is the value

119891 (119866) = sum

119909isin119881(119866)

1003816100381610038161003816119891 (119909)1003816100381610038161003816 (73)

The 119896-rainbow domination number of a graph 119866 denoted by120574119903119896(119866) is the minimum weight of a 119896-rainbow dominating

function 119891 of 119866Note that 120574

1199031(119866) is the classical domination number 120574(119866)

The 119896-rainbow domination number is introduced by Bresaret al [126] Rainbow domination of a graph 119866 coincides withordinary domination of the Cartesian product of 119866 with thecomplete graph in particular 120574

119903119896(119866) = 120574(119866 times 119870

119896) for any

positive integer 119896 and any graph 119866 [126] Hartnell and Rall[127] proved that 120574(119879) lt 120574

1199032(119879) for any tree 119879 no graph has

120574 = 120574119903119896for 119896 ⩾ 3 for any 119896 ⩾ 2 and any graph 119866 of order 119899

min 119899 120574 (119866) + 119896 minus 2 ⩽ 120574119903119896(119866) ⩽ 119896120574 (119866) (74)

The introduction of rainbow domination is motivatedby the study of paired domination in Cartesian productsof graphs where certain upper bounds can be expressed interms of rainbow domination that is for any graph 119866 andany graph 119867 if 119881(119867) can be partitioned into 119896120574

119875-sets then

120574119875(119866 times 119867) ⩽ (1119896)120592(119867)120574

119903119896(119866) proved by Bresar et al [126]

Dehgardi et al [128] introduced the concept of 119896-rainbowbondage number of graphs Let 119866 be a graph with maximumdegree at least two The 119896-rainbow bondage number 119887

119903119896(119866)

of 119866 is the minimum cardinality of all sets 119861 sube 119864(119866) forwhich 120574

119903119896(119866 minus 119861) gt 120574

119903119896(119866) Since in the study of 119896-rainbow

bondage number the assumption Δ(119866) ⩾ 2 is necessarywe always assume that when we discuss 119887

119903119896(119866) all graphs

involved satisfy Δ(119866) ⩾ 2The 2-rainbow bondage number of some special families

of graphs has been determined

Theorem 172 (Dehgardi et al [128]) For any integer 119899 ⩾ 3

1198871199032(119875

119899) = 1 119891119900119903 119899 ⩾ 2

1198871199032(119862

119899) =

1 119894119891 119899 equiv 0 (mod 4) 2 119900119905ℎ119890119903119908119894119904119890

1198871199032(119870

119899119899) =

1 119894119891 119899 = 2

3 119894119891 119899 = 3

6 119894119891 119899 = 4

119898 119894119891 119899 ⩾ 5

(75)

Theorem 173 (Dehgardi et al [128]) For any connected graph119866 of order 119899 ⩾ 3 119887

1199032(119866) ⩽ 120582(119866) + 2Δ(119866) minus 3

Theorem 174 (Dehgardi et al [128]) For any tree 119879 of order119899 ⩾ 3 119887

1199032(119879) ⩽ 2

Theorem 175 (Dehgardi et al [128]) If 119866 is a planar graphwithmaximumdegree at least two then 119887

1199032(119866) ⩽ 15Moreover

if the girth 119892(119866) ⩾ 4 then 1198871199032(119866) ⩽ 11 and if 119892(119866) ⩾ 6 then

1198871199032(119866) ⩽ 9

Theorem 176 (Dehgardi et al [128]) For a complete bipartitegraph 119870

119901119902 if 2119896 + 1 ⩽ 119901 ⩽ 119902 then 119887

119903119896(119870

119901119902) = 119901

Theorem177 (Dehgardi et al [128]) For a complete graph119870119899

if 119899 ⩾ 119896 + 1 then 119887119903119896(119870

119899) = lceil119896119899(119896 + 1)rceil

The special case 119896 = 1 of Theorem 177 can be found inTheorem 1 The following is a simple observation similar toObservation 1

Observation 6 (Dehgardi et al [128]) Let 119866 be a graphwith maximum degree at least two 119867 a spanning subgraphobtained from 119866 by removing 119896 edges If 120574

119903119896(119867) = 120574

119903119896(119866)

then 119887119903119896(119867) ⩽ 119887

119903119896(119866) ⩽ 119887

119903119896(119867) + 119896

Theorem 178 (Dehgardi et al [128]) For every graph 119866 with120574119903119896(119866) = 119896120574(119866) 119887

119903119896(119866) ⩾ 119887(119866)

A graph 119866 is said to be vertex 119896-rainbow domination-critical if 120574

119903119896(119866 minus 119909) lt 120574

119903119896(119866) for every vertex 119909 in 119866 It is

clear that if 119866 has no isolated vertices then 120574119903119896(119866) ⩽ 119896120574(119866) ⩽

119896120573(119866) where 120573(119866) is the vertex-covering number of119866 Thusif 120574

119903119896(119866) = 119896120573(119866) then 120574

119903119896(119866) = 119896120574(119866)

Theorem 179 (Dehgardi et al [128]) For any graph 119866 if120574119903119896(119866) = 119896120573(119866) then

(a) 119887119903119896(119866) ⩾ 120575(119866)

(b) 119887119903119896(119866) ⩾ 120575(119866)+1 if119866 is vertex 119896-rainbow domination-

critical

As far as we know there are no more results on the 119896-rainbow bondage number of graphs in the literature

28 International Journal of Combinatorics

10 Results on Digraphs

Although domination has been extensively studied in undi-rected graphs it is natural to think of a dominating setas a one-way relationship between vertices of the graphIndeed among the earliest literatures on this subject vonNeumann and Morgenstern [129] used what is now calleddomination in digraphs to find solution (or kernels whichare independent dominating sets) for cooperative 119899-persongames Most likely the first formulation of domination byBerge [130] in 1958 was given in the context of digraphs andonly some years latter by Ore [131] in 1962 for undirectedgraphs Despite this history examination of domination andits variants in digraphs has been essentially overlooked (see[4] for an overview of the domination literature) Thus thereare few if any such results on domination for digraphs in theliterature

The bondage number and its related topics for undirectedgraph have become one of major areas both in theoreticaland applied researches However until recently Carlsonand Develin [42] Shan and Kang [132] and Huang andXu [133 134] studied the bondage number for digraphsindependently In this section we introduce their resultsfor general digraphs Results for some special digraphs suchas vertex-transitive digraphs are introduced in the nextsection

101 Upper Bounds for General Digraphs Let 119866 = (119881 119864)

be a digraph without loops and parallel edges A digraph 119866is called to be asymmetric if whenever (119909 119910) isin 119864(119866) then(119910 119909) notin 119864(119866) and to be symmetric if (119909 119910) isin 119864(119866) implies(119910 119909) isin 119864(119866) For a vertex 119909 of 119881(119866) the sets of out-neighbors and in-neighbors of 119909 are respectively defined as119873

+

119866(119909) = 119910 isin 119881(119866) (119909 119910) isin 119864(119866) and 119873minus

119866(119909) = 119909 isin

119881(119866) (119909 119910) isin 119864(119866) the out-degree and the in-degree of119909 are respectively defined as 119889+

119866(119909) = |119873

+

119866(119909)| and 119889minus

119866(119909) =

|119873+

119866(119909)| Denote themaximum and theminimum out-degree

(resp in-degree) of 119866 by Δ+(119866) and 120575+(119866) (resp Δminus(119866) and120575minus

(119866))The degree 119889119866(119909) of 119909 is defined as 119889+

119866(119909)+119889

minus

119866(119909) and

the maximum and the minimum degrees of119866 are denoted byΔ(119866) and 120575(119866) respectively that is Δ(119866) = max119889

119866(119909) 119909 isin

119881(119866) and 120575(119866) = min119889119866(119909) 119909 isin 119881(119866) Note that the

definitions here are different from the ones in the textbookon digraphs see for example Xu [1]

A subset 119878 of119881(119866) is called a dominating set if119881(119866) = 119878cup119873

+

119866(119878) where119873+

119866(119878) is the set of out-neighbors of 119878Then just

as for undirected graphs 120574(119866) is the minimum cardinalityof a dominating set and the bondage number 119887(119866) is thesmallest cardinality of a set 119861 of edges such that 120574(119866 minus 119861) gt120574(119866) if such a subset 119861 sube 119864(119866) exists Otherwise we put119887(119866) = infin

Some basic results for undirected graphs stated inSection 3 can be generalized to digraphs For exampleTheorem 18 is generalized by Carlson and Develin [42]Huang andXu [133] and Shan andKang [132] independentlyas follows

Theorem 180 Let 119866 be a digraph and (119909 119910) isin 119864(119866) Then119887(119866) ⩽ 119889

119866(119910) + 119889

minus

119866(119909) minus |119873

minus

119866(119909) cap 119873

minus

119866(119910)|

Corollary 181 For a digraph 119866 119887(119866) ⩽ 120575minus(119866) + Δ(119866)

Since 120575minus

(119866) ⩽ (12)Δ(119866) from Corollary 181Conjecture 53 is valid for digraphs

Corollary 182 For a digraph 119866 119887(119866) ⩽ (32)Δ(119866)

In the case of undirected graphs the bondage number of119866119899= 119870

119899times 119870

119899achieves this bound However it was shown

by Carlson and Develin in [42] that if we take the symmetricdigraph119866

119899 we haveΔ(119866

119899) = 4(119899minus1) 120574(119866

119899) = 119899 and 119887(119866

119899) ⩽

4(119899 minus 1) + 1 = Δ(119866119899) + 1 So this family of digraphs cannot

show that the bound in Corollary 182 is tightCorresponding to Conjecture 51 which is discredited

for undirected graphs and is valid for digraphs the sameconjecture for digraphs can be proposed as follows

Conjecture 183 (Carlson and Develin [42] 2006) 119887(119866) ⩽Δ(119866) + 1 for any digraph 119866

If Conjecture 183 is true then the following results showsthat this upper bound is tight since 119887(119870

119899times119870

119899) = Δ(119870

119899times119870

119899)+1

for a complete digraph 119870119899

In 2007 Shan and Kang [132] gave some tight upperbounds on the bondage numbers for some asymmetricdigraphs For example 119887(119879) ⩽ Δ(119879) for any asymmetricdirected tree 119879 119887(119866) ⩽ Δ(119866) for any asymmetric digraph 119866with order at least 4 and 120574(119866) ⩽ 2 For planar digraphs theyobtained the following results

Theorem 184 (Shan and Kang [132] 2007) Let 119866 be aasymmetric planar digraphThen 119887(119866) ⩽ Δ(119866)+2 and 119887(119866) ⩽Δ(119866) + 1 if Δ(119866) ⩾ 5 and 119889minus

119866(119909) ⩾ 3 for every vertex 119909 with

119889119866(119909) ⩾ 4

102 Results for Some Special Digraphs The exact values andbounds on 119887(119866) for some standard digraphs are determined

Theorem185 (Huang andXu [133] 2006) For a directed cycle119862119899and a directed path 119875

119899

119887 (119862119899) =

3 119894119891 119899 119894119904 119900119889119889

2 119894119891 119899 119894119904 119890V119890119899

119887 (119875119899) =

2 119894119891 119899 119894119904 119900119889119889

1 119894119891 119899 119894119904 119890V119890119899

(76)

For the de Bruijn digraph 119861(119889 119899) and the Kautz digraph119870(119889 119899)

119887 (119861 (119889 119899)) = 119889 119894119891 119899 119894119904 119900119889119889

119889 ⩽ 119887 (119861 (119889 119899)) ⩽ 2119889 119894119891 119899 119894119904 119890V119890119899

119887 (119870 (119889 119899)) = 119889 + 1

(77)

Like undirected graphs we can define the total domi-nation number and the total bondage number On the totalbondage numbers for some special digraphs the knownresults are as follows

International Journal of Combinatorics 29

Theorem 186 (Huang and Xu [134] 2007) For a directedcycle 119862

119899and a directed path 119875

119899 119887

119879(119875

119899) and 119887

119879(119862

119899) all do not

exist For a complete digraph 119870119899

119887119879(119870

119899) = infin 119894119891 119899 = 1 2

119887119879(119870

119899) = 3 119894119891 119899 = 3

119899 ⩽ 119887119879(119870

119899) ⩽ 2119899 minus 3 119894119891 119899 ⩾ 4

(78)

The extended de Bruijn digraph EB(119889 119899 1199021 119902

119901) and

the extended Kautz digraph EK(119889 119899 1199021 119902

119901) are intro-

duced by Shibata and Gonda [135] If 119901 = 1 then theyare the de Bruijn digraph B(119889 119899) and the Kautz digraphK(119889 119899) respectively Huang and Xu [134] determined theirtotal domination numbers In particular their total bondagenumbers for general cases are determined as follows

Theorem 187 (Huang andXu [134] 2007) If 119889 ⩾ 2 and 119902119894⩾ 2

for each 119894 = 1 2 119901 then

119887119879(EB(119889 119899 119902

1 119902

119901)) = 119889

119901

minus 1

119887119879(EK(119889 119899 119902

1 119902

119901)) = 119889

119901

(79)

In particular for the de Bruijn digraph B(119889 119899) and the Kautzdigraph K(119889 119899)

119887119879(B (119889 119899)) = 119889 minus 1 119887

119879(K (119889 119899)) = 119889 (80)

Zhang et al [136] determined the bondage number incomplete 119905-partite digraphs

Theorem 188 (Zhang et al [136] 2009) For a complete 119905-partite digraph 119870

11989911198992119899119905

where 1198991⩽ 119899

2⩽ sdot sdot sdot ⩽ 119899

119905

119887 (11987011989911198992119899119905

)

=

119898 119894119891 119899119898= 1 119899

119898+1⩾ 2 119891119900119903 119904119900119898119890

119898 (1 ⩽ 119898 lt 119905)

4119905 minus 3 119894119891 1198991= 119899

2= sdot sdot sdot = 119899

119905= 2

119905minus1

sum

119894=1

119899119894+ 2 (119905 minus 1) 119900119905ℎ119890119903119908119894119904119890

(81)

Since an undirected graph can be thought of as a sym-metric digraph any result for digraphs has an analogy forundirected graphs in general In view of this point studyingthe bondage number for digraphs is more significant thanfor undirected graphs Thus we should further study thebondage number of digraphs and try to generalize knownresults on the bondage number and related variants for undi-rected graphs to digraphs prove or disprove Conjecture 183In particular determine the exact values of 119887(B(119889 119899)) for aneven 119899 and 119887

119879(119870

119899) for 119899 ⩾ 4

11 Efficient Dominating Sets

A dominating set 119878 of a graph 119866 is called to be efficient if forevery vertex 119909 in119866 |119873

119866[119909]cap119878| = 1 if119866 is a undirected graph

or |119873minus

119866[119909] cap 119878| = 1 if 119866 is a directed graph The concept of an

efficient set was proposed by Bange et al [137] in 1988 In theliterature an efficient set is also called a perfect dominatingset (see Livingston and Stout [138] for an explanation)

From definition if 119878 is an efficient dominating set of agraph 119866 then 119878 is certainly an independent set and everyvertex not in 119878 is adjacent to exactly one vertex in 119878

It is also clear from definition that a dominating set 119878 isefficient if and only if N

119866[119878] = 119873

119866[119909] 119909 isin 119878 for the

undirected graph 119866 or N+

119866[119878] = 119873

+

119866[119909] 119909 isin 119878 for the

digraph119866 is a partition of119881(119866) where the induced subgraphby119873

119866[119909] or119873+

119866[119909] is a star or an out-star with the root 119909

The efficient domination has important applications inmany areas such as error-correcting codes and receivesmuch attention in the late years

The concept of efficient dominating sets is a measureof the efficiency of domination in graphs Unfortunately asshown in [137] not every graph has an efficient dominatingset andmoreover it is anNP-complete problem to determinewhether a given graph has an efficient dominating set Inaddition it was shown by Clark [139] in 1993 that for a widerange of 119901 almost every random undirected graph 119866 isin

G(120592 119901) has no efficient dominating sets This means thatundirected graphs possessing an efficient dominating set arerare However it is easy to show that every undirected graphhas an orientationwith an efficient dominating set (see Bangeet al [140])

In 1993 Barkauskas and Host [141] showed that deter-mining whether an arbitrary oriented graph has an efficientdominating set is NP-complete Even so the existence ofefficient dominating sets for some special classes of graphs hasbeen examined see for example Dejter and Serra [142] andLee [143] for Cayley graph Gu et al [144] formeshes and toriObradovic et al [145] Huang and Xu [36] Reji Kumar andMacGillivray [146] for circulant graphs Harary graphs andtori and Van Wieren et al [147] for cube-connected cycles

In this section we introduce results of the bondagenumber for some graphs with efficient dominating sets dueto Huang and Xu [36]

111 Results for General Graphs In this subsection we intro-duce some results on bondage numbers obtained by applyingefficient dominating sets We first state the two followinglemmas

Lemma 189 For any 119896-regular graph or digraph 119866 of order 119899120574(119866) ⩾ 119899(119896 + 1) with equality if and only if 119866 has an efficientdominating set In addition if 119866 has an efficient dominatingset then every efficient dominating set is certainly a 120574-set andvice versa

Let 119890 be an edge and 119878 a dominating set in 119866 We say that119890 supports 119878 if 119890 isin (119878 119878) where (119878 119878) = (119909 119910) isin 119864(119866) 119909 isin 119878 119910 isin 119878 Denote by 119904(119866) the minimum number of edgeswhich support all 120574-sets in 119866

Lemma 190 For any graph or digraph 119866 119887(119866) ⩾ 119904(119866) withequality if 119866 is regular and has an efficient dominating set

30 International Journal of Combinatorics

A graph 119866 is called to be vertex-transitive if its automor-phism group Aut(119866) acts transitively on its vertex-set 119881(119866)A vertex-transitive graph is regular Applying Lemmas 189and 190 Huang and Xu obtained some results on bondagenumbers for vertex-transitive graphs or digraphs

Theorem 191 (Huang and Xu [36] 2008) For any vertex-transitive graph or digraph 119866 of order 119899

119887 (119866) ⩾

lceil119899

2120574 (119866)rceil 119894119891 119866 119894119904 119906119899119889119894119903119890119888119905119890119889

lceil119899

120574 (119866)rceil 119894119891 119866 119894119904 119889119894119903119890119888119905119890119889

(82)

Theorem192 (Huang andXu [36] 2008) Let119866 be a 119896-regulargraph of order 119899 Then

119887(119866) ⩽ 119896 if119866 is undirected and 119899 ⩾ 120574(119866)(119896+1)minus119896+1119887(119866) ⩽ 119896+1+ℓ if119866 is directed and 119899 ⩾ 120574(119866)(119896+1)minusℓwith 0 ⩽ ℓ ⩽ 119896 minus 1

Next we introduce a better upper bound on 119887(119866) To thisaim we need the following concept which is a generalizationof the concept of the edge-covering of a graph 119866 For 1198811015840

sube

119881(119866) and 1198641015840 sube 119864(119866) we say that 1198641015840covers 1198811015840 and call 1198641015840an edge-covering for 1198811015840 if there exists an edge (119909 119910) isin 1198641015840 forany vertex 119909 isin 1198811015840 For 119910 isin 119881(119866) let 1205731015840[119910] be the minimumcardinality over all edge-coverings for119873minus

119866[119910]

Theorem 193 (Huang and Xu [36] 2008) If 119866 is a 119896-regulargraph with order 120574(119866)(119896 + 1) then 119887(119866) ⩽ 1205731015840[119910] for any 119910 isin119881(119866)

Theupper bound on 119887(119866) given inTheorem 193 is tight inview of 119887(119862

119899) for a cycle or a directed cycle 119862

119899(seeTheorems

1 and 185 resp)It is easy to see that for a 119896-regular graph 119866 lceil(119896 + 1)2rceil ⩽

1205731015840

[119910] ⩽ 119896 when 119866 is undirected and 1205731015840[119910] = 119896 + 1 when 119866 isdirected By this fact and Lemma 189 the following theoremis merely a simple combination of Theorems 191 and 193

Theorem 194 (Huang and Xu [36] 2008) Let 119866 be a vertex-transitive graph of degree 119896 If 119866 has an efficient dominatingset then

lceil119896 + 1

2rceil ⩽ 119887 (119866) ⩽ 119896 119894119891 119866 119894119904 119906119899119889119894119903119890119888119905119890119889

119887 (119866) = 119896 + 1 119894119891 119866 119894119904 119889119894119903119890119888119905119890119889

(83)

Theorem 195 (Huang and Xu [36] 2008) If 119866 is an undi-rected vertex-transitive cubic graph with order 4120574(119866) and girth119892(119866) ⩽ 5 then 119887(119866) = 2

The proof of Theorem 195 leads to a byproduct If 119892 =5 let (119906

1 119906

2 119906

3 119906

4 119906

5) be a cycle and let D

119894be the family

of efficient dominating sets containing 119906119894for each 119894 isin

1 2 3 4 5 ThenD119894capD

119895= 0 for 119894 = 1 2 5 Then 119866 has

at least 5119904 efficient dominating sets where 119904 = 119904(119866) But thereare only 4s (=ns120574) distinct efficient dominating sets in119866This

contradiction implies that an undirected vertex-transitivecubic graph with girth five has no efficient dominating setsBut a similar argument for 119892(119866) = 3 4 or 119892(119866) ⩾ 6 couldnot give any contradiction This is consistent with the resultthat CCC(119899) a vertex-transitive cubic graph with girth 119899 if3 ⩽ 119899 ⩽ 8 or girth 8 if 119899 ⩾ 9 has efficient dominating setsfor all 119899 ⩾ 3 except 119899 = 5 (see Theorem 199 in the followingsubsection)

112 Results for Cayley Graphs In this subsection we useTheorem 194 to determine the exact values or approximativevalues of bondage numbers for some special vertex-transitivegraphs by characterizing the existence of efficient dominatingsets in these graphs

Let Γ be a nontrivial finite group 119878 a nonempty subset ofΓ without the identity element of Γ A digraph 119866 is defined asfollows

119881 (119866) = Γ (119909 119910) isin 119864 (119866) lArrrArr 119909minus1

119910 isin 119878

for any 119909 119910 isin Γ(84)

is called a Cayley digraph of the group Γ with respect to119878 denoted by 119862

Γ(119878) If 119878minus1 = 119904minus1 119904 isin 119878 = 119878 then

119862Γ(119878) is symmetric and is called a Cayley undirected graph

a Cayley graph for short Cayley graphs or digraphs arecertainly vertex-transitive (see Xu [1])

A circulant graph 119866(119899 119878) of order 119899 is a Cayley graph119862119885119899

(119878) where 119885119899= 0 119899 minus 1 is the addition group of

order 119899 and 119878 is a nonempty subset of119885119899without the identity

element and hence is a vertex-transitive digraph of degree|119878| If 119878minus1 = 119878 then119866(119899 119878) is an undirected graph If 119878 = 1 119896where 2 ⩽ 119896 ⩽ 119899 minus 2 we write 119866(119899 1 119896) for 119866(119899 1 119896) or119866(119899 plusmn1 plusmn119896) and call it a double-loop circulant graph

Huang and Xu [36] showed that for directed 119866 =

119866(119899 1 119896) lceil1198993rceil ⩽ 120574(119866) ⩽ lceil1198992rceil and 119866 has an efficientdominating set if and only if 3 | 119899 and 119896 equiv 2 (mod 3) fordirected 119866 = 119866(119899 1 119896) and 119896 = 1198992 lceil1198995rceil ⩽ 120574(119866) ⩽ lceil1198993rceiland 119866 has an efficient dominating set if and only if 5 | 119899 and119896 equiv plusmn2 (mod 5) By Theorem 194 we obtain the bondagenumber of a double loop circulant graph if it has an efficientdominating set

Theorem 196 (Huang and Xu [36] 2008) Let 119866 be a double-loop circulant graph 119866(119899 1 119896) If 119866 is directed with 3 | 119899and 119896 equiv 2 (mod 3) or 119866 is undirected with 5 | 119899 and119896 equiv plusmn2 (mod 5) then 119887(119866) = 3

The 119898 times 119899 torus is the Cartesian product 119862119898times 119862

119899of

two cycles and is a Cayley graph 119862119885119898times119885119899

(119878) where 119878 =

(0 1) (1 0) for directed cycles and 119878 = (0 plusmn1) (plusmn1 0) forundirected cycles and hence is vertex-transitive Gu et al[144] showed that the undirected torus119862

119898times119862

119899has an efficient

dominating set if and only if both 119898 and 119899 are multiplesof 5 Huang and Xu [36] showed that the directed torus119862119898times 119862

119899has an efficient dominating set if and only if both

119898 and 119899 are multiples of 3 Moreover they found a necessarycondition for a dominating set containing the vertex (0 0)in 119862

119898times 119862

119899to be efficient and obtained the following

result

International Journal of Combinatorics 31

Theorem 197 (Huang and Xu [36] 2008) Let 119866 = 119862119898times 119862

119899

If 119866 is undirected and both119898 and 119899 are multiples of 5 or if 119866 isdirected and both119898 and 119899 are multiples of 3 then 119887(119866) = 3

The hypercube 119876119899

is the Cayley graph 119862Γ(119878)

where Γ = 1198852times sdot sdot sdot times 119885

2= (119885

2)119899 and 119878 =

100 sdot sdot sdot 0 010 sdot sdot sdot 0 00 sdot sdot sdot 01 Lee [143] showed that119876119899has an efficient dominating set if and only if 119899 = 2119898 minus 1

for a positive integer119898 ByTheorem 194 the following resultis obtained immediately

Theorem 198 (Huang and Xu [36] 2008) If 119899 = 2119898 minus 1 for apositive integer119898 then 2119898minus1

⩽ 119887(119876119899) ⩽ 2

119898

minus 1

Problem 10 Determine the exact value of 119887(119876119899) for 119899 = 2119898minus1

The 119899-dimensional cube-connected cycle denoted byCCC(119899) is constructed from the 119899-dimensional hypercube119876119899by replacing each vertex 119909 in 119876

119899with an undirected cycle

119862119899of length 119899 and linking the 119894th vertex of the 119862

119899to the 119894th

neighbor of 119909 It has been proved that CCC(119899) is a Cayleygraph and hence is a vertex-transitive graph with degree 3Van Wieren et al [147] proved that CCC(119899) has an efficientdominating set if and only if 119899 = 5 From Theorems 194 and195 the following result can be derived

Theorem 199 (Huang and Xu [36] 2008) Let 119866 = 119862119862119862(119899)be the 119899-dimensional cube-connected cycles with 119899 ⩾ 3 and119899 = 5 Then 120574(119866) = 1198992119899minus2 and 2 ⩽ 119887(119866) ⩽ 3 In addition119887(119862119862119862(3)) = 119887(119862119862119862(4)) = 2

Problem 11 Determine the exact value of 119887(CCC(119899)) for 119899 ⩾5

Acknowledgments

This work was supported by NNSF of China (Grants nos11071233 61272008) The author would like to express hisgratitude to the anonymous referees for their kind sugges-tions and comments on the original paper to ProfessorSheikholeslami and Professor Jafari Rad for sending thereferences [128] and [81] which resulted in this version

References

[1] J-M Xu Theory and Application of Graphs vol 10 of NetworkTheory and Applications Kluwer Academic Dordrecht TheNetherlands 2003

[2] S THedetniemi andR C Laskar ldquoBibliography on dominationin graphs and some basic definitions of domination parame-tersrdquo Discrete Mathematics vol 86 no 1ndash3 pp 257ndash277 1990

[3] TW Haynes S T Hedetniemi and P J Slater Fundamentals ofDomination in Graphs vol 208 ofMonographs and Textbooks inPure and Applied Mathematics Marcel Dekker New York NYUSA 1998

[4] TWHaynes S THedetniemi and P J Slater EdsDominationin Graphs vol 209 of Monographs and Textbooks in Pure andAppliedMathematicsMarcelDekker NewYorkNYUSA 1998

[5] M R Garey andD S JohnsonComputers and Intractability WH Freeman and Company San Francisco Calif USA 1979

[6] H B Walikar and B D Acharya ldquoDomination critical graphsrdquoNational Academy Science Letters vol 2 pp 70ndash72 1979

[7] D Bauer F Harary J Nieminen and C L Suffel ldquoDominationalteration sets in graphsrdquo Discrete Mathematics vol 47 no 2-3pp 153ndash161 1983

[8] J F Fink M S Jacobson L F Kinch and J Roberts ldquoThebondage number of a graphrdquo Discrete Mathematics vol 86 no1ndash3 pp 47ndash57 1990

[9] F-T Hu and J-M Xu ldquoThe bondage number of (119899 minus 3)-regulargraph G of order nrdquo Ars Combinatoria to appear

[10] U Teschner ldquoNew results about the bondage number of agraphrdquoDiscreteMathematics vol 171 no 1ndash3 pp 249ndash259 1997

[11] B L Hartnell and D F Rall ldquoA bound on the size of a graphwith given order and bondage numberrdquo Discrete Mathematicsvol 197-198 pp 409ndash413 1999 16th British CombinatorialConference (London 1997)

[12] B L Hartnell and D F Rall ldquoA characterization of trees inwhich no edge is essential to the domination numberrdquo ArsCombinatoria vol 33 pp 65ndash76 1992

[13] J Topp and P D Vestergaard ldquo120572119896- and 120574

119896-stable graphsrdquo

Discrete Mathematics vol 212 no 1-2 pp 149ndash160 2000[14] A Meir and J W Moon ldquoRelations between packing and

covering numbers of a treerdquo Pacific Journal of Mathematics vol61 no 1 pp 225ndash233 1975

[15] B L Hartnell L K Joslashrgensen P D Vestergaard and CA Whitehead ldquoEdge stability of the k-domination numberof treesrdquo Bulletin of the Institute of Combinatorics and ItsApplications vol 22 pp 31ndash40 1998

[16] F-T Hu and J-M Xu ldquoOn the complexity of the bondage andreinforcement problemsrdquo Journal of Complexity vol 28 no 2pp 192ndash201 2012

[17] B L Hartnell and D F Rall ldquoBounds on the bondage numberof a graphrdquo Discrete Mathematics vol 128 no 1ndash3 pp 173ndash1771994

[18] Y-L Wang ldquoOn the bondage number of a graphrdquo DiscreteMathematics vol 159 no 1ndash3 pp 291ndash294 1996

[19] G H Fricke TW Haynes S M Hedetniemi S T Hedetniemiand R C Laskar ldquoExcellent treesrdquo Bulletin of the Institute ofCombinatorics and Its Applications vol 34 pp 27ndash38 2002

[20] V Samodivkin ldquoThe bondage number of graphs good and badverticesrdquoDiscussiones Mathematicae GraphTheory vol 28 no3 pp 453ndash462 2008

[21] J E Dunbar T W Haynes U Teschner and L VolkmannldquoBondage insensitivity and reinforcementrdquo in Domination inGraphs vol 209 of Monographs and Textbooks in Pure andAppliedMathematics pp 471ndash489 Dekker NewYork NY USA1998

[22] H Liu and L Sun ldquoThe bondage and connectivity of a graphrdquoDiscrete Mathematics vol 263 no 1ndash3 pp 289ndash293 2003

[23] A-H Esfahanian and S L Hakimi ldquoOn computing a con-ditional edge-connectivity of a graphrdquo Information ProcessingLetters vol 27 no 4 pp 195ndash199 1988

[24] R C Brigham P Z Chinn and R D Dutton ldquoVertexdomination-critical graphsrdquoNetworks vol 18 no 3 pp 173ndash1791988

[25] V Samodivkin ldquoDomination with respect to nondegenerateproperties bondage numberrdquoThe Australasian Journal of Com-binatorics vol 45 pp 217ndash226 2009

[26] U Teschner ldquoA new upper bound for the bondage numberof graphs with small domination numberrdquo The AustralasianJournal of Combinatorics vol 12 pp 27ndash35 1995

32 International Journal of Combinatorics

[27] J Fulman D Hanson and G MacGillivray ldquoVertex dom-ination-critical graphsrdquoNetworks vol 25 no 2 pp 41ndash43 1995

[28] U Teschner ldquoA counterexample to a conjecture on the bondagenumber of a graphrdquo Discrete Mathematics vol 122 no 1ndash3 pp393ndash395 1993

[29] U Teschner ldquoThe bondage number of a graph G can be muchgreater than Δ(G)rdquo Ars Combinatoria vol 43 pp 81ndash87 1996

[30] L Volkmann ldquoOn graphs with equal domination and coveringnumbersrdquoDiscrete AppliedMathematics vol 51 no 1-2 pp 211ndash217 1994

[31] L A Sanchis ldquoMaximum number of edges in connected graphswith a given domination numberrdquoDiscreteMathematics vol 87no 1 pp 65ndash72 1991

[32] S Klavzar and N Seifter ldquoDominating Cartesian products ofcyclesrdquoDiscrete AppliedMathematics vol 59 no 2 pp 129ndash1361995

[33] H K Kim ldquoA note on Vizingrsquos conjecture about the bondagenumberrdquo Korean Journal of Mathematics vol 10 no 2 pp 39ndash42 2003

[34] M Y Sohn X Yuan and H S Jeong ldquoThe bondage number of1198623times119862

119899rdquo Journal of the KoreanMathematical Society vol 44 no

6 pp 1213ndash1231 2007[35] L Kang M Y Sohn and H K Kim ldquoBondage number of the

discrete torus 119862119899times 119862

4rdquo Discrete Mathematics vol 303 no 1ndash3

pp 80ndash86 2005[36] J Huang and J-M Xu ldquoThe bondage numbers and efficient

dominations of vertex-transitive graphsrdquoDiscrete Mathematicsvol 308 no 4 pp 571ndash582 2008

[37] C J Xiang X Yuan andM Y Sohn ldquoDomination and bondagenumber of119862

3times119862

119899rdquoArs Combinatoria vol 97 pp 299ndash310 2010

[38] M S Jacobson and L F Kinch ldquoOn the domination numberof products of graphs Irdquo Ars Combinatoria vol 18 pp 33ndash441984

[39] Y-Q Li ldquoA note on the bondage number of a graphrdquo ChineseQuarterly Journal of Mathematics vol 9 no 4 pp 1ndash3 1994

[40] F-T Hu and J-M Xu ldquoBondage number of mesh networksrdquoFrontiers ofMathematics inChina vol 7 no 5 pp 813ndash826 2012

[41] R Frucht and F Harary ldquoOn the corona of two graphsrdquoAequationes Mathematicae vol 4 pp 322ndash325 1970

[42] K Carlson and M Develin ldquoOn the bondage number of planarand directed graphsrdquoDiscreteMathematics vol 306 no 8-9 pp820ndash826 2006

[43] U Teschner ldquoOn the bondage number of block graphsrdquo ArsCombinatoria vol 46 pp 25ndash32 1997

[44] J Huang and J-M Xu ldquoNote on conjectures of bondagenumbers of planar graphsrdquo Applied Mathematical Sciences vol6 no 65ndash68 pp 3277ndash3287 2012

[45] L Kang and J Yuan ldquoBondage number of planar graphsrdquoDiscrete Mathematics vol 222 no 1ndash3 pp 191ndash198 2000

[46] M Fischermann D Rautenbach and L Volkmann ldquoRemarkson the bondage number of planar graphsrdquo Discrete Mathemat-ics vol 260 no 1ndash3 pp 57ndash67 2003

[47] J Hou G Liu and J Wu ldquoBondage number of planar graphswithout small cyclesrdquoUtilitasMathematica vol 84 pp 189ndash1992011

[48] J Huang and J-M Xu ldquoThe bondage numbers of graphs withsmall crossing numbersrdquo Discrete Mathematics vol 307 no 15pp 1881ndash1897 2007

[49] Q Ma S Zhang and J Wang ldquoBondage number of 1-planargraphrdquo Applied Mathematics vol 1 no 2 pp 101ndash103 2010

[50] J Hou and G Liu ldquoA bound on the bondage number of toroidalgraphsrdquoDiscreteMathematics Algorithms and Applications vol4 no 3 Article ID 1250046 5 pages 2012

[51] Y-C Cao J-M Xu and X-R Xu ldquoOn bondage number oftoroidal graphsrdquo Journal of University of Science and Technologyof China vol 39 no 3 pp 225ndash228 2009

[52] Y-C Cao J Huang and J-M Xu ldquoThe bondage number ofgraphs with crossing number less than fourrdquo Ars Combinatoriato appear

[53] H K Kim ldquoOn the bondage numbers of some graphsrdquo KoreanJournal of Mathematics vol 11 no 2 pp 11ndash15 2004

[54] A Gagarin and V Zverovich ldquoUpper bounds for the bondagenumber of graphs on topological surfacesrdquo Discrete Mathemat-ics 2012

[55] J Huang ldquoAn improved upper bound for the bondage numberof graphs on surfacesrdquoDiscrete Mathematics vol 312 no 18 pp2776ndash2781 2012

[56] A Gagarin and V Zverovich ldquoThe bondage number of graphson topological surfaces and Teschnerrsquos conjecturerdquo DiscreteMathematics vol 313 no 6 pp 796ndash808 2013

[57] V Samodivkin ldquoNote on the bondage number of graphs ontopological surfacesrdquo httparxivorgabs12086203

[58] E J Cockayne R M Dawes and S T Hedetniemi ldquoTotaldomination in graphsrdquoNetworks vol 10 no 3 pp 211ndash219 1980

[59] R Laskar J Pfaff S M Hedetniemi and S T HedetniemildquoOn the algorithmic complexity of total dominationrdquo Society forIndustrial and Applied Mathematics vol 5 no 3 pp 420ndash4251984

[60] J Pfaff R C Laskar and S T Hedetniemi ldquoNP-completeness oftotal and connected domination and irredundance for bipartitegraphsrdquo Tech Rep 428 Department of Mathematical SciencesClemson University 1983

[61] M A Henning ldquoA survey of selected recent results on totaldomination in graphsrdquoDiscrete Mathematics vol 309 no 1 pp32ndash63 2009

[62] V R Kulli and D K Patwari ldquoThe total bondage number ofa graphrdquo in Advances in Graph Theory pp 227ndash235 VishwaGulbarga India 1991

[63] F-T Hu Y Lu and J-M Xu ldquoThe total bondage number ofgrid graphsrdquo Discrete Applied Mathematics vol 160 no 16-17pp 2408ndash2418 2012

[64] J Cao M Shi and B Wu ldquoResearch on the total bondagenumber of a spectial networkrdquo in Proceedings of the 3rdInternational Joint Conference on Computational Sciences andOptimization (CSO rsquo10) pp 456ndash458 May 2010

[65] N Sridharan MD Elias and V S A Subramanian ldquoTotalbondage number of a graphrdquo AKCE International Journal ofGraphs and Combinatorics vol 4 no 2 pp 203ndash209 2007

[66] N Jafari Rad and J Raczek ldquoSome progress on total bondage ingraphsrdquo Graphs and Combinatorics 2011

[67] T W Haynes and P J Slater ldquoPaired-domination and thepaired-domatic numberrdquoCongressusNumerantium vol 109 pp65ndash72 1995

[68] T W Haynes and P J Slater ldquoPaired-domination in graphsrdquoNetworks vol 32 no 3 pp 199ndash206 1998

[69] X Hou ldquoA characterization of 2120574 120574119875-treesrdquoDiscrete Mathemat-

ics vol 308 no 15 pp 3420ndash3426 2008[70] K E Proffitt T W Haynes and P J Slater ldquoPaired-domination

in grid graphsrdquo Congressus Numerantium vol 150 pp 161ndash1722001

International Journal of Combinatorics 33

[71] H Qiao L KangM Cardei andD-Z Du ldquoPaired-dominationof treesrdquo Journal of Global Optimization vol 25 no 1 pp 43ndash542003

[72] E Shan L Kang and M A Henning ldquoA characterizationof trees with equal total domination and paired-dominationnumbersrdquo The Australasian Journal of Combinatorics vol 30pp 31ndash39 2004

[73] J Raczek ldquoPaired bondage in treesrdquo Discrete Mathematics vol308 no 23 pp 5570ndash5575 2008

[74] J A Telle and A Proskurowski ldquoAlgorithms for vertex parti-tioning problems on partial k-treesrdquo SIAM Journal on DiscreteMathematics vol 10 no 4 pp 529ndash550 1997

[75] G S Domke J H Hattingh M A Henning and L R MarkusldquoRestrained domination in treesrdquoDiscreteMathematics vol 211no 1ndash3 pp 1ndash9 2000

[76] G S Domke J H Hattingh M A Henning and L R MarkusldquoRestrained domination in graphs with minimum degree twordquoJournal of Combinatorial Mathematics and Combinatorial Com-puting vol 35 pp 239ndash254 2000

[77] G S Domke J H Hattingh S T Hedetniemi R C Laskarand L R Markus ldquoRestrained domination in graphsrdquo DiscreteMathematics vol 203 no 1ndash3 pp 61ndash69 1999

[78] J H Hattingh and E J Joubert ldquoAn upper bound for therestrained domination number of a graph with minimumdegree at least two in terms of order and minimum degreerdquoDiscrete Applied Mathematics vol 157 no 13 pp 2846ndash28582009

[79] M A Henning ldquoGraphs with large restrained dominationnumberrdquo Discrete Mathematics vol 197-198 pp 415ndash429 199916th British Combinatorial Conference (London 1997)

[80] J H Hattingh and A R Plummer ldquoRestrained bondage ingraphsrdquo Discrete Mathematics vol 308 no 23 pp 5446ndash54532008

[81] N Jafari Rad R Hasni J Raczek and L Volkmann ldquoTotalrestrained bondage in graphsrdquoActaMathematica Sinica vol 29no 6 pp 1033ndash1042 2013

[82] J Cyman and J Raczek ldquoOn the total restrained dominationnumber of a graphrdquoThe Australasian Journal of Combinatoricsvol 36 pp 91ndash100 2006

[83] M A Henning ldquoGraphs with large total domination numberrdquoJournal of Graph Theory vol 35 no 1 pp 21ndash45 2000

[84] D-X Ma X-G Chen and L Sun ldquoOn total restraineddomination in graphsrdquoCzechoslovakMathematical Journal vol55 no 1 pp 165ndash173 2005

[85] B Zelinka ldquoRemarks on restrained domination and totalrestrained domination in graphsrdquo Czechoslovak MathematicalJournal vol 55 no 2 pp 393ndash396 2005

[86] W Goddard and M A Henning ldquoIndependent domination ingraphs a survey and recent resultsrdquo Discrete Mathematics vol313 no 7 pp 839ndash854 2013

[87] J H Zhang H L Liu and L Sun ldquoIndependence bondagenumber and reinforcement number of some graphsrdquo Transac-tions of Beijing Institute of Technology vol 23 no 2 pp 140ndash1422003

[88] K D Dharmalingam Studies in Graph Theorymdashequitabledomination and bottleneck domination [PhD thesis] MaduraiKamaraj University Madurai India 2006

[89] GDeepakND Soner andAAlwardi ldquoThe equitable bondagenumber of a graphrdquo Research Journal of Pure Algebra vol 1 no9 pp 209ndash212 2011

[90] E Sampathkumar and H B Walikar ldquoThe connected domina-tion number of a graphrdquo Journal of Mathematical and PhysicalSciences vol 13 no 6 pp 607ndash613 1979

[91] K White M Farber and W Pulleyblank ldquoSteiner trees con-nected domination and strongly chordal graphsrdquoNetworks vol15 no 1 pp 109ndash124 1985

[92] M Cadei M X Cheng X Cheng and D-Z Du ldquoConnecteddomination in Ad Hoc wireless networksrdquo in Proceedings ofthe 6th International Conference on Computer Science andInformatics (CSampI rsquo02) 2002

[93] J F Fink and M S Jacobson ldquon-domination in graphsrdquo inGraph Theory with Applications to Algorithms and ComputerScience Wiley-Interscience Publications pp 283ndash300 WileyNew York NY USA 1985

[94] M Chellali O Favaron A Hansberg and L Volkmann ldquok-domination and k-independence in graphs a surveyrdquo Graphsand Combinatorics vol 28 no 1 pp 1ndash55 2012

[95] Y Lu X Hou J-M Xu andN Li ldquoTrees with uniqueminimump-dominating setsrdquo Utilitas Mathematica vol 86 pp 193ndash2052011

[96] Y Lu and J-M Xu ldquoThe p-bondage number of treesrdquo Graphsand Combinatorics vol 27 no 1 pp 129ndash141 2011

[97] Y Lu and J-M Xu ldquoThe 2-domination and 2-bondage numbersof grid graphs A manuscriptrdquo httparxivorgabs12044514

[98] M Krzywkowski ldquoNon-isolating 2-bondage in graphsrdquo TheJournal of the Mathematical Society of Japan vol 64 no 4 2012

[99] M A Henning O R Oellermann and H C Swart ldquoBoundson distance domination parametersrdquo Journal of CombinatoricsInformation amp System Sciences vol 16 no 1 pp 11ndash18 1991

[100] F Tian and J-M Xu ldquoBounds for distance domination numbersof graphsrdquo Journal of University of Science and Technology ofChina vol 34 no 5 pp 529ndash534 2004

[101] F Tian and J-M Xu ldquoDistance domination-critical graphsrdquoApplied Mathematics Letters vol 21 no 4 pp 416ndash420 2008

[102] F Tian and J-M Xu ldquoA note on distance domination numbersof graphsrdquo The Australasian Journal of Combinatorics vol 43pp 181ndash190 2009

[103] F Tian and J-M Xu ldquoAverage distances and distance domina-tion numbersrdquo Discrete Applied Mathematics vol 157 no 5 pp1113ndash1127 2009

[104] Z-L Liu F Tian and J-M Xu ldquoProbabilistic analysis of upperbounds for 2-connected distance k-dominating sets in graphsrdquoTheoretical Computer Science vol 410 no 38ndash40 pp 3804ndash3813 2009

[105] M A Henning O R Oellermann and H C Swart ldquoDistancedomination critical graphsrdquo Journal of Combinatorial Mathe-matics and Combinatorial Computing vol 44 pp 33ndash45 2003

[106] T J Bean M A Henning and H C Swart ldquoOn the integrityof distance domination in graphsrdquo The Australasian Journal ofCombinatorics vol 10 pp 29ndash43 1994

[107] M Fischermann and L Volkmann ldquoA remark on a conjecturefor the (kp)-domination numberrdquoUtilitasMathematica vol 67pp 223ndash227 2005

[108] T Korneffel D Meierling and L Volkmann ldquoA remark on the(22)-domination numberrdquo Discussiones Mathematicae GraphTheory vol 28 no 2 pp 361ndash366 2008

[109] Y Lu X Hou and J-M Xu ldquoOn the (22)-domination numberof treesrdquo Discussiones Mathematicae Graph Theory vol 30 no2 pp 185ndash199 2010

34 International Journal of Combinatorics

[110] H Li and J-M Xu ldquo(dm)-domination numbers of m-connected graphsrdquo Rapports de Recherche 1130 LRI UniversiteParis-Sud 1997

[111] S M Hedetniemi S T Hedetniemi and T V Wimer ldquoLineartime resource allocation algorithms for treesrdquo Tech Rep URI-014 Department of Mathematical Sciences Clemson Univer-sity 1987

[112] M Farber ldquoDomination independent domination and dualityin strongly chordal graphsrdquo Discrete Applied Mathematics vol7 no 2 pp 115ndash130 1984

[113] G S Domke and R C Laskar ldquoThe bondage and reinforcementnumbers of 120574

119891for some graphsrdquoDiscrete Mathematics vol 167-

168 pp 249ndash259 1997[114] G S Domke S T Hedetniemi and R C Laskar ldquoFractional

packings coverings and irredundance in graphsrdquo vol 66 pp227ndash238 1988

[115] E J Cockayne P A Dreyer Jr S M Hedetniemi andS T Hedetniemi ldquoRoman domination in graphsrdquo DiscreteMathematics vol 278 no 1ndash3 pp 11ndash22 2004

[116] I Stewart ldquoDefend the roman empirerdquo Scientific American vol281 pp 136ndash139 1999

[117] C S ReVelle and K E Rosing ldquoDefendens imperiumromanum a classical problem in military strategyrdquoThe Ameri-can Mathematical Monthly vol 107 no 7 pp 585ndash594 2000

[118] A Bahremandpour F-T Hu S M Sheikholeslami and J-MXu ldquoOn the Roman bondage number of a graphrdquo DiscreteMathematics Algorithms and Applications vol 5 no 1 ArticleID 1350001 15 pages 2013

[119] N Jafari Rad and L Volkmann ldquoRoman bondage in graphsrdquoDiscussiones Mathematicae Graph Theory vol 31 no 4 pp763ndash773 2011

[120] K Ebadi and L PushpaLatha ldquoRoman bondage and Romanreinforcement numbers of a graphrdquo International Journal ofContemporary Mathematical Sciences vol 5 pp 1487ndash14972010

[121] N Dehgardi S M Sheikholeslami and L Volkman ldquoOn theRoman k-bondage number of a graphrdquo AKCE InternationalJournal of Graphs and Combinatorics vol 8 pp 169ndash180 2011

[122] S Akbari and S Qajar ldquoA note on Roman bondage number ofgraphsrdquo Ars Combinatoria to Appear

[123] F-T Hu and J-M Xu ldquoRoman bondage numbers of somegraphsrdquo httparxivorgabs11093933

[124] N Jafari Rad and L Volkmann ldquoOn the Roman bondagenumber of planar graphsrdquo Graphs and Combinatorics vol 27no 4 pp 531ndash538 2011

[125] S Akbari M Khatirinejad and S Qajar ldquoA note on the romanbondage number of planar graphsrdquo Graphs and Combinatorics2012

[126] B Bresar M A Henning and D F Rall ldquoRainbow dominationin graphsrdquo Taiwanese Journal of Mathematics vol 12 no 1 pp213ndash225 2008

[127] B L Hartnell and D F Rall ldquoOn dominating the Cartesianproduct of a graph and 120572krdquo Discussiones Mathematicae GraphTheory vol 24 no 3 pp 389ndash402 2004

[128] N Dehgardi S M Sheikholeslami and L Volkman ldquoThe k-rainbow bondage number of a graphrdquo to appear in DiscreteApplied Mathematics

[129] J von Neumann and O Morgenstern Theory of Games andEconomic Behavior Princeton University Press Princeton NJUSA 1944

[130] C BergeTheorıe des Graphes et ses Applications 1958[131] O Ore Theory of Graphs vol 38 American Mathematical

Society Colloquium Publications 1962[132] E Shan and L Kang ldquoBondage number in oriented graphsrdquoArs

Combinatoria vol 84 pp 319ndash331 2007[133] J Huang and J-M Xu ldquoThe bondage numbers of extended

de Bruijn and Kautz digraphsrdquo Computers amp Mathematics withApplications vol 51 no 6-7 pp 1137ndash1147 2006

[134] J Huang and J-M Xu ldquoThe total domination and total bondagenumbers of extended de Bruijn and Kautz digraphsrdquoComputersamp Mathematics with Applications vol 53 no 8 pp 1206ndash12132007

[135] Y Shibata and Y Gonda ldquoExtension of de Bruijn graph andKautz graphrdquo Computers amp Mathematics with Applications vol30 no 9 pp 51ndash61 1995

[136] X Zhang J Liu and JMeng ldquoThebondage number in completet-partite digraphsrdquo Information Processing Letters vol 109 no17 pp 997ndash1000 2009

[137] D W Bange A E Barkauskas and P J Slater ldquoEfficient dom-inating sets in graphsrdquo in Applications of Discrete Mathematicspp 189ndash199 SIAM Philadelphia Pa USA 1988

[138] M Livingston and Q F Stout ldquoPerfect dominating setsrdquoCongressus Numerantium vol 79 pp 187ndash203 1990

[139] L Clark ldquoPerfect domination in random graphsrdquo Journal ofCombinatorial Mathematics and Combinatorial Computing vol14 pp 173ndash182 1993

[140] D W Bange A E Barkauskas L H Host and L H ClarkldquoEfficient domination of the orientations of a graphrdquo DiscreteMathematics vol 178 no 1ndash3 pp 1ndash14 1998

[141] A E Barkauskas and L H Host ldquoFinding efficient dominatingsets in oriented graphsrdquo Congressus Numerantium vol 98 pp27ndash32 1993

[142] I J Dejter and O Serra ldquoEfficient dominating sets in CayleygraphsrdquoDiscrete AppliedMathematics vol 129 no 2-3 pp 319ndash328 2003

[143] J Lee ldquoIndependent perfect domination sets in Cayley graphsrdquoJournal of Graph Theory vol 37 no 4 pp 213ndash219 2001

[144] WGu X Jia and J Shen ldquoIndependent perfect domination setsin meshes tori and treesrdquo Unpublished

[145] N Obradovic J Peters and G Ruzic ldquoEfficient domination incirculant graphswith two chord lengthsrdquo Information ProcessingLetters vol 102 no 6 pp 253ndash258 2007

[146] K Reji Kumar and G MacGillivray ldquoEfficient domination incirculant graphsrdquo Discrete Mathematics vol 313 no 6 pp 767ndash771 2013

[147] D Van Wieren M Livingston and Q F Stout ldquoPerfectdominating sets on cube-connected cyclesrdquo Congressus Numer-antium vol 97 pp 51ndash70 1993

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

14 International Journal of Combinatorics

It follows fromTheorems 16 and 18 that

119887 (119866) ⩽ 119861 (119866) (24)

The proofs given in Theorems 74 and 79 indeed imply thefollowing stronger results

Theorem 83 If 119866 is a connected planar graph then

119861 (119866) ⩽

min 8 Δ (119866) + 26 119894119891 119892 (119866) ⩾ 4

5 119894119891 119892 (119866) ⩾ 5

4 119894119891 119892 (119866) ⩾ 6

3 119894119891 119892 (119866) ⩾ 8

(25)

Thus using Theorem 16 or Theorem 18 if we can proveConjectures 73 and 82 then we can prove the followingstatement

Statement If 119866 is a connected planar graph then

119861 (119866) ⩽

Δ (119866) + 1

7

5 if 119892 (119866) ⩾ 44 if 119892 (119866) ⩾ 5

(26)

It follows from (24) that Statement implies Conjectures 73and 82 However Huang and Xu [44] gave examples to showthat none of conclusions in Statement is true As a result theystated the following conclusion

Theorem 84 (Huang and Xu [44] 2012) It is not possible toprove Conjectures 73 and 82 if they are right using Theorems16 and 18

Therefore a new method is needed to prove or disprovethese conjectures At the present time one cannot prove theseconjectures and may consider to disprove them If one ofthese conjectures is invalid then there exists a minimumcounterexample 119866 with respect to 120592(119866) + 120576(119866) In order toobtain a minimum counterexample one may consider twosimple operations satisfying these requirements the edge-deletion and the edge-contractionwhich decrease 120592(119866)+120576(119866)and preserve planarity However applying Theorems 71 and72 Huang and Xu [44] presented the following results

Theorem 85 (Huang and Xu [44] 2012) It is not possible toconstruct minimum counterexamples to Conjectures 73 and 82by the operation of an edge-deletion or an edge-contraction

7 Results on Crossing Number Restraints

It is quite natural to generalize the known results on thebondage number for planar graphs to formore general graphsin terms of graph-theoretical parameters In this section weconsider graphs with crossing number restraints

The crossing number cr(119866) of 119866 is the smallest numberof pairwise intersections of its edges when 119866 is drawn in theplane If cr(119866) = 0 then 119866 is a planar graph

71 General Methods Use 119899119894to denote the number of vertices

of degree 119894 in119866 for 119894 = 1 2 Δ(119866) Huang and Xu obtainedthe following results

Theorem 86 (Huang and Xu [48] 2007) For any connectedgraph 119866

119887 (119866) ⩽

6 119894119891 119892 (119866) ⩾ 4 119886119899119889 2cr (119866) lt 121198861

5 119894119891 119892 (119866) ⩾ 5 119886119899119889 6cr (119866) lt 161198862

4 119894119891 119892 (119866) ⩾ 6 119886119899119889 4cr (119866) lt 141198863

3 119894119891 119892 (119866) ⩾ 8 119886119899119889 6cr (119866) lt 161198864

(27)

where 1198861= 119899

1+ 2119899

2+ 2119899

3+sum

Δ

119894=8(119894 minus 7)119899

119894+ 8 119886

2= 3119899

1+ 6119899

2+

51198993+sum

Δ

119894=7(3119894minus18)119899

119894+20 119886

3= 119899

1+2119899

2+sum

Δ

119894=6(2119894minus10)119899

119894+12

and 1198864= sum

Δ

119894=5(3119894 minus 12)119899

119894+ 16

Simplifying the conditions in Theorem 86 we obtain thefollowing corollaries immediately

Corollary 87 (Huang and Xu [48] 2007) For any connectedgraph 119866

119887 (119866) ⩽

6 119894119891 119892 (119866) ⩾ 4 119886119899119889 cr (119866) ⩽ 3

5 119894119891 119892 (119866) ⩾ 5 119886119899119889 cr (119866) ⩽ 4

4 119894119891 119892 (119866) ⩾ 6 119886119899119889 cr (119866) ⩽ 2

3 119894119891 119892 (119866) ⩾ 8 119886119899119889 cr (119866) ⩽ 2

(28)

Corollary 88 (Huang and Xu [48] 2007) For any connectedgraph 119866

(a) 119887(119866) ⩽ 6 if 119866 is not 4-regular cr(119866) = 4 and 119892(119866) ⩾ 4

(b) 119887(119866) ⩽ Δ(119866) + 1 if 119866 is not 3-regular cr(119866) ⩽ 4 and119892(119866) ⩾ 5

(c) 119887(119866) ⩽ 4 if 119866 is not 3-regular cr(119866) = 3 and 119892(119866) ⩾ 6

(d) 119887(119866) ⩽ 3 if cr(119866) = 3 119892(119866) ⩾ 8 and Δ(119866) ⩾ 5

These corollaries generalize some known results for pla-nar graphs For example Corollary 87 contains Theorem 79Corollary 88 (b) contains Corollary 80

Theorem 89 (Huang and Xu [48] 2007) For any connectedgraph 119866 if cr(119866) ⩽ 119899

3(119866) + 119899

4(119866) + 3 then 119887(119866) ⩽ 8

Corollary 90 (Huang and Xu [48] 2007) For any connectedgraph 119866 if cr(119866) ⩽ 3 then 119887(119866) ⩽ 8

Perhaps being unaware of this result in 2010 Ma et al[49] proved that 119887(119866) ⩽ 12 for any graph 119866 with cr(119866) = 1

Theorem91 (Huang and Xu [48] 2007) Let119866 be a connectedgraph and 119868 = 119909 isin 119881(119866) 119889

119866(119909) = 5 119889

119866(119909 119910) ⩾ 3 if

International Journal of Combinatorics 15

119889119866(119910) ⩽ 3 and 119889

119866(119910) = 4 for every 119910 isin 119873

119866(119909) If 119868 is

independent no vertices adjacent to vertices of degree 6 and

cr (119866) lt max5119899

3+ |119868| minus 2119899

4+ 28

117119899

3+ 40

16 (29)

then 119887(119866) ⩽ 7

Corollary 92 (Huang and Xu [48] 2007) Let 119866 be aconnected graph with cr(119866) ⩽ 2 If 119868 = 119909 isin 119881(119866) 119889

119866(119909) =

5 119889119866(119910 119909) ⩾ 3 if 119889

119866(119910) ⩽ 3 and 119889

119866(119910) = 4 for every 119910 isin

119873119866(119909) is independent and no vertices adjacent to vertices of

degree 6 then 119887(119866) ⩽ 7

Theorem93 (Huang andXu [48] 2007) Let119866 be a connectedgraph If 119866 satisfies

(a) 5cr(119866) + 1198995lt 2119899

2+ 3119899

3+ 2119899

4+ 12 or

(b) 7cr(119866) + 21198995lt 3119899

2+ 4119899

4+ 24

then 119887(119866) ⩽ 7

Proposition 94 (Huang and Xu [48] 2007) Let 119866 be aconnected graph with no vertices of degrees four and five Ifcr(119866) ⩽ 2 then 119887(119866) ⩽ 6

Theorem 95 (Huang and Xu [48] 2007) If 119866 is a connectedgraph with cr(119866) ⩽ 4 and not 4-regular when cr(119866) = 4 then119887(119866) ⩽ Δ(119866) + 2

The above results generalize some known results forplanar graphs For example Corollary 90 and Theorem 95contain Theorem 74 the first condition in Theorem 93 con-tains the second condition inTheorem 77

From Corollary 88 and Theorem 95 we suggest the fol-lowing questions

Question 2 Is there a

(a) 4-regular graph 119866 with cr(119866) = 4 such that 119887(119866) ⩾ 7

(b) 3-regular graph 119866 with cr ⩽ 4 and 119892(119866) ⩾ 5 such that119887(119866) = 5

(c) 3-regular graph 119866 with cr = 3 and 119892(119866) = 6 or 7 suchthat 119887(119866) = 5

72 Carlson-Develin Methods In this subsection we intro-duce an elegant method presented by Carlson and Develin[42] to obtain some upper bounds for the bondage numberof a graph

Suppose that 119866 is a connected graph We say that 119866 hasgenus 120588 if 119866 can be embedded in a surface 119878 with 120588 handlessuch that edges are pairwise disjoint except possibly for end-vertices Let119866 be an embedding of119866 in surface 119878 and let120601(119866)denote the number of regions in 119866 The boundary of everyregion contains at least three edges and every edge is on theboundary of atmost two regions (the two regions are identicalwhen 119890 is a cut-edge) For any edge 119890 of 119866 let 1199031

119866(119890) and 1199032

119866(119890)

be the numbers of edges comprising the regions in 119866 whichthe edge 119890 borders It is clear that every 119890 = 119909119910 isin 119864(119866)

1199032

119866(119890) ⩾ 119903

1

119866(119890) ⩾ 4 if 1003816100381610038161003816119873119866

(119909) cap 119873119866(119910)1003816100381610038161003816 = 0

1199032

119866(119890) ⩾ 4 119903

1

119866(119890) ⩾ 3 if 1003816100381610038161003816119873119866

(119909) cap 119873119866(119910)1003816100381610038161003816 = 1

1199032

119866(119890) ⩾ 119903

1

119866(119890) ⩾ 3 if 1003816100381610038161003816119873119866

(119909) cap 119873119866(119910)1003816100381610038161003816 ⩾ 2

(30)

Following Carlson and Develin [42] for any edge 119890 = 119909119910 of119866 we define

119863119866(119890) =

1

119889119866(119909)+

1

119889119866(119910)

+1

1199031

119866(119890)+

1

1199032

119866(119890)minus 1 (31)

By the well-known Eulerrsquos formula

120592 (119866) minus 120576 (119866) + 120601 (119866) = 2 minus 2120588 (119866) (32)

it is easy to see that

sum

119890isin119864(119866)

119863119866(119890) = 120592 (119866) minus 120576 (119866) + 120601 (119866) = 2 minus 2120588 (119866) (33)

If 119866 is a planar graph that is 120588(119866) = 0 then

sum

119890isin119864(119866)

119863119866(119890) = 120592 (119866) minus 120576 (119866) + 120601 (119866) = 2 (34)

Combining these formulas withTheorem 18 Carlson andDevelin [42] gave a simple and intuitive proof ofTheorem 74and obtained the following result

Theorem 96 (Carlson and Develin [42] 2006) Let 119866 be aconnected graph which can be embedded on a torus Then119887(119866) ⩽ Δ(119866) + 3

Recently Hou and Liu [50] improved Theorem 96 asfollows

Theorem 97 (Hou and Liu [50] 2012) Let 119866 be a connectedgraph which can be embedded on a torus Then 119887(119866) ⩽ 9Moreover if Δ(119866) = 6 then 119887(119866) ⩽ 8

By the way Cao et al [51] generalized the result inTheorem 79 to a connected graph 119866 that can be embeddedon a torus that is

119887 (119866) le

6 if 119892 (119866) ⩾ 4 and 119866 is not 4-regular5 if 119892 (119866) ⩾ 54 if 119892 (119866) ⩾ 6 and 119866 is not 3-regular3 if 119892 (119866) ⩾ 8

(35)

Several authors used this method to obtain some resultson the bondage number For example Fischermann et al[46] used this method to prove the second conclusion inTheorem 78 Recently Cao et al [52] have used thismethod todeal with more general graphs with small crossing numbersFirst they found the following property

Lemma 98 120575(119866) ⩽ 5 for any graph 119866 with cr(119866) ⩽ 5

16 International Journal of Combinatorics

This result implies that 119887(119866) ⩽ Δ(119866) + 4 by Corollary 15for any graph 119866 with cr(119866) ⩽ 5 Cao et al established thefollowing relation in terms of maximum planar subgraphs

Lemma 99 (Cao et al [52]) Let 119866 be a connected graph withcrossing number cr(119866) For any maximum planar subgraph119867of 119866

sum

119890isin119864(119867)

119863119866(119890) ⩾ 2 minus

2cr (119866)120575 (119866)

(36)

Using Carlson-Develin method and combiningLemma 98 with Lemma 99 Cao et al proved the followingresults

Theorem 100 (Cao et al [52]) For any connected graph 119866119887(119866) ⩽ Δ(119866) + 2 if 119866 satisfies one of the following conditions

(a) cr(119866) ⩽ 3(b) cr(119866) = 4 and 119866 is not 4-regular(c) cr(119866) = 5 and 119866 contains no vertices of degree 4

Theorem101 (Cao et al [52]) Let119866 be a connected graphwithΔ(119866) = 5 and cr(119866) ⩽ 4 If no triangles contain two vertices ofdegree 5 then 119887(119866) ⩽ 6 = Δ(119866) + 1

Theorem 102 (Cao et al [52]) Let 119866 be a connected graphwith Δ(119866) ⩾ 6 and cr(119866) ⩽ 3 If Δ(119866) ⩾ 7 or if Δ(119866) = 6120575(119866) = 3 and every edge 119890 = 119909119910 with 119889

119866(119909) = 5 and 119889

119866(119910) = 6

is contained in atmost one triangle then 119887(119866) ⩽ min8 Δ(119866)+1

Using Carlson-Develin method it can be proved that119887(119866) ⩽ Δ(119866) + 2 if cr(119866) ⩽ 3 (see the first conclusion inTheorem 100) and not yet proved that 119887(119866) ⩽ 8 if cr(119866) ⩽3 although it has been proved by using other method (seeCorollary 90)

By using Carlson-Develin method Samodivkin [25]obtained some results on the bondage number for graphswithsome given properties

Kim [53] showed that 119887(119866) ⩽ Δ(119866) + 2 for a connectedgraph 119866 with genus 1 Δ(119866) ⩽ 5 and having a toroidalembedding of which at least one region is not 4-sidedRecently Gagarin and Zverovich [54] further have extendedCarlson-Develin ideas to establish a nice upper bound forarbitrary graphs that can be embedded on orientable ornonorientable topological surface

Theorem 103 (Gagarin and Zverovich [54] 2012) Let 119866 bea graph embeddable on an orientable surface of genus ℎ and anonorientable surface of genus 119896 Then 119887(119866) ⩽ minΔ(119866)+ℎ+2 Δ(119866) + 119896 + 1

This result generalizes the corresponding upper boundsin Theorems 74 and 96 for any orientable or nonori-entable topological surface By investigating the proof ofTheorem 103 Huang [55] found that the issue of the ori-entability can be avoided by using the Euler characteristic120594(= 120592(119866) minus 120576(119866) + 120601(119866)) instead of the genera ℎ and 119896 the

relations are 120594 = 2minus2ℎ and 120594 = 2minus119896 To obtain the best resultfromTheorem 103 onewants ℎ and 119896 as small as possible thisis equivalent to having 120594 as large as possible

According toTheorem 103 if119866 is planar (ℎ = 0 120594 = 2) orcan be embedded on the real projective plane (119896 = 1 120594 = 1)then 119887(119866) ⩽ Δ(119866) + 2 In all other cases Huang [55] had thefollowing improvement for Theorem 103 the proof is basedon the technique developed byCarlson-Develin andGagarin-Zverovich and includes some elementary calculus as a newingredient mainly the intermediate-value theorem and themean-value theorem

Theorem 104 (Huang [55] 2012) Let 119866 be a graph embed-dable on a surface whose Euler characteristic 120594 is as large aspossible If 120594 ⩽ 0 then 119887(119866) ⩽ Δ(119866)+lfloor119903rfloor where 119903 is the largestreal root of the following cubic equation in 119911

1199113

+ 21199112

+ (6120594 minus 7) 119911 + 18120594 minus 24 = 0 (37)

In addition if 120594 decreases then 119903 increases

The following result is asymptotically equivalent toTheorem 104

Theorem 105 (Huang [55] 2012) Let 119866 be a graph embed-dable on a surface whose Euler characteristic 120594 is as large aspossible If 120594 ⩽ 0 then 119887(119866) lt Δ(119866) + radic12 minus 6120594 + 12 orequivalently 119887(119866) ⩽ Δ(119866) + lceilradic12 minus 6120594 minus 12rceil

Also Gagarin andZverovich [54] indicated that the upperbound in Theorem 103 can be improved for larger values ofthe genera ℎ and 119896 by adjusting the proofs and stated thefollowing general conjecture

Conjecture 106 (Gagarin and Zverovich [54] 2012) For aconnected graph G of orientable genus ℎ and nonorientablegenus 119896

119887 (119866) ⩽ min 119888ℎ 119888

1015840

119896 Δ (119866) + 119900 (ℎ) Δ (119866) + 119900 (119896) (38)

where 119888ℎand 1198881015840

119896are constants depending respectively on the

orientable and nonorientable genera of 119866

In the recent paper Gagarin and Zverovich [56] providedconstant upper bounds for the bondage number of graphs ontopological surfaces which can be used as the first estimationfor the constants 119888

ℎand 1198881015840

119896of Conjecture 106

Theorem 107 (Gagarin and Zverovich [56] 2013) Let 119866 bea connected graph of order 119899 If 119866 can be embedded on thesurface of its orientable or nonorientable genus of the Eulercharacteristic 120594 then

(a) 120594 ⩾ 1 implies 119887(119866) ⩽ 10(b) 120594 ⩽ 0 and 119899 gt minus12120594 imply 119887(119866) ⩽ 11(c) 120594 ⩽ minus1 and 119899 ⩽ minus12120594 imply 119887(119866) ⩽ 11 +

3120594(radic17 minus 8120594 minus 3)(120594 minus 1) = 11 + 119874(radicminus120594)

Theorem 79 shows that a connected planar triangle-freegraph 119866 has 119887(119866) ⩽ 6 The following result generalizes to itall the other topological surfaces

International Journal of Combinatorics 17

Theorem 108 (Gagarin and Zverovich [56] 2013) Let 119866 be aconnected triangle-free graph of order 119899 If 119866 can be embeddedon the surface of its orientable or nonorientable genus of theEuler characteristic 120594 then

(a) 120594 ⩾ 1 implies 119887(119866) ⩽ 6(b) 120594 ⩽ 0 and 119899 gt minus8120594 imply 119887(119866) ⩽ 7(c) 120594 ⩽ minus1 and 119899 ⩽ minus8120594 imply 119887(119866) ⩽ 7minus4120594(1+radic2 minus 120594)

In [56] the authors also explicitly improved upperbounds in Theorem 103 and gave tight lower bounds forthe number of vertices of graphs 2-cell embeddable ontopological surfaces of a given genusThe interested reader isreferred to the original paper for details The following resultis interesting although its proof is easy

Theorem 109 (Samodivkin [57]) Let119866 be a graph with order119899 and girth 119892 If 119866 is embeddable on a surface whose Eulercharacteristic 120594 is as large as possible then

119887 (119866) ⩽ 3 +8

119892 minus 2minus

4120594119892

119899 (119892 minus 2) (39)

If 119866 is a planar graph with girth 119892 ⩾ 4 + 119894 for any 119894 isin0 1 2 then Theorem 109 leads to 119887(119866) ⩽ 6 minus 119894 which isthe result inTheorem 79 obtained by Fischermann et al [46]Also inmany cases the bound stated inTheorem 109 is betterthan those given byTheorems 103 and 105

8 Conditional Bondage Numbers

Since the concept of the bondage number is based upon thedomination all sorts of dominations which are variationsof the normal domination by adding restricted conditionsto the normal dominating set can develop a new ldquobondagenumberrdquo as long as a variation of domination number isgiven In this section we survey results on the bondagenumber under some restricted conditions

81 Total Bondage Numbers A dominating set 119878 of a graph119866 without isolated vertices is called total if the subgraphinduced by 119878 contains no isolated vertices The minimumcardinality of a total dominating set is called the totaldomination number of 119866 and denoted by 120574

119879(119866) It is clear

that 120574(119866) ⩽ 120574119879(119866) ⩽ 2120574(119866) for any graph 119866 without isolated

verticesThe total domination in graphs was introduced by Cock-

ayne et al [58] in 1980 Pfaff et al [59 60] in 1983 showedthat the problem determining total domination number forgeneral graphs is NP-complete even for bipartite graphs andchordal graphs Even now total domination in graphs hasbeen extensively studied in the literature In 2009 Henning[61] gave a survey of selected recent results on total domina-tion in graphs

The total bondage number of 119866 denoted by 119887119879(119866) is the

smallest cardinality of a subset 119861 sube 119864(119866) with the propertythat119866minus119861 contains no isolated vertices and 120574

119879(119866minus119861) gt 120574

119879(119866)

From definition 119887119879(119866)may not exist for some graphs for

example119866 = 1198701119899 We put 119887

119879(119866) = infin if 119887

119879(119866) does not exist

It is easy to see that 119887119879(119866) is finite for any connected graph 119866

other than 1198701 119870

2 119870

3 and 119870

1119899 In fact for such a graph 119866

since there is a path of length 3 in 119866 we can find 1198611sube 119864(119866)

such that 119866 minus 1198611is a spanning tree 119879 of 119866 containing a path

of length 3 So 120574119879(119879) = 120574

119879(119866minus119861

1) ⩾ 120574

119879(119866) For the tree119879 we

find 1198612sube 119864(119879) such that 120574

119879(119879 minus 119861

2) gt 120574

119879(119879) ⩾ 120574

119879(119866) Thus

we have 120574119879(119866minus119861

2minus119861

1) gt 120574

119879(119866) and so 119887

119879(119866) ⩽ |119861

1|+|119861

2| In

the following discussion we always assume that 119887119879(119866) exists

when a graph 119866 is mentionedIn 1991 Kulli and Patwari [62] first studied the total

bondage number of a graph and calculated the exact valuesof 119887

119879(119866) for some standard graphs

Theorem 110 (Kulli and Patwari [62] 1991) For 119899 ⩾ 4

119887119879(119862

119899) =

3 119894119891 119899 equiv 2 (mod 4) 2 119900119905ℎ119890119903119908119894119904119890

119887119879(119875

119899) =

2 119894119891 119899 equiv 2 (mod 4) 1 119900119905ℎ119890119903119908119894119904119890

119887119879(119870

119898119899) = 119898 119908119894119905ℎ 2 ⩽ 119898 ⩽ 119899

119887119879(119870

119899) =

4 119891119900119903 119899 = 4

2119899 minus 5 119891119900119903 119899 ⩾ 5

(40)

Recently Hu et al [63] have obtained some results on thetotal bondage number of the Cartesian product119875

119898times119875

119899of two

paths 119875119898and 119875

119899

Theorem 111 (Hu et al [63] 2012) For the Cartesian product119875119898times 119875

119899of two paths 119875

119898and 119875

119899

119887119879(119875

2times 119875

119899) =

1 119894119891 119899 equiv 0 (mod 3) 2 119894119891 119899 equiv 2 (mod 3) 3 119894119891 119899 equiv 1 (mod 3)

119887119879(119875

3times 119875

119899) = 1

119887119879(119875

4times 119875

119899)

= 1 119894119891 119899 equiv 1 (mod 5) = 2 119894119891 119899 equiv 4 (mod 5) ⩽ 3 119894119891 119899 equiv 2 (mod 5) ⩽ 4 119894119891 119899 equiv 0 3 (mod 5)

(41)

Generalized Petersen graphs are an important class ofcommonly used interconnection networks and have beenstudied recently By constructing a family of minimum totaldominating sets Cao et al [64] determined the total bondagenumber of the generalized Petersen graphs

FromTheorem 6 we know that 119887(119879) ⩽ 2 for a nontrivialtree 119879 But given any positive integer 119896 Sridharan et al [65]constructed a tree 119879 for which 119887

119879(119879) = 119896 Let119867

119896be the tree

obtained from a star1198701119896+1

by subdividing 119896 edges twice Thetree 119867

7is shown in Figure 6 It can be easily verified that

119887119879(119867

119896) = 119896 This fact can be stated as the following theorem

Theorem 112 (Sridharan et al [65] 2007) For any positiveinteger 119896 there exists a tree 119879 with 119887

119879(119879) = 119896

18 International Journal of Combinatorics

Figure 61198677

Combining Theorem 1 with Theorem 112 we have whatfollows

Corollary 113 (Sridharan et al [65] 2007) The bondagenumber and the total bondage number are unrelated even fortrees

However Sridharan et al [65] gave an upper bound onthe total bondage number of a tree in terms of its maximumdegree For any tree 119879 of order 119899 if 119879 =119870

1119899minus1 then 119887

119879(119879) =

minΔ(119879) (13)(119899 minus 1) Rad and Raczek [66] improved thisupper bound and gave a constructive characterization of acertain class of trees attaching the upper bound

Theorem 114 (Rad and Raczek [66] 2011) 119887119879(119879) ⩽ Δ(119879) minus 1

for any tree 119879 with maximum degree at least three

In general the decision problem for 119887119879(119866) is NP-complete

for any graph 119866 We state the decision problem for the totalbondage as follows

Problem 3 Consider the decision problemTotal Bondage ProblemInstance a graph 119866 and a positive integer 119887Question is 119887

119879(119866) ⩽ 119887

Theorem 115 (Hu and Xu [16] 2012) The total bondageproblem is NP-complete

Consequently in view of its computational hardnessit is significative to establish some sharp bounds on thetotal bondage number of a graph in terms of other graphicparameters In 1991 Kulli and Patwari [62] showed that119887119879(119866) ⩽ 2119899 minus 5 for any graph 119866 with order 119899 ⩾ 5 Sridharan

et al [65] improved this result as follows

Theorem 116 (Sridharan et al [65] 2007) For any graph 119866with order 119899 if 119899 ⩾ 5 then

119887119879(119866) ⩽

1

3(119899 minus 1) 119894119891 119866 119888119900119899119905119886119894119899119904 119899119900 119888119910119888119897119890119904

119899 minus 1 119894119891 119892 (119866) ⩾ 5

119899 minus 2 119894119891 119892 (119866) = 4

(42)

Rad and Raczek [66] also established some upperbounds on 119887

119879(119866) for a general graph 119866 In particular they

gave an upper bound on 119887119879(119866) in terms of the girth of

119866

Theorem 117 (Rad and Raczek [66] 2011) 119887119879(119866)⩽4Δ(119866) minus 5

for any graph 119866 with 119892(119866) ⩾ 4

82 Paired Bondage Numbers A dominating set 119878 of 119866 iscalled to be paired if the subgraph induced by 119878 containsa perfect matching The paired domination number of 119866denoted by 120574

119875(119866) is the minimum cardinality of a paired

dominating set of 119866 Clearly 120574119879(119866) ⩽ 120574

119875(119866) for every

connected graph 119866 with order at least two where 120574119879(119866) is

the total domination number of 119866 and 120574119875(119866) ⩽ 2120574(119866) for

any graph119866without isolated vertices Paired domination wasintroduced by Haynes and Slater [67 68] and further studiedin [69ndash72]

The paired bondage number of 119866 with 120575(119866) ⩾ 1 denotedby 119887P(119866) is the minimum cardinality among all sets of edges119861 sube 119864 such that 120575(119866 minus 119861) ⩾ 1 and 120574

119875(119866 minus 119861) gt 120574

119875(119866)

The concept of the paired bondage number was firstproposed by Raczek [73] in 2008The following observationsfollow immediately from the definition of the paired bondagenumber

Observation 2 Let 119866 be a graph with 120575(119866) ⩾ 1

(a) If 119867 is a subgraph of 119866 such that 120575(119867) ⩾ 1 then119887119875(119867) ⩽ 119887

119875(119866)

(b) If 119867 is a subgraph of 119866 such that 119887119875(119867) = 1 and 119896

is the number of edges removed to form119867 then 1 ⩽119887119875(119866) ⩽ 119896 + 1

(c) If 119909119910 isin 119864(119866) such that 119889119866(119909) 119889

119866(119910) gt 1 and 119909119910

belongs to each perfect matching of each minimumpaired dominating set of 119866 then 119887

119875(119866) = 1

Based on these observations 119887119875(119875

119899) ⩽ 2 In fact the

paired bondage number of a path has been determined

Theorem 118 (Raczek [73] 2008) For any positive integer 119896if 119899 ⩾ 2 then

119887119875(119875

119899) =

0 119894119891 119899 = 2 3 119900119903 5

1 119894119891 119899 = 4119896 4119896 + 3 119900119903 4119896 + 6

2 119900119905ℎ119890119903119908119894119904119890

(43)

For a cycle 119862119899of order 119899 ⩾ 3 since 120574

119875(119862

119899) = 120574

119875(119875

119899) from

Theorem 118 the following result holds immediately

Corollary 119 (Raczek [73] 2008) For any positive integer 119896if 119899 ⩾ 3 then

119887119875(119862

119899) =

0 119894119891 119899 = 3 119900119903 5

2 119894119891 119899 = 4119896 4119896 + 3 119900119903 4119896 + 6

3 119900119905ℎ119890119903119908119894119904119890

(44)

A wheel 119882119899 where 119899 ⩾ 4 is a graph with 119899 vertices

formed by connecting a single vertex to all vertices of a cycle119862119899minus1

Of course 120574119875(119882

119899) = 2

International Journal of Combinatorics 19

119896

Figure 7 A tree 119879 with 119887119875(119879) = 119896

Theorem 120 (Raczek [73] 2008) For positive integers 119899 and119898

119887119875(119870

119898119899) = 119898 119891119900119903 1 lt 119898 ⩽ 119899

119887119875(119882

119899) =

4 119894119891 119899 = 4

3 119894119891 119899 = 5

2 119900119905ℎ119890119903119908119894119904119890

(45)

Theorem 16 shows that if 119879 is a nontrivial tree then119887(119879) ⩽ 2 However no similar result exists for pairedbondage For any nonnegative integer 119896 let 119879

119896be a tree

obtained by subdividing all but one edge of a star 1198701119896+1

(asshown in Figure 7) It is easy to see that 119887

119875(119879

119896) = 119896 We

state this fact as the following theorem which is similar toTheorem 112

Theorem121 (Raczek [73] 2008) For any nonnegative integer119896 there exists a tree 119879 with 119887

119875(119879) = 119896

Consider the tree defined in Figure 5 for 119896 = 3 Then119887(119879

3) ⩽ 2 and 119887

119875(119879

3) = 3 On the other hand 119887(119875

4) = 2

and 119887119875(119875

4) = 1 Thus we have what follows which is similar

to Corollary 113

Corollary 122 The bondage number and the paired bondagenumber are unrelated even for trees

A constructive characterization of trees with 119887(119879) = 2is given by Hartnell and Rall in [12] Raczek [73] provideda constructive characterization of trees with 119887

119875(119879) = 0 In

order to state the characterization we define a labeling andthree simple operations on a tree119879 Let 119910 isin 119881(119879) and let ℓ(119910)be the label assigned to 119910

Operation T1 If ℓ(119910) = 119861 add a vertex 119909 and the

edge 119910119909 and let ℓ(119909) = 119860OperationT

2 If ℓ(119910) = 119862 add a path (119909

1 119909

2) and the

edge 1199101199091 and let ℓ(119909

1) = 119861 ℓ(119909

2) = 119860

Operation T3 If ℓ(119910) = 119861 add a path (119909

1 119909

2 119909

3)

and the edge 1199101199091 and let ℓ(119909

1) = 119862 ℓ(119909

2) = 119861 and

ℓ(1199093) = 119860

Let 1198752= (119906 V) with ℓ(119906) = 119860 and ℓ(V) = 119861 Let T be

the class of all trees obtained from the labeled 1198752by a finite

sequence of operationsT1 T

2 T

3

A tree 119879 in Figure 8 belongs to the familyTRaczek [73] obtained the following characterization of all

trees 119879 with 119887119875(119879) = 0

119860

119860

119860

119860

119860

119861 119862

119861 119862 119861

119861119860

Figure 8 A tree 119879 belong to the familyT

Theorem 123 (Raczek [73] 2008) For a tree 119879 119887119875(119879) = 0 if

and only if 119879 is inT

We state the decision problem for the paired bondage asfollows

Problem 4 Consider the decision problem

Paired Bondage ProblemInstance a graph 119866 and a positive integer 119887Question is 119887

119875(119866) ⩽ 119887

Conjecture 124 The paired bondage problem is NP-complete

83 Restrained Bondage Numbers A dominating set 119878 of 119866 iscalled to be restrained if the subgraph induced by 119878 containsno isolated vertices where 119878 = 119881(119866) 119878 The restraineddomination number of 119866 denoted by 120574

119877(119866) is the smallest

cardinality of a restrained dominating set of 119866 It is clear that120574119877(119866) exists and 120574(119866) ⩽ 120574

119877(119866) for any nonempty graph 119866

The concept of restrained domination was introducedby Telle and Proskurowski [74] in 1997 albeit indirectly asa vertex partitioning problem Concerning the study of therestrained domination numbers the reader is referred to [74ndash79]

In 2008 Hattingh and Plummer [80] defined therestrained bondage number 119887R(119866) of a nonempty graph 119866 tobe the minimum cardinality among all subsets 119861 sube 119864(119866) forwhich 120574

119877(119866 minus 119861) gt 120574

119877(119866)

For some simple graphs their restrained dominationnumbers can be easily determined and so restrained bondagenumbers have been also determined For example it is clearthat 120574

119877(119870

119899) = 1 Domke et al [77] showed that 120574

119877(119862

119899) =

119899 minus 2lfloor1198993rfloor for 119899 ⩾ 3 and 120574119877(119875

119899) = 119899 minus 2lfloor(119899 minus 1)3rfloor for 119899 ⩾ 1

Using these results Hattingh and Plummer [80] obtained therestricted bondage numbers for119870

119899 119862

119899 119875

119899 and119866 = 119870

11989911198992119899119905

as follows

Theorem 125 (Hattingh and Plummer [80] 2008) For 119899 ⩾ 3

119887R (119870119899) =

1 119894119891 119899 = 3

lceil119899

2rceil 119900119905ℎ119890119903119908119894119904119890

119887R (119862119899) =

1 119894119891 119899 equiv 0 (mod 3) 2 119900119905ℎ119890119903119908119894119904119890

119887R (119875119899) = 1 119899 ⩾ 4

20 International Journal of Combinatorics

119887R (119866)

=

lceil119898

2rceil 119894119891 119899

119898= 1 119886119899119889 119899

119898+1⩾ 2 (1 ⩽ 119898 lt 119905)

2119905 minus 2 119894119891 1198991= 119899

2= sdot sdot sdot = 119899

119905= 2 (119905 ⩾ 2)

2 119894119891 1198991= 2 119886119899119889 119899

2⩾ 3 (119905 = 2)

119905minus1

sum

119894=1

119899119894minus 1 119900119905ℎ119890119903119908119894119904119890

(46)

Theorem 126 (Hattingh and Plummer [80] 2008) For a tree119879 of order 119899 (⩾ 4) 119879 ≇ 119870

1119899minus1if and only if 119887R(119879) = 1

Theorem 126 shows that the restrained bondage numberof a tree can be computed in constant time However thedecision problem for 119887R(119866) is NP-complete even for bipartitegraphs

Problem 5 Consider the decision problem

Restrained Bondage Problem

Instance a graph 119866 and a positive integer 119887

Question is 119887R(119866) ⩽ 119887

Theorem 127 (Hattingh and Plummer [80] 2008) Therestrained bondage problem is NP-complete even for bipartitegraphs

Consequently in view of its computational hardnessit is significative to establish some sharp bounds on therestrained bondage number of a graph in terms of othergraphic parameters

Theorem 128 (Hattingh and Plummer [80] 2008) For anygraph 119866 with 120575(119866) ⩾ 2

119887R (119866) ⩽ min119909119910isin119864(119866)

119889119866(119909) + 119889

119866(119910) minus 2 (47)

Corollary 129 119887R(119866) ⩽ Δ(119866)+120575(119866)minus2 for any graph119866 with120575(119866) ⩾ 2

Notice that the bounds stated in Theorem 128 andCorollary 129 are sharp Indeed the class of cycles whoseorders are congruent to 1 2 (mod 3) have a restrained bondagenumber achieving these bounds (see Theorem 125)

Theorem 128 is an analogue of Theorem 14 A quitenatural problem is whether or not for restricted bondagenumber there are analogues of Theorem 16 Theorem 18Theorem 29 and so on Indeed we should consider all resultsfor the bondage number whether or not there are analoguesfor restricted bondage number

Theorem 130 (Hattingh and Plummer [80] 2008) For anygraph 119866 with 120574

119877(119866) = 2

119887R (119866) ⩽ Δ (119866) + 1 (48)

We now consider the relation between 119887119877(119866) and 119887(119866)

Theorem 131 (Hattingh and Plummer [80] 2008) If 120574119877(119866) =

120574(119866) for some graph 119866 then 119887R(119866) ⩽ 119887(119866)

Corollary 129 and Theorem 131 provide a possibility toattack Conjecture 73 by characterizing planar graphs with120574119877= 120574 and 119887R = 119887 when 3 ⩽ Δ ⩽ 6However we do not have 119887R(119866) = 119887(119866) for any graph

119866 even if 120574119877(119866) = 120574(119866) Observe that 120574

119877(119870

3) = 120574(119870

3) yet

119887119877(119870

3) = 1 and 119887(119870

3) = 2 We still may not claim that

119887119877(119866) = 119887(119866) even in the case that every 120574-set is a 120574

119877-set

The example 1198703again demonstrates this

Theorem 132 (Hattingh and Plummer [80] 2008) There isan infinite class of graphs inwhich each graph119866 satisfies 119887(119866) lt119887R(119866)

In fact such an infinite class of graphs can be obtained asfollows Let119867 be a connected graph BC(119867) a graph obtainedfrom 119867 by attaching ℓ (⩾ 2) vertices of degree 1 to eachvertex in 119867 and B = 119866 119866 = BC(119867) for some graph 119867such that 120575(119867) ⩾ 2 By Theorem 3 we can easily check that119887(119866) = 1 lt 2 ⩽ min120575(119867) ℓ = 119887R(119866) for any 119866 isin BMoreover there exists a graph 119866 isin B such that 119887R(119866) canbe much larger than 119887(119866) which is stated as the followingtheorem

Theorem 133 (Hattingh and Plummer [80] 2008) For eachpositive integer 119896 there is a graph119866 such that 119896 = 119887R(119866)minus119887(119866)

Combining Theorem 133 with Theorem 132 we have thefollowing corollary

Corollary 134 The bondage number and the restrainedbondage number are unrelated

Very recently Jafari Rad et al [81] have consideredthe total restrained bondage in graphs based on both totaldominating sets and restrained dominating sets and obtainedseveral properties exact values and bounds for the totalrestrained bondage number of a graph

A restrained dominating set 119878 of a graph 119866 without iso-lated vertices is called to be a total restrained dominating setif 119878 is a total dominating set The total restrained dominationnumber of119866 denoted by 120574TR (119866) is theminimum cardinalityof a total restrained dominating set of 119866 For referenceson this domination in graphs see [3 61 82ndash85] The totalrestrained bondage number 119887TR (119866) of a graph 119866 with noisolated vertex is the cardinality of a smallest set of edges 119861 sube119864(119866) for which119866minus119861 has no isolated vertex and 120574TR (119866minus119861) gt120574TR (119866) In the case that there is no such subset 119861 we define119887TR (119866) = infin

Ma et al [84] determined that 120574TR (119875119899) = 119899minus2lfloor(119899minus2)4rfloorfor 119899 ⩾ 3 120574TR (119862119899

) = 119899 minus 2lfloor1198994rfloor for 119899 ⩾ 3 120574TR (1198703) =

3 and 120574TR (119870119899) = 2 for 119899 = 3 and 120574TR (1198701119899

) = 1 + 119899

and 120574TR (119870119898119899) = 2 for 2 ⩽ 119898 ⩽ 119899 According to these

results Jafari Rad et al [81] determined the exact valuesof total restrained bondage numbers for the correspondinggraphs

International Journal of Combinatorics 21

Theorem 135 (Jafari Rad et al [81]) For an integer 119899

119887TR (119875119899) = infin 119894119891 2 ⩽ 119899 ⩽ 5

1 119894119891 119899 ⩾ 5

119887TR (119862119899) =

infin 119894119891 119899 = 3

1 119894119891 3 lt 119899 119899 equiv 0 1 (mod 4)2 119894119891 3 lt 119899 119899 equiv 2 3 (mod 4)

119887TR (119882119899) = 2 119891119900119903 119899 ⩾ 6

119887TR (119870119899) = 119899 minus 1 119891119900119903 119899 ⩾ 4

119887TR (119870119898119899) = 119898 minus 1 119891119900119903 2 ⩽ 119898 ⩽ 119899

(49)

119887TR (119870119899) = 119899minus1 shows the fact that for any integer 119899 ⩾ 3 there

exists a connected graph namely119870119899+1

such that 119887TR (119870119899+1) =

119899 that is the total restrained bondage number is unboundedfrom the above

A vertex is said to be a support vertex if it is adjacent toa vertex of degree one Let A be the family of all connectedgraphs such that119866 belongs toA if and only if every edge of119866is incident to a support vertex or 119866 is a cycle of length three

Proposition 136 (Cyman and Raczek [82] 2006) For aconnected graph119866 of order 119899 120574TR (119866) = 119899 if and only if119866 isin A

Hence if 119866 isin A then the removal of any subset of edgesof119866 cannot increase the total restrained domination numberof 119866 and thus 119887TR (119866) = infin When 119866 notin A a result similar toTheorem 6 is obtained

Theorem 137 (Jafari Rad et al [81]) For any tree 119879 notin A119887TR (119879) ⩽ 2 This bound is sharp

In the samepaper the authors provided a characterizationfor trees 119879 notin A with 119887TR (119879) = 1 or 2 The following result issimilar to Observation 1

Observation 3 (Jafari Rad et al [81]) If 119896 edges can beremoved from a graph 119866 to obtain a graph 119867 without anisolated vertex and with 119887TR (119867) = 119905 then 119887TR (119866) ⩽ 119896 + 119905

ApplyingTheorem 137 andObservation 3 Jafari Rad et alcharacterized all connected graphs 119866 with 119887TR (119866) = infin

Theorem 138 (Jafari Rad et al [81]) For a connected graph119866119887TR (119866) = infin if and only if 119866 isin A

84 Other Conditional Bondage Numbers A subset 119868 sube 119881(119866)is called an independent set if no two vertices in 119868 are adjacentin 119866 The maximum cardinality among all independent setsis called the independence number of 119866 denoted by 120572(119866)

A dominating set 119878 of a graph 119866 is called to be inde-pendent if 119878 is an independent set of 119866 The minimumcardinality among all independent dominating set is calledthe independence domination number of 119866 and denoted by120574119868(119866)

Since an independent dominating set is not only adominating set but also an independent set 120574(119866) ⩽ 120574

119868(119866) ⩽

120572(119866) for any graph 119866It is clear that a maximal independent set is certainly a

dominating set Thus an independent set is maximal if andonly if it is an independent dominating set and so 120574

119868(119866) is the

minimum cardinality among all maximal independent setsof 119866 This graph-theoretical invariant has been well studiedin the literature see for example Haynes et al [3] for earlyresults Goddard and Henning [86] for recent results onindependent domination in graphs

In 2003 Zhang et al [87] defined the independencebondage number 119887

119868(119866) of a nonempty graph 119866 to be the

minimum cardinality among all subsets 119861 sube 119864(119866) for which120574119868(119866 minus 119861) gt 120574

119868(119866) For some ordinary graphs their indepen-

dence domination numbers can be easily determined and soindependence bondage numbers have been also determinedClearly 119887

119868(119870

119899) = 1 if 119899 ⩾ 2

Theorem 139 (Zhang et al [87] 2003) For any integer 119899 ⩾ 4

119887119868(119862

119899) =

1 119894119891 119899 equiv 1 (mod 2) 2 119894119891 119899 equiv 0 (mod 2)

119887119868(119875

119899) =

1 119894119891 119899 equiv 0 (mod 2) 2 119894119891 119899 equiv 1 (mod 2)

119887119868(119870

119898119899) = 119899 119891119900119903 119898 ⩽ 119899

(50)

Apart from the above-mentioned results as far as we findno other results on the independence bondage number in thepresent literature we never knew much of any result on thisparameter for a tree

A subset 119878 of 119881(119866) is called an equitable dominatingset if for every 119910 isin 119881(119866 minus 119878) there exists a vertex 119909 isin 119878such that 119909119910 isin 119864(119866) and |119889

119866(119909) minus 119889

119866(119910)| ⩽ 1 proposed

by Dharmalingam [88] The minimum cardinality of such adominating set is called the equitable domination number andis denoted by 120574

119864(119866) Deepak et al [89] defined the equitable

bondage number 119887119864(119866) of a graph 119866 to be the cardinality of a

smallest set 119861 sube 119864(119866) of edges for which 120574119864(119866 minus 119861) gt 120574

119864(119866)

and determined 119887119864(119866) for some special graphs

Theorem 140 (Deepak et al [89] 2011) For any integer 119899 ⩾ 2

119887119864(119870

119899) = lceil

119899

2rceil

119887119864(119870

119898119899) = 119898 119891119900119903 0 ⩽ 119899 minus 119898 ⩽ 1

119887119864(119862

119899) =

3 119894119891 119899 equiv 1 (mod 3)2 119900119905ℎ119890119903119908119894119904119890

119887119864(119875

119899) =

2 119894119891 119899 equiv 1 (mod 3)1 119900119905ℎ119890119903119908119894119904119890

119887119864(119879) ⩽ 2 119891119900119903 119886119899119910 119899119900119899119905119903119894119907119894119886119897 119905119903119890119890 119879

119887119864(119866) ⩽ 119899 minus 1 119891119900119903 119886119899119910 119888119900119899119899119890119888119905119890119889 119892119903119886119901ℎ 119866 119900119891 119900119903119889119890119903 119899

(51)

22 International Journal of Combinatorics

As far as we know there are no more results on theequitable bondage number of graphs in the present literature

There are other variations of the normal domination byadding restricted conditions to the normal dominating setFor example the connected domination set of a graph119866 pro-posed by Sampathkumar and Walikar [90] is a dominatingset 119878 such that the subgraph induced by 119878 is connected Theproblem of finding a minimum connected domination set ina graph is equivalent to the problemof finding a spanning treewithmaximumnumber of vertices of degree one [91] and hassome important applications in wireless networks [92] thusit has received much research attention However for suchan important domination there is no result on its bondagenumber We believe that the topic is of significance for futureresearch

9 Generalized Bondage Numbers

There are various generalizations of the classical dominationsuch as 119901-domination distance domination fractional dom-ination Roman domination and rainbow domination Everysuch a generalization can lead to a corresponding bondageIn this section we introduce some of them

91 119901-Bondage Numbers In 1985 Fink and Jacobson [93]introduced the concept of 119901-domination Let 119901 be a positiveinteger A subset 119878 of 119881(119866) is a 119901-dominating set of 119866 if|119878 cap 119873

119866(119910)| ⩾ 119901 for every 119910 isin 119878 The 119901-domination number

120574119901(119866) is the minimum cardinality among all 119901-dominating

sets of 119866 Any 119901-dominating set of 119866 with cardinality 120574119901(119866)

is called a 120574119901-set of 119866 Note that a 120574

1-set is a classic minimum

dominating set Notice that every graph has a 119901-dominatingset since the vertex-set119881(119866) is such a setWe also note that a 1-dominating set is a dominating set and so 120574(119866) = 120574

1(119866) The

119901-domination number has received much research attentionsee a state-of-the-art survey article by Chellali et al [94]

It is clear from definition that every 119901-dominating setof a graph certainly contains all vertices of degree at most119901 minus 1 By this simple observation to avoid the trivial caseoccurrence we always assume that Δ(119866) ⩾ 119901 For 119901 ⩾ 2 Luet al [95] gave a constructive characterization of trees withunique minimum 119901-dominating sets

Recently Lu and Xu [96] have introduced the conceptto the 119901-bondage number of 119866 denoted by 119887

119901(119866) as the

minimum cardinality among all sets of edges 119861 sube 119864(119866) suchthat 120574

119901(119866 minus 119861) gt 120574

119901(119866) Clearly 119887

1(119866) = 119887(119866)

Lu and Xu [96] established a lower bound and an upperbound on 119887

119901(119879) for any integer 119901 ⩾ 2 and any tree 119879 with

Δ(119879) ⩾ 119901 and characterized all trees achieving the lowerbound and the upper bound respectively

Theorem 141 (Lu and Xu [96] 2011) For any integer 119901 ⩾ 2and any tree 119879 with Δ(119879) ⩾ 119901

1 ⩽ 119887119901(119879) ⩽ Δ (119879) minus 119901 + 1 (52)

Let 119878 be a given subset of 119881(119866) and for any 119909 isin 119878 let

119873119901(119909 119878 119866) = 119910 isin 119878 cap 119873

119866(119909)

1003816100381610038161003816119873119866(119910) cap 119878

1003816100381610038161003816 = 119901

(53)

Then all trees achieving the lower bound 1 can be character-ized as follows

Theorem 142 (Lu and Xu [96] 2011) Let 119879 be a tree withΔ(119879) ⩾ 119901 ⩾ 2 Then 119887

119901(119879) = 1 if and only if for any 120574

119901-set

119878 of 119879 there exists an edge 119909119910 isin (119878 119878) such that 119910 isin 119873119901(119909 119878 119879)

The notation 119878(119886 119887) denotes a double star obtained byadding an edge between the central vertices of two stars119870

1119886minus1

and1198701119887minus1

And the vertex with degree 119886 (resp 119887) in 119878(119886 119887) iscalled the 119871-central vertex (resp 119877-central vertex) of 119878(119886 119887)

To characterize all trees attaining the upper bound givenin Theorem 141 we define three types of operations on a tree119879 with Δ(119879) = Δ ⩾ 119901 + 1

Type 1 attach a pendant edge to a vertex 119910with 119889119879(119910) ⩽ 119901minus

2 in 119879

Type 2 attach a star 1198701Δminus1

to a vertex 119910 of 119879 by joining itscentral vertex to 119910 where 119910 in a 120574

119901-set of 119879 and

119889119879(119910) ⩽ Δ minus 1

Type 3 attach a double star 119878(119901 Δ minus 1) to a pendant vertex 119910of 119879 by coinciding its 119877-central vertex with 119910 wherethe unique neighbor of 119910 is in a 120574

119901-set of 119879

Let B = 119879 a tree obtained from 1198701Δ

or 119878(119901 Δ) by afinite sequence of operations of Types 1 2 and 3

Theorem 143 (Lu and Xu [96] 2011) A tree with the maxi-mum Δ ⩾ 119901 + 1 has 119901-bondage number Δ minus 119901 + 1 if and onlyif it belongs toB

Theorem 144 (Lu and Xu [97] 2012) Let 119866119898119899= 119875

119898times 119875

119899

Then

1198872(119866

2119899) = 1 119891119900119903 119899 ⩾ 2

1198872(119866

3119899) =

2 119894119891 119899 equiv 1 (mod 3) 1 119900119905ℎ119890119903119908119894119904119890

for 119899 ⩾ 2

1198872(119866

4119899) =

1 119894119891 119899 equiv 3 (mod 4) 2 119900119905ℎ119890119903119908119894119904119890

for 119899 ⩾ 7

(54)

Recently Krzywkowski [98] has proposed the conceptof the nonisolating 2-bondage of a graph The nonisolating2-bondage number of a graph 119866 denoted by 1198871015840

2(119866) is the

minimum cardinality among all sets of edges 119861 sube 119864(119866) suchthat 119866 minus 119861 has no isolated vertices and 120574

2(119866 minus 119861) gt 120574

2(119866) If

for every 119861 sube 119864(119866) either 1205742(119866 minus 119861) = 120574

2(119866) or 119866 minus 119861 has

isolated vertices then define 11988710158402(119866) = 0 and say that 119866 is a 2-

nonisolatedly strongly stable graph Krzywkowski presentedsome basic properties of nonisolating 2-bondage in graphsshowed that for every nonnegative integer 119896 there exists atree 119879 such 1198871015840

2(119879) = 119896 characterized all 2-non-isolatedly

strongly stable trees and determined 11988710158402(119866) for several special

graphs

International Journal of Combinatorics 23

Theorem 145 (Krzywkowski [98] 2012) For any positiveintegers 119899 and119898 with119898 ⩽ 119899

1198871015840

2(119870

119899) =

0 119894119891 119899 = 1 2 3

lfloor2119899

3rfloor 119900119905ℎ119890119903119908119894119904119890

1198871015840

2(119875

119899) =

0 119894119891 119899 = 1 2 3

1 119900119905ℎ119890119903119908119894119904119890

1198871015840

2(119862

119899) =

0 119894119891 119899 = 3

1 119894119891 119899 119894119904 119890V119890119899

2 119894119891 119899 119894119904 119900119889119889

1198871015840

2(119870

119898119899) =

3 119894119891 119898 = 119899 = 3

1 119894119891 119898 = 119899 = 4

2 119900119905ℎ119890119903119908119894119904119890

(55)

92 Distance Bondage Numbers A subset 119878 of vertices of agraph119866 is said to be a distance 119896-dominating set for119866 if everyvertex in 119866 not in 119878 is at distance at most 119896 from some vertexof 119878 The minimum cardinality of all distance 119896-dominatingsets is called the distance 119896-domination number of 119866 anddenoted by 120574

119896(119866) (do not confuse with the above-mentioned

120574119901(119866)) When 119896 = 1 a distance 1-dominating set is a normal

dominating set and so 1205741(119866) = 120574(119866) for any graph 119866 Thus

the distance 119896-domination is a generalization of the classicaldomination

The relation between 120574119896and 120572

119896(the 119896-independence

number defined in Section 22) for a tree obtained by Meirand Moon [14] who proved that 120574

119896(119879) = 120572

2119896(119879) for any tree

119879 (a special case for 119896 = 1 is stated in Proposition 10) Thefurther research results can be found in Henning et al [99]Tian and Xu [100ndash103] and Liu et al [104]

In 1998 Hartnell et al [15] defined the distance 119896-bondagenumber of 119866 denoted by 119887

119896(119866) to be the cardinality of a

smallest subset 119861 of edges of 119866 with the property that 120574119896(119866 minus

119861) gt 120574119896(119866) FromTheorem 6 it is clear that if119879 is a nontrivial

tree then 1 ⩽ 1198871(119879) ⩽ 2 Hartnell et al [15] generalized this

result to any integer 119896 ⩾ 1

Theorem 146 (Hartnell et al [15] 1998) For every nontrivialtree 119879 and positive integer 119896 1 ⩽ 119887

119896(119879) ⩽ 2

Hartnell et al [15] and Topp and Vestergaard [13] alsocharacterized the trees having distance 119896-bondage number 2In particular the class of trees for which 119887

1(119879) = 2 are just

those which have a unique maximum 2-independent set (seeTheorem 11)

Since when 119896 = 1 the distance 1-bondage number 1198871(119866)

is the classical bondage number 119887(119866) Theorem 12 gives theNP-hardness of deciding the distance 119896-bondage number ofgeneral graphs

Theorem 147 Given a nonempty undirected graph 119866 andpositive integers 119896 and 119887 with 119887 ⩽ 120576(119866) determining wetheror not 119887

119896(119866) ⩽ 119887 is NP-hard

For a vertex 119909 in119866 the open 119896-neighborhood119873119896(119909) of 119909 is

defined as119873119896(119909) = 119910 isin 119881(119866) 1 ⩽ 119889

119866(119909 119910) le 119896The closed

119896-neighborhood119873119896[119909] of 119909 in 119866 is defined as119873

119896(119909) cup 119909 Let

Δ119896(119866) = max 1003816100381610038161003816119873119896

(119909)1003816100381610038161003816 for any 119909 isin 119881 (119866) (56)

Clearly Δ1(119866) = Δ(119866) The 119896th power of a graph 119866 is the

graph119866119896 with vertex-set119881(119866119896

) = 119881(119866) and edge-set119864(119866119896

) =

119909119910 1 ⩽ 119889119866(119909 119910) ⩽ 119896 The following lemma holds directly

from the definition of 119866119896

Proposition 148 (Tian and Xu [101]) Δ(119866119896

) = Δ119896(119866) and

120574(119866119896

) = 120574119896(119866) for any graph 119866 and each 119896 ⩾ 1

A graph 119866 is 119896-distance domination-critical or 120574119896-critical

for short if 120574119896(119866 minus 119909) lt 120574

119896(119866) for every vertex 119909 in 119866

proposed by Henning et al [105]

Proposition 149 (Tian and Xu [101]) For each 119896 ⩾ 1 a graph119866 is 120574

119896-critical if and only if 119866119896 is 120574

119896-critical

By the above facts we suggest the following worthwhileproblem

Problem 6 Can we generalize the results on the bondage for119866 to 119866119896 In particular do the following propositions hold

(a) 119887(119866119896

) = 119887119896(119866) for any graph 119866 and each 119896 ⩾ 1

(b) 119887(119866119896

) ⩽ Δ119896(119866) if 119866 is not 120574

119896-critical

Let 119896 and 119901 be positive integers A subset 119878 of 119881(119866) isdefined to be a (119896 119901)-dominating set of 119866 if for any vertex119909 isin 119878 |119873

119896(119909) cap 119878| ⩾ 119901 The (119896 119901)-domination number of

119866 denoted by 120574119896119901(119866) is the minimum cardinality among

all (119896 119901)-dominating sets of 119866 Clearly for a graph 119866 a(1 1)-dominating set is a classical dominating set a (119896 1)-dominating set is a distance 119896-dominating set and a (1 119901)-dominating set is the above-mentioned 119901-dominating setThat is 120574

11(119866) = 120574(119866) 120574

1198961(119866) = 120574

119896(119866) and 120574

1119901(119866) = 120574

119901(119866)

The concept of (119896 119901)-domination in a graph 119866 is a gen-eralized domination which combines distance 119896-dominationand 119901-domination in 119866 So the investigation of (119896 119901)-domination of 119866 is more interesting and has received theattention of many researchers see for example [106ndash109]

More general Li and Xu [110] proposed the concept of the(ℓ 119908)-domination number of a119908-connected graph A subset119878 of 119881(119866) is defined to be an (ℓ 119908)-dominating set of 119866 iffor any vertex 119909 isin 119878 there are 119908 internally disjoint pathsof length at most ℓ from 119909 to some vertex in 119878 The (ℓ 119908)-domination number of119866 denoted by 120574

ℓ119908(119866) is theminimum

cardinality among all (ℓ 119908)-dominating sets of 119866 Clearly1205741198961(119866) = 120574

119896(119866) and 120574

1119901(119866) = 120574

119901(119866)

It is quite natural to propose the concept of bondage num-ber for (119896 119901)-domination or (ℓ 119908)-domination However asfar as we know none has proposed this concept until todayThis is a worthwhile topic for further research

24 International Journal of Combinatorics

93 Fractional Bondage Numbers If 120590 is a function mappingthe vertex-set 119881 into some set of real numbers then for anysubset 119878 sube 119881 let 120590(119878) = sum

119909isin119878120590(119909) Also let |120590| = 120590(119881)

A real-value function 120590 119881 rarr [0 1] is a dominatingfunction of a graph 119866 if for every 119909 isin 119881(119866) 120590(119873

119866[119909]) ⩾ 1

Thus if 119878 is a dominating set of 119866 120590 is a function where

120590 (119909) = 1 if 119909 isin 1198780 otherwise

(57)

then 120590 is a dominating function of119866 The fractional domina-tion number of 119866 denoted by 120574

119865(119866) is defined as follows

120574119865(119866) = min |120590| 120590 is a domination function of 119866

(58)

The 120574119865-bondage number of 119866 denoted by 119887

119865(119866) is

defined as the minimum cardinality of a subset 119861 sube 119864 whoseremoval results in 120574

119865(119866 minus 119861) gt 120574

119865(119866)

Hedetniemi et al [111] are the first to study fractionaldomination although Farber [112] introduced the idea indi-rectly The concept of the fractional domination numberwas proposed by Domke and Laskar [113] in 1997 Thefractional domination numbers for some ordinary graphs aredetermined

Proposition 150 (Domke et al [114] 1998 Domke and Laskar[113] 1997) (a) If 119866 is a 119896-regular graph with order 119899 then120574119865(119866) = 119899119896 + 1(b) 120574

119865(119862

119899) = 1198993 120574

119865(119875

119899) = lceil1198993rceil 120574

119865(119870

119899) = 1

(c) 120574119865(119866) = 1 if and only if Δ(119866) = 119899 minus 1

(d) 120574119865(119879) = 120574(119879) for any tree 119879

(e) 120574119865(119870

119899119898) = (119899(119898 + 1) + 119898(119899 + 1))(119899119898 minus 1) where

max119899119898 gt 1

The assertions (a)ndash(d) are due to Domke et al [114] andthe assertion (e) is due to Domke and Laskar [113] Accordingto these results Domke and Laskar [113] determined the 120574

119865-

bondage numbers for these graphs

Theorem 151 (Domke and Laskar [113] 1997) 119887119865(119870

119899) =

lceil1198992rceil 119887119865(119870

119899119898) = 1 wheremax119899119898 gt 1

119887119865(119862

119899) =

2 119894119891 119899 equiv 0 (mod 3) 1 119900119905ℎ119890119903119908119894119904119890

119887119865(119875

119899) =

2 119894119891 119899 equiv 1 (mod 3) 1 119900119905ℎ119890119903119908119894119904119890

(59)

It is easy to see that for any tree119879 119887119865(119879) ⩽ 2 In fact since

120574119865(119879) = 120574(119879) for any tree 119879 120574

119865(119879

1015840

) = 120574(1198791015840

) for any subgraph1198791015840 of 119879 It follows that

119887119865(119879) = min 119861 sube 119864 (119879) 120574

119865(119879 minus 119861) gt 120574

119865(119879)

= min 119861 sube 119864 (119879) 120574 (119879 minus 119861) gt 120574 (119879)

= 119887 (119879) ⩽ 2

(60)

Except for these no results on 119887119865(119866) have been known as

yet

94 Roman Bondage Numbers A Roman dominating func-tion on a graph 119866 is a labeling 119891 119881 rarr 0 1 2 such thatevery vertex with label 0 has at least one neighbor with label2 The weight of a Roman dominating function 119891 on 119866 is thevalue

119891 (119866) = sum

119906isin119881(119866)

119891 (119906) (61)

The minimum weight of a Roman dominating function ona graph 119866 is called the Roman domination number of 119866denoted by 120574RM (119866)

A Roman dominating function 119891 119881 rarr 0 1 2

can be represented by the ordered partition (1198810 119881

1 119881

2) (or

(119881119891

0 119881

119891

1 119881

119891

2) to refer to119891) of119881 where119881

119894= V isin 119881 | 119891(V) = 119894

In this representation its weight 119891(119866) = |1198811| + 2|119881

2| It is

clear that119881119891

1cup119881

119891

2is a dominating set of 119866 called the Roman

dominating set denoted by 119863119891

R = (1198811 1198812) Since 119881119891

1cup 119881

119891

2is

a dominating set when 119891 is a Roman dominating functionand since placing weight 2 at the vertices of a dominating setyields a Roman dominating function in [115] it is observedthat

120574 (119866) ⩽ 120574RM (119866) ⩽ 2120574 (119866) (62)

A graph 119866 is called to be Roman if 120574RM (119866) = 2120574(119866)The definition of the Roman domination function is

proposed implicitly by Stewart [116] and ReVelle and Rosing[117] Roman dominating numbers have been deeply studiedIn particular Bahremandpour et al [118] showed that theproblem determining the Roman domination number is NP-complete even for bipartite graphs

Let 119866 be a graph with maximum degree at least twoThe Roman bondage number 119887RM (119866) of 119866 is the minimumcardinality of all sets 1198641015840 sube 119864 for which 120574RM (119866 minus 119864

1015840

) gt

120574RM (119866) Since in study of Roman bondage number theassumption Δ(119866) ⩾ 2 is necessary we always assume thatwhen we discuss 119887RM (119866) all graphs involved satisfy Δ(119866) ⩾2 The Roman bondage number 119887RM (119866) is introduced byJafari Rad and Volkmann in [119]

Recently Bahremandpour et al [118] have shown that theproblem determining the Roman bondage number is NP-hard even for bipartite graphs

Problem 7 Consider the decision problem

Roman Bondage Problem

Instance a nonempty bipartite graph119866 and a positiveinteger 119896

Question is 119887RM (119866) ⩽ 119896

Theorem 152 (Bahremandpour et al [118] 2013) TheRomanbondage problem is NP-hard even for bipartite graphs

The exact value of 119887RM(119866) is known only for a few familyof graphs including the complete graphs cycles and paths

International Journal of Combinatorics 25

Theorem 153 (Jafari Rad and Volkmann [119] 2011) For anypositive integers 119899 and119898 with119898 ⩽ 119899

119887RM (119875119899) = 2 119894119891 119899 equiv 2 (mod 3) 1 119900119905ℎ119890119903119908119894119904119890

119887RM (119862119899) =

3 119894119891 119899 = 2 (mod 3) 2 119900119905ℎ119890119903119908119894119904119890

119887RM (119870119899) = lceil

119899

2rceil 119891119900119903 119899 ⩾ 3

119887RM (119870119898119899) =

1 119894119891 119898 = 1 119886119899119889 119899 = 1

4 119894119891 119898 = 119899 = 3

119898 119900119905ℎ119890119903119908119894119904119890

(63)

Lemma 154 (Cockayne et al [115] 2004) If 119866 is a graph oforder 119899 and contains vertices of degree 119899 minus 1 then 120574RM(119866) = 2

Using Lemma 154 the third conclusion in Theorem 153can be generalized to more general case which is similar toLemma 49

Theorem 155 (Jafari Rad and Volkmann [119] 2011) Let119866 bea graph with order 119899 ge 3 and 119905 the number of vertices of degree119899 minus 1 in 119866 If 119905 ⩾ 1 then 119887RM(119866) = lceil1199052rceil

Ebadi and PushpaLatha [120] conjectured that 119887RM(119866) ⩽119899 minus 1 for any graph 119866 of order 119899 ⩾ 3 Dehgardi et al[121] Akbari andQajar [122] independently showed that thisconjecture is true

Theorem 156 119887RM(119866) ⩽ 119899 minus 1 for any connected graph 119866 oforder 119899 ⩾ 3

For a complete 119905-partite graph we have the followingresult

Theorem 157 (Hu and Xu [123] 2011) Let 119866 = 11987011989811198982119898119905

bea complete 119905-partite graph with 119898

1= sdot sdot sdot = 119898

119894lt 119898

119894+1le sdot sdot sdot ⩽

119898119905 119905 ⩾ 2 and 119899 = sum119905

119895=1119898

119895 Then

119887RM (119866) =

lceil119894

2rceil 119894119891 119898

119894= 1 119886119899119889 119899 ⩾ 3

2 119894119891 119898119894= 2 119886119899119889 119894 = 1

119894 119894119891 119898119894= 2 119886119899119889 119894 ⩾ 2

4 119894119891 119898119894= 3 119886119899119889 119894 = 119905 = 2

119899 minus 1 119894119891 119898119894= 3 119886119899119889 119894 = 119905 ⩾ 3

119899 minus 119898119905

119894119891 119898119894⩾ 3 119886119899119889 119898

119905⩾ 4

(64)

Consider a complete 119905-partite graph 119870119905(3) for 119905 ⩾ 3

119887RM(119870119905(3)) = 119899 minus 1 by Theorem 157 where 119899 = 3119905

which shows that this upper bound of 119899 minus 1 on 119887RM(119866) inTheorem 156 is sharp This fact and 120574

119877(119870

119905(3)) = 4 give

a negative answer to the following two problems posed byDehgardi et al [121]Prove or Disprove for any connected graph 119866 of order 119899 ge 3(i) 119887RM(119866) = 119899 minus 1 if and only if 119866 cong 119870

3 (ii) 119887RM(119866) ⩽ 119899 minus

120574119877(119866) + 1Note that a complete 119905-partite graph 119870

119905(3) is (119899 minus 3)-

regular and 119887RM(119870119905(3)) = 119899 minus 1 for 119905 ⩾ 3 where 119899 = 3119905

Hu and Xu further determined that 119887119877(119866) = 119899 minus 2 for any

(119899 minus 3)-regular graph 119866 of order 119899 ⩽ 5 except 119870119905(3)

Theorem 158 (Hu and Xu [123] 2011) For any (119899minus3)-regulargraph 119866 of order 119899 other than119870

119905(3) 119887RM(119866) = 119899 minus 2 for 119899 ⩾ 5

For a tree 119879 with order 119899 ⩾ 3 Ebadi and PushpaLatha[120] and Jafari Rad and Volkmann [119] independentlyobtained an upper bound on 119887RM(119879)

Theorem 159 119887RM(119879) ⩽ 3 for any tree 119879 with order 119899 ⩾ 3

Theorem 160 (Bahremandpour et al [118] 2013) 119887RM(1198752 times119875119899) = 2 for 119899 ⩾ 2

Theorem 161 (Jafari Rad and Volkmann [119] 2011) Let119866 bea graph with order at least three

(a) If (119909 119910 119911) is a path of length 2 in 119866 then 119887RM(119866) ⩽119889119866(119909) + 119889

119866(119910) + 119889

119866(119911) minus 3 minus |119873

119866(119909) cap 119873

119866(119910)|

(b) If 119866 is connected then 119887RM(119866) ⩽ 120582(119866) + 2Δ(119866) minus 3

Theorem 161 (b) implies 119887RM(119866) ⩽ 120575(119866) + 2Δ(119866) minus 3 for aconnected graph 119866 Note that for a planar graph 119866 120575(119866) ⩽ 5moreover 120575(119866) ⩽ 3 if the girth at least 4 and 120575(119866) ⩽ 2 if thegirth at least 6 These two facts show that 119887RM(119866) ⩽ 2Δ(119866) +2 for a connected planar graph 119866 Jafari Rad and Volkmann[124] improved this bound

Theorem 162 (Jafari Rad and Volkmann [124] 2011) For anyconnected planar graph 119866 of order 119899 ⩾ 3 with girth 119892(119866)

119887RM (119866)

⩽

2Δ (119866)

Δ (119866) + 6

Δ (119866) + 5 119894119891 119866 119888119900119899119905119886119894119899119904 119899119900 V119890119903119905119894119888119890119904 119900119891 119889119890119892119903119890119890 119891119894V119890

Δ (119866) + 4 119894119891 119892 (119866) ⩾ 4

Δ (119866) + 3 119894119891 119892 (119866) ⩾ 5

Δ (119866) + 2 119894119891 119892 (119866) ⩾ 6

Δ (119866) + 1 119894119891 119892 (119866) ⩾ 8

(65)

According toTheorem 157 119887RM(119862119899) = 3 = Δ(119862

119899)+1 for a

cycle 119862119899of length 119899 ⩾ 8 with 119899 equiv 2 (mod 3) and therefore

the last upper bound inTheorem 162 is sharp at least for Δ =2

Combining the fact that every planar graph 119866 withminimum degree 5 contains an edge 119909119910 with 119889

119866(119909) = 5

and 119889119866(119910) isin 5 6 with Theorem 161 (a) Akbari et al [125]

obtained the following result

26 International Journal of Combinatorics

middot middot middot

middot middot middot

middot middot middot

middot middot middot

119866

Figure 9 The graph 119866 is constructed from 119866

Theorem 163 (Akbari et al [125] 2012) 119887RM(119866) ⩽ 15 forevery planar graph 119866

It remains open to show whether the bound inTheorem 163 is sharp or not Though finding a planargraph 119866 with 119887RM(119866) = 15 seems to be difficult Akbari etal [125] constructed an infinite family of planar graphs withRoman bondage number equal to 7 by proving the followingresult

Theorem 164 (Akbari et al [125] 2012) Let 119866 be a graph oforder 119899 and 119866 is the graph of order 5119899 obtained from 119866 byattaching the central vertex of a copy of a path 119875

5to each vertex

of119866 (see Figure 9) Then 120574RM(119866) = 4119899 and 119887RM(119866) = 120575(119866)+2

ByTheorem 164 infinitemany planar graphswith Romanbondage number 7 can be obtained by considering any planargraph 119866 with 120575(119866) = 5 (eg the icosahedron graph)

Conjecture 165 (Akbari et al [125] 2012) The Romanbondage number of every planar graph is at most 7

For general bounds the following observation is directlyobtained which is similar to Observation 1

Observation 4 Let 119866 be a graph of order 119899 with maximumdegree at least two Assume that 119867 is a spanning subgraphobtained from 119866 by removing 119896 edges If 120574RM(119867) = 120574RM(119866)then 119887

119877(119867) ⩽ 119887RM(119866) ⩽ 119887RM(119867) + 119896

Theorem 166 (Bahremandpour et al [118] 2013) For anyconnected graph 119866 of order 119899 if 119899 ⩾ 3 and 120574RM(119866) = 120574(119866) + 1then

119887RM (119866) ⩽ min 119887 (119866) 119899Δ (66)

where 119899Δis the number of vertices with maximum degree Δ in

119866

Observation 5 For any graph 119866 if 120574(119866) = 120574RM(119866) then119887RM(119866) ⩽ 119887(119866)

Theorem 167 (Bahremandpour et al [118] 2013) For everyRoman graph 119866

119887RM (119866) ⩾ 119887 (119866) (67)

The bound is sharp for cycles on 119899 vertices where 119899 equiv 0 (mod3)

The strict inequality in Theorem 167 can hold for exam-ple 119887(119862

3119896+2) = 2 lt 3 = 119887RM(1198623119896+2

) byTheorem 157A graph 119866 is called to be vertex Roman domination-

critical if 120574RM(119866 minus 119909) lt 120574RM(119866) for every vertex 119909 in 119866 If119866 has no isolated vertices then 120574RM(119866) ⩽ 2120574(119866) ⩽ 2120573(119866)If 120574RM(119866) = 2120573(119866) then 120574RM(119866) = 2120574(119866) and hence 119866 is aRoman graph In [30] Volkmann gave a lot of graphs with120574(119866) = 120573(119866) The following result is similar to Theorem 57

Theorem 168 (Bahremandpour et al [118] 2013) For anygraph 119866 if 120574RM(119866) = 2120573(119866) then

(a) 119887RM(119866) ⩾ 120575(119866)

(b) 119887RM(119866) ⩾ 120575(119866)+1 if119866 is a vertex Roman domination-critical graph

If 120574RM(119866) = 2 then obviously 119887RM(119866) ⩽ 120575(119866) So weassume 120574RM(119866) ⩾ 3 Dehgardi et al [121] proved 119887RM(119866) ⩽(120574RM(119866) minus 2)Δ(119866) + 1 and posed the following problem if 119866is a connected graph of order 119899 ⩾ 4 with 120574RM(119866) ⩾ 3 then

119887RM (119866) ⩽ (120574RM (119866) minus 2) Δ (119866) (68)

Theorem 161 (a) shows that the inequality (68) holds if120574RM(119866) ⩾ 5 Thus the bound in (68) is of interest only when120574RM(119866) is 3 or 4

Lemma 169 (Bahremandpour et al [118] 2013) For anynonempty graph 119866 of order 119899 ⩾ 3 120574RM(119866) = 3 if and onlyif Δ(119866) = 119899 minus 2

The following result shows that (68) holds for all graphs119866 of order 119899 ⩾ 4 with 120574RM(119866) = 3 or 4 which improvesTheorem 161 (b)

Theorem 170 (Bahremandpour et al [118] 2013) For anyconnected graph 119866 of order 119899 ⩾ 4

119887RM (119866) le Δ (119866) = 119899 minus 2 119894119891 120574RM (119866) = 3

Δ (119866) + 120575 (119866) minus 1 119894119891 120574RM (119866) = 4(69)

with the first equality if and only if 119866 cong 1198624

Recently Akbari and Qajar [122] have proved the follow-ing result

Theorem 171 (Akbari and Qajar [122]) For any connectedgraph 119866 of order 119899 ⩾ 3

119887RM (119866) ⩽ 119899 minus 120574RM (119866) + 5 (70)

International Journal of Combinatorics 27

We conclude this section with the following problems

Problem 8 Characterize all connected graphs 119866 of order 119899 ⩾3 for which 119887RM(119866) = 119899 minus 1

Problem 9 Prove or disprove if 119866 is a connected graph oforder 119899 ⩾ 3 then

119887RM (119866) ⩽ 119899 minus 120574RM (119866) + 3 (71)

Note that 120574119877(119870

119905(3)) = 4 and 119887RM(119870119905

(3)) = 119899minus1 where 119899 =3119905 for 119905 ⩾ 3 The upper bound on 119887RM(119866) given in Problem 9is sharp if Problem 9 has positive answer

95 Rainbow Bondage Numbers For a positive integer 119896 a119896-rainbow dominating function of a graph 119866 is a function 119891from the vertex-set 119881(119866) to the set of all subsets of the set1 2 119896 such that for any vertex 119909 in 119866 with 119891(119909) = 0 thecondition

⋃

119910isin119873119866(119909)

119891 (119910) = 1 2 119896 (72)

is fulfilledThe weight of a 119896-rainbow dominating function 119891on 119866 is the value

119891 (119866) = sum

119909isin119881(119866)

1003816100381610038161003816119891 (119909)1003816100381610038161003816 (73)

The 119896-rainbow domination number of a graph 119866 denoted by120574119903119896(119866) is the minimum weight of a 119896-rainbow dominating

function 119891 of 119866Note that 120574

1199031(119866) is the classical domination number 120574(119866)

The 119896-rainbow domination number is introduced by Bresaret al [126] Rainbow domination of a graph 119866 coincides withordinary domination of the Cartesian product of 119866 with thecomplete graph in particular 120574

119903119896(119866) = 120574(119866 times 119870

119896) for any

positive integer 119896 and any graph 119866 [126] Hartnell and Rall[127] proved that 120574(119879) lt 120574

1199032(119879) for any tree 119879 no graph has

120574 = 120574119903119896for 119896 ⩾ 3 for any 119896 ⩾ 2 and any graph 119866 of order 119899

min 119899 120574 (119866) + 119896 minus 2 ⩽ 120574119903119896(119866) ⩽ 119896120574 (119866) (74)

The introduction of rainbow domination is motivatedby the study of paired domination in Cartesian productsof graphs where certain upper bounds can be expressed interms of rainbow domination that is for any graph 119866 andany graph 119867 if 119881(119867) can be partitioned into 119896120574

119875-sets then

120574119875(119866 times 119867) ⩽ (1119896)120592(119867)120574

119903119896(119866) proved by Bresar et al [126]

Dehgardi et al [128] introduced the concept of 119896-rainbowbondage number of graphs Let 119866 be a graph with maximumdegree at least two The 119896-rainbow bondage number 119887

119903119896(119866)

of 119866 is the minimum cardinality of all sets 119861 sube 119864(119866) forwhich 120574

119903119896(119866 minus 119861) gt 120574

119903119896(119866) Since in the study of 119896-rainbow

bondage number the assumption Δ(119866) ⩾ 2 is necessarywe always assume that when we discuss 119887

119903119896(119866) all graphs

involved satisfy Δ(119866) ⩾ 2The 2-rainbow bondage number of some special families

of graphs has been determined

Theorem 172 (Dehgardi et al [128]) For any integer 119899 ⩾ 3

1198871199032(119875

119899) = 1 119891119900119903 119899 ⩾ 2

1198871199032(119862

119899) =

1 119894119891 119899 equiv 0 (mod 4) 2 119900119905ℎ119890119903119908119894119904119890

1198871199032(119870

119899119899) =

1 119894119891 119899 = 2

3 119894119891 119899 = 3

6 119894119891 119899 = 4

119898 119894119891 119899 ⩾ 5

(75)

Theorem 173 (Dehgardi et al [128]) For any connected graph119866 of order 119899 ⩾ 3 119887

1199032(119866) ⩽ 120582(119866) + 2Δ(119866) minus 3

Theorem 174 (Dehgardi et al [128]) For any tree 119879 of order119899 ⩾ 3 119887

1199032(119879) ⩽ 2

Theorem 175 (Dehgardi et al [128]) If 119866 is a planar graphwithmaximumdegree at least two then 119887

1199032(119866) ⩽ 15Moreover

if the girth 119892(119866) ⩾ 4 then 1198871199032(119866) ⩽ 11 and if 119892(119866) ⩾ 6 then

1198871199032(119866) ⩽ 9

Theorem 176 (Dehgardi et al [128]) For a complete bipartitegraph 119870

119901119902 if 2119896 + 1 ⩽ 119901 ⩽ 119902 then 119887

119903119896(119870

119901119902) = 119901

Theorem177 (Dehgardi et al [128]) For a complete graph119870119899

if 119899 ⩾ 119896 + 1 then 119887119903119896(119870

119899) = lceil119896119899(119896 + 1)rceil

The special case 119896 = 1 of Theorem 177 can be found inTheorem 1 The following is a simple observation similar toObservation 1

Observation 6 (Dehgardi et al [128]) Let 119866 be a graphwith maximum degree at least two 119867 a spanning subgraphobtained from 119866 by removing 119896 edges If 120574

119903119896(119867) = 120574

119903119896(119866)

then 119887119903119896(119867) ⩽ 119887

119903119896(119866) ⩽ 119887

119903119896(119867) + 119896

Theorem 178 (Dehgardi et al [128]) For every graph 119866 with120574119903119896(119866) = 119896120574(119866) 119887

119903119896(119866) ⩾ 119887(119866)

A graph 119866 is said to be vertex 119896-rainbow domination-critical if 120574

119903119896(119866 minus 119909) lt 120574

119903119896(119866) for every vertex 119909 in 119866 It is

clear that if 119866 has no isolated vertices then 120574119903119896(119866) ⩽ 119896120574(119866) ⩽

119896120573(119866) where 120573(119866) is the vertex-covering number of119866 Thusif 120574

119903119896(119866) = 119896120573(119866) then 120574

119903119896(119866) = 119896120574(119866)

Theorem 179 (Dehgardi et al [128]) For any graph 119866 if120574119903119896(119866) = 119896120573(119866) then

(a) 119887119903119896(119866) ⩾ 120575(119866)

(b) 119887119903119896(119866) ⩾ 120575(119866)+1 if119866 is vertex 119896-rainbow domination-

critical

As far as we know there are no more results on the 119896-rainbow bondage number of graphs in the literature

28 International Journal of Combinatorics

10 Results on Digraphs

Although domination has been extensively studied in undi-rected graphs it is natural to think of a dominating setas a one-way relationship between vertices of the graphIndeed among the earliest literatures on this subject vonNeumann and Morgenstern [129] used what is now calleddomination in digraphs to find solution (or kernels whichare independent dominating sets) for cooperative 119899-persongames Most likely the first formulation of domination byBerge [130] in 1958 was given in the context of digraphs andonly some years latter by Ore [131] in 1962 for undirectedgraphs Despite this history examination of domination andits variants in digraphs has been essentially overlooked (see[4] for an overview of the domination literature) Thus thereare few if any such results on domination for digraphs in theliterature

The bondage number and its related topics for undirectedgraph have become one of major areas both in theoreticaland applied researches However until recently Carlsonand Develin [42] Shan and Kang [132] and Huang andXu [133 134] studied the bondage number for digraphsindependently In this section we introduce their resultsfor general digraphs Results for some special digraphs suchas vertex-transitive digraphs are introduced in the nextsection

101 Upper Bounds for General Digraphs Let 119866 = (119881 119864)

be a digraph without loops and parallel edges A digraph 119866is called to be asymmetric if whenever (119909 119910) isin 119864(119866) then(119910 119909) notin 119864(119866) and to be symmetric if (119909 119910) isin 119864(119866) implies(119910 119909) isin 119864(119866) For a vertex 119909 of 119881(119866) the sets of out-neighbors and in-neighbors of 119909 are respectively defined as119873

+

119866(119909) = 119910 isin 119881(119866) (119909 119910) isin 119864(119866) and 119873minus

119866(119909) = 119909 isin

119881(119866) (119909 119910) isin 119864(119866) the out-degree and the in-degree of119909 are respectively defined as 119889+

119866(119909) = |119873

+

119866(119909)| and 119889minus

119866(119909) =

|119873+

119866(119909)| Denote themaximum and theminimum out-degree

(resp in-degree) of 119866 by Δ+(119866) and 120575+(119866) (resp Δminus(119866) and120575minus

(119866))The degree 119889119866(119909) of 119909 is defined as 119889+

119866(119909)+119889

minus

119866(119909) and

the maximum and the minimum degrees of119866 are denoted byΔ(119866) and 120575(119866) respectively that is Δ(119866) = max119889

119866(119909) 119909 isin

119881(119866) and 120575(119866) = min119889119866(119909) 119909 isin 119881(119866) Note that the

definitions here are different from the ones in the textbookon digraphs see for example Xu [1]

A subset 119878 of119881(119866) is called a dominating set if119881(119866) = 119878cup119873

+

119866(119878) where119873+

119866(119878) is the set of out-neighbors of 119878Then just

as for undirected graphs 120574(119866) is the minimum cardinalityof a dominating set and the bondage number 119887(119866) is thesmallest cardinality of a set 119861 of edges such that 120574(119866 minus 119861) gt120574(119866) if such a subset 119861 sube 119864(119866) exists Otherwise we put119887(119866) = infin

Some basic results for undirected graphs stated inSection 3 can be generalized to digraphs For exampleTheorem 18 is generalized by Carlson and Develin [42]Huang andXu [133] and Shan andKang [132] independentlyas follows

Theorem 180 Let 119866 be a digraph and (119909 119910) isin 119864(119866) Then119887(119866) ⩽ 119889

119866(119910) + 119889

minus

119866(119909) minus |119873

minus

119866(119909) cap 119873

minus

119866(119910)|

Corollary 181 For a digraph 119866 119887(119866) ⩽ 120575minus(119866) + Δ(119866)

Since 120575minus

(119866) ⩽ (12)Δ(119866) from Corollary 181Conjecture 53 is valid for digraphs

Corollary 182 For a digraph 119866 119887(119866) ⩽ (32)Δ(119866)

In the case of undirected graphs the bondage number of119866119899= 119870

119899times 119870

119899achieves this bound However it was shown

by Carlson and Develin in [42] that if we take the symmetricdigraph119866

119899 we haveΔ(119866

119899) = 4(119899minus1) 120574(119866

119899) = 119899 and 119887(119866

119899) ⩽

4(119899 minus 1) + 1 = Δ(119866119899) + 1 So this family of digraphs cannot

show that the bound in Corollary 182 is tightCorresponding to Conjecture 51 which is discredited

for undirected graphs and is valid for digraphs the sameconjecture for digraphs can be proposed as follows

Conjecture 183 (Carlson and Develin [42] 2006) 119887(119866) ⩽Δ(119866) + 1 for any digraph 119866

If Conjecture 183 is true then the following results showsthat this upper bound is tight since 119887(119870

119899times119870

119899) = Δ(119870

119899times119870

119899)+1

for a complete digraph 119870119899

In 2007 Shan and Kang [132] gave some tight upperbounds on the bondage numbers for some asymmetricdigraphs For example 119887(119879) ⩽ Δ(119879) for any asymmetricdirected tree 119879 119887(119866) ⩽ Δ(119866) for any asymmetric digraph 119866with order at least 4 and 120574(119866) ⩽ 2 For planar digraphs theyobtained the following results

Theorem 184 (Shan and Kang [132] 2007) Let 119866 be aasymmetric planar digraphThen 119887(119866) ⩽ Δ(119866)+2 and 119887(119866) ⩽Δ(119866) + 1 if Δ(119866) ⩾ 5 and 119889minus

119866(119909) ⩾ 3 for every vertex 119909 with

119889119866(119909) ⩾ 4

102 Results for Some Special Digraphs The exact values andbounds on 119887(119866) for some standard digraphs are determined

Theorem185 (Huang andXu [133] 2006) For a directed cycle119862119899and a directed path 119875

119899

119887 (119862119899) =

3 119894119891 119899 119894119904 119900119889119889

2 119894119891 119899 119894119904 119890V119890119899

119887 (119875119899) =

2 119894119891 119899 119894119904 119900119889119889

1 119894119891 119899 119894119904 119890V119890119899

(76)

For the de Bruijn digraph 119861(119889 119899) and the Kautz digraph119870(119889 119899)

119887 (119861 (119889 119899)) = 119889 119894119891 119899 119894119904 119900119889119889

119889 ⩽ 119887 (119861 (119889 119899)) ⩽ 2119889 119894119891 119899 119894119904 119890V119890119899

119887 (119870 (119889 119899)) = 119889 + 1

(77)

Like undirected graphs we can define the total domi-nation number and the total bondage number On the totalbondage numbers for some special digraphs the knownresults are as follows

International Journal of Combinatorics 29

Theorem 186 (Huang and Xu [134] 2007) For a directedcycle 119862

119899and a directed path 119875

119899 119887

119879(119875

119899) and 119887

119879(119862

119899) all do not

exist For a complete digraph 119870119899

119887119879(119870

119899) = infin 119894119891 119899 = 1 2

119887119879(119870

119899) = 3 119894119891 119899 = 3

119899 ⩽ 119887119879(119870

119899) ⩽ 2119899 minus 3 119894119891 119899 ⩾ 4

(78)

The extended de Bruijn digraph EB(119889 119899 1199021 119902

119901) and

the extended Kautz digraph EK(119889 119899 1199021 119902

119901) are intro-

duced by Shibata and Gonda [135] If 119901 = 1 then theyare the de Bruijn digraph B(119889 119899) and the Kautz digraphK(119889 119899) respectively Huang and Xu [134] determined theirtotal domination numbers In particular their total bondagenumbers for general cases are determined as follows

Theorem 187 (Huang andXu [134] 2007) If 119889 ⩾ 2 and 119902119894⩾ 2

for each 119894 = 1 2 119901 then

119887119879(EB(119889 119899 119902

1 119902

119901)) = 119889

119901

minus 1

119887119879(EK(119889 119899 119902

1 119902

119901)) = 119889

119901

(79)

In particular for the de Bruijn digraph B(119889 119899) and the Kautzdigraph K(119889 119899)

119887119879(B (119889 119899)) = 119889 minus 1 119887

119879(K (119889 119899)) = 119889 (80)

Zhang et al [136] determined the bondage number incomplete 119905-partite digraphs

Theorem 188 (Zhang et al [136] 2009) For a complete 119905-partite digraph 119870

11989911198992119899119905

where 1198991⩽ 119899

2⩽ sdot sdot sdot ⩽ 119899

119905

119887 (11987011989911198992119899119905

)

=

119898 119894119891 119899119898= 1 119899

119898+1⩾ 2 119891119900119903 119904119900119898119890

119898 (1 ⩽ 119898 lt 119905)

4119905 minus 3 119894119891 1198991= 119899

2= sdot sdot sdot = 119899

119905= 2

119905minus1

sum

119894=1

119899119894+ 2 (119905 minus 1) 119900119905ℎ119890119903119908119894119904119890

(81)

Since an undirected graph can be thought of as a sym-metric digraph any result for digraphs has an analogy forundirected graphs in general In view of this point studyingthe bondage number for digraphs is more significant thanfor undirected graphs Thus we should further study thebondage number of digraphs and try to generalize knownresults on the bondage number and related variants for undi-rected graphs to digraphs prove or disprove Conjecture 183In particular determine the exact values of 119887(B(119889 119899)) for aneven 119899 and 119887

119879(119870

119899) for 119899 ⩾ 4

11 Efficient Dominating Sets

A dominating set 119878 of a graph 119866 is called to be efficient if forevery vertex 119909 in119866 |119873

119866[119909]cap119878| = 1 if119866 is a undirected graph

or |119873minus

119866[119909] cap 119878| = 1 if 119866 is a directed graph The concept of an

efficient set was proposed by Bange et al [137] in 1988 In theliterature an efficient set is also called a perfect dominatingset (see Livingston and Stout [138] for an explanation)

From definition if 119878 is an efficient dominating set of agraph 119866 then 119878 is certainly an independent set and everyvertex not in 119878 is adjacent to exactly one vertex in 119878

It is also clear from definition that a dominating set 119878 isefficient if and only if N

119866[119878] = 119873

119866[119909] 119909 isin 119878 for the

undirected graph 119866 or N+

119866[119878] = 119873

+

119866[119909] 119909 isin 119878 for the

digraph119866 is a partition of119881(119866) where the induced subgraphby119873

119866[119909] or119873+

119866[119909] is a star or an out-star with the root 119909

The efficient domination has important applications inmany areas such as error-correcting codes and receivesmuch attention in the late years

The concept of efficient dominating sets is a measureof the efficiency of domination in graphs Unfortunately asshown in [137] not every graph has an efficient dominatingset andmoreover it is anNP-complete problem to determinewhether a given graph has an efficient dominating set Inaddition it was shown by Clark [139] in 1993 that for a widerange of 119901 almost every random undirected graph 119866 isin

G(120592 119901) has no efficient dominating sets This means thatundirected graphs possessing an efficient dominating set arerare However it is easy to show that every undirected graphhas an orientationwith an efficient dominating set (see Bangeet al [140])

In 1993 Barkauskas and Host [141] showed that deter-mining whether an arbitrary oriented graph has an efficientdominating set is NP-complete Even so the existence ofefficient dominating sets for some special classes of graphs hasbeen examined see for example Dejter and Serra [142] andLee [143] for Cayley graph Gu et al [144] formeshes and toriObradovic et al [145] Huang and Xu [36] Reji Kumar andMacGillivray [146] for circulant graphs Harary graphs andtori and Van Wieren et al [147] for cube-connected cycles

In this section we introduce results of the bondagenumber for some graphs with efficient dominating sets dueto Huang and Xu [36]

111 Results for General Graphs In this subsection we intro-duce some results on bondage numbers obtained by applyingefficient dominating sets We first state the two followinglemmas

Lemma 189 For any 119896-regular graph or digraph 119866 of order 119899120574(119866) ⩾ 119899(119896 + 1) with equality if and only if 119866 has an efficientdominating set In addition if 119866 has an efficient dominatingset then every efficient dominating set is certainly a 120574-set andvice versa

Let 119890 be an edge and 119878 a dominating set in 119866 We say that119890 supports 119878 if 119890 isin (119878 119878) where (119878 119878) = (119909 119910) isin 119864(119866) 119909 isin 119878 119910 isin 119878 Denote by 119904(119866) the minimum number of edgeswhich support all 120574-sets in 119866

Lemma 190 For any graph or digraph 119866 119887(119866) ⩾ 119904(119866) withequality if 119866 is regular and has an efficient dominating set

30 International Journal of Combinatorics

A graph 119866 is called to be vertex-transitive if its automor-phism group Aut(119866) acts transitively on its vertex-set 119881(119866)A vertex-transitive graph is regular Applying Lemmas 189and 190 Huang and Xu obtained some results on bondagenumbers for vertex-transitive graphs or digraphs

Theorem 191 (Huang and Xu [36] 2008) For any vertex-transitive graph or digraph 119866 of order 119899

119887 (119866) ⩾

lceil119899

2120574 (119866)rceil 119894119891 119866 119894119904 119906119899119889119894119903119890119888119905119890119889

lceil119899

120574 (119866)rceil 119894119891 119866 119894119904 119889119894119903119890119888119905119890119889

(82)

Theorem192 (Huang andXu [36] 2008) Let119866 be a 119896-regulargraph of order 119899 Then

119887(119866) ⩽ 119896 if119866 is undirected and 119899 ⩾ 120574(119866)(119896+1)minus119896+1119887(119866) ⩽ 119896+1+ℓ if119866 is directed and 119899 ⩾ 120574(119866)(119896+1)minusℓwith 0 ⩽ ℓ ⩽ 119896 minus 1

Next we introduce a better upper bound on 119887(119866) To thisaim we need the following concept which is a generalizationof the concept of the edge-covering of a graph 119866 For 1198811015840

sube

119881(119866) and 1198641015840 sube 119864(119866) we say that 1198641015840covers 1198811015840 and call 1198641015840an edge-covering for 1198811015840 if there exists an edge (119909 119910) isin 1198641015840 forany vertex 119909 isin 1198811015840 For 119910 isin 119881(119866) let 1205731015840[119910] be the minimumcardinality over all edge-coverings for119873minus

119866[119910]

Theorem 193 (Huang and Xu [36] 2008) If 119866 is a 119896-regulargraph with order 120574(119866)(119896 + 1) then 119887(119866) ⩽ 1205731015840[119910] for any 119910 isin119881(119866)

Theupper bound on 119887(119866) given inTheorem 193 is tight inview of 119887(119862

119899) for a cycle or a directed cycle 119862

119899(seeTheorems

1 and 185 resp)It is easy to see that for a 119896-regular graph 119866 lceil(119896 + 1)2rceil ⩽

1205731015840

[119910] ⩽ 119896 when 119866 is undirected and 1205731015840[119910] = 119896 + 1 when 119866 isdirected By this fact and Lemma 189 the following theoremis merely a simple combination of Theorems 191 and 193

Theorem 194 (Huang and Xu [36] 2008) Let 119866 be a vertex-transitive graph of degree 119896 If 119866 has an efficient dominatingset then

lceil119896 + 1

2rceil ⩽ 119887 (119866) ⩽ 119896 119894119891 119866 119894119904 119906119899119889119894119903119890119888119905119890119889

119887 (119866) = 119896 + 1 119894119891 119866 119894119904 119889119894119903119890119888119905119890119889

(83)

Theorem 195 (Huang and Xu [36] 2008) If 119866 is an undi-rected vertex-transitive cubic graph with order 4120574(119866) and girth119892(119866) ⩽ 5 then 119887(119866) = 2

The proof of Theorem 195 leads to a byproduct If 119892 =5 let (119906

1 119906

2 119906

3 119906

4 119906

5) be a cycle and let D

119894be the family

of efficient dominating sets containing 119906119894for each 119894 isin

1 2 3 4 5 ThenD119894capD

119895= 0 for 119894 = 1 2 5 Then 119866 has

at least 5119904 efficient dominating sets where 119904 = 119904(119866) But thereare only 4s (=ns120574) distinct efficient dominating sets in119866This

contradiction implies that an undirected vertex-transitivecubic graph with girth five has no efficient dominating setsBut a similar argument for 119892(119866) = 3 4 or 119892(119866) ⩾ 6 couldnot give any contradiction This is consistent with the resultthat CCC(119899) a vertex-transitive cubic graph with girth 119899 if3 ⩽ 119899 ⩽ 8 or girth 8 if 119899 ⩾ 9 has efficient dominating setsfor all 119899 ⩾ 3 except 119899 = 5 (see Theorem 199 in the followingsubsection)

112 Results for Cayley Graphs In this subsection we useTheorem 194 to determine the exact values or approximativevalues of bondage numbers for some special vertex-transitivegraphs by characterizing the existence of efficient dominatingsets in these graphs

Let Γ be a nontrivial finite group 119878 a nonempty subset ofΓ without the identity element of Γ A digraph 119866 is defined asfollows

119881 (119866) = Γ (119909 119910) isin 119864 (119866) lArrrArr 119909minus1

119910 isin 119878

for any 119909 119910 isin Γ(84)

is called a Cayley digraph of the group Γ with respect to119878 denoted by 119862

Γ(119878) If 119878minus1 = 119904minus1 119904 isin 119878 = 119878 then

119862Γ(119878) is symmetric and is called a Cayley undirected graph

a Cayley graph for short Cayley graphs or digraphs arecertainly vertex-transitive (see Xu [1])

A circulant graph 119866(119899 119878) of order 119899 is a Cayley graph119862119885119899

(119878) where 119885119899= 0 119899 minus 1 is the addition group of

order 119899 and 119878 is a nonempty subset of119885119899without the identity

element and hence is a vertex-transitive digraph of degree|119878| If 119878minus1 = 119878 then119866(119899 119878) is an undirected graph If 119878 = 1 119896where 2 ⩽ 119896 ⩽ 119899 minus 2 we write 119866(119899 1 119896) for 119866(119899 1 119896) or119866(119899 plusmn1 plusmn119896) and call it a double-loop circulant graph

Huang and Xu [36] showed that for directed 119866 =

119866(119899 1 119896) lceil1198993rceil ⩽ 120574(119866) ⩽ lceil1198992rceil and 119866 has an efficientdominating set if and only if 3 | 119899 and 119896 equiv 2 (mod 3) fordirected 119866 = 119866(119899 1 119896) and 119896 = 1198992 lceil1198995rceil ⩽ 120574(119866) ⩽ lceil1198993rceiland 119866 has an efficient dominating set if and only if 5 | 119899 and119896 equiv plusmn2 (mod 5) By Theorem 194 we obtain the bondagenumber of a double loop circulant graph if it has an efficientdominating set

Theorem 196 (Huang and Xu [36] 2008) Let 119866 be a double-loop circulant graph 119866(119899 1 119896) If 119866 is directed with 3 | 119899and 119896 equiv 2 (mod 3) or 119866 is undirected with 5 | 119899 and119896 equiv plusmn2 (mod 5) then 119887(119866) = 3

The 119898 times 119899 torus is the Cartesian product 119862119898times 119862

119899of

two cycles and is a Cayley graph 119862119885119898times119885119899

(119878) where 119878 =

(0 1) (1 0) for directed cycles and 119878 = (0 plusmn1) (plusmn1 0) forundirected cycles and hence is vertex-transitive Gu et al[144] showed that the undirected torus119862

119898times119862

119899has an efficient

dominating set if and only if both 119898 and 119899 are multiplesof 5 Huang and Xu [36] showed that the directed torus119862119898times 119862

119899has an efficient dominating set if and only if both

119898 and 119899 are multiples of 3 Moreover they found a necessarycondition for a dominating set containing the vertex (0 0)in 119862

119898times 119862

119899to be efficient and obtained the following

result

International Journal of Combinatorics 31

Theorem 197 (Huang and Xu [36] 2008) Let 119866 = 119862119898times 119862

119899

If 119866 is undirected and both119898 and 119899 are multiples of 5 or if 119866 isdirected and both119898 and 119899 are multiples of 3 then 119887(119866) = 3

The hypercube 119876119899

is the Cayley graph 119862Γ(119878)

where Γ = 1198852times sdot sdot sdot times 119885

2= (119885

2)119899 and 119878 =

100 sdot sdot sdot 0 010 sdot sdot sdot 0 00 sdot sdot sdot 01 Lee [143] showed that119876119899has an efficient dominating set if and only if 119899 = 2119898 minus 1

for a positive integer119898 ByTheorem 194 the following resultis obtained immediately

Theorem 198 (Huang and Xu [36] 2008) If 119899 = 2119898 minus 1 for apositive integer119898 then 2119898minus1

⩽ 119887(119876119899) ⩽ 2

119898

minus 1

Problem 10 Determine the exact value of 119887(119876119899) for 119899 = 2119898minus1

The 119899-dimensional cube-connected cycle denoted byCCC(119899) is constructed from the 119899-dimensional hypercube119876119899by replacing each vertex 119909 in 119876

119899with an undirected cycle

119862119899of length 119899 and linking the 119894th vertex of the 119862

119899to the 119894th

neighbor of 119909 It has been proved that CCC(119899) is a Cayleygraph and hence is a vertex-transitive graph with degree 3Van Wieren et al [147] proved that CCC(119899) has an efficientdominating set if and only if 119899 = 5 From Theorems 194 and195 the following result can be derived

Theorem 199 (Huang and Xu [36] 2008) Let 119866 = 119862119862119862(119899)be the 119899-dimensional cube-connected cycles with 119899 ⩾ 3 and119899 = 5 Then 120574(119866) = 1198992119899minus2 and 2 ⩽ 119887(119866) ⩽ 3 In addition119887(119862119862119862(3)) = 119887(119862119862119862(4)) = 2

Problem 11 Determine the exact value of 119887(CCC(119899)) for 119899 ⩾5

Acknowledgments

This work was supported by NNSF of China (Grants nos11071233 61272008) The author would like to express hisgratitude to the anonymous referees for their kind sugges-tions and comments on the original paper to ProfessorSheikholeslami and Professor Jafari Rad for sending thereferences [128] and [81] which resulted in this version

References

[1] J-M Xu Theory and Application of Graphs vol 10 of NetworkTheory and Applications Kluwer Academic Dordrecht TheNetherlands 2003

[2] S THedetniemi andR C Laskar ldquoBibliography on dominationin graphs and some basic definitions of domination parame-tersrdquo Discrete Mathematics vol 86 no 1ndash3 pp 257ndash277 1990

[3] TW Haynes S T Hedetniemi and P J Slater Fundamentals ofDomination in Graphs vol 208 ofMonographs and Textbooks inPure and Applied Mathematics Marcel Dekker New York NYUSA 1998

[4] TWHaynes S THedetniemi and P J Slater EdsDominationin Graphs vol 209 of Monographs and Textbooks in Pure andAppliedMathematicsMarcelDekker NewYorkNYUSA 1998

[5] M R Garey andD S JohnsonComputers and Intractability WH Freeman and Company San Francisco Calif USA 1979

[6] H B Walikar and B D Acharya ldquoDomination critical graphsrdquoNational Academy Science Letters vol 2 pp 70ndash72 1979

[7] D Bauer F Harary J Nieminen and C L Suffel ldquoDominationalteration sets in graphsrdquo Discrete Mathematics vol 47 no 2-3pp 153ndash161 1983

[8] J F Fink M S Jacobson L F Kinch and J Roberts ldquoThebondage number of a graphrdquo Discrete Mathematics vol 86 no1ndash3 pp 47ndash57 1990

[9] F-T Hu and J-M Xu ldquoThe bondage number of (119899 minus 3)-regulargraph G of order nrdquo Ars Combinatoria to appear

[10] U Teschner ldquoNew results about the bondage number of agraphrdquoDiscreteMathematics vol 171 no 1ndash3 pp 249ndash259 1997

[11] B L Hartnell and D F Rall ldquoA bound on the size of a graphwith given order and bondage numberrdquo Discrete Mathematicsvol 197-198 pp 409ndash413 1999 16th British CombinatorialConference (London 1997)

[12] B L Hartnell and D F Rall ldquoA characterization of trees inwhich no edge is essential to the domination numberrdquo ArsCombinatoria vol 33 pp 65ndash76 1992

[13] J Topp and P D Vestergaard ldquo120572119896- and 120574

119896-stable graphsrdquo

Discrete Mathematics vol 212 no 1-2 pp 149ndash160 2000[14] A Meir and J W Moon ldquoRelations between packing and

covering numbers of a treerdquo Pacific Journal of Mathematics vol61 no 1 pp 225ndash233 1975

[15] B L Hartnell L K Joslashrgensen P D Vestergaard and CA Whitehead ldquoEdge stability of the k-domination numberof treesrdquo Bulletin of the Institute of Combinatorics and ItsApplications vol 22 pp 31ndash40 1998

[16] F-T Hu and J-M Xu ldquoOn the complexity of the bondage andreinforcement problemsrdquo Journal of Complexity vol 28 no 2pp 192ndash201 2012

[17] B L Hartnell and D F Rall ldquoBounds on the bondage numberof a graphrdquo Discrete Mathematics vol 128 no 1ndash3 pp 173ndash1771994

[18] Y-L Wang ldquoOn the bondage number of a graphrdquo DiscreteMathematics vol 159 no 1ndash3 pp 291ndash294 1996

[19] G H Fricke TW Haynes S M Hedetniemi S T Hedetniemiand R C Laskar ldquoExcellent treesrdquo Bulletin of the Institute ofCombinatorics and Its Applications vol 34 pp 27ndash38 2002

[20] V Samodivkin ldquoThe bondage number of graphs good and badverticesrdquoDiscussiones Mathematicae GraphTheory vol 28 no3 pp 453ndash462 2008

[21] J E Dunbar T W Haynes U Teschner and L VolkmannldquoBondage insensitivity and reinforcementrdquo in Domination inGraphs vol 209 of Monographs and Textbooks in Pure andAppliedMathematics pp 471ndash489 Dekker NewYork NY USA1998

[22] H Liu and L Sun ldquoThe bondage and connectivity of a graphrdquoDiscrete Mathematics vol 263 no 1ndash3 pp 289ndash293 2003

[23] A-H Esfahanian and S L Hakimi ldquoOn computing a con-ditional edge-connectivity of a graphrdquo Information ProcessingLetters vol 27 no 4 pp 195ndash199 1988

[24] R C Brigham P Z Chinn and R D Dutton ldquoVertexdomination-critical graphsrdquoNetworks vol 18 no 3 pp 173ndash1791988

[25] V Samodivkin ldquoDomination with respect to nondegenerateproperties bondage numberrdquoThe Australasian Journal of Com-binatorics vol 45 pp 217ndash226 2009

[26] U Teschner ldquoA new upper bound for the bondage numberof graphs with small domination numberrdquo The AustralasianJournal of Combinatorics vol 12 pp 27ndash35 1995

32 International Journal of Combinatorics

[27] J Fulman D Hanson and G MacGillivray ldquoVertex dom-ination-critical graphsrdquoNetworks vol 25 no 2 pp 41ndash43 1995

[28] U Teschner ldquoA counterexample to a conjecture on the bondagenumber of a graphrdquo Discrete Mathematics vol 122 no 1ndash3 pp393ndash395 1993

[29] U Teschner ldquoThe bondage number of a graph G can be muchgreater than Δ(G)rdquo Ars Combinatoria vol 43 pp 81ndash87 1996

[30] L Volkmann ldquoOn graphs with equal domination and coveringnumbersrdquoDiscrete AppliedMathematics vol 51 no 1-2 pp 211ndash217 1994

[31] L A Sanchis ldquoMaximum number of edges in connected graphswith a given domination numberrdquoDiscreteMathematics vol 87no 1 pp 65ndash72 1991

[32] S Klavzar and N Seifter ldquoDominating Cartesian products ofcyclesrdquoDiscrete AppliedMathematics vol 59 no 2 pp 129ndash1361995

[33] H K Kim ldquoA note on Vizingrsquos conjecture about the bondagenumberrdquo Korean Journal of Mathematics vol 10 no 2 pp 39ndash42 2003

[34] M Y Sohn X Yuan and H S Jeong ldquoThe bondage number of1198623times119862

119899rdquo Journal of the KoreanMathematical Society vol 44 no

6 pp 1213ndash1231 2007[35] L Kang M Y Sohn and H K Kim ldquoBondage number of the

discrete torus 119862119899times 119862

4rdquo Discrete Mathematics vol 303 no 1ndash3

pp 80ndash86 2005[36] J Huang and J-M Xu ldquoThe bondage numbers and efficient

dominations of vertex-transitive graphsrdquoDiscrete Mathematicsvol 308 no 4 pp 571ndash582 2008

[37] C J Xiang X Yuan andM Y Sohn ldquoDomination and bondagenumber of119862

3times119862

119899rdquoArs Combinatoria vol 97 pp 299ndash310 2010

[38] M S Jacobson and L F Kinch ldquoOn the domination numberof products of graphs Irdquo Ars Combinatoria vol 18 pp 33ndash441984

[39] Y-Q Li ldquoA note on the bondage number of a graphrdquo ChineseQuarterly Journal of Mathematics vol 9 no 4 pp 1ndash3 1994

[40] F-T Hu and J-M Xu ldquoBondage number of mesh networksrdquoFrontiers ofMathematics inChina vol 7 no 5 pp 813ndash826 2012

[41] R Frucht and F Harary ldquoOn the corona of two graphsrdquoAequationes Mathematicae vol 4 pp 322ndash325 1970

[42] K Carlson and M Develin ldquoOn the bondage number of planarand directed graphsrdquoDiscreteMathematics vol 306 no 8-9 pp820ndash826 2006

[43] U Teschner ldquoOn the bondage number of block graphsrdquo ArsCombinatoria vol 46 pp 25ndash32 1997

[44] J Huang and J-M Xu ldquoNote on conjectures of bondagenumbers of planar graphsrdquo Applied Mathematical Sciences vol6 no 65ndash68 pp 3277ndash3287 2012

[45] L Kang and J Yuan ldquoBondage number of planar graphsrdquoDiscrete Mathematics vol 222 no 1ndash3 pp 191ndash198 2000

[46] M Fischermann D Rautenbach and L Volkmann ldquoRemarkson the bondage number of planar graphsrdquo Discrete Mathemat-ics vol 260 no 1ndash3 pp 57ndash67 2003

[47] J Hou G Liu and J Wu ldquoBondage number of planar graphswithout small cyclesrdquoUtilitasMathematica vol 84 pp 189ndash1992011

[48] J Huang and J-M Xu ldquoThe bondage numbers of graphs withsmall crossing numbersrdquo Discrete Mathematics vol 307 no 15pp 1881ndash1897 2007

[49] Q Ma S Zhang and J Wang ldquoBondage number of 1-planargraphrdquo Applied Mathematics vol 1 no 2 pp 101ndash103 2010

[50] J Hou and G Liu ldquoA bound on the bondage number of toroidalgraphsrdquoDiscreteMathematics Algorithms and Applications vol4 no 3 Article ID 1250046 5 pages 2012

[51] Y-C Cao J-M Xu and X-R Xu ldquoOn bondage number oftoroidal graphsrdquo Journal of University of Science and Technologyof China vol 39 no 3 pp 225ndash228 2009

[52] Y-C Cao J Huang and J-M Xu ldquoThe bondage number ofgraphs with crossing number less than fourrdquo Ars Combinatoriato appear

[53] H K Kim ldquoOn the bondage numbers of some graphsrdquo KoreanJournal of Mathematics vol 11 no 2 pp 11ndash15 2004

[54] A Gagarin and V Zverovich ldquoUpper bounds for the bondagenumber of graphs on topological surfacesrdquo Discrete Mathemat-ics 2012

[55] J Huang ldquoAn improved upper bound for the bondage numberof graphs on surfacesrdquoDiscrete Mathematics vol 312 no 18 pp2776ndash2781 2012

[56] A Gagarin and V Zverovich ldquoThe bondage number of graphson topological surfaces and Teschnerrsquos conjecturerdquo DiscreteMathematics vol 313 no 6 pp 796ndash808 2013

[57] V Samodivkin ldquoNote on the bondage number of graphs ontopological surfacesrdquo httparxivorgabs12086203

[58] E J Cockayne R M Dawes and S T Hedetniemi ldquoTotaldomination in graphsrdquoNetworks vol 10 no 3 pp 211ndash219 1980

[59] R Laskar J Pfaff S M Hedetniemi and S T HedetniemildquoOn the algorithmic complexity of total dominationrdquo Society forIndustrial and Applied Mathematics vol 5 no 3 pp 420ndash4251984

[60] J Pfaff R C Laskar and S T Hedetniemi ldquoNP-completeness oftotal and connected domination and irredundance for bipartitegraphsrdquo Tech Rep 428 Department of Mathematical SciencesClemson University 1983

[61] M A Henning ldquoA survey of selected recent results on totaldomination in graphsrdquoDiscrete Mathematics vol 309 no 1 pp32ndash63 2009

[62] V R Kulli and D K Patwari ldquoThe total bondage number ofa graphrdquo in Advances in Graph Theory pp 227ndash235 VishwaGulbarga India 1991

[63] F-T Hu Y Lu and J-M Xu ldquoThe total bondage number ofgrid graphsrdquo Discrete Applied Mathematics vol 160 no 16-17pp 2408ndash2418 2012

[64] J Cao M Shi and B Wu ldquoResearch on the total bondagenumber of a spectial networkrdquo in Proceedings of the 3rdInternational Joint Conference on Computational Sciences andOptimization (CSO rsquo10) pp 456ndash458 May 2010

[65] N Sridharan MD Elias and V S A Subramanian ldquoTotalbondage number of a graphrdquo AKCE International Journal ofGraphs and Combinatorics vol 4 no 2 pp 203ndash209 2007

[66] N Jafari Rad and J Raczek ldquoSome progress on total bondage ingraphsrdquo Graphs and Combinatorics 2011

[67] T W Haynes and P J Slater ldquoPaired-domination and thepaired-domatic numberrdquoCongressusNumerantium vol 109 pp65ndash72 1995

[68] T W Haynes and P J Slater ldquoPaired-domination in graphsrdquoNetworks vol 32 no 3 pp 199ndash206 1998

[69] X Hou ldquoA characterization of 2120574 120574119875-treesrdquoDiscrete Mathemat-

ics vol 308 no 15 pp 3420ndash3426 2008[70] K E Proffitt T W Haynes and P J Slater ldquoPaired-domination

in grid graphsrdquo Congressus Numerantium vol 150 pp 161ndash1722001

International Journal of Combinatorics 33

[71] H Qiao L KangM Cardei andD-Z Du ldquoPaired-dominationof treesrdquo Journal of Global Optimization vol 25 no 1 pp 43ndash542003

[72] E Shan L Kang and M A Henning ldquoA characterizationof trees with equal total domination and paired-dominationnumbersrdquo The Australasian Journal of Combinatorics vol 30pp 31ndash39 2004

[73] J Raczek ldquoPaired bondage in treesrdquo Discrete Mathematics vol308 no 23 pp 5570ndash5575 2008

[74] J A Telle and A Proskurowski ldquoAlgorithms for vertex parti-tioning problems on partial k-treesrdquo SIAM Journal on DiscreteMathematics vol 10 no 4 pp 529ndash550 1997

[75] G S Domke J H Hattingh M A Henning and L R MarkusldquoRestrained domination in treesrdquoDiscreteMathematics vol 211no 1ndash3 pp 1ndash9 2000

[76] G S Domke J H Hattingh M A Henning and L R MarkusldquoRestrained domination in graphs with minimum degree twordquoJournal of Combinatorial Mathematics and Combinatorial Com-puting vol 35 pp 239ndash254 2000

[77] G S Domke J H Hattingh S T Hedetniemi R C Laskarand L R Markus ldquoRestrained domination in graphsrdquo DiscreteMathematics vol 203 no 1ndash3 pp 61ndash69 1999

[78] J H Hattingh and E J Joubert ldquoAn upper bound for therestrained domination number of a graph with minimumdegree at least two in terms of order and minimum degreerdquoDiscrete Applied Mathematics vol 157 no 13 pp 2846ndash28582009

[79] M A Henning ldquoGraphs with large restrained dominationnumberrdquo Discrete Mathematics vol 197-198 pp 415ndash429 199916th British Combinatorial Conference (London 1997)

[80] J H Hattingh and A R Plummer ldquoRestrained bondage ingraphsrdquo Discrete Mathematics vol 308 no 23 pp 5446ndash54532008

[81] N Jafari Rad R Hasni J Raczek and L Volkmann ldquoTotalrestrained bondage in graphsrdquoActaMathematica Sinica vol 29no 6 pp 1033ndash1042 2013

[82] J Cyman and J Raczek ldquoOn the total restrained dominationnumber of a graphrdquoThe Australasian Journal of Combinatoricsvol 36 pp 91ndash100 2006

[83] M A Henning ldquoGraphs with large total domination numberrdquoJournal of Graph Theory vol 35 no 1 pp 21ndash45 2000

[84] D-X Ma X-G Chen and L Sun ldquoOn total restraineddomination in graphsrdquoCzechoslovakMathematical Journal vol55 no 1 pp 165ndash173 2005

[85] B Zelinka ldquoRemarks on restrained domination and totalrestrained domination in graphsrdquo Czechoslovak MathematicalJournal vol 55 no 2 pp 393ndash396 2005

[86] W Goddard and M A Henning ldquoIndependent domination ingraphs a survey and recent resultsrdquo Discrete Mathematics vol313 no 7 pp 839ndash854 2013

[87] J H Zhang H L Liu and L Sun ldquoIndependence bondagenumber and reinforcement number of some graphsrdquo Transac-tions of Beijing Institute of Technology vol 23 no 2 pp 140ndash1422003

[88] K D Dharmalingam Studies in Graph Theorymdashequitabledomination and bottleneck domination [PhD thesis] MaduraiKamaraj University Madurai India 2006

[89] GDeepakND Soner andAAlwardi ldquoThe equitable bondagenumber of a graphrdquo Research Journal of Pure Algebra vol 1 no9 pp 209ndash212 2011

[90] E Sampathkumar and H B Walikar ldquoThe connected domina-tion number of a graphrdquo Journal of Mathematical and PhysicalSciences vol 13 no 6 pp 607ndash613 1979

[91] K White M Farber and W Pulleyblank ldquoSteiner trees con-nected domination and strongly chordal graphsrdquoNetworks vol15 no 1 pp 109ndash124 1985

[92] M Cadei M X Cheng X Cheng and D-Z Du ldquoConnecteddomination in Ad Hoc wireless networksrdquo in Proceedings ofthe 6th International Conference on Computer Science andInformatics (CSampI rsquo02) 2002

[93] J F Fink and M S Jacobson ldquon-domination in graphsrdquo inGraph Theory with Applications to Algorithms and ComputerScience Wiley-Interscience Publications pp 283ndash300 WileyNew York NY USA 1985

[94] M Chellali O Favaron A Hansberg and L Volkmann ldquok-domination and k-independence in graphs a surveyrdquo Graphsand Combinatorics vol 28 no 1 pp 1ndash55 2012

[95] Y Lu X Hou J-M Xu andN Li ldquoTrees with uniqueminimump-dominating setsrdquo Utilitas Mathematica vol 86 pp 193ndash2052011

[96] Y Lu and J-M Xu ldquoThe p-bondage number of treesrdquo Graphsand Combinatorics vol 27 no 1 pp 129ndash141 2011

[97] Y Lu and J-M Xu ldquoThe 2-domination and 2-bondage numbersof grid graphs A manuscriptrdquo httparxivorgabs12044514

[98] M Krzywkowski ldquoNon-isolating 2-bondage in graphsrdquo TheJournal of the Mathematical Society of Japan vol 64 no 4 2012

[99] M A Henning O R Oellermann and H C Swart ldquoBoundson distance domination parametersrdquo Journal of CombinatoricsInformation amp System Sciences vol 16 no 1 pp 11ndash18 1991

[100] F Tian and J-M Xu ldquoBounds for distance domination numbersof graphsrdquo Journal of University of Science and Technology ofChina vol 34 no 5 pp 529ndash534 2004

[101] F Tian and J-M Xu ldquoDistance domination-critical graphsrdquoApplied Mathematics Letters vol 21 no 4 pp 416ndash420 2008

[102] F Tian and J-M Xu ldquoA note on distance domination numbersof graphsrdquo The Australasian Journal of Combinatorics vol 43pp 181ndash190 2009

[103] F Tian and J-M Xu ldquoAverage distances and distance domina-tion numbersrdquo Discrete Applied Mathematics vol 157 no 5 pp1113ndash1127 2009

[104] Z-L Liu F Tian and J-M Xu ldquoProbabilistic analysis of upperbounds for 2-connected distance k-dominating sets in graphsrdquoTheoretical Computer Science vol 410 no 38ndash40 pp 3804ndash3813 2009

[105] M A Henning O R Oellermann and H C Swart ldquoDistancedomination critical graphsrdquo Journal of Combinatorial Mathe-matics and Combinatorial Computing vol 44 pp 33ndash45 2003

[106] T J Bean M A Henning and H C Swart ldquoOn the integrityof distance domination in graphsrdquo The Australasian Journal ofCombinatorics vol 10 pp 29ndash43 1994

[107] M Fischermann and L Volkmann ldquoA remark on a conjecturefor the (kp)-domination numberrdquoUtilitasMathematica vol 67pp 223ndash227 2005

[108] T Korneffel D Meierling and L Volkmann ldquoA remark on the(22)-domination numberrdquo Discussiones Mathematicae GraphTheory vol 28 no 2 pp 361ndash366 2008

[109] Y Lu X Hou and J-M Xu ldquoOn the (22)-domination numberof treesrdquo Discussiones Mathematicae Graph Theory vol 30 no2 pp 185ndash199 2010

34 International Journal of Combinatorics

[110] H Li and J-M Xu ldquo(dm)-domination numbers of m-connected graphsrdquo Rapports de Recherche 1130 LRI UniversiteParis-Sud 1997

[111] S M Hedetniemi S T Hedetniemi and T V Wimer ldquoLineartime resource allocation algorithms for treesrdquo Tech Rep URI-014 Department of Mathematical Sciences Clemson Univer-sity 1987

[112] M Farber ldquoDomination independent domination and dualityin strongly chordal graphsrdquo Discrete Applied Mathematics vol7 no 2 pp 115ndash130 1984

[113] G S Domke and R C Laskar ldquoThe bondage and reinforcementnumbers of 120574

119891for some graphsrdquoDiscrete Mathematics vol 167-

168 pp 249ndash259 1997[114] G S Domke S T Hedetniemi and R C Laskar ldquoFractional

packings coverings and irredundance in graphsrdquo vol 66 pp227ndash238 1988

[115] E J Cockayne P A Dreyer Jr S M Hedetniemi andS T Hedetniemi ldquoRoman domination in graphsrdquo DiscreteMathematics vol 278 no 1ndash3 pp 11ndash22 2004

[116] I Stewart ldquoDefend the roman empirerdquo Scientific American vol281 pp 136ndash139 1999

[117] C S ReVelle and K E Rosing ldquoDefendens imperiumromanum a classical problem in military strategyrdquoThe Ameri-can Mathematical Monthly vol 107 no 7 pp 585ndash594 2000

[118] A Bahremandpour F-T Hu S M Sheikholeslami and J-MXu ldquoOn the Roman bondage number of a graphrdquo DiscreteMathematics Algorithms and Applications vol 5 no 1 ArticleID 1350001 15 pages 2013

[119] N Jafari Rad and L Volkmann ldquoRoman bondage in graphsrdquoDiscussiones Mathematicae Graph Theory vol 31 no 4 pp763ndash773 2011

[120] K Ebadi and L PushpaLatha ldquoRoman bondage and Romanreinforcement numbers of a graphrdquo International Journal ofContemporary Mathematical Sciences vol 5 pp 1487ndash14972010

[121] N Dehgardi S M Sheikholeslami and L Volkman ldquoOn theRoman k-bondage number of a graphrdquo AKCE InternationalJournal of Graphs and Combinatorics vol 8 pp 169ndash180 2011

[122] S Akbari and S Qajar ldquoA note on Roman bondage number ofgraphsrdquo Ars Combinatoria to Appear

[123] F-T Hu and J-M Xu ldquoRoman bondage numbers of somegraphsrdquo httparxivorgabs11093933

[124] N Jafari Rad and L Volkmann ldquoOn the Roman bondagenumber of planar graphsrdquo Graphs and Combinatorics vol 27no 4 pp 531ndash538 2011

[125] S Akbari M Khatirinejad and S Qajar ldquoA note on the romanbondage number of planar graphsrdquo Graphs and Combinatorics2012

[126] B Bresar M A Henning and D F Rall ldquoRainbow dominationin graphsrdquo Taiwanese Journal of Mathematics vol 12 no 1 pp213ndash225 2008

[127] B L Hartnell and D F Rall ldquoOn dominating the Cartesianproduct of a graph and 120572krdquo Discussiones Mathematicae GraphTheory vol 24 no 3 pp 389ndash402 2004

[128] N Dehgardi S M Sheikholeslami and L Volkman ldquoThe k-rainbow bondage number of a graphrdquo to appear in DiscreteApplied Mathematics

[129] J von Neumann and O Morgenstern Theory of Games andEconomic Behavior Princeton University Press Princeton NJUSA 1944

[130] C BergeTheorıe des Graphes et ses Applications 1958[131] O Ore Theory of Graphs vol 38 American Mathematical

Society Colloquium Publications 1962[132] E Shan and L Kang ldquoBondage number in oriented graphsrdquoArs

Combinatoria vol 84 pp 319ndash331 2007[133] J Huang and J-M Xu ldquoThe bondage numbers of extended

de Bruijn and Kautz digraphsrdquo Computers amp Mathematics withApplications vol 51 no 6-7 pp 1137ndash1147 2006

[134] J Huang and J-M Xu ldquoThe total domination and total bondagenumbers of extended de Bruijn and Kautz digraphsrdquoComputersamp Mathematics with Applications vol 53 no 8 pp 1206ndash12132007

[135] Y Shibata and Y Gonda ldquoExtension of de Bruijn graph andKautz graphrdquo Computers amp Mathematics with Applications vol30 no 9 pp 51ndash61 1995

[136] X Zhang J Liu and JMeng ldquoThebondage number in completet-partite digraphsrdquo Information Processing Letters vol 109 no17 pp 997ndash1000 2009

[137] D W Bange A E Barkauskas and P J Slater ldquoEfficient dom-inating sets in graphsrdquo in Applications of Discrete Mathematicspp 189ndash199 SIAM Philadelphia Pa USA 1988

[138] M Livingston and Q F Stout ldquoPerfect dominating setsrdquoCongressus Numerantium vol 79 pp 187ndash203 1990

[139] L Clark ldquoPerfect domination in random graphsrdquo Journal ofCombinatorial Mathematics and Combinatorial Computing vol14 pp 173ndash182 1993

[140] D W Bange A E Barkauskas L H Host and L H ClarkldquoEfficient domination of the orientations of a graphrdquo DiscreteMathematics vol 178 no 1ndash3 pp 1ndash14 1998

[141] A E Barkauskas and L H Host ldquoFinding efficient dominatingsets in oriented graphsrdquo Congressus Numerantium vol 98 pp27ndash32 1993

[142] I J Dejter and O Serra ldquoEfficient dominating sets in CayleygraphsrdquoDiscrete AppliedMathematics vol 129 no 2-3 pp 319ndash328 2003

[143] J Lee ldquoIndependent perfect domination sets in Cayley graphsrdquoJournal of Graph Theory vol 37 no 4 pp 213ndash219 2001

[144] WGu X Jia and J Shen ldquoIndependent perfect domination setsin meshes tori and treesrdquo Unpublished

[145] N Obradovic J Peters and G Ruzic ldquoEfficient domination incirculant graphswith two chord lengthsrdquo Information ProcessingLetters vol 102 no 6 pp 253ndash258 2007

[146] K Reji Kumar and G MacGillivray ldquoEfficient domination incirculant graphsrdquo Discrete Mathematics vol 313 no 6 pp 767ndash771 2013

[147] D Van Wieren M Livingston and Q F Stout ldquoPerfectdominating sets on cube-connected cyclesrdquo Congressus Numer-antium vol 97 pp 51ndash70 1993

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Mathematical PhysicsAdvances in

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

International Journal of Combinatorics 15

119889119866(119910) ⩽ 3 and 119889

119866(119910) = 4 for every 119910 isin 119873

119866(119909) If 119868 is

independent no vertices adjacent to vertices of degree 6 and

cr (119866) lt max5119899

3+ |119868| minus 2119899

4+ 28

117119899

3+ 40

16 (29)

then 119887(119866) ⩽ 7

Corollary 92 (Huang and Xu [48] 2007) Let 119866 be aconnected graph with cr(119866) ⩽ 2 If 119868 = 119909 isin 119881(119866) 119889

119866(119909) =

5 119889119866(119910 119909) ⩾ 3 if 119889

119866(119910) ⩽ 3 and 119889

119866(119910) = 4 for every 119910 isin

119873119866(119909) is independent and no vertices adjacent to vertices of

degree 6 then 119887(119866) ⩽ 7

Theorem93 (Huang andXu [48] 2007) Let119866 be a connectedgraph If 119866 satisfies

(a) 5cr(119866) + 1198995lt 2119899

2+ 3119899

3+ 2119899

4+ 12 or

(b) 7cr(119866) + 21198995lt 3119899

2+ 4119899

4+ 24

then 119887(119866) ⩽ 7

Proposition 94 (Huang and Xu [48] 2007) Let 119866 be aconnected graph with no vertices of degrees four and five Ifcr(119866) ⩽ 2 then 119887(119866) ⩽ 6

Theorem 95 (Huang and Xu [48] 2007) If 119866 is a connectedgraph with cr(119866) ⩽ 4 and not 4-regular when cr(119866) = 4 then119887(119866) ⩽ Δ(119866) + 2

The above results generalize some known results forplanar graphs For example Corollary 90 and Theorem 95contain Theorem 74 the first condition in Theorem 93 con-tains the second condition inTheorem 77

From Corollary 88 and Theorem 95 we suggest the fol-lowing questions

Question 2 Is there a

(a) 4-regular graph 119866 with cr(119866) = 4 such that 119887(119866) ⩾ 7

(b) 3-regular graph 119866 with cr ⩽ 4 and 119892(119866) ⩾ 5 such that119887(119866) = 5

(c) 3-regular graph 119866 with cr = 3 and 119892(119866) = 6 or 7 suchthat 119887(119866) = 5

72 Carlson-Develin Methods In this subsection we intro-duce an elegant method presented by Carlson and Develin[42] to obtain some upper bounds for the bondage numberof a graph

Suppose that 119866 is a connected graph We say that 119866 hasgenus 120588 if 119866 can be embedded in a surface 119878 with 120588 handlessuch that edges are pairwise disjoint except possibly for end-vertices Let119866 be an embedding of119866 in surface 119878 and let120601(119866)denote the number of regions in 119866 The boundary of everyregion contains at least three edges and every edge is on theboundary of atmost two regions (the two regions are identicalwhen 119890 is a cut-edge) For any edge 119890 of 119866 let 1199031

119866(119890) and 1199032

119866(119890)

be the numbers of edges comprising the regions in 119866 whichthe edge 119890 borders It is clear that every 119890 = 119909119910 isin 119864(119866)

1199032

119866(119890) ⩾ 119903

1

119866(119890) ⩾ 4 if 1003816100381610038161003816119873119866

(119909) cap 119873119866(119910)1003816100381610038161003816 = 0

1199032

119866(119890) ⩾ 4 119903

1

119866(119890) ⩾ 3 if 1003816100381610038161003816119873119866

(119909) cap 119873119866(119910)1003816100381610038161003816 = 1

1199032

119866(119890) ⩾ 119903

1

119866(119890) ⩾ 3 if 1003816100381610038161003816119873119866

(119909) cap 119873119866(119910)1003816100381610038161003816 ⩾ 2

(30)

Following Carlson and Develin [42] for any edge 119890 = 119909119910 of119866 we define

119863119866(119890) =

1

119889119866(119909)+

1

119889119866(119910)

+1

1199031

119866(119890)+

1

1199032

119866(119890)minus 1 (31)

By the well-known Eulerrsquos formula

120592 (119866) minus 120576 (119866) + 120601 (119866) = 2 minus 2120588 (119866) (32)

it is easy to see that

sum

119890isin119864(119866)

119863119866(119890) = 120592 (119866) minus 120576 (119866) + 120601 (119866) = 2 minus 2120588 (119866) (33)

If 119866 is a planar graph that is 120588(119866) = 0 then

sum

119890isin119864(119866)

119863119866(119890) = 120592 (119866) minus 120576 (119866) + 120601 (119866) = 2 (34)

Combining these formulas withTheorem 18 Carlson andDevelin [42] gave a simple and intuitive proof ofTheorem 74and obtained the following result

Theorem 96 (Carlson and Develin [42] 2006) Let 119866 be aconnected graph which can be embedded on a torus Then119887(119866) ⩽ Δ(119866) + 3

Recently Hou and Liu [50] improved Theorem 96 asfollows

Theorem 97 (Hou and Liu [50] 2012) Let 119866 be a connectedgraph which can be embedded on a torus Then 119887(119866) ⩽ 9Moreover if Δ(119866) = 6 then 119887(119866) ⩽ 8

By the way Cao et al [51] generalized the result inTheorem 79 to a connected graph 119866 that can be embeddedon a torus that is

119887 (119866) le

6 if 119892 (119866) ⩾ 4 and 119866 is not 4-regular5 if 119892 (119866) ⩾ 54 if 119892 (119866) ⩾ 6 and 119866 is not 3-regular3 if 119892 (119866) ⩾ 8

(35)

Several authors used this method to obtain some resultson the bondage number For example Fischermann et al[46] used this method to prove the second conclusion inTheorem 78 Recently Cao et al [52] have used thismethod todeal with more general graphs with small crossing numbersFirst they found the following property

Lemma 98 120575(119866) ⩽ 5 for any graph 119866 with cr(119866) ⩽ 5

16 International Journal of Combinatorics

This result implies that 119887(119866) ⩽ Δ(119866) + 4 by Corollary 15for any graph 119866 with cr(119866) ⩽ 5 Cao et al established thefollowing relation in terms of maximum planar subgraphs

Lemma 99 (Cao et al [52]) Let 119866 be a connected graph withcrossing number cr(119866) For any maximum planar subgraph119867of 119866

sum

119890isin119864(119867)

119863119866(119890) ⩾ 2 minus

2cr (119866)120575 (119866)

(36)

Using Carlson-Develin method and combiningLemma 98 with Lemma 99 Cao et al proved the followingresults

Theorem 100 (Cao et al [52]) For any connected graph 119866119887(119866) ⩽ Δ(119866) + 2 if 119866 satisfies one of the following conditions

(a) cr(119866) ⩽ 3(b) cr(119866) = 4 and 119866 is not 4-regular(c) cr(119866) = 5 and 119866 contains no vertices of degree 4

Theorem101 (Cao et al [52]) Let119866 be a connected graphwithΔ(119866) = 5 and cr(119866) ⩽ 4 If no triangles contain two vertices ofdegree 5 then 119887(119866) ⩽ 6 = Δ(119866) + 1

Theorem 102 (Cao et al [52]) Let 119866 be a connected graphwith Δ(119866) ⩾ 6 and cr(119866) ⩽ 3 If Δ(119866) ⩾ 7 or if Δ(119866) = 6120575(119866) = 3 and every edge 119890 = 119909119910 with 119889

119866(119909) = 5 and 119889

119866(119910) = 6

is contained in atmost one triangle then 119887(119866) ⩽ min8 Δ(119866)+1

Using Carlson-Develin method it can be proved that119887(119866) ⩽ Δ(119866) + 2 if cr(119866) ⩽ 3 (see the first conclusion inTheorem 100) and not yet proved that 119887(119866) ⩽ 8 if cr(119866) ⩽3 although it has been proved by using other method (seeCorollary 90)

By using Carlson-Develin method Samodivkin [25]obtained some results on the bondage number for graphswithsome given properties

Kim [53] showed that 119887(119866) ⩽ Δ(119866) + 2 for a connectedgraph 119866 with genus 1 Δ(119866) ⩽ 5 and having a toroidalembedding of which at least one region is not 4-sidedRecently Gagarin and Zverovich [54] further have extendedCarlson-Develin ideas to establish a nice upper bound forarbitrary graphs that can be embedded on orientable ornonorientable topological surface

Theorem 103 (Gagarin and Zverovich [54] 2012) Let 119866 bea graph embeddable on an orientable surface of genus ℎ and anonorientable surface of genus 119896 Then 119887(119866) ⩽ minΔ(119866)+ℎ+2 Δ(119866) + 119896 + 1

This result generalizes the corresponding upper boundsin Theorems 74 and 96 for any orientable or nonori-entable topological surface By investigating the proof ofTheorem 103 Huang [55] found that the issue of the ori-entability can be avoided by using the Euler characteristic120594(= 120592(119866) minus 120576(119866) + 120601(119866)) instead of the genera ℎ and 119896 the

relations are 120594 = 2minus2ℎ and 120594 = 2minus119896 To obtain the best resultfromTheorem 103 onewants ℎ and 119896 as small as possible thisis equivalent to having 120594 as large as possible

According toTheorem 103 if119866 is planar (ℎ = 0 120594 = 2) orcan be embedded on the real projective plane (119896 = 1 120594 = 1)then 119887(119866) ⩽ Δ(119866) + 2 In all other cases Huang [55] had thefollowing improvement for Theorem 103 the proof is basedon the technique developed byCarlson-Develin andGagarin-Zverovich and includes some elementary calculus as a newingredient mainly the intermediate-value theorem and themean-value theorem

Theorem 104 (Huang [55] 2012) Let 119866 be a graph embed-dable on a surface whose Euler characteristic 120594 is as large aspossible If 120594 ⩽ 0 then 119887(119866) ⩽ Δ(119866)+lfloor119903rfloor where 119903 is the largestreal root of the following cubic equation in 119911

1199113

+ 21199112

+ (6120594 minus 7) 119911 + 18120594 minus 24 = 0 (37)

In addition if 120594 decreases then 119903 increases

The following result is asymptotically equivalent toTheorem 104

Theorem 105 (Huang [55] 2012) Let 119866 be a graph embed-dable on a surface whose Euler characteristic 120594 is as large aspossible If 120594 ⩽ 0 then 119887(119866) lt Δ(119866) + radic12 minus 6120594 + 12 orequivalently 119887(119866) ⩽ Δ(119866) + lceilradic12 minus 6120594 minus 12rceil

Also Gagarin andZverovich [54] indicated that the upperbound in Theorem 103 can be improved for larger values ofthe genera ℎ and 119896 by adjusting the proofs and stated thefollowing general conjecture

Conjecture 106 (Gagarin and Zverovich [54] 2012) For aconnected graph G of orientable genus ℎ and nonorientablegenus 119896

119887 (119866) ⩽ min 119888ℎ 119888

1015840

119896 Δ (119866) + 119900 (ℎ) Δ (119866) + 119900 (119896) (38)

where 119888ℎand 1198881015840

119896are constants depending respectively on the

orientable and nonorientable genera of 119866

In the recent paper Gagarin and Zverovich [56] providedconstant upper bounds for the bondage number of graphs ontopological surfaces which can be used as the first estimationfor the constants 119888

ℎand 1198881015840

119896of Conjecture 106

Theorem 107 (Gagarin and Zverovich [56] 2013) Let 119866 bea connected graph of order 119899 If 119866 can be embedded on thesurface of its orientable or nonorientable genus of the Eulercharacteristic 120594 then

(a) 120594 ⩾ 1 implies 119887(119866) ⩽ 10(b) 120594 ⩽ 0 and 119899 gt minus12120594 imply 119887(119866) ⩽ 11(c) 120594 ⩽ minus1 and 119899 ⩽ minus12120594 imply 119887(119866) ⩽ 11 +

3120594(radic17 minus 8120594 minus 3)(120594 minus 1) = 11 + 119874(radicminus120594)

Theorem 79 shows that a connected planar triangle-freegraph 119866 has 119887(119866) ⩽ 6 The following result generalizes to itall the other topological surfaces

International Journal of Combinatorics 17

Theorem 108 (Gagarin and Zverovich [56] 2013) Let 119866 be aconnected triangle-free graph of order 119899 If 119866 can be embeddedon the surface of its orientable or nonorientable genus of theEuler characteristic 120594 then

(a) 120594 ⩾ 1 implies 119887(119866) ⩽ 6(b) 120594 ⩽ 0 and 119899 gt minus8120594 imply 119887(119866) ⩽ 7(c) 120594 ⩽ minus1 and 119899 ⩽ minus8120594 imply 119887(119866) ⩽ 7minus4120594(1+radic2 minus 120594)

In [56] the authors also explicitly improved upperbounds in Theorem 103 and gave tight lower bounds forthe number of vertices of graphs 2-cell embeddable ontopological surfaces of a given genusThe interested reader isreferred to the original paper for details The following resultis interesting although its proof is easy

Theorem 109 (Samodivkin [57]) Let119866 be a graph with order119899 and girth 119892 If 119866 is embeddable on a surface whose Eulercharacteristic 120594 is as large as possible then

119887 (119866) ⩽ 3 +8

119892 minus 2minus

4120594119892

119899 (119892 minus 2) (39)

If 119866 is a planar graph with girth 119892 ⩾ 4 + 119894 for any 119894 isin0 1 2 then Theorem 109 leads to 119887(119866) ⩽ 6 minus 119894 which isthe result inTheorem 79 obtained by Fischermann et al [46]Also inmany cases the bound stated inTheorem 109 is betterthan those given byTheorems 103 and 105

8 Conditional Bondage Numbers

Since the concept of the bondage number is based upon thedomination all sorts of dominations which are variationsof the normal domination by adding restricted conditionsto the normal dominating set can develop a new ldquobondagenumberrdquo as long as a variation of domination number isgiven In this section we survey results on the bondagenumber under some restricted conditions

81 Total Bondage Numbers A dominating set 119878 of a graph119866 without isolated vertices is called total if the subgraphinduced by 119878 contains no isolated vertices The minimumcardinality of a total dominating set is called the totaldomination number of 119866 and denoted by 120574

119879(119866) It is clear

that 120574(119866) ⩽ 120574119879(119866) ⩽ 2120574(119866) for any graph 119866 without isolated

verticesThe total domination in graphs was introduced by Cock-

ayne et al [58] in 1980 Pfaff et al [59 60] in 1983 showedthat the problem determining total domination number forgeneral graphs is NP-complete even for bipartite graphs andchordal graphs Even now total domination in graphs hasbeen extensively studied in the literature In 2009 Henning[61] gave a survey of selected recent results on total domina-tion in graphs

The total bondage number of 119866 denoted by 119887119879(119866) is the

smallest cardinality of a subset 119861 sube 119864(119866) with the propertythat119866minus119861 contains no isolated vertices and 120574

119879(119866minus119861) gt 120574

119879(119866)

From definition 119887119879(119866)may not exist for some graphs for

example119866 = 1198701119899 We put 119887

119879(119866) = infin if 119887

119879(119866) does not exist

It is easy to see that 119887119879(119866) is finite for any connected graph 119866

other than 1198701 119870

2 119870

3 and 119870

1119899 In fact for such a graph 119866

since there is a path of length 3 in 119866 we can find 1198611sube 119864(119866)

such that 119866 minus 1198611is a spanning tree 119879 of 119866 containing a path

of length 3 So 120574119879(119879) = 120574

119879(119866minus119861

1) ⩾ 120574

119879(119866) For the tree119879 we

find 1198612sube 119864(119879) such that 120574

119879(119879 minus 119861

2) gt 120574

119879(119879) ⩾ 120574

119879(119866) Thus

we have 120574119879(119866minus119861

2minus119861

1) gt 120574

119879(119866) and so 119887

119879(119866) ⩽ |119861

1|+|119861

2| In

the following discussion we always assume that 119887119879(119866) exists

when a graph 119866 is mentionedIn 1991 Kulli and Patwari [62] first studied the total

bondage number of a graph and calculated the exact valuesof 119887

119879(119866) for some standard graphs

Theorem 110 (Kulli and Patwari [62] 1991) For 119899 ⩾ 4

119887119879(119862

119899) =

3 119894119891 119899 equiv 2 (mod 4) 2 119900119905ℎ119890119903119908119894119904119890

119887119879(119875

119899) =

2 119894119891 119899 equiv 2 (mod 4) 1 119900119905ℎ119890119903119908119894119904119890

119887119879(119870

119898119899) = 119898 119908119894119905ℎ 2 ⩽ 119898 ⩽ 119899

119887119879(119870

119899) =

4 119891119900119903 119899 = 4

2119899 minus 5 119891119900119903 119899 ⩾ 5

(40)

Recently Hu et al [63] have obtained some results on thetotal bondage number of the Cartesian product119875

119898times119875

119899of two

paths 119875119898and 119875

119899

Theorem 111 (Hu et al [63] 2012) For the Cartesian product119875119898times 119875

119899of two paths 119875

119898and 119875

119899

119887119879(119875

2times 119875

119899) =

1 119894119891 119899 equiv 0 (mod 3) 2 119894119891 119899 equiv 2 (mod 3) 3 119894119891 119899 equiv 1 (mod 3)

119887119879(119875

3times 119875

119899) = 1

119887119879(119875

4times 119875

119899)

= 1 119894119891 119899 equiv 1 (mod 5) = 2 119894119891 119899 equiv 4 (mod 5) ⩽ 3 119894119891 119899 equiv 2 (mod 5) ⩽ 4 119894119891 119899 equiv 0 3 (mod 5)

(41)

Generalized Petersen graphs are an important class ofcommonly used interconnection networks and have beenstudied recently By constructing a family of minimum totaldominating sets Cao et al [64] determined the total bondagenumber of the generalized Petersen graphs

FromTheorem 6 we know that 119887(119879) ⩽ 2 for a nontrivialtree 119879 But given any positive integer 119896 Sridharan et al [65]constructed a tree 119879 for which 119887

119879(119879) = 119896 Let119867

119896be the tree

obtained from a star1198701119896+1

by subdividing 119896 edges twice Thetree 119867

7is shown in Figure 6 It can be easily verified that

119887119879(119867

119896) = 119896 This fact can be stated as the following theorem

Theorem 112 (Sridharan et al [65] 2007) For any positiveinteger 119896 there exists a tree 119879 with 119887

119879(119879) = 119896

18 International Journal of Combinatorics

Figure 61198677

Combining Theorem 1 with Theorem 112 we have whatfollows

Corollary 113 (Sridharan et al [65] 2007) The bondagenumber and the total bondage number are unrelated even fortrees

However Sridharan et al [65] gave an upper bound onthe total bondage number of a tree in terms of its maximumdegree For any tree 119879 of order 119899 if 119879 =119870

1119899minus1 then 119887

119879(119879) =

minΔ(119879) (13)(119899 minus 1) Rad and Raczek [66] improved thisupper bound and gave a constructive characterization of acertain class of trees attaching the upper bound

Theorem 114 (Rad and Raczek [66] 2011) 119887119879(119879) ⩽ Δ(119879) minus 1

for any tree 119879 with maximum degree at least three

In general the decision problem for 119887119879(119866) is NP-complete

for any graph 119866 We state the decision problem for the totalbondage as follows

Problem 3 Consider the decision problemTotal Bondage ProblemInstance a graph 119866 and a positive integer 119887Question is 119887

119879(119866) ⩽ 119887

Theorem 115 (Hu and Xu [16] 2012) The total bondageproblem is NP-complete

Consequently in view of its computational hardnessit is significative to establish some sharp bounds on thetotal bondage number of a graph in terms of other graphicparameters In 1991 Kulli and Patwari [62] showed that119887119879(119866) ⩽ 2119899 minus 5 for any graph 119866 with order 119899 ⩾ 5 Sridharan

et al [65] improved this result as follows

Theorem 116 (Sridharan et al [65] 2007) For any graph 119866with order 119899 if 119899 ⩾ 5 then

119887119879(119866) ⩽

1

3(119899 minus 1) 119894119891 119866 119888119900119899119905119886119894119899119904 119899119900 119888119910119888119897119890119904

119899 minus 1 119894119891 119892 (119866) ⩾ 5

119899 minus 2 119894119891 119892 (119866) = 4

(42)

Rad and Raczek [66] also established some upperbounds on 119887

119879(119866) for a general graph 119866 In particular they

gave an upper bound on 119887119879(119866) in terms of the girth of

119866

Theorem 117 (Rad and Raczek [66] 2011) 119887119879(119866)⩽4Δ(119866) minus 5

for any graph 119866 with 119892(119866) ⩾ 4

82 Paired Bondage Numbers A dominating set 119878 of 119866 iscalled to be paired if the subgraph induced by 119878 containsa perfect matching The paired domination number of 119866denoted by 120574

119875(119866) is the minimum cardinality of a paired

dominating set of 119866 Clearly 120574119879(119866) ⩽ 120574

119875(119866) for every

connected graph 119866 with order at least two where 120574119879(119866) is

the total domination number of 119866 and 120574119875(119866) ⩽ 2120574(119866) for

any graph119866without isolated vertices Paired domination wasintroduced by Haynes and Slater [67 68] and further studiedin [69ndash72]

The paired bondage number of 119866 with 120575(119866) ⩾ 1 denotedby 119887P(119866) is the minimum cardinality among all sets of edges119861 sube 119864 such that 120575(119866 minus 119861) ⩾ 1 and 120574

119875(119866 minus 119861) gt 120574

119875(119866)

The concept of the paired bondage number was firstproposed by Raczek [73] in 2008The following observationsfollow immediately from the definition of the paired bondagenumber

Observation 2 Let 119866 be a graph with 120575(119866) ⩾ 1

(a) If 119867 is a subgraph of 119866 such that 120575(119867) ⩾ 1 then119887119875(119867) ⩽ 119887

119875(119866)

(b) If 119867 is a subgraph of 119866 such that 119887119875(119867) = 1 and 119896

is the number of edges removed to form119867 then 1 ⩽119887119875(119866) ⩽ 119896 + 1

(c) If 119909119910 isin 119864(119866) such that 119889119866(119909) 119889

119866(119910) gt 1 and 119909119910

belongs to each perfect matching of each minimumpaired dominating set of 119866 then 119887

119875(119866) = 1

Based on these observations 119887119875(119875

119899) ⩽ 2 In fact the

paired bondage number of a path has been determined

Theorem 118 (Raczek [73] 2008) For any positive integer 119896if 119899 ⩾ 2 then

119887119875(119875

119899) =

0 119894119891 119899 = 2 3 119900119903 5

1 119894119891 119899 = 4119896 4119896 + 3 119900119903 4119896 + 6

2 119900119905ℎ119890119903119908119894119904119890

(43)

For a cycle 119862119899of order 119899 ⩾ 3 since 120574

119875(119862

119899) = 120574

119875(119875

119899) from

Theorem 118 the following result holds immediately

Corollary 119 (Raczek [73] 2008) For any positive integer 119896if 119899 ⩾ 3 then

119887119875(119862

119899) =

0 119894119891 119899 = 3 119900119903 5

2 119894119891 119899 = 4119896 4119896 + 3 119900119903 4119896 + 6

3 119900119905ℎ119890119903119908119894119904119890

(44)

A wheel 119882119899 where 119899 ⩾ 4 is a graph with 119899 vertices

formed by connecting a single vertex to all vertices of a cycle119862119899minus1

Of course 120574119875(119882

119899) = 2

International Journal of Combinatorics 19

119896

Figure 7 A tree 119879 with 119887119875(119879) = 119896

Theorem 120 (Raczek [73] 2008) For positive integers 119899 and119898

119887119875(119870

119898119899) = 119898 119891119900119903 1 lt 119898 ⩽ 119899

119887119875(119882

119899) =

4 119894119891 119899 = 4

3 119894119891 119899 = 5

2 119900119905ℎ119890119903119908119894119904119890

(45)

Theorem 16 shows that if 119879 is a nontrivial tree then119887(119879) ⩽ 2 However no similar result exists for pairedbondage For any nonnegative integer 119896 let 119879

119896be a tree

obtained by subdividing all but one edge of a star 1198701119896+1

(asshown in Figure 7) It is easy to see that 119887

119875(119879

119896) = 119896 We

state this fact as the following theorem which is similar toTheorem 112

Theorem121 (Raczek [73] 2008) For any nonnegative integer119896 there exists a tree 119879 with 119887

119875(119879) = 119896

Consider the tree defined in Figure 5 for 119896 = 3 Then119887(119879

3) ⩽ 2 and 119887

119875(119879

3) = 3 On the other hand 119887(119875

4) = 2

and 119887119875(119875

4) = 1 Thus we have what follows which is similar

to Corollary 113

Corollary 122 The bondage number and the paired bondagenumber are unrelated even for trees

A constructive characterization of trees with 119887(119879) = 2is given by Hartnell and Rall in [12] Raczek [73] provideda constructive characterization of trees with 119887

119875(119879) = 0 In

order to state the characterization we define a labeling andthree simple operations on a tree119879 Let 119910 isin 119881(119879) and let ℓ(119910)be the label assigned to 119910

Operation T1 If ℓ(119910) = 119861 add a vertex 119909 and the

edge 119910119909 and let ℓ(119909) = 119860OperationT

2 If ℓ(119910) = 119862 add a path (119909

1 119909

2) and the

edge 1199101199091 and let ℓ(119909

1) = 119861 ℓ(119909

2) = 119860

Operation T3 If ℓ(119910) = 119861 add a path (119909

1 119909

2 119909

3)

and the edge 1199101199091 and let ℓ(119909

1) = 119862 ℓ(119909

2) = 119861 and

ℓ(1199093) = 119860

Let 1198752= (119906 V) with ℓ(119906) = 119860 and ℓ(V) = 119861 Let T be

the class of all trees obtained from the labeled 1198752by a finite

sequence of operationsT1 T

2 T

3

A tree 119879 in Figure 8 belongs to the familyTRaczek [73] obtained the following characterization of all

trees 119879 with 119887119875(119879) = 0

119860

119860

119860

119860

119860

119861 119862

119861 119862 119861

119861119860

Figure 8 A tree 119879 belong to the familyT

Theorem 123 (Raczek [73] 2008) For a tree 119879 119887119875(119879) = 0 if

and only if 119879 is inT

We state the decision problem for the paired bondage asfollows

Problem 4 Consider the decision problem

Paired Bondage ProblemInstance a graph 119866 and a positive integer 119887Question is 119887

119875(119866) ⩽ 119887

Conjecture 124 The paired bondage problem is NP-complete

83 Restrained Bondage Numbers A dominating set 119878 of 119866 iscalled to be restrained if the subgraph induced by 119878 containsno isolated vertices where 119878 = 119881(119866) 119878 The restraineddomination number of 119866 denoted by 120574

119877(119866) is the smallest

cardinality of a restrained dominating set of 119866 It is clear that120574119877(119866) exists and 120574(119866) ⩽ 120574

119877(119866) for any nonempty graph 119866

The concept of restrained domination was introducedby Telle and Proskurowski [74] in 1997 albeit indirectly asa vertex partitioning problem Concerning the study of therestrained domination numbers the reader is referred to [74ndash79]

In 2008 Hattingh and Plummer [80] defined therestrained bondage number 119887R(119866) of a nonempty graph 119866 tobe the minimum cardinality among all subsets 119861 sube 119864(119866) forwhich 120574

119877(119866 minus 119861) gt 120574

119877(119866)

For some simple graphs their restrained dominationnumbers can be easily determined and so restrained bondagenumbers have been also determined For example it is clearthat 120574

119877(119870

119899) = 1 Domke et al [77] showed that 120574

119877(119862

119899) =

119899 minus 2lfloor1198993rfloor for 119899 ⩾ 3 and 120574119877(119875

119899) = 119899 minus 2lfloor(119899 minus 1)3rfloor for 119899 ⩾ 1

Using these results Hattingh and Plummer [80] obtained therestricted bondage numbers for119870

119899 119862

119899 119875

119899 and119866 = 119870

11989911198992119899119905

as follows

Theorem 125 (Hattingh and Plummer [80] 2008) For 119899 ⩾ 3

119887R (119870119899) =

1 119894119891 119899 = 3

lceil119899

2rceil 119900119905ℎ119890119903119908119894119904119890

119887R (119862119899) =

1 119894119891 119899 equiv 0 (mod 3) 2 119900119905ℎ119890119903119908119894119904119890

119887R (119875119899) = 1 119899 ⩾ 4

20 International Journal of Combinatorics

119887R (119866)

=

lceil119898

2rceil 119894119891 119899

119898= 1 119886119899119889 119899

119898+1⩾ 2 (1 ⩽ 119898 lt 119905)

2119905 minus 2 119894119891 1198991= 119899

2= sdot sdot sdot = 119899

119905= 2 (119905 ⩾ 2)

2 119894119891 1198991= 2 119886119899119889 119899

2⩾ 3 (119905 = 2)

119905minus1

sum

119894=1

119899119894minus 1 119900119905ℎ119890119903119908119894119904119890

(46)

Theorem 126 (Hattingh and Plummer [80] 2008) For a tree119879 of order 119899 (⩾ 4) 119879 ≇ 119870

1119899minus1if and only if 119887R(119879) = 1

Theorem 126 shows that the restrained bondage numberof a tree can be computed in constant time However thedecision problem for 119887R(119866) is NP-complete even for bipartitegraphs

Problem 5 Consider the decision problem

Restrained Bondage Problem

Instance a graph 119866 and a positive integer 119887

Question is 119887R(119866) ⩽ 119887

Theorem 127 (Hattingh and Plummer [80] 2008) Therestrained bondage problem is NP-complete even for bipartitegraphs

Consequently in view of its computational hardnessit is significative to establish some sharp bounds on therestrained bondage number of a graph in terms of othergraphic parameters

Theorem 128 (Hattingh and Plummer [80] 2008) For anygraph 119866 with 120575(119866) ⩾ 2

119887R (119866) ⩽ min119909119910isin119864(119866)

119889119866(119909) + 119889

119866(119910) minus 2 (47)

Corollary 129 119887R(119866) ⩽ Δ(119866)+120575(119866)minus2 for any graph119866 with120575(119866) ⩾ 2

Notice that the bounds stated in Theorem 128 andCorollary 129 are sharp Indeed the class of cycles whoseorders are congruent to 1 2 (mod 3) have a restrained bondagenumber achieving these bounds (see Theorem 125)

Theorem 128 is an analogue of Theorem 14 A quitenatural problem is whether or not for restricted bondagenumber there are analogues of Theorem 16 Theorem 18Theorem 29 and so on Indeed we should consider all resultsfor the bondage number whether or not there are analoguesfor restricted bondage number

Theorem 130 (Hattingh and Plummer [80] 2008) For anygraph 119866 with 120574

119877(119866) = 2

119887R (119866) ⩽ Δ (119866) + 1 (48)

We now consider the relation between 119887119877(119866) and 119887(119866)

Theorem 131 (Hattingh and Plummer [80] 2008) If 120574119877(119866) =

120574(119866) for some graph 119866 then 119887R(119866) ⩽ 119887(119866)

Corollary 129 and Theorem 131 provide a possibility toattack Conjecture 73 by characterizing planar graphs with120574119877= 120574 and 119887R = 119887 when 3 ⩽ Δ ⩽ 6However we do not have 119887R(119866) = 119887(119866) for any graph

119866 even if 120574119877(119866) = 120574(119866) Observe that 120574

119877(119870

3) = 120574(119870

3) yet

119887119877(119870

3) = 1 and 119887(119870

3) = 2 We still may not claim that

119887119877(119866) = 119887(119866) even in the case that every 120574-set is a 120574

119877-set

The example 1198703again demonstrates this

Theorem 132 (Hattingh and Plummer [80] 2008) There isan infinite class of graphs inwhich each graph119866 satisfies 119887(119866) lt119887R(119866)

In fact such an infinite class of graphs can be obtained asfollows Let119867 be a connected graph BC(119867) a graph obtainedfrom 119867 by attaching ℓ (⩾ 2) vertices of degree 1 to eachvertex in 119867 and B = 119866 119866 = BC(119867) for some graph 119867such that 120575(119867) ⩾ 2 By Theorem 3 we can easily check that119887(119866) = 1 lt 2 ⩽ min120575(119867) ℓ = 119887R(119866) for any 119866 isin BMoreover there exists a graph 119866 isin B such that 119887R(119866) canbe much larger than 119887(119866) which is stated as the followingtheorem

Theorem 133 (Hattingh and Plummer [80] 2008) For eachpositive integer 119896 there is a graph119866 such that 119896 = 119887R(119866)minus119887(119866)

Combining Theorem 133 with Theorem 132 we have thefollowing corollary

Corollary 134 The bondage number and the restrainedbondage number are unrelated

Very recently Jafari Rad et al [81] have consideredthe total restrained bondage in graphs based on both totaldominating sets and restrained dominating sets and obtainedseveral properties exact values and bounds for the totalrestrained bondage number of a graph

A restrained dominating set 119878 of a graph 119866 without iso-lated vertices is called to be a total restrained dominating setif 119878 is a total dominating set The total restrained dominationnumber of119866 denoted by 120574TR (119866) is theminimum cardinalityof a total restrained dominating set of 119866 For referenceson this domination in graphs see [3 61 82ndash85] The totalrestrained bondage number 119887TR (119866) of a graph 119866 with noisolated vertex is the cardinality of a smallest set of edges 119861 sube119864(119866) for which119866minus119861 has no isolated vertex and 120574TR (119866minus119861) gt120574TR (119866) In the case that there is no such subset 119861 we define119887TR (119866) = infin

Ma et al [84] determined that 120574TR (119875119899) = 119899minus2lfloor(119899minus2)4rfloorfor 119899 ⩾ 3 120574TR (119862119899

) = 119899 minus 2lfloor1198994rfloor for 119899 ⩾ 3 120574TR (1198703) =

3 and 120574TR (119870119899) = 2 for 119899 = 3 and 120574TR (1198701119899

) = 1 + 119899

and 120574TR (119870119898119899) = 2 for 2 ⩽ 119898 ⩽ 119899 According to these

results Jafari Rad et al [81] determined the exact valuesof total restrained bondage numbers for the correspondinggraphs

International Journal of Combinatorics 21

Theorem 135 (Jafari Rad et al [81]) For an integer 119899

119887TR (119875119899) = infin 119894119891 2 ⩽ 119899 ⩽ 5

1 119894119891 119899 ⩾ 5

119887TR (119862119899) =

infin 119894119891 119899 = 3

1 119894119891 3 lt 119899 119899 equiv 0 1 (mod 4)2 119894119891 3 lt 119899 119899 equiv 2 3 (mod 4)

119887TR (119882119899) = 2 119891119900119903 119899 ⩾ 6

119887TR (119870119899) = 119899 minus 1 119891119900119903 119899 ⩾ 4

119887TR (119870119898119899) = 119898 minus 1 119891119900119903 2 ⩽ 119898 ⩽ 119899

(49)

119887TR (119870119899) = 119899minus1 shows the fact that for any integer 119899 ⩾ 3 there

exists a connected graph namely119870119899+1

such that 119887TR (119870119899+1) =

119899 that is the total restrained bondage number is unboundedfrom the above

A vertex is said to be a support vertex if it is adjacent toa vertex of degree one Let A be the family of all connectedgraphs such that119866 belongs toA if and only if every edge of119866is incident to a support vertex or 119866 is a cycle of length three

Proposition 136 (Cyman and Raczek [82] 2006) For aconnected graph119866 of order 119899 120574TR (119866) = 119899 if and only if119866 isin A

Hence if 119866 isin A then the removal of any subset of edgesof119866 cannot increase the total restrained domination numberof 119866 and thus 119887TR (119866) = infin When 119866 notin A a result similar toTheorem 6 is obtained

Theorem 137 (Jafari Rad et al [81]) For any tree 119879 notin A119887TR (119879) ⩽ 2 This bound is sharp

In the samepaper the authors provided a characterizationfor trees 119879 notin A with 119887TR (119879) = 1 or 2 The following result issimilar to Observation 1

Observation 3 (Jafari Rad et al [81]) If 119896 edges can beremoved from a graph 119866 to obtain a graph 119867 without anisolated vertex and with 119887TR (119867) = 119905 then 119887TR (119866) ⩽ 119896 + 119905

ApplyingTheorem 137 andObservation 3 Jafari Rad et alcharacterized all connected graphs 119866 with 119887TR (119866) = infin

Theorem 138 (Jafari Rad et al [81]) For a connected graph119866119887TR (119866) = infin if and only if 119866 isin A

84 Other Conditional Bondage Numbers A subset 119868 sube 119881(119866)is called an independent set if no two vertices in 119868 are adjacentin 119866 The maximum cardinality among all independent setsis called the independence number of 119866 denoted by 120572(119866)

A dominating set 119878 of a graph 119866 is called to be inde-pendent if 119878 is an independent set of 119866 The minimumcardinality among all independent dominating set is calledthe independence domination number of 119866 and denoted by120574119868(119866)

Since an independent dominating set is not only adominating set but also an independent set 120574(119866) ⩽ 120574

119868(119866) ⩽

120572(119866) for any graph 119866It is clear that a maximal independent set is certainly a

dominating set Thus an independent set is maximal if andonly if it is an independent dominating set and so 120574

119868(119866) is the

minimum cardinality among all maximal independent setsof 119866 This graph-theoretical invariant has been well studiedin the literature see for example Haynes et al [3] for earlyresults Goddard and Henning [86] for recent results onindependent domination in graphs

In 2003 Zhang et al [87] defined the independencebondage number 119887

119868(119866) of a nonempty graph 119866 to be the

minimum cardinality among all subsets 119861 sube 119864(119866) for which120574119868(119866 minus 119861) gt 120574

119868(119866) For some ordinary graphs their indepen-

dence domination numbers can be easily determined and soindependence bondage numbers have been also determinedClearly 119887

119868(119870

119899) = 1 if 119899 ⩾ 2

Theorem 139 (Zhang et al [87] 2003) For any integer 119899 ⩾ 4

119887119868(119862

119899) =

1 119894119891 119899 equiv 1 (mod 2) 2 119894119891 119899 equiv 0 (mod 2)

119887119868(119875

119899) =

1 119894119891 119899 equiv 0 (mod 2) 2 119894119891 119899 equiv 1 (mod 2)

119887119868(119870

119898119899) = 119899 119891119900119903 119898 ⩽ 119899

(50)

Apart from the above-mentioned results as far as we findno other results on the independence bondage number in thepresent literature we never knew much of any result on thisparameter for a tree

A subset 119878 of 119881(119866) is called an equitable dominatingset if for every 119910 isin 119881(119866 minus 119878) there exists a vertex 119909 isin 119878such that 119909119910 isin 119864(119866) and |119889

119866(119909) minus 119889

119866(119910)| ⩽ 1 proposed

by Dharmalingam [88] The minimum cardinality of such adominating set is called the equitable domination number andis denoted by 120574

119864(119866) Deepak et al [89] defined the equitable

bondage number 119887119864(119866) of a graph 119866 to be the cardinality of a

smallest set 119861 sube 119864(119866) of edges for which 120574119864(119866 minus 119861) gt 120574

119864(119866)

and determined 119887119864(119866) for some special graphs

Theorem 140 (Deepak et al [89] 2011) For any integer 119899 ⩾ 2

119887119864(119870

119899) = lceil

119899

2rceil

119887119864(119870

119898119899) = 119898 119891119900119903 0 ⩽ 119899 minus 119898 ⩽ 1

119887119864(119862

119899) =

3 119894119891 119899 equiv 1 (mod 3)2 119900119905ℎ119890119903119908119894119904119890

119887119864(119875

119899) =

2 119894119891 119899 equiv 1 (mod 3)1 119900119905ℎ119890119903119908119894119904119890

119887119864(119879) ⩽ 2 119891119900119903 119886119899119910 119899119900119899119905119903119894119907119894119886119897 119905119903119890119890 119879

119887119864(119866) ⩽ 119899 minus 1 119891119900119903 119886119899119910 119888119900119899119899119890119888119905119890119889 119892119903119886119901ℎ 119866 119900119891 119900119903119889119890119903 119899

(51)

22 International Journal of Combinatorics

As far as we know there are no more results on theequitable bondage number of graphs in the present literature

There are other variations of the normal domination byadding restricted conditions to the normal dominating setFor example the connected domination set of a graph119866 pro-posed by Sampathkumar and Walikar [90] is a dominatingset 119878 such that the subgraph induced by 119878 is connected Theproblem of finding a minimum connected domination set ina graph is equivalent to the problemof finding a spanning treewithmaximumnumber of vertices of degree one [91] and hassome important applications in wireless networks [92] thusit has received much research attention However for suchan important domination there is no result on its bondagenumber We believe that the topic is of significance for futureresearch

9 Generalized Bondage Numbers

There are various generalizations of the classical dominationsuch as 119901-domination distance domination fractional dom-ination Roman domination and rainbow domination Everysuch a generalization can lead to a corresponding bondageIn this section we introduce some of them

91 119901-Bondage Numbers In 1985 Fink and Jacobson [93]introduced the concept of 119901-domination Let 119901 be a positiveinteger A subset 119878 of 119881(119866) is a 119901-dominating set of 119866 if|119878 cap 119873

119866(119910)| ⩾ 119901 for every 119910 isin 119878 The 119901-domination number

120574119901(119866) is the minimum cardinality among all 119901-dominating

sets of 119866 Any 119901-dominating set of 119866 with cardinality 120574119901(119866)

is called a 120574119901-set of 119866 Note that a 120574

1-set is a classic minimum

dominating set Notice that every graph has a 119901-dominatingset since the vertex-set119881(119866) is such a setWe also note that a 1-dominating set is a dominating set and so 120574(119866) = 120574

1(119866) The

119901-domination number has received much research attentionsee a state-of-the-art survey article by Chellali et al [94]

It is clear from definition that every 119901-dominating setof a graph certainly contains all vertices of degree at most119901 minus 1 By this simple observation to avoid the trivial caseoccurrence we always assume that Δ(119866) ⩾ 119901 For 119901 ⩾ 2 Luet al [95] gave a constructive characterization of trees withunique minimum 119901-dominating sets

Recently Lu and Xu [96] have introduced the conceptto the 119901-bondage number of 119866 denoted by 119887

119901(119866) as the

minimum cardinality among all sets of edges 119861 sube 119864(119866) suchthat 120574

119901(119866 minus 119861) gt 120574

119901(119866) Clearly 119887

1(119866) = 119887(119866)

Lu and Xu [96] established a lower bound and an upperbound on 119887

119901(119879) for any integer 119901 ⩾ 2 and any tree 119879 with

Δ(119879) ⩾ 119901 and characterized all trees achieving the lowerbound and the upper bound respectively

Theorem 141 (Lu and Xu [96] 2011) For any integer 119901 ⩾ 2and any tree 119879 with Δ(119879) ⩾ 119901

1 ⩽ 119887119901(119879) ⩽ Δ (119879) minus 119901 + 1 (52)

Let 119878 be a given subset of 119881(119866) and for any 119909 isin 119878 let

119873119901(119909 119878 119866) = 119910 isin 119878 cap 119873

119866(119909)

1003816100381610038161003816119873119866(119910) cap 119878

1003816100381610038161003816 = 119901

(53)

Then all trees achieving the lower bound 1 can be character-ized as follows

Theorem 142 (Lu and Xu [96] 2011) Let 119879 be a tree withΔ(119879) ⩾ 119901 ⩾ 2 Then 119887

119901(119879) = 1 if and only if for any 120574

119901-set

119878 of 119879 there exists an edge 119909119910 isin (119878 119878) such that 119910 isin 119873119901(119909 119878 119879)

The notation 119878(119886 119887) denotes a double star obtained byadding an edge between the central vertices of two stars119870

1119886minus1

and1198701119887minus1

And the vertex with degree 119886 (resp 119887) in 119878(119886 119887) iscalled the 119871-central vertex (resp 119877-central vertex) of 119878(119886 119887)

To characterize all trees attaining the upper bound givenin Theorem 141 we define three types of operations on a tree119879 with Δ(119879) = Δ ⩾ 119901 + 1

Type 1 attach a pendant edge to a vertex 119910with 119889119879(119910) ⩽ 119901minus

2 in 119879

Type 2 attach a star 1198701Δminus1

to a vertex 119910 of 119879 by joining itscentral vertex to 119910 where 119910 in a 120574

119901-set of 119879 and

119889119879(119910) ⩽ Δ minus 1

Type 3 attach a double star 119878(119901 Δ minus 1) to a pendant vertex 119910of 119879 by coinciding its 119877-central vertex with 119910 wherethe unique neighbor of 119910 is in a 120574

119901-set of 119879

Let B = 119879 a tree obtained from 1198701Δ

or 119878(119901 Δ) by afinite sequence of operations of Types 1 2 and 3

Theorem 143 (Lu and Xu [96] 2011) A tree with the maxi-mum Δ ⩾ 119901 + 1 has 119901-bondage number Δ minus 119901 + 1 if and onlyif it belongs toB

Theorem 144 (Lu and Xu [97] 2012) Let 119866119898119899= 119875

119898times 119875

119899

Then

1198872(119866

2119899) = 1 119891119900119903 119899 ⩾ 2

1198872(119866

3119899) =

2 119894119891 119899 equiv 1 (mod 3) 1 119900119905ℎ119890119903119908119894119904119890

for 119899 ⩾ 2

1198872(119866

4119899) =

1 119894119891 119899 equiv 3 (mod 4) 2 119900119905ℎ119890119903119908119894119904119890

for 119899 ⩾ 7

(54)

Recently Krzywkowski [98] has proposed the conceptof the nonisolating 2-bondage of a graph The nonisolating2-bondage number of a graph 119866 denoted by 1198871015840

2(119866) is the

minimum cardinality among all sets of edges 119861 sube 119864(119866) suchthat 119866 minus 119861 has no isolated vertices and 120574

2(119866 minus 119861) gt 120574

2(119866) If

for every 119861 sube 119864(119866) either 1205742(119866 minus 119861) = 120574

2(119866) or 119866 minus 119861 has

isolated vertices then define 11988710158402(119866) = 0 and say that 119866 is a 2-

nonisolatedly strongly stable graph Krzywkowski presentedsome basic properties of nonisolating 2-bondage in graphsshowed that for every nonnegative integer 119896 there exists atree 119879 such 1198871015840

2(119879) = 119896 characterized all 2-non-isolatedly

strongly stable trees and determined 11988710158402(119866) for several special

graphs

International Journal of Combinatorics 23

Theorem 145 (Krzywkowski [98] 2012) For any positiveintegers 119899 and119898 with119898 ⩽ 119899

1198871015840

2(119870

119899) =

0 119894119891 119899 = 1 2 3

lfloor2119899

3rfloor 119900119905ℎ119890119903119908119894119904119890

1198871015840

2(119875

119899) =

0 119894119891 119899 = 1 2 3

1 119900119905ℎ119890119903119908119894119904119890

1198871015840

2(119862

119899) =

0 119894119891 119899 = 3

1 119894119891 119899 119894119904 119890V119890119899

2 119894119891 119899 119894119904 119900119889119889

1198871015840

2(119870

119898119899) =

3 119894119891 119898 = 119899 = 3

1 119894119891 119898 = 119899 = 4

2 119900119905ℎ119890119903119908119894119904119890

(55)

92 Distance Bondage Numbers A subset 119878 of vertices of agraph119866 is said to be a distance 119896-dominating set for119866 if everyvertex in 119866 not in 119878 is at distance at most 119896 from some vertexof 119878 The minimum cardinality of all distance 119896-dominatingsets is called the distance 119896-domination number of 119866 anddenoted by 120574

119896(119866) (do not confuse with the above-mentioned

120574119901(119866)) When 119896 = 1 a distance 1-dominating set is a normal

dominating set and so 1205741(119866) = 120574(119866) for any graph 119866 Thus

the distance 119896-domination is a generalization of the classicaldomination

The relation between 120574119896and 120572

119896(the 119896-independence

number defined in Section 22) for a tree obtained by Meirand Moon [14] who proved that 120574

119896(119879) = 120572

2119896(119879) for any tree

119879 (a special case for 119896 = 1 is stated in Proposition 10) Thefurther research results can be found in Henning et al [99]Tian and Xu [100ndash103] and Liu et al [104]

In 1998 Hartnell et al [15] defined the distance 119896-bondagenumber of 119866 denoted by 119887

119896(119866) to be the cardinality of a

smallest subset 119861 of edges of 119866 with the property that 120574119896(119866 minus

119861) gt 120574119896(119866) FromTheorem 6 it is clear that if119879 is a nontrivial

tree then 1 ⩽ 1198871(119879) ⩽ 2 Hartnell et al [15] generalized this

result to any integer 119896 ⩾ 1

Theorem 146 (Hartnell et al [15] 1998) For every nontrivialtree 119879 and positive integer 119896 1 ⩽ 119887

119896(119879) ⩽ 2

Hartnell et al [15] and Topp and Vestergaard [13] alsocharacterized the trees having distance 119896-bondage number 2In particular the class of trees for which 119887

1(119879) = 2 are just

those which have a unique maximum 2-independent set (seeTheorem 11)

Since when 119896 = 1 the distance 1-bondage number 1198871(119866)

is the classical bondage number 119887(119866) Theorem 12 gives theNP-hardness of deciding the distance 119896-bondage number ofgeneral graphs

Theorem 147 Given a nonempty undirected graph 119866 andpositive integers 119896 and 119887 with 119887 ⩽ 120576(119866) determining wetheror not 119887

119896(119866) ⩽ 119887 is NP-hard

For a vertex 119909 in119866 the open 119896-neighborhood119873119896(119909) of 119909 is

defined as119873119896(119909) = 119910 isin 119881(119866) 1 ⩽ 119889

119866(119909 119910) le 119896The closed

119896-neighborhood119873119896[119909] of 119909 in 119866 is defined as119873

119896(119909) cup 119909 Let

Δ119896(119866) = max 1003816100381610038161003816119873119896

(119909)1003816100381610038161003816 for any 119909 isin 119881 (119866) (56)

Clearly Δ1(119866) = Δ(119866) The 119896th power of a graph 119866 is the

graph119866119896 with vertex-set119881(119866119896

) = 119881(119866) and edge-set119864(119866119896

) =

119909119910 1 ⩽ 119889119866(119909 119910) ⩽ 119896 The following lemma holds directly

from the definition of 119866119896

Proposition 148 (Tian and Xu [101]) Δ(119866119896

) = Δ119896(119866) and

120574(119866119896

) = 120574119896(119866) for any graph 119866 and each 119896 ⩾ 1

A graph 119866 is 119896-distance domination-critical or 120574119896-critical

for short if 120574119896(119866 minus 119909) lt 120574

119896(119866) for every vertex 119909 in 119866

proposed by Henning et al [105]

Proposition 149 (Tian and Xu [101]) For each 119896 ⩾ 1 a graph119866 is 120574

119896-critical if and only if 119866119896 is 120574

119896-critical

By the above facts we suggest the following worthwhileproblem

Problem 6 Can we generalize the results on the bondage for119866 to 119866119896 In particular do the following propositions hold

(a) 119887(119866119896

) = 119887119896(119866) for any graph 119866 and each 119896 ⩾ 1

(b) 119887(119866119896

) ⩽ Δ119896(119866) if 119866 is not 120574

119896-critical

Let 119896 and 119901 be positive integers A subset 119878 of 119881(119866) isdefined to be a (119896 119901)-dominating set of 119866 if for any vertex119909 isin 119878 |119873

119896(119909) cap 119878| ⩾ 119901 The (119896 119901)-domination number of

119866 denoted by 120574119896119901(119866) is the minimum cardinality among

all (119896 119901)-dominating sets of 119866 Clearly for a graph 119866 a(1 1)-dominating set is a classical dominating set a (119896 1)-dominating set is a distance 119896-dominating set and a (1 119901)-dominating set is the above-mentioned 119901-dominating setThat is 120574

11(119866) = 120574(119866) 120574

1198961(119866) = 120574

119896(119866) and 120574

1119901(119866) = 120574

119901(119866)

The concept of (119896 119901)-domination in a graph 119866 is a gen-eralized domination which combines distance 119896-dominationand 119901-domination in 119866 So the investigation of (119896 119901)-domination of 119866 is more interesting and has received theattention of many researchers see for example [106ndash109]

More general Li and Xu [110] proposed the concept of the(ℓ 119908)-domination number of a119908-connected graph A subset119878 of 119881(119866) is defined to be an (ℓ 119908)-dominating set of 119866 iffor any vertex 119909 isin 119878 there are 119908 internally disjoint pathsof length at most ℓ from 119909 to some vertex in 119878 The (ℓ 119908)-domination number of119866 denoted by 120574

ℓ119908(119866) is theminimum

cardinality among all (ℓ 119908)-dominating sets of 119866 Clearly1205741198961(119866) = 120574

119896(119866) and 120574

1119901(119866) = 120574

119901(119866)

It is quite natural to propose the concept of bondage num-ber for (119896 119901)-domination or (ℓ 119908)-domination However asfar as we know none has proposed this concept until todayThis is a worthwhile topic for further research

24 International Journal of Combinatorics

93 Fractional Bondage Numbers If 120590 is a function mappingthe vertex-set 119881 into some set of real numbers then for anysubset 119878 sube 119881 let 120590(119878) = sum

119909isin119878120590(119909) Also let |120590| = 120590(119881)

A real-value function 120590 119881 rarr [0 1] is a dominatingfunction of a graph 119866 if for every 119909 isin 119881(119866) 120590(119873

119866[119909]) ⩾ 1

Thus if 119878 is a dominating set of 119866 120590 is a function where

120590 (119909) = 1 if 119909 isin 1198780 otherwise

(57)

then 120590 is a dominating function of119866 The fractional domina-tion number of 119866 denoted by 120574

119865(119866) is defined as follows

120574119865(119866) = min |120590| 120590 is a domination function of 119866

(58)

The 120574119865-bondage number of 119866 denoted by 119887

119865(119866) is

defined as the minimum cardinality of a subset 119861 sube 119864 whoseremoval results in 120574

119865(119866 minus 119861) gt 120574

119865(119866)

Hedetniemi et al [111] are the first to study fractionaldomination although Farber [112] introduced the idea indi-rectly The concept of the fractional domination numberwas proposed by Domke and Laskar [113] in 1997 Thefractional domination numbers for some ordinary graphs aredetermined

Proposition 150 (Domke et al [114] 1998 Domke and Laskar[113] 1997) (a) If 119866 is a 119896-regular graph with order 119899 then120574119865(119866) = 119899119896 + 1(b) 120574

119865(119862

119899) = 1198993 120574

119865(119875

119899) = lceil1198993rceil 120574

119865(119870

119899) = 1

(c) 120574119865(119866) = 1 if and only if Δ(119866) = 119899 minus 1

(d) 120574119865(119879) = 120574(119879) for any tree 119879

(e) 120574119865(119870

119899119898) = (119899(119898 + 1) + 119898(119899 + 1))(119899119898 minus 1) where

max119899119898 gt 1

The assertions (a)ndash(d) are due to Domke et al [114] andthe assertion (e) is due to Domke and Laskar [113] Accordingto these results Domke and Laskar [113] determined the 120574

119865-

bondage numbers for these graphs

Theorem 151 (Domke and Laskar [113] 1997) 119887119865(119870

119899) =

lceil1198992rceil 119887119865(119870

119899119898) = 1 wheremax119899119898 gt 1

119887119865(119862

119899) =

2 119894119891 119899 equiv 0 (mod 3) 1 119900119905ℎ119890119903119908119894119904119890

119887119865(119875

119899) =

2 119894119891 119899 equiv 1 (mod 3) 1 119900119905ℎ119890119903119908119894119904119890

(59)

It is easy to see that for any tree119879 119887119865(119879) ⩽ 2 In fact since

120574119865(119879) = 120574(119879) for any tree 119879 120574

119865(119879

1015840

) = 120574(1198791015840

) for any subgraph1198791015840 of 119879 It follows that

119887119865(119879) = min 119861 sube 119864 (119879) 120574

119865(119879 minus 119861) gt 120574

119865(119879)

= min 119861 sube 119864 (119879) 120574 (119879 minus 119861) gt 120574 (119879)

= 119887 (119879) ⩽ 2

(60)

Except for these no results on 119887119865(119866) have been known as

yet

94 Roman Bondage Numbers A Roman dominating func-tion on a graph 119866 is a labeling 119891 119881 rarr 0 1 2 such thatevery vertex with label 0 has at least one neighbor with label2 The weight of a Roman dominating function 119891 on 119866 is thevalue

119891 (119866) = sum

119906isin119881(119866)

119891 (119906) (61)

The minimum weight of a Roman dominating function ona graph 119866 is called the Roman domination number of 119866denoted by 120574RM (119866)

A Roman dominating function 119891 119881 rarr 0 1 2

can be represented by the ordered partition (1198810 119881

1 119881

2) (or

(119881119891

0 119881

119891

1 119881

119891

2) to refer to119891) of119881 where119881

119894= V isin 119881 | 119891(V) = 119894

In this representation its weight 119891(119866) = |1198811| + 2|119881

2| It is

clear that119881119891

1cup119881

119891

2is a dominating set of 119866 called the Roman

dominating set denoted by 119863119891

R = (1198811 1198812) Since 119881119891

1cup 119881

119891

2is

a dominating set when 119891 is a Roman dominating functionand since placing weight 2 at the vertices of a dominating setyields a Roman dominating function in [115] it is observedthat

120574 (119866) ⩽ 120574RM (119866) ⩽ 2120574 (119866) (62)

A graph 119866 is called to be Roman if 120574RM (119866) = 2120574(119866)The definition of the Roman domination function is

proposed implicitly by Stewart [116] and ReVelle and Rosing[117] Roman dominating numbers have been deeply studiedIn particular Bahremandpour et al [118] showed that theproblem determining the Roman domination number is NP-complete even for bipartite graphs

Let 119866 be a graph with maximum degree at least twoThe Roman bondage number 119887RM (119866) of 119866 is the minimumcardinality of all sets 1198641015840 sube 119864 for which 120574RM (119866 minus 119864

1015840

) gt

120574RM (119866) Since in study of Roman bondage number theassumption Δ(119866) ⩾ 2 is necessary we always assume thatwhen we discuss 119887RM (119866) all graphs involved satisfy Δ(119866) ⩾2 The Roman bondage number 119887RM (119866) is introduced byJafari Rad and Volkmann in [119]

Recently Bahremandpour et al [118] have shown that theproblem determining the Roman bondage number is NP-hard even for bipartite graphs

Problem 7 Consider the decision problem

Roman Bondage Problem

Instance a nonempty bipartite graph119866 and a positiveinteger 119896

Question is 119887RM (119866) ⩽ 119896

Theorem 152 (Bahremandpour et al [118] 2013) TheRomanbondage problem is NP-hard even for bipartite graphs

The exact value of 119887RM(119866) is known only for a few familyof graphs including the complete graphs cycles and paths

International Journal of Combinatorics 25

Theorem 153 (Jafari Rad and Volkmann [119] 2011) For anypositive integers 119899 and119898 with119898 ⩽ 119899

119887RM (119875119899) = 2 119894119891 119899 equiv 2 (mod 3) 1 119900119905ℎ119890119903119908119894119904119890

119887RM (119862119899) =

3 119894119891 119899 = 2 (mod 3) 2 119900119905ℎ119890119903119908119894119904119890

119887RM (119870119899) = lceil

119899

2rceil 119891119900119903 119899 ⩾ 3

119887RM (119870119898119899) =

1 119894119891 119898 = 1 119886119899119889 119899 = 1

4 119894119891 119898 = 119899 = 3

119898 119900119905ℎ119890119903119908119894119904119890

(63)

Lemma 154 (Cockayne et al [115] 2004) If 119866 is a graph oforder 119899 and contains vertices of degree 119899 minus 1 then 120574RM(119866) = 2

Using Lemma 154 the third conclusion in Theorem 153can be generalized to more general case which is similar toLemma 49

Theorem 155 (Jafari Rad and Volkmann [119] 2011) Let119866 bea graph with order 119899 ge 3 and 119905 the number of vertices of degree119899 minus 1 in 119866 If 119905 ⩾ 1 then 119887RM(119866) = lceil1199052rceil

Ebadi and PushpaLatha [120] conjectured that 119887RM(119866) ⩽119899 minus 1 for any graph 119866 of order 119899 ⩾ 3 Dehgardi et al[121] Akbari andQajar [122] independently showed that thisconjecture is true

Theorem 156 119887RM(119866) ⩽ 119899 minus 1 for any connected graph 119866 oforder 119899 ⩾ 3

For a complete 119905-partite graph we have the followingresult

Theorem 157 (Hu and Xu [123] 2011) Let 119866 = 11987011989811198982119898119905

bea complete 119905-partite graph with 119898

1= sdot sdot sdot = 119898

119894lt 119898

119894+1le sdot sdot sdot ⩽

119898119905 119905 ⩾ 2 and 119899 = sum119905

119895=1119898

119895 Then

119887RM (119866) =

lceil119894

2rceil 119894119891 119898

119894= 1 119886119899119889 119899 ⩾ 3

2 119894119891 119898119894= 2 119886119899119889 119894 = 1

119894 119894119891 119898119894= 2 119886119899119889 119894 ⩾ 2

4 119894119891 119898119894= 3 119886119899119889 119894 = 119905 = 2

119899 minus 1 119894119891 119898119894= 3 119886119899119889 119894 = 119905 ⩾ 3

119899 minus 119898119905

119894119891 119898119894⩾ 3 119886119899119889 119898

119905⩾ 4

(64)

Consider a complete 119905-partite graph 119870119905(3) for 119905 ⩾ 3

119887RM(119870119905(3)) = 119899 minus 1 by Theorem 157 where 119899 = 3119905

which shows that this upper bound of 119899 minus 1 on 119887RM(119866) inTheorem 156 is sharp This fact and 120574

119877(119870

119905(3)) = 4 give

a negative answer to the following two problems posed byDehgardi et al [121]Prove or Disprove for any connected graph 119866 of order 119899 ge 3(i) 119887RM(119866) = 119899 minus 1 if and only if 119866 cong 119870

3 (ii) 119887RM(119866) ⩽ 119899 minus

120574119877(119866) + 1Note that a complete 119905-partite graph 119870

119905(3) is (119899 minus 3)-

regular and 119887RM(119870119905(3)) = 119899 minus 1 for 119905 ⩾ 3 where 119899 = 3119905

Hu and Xu further determined that 119887119877(119866) = 119899 minus 2 for any

(119899 minus 3)-regular graph 119866 of order 119899 ⩽ 5 except 119870119905(3)

Theorem 158 (Hu and Xu [123] 2011) For any (119899minus3)-regulargraph 119866 of order 119899 other than119870

119905(3) 119887RM(119866) = 119899 minus 2 for 119899 ⩾ 5

For a tree 119879 with order 119899 ⩾ 3 Ebadi and PushpaLatha[120] and Jafari Rad and Volkmann [119] independentlyobtained an upper bound on 119887RM(119879)

Theorem 159 119887RM(119879) ⩽ 3 for any tree 119879 with order 119899 ⩾ 3

Theorem 160 (Bahremandpour et al [118] 2013) 119887RM(1198752 times119875119899) = 2 for 119899 ⩾ 2

Theorem 161 (Jafari Rad and Volkmann [119] 2011) Let119866 bea graph with order at least three

(a) If (119909 119910 119911) is a path of length 2 in 119866 then 119887RM(119866) ⩽119889119866(119909) + 119889

119866(119910) + 119889

119866(119911) minus 3 minus |119873

119866(119909) cap 119873

119866(119910)|

(b) If 119866 is connected then 119887RM(119866) ⩽ 120582(119866) + 2Δ(119866) minus 3

Theorem 161 (b) implies 119887RM(119866) ⩽ 120575(119866) + 2Δ(119866) minus 3 for aconnected graph 119866 Note that for a planar graph 119866 120575(119866) ⩽ 5moreover 120575(119866) ⩽ 3 if the girth at least 4 and 120575(119866) ⩽ 2 if thegirth at least 6 These two facts show that 119887RM(119866) ⩽ 2Δ(119866) +2 for a connected planar graph 119866 Jafari Rad and Volkmann[124] improved this bound

Theorem 162 (Jafari Rad and Volkmann [124] 2011) For anyconnected planar graph 119866 of order 119899 ⩾ 3 with girth 119892(119866)

119887RM (119866)

⩽

2Δ (119866)

Δ (119866) + 6

Δ (119866) + 5 119894119891 119866 119888119900119899119905119886119894119899119904 119899119900 V119890119903119905119894119888119890119904 119900119891 119889119890119892119903119890119890 119891119894V119890

Δ (119866) + 4 119894119891 119892 (119866) ⩾ 4

Δ (119866) + 3 119894119891 119892 (119866) ⩾ 5

Δ (119866) + 2 119894119891 119892 (119866) ⩾ 6

Δ (119866) + 1 119894119891 119892 (119866) ⩾ 8

(65)

According toTheorem 157 119887RM(119862119899) = 3 = Δ(119862

119899)+1 for a

cycle 119862119899of length 119899 ⩾ 8 with 119899 equiv 2 (mod 3) and therefore

the last upper bound inTheorem 162 is sharp at least for Δ =2

Combining the fact that every planar graph 119866 withminimum degree 5 contains an edge 119909119910 with 119889

119866(119909) = 5

and 119889119866(119910) isin 5 6 with Theorem 161 (a) Akbari et al [125]

obtained the following result

26 International Journal of Combinatorics

middot middot middot

middot middot middot

middot middot middot

middot middot middot

119866

Figure 9 The graph 119866 is constructed from 119866

Theorem 163 (Akbari et al [125] 2012) 119887RM(119866) ⩽ 15 forevery planar graph 119866

It remains open to show whether the bound inTheorem 163 is sharp or not Though finding a planargraph 119866 with 119887RM(119866) = 15 seems to be difficult Akbari etal [125] constructed an infinite family of planar graphs withRoman bondage number equal to 7 by proving the followingresult

Theorem 164 (Akbari et al [125] 2012) Let 119866 be a graph oforder 119899 and 119866 is the graph of order 5119899 obtained from 119866 byattaching the central vertex of a copy of a path 119875

5to each vertex

of119866 (see Figure 9) Then 120574RM(119866) = 4119899 and 119887RM(119866) = 120575(119866)+2

ByTheorem 164 infinitemany planar graphswith Romanbondage number 7 can be obtained by considering any planargraph 119866 with 120575(119866) = 5 (eg the icosahedron graph)

Conjecture 165 (Akbari et al [125] 2012) The Romanbondage number of every planar graph is at most 7

For general bounds the following observation is directlyobtained which is similar to Observation 1

Observation 4 Let 119866 be a graph of order 119899 with maximumdegree at least two Assume that 119867 is a spanning subgraphobtained from 119866 by removing 119896 edges If 120574RM(119867) = 120574RM(119866)then 119887

119877(119867) ⩽ 119887RM(119866) ⩽ 119887RM(119867) + 119896

Theorem 166 (Bahremandpour et al [118] 2013) For anyconnected graph 119866 of order 119899 if 119899 ⩾ 3 and 120574RM(119866) = 120574(119866) + 1then

119887RM (119866) ⩽ min 119887 (119866) 119899Δ (66)

where 119899Δis the number of vertices with maximum degree Δ in

119866

Observation 5 For any graph 119866 if 120574(119866) = 120574RM(119866) then119887RM(119866) ⩽ 119887(119866)

Theorem 167 (Bahremandpour et al [118] 2013) For everyRoman graph 119866

119887RM (119866) ⩾ 119887 (119866) (67)

The bound is sharp for cycles on 119899 vertices where 119899 equiv 0 (mod3)

The strict inequality in Theorem 167 can hold for exam-ple 119887(119862

3119896+2) = 2 lt 3 = 119887RM(1198623119896+2

) byTheorem 157A graph 119866 is called to be vertex Roman domination-

critical if 120574RM(119866 minus 119909) lt 120574RM(119866) for every vertex 119909 in 119866 If119866 has no isolated vertices then 120574RM(119866) ⩽ 2120574(119866) ⩽ 2120573(119866)If 120574RM(119866) = 2120573(119866) then 120574RM(119866) = 2120574(119866) and hence 119866 is aRoman graph In [30] Volkmann gave a lot of graphs with120574(119866) = 120573(119866) The following result is similar to Theorem 57

Theorem 168 (Bahremandpour et al [118] 2013) For anygraph 119866 if 120574RM(119866) = 2120573(119866) then

(a) 119887RM(119866) ⩾ 120575(119866)

(b) 119887RM(119866) ⩾ 120575(119866)+1 if119866 is a vertex Roman domination-critical graph

If 120574RM(119866) = 2 then obviously 119887RM(119866) ⩽ 120575(119866) So weassume 120574RM(119866) ⩾ 3 Dehgardi et al [121] proved 119887RM(119866) ⩽(120574RM(119866) minus 2)Δ(119866) + 1 and posed the following problem if 119866is a connected graph of order 119899 ⩾ 4 with 120574RM(119866) ⩾ 3 then

119887RM (119866) ⩽ (120574RM (119866) minus 2) Δ (119866) (68)

Theorem 161 (a) shows that the inequality (68) holds if120574RM(119866) ⩾ 5 Thus the bound in (68) is of interest only when120574RM(119866) is 3 or 4

Lemma 169 (Bahremandpour et al [118] 2013) For anynonempty graph 119866 of order 119899 ⩾ 3 120574RM(119866) = 3 if and onlyif Δ(119866) = 119899 minus 2

The following result shows that (68) holds for all graphs119866 of order 119899 ⩾ 4 with 120574RM(119866) = 3 or 4 which improvesTheorem 161 (b)

Theorem 170 (Bahremandpour et al [118] 2013) For anyconnected graph 119866 of order 119899 ⩾ 4

119887RM (119866) le Δ (119866) = 119899 minus 2 119894119891 120574RM (119866) = 3

Δ (119866) + 120575 (119866) minus 1 119894119891 120574RM (119866) = 4(69)

with the first equality if and only if 119866 cong 1198624

Recently Akbari and Qajar [122] have proved the follow-ing result

Theorem 171 (Akbari and Qajar [122]) For any connectedgraph 119866 of order 119899 ⩾ 3

119887RM (119866) ⩽ 119899 minus 120574RM (119866) + 5 (70)

International Journal of Combinatorics 27

We conclude this section with the following problems

Problem 8 Characterize all connected graphs 119866 of order 119899 ⩾3 for which 119887RM(119866) = 119899 minus 1

Problem 9 Prove or disprove if 119866 is a connected graph oforder 119899 ⩾ 3 then

119887RM (119866) ⩽ 119899 minus 120574RM (119866) + 3 (71)

Note that 120574119877(119870

119905(3)) = 4 and 119887RM(119870119905

(3)) = 119899minus1 where 119899 =3119905 for 119905 ⩾ 3 The upper bound on 119887RM(119866) given in Problem 9is sharp if Problem 9 has positive answer

95 Rainbow Bondage Numbers For a positive integer 119896 a119896-rainbow dominating function of a graph 119866 is a function 119891from the vertex-set 119881(119866) to the set of all subsets of the set1 2 119896 such that for any vertex 119909 in 119866 with 119891(119909) = 0 thecondition

⋃

119910isin119873119866(119909)

119891 (119910) = 1 2 119896 (72)

is fulfilledThe weight of a 119896-rainbow dominating function 119891on 119866 is the value

119891 (119866) = sum

119909isin119881(119866)

1003816100381610038161003816119891 (119909)1003816100381610038161003816 (73)

The 119896-rainbow domination number of a graph 119866 denoted by120574119903119896(119866) is the minimum weight of a 119896-rainbow dominating

function 119891 of 119866Note that 120574

1199031(119866) is the classical domination number 120574(119866)

The 119896-rainbow domination number is introduced by Bresaret al [126] Rainbow domination of a graph 119866 coincides withordinary domination of the Cartesian product of 119866 with thecomplete graph in particular 120574

119903119896(119866) = 120574(119866 times 119870

119896) for any

positive integer 119896 and any graph 119866 [126] Hartnell and Rall[127] proved that 120574(119879) lt 120574

1199032(119879) for any tree 119879 no graph has

120574 = 120574119903119896for 119896 ⩾ 3 for any 119896 ⩾ 2 and any graph 119866 of order 119899

min 119899 120574 (119866) + 119896 minus 2 ⩽ 120574119903119896(119866) ⩽ 119896120574 (119866) (74)

The introduction of rainbow domination is motivatedby the study of paired domination in Cartesian productsof graphs where certain upper bounds can be expressed interms of rainbow domination that is for any graph 119866 andany graph 119867 if 119881(119867) can be partitioned into 119896120574

119875-sets then

120574119875(119866 times 119867) ⩽ (1119896)120592(119867)120574

119903119896(119866) proved by Bresar et al [126]

Dehgardi et al [128] introduced the concept of 119896-rainbowbondage number of graphs Let 119866 be a graph with maximumdegree at least two The 119896-rainbow bondage number 119887

119903119896(119866)

of 119866 is the minimum cardinality of all sets 119861 sube 119864(119866) forwhich 120574

119903119896(119866 minus 119861) gt 120574

119903119896(119866) Since in the study of 119896-rainbow

bondage number the assumption Δ(119866) ⩾ 2 is necessarywe always assume that when we discuss 119887

119903119896(119866) all graphs

involved satisfy Δ(119866) ⩾ 2The 2-rainbow bondage number of some special families

of graphs has been determined

Theorem 172 (Dehgardi et al [128]) For any integer 119899 ⩾ 3

1198871199032(119875

119899) = 1 119891119900119903 119899 ⩾ 2

1198871199032(119862

119899) =

1 119894119891 119899 equiv 0 (mod 4) 2 119900119905ℎ119890119903119908119894119904119890

1198871199032(119870

119899119899) =

1 119894119891 119899 = 2

3 119894119891 119899 = 3

6 119894119891 119899 = 4

119898 119894119891 119899 ⩾ 5

(75)

Theorem 173 (Dehgardi et al [128]) For any connected graph119866 of order 119899 ⩾ 3 119887

1199032(119866) ⩽ 120582(119866) + 2Δ(119866) minus 3

Theorem 174 (Dehgardi et al [128]) For any tree 119879 of order119899 ⩾ 3 119887

1199032(119879) ⩽ 2

Theorem 175 (Dehgardi et al [128]) If 119866 is a planar graphwithmaximumdegree at least two then 119887

1199032(119866) ⩽ 15Moreover

if the girth 119892(119866) ⩾ 4 then 1198871199032(119866) ⩽ 11 and if 119892(119866) ⩾ 6 then

1198871199032(119866) ⩽ 9

Theorem 176 (Dehgardi et al [128]) For a complete bipartitegraph 119870

119901119902 if 2119896 + 1 ⩽ 119901 ⩽ 119902 then 119887

119903119896(119870

119901119902) = 119901

Theorem177 (Dehgardi et al [128]) For a complete graph119870119899

if 119899 ⩾ 119896 + 1 then 119887119903119896(119870

119899) = lceil119896119899(119896 + 1)rceil

The special case 119896 = 1 of Theorem 177 can be found inTheorem 1 The following is a simple observation similar toObservation 1

Observation 6 (Dehgardi et al [128]) Let 119866 be a graphwith maximum degree at least two 119867 a spanning subgraphobtained from 119866 by removing 119896 edges If 120574

119903119896(119867) = 120574

119903119896(119866)

then 119887119903119896(119867) ⩽ 119887

119903119896(119866) ⩽ 119887

119903119896(119867) + 119896

Theorem 178 (Dehgardi et al [128]) For every graph 119866 with120574119903119896(119866) = 119896120574(119866) 119887

119903119896(119866) ⩾ 119887(119866)

A graph 119866 is said to be vertex 119896-rainbow domination-critical if 120574

119903119896(119866 minus 119909) lt 120574

119903119896(119866) for every vertex 119909 in 119866 It is

clear that if 119866 has no isolated vertices then 120574119903119896(119866) ⩽ 119896120574(119866) ⩽

119896120573(119866) where 120573(119866) is the vertex-covering number of119866 Thusif 120574

119903119896(119866) = 119896120573(119866) then 120574

119903119896(119866) = 119896120574(119866)

Theorem 179 (Dehgardi et al [128]) For any graph 119866 if120574119903119896(119866) = 119896120573(119866) then

(a) 119887119903119896(119866) ⩾ 120575(119866)

(b) 119887119903119896(119866) ⩾ 120575(119866)+1 if119866 is vertex 119896-rainbow domination-

critical

As far as we know there are no more results on the 119896-rainbow bondage number of graphs in the literature

28 International Journal of Combinatorics

10 Results on Digraphs

Although domination has been extensively studied in undi-rected graphs it is natural to think of a dominating setas a one-way relationship between vertices of the graphIndeed among the earliest literatures on this subject vonNeumann and Morgenstern [129] used what is now calleddomination in digraphs to find solution (or kernels whichare independent dominating sets) for cooperative 119899-persongames Most likely the first formulation of domination byBerge [130] in 1958 was given in the context of digraphs andonly some years latter by Ore [131] in 1962 for undirectedgraphs Despite this history examination of domination andits variants in digraphs has been essentially overlooked (see[4] for an overview of the domination literature) Thus thereare few if any such results on domination for digraphs in theliterature

The bondage number and its related topics for undirectedgraph have become one of major areas both in theoreticaland applied researches However until recently Carlsonand Develin [42] Shan and Kang [132] and Huang andXu [133 134] studied the bondage number for digraphsindependently In this section we introduce their resultsfor general digraphs Results for some special digraphs suchas vertex-transitive digraphs are introduced in the nextsection

101 Upper Bounds for General Digraphs Let 119866 = (119881 119864)

be a digraph without loops and parallel edges A digraph 119866is called to be asymmetric if whenever (119909 119910) isin 119864(119866) then(119910 119909) notin 119864(119866) and to be symmetric if (119909 119910) isin 119864(119866) implies(119910 119909) isin 119864(119866) For a vertex 119909 of 119881(119866) the sets of out-neighbors and in-neighbors of 119909 are respectively defined as119873

+

119866(119909) = 119910 isin 119881(119866) (119909 119910) isin 119864(119866) and 119873minus

119866(119909) = 119909 isin

119881(119866) (119909 119910) isin 119864(119866) the out-degree and the in-degree of119909 are respectively defined as 119889+

119866(119909) = |119873

+

119866(119909)| and 119889minus

119866(119909) =

|119873+

119866(119909)| Denote themaximum and theminimum out-degree

(resp in-degree) of 119866 by Δ+(119866) and 120575+(119866) (resp Δminus(119866) and120575minus

(119866))The degree 119889119866(119909) of 119909 is defined as 119889+

119866(119909)+119889

minus

119866(119909) and

the maximum and the minimum degrees of119866 are denoted byΔ(119866) and 120575(119866) respectively that is Δ(119866) = max119889

119866(119909) 119909 isin

119881(119866) and 120575(119866) = min119889119866(119909) 119909 isin 119881(119866) Note that the

definitions here are different from the ones in the textbookon digraphs see for example Xu [1]

A subset 119878 of119881(119866) is called a dominating set if119881(119866) = 119878cup119873

+

119866(119878) where119873+

119866(119878) is the set of out-neighbors of 119878Then just

as for undirected graphs 120574(119866) is the minimum cardinalityof a dominating set and the bondage number 119887(119866) is thesmallest cardinality of a set 119861 of edges such that 120574(119866 minus 119861) gt120574(119866) if such a subset 119861 sube 119864(119866) exists Otherwise we put119887(119866) = infin

Some basic results for undirected graphs stated inSection 3 can be generalized to digraphs For exampleTheorem 18 is generalized by Carlson and Develin [42]Huang andXu [133] and Shan andKang [132] independentlyas follows

Theorem 180 Let 119866 be a digraph and (119909 119910) isin 119864(119866) Then119887(119866) ⩽ 119889

119866(119910) + 119889

minus

119866(119909) minus |119873

minus

119866(119909) cap 119873

minus

119866(119910)|

Corollary 181 For a digraph 119866 119887(119866) ⩽ 120575minus(119866) + Δ(119866)

Since 120575minus

(119866) ⩽ (12)Δ(119866) from Corollary 181Conjecture 53 is valid for digraphs

Corollary 182 For a digraph 119866 119887(119866) ⩽ (32)Δ(119866)

In the case of undirected graphs the bondage number of119866119899= 119870

119899times 119870

119899achieves this bound However it was shown

by Carlson and Develin in [42] that if we take the symmetricdigraph119866

119899 we haveΔ(119866

119899) = 4(119899minus1) 120574(119866

119899) = 119899 and 119887(119866

119899) ⩽

4(119899 minus 1) + 1 = Δ(119866119899) + 1 So this family of digraphs cannot

show that the bound in Corollary 182 is tightCorresponding to Conjecture 51 which is discredited

for undirected graphs and is valid for digraphs the sameconjecture for digraphs can be proposed as follows

Conjecture 183 (Carlson and Develin [42] 2006) 119887(119866) ⩽Δ(119866) + 1 for any digraph 119866

If Conjecture 183 is true then the following results showsthat this upper bound is tight since 119887(119870

119899times119870

119899) = Δ(119870

119899times119870

119899)+1

for a complete digraph 119870119899

In 2007 Shan and Kang [132] gave some tight upperbounds on the bondage numbers for some asymmetricdigraphs For example 119887(119879) ⩽ Δ(119879) for any asymmetricdirected tree 119879 119887(119866) ⩽ Δ(119866) for any asymmetric digraph 119866with order at least 4 and 120574(119866) ⩽ 2 For planar digraphs theyobtained the following results

Theorem 184 (Shan and Kang [132] 2007) Let 119866 be aasymmetric planar digraphThen 119887(119866) ⩽ Δ(119866)+2 and 119887(119866) ⩽Δ(119866) + 1 if Δ(119866) ⩾ 5 and 119889minus

119866(119909) ⩾ 3 for every vertex 119909 with

119889119866(119909) ⩾ 4

102 Results for Some Special Digraphs The exact values andbounds on 119887(119866) for some standard digraphs are determined

Theorem185 (Huang andXu [133] 2006) For a directed cycle119862119899and a directed path 119875

119899

119887 (119862119899) =

3 119894119891 119899 119894119904 119900119889119889

2 119894119891 119899 119894119904 119890V119890119899

119887 (119875119899) =

2 119894119891 119899 119894119904 119900119889119889

1 119894119891 119899 119894119904 119890V119890119899

(76)

For the de Bruijn digraph 119861(119889 119899) and the Kautz digraph119870(119889 119899)

119887 (119861 (119889 119899)) = 119889 119894119891 119899 119894119904 119900119889119889

119889 ⩽ 119887 (119861 (119889 119899)) ⩽ 2119889 119894119891 119899 119894119904 119890V119890119899

119887 (119870 (119889 119899)) = 119889 + 1

(77)

Like undirected graphs we can define the total domi-nation number and the total bondage number On the totalbondage numbers for some special digraphs the knownresults are as follows

International Journal of Combinatorics 29

Theorem 186 (Huang and Xu [134] 2007) For a directedcycle 119862

119899and a directed path 119875

119899 119887

119879(119875

119899) and 119887

119879(119862

119899) all do not

exist For a complete digraph 119870119899

119887119879(119870

119899) = infin 119894119891 119899 = 1 2

119887119879(119870

119899) = 3 119894119891 119899 = 3

119899 ⩽ 119887119879(119870

119899) ⩽ 2119899 minus 3 119894119891 119899 ⩾ 4

(78)

The extended de Bruijn digraph EB(119889 119899 1199021 119902

119901) and

the extended Kautz digraph EK(119889 119899 1199021 119902

119901) are intro-

duced by Shibata and Gonda [135] If 119901 = 1 then theyare the de Bruijn digraph B(119889 119899) and the Kautz digraphK(119889 119899) respectively Huang and Xu [134] determined theirtotal domination numbers In particular their total bondagenumbers for general cases are determined as follows

Theorem 187 (Huang andXu [134] 2007) If 119889 ⩾ 2 and 119902119894⩾ 2

for each 119894 = 1 2 119901 then

119887119879(EB(119889 119899 119902

1 119902

119901)) = 119889

119901

minus 1

119887119879(EK(119889 119899 119902

1 119902

119901)) = 119889

119901

(79)

In particular for the de Bruijn digraph B(119889 119899) and the Kautzdigraph K(119889 119899)

119887119879(B (119889 119899)) = 119889 minus 1 119887

119879(K (119889 119899)) = 119889 (80)

Zhang et al [136] determined the bondage number incomplete 119905-partite digraphs

Theorem 188 (Zhang et al [136] 2009) For a complete 119905-partite digraph 119870

11989911198992119899119905

where 1198991⩽ 119899

2⩽ sdot sdot sdot ⩽ 119899

119905

119887 (11987011989911198992119899119905

)

=

119898 119894119891 119899119898= 1 119899

119898+1⩾ 2 119891119900119903 119904119900119898119890

119898 (1 ⩽ 119898 lt 119905)

4119905 minus 3 119894119891 1198991= 119899

2= sdot sdot sdot = 119899

119905= 2

119905minus1

sum

119894=1

119899119894+ 2 (119905 minus 1) 119900119905ℎ119890119903119908119894119904119890

(81)

Since an undirected graph can be thought of as a sym-metric digraph any result for digraphs has an analogy forundirected graphs in general In view of this point studyingthe bondage number for digraphs is more significant thanfor undirected graphs Thus we should further study thebondage number of digraphs and try to generalize knownresults on the bondage number and related variants for undi-rected graphs to digraphs prove or disprove Conjecture 183In particular determine the exact values of 119887(B(119889 119899)) for aneven 119899 and 119887

119879(119870

119899) for 119899 ⩾ 4

11 Efficient Dominating Sets

A dominating set 119878 of a graph 119866 is called to be efficient if forevery vertex 119909 in119866 |119873

119866[119909]cap119878| = 1 if119866 is a undirected graph

or |119873minus

119866[119909] cap 119878| = 1 if 119866 is a directed graph The concept of an

efficient set was proposed by Bange et al [137] in 1988 In theliterature an efficient set is also called a perfect dominatingset (see Livingston and Stout [138] for an explanation)

From definition if 119878 is an efficient dominating set of agraph 119866 then 119878 is certainly an independent set and everyvertex not in 119878 is adjacent to exactly one vertex in 119878

It is also clear from definition that a dominating set 119878 isefficient if and only if N

119866[119878] = 119873

119866[119909] 119909 isin 119878 for the

undirected graph 119866 or N+

119866[119878] = 119873

+

119866[119909] 119909 isin 119878 for the

digraph119866 is a partition of119881(119866) where the induced subgraphby119873

119866[119909] or119873+

119866[119909] is a star or an out-star with the root 119909

The efficient domination has important applications inmany areas such as error-correcting codes and receivesmuch attention in the late years

The concept of efficient dominating sets is a measureof the efficiency of domination in graphs Unfortunately asshown in [137] not every graph has an efficient dominatingset andmoreover it is anNP-complete problem to determinewhether a given graph has an efficient dominating set Inaddition it was shown by Clark [139] in 1993 that for a widerange of 119901 almost every random undirected graph 119866 isin

G(120592 119901) has no efficient dominating sets This means thatundirected graphs possessing an efficient dominating set arerare However it is easy to show that every undirected graphhas an orientationwith an efficient dominating set (see Bangeet al [140])

In 1993 Barkauskas and Host [141] showed that deter-mining whether an arbitrary oriented graph has an efficientdominating set is NP-complete Even so the existence ofefficient dominating sets for some special classes of graphs hasbeen examined see for example Dejter and Serra [142] andLee [143] for Cayley graph Gu et al [144] formeshes and toriObradovic et al [145] Huang and Xu [36] Reji Kumar andMacGillivray [146] for circulant graphs Harary graphs andtori and Van Wieren et al [147] for cube-connected cycles

In this section we introduce results of the bondagenumber for some graphs with efficient dominating sets dueto Huang and Xu [36]

111 Results for General Graphs In this subsection we intro-duce some results on bondage numbers obtained by applyingefficient dominating sets We first state the two followinglemmas

Lemma 189 For any 119896-regular graph or digraph 119866 of order 119899120574(119866) ⩾ 119899(119896 + 1) with equality if and only if 119866 has an efficientdominating set In addition if 119866 has an efficient dominatingset then every efficient dominating set is certainly a 120574-set andvice versa

Let 119890 be an edge and 119878 a dominating set in 119866 We say that119890 supports 119878 if 119890 isin (119878 119878) where (119878 119878) = (119909 119910) isin 119864(119866) 119909 isin 119878 119910 isin 119878 Denote by 119904(119866) the minimum number of edgeswhich support all 120574-sets in 119866

Lemma 190 For any graph or digraph 119866 119887(119866) ⩾ 119904(119866) withequality if 119866 is regular and has an efficient dominating set

30 International Journal of Combinatorics

A graph 119866 is called to be vertex-transitive if its automor-phism group Aut(119866) acts transitively on its vertex-set 119881(119866)A vertex-transitive graph is regular Applying Lemmas 189and 190 Huang and Xu obtained some results on bondagenumbers for vertex-transitive graphs or digraphs

Theorem 191 (Huang and Xu [36] 2008) For any vertex-transitive graph or digraph 119866 of order 119899

119887 (119866) ⩾

lceil119899

2120574 (119866)rceil 119894119891 119866 119894119904 119906119899119889119894119903119890119888119905119890119889

lceil119899

120574 (119866)rceil 119894119891 119866 119894119904 119889119894119903119890119888119905119890119889

(82)

Theorem192 (Huang andXu [36] 2008) Let119866 be a 119896-regulargraph of order 119899 Then

119887(119866) ⩽ 119896 if119866 is undirected and 119899 ⩾ 120574(119866)(119896+1)minus119896+1119887(119866) ⩽ 119896+1+ℓ if119866 is directed and 119899 ⩾ 120574(119866)(119896+1)minusℓwith 0 ⩽ ℓ ⩽ 119896 minus 1

Next we introduce a better upper bound on 119887(119866) To thisaim we need the following concept which is a generalizationof the concept of the edge-covering of a graph 119866 For 1198811015840

sube

119881(119866) and 1198641015840 sube 119864(119866) we say that 1198641015840covers 1198811015840 and call 1198641015840an edge-covering for 1198811015840 if there exists an edge (119909 119910) isin 1198641015840 forany vertex 119909 isin 1198811015840 For 119910 isin 119881(119866) let 1205731015840[119910] be the minimumcardinality over all edge-coverings for119873minus

119866[119910]

Theorem 193 (Huang and Xu [36] 2008) If 119866 is a 119896-regulargraph with order 120574(119866)(119896 + 1) then 119887(119866) ⩽ 1205731015840[119910] for any 119910 isin119881(119866)

Theupper bound on 119887(119866) given inTheorem 193 is tight inview of 119887(119862

119899) for a cycle or a directed cycle 119862

119899(seeTheorems

1 and 185 resp)It is easy to see that for a 119896-regular graph 119866 lceil(119896 + 1)2rceil ⩽

1205731015840

[119910] ⩽ 119896 when 119866 is undirected and 1205731015840[119910] = 119896 + 1 when 119866 isdirected By this fact and Lemma 189 the following theoremis merely a simple combination of Theorems 191 and 193

Theorem 194 (Huang and Xu [36] 2008) Let 119866 be a vertex-transitive graph of degree 119896 If 119866 has an efficient dominatingset then

lceil119896 + 1

2rceil ⩽ 119887 (119866) ⩽ 119896 119894119891 119866 119894119904 119906119899119889119894119903119890119888119905119890119889

119887 (119866) = 119896 + 1 119894119891 119866 119894119904 119889119894119903119890119888119905119890119889

(83)

Theorem 195 (Huang and Xu [36] 2008) If 119866 is an undi-rected vertex-transitive cubic graph with order 4120574(119866) and girth119892(119866) ⩽ 5 then 119887(119866) = 2

The proof of Theorem 195 leads to a byproduct If 119892 =5 let (119906

1 119906

2 119906

3 119906

4 119906

5) be a cycle and let D

119894be the family

of efficient dominating sets containing 119906119894for each 119894 isin

1 2 3 4 5 ThenD119894capD

119895= 0 for 119894 = 1 2 5 Then 119866 has

at least 5119904 efficient dominating sets where 119904 = 119904(119866) But thereare only 4s (=ns120574) distinct efficient dominating sets in119866This

contradiction implies that an undirected vertex-transitivecubic graph with girth five has no efficient dominating setsBut a similar argument for 119892(119866) = 3 4 or 119892(119866) ⩾ 6 couldnot give any contradiction This is consistent with the resultthat CCC(119899) a vertex-transitive cubic graph with girth 119899 if3 ⩽ 119899 ⩽ 8 or girth 8 if 119899 ⩾ 9 has efficient dominating setsfor all 119899 ⩾ 3 except 119899 = 5 (see Theorem 199 in the followingsubsection)

112 Results for Cayley Graphs In this subsection we useTheorem 194 to determine the exact values or approximativevalues of bondage numbers for some special vertex-transitivegraphs by characterizing the existence of efficient dominatingsets in these graphs

Let Γ be a nontrivial finite group 119878 a nonempty subset ofΓ without the identity element of Γ A digraph 119866 is defined asfollows

119881 (119866) = Γ (119909 119910) isin 119864 (119866) lArrrArr 119909minus1

119910 isin 119878

for any 119909 119910 isin Γ(84)

is called a Cayley digraph of the group Γ with respect to119878 denoted by 119862

Γ(119878) If 119878minus1 = 119904minus1 119904 isin 119878 = 119878 then

119862Γ(119878) is symmetric and is called a Cayley undirected graph

a Cayley graph for short Cayley graphs or digraphs arecertainly vertex-transitive (see Xu [1])

A circulant graph 119866(119899 119878) of order 119899 is a Cayley graph119862119885119899

(119878) where 119885119899= 0 119899 minus 1 is the addition group of

order 119899 and 119878 is a nonempty subset of119885119899without the identity

element and hence is a vertex-transitive digraph of degree|119878| If 119878minus1 = 119878 then119866(119899 119878) is an undirected graph If 119878 = 1 119896where 2 ⩽ 119896 ⩽ 119899 minus 2 we write 119866(119899 1 119896) for 119866(119899 1 119896) or119866(119899 plusmn1 plusmn119896) and call it a double-loop circulant graph

Huang and Xu [36] showed that for directed 119866 =

119866(119899 1 119896) lceil1198993rceil ⩽ 120574(119866) ⩽ lceil1198992rceil and 119866 has an efficientdominating set if and only if 3 | 119899 and 119896 equiv 2 (mod 3) fordirected 119866 = 119866(119899 1 119896) and 119896 = 1198992 lceil1198995rceil ⩽ 120574(119866) ⩽ lceil1198993rceiland 119866 has an efficient dominating set if and only if 5 | 119899 and119896 equiv plusmn2 (mod 5) By Theorem 194 we obtain the bondagenumber of a double loop circulant graph if it has an efficientdominating set

Theorem 196 (Huang and Xu [36] 2008) Let 119866 be a double-loop circulant graph 119866(119899 1 119896) If 119866 is directed with 3 | 119899and 119896 equiv 2 (mod 3) or 119866 is undirected with 5 | 119899 and119896 equiv plusmn2 (mod 5) then 119887(119866) = 3

The 119898 times 119899 torus is the Cartesian product 119862119898times 119862

119899of

two cycles and is a Cayley graph 119862119885119898times119885119899

(119878) where 119878 =

(0 1) (1 0) for directed cycles and 119878 = (0 plusmn1) (plusmn1 0) forundirected cycles and hence is vertex-transitive Gu et al[144] showed that the undirected torus119862

119898times119862

119899has an efficient

dominating set if and only if both 119898 and 119899 are multiplesof 5 Huang and Xu [36] showed that the directed torus119862119898times 119862

119899has an efficient dominating set if and only if both

119898 and 119899 are multiples of 3 Moreover they found a necessarycondition for a dominating set containing the vertex (0 0)in 119862

119898times 119862

119899to be efficient and obtained the following

result

International Journal of Combinatorics 31

Theorem 197 (Huang and Xu [36] 2008) Let 119866 = 119862119898times 119862

119899

If 119866 is undirected and both119898 and 119899 are multiples of 5 or if 119866 isdirected and both119898 and 119899 are multiples of 3 then 119887(119866) = 3

The hypercube 119876119899

is the Cayley graph 119862Γ(119878)

where Γ = 1198852times sdot sdot sdot times 119885

2= (119885

2)119899 and 119878 =

100 sdot sdot sdot 0 010 sdot sdot sdot 0 00 sdot sdot sdot 01 Lee [143] showed that119876119899has an efficient dominating set if and only if 119899 = 2119898 minus 1

for a positive integer119898 ByTheorem 194 the following resultis obtained immediately

Theorem 198 (Huang and Xu [36] 2008) If 119899 = 2119898 minus 1 for apositive integer119898 then 2119898minus1

⩽ 119887(119876119899) ⩽ 2

119898

minus 1

Problem 10 Determine the exact value of 119887(119876119899) for 119899 = 2119898minus1

The 119899-dimensional cube-connected cycle denoted byCCC(119899) is constructed from the 119899-dimensional hypercube119876119899by replacing each vertex 119909 in 119876

119899with an undirected cycle

119862119899of length 119899 and linking the 119894th vertex of the 119862

119899to the 119894th

neighbor of 119909 It has been proved that CCC(119899) is a Cayleygraph and hence is a vertex-transitive graph with degree 3Van Wieren et al [147] proved that CCC(119899) has an efficientdominating set if and only if 119899 = 5 From Theorems 194 and195 the following result can be derived

Theorem 199 (Huang and Xu [36] 2008) Let 119866 = 119862119862119862(119899)be the 119899-dimensional cube-connected cycles with 119899 ⩾ 3 and119899 = 5 Then 120574(119866) = 1198992119899minus2 and 2 ⩽ 119887(119866) ⩽ 3 In addition119887(119862119862119862(3)) = 119887(119862119862119862(4)) = 2

Problem 11 Determine the exact value of 119887(CCC(119899)) for 119899 ⩾5

Acknowledgments

This work was supported by NNSF of China (Grants nos11071233 61272008) The author would like to express hisgratitude to the anonymous referees for their kind sugges-tions and comments on the original paper to ProfessorSheikholeslami and Professor Jafari Rad for sending thereferences [128] and [81] which resulted in this version

References

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[2] S THedetniemi andR C Laskar ldquoBibliography on dominationin graphs and some basic definitions of domination parame-tersrdquo Discrete Mathematics vol 86 no 1ndash3 pp 257ndash277 1990

[3] TW Haynes S T Hedetniemi and P J Slater Fundamentals ofDomination in Graphs vol 208 ofMonographs and Textbooks inPure and Applied Mathematics Marcel Dekker New York NYUSA 1998

[4] TWHaynes S THedetniemi and P J Slater EdsDominationin Graphs vol 209 of Monographs and Textbooks in Pure andAppliedMathematicsMarcelDekker NewYorkNYUSA 1998

[5] M R Garey andD S JohnsonComputers and Intractability WH Freeman and Company San Francisco Calif USA 1979

[6] H B Walikar and B D Acharya ldquoDomination critical graphsrdquoNational Academy Science Letters vol 2 pp 70ndash72 1979

[7] D Bauer F Harary J Nieminen and C L Suffel ldquoDominationalteration sets in graphsrdquo Discrete Mathematics vol 47 no 2-3pp 153ndash161 1983

[8] J F Fink M S Jacobson L F Kinch and J Roberts ldquoThebondage number of a graphrdquo Discrete Mathematics vol 86 no1ndash3 pp 47ndash57 1990

[9] F-T Hu and J-M Xu ldquoThe bondage number of (119899 minus 3)-regulargraph G of order nrdquo Ars Combinatoria to appear

[10] U Teschner ldquoNew results about the bondage number of agraphrdquoDiscreteMathematics vol 171 no 1ndash3 pp 249ndash259 1997

[11] B L Hartnell and D F Rall ldquoA bound on the size of a graphwith given order and bondage numberrdquo Discrete Mathematicsvol 197-198 pp 409ndash413 1999 16th British CombinatorialConference (London 1997)

[12] B L Hartnell and D F Rall ldquoA characterization of trees inwhich no edge is essential to the domination numberrdquo ArsCombinatoria vol 33 pp 65ndash76 1992

[13] J Topp and P D Vestergaard ldquo120572119896- and 120574

119896-stable graphsrdquo

Discrete Mathematics vol 212 no 1-2 pp 149ndash160 2000[14] A Meir and J W Moon ldquoRelations between packing and

covering numbers of a treerdquo Pacific Journal of Mathematics vol61 no 1 pp 225ndash233 1975

[15] B L Hartnell L K Joslashrgensen P D Vestergaard and CA Whitehead ldquoEdge stability of the k-domination numberof treesrdquo Bulletin of the Institute of Combinatorics and ItsApplications vol 22 pp 31ndash40 1998

[16] F-T Hu and J-M Xu ldquoOn the complexity of the bondage andreinforcement problemsrdquo Journal of Complexity vol 28 no 2pp 192ndash201 2012

[17] B L Hartnell and D F Rall ldquoBounds on the bondage numberof a graphrdquo Discrete Mathematics vol 128 no 1ndash3 pp 173ndash1771994

[18] Y-L Wang ldquoOn the bondage number of a graphrdquo DiscreteMathematics vol 159 no 1ndash3 pp 291ndash294 1996

[19] G H Fricke TW Haynes S M Hedetniemi S T Hedetniemiand R C Laskar ldquoExcellent treesrdquo Bulletin of the Institute ofCombinatorics and Its Applications vol 34 pp 27ndash38 2002

[20] V Samodivkin ldquoThe bondage number of graphs good and badverticesrdquoDiscussiones Mathematicae GraphTheory vol 28 no3 pp 453ndash462 2008

[21] J E Dunbar T W Haynes U Teschner and L VolkmannldquoBondage insensitivity and reinforcementrdquo in Domination inGraphs vol 209 of Monographs and Textbooks in Pure andAppliedMathematics pp 471ndash489 Dekker NewYork NY USA1998

[22] H Liu and L Sun ldquoThe bondage and connectivity of a graphrdquoDiscrete Mathematics vol 263 no 1ndash3 pp 289ndash293 2003

[23] A-H Esfahanian and S L Hakimi ldquoOn computing a con-ditional edge-connectivity of a graphrdquo Information ProcessingLetters vol 27 no 4 pp 195ndash199 1988

[24] R C Brigham P Z Chinn and R D Dutton ldquoVertexdomination-critical graphsrdquoNetworks vol 18 no 3 pp 173ndash1791988

[25] V Samodivkin ldquoDomination with respect to nondegenerateproperties bondage numberrdquoThe Australasian Journal of Com-binatorics vol 45 pp 217ndash226 2009

[26] U Teschner ldquoA new upper bound for the bondage numberof graphs with small domination numberrdquo The AustralasianJournal of Combinatorics vol 12 pp 27ndash35 1995

32 International Journal of Combinatorics

[27] J Fulman D Hanson and G MacGillivray ldquoVertex dom-ination-critical graphsrdquoNetworks vol 25 no 2 pp 41ndash43 1995

[28] U Teschner ldquoA counterexample to a conjecture on the bondagenumber of a graphrdquo Discrete Mathematics vol 122 no 1ndash3 pp393ndash395 1993

[29] U Teschner ldquoThe bondage number of a graph G can be muchgreater than Δ(G)rdquo Ars Combinatoria vol 43 pp 81ndash87 1996

[30] L Volkmann ldquoOn graphs with equal domination and coveringnumbersrdquoDiscrete AppliedMathematics vol 51 no 1-2 pp 211ndash217 1994

[31] L A Sanchis ldquoMaximum number of edges in connected graphswith a given domination numberrdquoDiscreteMathematics vol 87no 1 pp 65ndash72 1991

[32] S Klavzar and N Seifter ldquoDominating Cartesian products ofcyclesrdquoDiscrete AppliedMathematics vol 59 no 2 pp 129ndash1361995

[33] H K Kim ldquoA note on Vizingrsquos conjecture about the bondagenumberrdquo Korean Journal of Mathematics vol 10 no 2 pp 39ndash42 2003

[34] M Y Sohn X Yuan and H S Jeong ldquoThe bondage number of1198623times119862

119899rdquo Journal of the KoreanMathematical Society vol 44 no

6 pp 1213ndash1231 2007[35] L Kang M Y Sohn and H K Kim ldquoBondage number of the

discrete torus 119862119899times 119862

4rdquo Discrete Mathematics vol 303 no 1ndash3

pp 80ndash86 2005[36] J Huang and J-M Xu ldquoThe bondage numbers and efficient

dominations of vertex-transitive graphsrdquoDiscrete Mathematicsvol 308 no 4 pp 571ndash582 2008

[37] C J Xiang X Yuan andM Y Sohn ldquoDomination and bondagenumber of119862

3times119862

119899rdquoArs Combinatoria vol 97 pp 299ndash310 2010

[38] M S Jacobson and L F Kinch ldquoOn the domination numberof products of graphs Irdquo Ars Combinatoria vol 18 pp 33ndash441984

[39] Y-Q Li ldquoA note on the bondage number of a graphrdquo ChineseQuarterly Journal of Mathematics vol 9 no 4 pp 1ndash3 1994

[40] F-T Hu and J-M Xu ldquoBondage number of mesh networksrdquoFrontiers ofMathematics inChina vol 7 no 5 pp 813ndash826 2012

[41] R Frucht and F Harary ldquoOn the corona of two graphsrdquoAequationes Mathematicae vol 4 pp 322ndash325 1970

[42] K Carlson and M Develin ldquoOn the bondage number of planarand directed graphsrdquoDiscreteMathematics vol 306 no 8-9 pp820ndash826 2006

[43] U Teschner ldquoOn the bondage number of block graphsrdquo ArsCombinatoria vol 46 pp 25ndash32 1997

[44] J Huang and J-M Xu ldquoNote on conjectures of bondagenumbers of planar graphsrdquo Applied Mathematical Sciences vol6 no 65ndash68 pp 3277ndash3287 2012

[45] L Kang and J Yuan ldquoBondage number of planar graphsrdquoDiscrete Mathematics vol 222 no 1ndash3 pp 191ndash198 2000

[46] M Fischermann D Rautenbach and L Volkmann ldquoRemarkson the bondage number of planar graphsrdquo Discrete Mathemat-ics vol 260 no 1ndash3 pp 57ndash67 2003

[47] J Hou G Liu and J Wu ldquoBondage number of planar graphswithout small cyclesrdquoUtilitasMathematica vol 84 pp 189ndash1992011

[48] J Huang and J-M Xu ldquoThe bondage numbers of graphs withsmall crossing numbersrdquo Discrete Mathematics vol 307 no 15pp 1881ndash1897 2007

[49] Q Ma S Zhang and J Wang ldquoBondage number of 1-planargraphrdquo Applied Mathematics vol 1 no 2 pp 101ndash103 2010

[50] J Hou and G Liu ldquoA bound on the bondage number of toroidalgraphsrdquoDiscreteMathematics Algorithms and Applications vol4 no 3 Article ID 1250046 5 pages 2012

[51] Y-C Cao J-M Xu and X-R Xu ldquoOn bondage number oftoroidal graphsrdquo Journal of University of Science and Technologyof China vol 39 no 3 pp 225ndash228 2009

[52] Y-C Cao J Huang and J-M Xu ldquoThe bondage number ofgraphs with crossing number less than fourrdquo Ars Combinatoriato appear

[53] H K Kim ldquoOn the bondage numbers of some graphsrdquo KoreanJournal of Mathematics vol 11 no 2 pp 11ndash15 2004

[54] A Gagarin and V Zverovich ldquoUpper bounds for the bondagenumber of graphs on topological surfacesrdquo Discrete Mathemat-ics 2012

[55] J Huang ldquoAn improved upper bound for the bondage numberof graphs on surfacesrdquoDiscrete Mathematics vol 312 no 18 pp2776ndash2781 2012

[56] A Gagarin and V Zverovich ldquoThe bondage number of graphson topological surfaces and Teschnerrsquos conjecturerdquo DiscreteMathematics vol 313 no 6 pp 796ndash808 2013

[57] V Samodivkin ldquoNote on the bondage number of graphs ontopological surfacesrdquo httparxivorgabs12086203

[58] E J Cockayne R M Dawes and S T Hedetniemi ldquoTotaldomination in graphsrdquoNetworks vol 10 no 3 pp 211ndash219 1980

[59] R Laskar J Pfaff S M Hedetniemi and S T HedetniemildquoOn the algorithmic complexity of total dominationrdquo Society forIndustrial and Applied Mathematics vol 5 no 3 pp 420ndash4251984

[60] J Pfaff R C Laskar and S T Hedetniemi ldquoNP-completeness oftotal and connected domination and irredundance for bipartitegraphsrdquo Tech Rep 428 Department of Mathematical SciencesClemson University 1983

[61] M A Henning ldquoA survey of selected recent results on totaldomination in graphsrdquoDiscrete Mathematics vol 309 no 1 pp32ndash63 2009

[62] V R Kulli and D K Patwari ldquoThe total bondage number ofa graphrdquo in Advances in Graph Theory pp 227ndash235 VishwaGulbarga India 1991

[63] F-T Hu Y Lu and J-M Xu ldquoThe total bondage number ofgrid graphsrdquo Discrete Applied Mathematics vol 160 no 16-17pp 2408ndash2418 2012

[64] J Cao M Shi and B Wu ldquoResearch on the total bondagenumber of a spectial networkrdquo in Proceedings of the 3rdInternational Joint Conference on Computational Sciences andOptimization (CSO rsquo10) pp 456ndash458 May 2010

[65] N Sridharan MD Elias and V S A Subramanian ldquoTotalbondage number of a graphrdquo AKCE International Journal ofGraphs and Combinatorics vol 4 no 2 pp 203ndash209 2007

[66] N Jafari Rad and J Raczek ldquoSome progress on total bondage ingraphsrdquo Graphs and Combinatorics 2011

[67] T W Haynes and P J Slater ldquoPaired-domination and thepaired-domatic numberrdquoCongressusNumerantium vol 109 pp65ndash72 1995

[68] T W Haynes and P J Slater ldquoPaired-domination in graphsrdquoNetworks vol 32 no 3 pp 199ndash206 1998

[69] X Hou ldquoA characterization of 2120574 120574119875-treesrdquoDiscrete Mathemat-

ics vol 308 no 15 pp 3420ndash3426 2008[70] K E Proffitt T W Haynes and P J Slater ldquoPaired-domination

in grid graphsrdquo Congressus Numerantium vol 150 pp 161ndash1722001

International Journal of Combinatorics 33

[71] H Qiao L KangM Cardei andD-Z Du ldquoPaired-dominationof treesrdquo Journal of Global Optimization vol 25 no 1 pp 43ndash542003

[72] E Shan L Kang and M A Henning ldquoA characterizationof trees with equal total domination and paired-dominationnumbersrdquo The Australasian Journal of Combinatorics vol 30pp 31ndash39 2004

[73] J Raczek ldquoPaired bondage in treesrdquo Discrete Mathematics vol308 no 23 pp 5570ndash5575 2008

[74] J A Telle and A Proskurowski ldquoAlgorithms for vertex parti-tioning problems on partial k-treesrdquo SIAM Journal on DiscreteMathematics vol 10 no 4 pp 529ndash550 1997

[75] G S Domke J H Hattingh M A Henning and L R MarkusldquoRestrained domination in treesrdquoDiscreteMathematics vol 211no 1ndash3 pp 1ndash9 2000

[76] G S Domke J H Hattingh M A Henning and L R MarkusldquoRestrained domination in graphs with minimum degree twordquoJournal of Combinatorial Mathematics and Combinatorial Com-puting vol 35 pp 239ndash254 2000

[77] G S Domke J H Hattingh S T Hedetniemi R C Laskarand L R Markus ldquoRestrained domination in graphsrdquo DiscreteMathematics vol 203 no 1ndash3 pp 61ndash69 1999

[78] J H Hattingh and E J Joubert ldquoAn upper bound for therestrained domination number of a graph with minimumdegree at least two in terms of order and minimum degreerdquoDiscrete Applied Mathematics vol 157 no 13 pp 2846ndash28582009

[79] M A Henning ldquoGraphs with large restrained dominationnumberrdquo Discrete Mathematics vol 197-198 pp 415ndash429 199916th British Combinatorial Conference (London 1997)

[80] J H Hattingh and A R Plummer ldquoRestrained bondage ingraphsrdquo Discrete Mathematics vol 308 no 23 pp 5446ndash54532008

[81] N Jafari Rad R Hasni J Raczek and L Volkmann ldquoTotalrestrained bondage in graphsrdquoActaMathematica Sinica vol 29no 6 pp 1033ndash1042 2013

[82] J Cyman and J Raczek ldquoOn the total restrained dominationnumber of a graphrdquoThe Australasian Journal of Combinatoricsvol 36 pp 91ndash100 2006

[83] M A Henning ldquoGraphs with large total domination numberrdquoJournal of Graph Theory vol 35 no 1 pp 21ndash45 2000

[84] D-X Ma X-G Chen and L Sun ldquoOn total restraineddomination in graphsrdquoCzechoslovakMathematical Journal vol55 no 1 pp 165ndash173 2005

[85] B Zelinka ldquoRemarks on restrained domination and totalrestrained domination in graphsrdquo Czechoslovak MathematicalJournal vol 55 no 2 pp 393ndash396 2005

[86] W Goddard and M A Henning ldquoIndependent domination ingraphs a survey and recent resultsrdquo Discrete Mathematics vol313 no 7 pp 839ndash854 2013

[87] J H Zhang H L Liu and L Sun ldquoIndependence bondagenumber and reinforcement number of some graphsrdquo Transac-tions of Beijing Institute of Technology vol 23 no 2 pp 140ndash1422003

[88] K D Dharmalingam Studies in Graph Theorymdashequitabledomination and bottleneck domination [PhD thesis] MaduraiKamaraj University Madurai India 2006

[89] GDeepakND Soner andAAlwardi ldquoThe equitable bondagenumber of a graphrdquo Research Journal of Pure Algebra vol 1 no9 pp 209ndash212 2011

[90] E Sampathkumar and H B Walikar ldquoThe connected domina-tion number of a graphrdquo Journal of Mathematical and PhysicalSciences vol 13 no 6 pp 607ndash613 1979

[91] K White M Farber and W Pulleyblank ldquoSteiner trees con-nected domination and strongly chordal graphsrdquoNetworks vol15 no 1 pp 109ndash124 1985

[92] M Cadei M X Cheng X Cheng and D-Z Du ldquoConnecteddomination in Ad Hoc wireless networksrdquo in Proceedings ofthe 6th International Conference on Computer Science andInformatics (CSampI rsquo02) 2002

[93] J F Fink and M S Jacobson ldquon-domination in graphsrdquo inGraph Theory with Applications to Algorithms and ComputerScience Wiley-Interscience Publications pp 283ndash300 WileyNew York NY USA 1985

[94] M Chellali O Favaron A Hansberg and L Volkmann ldquok-domination and k-independence in graphs a surveyrdquo Graphsand Combinatorics vol 28 no 1 pp 1ndash55 2012

[95] Y Lu X Hou J-M Xu andN Li ldquoTrees with uniqueminimump-dominating setsrdquo Utilitas Mathematica vol 86 pp 193ndash2052011

[96] Y Lu and J-M Xu ldquoThe p-bondage number of treesrdquo Graphsand Combinatorics vol 27 no 1 pp 129ndash141 2011

[97] Y Lu and J-M Xu ldquoThe 2-domination and 2-bondage numbersof grid graphs A manuscriptrdquo httparxivorgabs12044514

[98] M Krzywkowski ldquoNon-isolating 2-bondage in graphsrdquo TheJournal of the Mathematical Society of Japan vol 64 no 4 2012

[99] M A Henning O R Oellermann and H C Swart ldquoBoundson distance domination parametersrdquo Journal of CombinatoricsInformation amp System Sciences vol 16 no 1 pp 11ndash18 1991

[100] F Tian and J-M Xu ldquoBounds for distance domination numbersof graphsrdquo Journal of University of Science and Technology ofChina vol 34 no 5 pp 529ndash534 2004

[101] F Tian and J-M Xu ldquoDistance domination-critical graphsrdquoApplied Mathematics Letters vol 21 no 4 pp 416ndash420 2008

[102] F Tian and J-M Xu ldquoA note on distance domination numbersof graphsrdquo The Australasian Journal of Combinatorics vol 43pp 181ndash190 2009

[103] F Tian and J-M Xu ldquoAverage distances and distance domina-tion numbersrdquo Discrete Applied Mathematics vol 157 no 5 pp1113ndash1127 2009

[104] Z-L Liu F Tian and J-M Xu ldquoProbabilistic analysis of upperbounds for 2-connected distance k-dominating sets in graphsrdquoTheoretical Computer Science vol 410 no 38ndash40 pp 3804ndash3813 2009

[105] M A Henning O R Oellermann and H C Swart ldquoDistancedomination critical graphsrdquo Journal of Combinatorial Mathe-matics and Combinatorial Computing vol 44 pp 33ndash45 2003

[106] T J Bean M A Henning and H C Swart ldquoOn the integrityof distance domination in graphsrdquo The Australasian Journal ofCombinatorics vol 10 pp 29ndash43 1994

[107] M Fischermann and L Volkmann ldquoA remark on a conjecturefor the (kp)-domination numberrdquoUtilitasMathematica vol 67pp 223ndash227 2005

[108] T Korneffel D Meierling and L Volkmann ldquoA remark on the(22)-domination numberrdquo Discussiones Mathematicae GraphTheory vol 28 no 2 pp 361ndash366 2008

[109] Y Lu X Hou and J-M Xu ldquoOn the (22)-domination numberof treesrdquo Discussiones Mathematicae Graph Theory vol 30 no2 pp 185ndash199 2010

34 International Journal of Combinatorics

[110] H Li and J-M Xu ldquo(dm)-domination numbers of m-connected graphsrdquo Rapports de Recherche 1130 LRI UniversiteParis-Sud 1997

[111] S M Hedetniemi S T Hedetniemi and T V Wimer ldquoLineartime resource allocation algorithms for treesrdquo Tech Rep URI-014 Department of Mathematical Sciences Clemson Univer-sity 1987

[112] M Farber ldquoDomination independent domination and dualityin strongly chordal graphsrdquo Discrete Applied Mathematics vol7 no 2 pp 115ndash130 1984

[113] G S Domke and R C Laskar ldquoThe bondage and reinforcementnumbers of 120574

119891for some graphsrdquoDiscrete Mathematics vol 167-

168 pp 249ndash259 1997[114] G S Domke S T Hedetniemi and R C Laskar ldquoFractional

packings coverings and irredundance in graphsrdquo vol 66 pp227ndash238 1988

[115] E J Cockayne P A Dreyer Jr S M Hedetniemi andS T Hedetniemi ldquoRoman domination in graphsrdquo DiscreteMathematics vol 278 no 1ndash3 pp 11ndash22 2004

[116] I Stewart ldquoDefend the roman empirerdquo Scientific American vol281 pp 136ndash139 1999

[117] C S ReVelle and K E Rosing ldquoDefendens imperiumromanum a classical problem in military strategyrdquoThe Ameri-can Mathematical Monthly vol 107 no 7 pp 585ndash594 2000

[118] A Bahremandpour F-T Hu S M Sheikholeslami and J-MXu ldquoOn the Roman bondage number of a graphrdquo DiscreteMathematics Algorithms and Applications vol 5 no 1 ArticleID 1350001 15 pages 2013

[119] N Jafari Rad and L Volkmann ldquoRoman bondage in graphsrdquoDiscussiones Mathematicae Graph Theory vol 31 no 4 pp763ndash773 2011

[120] K Ebadi and L PushpaLatha ldquoRoman bondage and Romanreinforcement numbers of a graphrdquo International Journal ofContemporary Mathematical Sciences vol 5 pp 1487ndash14972010

[121] N Dehgardi S M Sheikholeslami and L Volkman ldquoOn theRoman k-bondage number of a graphrdquo AKCE InternationalJournal of Graphs and Combinatorics vol 8 pp 169ndash180 2011

[122] S Akbari and S Qajar ldquoA note on Roman bondage number ofgraphsrdquo Ars Combinatoria to Appear

[123] F-T Hu and J-M Xu ldquoRoman bondage numbers of somegraphsrdquo httparxivorgabs11093933

[124] N Jafari Rad and L Volkmann ldquoOn the Roman bondagenumber of planar graphsrdquo Graphs and Combinatorics vol 27no 4 pp 531ndash538 2011

[125] S Akbari M Khatirinejad and S Qajar ldquoA note on the romanbondage number of planar graphsrdquo Graphs and Combinatorics2012

[126] B Bresar M A Henning and D F Rall ldquoRainbow dominationin graphsrdquo Taiwanese Journal of Mathematics vol 12 no 1 pp213ndash225 2008

[127] B L Hartnell and D F Rall ldquoOn dominating the Cartesianproduct of a graph and 120572krdquo Discussiones Mathematicae GraphTheory vol 24 no 3 pp 389ndash402 2004

[128] N Dehgardi S M Sheikholeslami and L Volkman ldquoThe k-rainbow bondage number of a graphrdquo to appear in DiscreteApplied Mathematics

[129] J von Neumann and O Morgenstern Theory of Games andEconomic Behavior Princeton University Press Princeton NJUSA 1944

[130] C BergeTheorıe des Graphes et ses Applications 1958[131] O Ore Theory of Graphs vol 38 American Mathematical

Society Colloquium Publications 1962[132] E Shan and L Kang ldquoBondage number in oriented graphsrdquoArs

Combinatoria vol 84 pp 319ndash331 2007[133] J Huang and J-M Xu ldquoThe bondage numbers of extended

de Bruijn and Kautz digraphsrdquo Computers amp Mathematics withApplications vol 51 no 6-7 pp 1137ndash1147 2006

[134] J Huang and J-M Xu ldquoThe total domination and total bondagenumbers of extended de Bruijn and Kautz digraphsrdquoComputersamp Mathematics with Applications vol 53 no 8 pp 1206ndash12132007

[135] Y Shibata and Y Gonda ldquoExtension of de Bruijn graph andKautz graphrdquo Computers amp Mathematics with Applications vol30 no 9 pp 51ndash61 1995

[136] X Zhang J Liu and JMeng ldquoThebondage number in completet-partite digraphsrdquo Information Processing Letters vol 109 no17 pp 997ndash1000 2009

[137] D W Bange A E Barkauskas and P J Slater ldquoEfficient dom-inating sets in graphsrdquo in Applications of Discrete Mathematicspp 189ndash199 SIAM Philadelphia Pa USA 1988

[138] M Livingston and Q F Stout ldquoPerfect dominating setsrdquoCongressus Numerantium vol 79 pp 187ndash203 1990

[139] L Clark ldquoPerfect domination in random graphsrdquo Journal ofCombinatorial Mathematics and Combinatorial Computing vol14 pp 173ndash182 1993

[140] D W Bange A E Barkauskas L H Host and L H ClarkldquoEfficient domination of the orientations of a graphrdquo DiscreteMathematics vol 178 no 1ndash3 pp 1ndash14 1998

[141] A E Barkauskas and L H Host ldquoFinding efficient dominatingsets in oriented graphsrdquo Congressus Numerantium vol 98 pp27ndash32 1993

[142] I J Dejter and O Serra ldquoEfficient dominating sets in CayleygraphsrdquoDiscrete AppliedMathematics vol 129 no 2-3 pp 319ndash328 2003

[143] J Lee ldquoIndependent perfect domination sets in Cayley graphsrdquoJournal of Graph Theory vol 37 no 4 pp 213ndash219 2001

[144] WGu X Jia and J Shen ldquoIndependent perfect domination setsin meshes tori and treesrdquo Unpublished

[145] N Obradovic J Peters and G Ruzic ldquoEfficient domination incirculant graphswith two chord lengthsrdquo Information ProcessingLetters vol 102 no 6 pp 253ndash258 2007

[146] K Reji Kumar and G MacGillivray ldquoEfficient domination incirculant graphsrdquo Discrete Mathematics vol 313 no 6 pp 767ndash771 2013

[147] D Van Wieren M Livingston and Q F Stout ldquoPerfectdominating sets on cube-connected cyclesrdquo Congressus Numer-antium vol 97 pp 51ndash70 1993

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

16 International Journal of Combinatorics

This result implies that 119887(119866) ⩽ Δ(119866) + 4 by Corollary 15for any graph 119866 with cr(119866) ⩽ 5 Cao et al established thefollowing relation in terms of maximum planar subgraphs

Lemma 99 (Cao et al [52]) Let 119866 be a connected graph withcrossing number cr(119866) For any maximum planar subgraph119867of 119866

sum

119890isin119864(119867)

119863119866(119890) ⩾ 2 minus

2cr (119866)120575 (119866)

(36)

Using Carlson-Develin method and combiningLemma 98 with Lemma 99 Cao et al proved the followingresults

Theorem 100 (Cao et al [52]) For any connected graph 119866119887(119866) ⩽ Δ(119866) + 2 if 119866 satisfies one of the following conditions

(a) cr(119866) ⩽ 3(b) cr(119866) = 4 and 119866 is not 4-regular(c) cr(119866) = 5 and 119866 contains no vertices of degree 4

Theorem101 (Cao et al [52]) Let119866 be a connected graphwithΔ(119866) = 5 and cr(119866) ⩽ 4 If no triangles contain two vertices ofdegree 5 then 119887(119866) ⩽ 6 = Δ(119866) + 1

Theorem 102 (Cao et al [52]) Let 119866 be a connected graphwith Δ(119866) ⩾ 6 and cr(119866) ⩽ 3 If Δ(119866) ⩾ 7 or if Δ(119866) = 6120575(119866) = 3 and every edge 119890 = 119909119910 with 119889

119866(119909) = 5 and 119889

119866(119910) = 6

is contained in atmost one triangle then 119887(119866) ⩽ min8 Δ(119866)+1

Using Carlson-Develin method it can be proved that119887(119866) ⩽ Δ(119866) + 2 if cr(119866) ⩽ 3 (see the first conclusion inTheorem 100) and not yet proved that 119887(119866) ⩽ 8 if cr(119866) ⩽3 although it has been proved by using other method (seeCorollary 90)

By using Carlson-Develin method Samodivkin [25]obtained some results on the bondage number for graphswithsome given properties

Kim [53] showed that 119887(119866) ⩽ Δ(119866) + 2 for a connectedgraph 119866 with genus 1 Δ(119866) ⩽ 5 and having a toroidalembedding of which at least one region is not 4-sidedRecently Gagarin and Zverovich [54] further have extendedCarlson-Develin ideas to establish a nice upper bound forarbitrary graphs that can be embedded on orientable ornonorientable topological surface

Theorem 103 (Gagarin and Zverovich [54] 2012) Let 119866 bea graph embeddable on an orientable surface of genus ℎ and anonorientable surface of genus 119896 Then 119887(119866) ⩽ minΔ(119866)+ℎ+2 Δ(119866) + 119896 + 1

This result generalizes the corresponding upper boundsin Theorems 74 and 96 for any orientable or nonori-entable topological surface By investigating the proof ofTheorem 103 Huang [55] found that the issue of the ori-entability can be avoided by using the Euler characteristic120594(= 120592(119866) minus 120576(119866) + 120601(119866)) instead of the genera ℎ and 119896 the

relations are 120594 = 2minus2ℎ and 120594 = 2minus119896 To obtain the best resultfromTheorem 103 onewants ℎ and 119896 as small as possible thisis equivalent to having 120594 as large as possible

According toTheorem 103 if119866 is planar (ℎ = 0 120594 = 2) orcan be embedded on the real projective plane (119896 = 1 120594 = 1)then 119887(119866) ⩽ Δ(119866) + 2 In all other cases Huang [55] had thefollowing improvement for Theorem 103 the proof is basedon the technique developed byCarlson-Develin andGagarin-Zverovich and includes some elementary calculus as a newingredient mainly the intermediate-value theorem and themean-value theorem

Theorem 104 (Huang [55] 2012) Let 119866 be a graph embed-dable on a surface whose Euler characteristic 120594 is as large aspossible If 120594 ⩽ 0 then 119887(119866) ⩽ Δ(119866)+lfloor119903rfloor where 119903 is the largestreal root of the following cubic equation in 119911

1199113

+ 21199112

+ (6120594 minus 7) 119911 + 18120594 minus 24 = 0 (37)

In addition if 120594 decreases then 119903 increases

The following result is asymptotically equivalent toTheorem 104

Theorem 105 (Huang [55] 2012) Let 119866 be a graph embed-dable on a surface whose Euler characteristic 120594 is as large aspossible If 120594 ⩽ 0 then 119887(119866) lt Δ(119866) + radic12 minus 6120594 + 12 orequivalently 119887(119866) ⩽ Δ(119866) + lceilradic12 minus 6120594 minus 12rceil

Also Gagarin andZverovich [54] indicated that the upperbound in Theorem 103 can be improved for larger values ofthe genera ℎ and 119896 by adjusting the proofs and stated thefollowing general conjecture

Conjecture 106 (Gagarin and Zverovich [54] 2012) For aconnected graph G of orientable genus ℎ and nonorientablegenus 119896

119887 (119866) ⩽ min 119888ℎ 119888

1015840

119896 Δ (119866) + 119900 (ℎ) Δ (119866) + 119900 (119896) (38)

where 119888ℎand 1198881015840

119896are constants depending respectively on the

orientable and nonorientable genera of 119866

In the recent paper Gagarin and Zverovich [56] providedconstant upper bounds for the bondage number of graphs ontopological surfaces which can be used as the first estimationfor the constants 119888

ℎand 1198881015840

119896of Conjecture 106

Theorem 107 (Gagarin and Zverovich [56] 2013) Let 119866 bea connected graph of order 119899 If 119866 can be embedded on thesurface of its orientable or nonorientable genus of the Eulercharacteristic 120594 then

(a) 120594 ⩾ 1 implies 119887(119866) ⩽ 10(b) 120594 ⩽ 0 and 119899 gt minus12120594 imply 119887(119866) ⩽ 11(c) 120594 ⩽ minus1 and 119899 ⩽ minus12120594 imply 119887(119866) ⩽ 11 +

3120594(radic17 minus 8120594 minus 3)(120594 minus 1) = 11 + 119874(radicminus120594)

Theorem 79 shows that a connected planar triangle-freegraph 119866 has 119887(119866) ⩽ 6 The following result generalizes to itall the other topological surfaces

International Journal of Combinatorics 17

Theorem 108 (Gagarin and Zverovich [56] 2013) Let 119866 be aconnected triangle-free graph of order 119899 If 119866 can be embeddedon the surface of its orientable or nonorientable genus of theEuler characteristic 120594 then

(a) 120594 ⩾ 1 implies 119887(119866) ⩽ 6(b) 120594 ⩽ 0 and 119899 gt minus8120594 imply 119887(119866) ⩽ 7(c) 120594 ⩽ minus1 and 119899 ⩽ minus8120594 imply 119887(119866) ⩽ 7minus4120594(1+radic2 minus 120594)

In [56] the authors also explicitly improved upperbounds in Theorem 103 and gave tight lower bounds forthe number of vertices of graphs 2-cell embeddable ontopological surfaces of a given genusThe interested reader isreferred to the original paper for details The following resultis interesting although its proof is easy

Theorem 109 (Samodivkin [57]) Let119866 be a graph with order119899 and girth 119892 If 119866 is embeddable on a surface whose Eulercharacteristic 120594 is as large as possible then

119887 (119866) ⩽ 3 +8

119892 minus 2minus

4120594119892

119899 (119892 minus 2) (39)

If 119866 is a planar graph with girth 119892 ⩾ 4 + 119894 for any 119894 isin0 1 2 then Theorem 109 leads to 119887(119866) ⩽ 6 minus 119894 which isthe result inTheorem 79 obtained by Fischermann et al [46]Also inmany cases the bound stated inTheorem 109 is betterthan those given byTheorems 103 and 105

8 Conditional Bondage Numbers

Since the concept of the bondage number is based upon thedomination all sorts of dominations which are variationsof the normal domination by adding restricted conditionsto the normal dominating set can develop a new ldquobondagenumberrdquo as long as a variation of domination number isgiven In this section we survey results on the bondagenumber under some restricted conditions

81 Total Bondage Numbers A dominating set 119878 of a graph119866 without isolated vertices is called total if the subgraphinduced by 119878 contains no isolated vertices The minimumcardinality of a total dominating set is called the totaldomination number of 119866 and denoted by 120574

119879(119866) It is clear

that 120574(119866) ⩽ 120574119879(119866) ⩽ 2120574(119866) for any graph 119866 without isolated

verticesThe total domination in graphs was introduced by Cock-

ayne et al [58] in 1980 Pfaff et al [59 60] in 1983 showedthat the problem determining total domination number forgeneral graphs is NP-complete even for bipartite graphs andchordal graphs Even now total domination in graphs hasbeen extensively studied in the literature In 2009 Henning[61] gave a survey of selected recent results on total domina-tion in graphs

The total bondage number of 119866 denoted by 119887119879(119866) is the

smallest cardinality of a subset 119861 sube 119864(119866) with the propertythat119866minus119861 contains no isolated vertices and 120574

119879(119866minus119861) gt 120574

119879(119866)

From definition 119887119879(119866)may not exist for some graphs for

example119866 = 1198701119899 We put 119887

119879(119866) = infin if 119887

119879(119866) does not exist

It is easy to see that 119887119879(119866) is finite for any connected graph 119866

other than 1198701 119870

2 119870

3 and 119870

1119899 In fact for such a graph 119866

since there is a path of length 3 in 119866 we can find 1198611sube 119864(119866)

such that 119866 minus 1198611is a spanning tree 119879 of 119866 containing a path

of length 3 So 120574119879(119879) = 120574

119879(119866minus119861

1) ⩾ 120574

119879(119866) For the tree119879 we

find 1198612sube 119864(119879) such that 120574

119879(119879 minus 119861

2) gt 120574

119879(119879) ⩾ 120574

119879(119866) Thus

we have 120574119879(119866minus119861

2minus119861

1) gt 120574

119879(119866) and so 119887

119879(119866) ⩽ |119861

1|+|119861

2| In

the following discussion we always assume that 119887119879(119866) exists

when a graph 119866 is mentionedIn 1991 Kulli and Patwari [62] first studied the total

bondage number of a graph and calculated the exact valuesof 119887

119879(119866) for some standard graphs

Theorem 110 (Kulli and Patwari [62] 1991) For 119899 ⩾ 4

119887119879(119862

119899) =

3 119894119891 119899 equiv 2 (mod 4) 2 119900119905ℎ119890119903119908119894119904119890

119887119879(119875

119899) =

2 119894119891 119899 equiv 2 (mod 4) 1 119900119905ℎ119890119903119908119894119904119890

119887119879(119870

119898119899) = 119898 119908119894119905ℎ 2 ⩽ 119898 ⩽ 119899

119887119879(119870

119899) =

4 119891119900119903 119899 = 4

2119899 minus 5 119891119900119903 119899 ⩾ 5

(40)

Recently Hu et al [63] have obtained some results on thetotal bondage number of the Cartesian product119875

119898times119875

119899of two

paths 119875119898and 119875

119899

Theorem 111 (Hu et al [63] 2012) For the Cartesian product119875119898times 119875

119899of two paths 119875

119898and 119875

119899

119887119879(119875

2times 119875

119899) =

1 119894119891 119899 equiv 0 (mod 3) 2 119894119891 119899 equiv 2 (mod 3) 3 119894119891 119899 equiv 1 (mod 3)

119887119879(119875

3times 119875

119899) = 1

119887119879(119875

4times 119875

119899)

= 1 119894119891 119899 equiv 1 (mod 5) = 2 119894119891 119899 equiv 4 (mod 5) ⩽ 3 119894119891 119899 equiv 2 (mod 5) ⩽ 4 119894119891 119899 equiv 0 3 (mod 5)

(41)

Generalized Petersen graphs are an important class ofcommonly used interconnection networks and have beenstudied recently By constructing a family of minimum totaldominating sets Cao et al [64] determined the total bondagenumber of the generalized Petersen graphs

FromTheorem 6 we know that 119887(119879) ⩽ 2 for a nontrivialtree 119879 But given any positive integer 119896 Sridharan et al [65]constructed a tree 119879 for which 119887

119879(119879) = 119896 Let119867

119896be the tree

obtained from a star1198701119896+1

by subdividing 119896 edges twice Thetree 119867

7is shown in Figure 6 It can be easily verified that

119887119879(119867

119896) = 119896 This fact can be stated as the following theorem

Theorem 112 (Sridharan et al [65] 2007) For any positiveinteger 119896 there exists a tree 119879 with 119887

119879(119879) = 119896

18 International Journal of Combinatorics

Figure 61198677

Combining Theorem 1 with Theorem 112 we have whatfollows

Corollary 113 (Sridharan et al [65] 2007) The bondagenumber and the total bondage number are unrelated even fortrees

However Sridharan et al [65] gave an upper bound onthe total bondage number of a tree in terms of its maximumdegree For any tree 119879 of order 119899 if 119879 =119870

1119899minus1 then 119887

119879(119879) =

minΔ(119879) (13)(119899 minus 1) Rad and Raczek [66] improved thisupper bound and gave a constructive characterization of acertain class of trees attaching the upper bound

Theorem 114 (Rad and Raczek [66] 2011) 119887119879(119879) ⩽ Δ(119879) minus 1

for any tree 119879 with maximum degree at least three

In general the decision problem for 119887119879(119866) is NP-complete

for any graph 119866 We state the decision problem for the totalbondage as follows

Problem 3 Consider the decision problemTotal Bondage ProblemInstance a graph 119866 and a positive integer 119887Question is 119887

119879(119866) ⩽ 119887

Theorem 115 (Hu and Xu [16] 2012) The total bondageproblem is NP-complete

Consequently in view of its computational hardnessit is significative to establish some sharp bounds on thetotal bondage number of a graph in terms of other graphicparameters In 1991 Kulli and Patwari [62] showed that119887119879(119866) ⩽ 2119899 minus 5 for any graph 119866 with order 119899 ⩾ 5 Sridharan

et al [65] improved this result as follows

Theorem 116 (Sridharan et al [65] 2007) For any graph 119866with order 119899 if 119899 ⩾ 5 then

119887119879(119866) ⩽

1

3(119899 minus 1) 119894119891 119866 119888119900119899119905119886119894119899119904 119899119900 119888119910119888119897119890119904

119899 minus 1 119894119891 119892 (119866) ⩾ 5

119899 minus 2 119894119891 119892 (119866) = 4

(42)

Rad and Raczek [66] also established some upperbounds on 119887

119879(119866) for a general graph 119866 In particular they

gave an upper bound on 119887119879(119866) in terms of the girth of

119866

Theorem 117 (Rad and Raczek [66] 2011) 119887119879(119866)⩽4Δ(119866) minus 5

for any graph 119866 with 119892(119866) ⩾ 4

82 Paired Bondage Numbers A dominating set 119878 of 119866 iscalled to be paired if the subgraph induced by 119878 containsa perfect matching The paired domination number of 119866denoted by 120574

119875(119866) is the minimum cardinality of a paired

dominating set of 119866 Clearly 120574119879(119866) ⩽ 120574

119875(119866) for every

connected graph 119866 with order at least two where 120574119879(119866) is

the total domination number of 119866 and 120574119875(119866) ⩽ 2120574(119866) for

any graph119866without isolated vertices Paired domination wasintroduced by Haynes and Slater [67 68] and further studiedin [69ndash72]

The paired bondage number of 119866 with 120575(119866) ⩾ 1 denotedby 119887P(119866) is the minimum cardinality among all sets of edges119861 sube 119864 such that 120575(119866 minus 119861) ⩾ 1 and 120574

119875(119866 minus 119861) gt 120574

119875(119866)

The concept of the paired bondage number was firstproposed by Raczek [73] in 2008The following observationsfollow immediately from the definition of the paired bondagenumber

Observation 2 Let 119866 be a graph with 120575(119866) ⩾ 1

(a) If 119867 is a subgraph of 119866 such that 120575(119867) ⩾ 1 then119887119875(119867) ⩽ 119887

119875(119866)

(b) If 119867 is a subgraph of 119866 such that 119887119875(119867) = 1 and 119896

is the number of edges removed to form119867 then 1 ⩽119887119875(119866) ⩽ 119896 + 1

(c) If 119909119910 isin 119864(119866) such that 119889119866(119909) 119889

119866(119910) gt 1 and 119909119910

belongs to each perfect matching of each minimumpaired dominating set of 119866 then 119887

119875(119866) = 1

Based on these observations 119887119875(119875

119899) ⩽ 2 In fact the

paired bondage number of a path has been determined

Theorem 118 (Raczek [73] 2008) For any positive integer 119896if 119899 ⩾ 2 then

119887119875(119875

119899) =

0 119894119891 119899 = 2 3 119900119903 5

1 119894119891 119899 = 4119896 4119896 + 3 119900119903 4119896 + 6

2 119900119905ℎ119890119903119908119894119904119890

(43)

For a cycle 119862119899of order 119899 ⩾ 3 since 120574

119875(119862

119899) = 120574

119875(119875

119899) from

Theorem 118 the following result holds immediately

Corollary 119 (Raczek [73] 2008) For any positive integer 119896if 119899 ⩾ 3 then

119887119875(119862

119899) =

0 119894119891 119899 = 3 119900119903 5

2 119894119891 119899 = 4119896 4119896 + 3 119900119903 4119896 + 6

3 119900119905ℎ119890119903119908119894119904119890

(44)

A wheel 119882119899 where 119899 ⩾ 4 is a graph with 119899 vertices

formed by connecting a single vertex to all vertices of a cycle119862119899minus1

Of course 120574119875(119882

119899) = 2

International Journal of Combinatorics 19

119896

Figure 7 A tree 119879 with 119887119875(119879) = 119896

Theorem 120 (Raczek [73] 2008) For positive integers 119899 and119898

119887119875(119870

119898119899) = 119898 119891119900119903 1 lt 119898 ⩽ 119899

119887119875(119882

119899) =

4 119894119891 119899 = 4

3 119894119891 119899 = 5

2 119900119905ℎ119890119903119908119894119904119890

(45)

Theorem 16 shows that if 119879 is a nontrivial tree then119887(119879) ⩽ 2 However no similar result exists for pairedbondage For any nonnegative integer 119896 let 119879

119896be a tree

obtained by subdividing all but one edge of a star 1198701119896+1

(asshown in Figure 7) It is easy to see that 119887

119875(119879

119896) = 119896 We

state this fact as the following theorem which is similar toTheorem 112

Theorem121 (Raczek [73] 2008) For any nonnegative integer119896 there exists a tree 119879 with 119887

119875(119879) = 119896

Consider the tree defined in Figure 5 for 119896 = 3 Then119887(119879

3) ⩽ 2 and 119887

119875(119879

3) = 3 On the other hand 119887(119875

4) = 2

and 119887119875(119875

4) = 1 Thus we have what follows which is similar

to Corollary 113

Corollary 122 The bondage number and the paired bondagenumber are unrelated even for trees

A constructive characterization of trees with 119887(119879) = 2is given by Hartnell and Rall in [12] Raczek [73] provideda constructive characterization of trees with 119887

119875(119879) = 0 In

order to state the characterization we define a labeling andthree simple operations on a tree119879 Let 119910 isin 119881(119879) and let ℓ(119910)be the label assigned to 119910

Operation T1 If ℓ(119910) = 119861 add a vertex 119909 and the

edge 119910119909 and let ℓ(119909) = 119860OperationT

2 If ℓ(119910) = 119862 add a path (119909

1 119909

2) and the

edge 1199101199091 and let ℓ(119909

1) = 119861 ℓ(119909

2) = 119860

Operation T3 If ℓ(119910) = 119861 add a path (119909

1 119909

2 119909

3)

and the edge 1199101199091 and let ℓ(119909

1) = 119862 ℓ(119909

2) = 119861 and

ℓ(1199093) = 119860

Let 1198752= (119906 V) with ℓ(119906) = 119860 and ℓ(V) = 119861 Let T be

the class of all trees obtained from the labeled 1198752by a finite

sequence of operationsT1 T

2 T

3

A tree 119879 in Figure 8 belongs to the familyTRaczek [73] obtained the following characterization of all

trees 119879 with 119887119875(119879) = 0

119860

119860

119860

119860

119860

119861 119862

119861 119862 119861

119861119860

Figure 8 A tree 119879 belong to the familyT

Theorem 123 (Raczek [73] 2008) For a tree 119879 119887119875(119879) = 0 if

and only if 119879 is inT

We state the decision problem for the paired bondage asfollows

Problem 4 Consider the decision problem

Paired Bondage ProblemInstance a graph 119866 and a positive integer 119887Question is 119887

119875(119866) ⩽ 119887

Conjecture 124 The paired bondage problem is NP-complete

83 Restrained Bondage Numbers A dominating set 119878 of 119866 iscalled to be restrained if the subgraph induced by 119878 containsno isolated vertices where 119878 = 119881(119866) 119878 The restraineddomination number of 119866 denoted by 120574

119877(119866) is the smallest

cardinality of a restrained dominating set of 119866 It is clear that120574119877(119866) exists and 120574(119866) ⩽ 120574

119877(119866) for any nonempty graph 119866

The concept of restrained domination was introducedby Telle and Proskurowski [74] in 1997 albeit indirectly asa vertex partitioning problem Concerning the study of therestrained domination numbers the reader is referred to [74ndash79]

In 2008 Hattingh and Plummer [80] defined therestrained bondage number 119887R(119866) of a nonempty graph 119866 tobe the minimum cardinality among all subsets 119861 sube 119864(119866) forwhich 120574

119877(119866 minus 119861) gt 120574

119877(119866)

For some simple graphs their restrained dominationnumbers can be easily determined and so restrained bondagenumbers have been also determined For example it is clearthat 120574

119877(119870

119899) = 1 Domke et al [77] showed that 120574

119877(119862

119899) =

119899 minus 2lfloor1198993rfloor for 119899 ⩾ 3 and 120574119877(119875

119899) = 119899 minus 2lfloor(119899 minus 1)3rfloor for 119899 ⩾ 1

Using these results Hattingh and Plummer [80] obtained therestricted bondage numbers for119870

119899 119862

119899 119875

119899 and119866 = 119870

11989911198992119899119905

as follows

Theorem 125 (Hattingh and Plummer [80] 2008) For 119899 ⩾ 3

119887R (119870119899) =

1 119894119891 119899 = 3

lceil119899

2rceil 119900119905ℎ119890119903119908119894119904119890

119887R (119862119899) =

1 119894119891 119899 equiv 0 (mod 3) 2 119900119905ℎ119890119903119908119894119904119890

119887R (119875119899) = 1 119899 ⩾ 4

20 International Journal of Combinatorics

119887R (119866)

=

lceil119898

2rceil 119894119891 119899

119898= 1 119886119899119889 119899

119898+1⩾ 2 (1 ⩽ 119898 lt 119905)

2119905 minus 2 119894119891 1198991= 119899

2= sdot sdot sdot = 119899

119905= 2 (119905 ⩾ 2)

2 119894119891 1198991= 2 119886119899119889 119899

2⩾ 3 (119905 = 2)

119905minus1

sum

119894=1

119899119894minus 1 119900119905ℎ119890119903119908119894119904119890

(46)

Theorem 126 (Hattingh and Plummer [80] 2008) For a tree119879 of order 119899 (⩾ 4) 119879 ≇ 119870

1119899minus1if and only if 119887R(119879) = 1

Theorem 126 shows that the restrained bondage numberof a tree can be computed in constant time However thedecision problem for 119887R(119866) is NP-complete even for bipartitegraphs

Problem 5 Consider the decision problem

Restrained Bondage Problem

Instance a graph 119866 and a positive integer 119887

Question is 119887R(119866) ⩽ 119887

Theorem 127 (Hattingh and Plummer [80] 2008) Therestrained bondage problem is NP-complete even for bipartitegraphs

Consequently in view of its computational hardnessit is significative to establish some sharp bounds on therestrained bondage number of a graph in terms of othergraphic parameters

Theorem 128 (Hattingh and Plummer [80] 2008) For anygraph 119866 with 120575(119866) ⩾ 2

119887R (119866) ⩽ min119909119910isin119864(119866)

119889119866(119909) + 119889

119866(119910) minus 2 (47)

Corollary 129 119887R(119866) ⩽ Δ(119866)+120575(119866)minus2 for any graph119866 with120575(119866) ⩾ 2

Notice that the bounds stated in Theorem 128 andCorollary 129 are sharp Indeed the class of cycles whoseorders are congruent to 1 2 (mod 3) have a restrained bondagenumber achieving these bounds (see Theorem 125)

Theorem 128 is an analogue of Theorem 14 A quitenatural problem is whether or not for restricted bondagenumber there are analogues of Theorem 16 Theorem 18Theorem 29 and so on Indeed we should consider all resultsfor the bondage number whether or not there are analoguesfor restricted bondage number

Theorem 130 (Hattingh and Plummer [80] 2008) For anygraph 119866 with 120574

119877(119866) = 2

119887R (119866) ⩽ Δ (119866) + 1 (48)

We now consider the relation between 119887119877(119866) and 119887(119866)

Theorem 131 (Hattingh and Plummer [80] 2008) If 120574119877(119866) =

120574(119866) for some graph 119866 then 119887R(119866) ⩽ 119887(119866)

Corollary 129 and Theorem 131 provide a possibility toattack Conjecture 73 by characterizing planar graphs with120574119877= 120574 and 119887R = 119887 when 3 ⩽ Δ ⩽ 6However we do not have 119887R(119866) = 119887(119866) for any graph

119866 even if 120574119877(119866) = 120574(119866) Observe that 120574

119877(119870

3) = 120574(119870

3) yet

119887119877(119870

3) = 1 and 119887(119870

3) = 2 We still may not claim that

119887119877(119866) = 119887(119866) even in the case that every 120574-set is a 120574

119877-set

The example 1198703again demonstrates this

Theorem 132 (Hattingh and Plummer [80] 2008) There isan infinite class of graphs inwhich each graph119866 satisfies 119887(119866) lt119887R(119866)

In fact such an infinite class of graphs can be obtained asfollows Let119867 be a connected graph BC(119867) a graph obtainedfrom 119867 by attaching ℓ (⩾ 2) vertices of degree 1 to eachvertex in 119867 and B = 119866 119866 = BC(119867) for some graph 119867such that 120575(119867) ⩾ 2 By Theorem 3 we can easily check that119887(119866) = 1 lt 2 ⩽ min120575(119867) ℓ = 119887R(119866) for any 119866 isin BMoreover there exists a graph 119866 isin B such that 119887R(119866) canbe much larger than 119887(119866) which is stated as the followingtheorem

Theorem 133 (Hattingh and Plummer [80] 2008) For eachpositive integer 119896 there is a graph119866 such that 119896 = 119887R(119866)minus119887(119866)

Combining Theorem 133 with Theorem 132 we have thefollowing corollary

Corollary 134 The bondage number and the restrainedbondage number are unrelated

Very recently Jafari Rad et al [81] have consideredthe total restrained bondage in graphs based on both totaldominating sets and restrained dominating sets and obtainedseveral properties exact values and bounds for the totalrestrained bondage number of a graph

A restrained dominating set 119878 of a graph 119866 without iso-lated vertices is called to be a total restrained dominating setif 119878 is a total dominating set The total restrained dominationnumber of119866 denoted by 120574TR (119866) is theminimum cardinalityof a total restrained dominating set of 119866 For referenceson this domination in graphs see [3 61 82ndash85] The totalrestrained bondage number 119887TR (119866) of a graph 119866 with noisolated vertex is the cardinality of a smallest set of edges 119861 sube119864(119866) for which119866minus119861 has no isolated vertex and 120574TR (119866minus119861) gt120574TR (119866) In the case that there is no such subset 119861 we define119887TR (119866) = infin

Ma et al [84] determined that 120574TR (119875119899) = 119899minus2lfloor(119899minus2)4rfloorfor 119899 ⩾ 3 120574TR (119862119899

) = 119899 minus 2lfloor1198994rfloor for 119899 ⩾ 3 120574TR (1198703) =

3 and 120574TR (119870119899) = 2 for 119899 = 3 and 120574TR (1198701119899

) = 1 + 119899

and 120574TR (119870119898119899) = 2 for 2 ⩽ 119898 ⩽ 119899 According to these

results Jafari Rad et al [81] determined the exact valuesof total restrained bondage numbers for the correspondinggraphs

International Journal of Combinatorics 21

Theorem 135 (Jafari Rad et al [81]) For an integer 119899

119887TR (119875119899) = infin 119894119891 2 ⩽ 119899 ⩽ 5

1 119894119891 119899 ⩾ 5

119887TR (119862119899) =

infin 119894119891 119899 = 3

1 119894119891 3 lt 119899 119899 equiv 0 1 (mod 4)2 119894119891 3 lt 119899 119899 equiv 2 3 (mod 4)

119887TR (119882119899) = 2 119891119900119903 119899 ⩾ 6

119887TR (119870119899) = 119899 minus 1 119891119900119903 119899 ⩾ 4

119887TR (119870119898119899) = 119898 minus 1 119891119900119903 2 ⩽ 119898 ⩽ 119899

(49)

119887TR (119870119899) = 119899minus1 shows the fact that for any integer 119899 ⩾ 3 there

exists a connected graph namely119870119899+1

such that 119887TR (119870119899+1) =

119899 that is the total restrained bondage number is unboundedfrom the above

A vertex is said to be a support vertex if it is adjacent toa vertex of degree one Let A be the family of all connectedgraphs such that119866 belongs toA if and only if every edge of119866is incident to a support vertex or 119866 is a cycle of length three

Proposition 136 (Cyman and Raczek [82] 2006) For aconnected graph119866 of order 119899 120574TR (119866) = 119899 if and only if119866 isin A

Hence if 119866 isin A then the removal of any subset of edgesof119866 cannot increase the total restrained domination numberof 119866 and thus 119887TR (119866) = infin When 119866 notin A a result similar toTheorem 6 is obtained

Theorem 137 (Jafari Rad et al [81]) For any tree 119879 notin A119887TR (119879) ⩽ 2 This bound is sharp

In the samepaper the authors provided a characterizationfor trees 119879 notin A with 119887TR (119879) = 1 or 2 The following result issimilar to Observation 1

Observation 3 (Jafari Rad et al [81]) If 119896 edges can beremoved from a graph 119866 to obtain a graph 119867 without anisolated vertex and with 119887TR (119867) = 119905 then 119887TR (119866) ⩽ 119896 + 119905

ApplyingTheorem 137 andObservation 3 Jafari Rad et alcharacterized all connected graphs 119866 with 119887TR (119866) = infin

Theorem 138 (Jafari Rad et al [81]) For a connected graph119866119887TR (119866) = infin if and only if 119866 isin A

84 Other Conditional Bondage Numbers A subset 119868 sube 119881(119866)is called an independent set if no two vertices in 119868 are adjacentin 119866 The maximum cardinality among all independent setsis called the independence number of 119866 denoted by 120572(119866)

A dominating set 119878 of a graph 119866 is called to be inde-pendent if 119878 is an independent set of 119866 The minimumcardinality among all independent dominating set is calledthe independence domination number of 119866 and denoted by120574119868(119866)

Since an independent dominating set is not only adominating set but also an independent set 120574(119866) ⩽ 120574

119868(119866) ⩽

120572(119866) for any graph 119866It is clear that a maximal independent set is certainly a

dominating set Thus an independent set is maximal if andonly if it is an independent dominating set and so 120574

119868(119866) is the

minimum cardinality among all maximal independent setsof 119866 This graph-theoretical invariant has been well studiedin the literature see for example Haynes et al [3] for earlyresults Goddard and Henning [86] for recent results onindependent domination in graphs

In 2003 Zhang et al [87] defined the independencebondage number 119887

119868(119866) of a nonempty graph 119866 to be the

minimum cardinality among all subsets 119861 sube 119864(119866) for which120574119868(119866 minus 119861) gt 120574

119868(119866) For some ordinary graphs their indepen-

dence domination numbers can be easily determined and soindependence bondage numbers have been also determinedClearly 119887

119868(119870

119899) = 1 if 119899 ⩾ 2

Theorem 139 (Zhang et al [87] 2003) For any integer 119899 ⩾ 4

119887119868(119862

119899) =

1 119894119891 119899 equiv 1 (mod 2) 2 119894119891 119899 equiv 0 (mod 2)

119887119868(119875

119899) =

1 119894119891 119899 equiv 0 (mod 2) 2 119894119891 119899 equiv 1 (mod 2)

119887119868(119870

119898119899) = 119899 119891119900119903 119898 ⩽ 119899

(50)

Apart from the above-mentioned results as far as we findno other results on the independence bondage number in thepresent literature we never knew much of any result on thisparameter for a tree

A subset 119878 of 119881(119866) is called an equitable dominatingset if for every 119910 isin 119881(119866 minus 119878) there exists a vertex 119909 isin 119878such that 119909119910 isin 119864(119866) and |119889

119866(119909) minus 119889

119866(119910)| ⩽ 1 proposed

by Dharmalingam [88] The minimum cardinality of such adominating set is called the equitable domination number andis denoted by 120574

119864(119866) Deepak et al [89] defined the equitable

bondage number 119887119864(119866) of a graph 119866 to be the cardinality of a

smallest set 119861 sube 119864(119866) of edges for which 120574119864(119866 minus 119861) gt 120574

119864(119866)

and determined 119887119864(119866) for some special graphs

Theorem 140 (Deepak et al [89] 2011) For any integer 119899 ⩾ 2

119887119864(119870

119899) = lceil

119899

2rceil

119887119864(119870

119898119899) = 119898 119891119900119903 0 ⩽ 119899 minus 119898 ⩽ 1

119887119864(119862

119899) =

3 119894119891 119899 equiv 1 (mod 3)2 119900119905ℎ119890119903119908119894119904119890

119887119864(119875

119899) =

2 119894119891 119899 equiv 1 (mod 3)1 119900119905ℎ119890119903119908119894119904119890

119887119864(119879) ⩽ 2 119891119900119903 119886119899119910 119899119900119899119905119903119894119907119894119886119897 119905119903119890119890 119879

119887119864(119866) ⩽ 119899 minus 1 119891119900119903 119886119899119910 119888119900119899119899119890119888119905119890119889 119892119903119886119901ℎ 119866 119900119891 119900119903119889119890119903 119899

(51)

22 International Journal of Combinatorics

As far as we know there are no more results on theequitable bondage number of graphs in the present literature

There are other variations of the normal domination byadding restricted conditions to the normal dominating setFor example the connected domination set of a graph119866 pro-posed by Sampathkumar and Walikar [90] is a dominatingset 119878 such that the subgraph induced by 119878 is connected Theproblem of finding a minimum connected domination set ina graph is equivalent to the problemof finding a spanning treewithmaximumnumber of vertices of degree one [91] and hassome important applications in wireless networks [92] thusit has received much research attention However for suchan important domination there is no result on its bondagenumber We believe that the topic is of significance for futureresearch

9 Generalized Bondage Numbers

There are various generalizations of the classical dominationsuch as 119901-domination distance domination fractional dom-ination Roman domination and rainbow domination Everysuch a generalization can lead to a corresponding bondageIn this section we introduce some of them

91 119901-Bondage Numbers In 1985 Fink and Jacobson [93]introduced the concept of 119901-domination Let 119901 be a positiveinteger A subset 119878 of 119881(119866) is a 119901-dominating set of 119866 if|119878 cap 119873

119866(119910)| ⩾ 119901 for every 119910 isin 119878 The 119901-domination number

120574119901(119866) is the minimum cardinality among all 119901-dominating

sets of 119866 Any 119901-dominating set of 119866 with cardinality 120574119901(119866)

is called a 120574119901-set of 119866 Note that a 120574

1-set is a classic minimum

dominating set Notice that every graph has a 119901-dominatingset since the vertex-set119881(119866) is such a setWe also note that a 1-dominating set is a dominating set and so 120574(119866) = 120574

1(119866) The

119901-domination number has received much research attentionsee a state-of-the-art survey article by Chellali et al [94]

It is clear from definition that every 119901-dominating setof a graph certainly contains all vertices of degree at most119901 minus 1 By this simple observation to avoid the trivial caseoccurrence we always assume that Δ(119866) ⩾ 119901 For 119901 ⩾ 2 Luet al [95] gave a constructive characterization of trees withunique minimum 119901-dominating sets

Recently Lu and Xu [96] have introduced the conceptto the 119901-bondage number of 119866 denoted by 119887

119901(119866) as the

minimum cardinality among all sets of edges 119861 sube 119864(119866) suchthat 120574

119901(119866 minus 119861) gt 120574

119901(119866) Clearly 119887

1(119866) = 119887(119866)

Lu and Xu [96] established a lower bound and an upperbound on 119887

119901(119879) for any integer 119901 ⩾ 2 and any tree 119879 with

Δ(119879) ⩾ 119901 and characterized all trees achieving the lowerbound and the upper bound respectively

Theorem 141 (Lu and Xu [96] 2011) For any integer 119901 ⩾ 2and any tree 119879 with Δ(119879) ⩾ 119901

1 ⩽ 119887119901(119879) ⩽ Δ (119879) minus 119901 + 1 (52)

Let 119878 be a given subset of 119881(119866) and for any 119909 isin 119878 let

119873119901(119909 119878 119866) = 119910 isin 119878 cap 119873

119866(119909)

1003816100381610038161003816119873119866(119910) cap 119878

1003816100381610038161003816 = 119901

(53)

Then all trees achieving the lower bound 1 can be character-ized as follows

Theorem 142 (Lu and Xu [96] 2011) Let 119879 be a tree withΔ(119879) ⩾ 119901 ⩾ 2 Then 119887

119901(119879) = 1 if and only if for any 120574

119901-set

119878 of 119879 there exists an edge 119909119910 isin (119878 119878) such that 119910 isin 119873119901(119909 119878 119879)

The notation 119878(119886 119887) denotes a double star obtained byadding an edge between the central vertices of two stars119870

1119886minus1

and1198701119887minus1

And the vertex with degree 119886 (resp 119887) in 119878(119886 119887) iscalled the 119871-central vertex (resp 119877-central vertex) of 119878(119886 119887)

To characterize all trees attaining the upper bound givenin Theorem 141 we define three types of operations on a tree119879 with Δ(119879) = Δ ⩾ 119901 + 1

Type 1 attach a pendant edge to a vertex 119910with 119889119879(119910) ⩽ 119901minus

2 in 119879

Type 2 attach a star 1198701Δminus1

to a vertex 119910 of 119879 by joining itscentral vertex to 119910 where 119910 in a 120574

119901-set of 119879 and

119889119879(119910) ⩽ Δ minus 1

Type 3 attach a double star 119878(119901 Δ minus 1) to a pendant vertex 119910of 119879 by coinciding its 119877-central vertex with 119910 wherethe unique neighbor of 119910 is in a 120574

119901-set of 119879

Let B = 119879 a tree obtained from 1198701Δ

or 119878(119901 Δ) by afinite sequence of operations of Types 1 2 and 3

Theorem 143 (Lu and Xu [96] 2011) A tree with the maxi-mum Δ ⩾ 119901 + 1 has 119901-bondage number Δ minus 119901 + 1 if and onlyif it belongs toB

Theorem 144 (Lu and Xu [97] 2012) Let 119866119898119899= 119875

119898times 119875

119899

Then

1198872(119866

2119899) = 1 119891119900119903 119899 ⩾ 2

1198872(119866

3119899) =

2 119894119891 119899 equiv 1 (mod 3) 1 119900119905ℎ119890119903119908119894119904119890

for 119899 ⩾ 2

1198872(119866

4119899) =

1 119894119891 119899 equiv 3 (mod 4) 2 119900119905ℎ119890119903119908119894119904119890

for 119899 ⩾ 7

(54)

Recently Krzywkowski [98] has proposed the conceptof the nonisolating 2-bondage of a graph The nonisolating2-bondage number of a graph 119866 denoted by 1198871015840

2(119866) is the

minimum cardinality among all sets of edges 119861 sube 119864(119866) suchthat 119866 minus 119861 has no isolated vertices and 120574

2(119866 minus 119861) gt 120574

2(119866) If

for every 119861 sube 119864(119866) either 1205742(119866 minus 119861) = 120574

2(119866) or 119866 minus 119861 has

isolated vertices then define 11988710158402(119866) = 0 and say that 119866 is a 2-

nonisolatedly strongly stable graph Krzywkowski presentedsome basic properties of nonisolating 2-bondage in graphsshowed that for every nonnegative integer 119896 there exists atree 119879 such 1198871015840

2(119879) = 119896 characterized all 2-non-isolatedly

strongly stable trees and determined 11988710158402(119866) for several special

graphs

International Journal of Combinatorics 23

Theorem 145 (Krzywkowski [98] 2012) For any positiveintegers 119899 and119898 with119898 ⩽ 119899

1198871015840

2(119870

119899) =

0 119894119891 119899 = 1 2 3

lfloor2119899

3rfloor 119900119905ℎ119890119903119908119894119904119890

1198871015840

2(119875

119899) =

0 119894119891 119899 = 1 2 3

1 119900119905ℎ119890119903119908119894119904119890

1198871015840

2(119862

119899) =

0 119894119891 119899 = 3

1 119894119891 119899 119894119904 119890V119890119899

2 119894119891 119899 119894119904 119900119889119889

1198871015840

2(119870

119898119899) =

3 119894119891 119898 = 119899 = 3

1 119894119891 119898 = 119899 = 4

2 119900119905ℎ119890119903119908119894119904119890

(55)

92 Distance Bondage Numbers A subset 119878 of vertices of agraph119866 is said to be a distance 119896-dominating set for119866 if everyvertex in 119866 not in 119878 is at distance at most 119896 from some vertexof 119878 The minimum cardinality of all distance 119896-dominatingsets is called the distance 119896-domination number of 119866 anddenoted by 120574

119896(119866) (do not confuse with the above-mentioned

120574119901(119866)) When 119896 = 1 a distance 1-dominating set is a normal

dominating set and so 1205741(119866) = 120574(119866) for any graph 119866 Thus

the distance 119896-domination is a generalization of the classicaldomination

The relation between 120574119896and 120572

119896(the 119896-independence

number defined in Section 22) for a tree obtained by Meirand Moon [14] who proved that 120574

119896(119879) = 120572

2119896(119879) for any tree

119879 (a special case for 119896 = 1 is stated in Proposition 10) Thefurther research results can be found in Henning et al [99]Tian and Xu [100ndash103] and Liu et al [104]

In 1998 Hartnell et al [15] defined the distance 119896-bondagenumber of 119866 denoted by 119887

119896(119866) to be the cardinality of a

smallest subset 119861 of edges of 119866 with the property that 120574119896(119866 minus

119861) gt 120574119896(119866) FromTheorem 6 it is clear that if119879 is a nontrivial

tree then 1 ⩽ 1198871(119879) ⩽ 2 Hartnell et al [15] generalized this

result to any integer 119896 ⩾ 1

Theorem 146 (Hartnell et al [15] 1998) For every nontrivialtree 119879 and positive integer 119896 1 ⩽ 119887

119896(119879) ⩽ 2

Hartnell et al [15] and Topp and Vestergaard [13] alsocharacterized the trees having distance 119896-bondage number 2In particular the class of trees for which 119887

1(119879) = 2 are just

those which have a unique maximum 2-independent set (seeTheorem 11)

Since when 119896 = 1 the distance 1-bondage number 1198871(119866)

is the classical bondage number 119887(119866) Theorem 12 gives theNP-hardness of deciding the distance 119896-bondage number ofgeneral graphs

Theorem 147 Given a nonempty undirected graph 119866 andpositive integers 119896 and 119887 with 119887 ⩽ 120576(119866) determining wetheror not 119887

119896(119866) ⩽ 119887 is NP-hard

For a vertex 119909 in119866 the open 119896-neighborhood119873119896(119909) of 119909 is

defined as119873119896(119909) = 119910 isin 119881(119866) 1 ⩽ 119889

119866(119909 119910) le 119896The closed

119896-neighborhood119873119896[119909] of 119909 in 119866 is defined as119873

119896(119909) cup 119909 Let

Δ119896(119866) = max 1003816100381610038161003816119873119896

(119909)1003816100381610038161003816 for any 119909 isin 119881 (119866) (56)

Clearly Δ1(119866) = Δ(119866) The 119896th power of a graph 119866 is the

graph119866119896 with vertex-set119881(119866119896

) = 119881(119866) and edge-set119864(119866119896

) =

119909119910 1 ⩽ 119889119866(119909 119910) ⩽ 119896 The following lemma holds directly

from the definition of 119866119896

Proposition 148 (Tian and Xu [101]) Δ(119866119896

) = Δ119896(119866) and

120574(119866119896

) = 120574119896(119866) for any graph 119866 and each 119896 ⩾ 1

A graph 119866 is 119896-distance domination-critical or 120574119896-critical

for short if 120574119896(119866 minus 119909) lt 120574

119896(119866) for every vertex 119909 in 119866

proposed by Henning et al [105]

Proposition 149 (Tian and Xu [101]) For each 119896 ⩾ 1 a graph119866 is 120574

119896-critical if and only if 119866119896 is 120574

119896-critical

By the above facts we suggest the following worthwhileproblem

Problem 6 Can we generalize the results on the bondage for119866 to 119866119896 In particular do the following propositions hold

(a) 119887(119866119896

) = 119887119896(119866) for any graph 119866 and each 119896 ⩾ 1

(b) 119887(119866119896

) ⩽ Δ119896(119866) if 119866 is not 120574

119896-critical

Let 119896 and 119901 be positive integers A subset 119878 of 119881(119866) isdefined to be a (119896 119901)-dominating set of 119866 if for any vertex119909 isin 119878 |119873

119896(119909) cap 119878| ⩾ 119901 The (119896 119901)-domination number of

119866 denoted by 120574119896119901(119866) is the minimum cardinality among

all (119896 119901)-dominating sets of 119866 Clearly for a graph 119866 a(1 1)-dominating set is a classical dominating set a (119896 1)-dominating set is a distance 119896-dominating set and a (1 119901)-dominating set is the above-mentioned 119901-dominating setThat is 120574

11(119866) = 120574(119866) 120574

1198961(119866) = 120574

119896(119866) and 120574

1119901(119866) = 120574

119901(119866)

The concept of (119896 119901)-domination in a graph 119866 is a gen-eralized domination which combines distance 119896-dominationand 119901-domination in 119866 So the investigation of (119896 119901)-domination of 119866 is more interesting and has received theattention of many researchers see for example [106ndash109]

More general Li and Xu [110] proposed the concept of the(ℓ 119908)-domination number of a119908-connected graph A subset119878 of 119881(119866) is defined to be an (ℓ 119908)-dominating set of 119866 iffor any vertex 119909 isin 119878 there are 119908 internally disjoint pathsof length at most ℓ from 119909 to some vertex in 119878 The (ℓ 119908)-domination number of119866 denoted by 120574

ℓ119908(119866) is theminimum

cardinality among all (ℓ 119908)-dominating sets of 119866 Clearly1205741198961(119866) = 120574

119896(119866) and 120574

1119901(119866) = 120574

119901(119866)

It is quite natural to propose the concept of bondage num-ber for (119896 119901)-domination or (ℓ 119908)-domination However asfar as we know none has proposed this concept until todayThis is a worthwhile topic for further research

24 International Journal of Combinatorics

93 Fractional Bondage Numbers If 120590 is a function mappingthe vertex-set 119881 into some set of real numbers then for anysubset 119878 sube 119881 let 120590(119878) = sum

119909isin119878120590(119909) Also let |120590| = 120590(119881)

A real-value function 120590 119881 rarr [0 1] is a dominatingfunction of a graph 119866 if for every 119909 isin 119881(119866) 120590(119873

119866[119909]) ⩾ 1

Thus if 119878 is a dominating set of 119866 120590 is a function where

120590 (119909) = 1 if 119909 isin 1198780 otherwise

(57)

then 120590 is a dominating function of119866 The fractional domina-tion number of 119866 denoted by 120574

119865(119866) is defined as follows

120574119865(119866) = min |120590| 120590 is a domination function of 119866

(58)

The 120574119865-bondage number of 119866 denoted by 119887

119865(119866) is

defined as the minimum cardinality of a subset 119861 sube 119864 whoseremoval results in 120574

119865(119866 minus 119861) gt 120574

119865(119866)

Hedetniemi et al [111] are the first to study fractionaldomination although Farber [112] introduced the idea indi-rectly The concept of the fractional domination numberwas proposed by Domke and Laskar [113] in 1997 Thefractional domination numbers for some ordinary graphs aredetermined

Proposition 150 (Domke et al [114] 1998 Domke and Laskar[113] 1997) (a) If 119866 is a 119896-regular graph with order 119899 then120574119865(119866) = 119899119896 + 1(b) 120574

119865(119862

119899) = 1198993 120574

119865(119875

119899) = lceil1198993rceil 120574

119865(119870

119899) = 1

(c) 120574119865(119866) = 1 if and only if Δ(119866) = 119899 minus 1

(d) 120574119865(119879) = 120574(119879) for any tree 119879

(e) 120574119865(119870

119899119898) = (119899(119898 + 1) + 119898(119899 + 1))(119899119898 minus 1) where

max119899119898 gt 1

The assertions (a)ndash(d) are due to Domke et al [114] andthe assertion (e) is due to Domke and Laskar [113] Accordingto these results Domke and Laskar [113] determined the 120574

119865-

bondage numbers for these graphs

Theorem 151 (Domke and Laskar [113] 1997) 119887119865(119870

119899) =

lceil1198992rceil 119887119865(119870

119899119898) = 1 wheremax119899119898 gt 1

119887119865(119862

119899) =

2 119894119891 119899 equiv 0 (mod 3) 1 119900119905ℎ119890119903119908119894119904119890

119887119865(119875

119899) =

2 119894119891 119899 equiv 1 (mod 3) 1 119900119905ℎ119890119903119908119894119904119890

(59)

It is easy to see that for any tree119879 119887119865(119879) ⩽ 2 In fact since

120574119865(119879) = 120574(119879) for any tree 119879 120574

119865(119879

1015840

) = 120574(1198791015840

) for any subgraph1198791015840 of 119879 It follows that

119887119865(119879) = min 119861 sube 119864 (119879) 120574

119865(119879 minus 119861) gt 120574

119865(119879)

= min 119861 sube 119864 (119879) 120574 (119879 minus 119861) gt 120574 (119879)

= 119887 (119879) ⩽ 2

(60)

Except for these no results on 119887119865(119866) have been known as

yet

94 Roman Bondage Numbers A Roman dominating func-tion on a graph 119866 is a labeling 119891 119881 rarr 0 1 2 such thatevery vertex with label 0 has at least one neighbor with label2 The weight of a Roman dominating function 119891 on 119866 is thevalue

119891 (119866) = sum

119906isin119881(119866)

119891 (119906) (61)

The minimum weight of a Roman dominating function ona graph 119866 is called the Roman domination number of 119866denoted by 120574RM (119866)

A Roman dominating function 119891 119881 rarr 0 1 2

can be represented by the ordered partition (1198810 119881

1 119881

2) (or

(119881119891

0 119881

119891

1 119881

119891

2) to refer to119891) of119881 where119881

119894= V isin 119881 | 119891(V) = 119894

In this representation its weight 119891(119866) = |1198811| + 2|119881

2| It is

clear that119881119891

1cup119881

119891

2is a dominating set of 119866 called the Roman

dominating set denoted by 119863119891

R = (1198811 1198812) Since 119881119891

1cup 119881

119891

2is

a dominating set when 119891 is a Roman dominating functionand since placing weight 2 at the vertices of a dominating setyields a Roman dominating function in [115] it is observedthat

120574 (119866) ⩽ 120574RM (119866) ⩽ 2120574 (119866) (62)

A graph 119866 is called to be Roman if 120574RM (119866) = 2120574(119866)The definition of the Roman domination function is

proposed implicitly by Stewart [116] and ReVelle and Rosing[117] Roman dominating numbers have been deeply studiedIn particular Bahremandpour et al [118] showed that theproblem determining the Roman domination number is NP-complete even for bipartite graphs

Let 119866 be a graph with maximum degree at least twoThe Roman bondage number 119887RM (119866) of 119866 is the minimumcardinality of all sets 1198641015840 sube 119864 for which 120574RM (119866 minus 119864

1015840

) gt

120574RM (119866) Since in study of Roman bondage number theassumption Δ(119866) ⩾ 2 is necessary we always assume thatwhen we discuss 119887RM (119866) all graphs involved satisfy Δ(119866) ⩾2 The Roman bondage number 119887RM (119866) is introduced byJafari Rad and Volkmann in [119]

Recently Bahremandpour et al [118] have shown that theproblem determining the Roman bondage number is NP-hard even for bipartite graphs

Problem 7 Consider the decision problem

Roman Bondage Problem

Instance a nonempty bipartite graph119866 and a positiveinteger 119896

Question is 119887RM (119866) ⩽ 119896

Theorem 152 (Bahremandpour et al [118] 2013) TheRomanbondage problem is NP-hard even for bipartite graphs

The exact value of 119887RM(119866) is known only for a few familyof graphs including the complete graphs cycles and paths

International Journal of Combinatorics 25

Theorem 153 (Jafari Rad and Volkmann [119] 2011) For anypositive integers 119899 and119898 with119898 ⩽ 119899

119887RM (119875119899) = 2 119894119891 119899 equiv 2 (mod 3) 1 119900119905ℎ119890119903119908119894119904119890

119887RM (119862119899) =

3 119894119891 119899 = 2 (mod 3) 2 119900119905ℎ119890119903119908119894119904119890

119887RM (119870119899) = lceil

119899

2rceil 119891119900119903 119899 ⩾ 3

119887RM (119870119898119899) =

1 119894119891 119898 = 1 119886119899119889 119899 = 1

4 119894119891 119898 = 119899 = 3

119898 119900119905ℎ119890119903119908119894119904119890

(63)

Lemma 154 (Cockayne et al [115] 2004) If 119866 is a graph oforder 119899 and contains vertices of degree 119899 minus 1 then 120574RM(119866) = 2

Using Lemma 154 the third conclusion in Theorem 153can be generalized to more general case which is similar toLemma 49

Theorem 155 (Jafari Rad and Volkmann [119] 2011) Let119866 bea graph with order 119899 ge 3 and 119905 the number of vertices of degree119899 minus 1 in 119866 If 119905 ⩾ 1 then 119887RM(119866) = lceil1199052rceil

Ebadi and PushpaLatha [120] conjectured that 119887RM(119866) ⩽119899 minus 1 for any graph 119866 of order 119899 ⩾ 3 Dehgardi et al[121] Akbari andQajar [122] independently showed that thisconjecture is true

Theorem 156 119887RM(119866) ⩽ 119899 minus 1 for any connected graph 119866 oforder 119899 ⩾ 3

For a complete 119905-partite graph we have the followingresult

Theorem 157 (Hu and Xu [123] 2011) Let 119866 = 11987011989811198982119898119905

bea complete 119905-partite graph with 119898

1= sdot sdot sdot = 119898

119894lt 119898

119894+1le sdot sdot sdot ⩽

119898119905 119905 ⩾ 2 and 119899 = sum119905

119895=1119898

119895 Then

119887RM (119866) =

lceil119894

2rceil 119894119891 119898

119894= 1 119886119899119889 119899 ⩾ 3

2 119894119891 119898119894= 2 119886119899119889 119894 = 1

119894 119894119891 119898119894= 2 119886119899119889 119894 ⩾ 2

4 119894119891 119898119894= 3 119886119899119889 119894 = 119905 = 2

119899 minus 1 119894119891 119898119894= 3 119886119899119889 119894 = 119905 ⩾ 3

119899 minus 119898119905

119894119891 119898119894⩾ 3 119886119899119889 119898

119905⩾ 4

(64)

Consider a complete 119905-partite graph 119870119905(3) for 119905 ⩾ 3

119887RM(119870119905(3)) = 119899 minus 1 by Theorem 157 where 119899 = 3119905

which shows that this upper bound of 119899 minus 1 on 119887RM(119866) inTheorem 156 is sharp This fact and 120574

119877(119870

119905(3)) = 4 give

a negative answer to the following two problems posed byDehgardi et al [121]Prove or Disprove for any connected graph 119866 of order 119899 ge 3(i) 119887RM(119866) = 119899 minus 1 if and only if 119866 cong 119870

3 (ii) 119887RM(119866) ⩽ 119899 minus

120574119877(119866) + 1Note that a complete 119905-partite graph 119870

119905(3) is (119899 minus 3)-

regular and 119887RM(119870119905(3)) = 119899 minus 1 for 119905 ⩾ 3 where 119899 = 3119905

Hu and Xu further determined that 119887119877(119866) = 119899 minus 2 for any

(119899 minus 3)-regular graph 119866 of order 119899 ⩽ 5 except 119870119905(3)

Theorem 158 (Hu and Xu [123] 2011) For any (119899minus3)-regulargraph 119866 of order 119899 other than119870

119905(3) 119887RM(119866) = 119899 minus 2 for 119899 ⩾ 5

For a tree 119879 with order 119899 ⩾ 3 Ebadi and PushpaLatha[120] and Jafari Rad and Volkmann [119] independentlyobtained an upper bound on 119887RM(119879)

Theorem 159 119887RM(119879) ⩽ 3 for any tree 119879 with order 119899 ⩾ 3

Theorem 160 (Bahremandpour et al [118] 2013) 119887RM(1198752 times119875119899) = 2 for 119899 ⩾ 2

Theorem 161 (Jafari Rad and Volkmann [119] 2011) Let119866 bea graph with order at least three

(a) If (119909 119910 119911) is a path of length 2 in 119866 then 119887RM(119866) ⩽119889119866(119909) + 119889

119866(119910) + 119889

119866(119911) minus 3 minus |119873

119866(119909) cap 119873

119866(119910)|

(b) If 119866 is connected then 119887RM(119866) ⩽ 120582(119866) + 2Δ(119866) minus 3

Theorem 161 (b) implies 119887RM(119866) ⩽ 120575(119866) + 2Δ(119866) minus 3 for aconnected graph 119866 Note that for a planar graph 119866 120575(119866) ⩽ 5moreover 120575(119866) ⩽ 3 if the girth at least 4 and 120575(119866) ⩽ 2 if thegirth at least 6 These two facts show that 119887RM(119866) ⩽ 2Δ(119866) +2 for a connected planar graph 119866 Jafari Rad and Volkmann[124] improved this bound

Theorem 162 (Jafari Rad and Volkmann [124] 2011) For anyconnected planar graph 119866 of order 119899 ⩾ 3 with girth 119892(119866)

119887RM (119866)

⩽

2Δ (119866)

Δ (119866) + 6

Δ (119866) + 5 119894119891 119866 119888119900119899119905119886119894119899119904 119899119900 V119890119903119905119894119888119890119904 119900119891 119889119890119892119903119890119890 119891119894V119890

Δ (119866) + 4 119894119891 119892 (119866) ⩾ 4

Δ (119866) + 3 119894119891 119892 (119866) ⩾ 5

Δ (119866) + 2 119894119891 119892 (119866) ⩾ 6

Δ (119866) + 1 119894119891 119892 (119866) ⩾ 8

(65)

According toTheorem 157 119887RM(119862119899) = 3 = Δ(119862

119899)+1 for a

cycle 119862119899of length 119899 ⩾ 8 with 119899 equiv 2 (mod 3) and therefore

the last upper bound inTheorem 162 is sharp at least for Δ =2

Combining the fact that every planar graph 119866 withminimum degree 5 contains an edge 119909119910 with 119889

119866(119909) = 5

and 119889119866(119910) isin 5 6 with Theorem 161 (a) Akbari et al [125]

obtained the following result

26 International Journal of Combinatorics

middot middot middot

middot middot middot

middot middot middot

middot middot middot

119866

Figure 9 The graph 119866 is constructed from 119866

Theorem 163 (Akbari et al [125] 2012) 119887RM(119866) ⩽ 15 forevery planar graph 119866

It remains open to show whether the bound inTheorem 163 is sharp or not Though finding a planargraph 119866 with 119887RM(119866) = 15 seems to be difficult Akbari etal [125] constructed an infinite family of planar graphs withRoman bondage number equal to 7 by proving the followingresult

Theorem 164 (Akbari et al [125] 2012) Let 119866 be a graph oforder 119899 and 119866 is the graph of order 5119899 obtained from 119866 byattaching the central vertex of a copy of a path 119875

5to each vertex

of119866 (see Figure 9) Then 120574RM(119866) = 4119899 and 119887RM(119866) = 120575(119866)+2

ByTheorem 164 infinitemany planar graphswith Romanbondage number 7 can be obtained by considering any planargraph 119866 with 120575(119866) = 5 (eg the icosahedron graph)

Conjecture 165 (Akbari et al [125] 2012) The Romanbondage number of every planar graph is at most 7

For general bounds the following observation is directlyobtained which is similar to Observation 1

Observation 4 Let 119866 be a graph of order 119899 with maximumdegree at least two Assume that 119867 is a spanning subgraphobtained from 119866 by removing 119896 edges If 120574RM(119867) = 120574RM(119866)then 119887

119877(119867) ⩽ 119887RM(119866) ⩽ 119887RM(119867) + 119896

Theorem 166 (Bahremandpour et al [118] 2013) For anyconnected graph 119866 of order 119899 if 119899 ⩾ 3 and 120574RM(119866) = 120574(119866) + 1then

119887RM (119866) ⩽ min 119887 (119866) 119899Δ (66)

where 119899Δis the number of vertices with maximum degree Δ in

119866

Observation 5 For any graph 119866 if 120574(119866) = 120574RM(119866) then119887RM(119866) ⩽ 119887(119866)

Theorem 167 (Bahremandpour et al [118] 2013) For everyRoman graph 119866

119887RM (119866) ⩾ 119887 (119866) (67)

The bound is sharp for cycles on 119899 vertices where 119899 equiv 0 (mod3)

The strict inequality in Theorem 167 can hold for exam-ple 119887(119862

3119896+2) = 2 lt 3 = 119887RM(1198623119896+2

) byTheorem 157A graph 119866 is called to be vertex Roman domination-

critical if 120574RM(119866 minus 119909) lt 120574RM(119866) for every vertex 119909 in 119866 If119866 has no isolated vertices then 120574RM(119866) ⩽ 2120574(119866) ⩽ 2120573(119866)If 120574RM(119866) = 2120573(119866) then 120574RM(119866) = 2120574(119866) and hence 119866 is aRoman graph In [30] Volkmann gave a lot of graphs with120574(119866) = 120573(119866) The following result is similar to Theorem 57

Theorem 168 (Bahremandpour et al [118] 2013) For anygraph 119866 if 120574RM(119866) = 2120573(119866) then

(a) 119887RM(119866) ⩾ 120575(119866)

(b) 119887RM(119866) ⩾ 120575(119866)+1 if119866 is a vertex Roman domination-critical graph

If 120574RM(119866) = 2 then obviously 119887RM(119866) ⩽ 120575(119866) So weassume 120574RM(119866) ⩾ 3 Dehgardi et al [121] proved 119887RM(119866) ⩽(120574RM(119866) minus 2)Δ(119866) + 1 and posed the following problem if 119866is a connected graph of order 119899 ⩾ 4 with 120574RM(119866) ⩾ 3 then

119887RM (119866) ⩽ (120574RM (119866) minus 2) Δ (119866) (68)

Theorem 161 (a) shows that the inequality (68) holds if120574RM(119866) ⩾ 5 Thus the bound in (68) is of interest only when120574RM(119866) is 3 or 4

Lemma 169 (Bahremandpour et al [118] 2013) For anynonempty graph 119866 of order 119899 ⩾ 3 120574RM(119866) = 3 if and onlyif Δ(119866) = 119899 minus 2

The following result shows that (68) holds for all graphs119866 of order 119899 ⩾ 4 with 120574RM(119866) = 3 or 4 which improvesTheorem 161 (b)

Theorem 170 (Bahremandpour et al [118] 2013) For anyconnected graph 119866 of order 119899 ⩾ 4

119887RM (119866) le Δ (119866) = 119899 minus 2 119894119891 120574RM (119866) = 3

Δ (119866) + 120575 (119866) minus 1 119894119891 120574RM (119866) = 4(69)

with the first equality if and only if 119866 cong 1198624

Recently Akbari and Qajar [122] have proved the follow-ing result

Theorem 171 (Akbari and Qajar [122]) For any connectedgraph 119866 of order 119899 ⩾ 3

119887RM (119866) ⩽ 119899 minus 120574RM (119866) + 5 (70)

International Journal of Combinatorics 27

We conclude this section with the following problems

Problem 8 Characterize all connected graphs 119866 of order 119899 ⩾3 for which 119887RM(119866) = 119899 minus 1

Problem 9 Prove or disprove if 119866 is a connected graph oforder 119899 ⩾ 3 then

119887RM (119866) ⩽ 119899 minus 120574RM (119866) + 3 (71)

Note that 120574119877(119870

119905(3)) = 4 and 119887RM(119870119905

(3)) = 119899minus1 where 119899 =3119905 for 119905 ⩾ 3 The upper bound on 119887RM(119866) given in Problem 9is sharp if Problem 9 has positive answer

95 Rainbow Bondage Numbers For a positive integer 119896 a119896-rainbow dominating function of a graph 119866 is a function 119891from the vertex-set 119881(119866) to the set of all subsets of the set1 2 119896 such that for any vertex 119909 in 119866 with 119891(119909) = 0 thecondition

⋃

119910isin119873119866(119909)

119891 (119910) = 1 2 119896 (72)

is fulfilledThe weight of a 119896-rainbow dominating function 119891on 119866 is the value

119891 (119866) = sum

119909isin119881(119866)

1003816100381610038161003816119891 (119909)1003816100381610038161003816 (73)

The 119896-rainbow domination number of a graph 119866 denoted by120574119903119896(119866) is the minimum weight of a 119896-rainbow dominating

function 119891 of 119866Note that 120574

1199031(119866) is the classical domination number 120574(119866)

The 119896-rainbow domination number is introduced by Bresaret al [126] Rainbow domination of a graph 119866 coincides withordinary domination of the Cartesian product of 119866 with thecomplete graph in particular 120574

119903119896(119866) = 120574(119866 times 119870

119896) for any

positive integer 119896 and any graph 119866 [126] Hartnell and Rall[127] proved that 120574(119879) lt 120574

1199032(119879) for any tree 119879 no graph has

120574 = 120574119903119896for 119896 ⩾ 3 for any 119896 ⩾ 2 and any graph 119866 of order 119899

min 119899 120574 (119866) + 119896 minus 2 ⩽ 120574119903119896(119866) ⩽ 119896120574 (119866) (74)

The introduction of rainbow domination is motivatedby the study of paired domination in Cartesian productsof graphs where certain upper bounds can be expressed interms of rainbow domination that is for any graph 119866 andany graph 119867 if 119881(119867) can be partitioned into 119896120574

119875-sets then

120574119875(119866 times 119867) ⩽ (1119896)120592(119867)120574

119903119896(119866) proved by Bresar et al [126]

Dehgardi et al [128] introduced the concept of 119896-rainbowbondage number of graphs Let 119866 be a graph with maximumdegree at least two The 119896-rainbow bondage number 119887

119903119896(119866)

of 119866 is the minimum cardinality of all sets 119861 sube 119864(119866) forwhich 120574

119903119896(119866 minus 119861) gt 120574

119903119896(119866) Since in the study of 119896-rainbow

bondage number the assumption Δ(119866) ⩾ 2 is necessarywe always assume that when we discuss 119887

119903119896(119866) all graphs

involved satisfy Δ(119866) ⩾ 2The 2-rainbow bondage number of some special families

of graphs has been determined

Theorem 172 (Dehgardi et al [128]) For any integer 119899 ⩾ 3

1198871199032(119875

119899) = 1 119891119900119903 119899 ⩾ 2

1198871199032(119862

119899) =

1 119894119891 119899 equiv 0 (mod 4) 2 119900119905ℎ119890119903119908119894119904119890

1198871199032(119870

119899119899) =

1 119894119891 119899 = 2

3 119894119891 119899 = 3

6 119894119891 119899 = 4

119898 119894119891 119899 ⩾ 5

(75)

Theorem 173 (Dehgardi et al [128]) For any connected graph119866 of order 119899 ⩾ 3 119887

1199032(119866) ⩽ 120582(119866) + 2Δ(119866) minus 3

Theorem 174 (Dehgardi et al [128]) For any tree 119879 of order119899 ⩾ 3 119887

1199032(119879) ⩽ 2

Theorem 175 (Dehgardi et al [128]) If 119866 is a planar graphwithmaximumdegree at least two then 119887

1199032(119866) ⩽ 15Moreover

if the girth 119892(119866) ⩾ 4 then 1198871199032(119866) ⩽ 11 and if 119892(119866) ⩾ 6 then

1198871199032(119866) ⩽ 9

Theorem 176 (Dehgardi et al [128]) For a complete bipartitegraph 119870

119901119902 if 2119896 + 1 ⩽ 119901 ⩽ 119902 then 119887

119903119896(119870

119901119902) = 119901

Theorem177 (Dehgardi et al [128]) For a complete graph119870119899

if 119899 ⩾ 119896 + 1 then 119887119903119896(119870

119899) = lceil119896119899(119896 + 1)rceil

The special case 119896 = 1 of Theorem 177 can be found inTheorem 1 The following is a simple observation similar toObservation 1

Observation 6 (Dehgardi et al [128]) Let 119866 be a graphwith maximum degree at least two 119867 a spanning subgraphobtained from 119866 by removing 119896 edges If 120574

119903119896(119867) = 120574

119903119896(119866)

then 119887119903119896(119867) ⩽ 119887

119903119896(119866) ⩽ 119887

119903119896(119867) + 119896

Theorem 178 (Dehgardi et al [128]) For every graph 119866 with120574119903119896(119866) = 119896120574(119866) 119887

119903119896(119866) ⩾ 119887(119866)

A graph 119866 is said to be vertex 119896-rainbow domination-critical if 120574

119903119896(119866 minus 119909) lt 120574

119903119896(119866) for every vertex 119909 in 119866 It is

clear that if 119866 has no isolated vertices then 120574119903119896(119866) ⩽ 119896120574(119866) ⩽

119896120573(119866) where 120573(119866) is the vertex-covering number of119866 Thusif 120574

119903119896(119866) = 119896120573(119866) then 120574

119903119896(119866) = 119896120574(119866)

Theorem 179 (Dehgardi et al [128]) For any graph 119866 if120574119903119896(119866) = 119896120573(119866) then

(a) 119887119903119896(119866) ⩾ 120575(119866)

(b) 119887119903119896(119866) ⩾ 120575(119866)+1 if119866 is vertex 119896-rainbow domination-

critical

As far as we know there are no more results on the 119896-rainbow bondage number of graphs in the literature

28 International Journal of Combinatorics

10 Results on Digraphs

Although domination has been extensively studied in undi-rected graphs it is natural to think of a dominating setas a one-way relationship between vertices of the graphIndeed among the earliest literatures on this subject vonNeumann and Morgenstern [129] used what is now calleddomination in digraphs to find solution (or kernels whichare independent dominating sets) for cooperative 119899-persongames Most likely the first formulation of domination byBerge [130] in 1958 was given in the context of digraphs andonly some years latter by Ore [131] in 1962 for undirectedgraphs Despite this history examination of domination andits variants in digraphs has been essentially overlooked (see[4] for an overview of the domination literature) Thus thereare few if any such results on domination for digraphs in theliterature

The bondage number and its related topics for undirectedgraph have become one of major areas both in theoreticaland applied researches However until recently Carlsonand Develin [42] Shan and Kang [132] and Huang andXu [133 134] studied the bondage number for digraphsindependently In this section we introduce their resultsfor general digraphs Results for some special digraphs suchas vertex-transitive digraphs are introduced in the nextsection

101 Upper Bounds for General Digraphs Let 119866 = (119881 119864)

be a digraph without loops and parallel edges A digraph 119866is called to be asymmetric if whenever (119909 119910) isin 119864(119866) then(119910 119909) notin 119864(119866) and to be symmetric if (119909 119910) isin 119864(119866) implies(119910 119909) isin 119864(119866) For a vertex 119909 of 119881(119866) the sets of out-neighbors and in-neighbors of 119909 are respectively defined as119873

+

119866(119909) = 119910 isin 119881(119866) (119909 119910) isin 119864(119866) and 119873minus

119866(119909) = 119909 isin

119881(119866) (119909 119910) isin 119864(119866) the out-degree and the in-degree of119909 are respectively defined as 119889+

119866(119909) = |119873

+

119866(119909)| and 119889minus

119866(119909) =

|119873+

119866(119909)| Denote themaximum and theminimum out-degree

(resp in-degree) of 119866 by Δ+(119866) and 120575+(119866) (resp Δminus(119866) and120575minus

(119866))The degree 119889119866(119909) of 119909 is defined as 119889+

119866(119909)+119889

minus

119866(119909) and

the maximum and the minimum degrees of119866 are denoted byΔ(119866) and 120575(119866) respectively that is Δ(119866) = max119889

119866(119909) 119909 isin

119881(119866) and 120575(119866) = min119889119866(119909) 119909 isin 119881(119866) Note that the

definitions here are different from the ones in the textbookon digraphs see for example Xu [1]

A subset 119878 of119881(119866) is called a dominating set if119881(119866) = 119878cup119873

+

119866(119878) where119873+

119866(119878) is the set of out-neighbors of 119878Then just

as for undirected graphs 120574(119866) is the minimum cardinalityof a dominating set and the bondage number 119887(119866) is thesmallest cardinality of a set 119861 of edges such that 120574(119866 minus 119861) gt120574(119866) if such a subset 119861 sube 119864(119866) exists Otherwise we put119887(119866) = infin

Some basic results for undirected graphs stated inSection 3 can be generalized to digraphs For exampleTheorem 18 is generalized by Carlson and Develin [42]Huang andXu [133] and Shan andKang [132] independentlyas follows

Theorem 180 Let 119866 be a digraph and (119909 119910) isin 119864(119866) Then119887(119866) ⩽ 119889

119866(119910) + 119889

minus

119866(119909) minus |119873

minus

119866(119909) cap 119873

minus

119866(119910)|

Corollary 181 For a digraph 119866 119887(119866) ⩽ 120575minus(119866) + Δ(119866)

Since 120575minus

(119866) ⩽ (12)Δ(119866) from Corollary 181Conjecture 53 is valid for digraphs

Corollary 182 For a digraph 119866 119887(119866) ⩽ (32)Δ(119866)

In the case of undirected graphs the bondage number of119866119899= 119870

119899times 119870

119899achieves this bound However it was shown

by Carlson and Develin in [42] that if we take the symmetricdigraph119866

119899 we haveΔ(119866

119899) = 4(119899minus1) 120574(119866

119899) = 119899 and 119887(119866

119899) ⩽

4(119899 minus 1) + 1 = Δ(119866119899) + 1 So this family of digraphs cannot

show that the bound in Corollary 182 is tightCorresponding to Conjecture 51 which is discredited

for undirected graphs and is valid for digraphs the sameconjecture for digraphs can be proposed as follows

Conjecture 183 (Carlson and Develin [42] 2006) 119887(119866) ⩽Δ(119866) + 1 for any digraph 119866

If Conjecture 183 is true then the following results showsthat this upper bound is tight since 119887(119870

119899times119870

119899) = Δ(119870

119899times119870

119899)+1

for a complete digraph 119870119899

In 2007 Shan and Kang [132] gave some tight upperbounds on the bondage numbers for some asymmetricdigraphs For example 119887(119879) ⩽ Δ(119879) for any asymmetricdirected tree 119879 119887(119866) ⩽ Δ(119866) for any asymmetric digraph 119866with order at least 4 and 120574(119866) ⩽ 2 For planar digraphs theyobtained the following results

Theorem 184 (Shan and Kang [132] 2007) Let 119866 be aasymmetric planar digraphThen 119887(119866) ⩽ Δ(119866)+2 and 119887(119866) ⩽Δ(119866) + 1 if Δ(119866) ⩾ 5 and 119889minus

119866(119909) ⩾ 3 for every vertex 119909 with

119889119866(119909) ⩾ 4

102 Results for Some Special Digraphs The exact values andbounds on 119887(119866) for some standard digraphs are determined

Theorem185 (Huang andXu [133] 2006) For a directed cycle119862119899and a directed path 119875

119899

119887 (119862119899) =

3 119894119891 119899 119894119904 119900119889119889

2 119894119891 119899 119894119904 119890V119890119899

119887 (119875119899) =

2 119894119891 119899 119894119904 119900119889119889

1 119894119891 119899 119894119904 119890V119890119899

(76)

For the de Bruijn digraph 119861(119889 119899) and the Kautz digraph119870(119889 119899)

119887 (119861 (119889 119899)) = 119889 119894119891 119899 119894119904 119900119889119889

119889 ⩽ 119887 (119861 (119889 119899)) ⩽ 2119889 119894119891 119899 119894119904 119890V119890119899

119887 (119870 (119889 119899)) = 119889 + 1

(77)

Like undirected graphs we can define the total domi-nation number and the total bondage number On the totalbondage numbers for some special digraphs the knownresults are as follows

International Journal of Combinatorics 29

Theorem 186 (Huang and Xu [134] 2007) For a directedcycle 119862

119899and a directed path 119875

119899 119887

119879(119875

119899) and 119887

119879(119862

119899) all do not

exist For a complete digraph 119870119899

119887119879(119870

119899) = infin 119894119891 119899 = 1 2

119887119879(119870

119899) = 3 119894119891 119899 = 3

119899 ⩽ 119887119879(119870

119899) ⩽ 2119899 minus 3 119894119891 119899 ⩾ 4

(78)

The extended de Bruijn digraph EB(119889 119899 1199021 119902

119901) and

the extended Kautz digraph EK(119889 119899 1199021 119902

119901) are intro-

duced by Shibata and Gonda [135] If 119901 = 1 then theyare the de Bruijn digraph B(119889 119899) and the Kautz digraphK(119889 119899) respectively Huang and Xu [134] determined theirtotal domination numbers In particular their total bondagenumbers for general cases are determined as follows

Theorem 187 (Huang andXu [134] 2007) If 119889 ⩾ 2 and 119902119894⩾ 2

for each 119894 = 1 2 119901 then

119887119879(EB(119889 119899 119902

1 119902

119901)) = 119889

119901

minus 1

119887119879(EK(119889 119899 119902

1 119902

119901)) = 119889

119901

(79)

In particular for the de Bruijn digraph B(119889 119899) and the Kautzdigraph K(119889 119899)

119887119879(B (119889 119899)) = 119889 minus 1 119887

119879(K (119889 119899)) = 119889 (80)

Zhang et al [136] determined the bondage number incomplete 119905-partite digraphs

Theorem 188 (Zhang et al [136] 2009) For a complete 119905-partite digraph 119870

11989911198992119899119905

where 1198991⩽ 119899

2⩽ sdot sdot sdot ⩽ 119899

119905

119887 (11987011989911198992119899119905

)

=

119898 119894119891 119899119898= 1 119899

119898+1⩾ 2 119891119900119903 119904119900119898119890

119898 (1 ⩽ 119898 lt 119905)

4119905 minus 3 119894119891 1198991= 119899

2= sdot sdot sdot = 119899

119905= 2

119905minus1

sum

119894=1

119899119894+ 2 (119905 minus 1) 119900119905ℎ119890119903119908119894119904119890

(81)

Since an undirected graph can be thought of as a sym-metric digraph any result for digraphs has an analogy forundirected graphs in general In view of this point studyingthe bondage number for digraphs is more significant thanfor undirected graphs Thus we should further study thebondage number of digraphs and try to generalize knownresults on the bondage number and related variants for undi-rected graphs to digraphs prove or disprove Conjecture 183In particular determine the exact values of 119887(B(119889 119899)) for aneven 119899 and 119887

119879(119870

119899) for 119899 ⩾ 4

11 Efficient Dominating Sets

A dominating set 119878 of a graph 119866 is called to be efficient if forevery vertex 119909 in119866 |119873

119866[119909]cap119878| = 1 if119866 is a undirected graph

or |119873minus

119866[119909] cap 119878| = 1 if 119866 is a directed graph The concept of an

efficient set was proposed by Bange et al [137] in 1988 In theliterature an efficient set is also called a perfect dominatingset (see Livingston and Stout [138] for an explanation)

From definition if 119878 is an efficient dominating set of agraph 119866 then 119878 is certainly an independent set and everyvertex not in 119878 is adjacent to exactly one vertex in 119878

It is also clear from definition that a dominating set 119878 isefficient if and only if N

119866[119878] = 119873

119866[119909] 119909 isin 119878 for the

undirected graph 119866 or N+

119866[119878] = 119873

+

119866[119909] 119909 isin 119878 for the

digraph119866 is a partition of119881(119866) where the induced subgraphby119873

119866[119909] or119873+

119866[119909] is a star or an out-star with the root 119909

The efficient domination has important applications inmany areas such as error-correcting codes and receivesmuch attention in the late years

The concept of efficient dominating sets is a measureof the efficiency of domination in graphs Unfortunately asshown in [137] not every graph has an efficient dominatingset andmoreover it is anNP-complete problem to determinewhether a given graph has an efficient dominating set Inaddition it was shown by Clark [139] in 1993 that for a widerange of 119901 almost every random undirected graph 119866 isin

G(120592 119901) has no efficient dominating sets This means thatundirected graphs possessing an efficient dominating set arerare However it is easy to show that every undirected graphhas an orientationwith an efficient dominating set (see Bangeet al [140])

In 1993 Barkauskas and Host [141] showed that deter-mining whether an arbitrary oriented graph has an efficientdominating set is NP-complete Even so the existence ofefficient dominating sets for some special classes of graphs hasbeen examined see for example Dejter and Serra [142] andLee [143] for Cayley graph Gu et al [144] formeshes and toriObradovic et al [145] Huang and Xu [36] Reji Kumar andMacGillivray [146] for circulant graphs Harary graphs andtori and Van Wieren et al [147] for cube-connected cycles

In this section we introduce results of the bondagenumber for some graphs with efficient dominating sets dueto Huang and Xu [36]

111 Results for General Graphs In this subsection we intro-duce some results on bondage numbers obtained by applyingefficient dominating sets We first state the two followinglemmas

Lemma 189 For any 119896-regular graph or digraph 119866 of order 119899120574(119866) ⩾ 119899(119896 + 1) with equality if and only if 119866 has an efficientdominating set In addition if 119866 has an efficient dominatingset then every efficient dominating set is certainly a 120574-set andvice versa

Let 119890 be an edge and 119878 a dominating set in 119866 We say that119890 supports 119878 if 119890 isin (119878 119878) where (119878 119878) = (119909 119910) isin 119864(119866) 119909 isin 119878 119910 isin 119878 Denote by 119904(119866) the minimum number of edgeswhich support all 120574-sets in 119866

Lemma 190 For any graph or digraph 119866 119887(119866) ⩾ 119904(119866) withequality if 119866 is regular and has an efficient dominating set

30 International Journal of Combinatorics

A graph 119866 is called to be vertex-transitive if its automor-phism group Aut(119866) acts transitively on its vertex-set 119881(119866)A vertex-transitive graph is regular Applying Lemmas 189and 190 Huang and Xu obtained some results on bondagenumbers for vertex-transitive graphs or digraphs

Theorem 191 (Huang and Xu [36] 2008) For any vertex-transitive graph or digraph 119866 of order 119899

119887 (119866) ⩾

lceil119899

2120574 (119866)rceil 119894119891 119866 119894119904 119906119899119889119894119903119890119888119905119890119889

lceil119899

120574 (119866)rceil 119894119891 119866 119894119904 119889119894119903119890119888119905119890119889

(82)

Theorem192 (Huang andXu [36] 2008) Let119866 be a 119896-regulargraph of order 119899 Then

119887(119866) ⩽ 119896 if119866 is undirected and 119899 ⩾ 120574(119866)(119896+1)minus119896+1119887(119866) ⩽ 119896+1+ℓ if119866 is directed and 119899 ⩾ 120574(119866)(119896+1)minusℓwith 0 ⩽ ℓ ⩽ 119896 minus 1

Next we introduce a better upper bound on 119887(119866) To thisaim we need the following concept which is a generalizationof the concept of the edge-covering of a graph 119866 For 1198811015840

sube

119881(119866) and 1198641015840 sube 119864(119866) we say that 1198641015840covers 1198811015840 and call 1198641015840an edge-covering for 1198811015840 if there exists an edge (119909 119910) isin 1198641015840 forany vertex 119909 isin 1198811015840 For 119910 isin 119881(119866) let 1205731015840[119910] be the minimumcardinality over all edge-coverings for119873minus

119866[119910]

Theorem 193 (Huang and Xu [36] 2008) If 119866 is a 119896-regulargraph with order 120574(119866)(119896 + 1) then 119887(119866) ⩽ 1205731015840[119910] for any 119910 isin119881(119866)

Theupper bound on 119887(119866) given inTheorem 193 is tight inview of 119887(119862

119899) for a cycle or a directed cycle 119862

119899(seeTheorems

1 and 185 resp)It is easy to see that for a 119896-regular graph 119866 lceil(119896 + 1)2rceil ⩽

1205731015840

[119910] ⩽ 119896 when 119866 is undirected and 1205731015840[119910] = 119896 + 1 when 119866 isdirected By this fact and Lemma 189 the following theoremis merely a simple combination of Theorems 191 and 193

Theorem 194 (Huang and Xu [36] 2008) Let 119866 be a vertex-transitive graph of degree 119896 If 119866 has an efficient dominatingset then

lceil119896 + 1

2rceil ⩽ 119887 (119866) ⩽ 119896 119894119891 119866 119894119904 119906119899119889119894119903119890119888119905119890119889

119887 (119866) = 119896 + 1 119894119891 119866 119894119904 119889119894119903119890119888119905119890119889

(83)

Theorem 195 (Huang and Xu [36] 2008) If 119866 is an undi-rected vertex-transitive cubic graph with order 4120574(119866) and girth119892(119866) ⩽ 5 then 119887(119866) = 2

The proof of Theorem 195 leads to a byproduct If 119892 =5 let (119906

1 119906

2 119906

3 119906

4 119906

5) be a cycle and let D

119894be the family

of efficient dominating sets containing 119906119894for each 119894 isin

1 2 3 4 5 ThenD119894capD

119895= 0 for 119894 = 1 2 5 Then 119866 has

at least 5119904 efficient dominating sets where 119904 = 119904(119866) But thereare only 4s (=ns120574) distinct efficient dominating sets in119866This

contradiction implies that an undirected vertex-transitivecubic graph with girth five has no efficient dominating setsBut a similar argument for 119892(119866) = 3 4 or 119892(119866) ⩾ 6 couldnot give any contradiction This is consistent with the resultthat CCC(119899) a vertex-transitive cubic graph with girth 119899 if3 ⩽ 119899 ⩽ 8 or girth 8 if 119899 ⩾ 9 has efficient dominating setsfor all 119899 ⩾ 3 except 119899 = 5 (see Theorem 199 in the followingsubsection)

112 Results for Cayley Graphs In this subsection we useTheorem 194 to determine the exact values or approximativevalues of bondage numbers for some special vertex-transitivegraphs by characterizing the existence of efficient dominatingsets in these graphs

Let Γ be a nontrivial finite group 119878 a nonempty subset ofΓ without the identity element of Γ A digraph 119866 is defined asfollows

119881 (119866) = Γ (119909 119910) isin 119864 (119866) lArrrArr 119909minus1

119910 isin 119878

for any 119909 119910 isin Γ(84)

is called a Cayley digraph of the group Γ with respect to119878 denoted by 119862

Γ(119878) If 119878minus1 = 119904minus1 119904 isin 119878 = 119878 then

119862Γ(119878) is symmetric and is called a Cayley undirected graph

a Cayley graph for short Cayley graphs or digraphs arecertainly vertex-transitive (see Xu [1])

A circulant graph 119866(119899 119878) of order 119899 is a Cayley graph119862119885119899

(119878) where 119885119899= 0 119899 minus 1 is the addition group of

order 119899 and 119878 is a nonempty subset of119885119899without the identity

element and hence is a vertex-transitive digraph of degree|119878| If 119878minus1 = 119878 then119866(119899 119878) is an undirected graph If 119878 = 1 119896where 2 ⩽ 119896 ⩽ 119899 minus 2 we write 119866(119899 1 119896) for 119866(119899 1 119896) or119866(119899 plusmn1 plusmn119896) and call it a double-loop circulant graph

Huang and Xu [36] showed that for directed 119866 =

119866(119899 1 119896) lceil1198993rceil ⩽ 120574(119866) ⩽ lceil1198992rceil and 119866 has an efficientdominating set if and only if 3 | 119899 and 119896 equiv 2 (mod 3) fordirected 119866 = 119866(119899 1 119896) and 119896 = 1198992 lceil1198995rceil ⩽ 120574(119866) ⩽ lceil1198993rceiland 119866 has an efficient dominating set if and only if 5 | 119899 and119896 equiv plusmn2 (mod 5) By Theorem 194 we obtain the bondagenumber of a double loop circulant graph if it has an efficientdominating set

Theorem 196 (Huang and Xu [36] 2008) Let 119866 be a double-loop circulant graph 119866(119899 1 119896) If 119866 is directed with 3 | 119899and 119896 equiv 2 (mod 3) or 119866 is undirected with 5 | 119899 and119896 equiv plusmn2 (mod 5) then 119887(119866) = 3

The 119898 times 119899 torus is the Cartesian product 119862119898times 119862

119899of

two cycles and is a Cayley graph 119862119885119898times119885119899

(119878) where 119878 =

(0 1) (1 0) for directed cycles and 119878 = (0 plusmn1) (plusmn1 0) forundirected cycles and hence is vertex-transitive Gu et al[144] showed that the undirected torus119862

119898times119862

119899has an efficient

dominating set if and only if both 119898 and 119899 are multiplesof 5 Huang and Xu [36] showed that the directed torus119862119898times 119862

119899has an efficient dominating set if and only if both

119898 and 119899 are multiples of 3 Moreover they found a necessarycondition for a dominating set containing the vertex (0 0)in 119862

119898times 119862

119899to be efficient and obtained the following

result

International Journal of Combinatorics 31

Theorem 197 (Huang and Xu [36] 2008) Let 119866 = 119862119898times 119862

119899

If 119866 is undirected and both119898 and 119899 are multiples of 5 or if 119866 isdirected and both119898 and 119899 are multiples of 3 then 119887(119866) = 3

The hypercube 119876119899

is the Cayley graph 119862Γ(119878)

where Γ = 1198852times sdot sdot sdot times 119885

2= (119885

2)119899 and 119878 =

100 sdot sdot sdot 0 010 sdot sdot sdot 0 00 sdot sdot sdot 01 Lee [143] showed that119876119899has an efficient dominating set if and only if 119899 = 2119898 minus 1

for a positive integer119898 ByTheorem 194 the following resultis obtained immediately

Theorem 198 (Huang and Xu [36] 2008) If 119899 = 2119898 minus 1 for apositive integer119898 then 2119898minus1

⩽ 119887(119876119899) ⩽ 2

119898

minus 1

Problem 10 Determine the exact value of 119887(119876119899) for 119899 = 2119898minus1

The 119899-dimensional cube-connected cycle denoted byCCC(119899) is constructed from the 119899-dimensional hypercube119876119899by replacing each vertex 119909 in 119876

119899with an undirected cycle

119862119899of length 119899 and linking the 119894th vertex of the 119862

119899to the 119894th

neighbor of 119909 It has been proved that CCC(119899) is a Cayleygraph and hence is a vertex-transitive graph with degree 3Van Wieren et al [147] proved that CCC(119899) has an efficientdominating set if and only if 119899 = 5 From Theorems 194 and195 the following result can be derived

Theorem 199 (Huang and Xu [36] 2008) Let 119866 = 119862119862119862(119899)be the 119899-dimensional cube-connected cycles with 119899 ⩾ 3 and119899 = 5 Then 120574(119866) = 1198992119899minus2 and 2 ⩽ 119887(119866) ⩽ 3 In addition119887(119862119862119862(3)) = 119887(119862119862119862(4)) = 2

Problem 11 Determine the exact value of 119887(CCC(119899)) for 119899 ⩾5

Acknowledgments

This work was supported by NNSF of China (Grants nos11071233 61272008) The author would like to express hisgratitude to the anonymous referees for their kind sugges-tions and comments on the original paper to ProfessorSheikholeslami and Professor Jafari Rad for sending thereferences [128] and [81] which resulted in this version

References

[1] J-M Xu Theory and Application of Graphs vol 10 of NetworkTheory and Applications Kluwer Academic Dordrecht TheNetherlands 2003

[2] S THedetniemi andR C Laskar ldquoBibliography on dominationin graphs and some basic definitions of domination parame-tersrdquo Discrete Mathematics vol 86 no 1ndash3 pp 257ndash277 1990

[3] TW Haynes S T Hedetniemi and P J Slater Fundamentals ofDomination in Graphs vol 208 ofMonographs and Textbooks inPure and Applied Mathematics Marcel Dekker New York NYUSA 1998

[4] TWHaynes S THedetniemi and P J Slater EdsDominationin Graphs vol 209 of Monographs and Textbooks in Pure andAppliedMathematicsMarcelDekker NewYorkNYUSA 1998

[5] M R Garey andD S JohnsonComputers and Intractability WH Freeman and Company San Francisco Calif USA 1979

[6] H B Walikar and B D Acharya ldquoDomination critical graphsrdquoNational Academy Science Letters vol 2 pp 70ndash72 1979

[7] D Bauer F Harary J Nieminen and C L Suffel ldquoDominationalteration sets in graphsrdquo Discrete Mathematics vol 47 no 2-3pp 153ndash161 1983

[8] J F Fink M S Jacobson L F Kinch and J Roberts ldquoThebondage number of a graphrdquo Discrete Mathematics vol 86 no1ndash3 pp 47ndash57 1990

[9] F-T Hu and J-M Xu ldquoThe bondage number of (119899 minus 3)-regulargraph G of order nrdquo Ars Combinatoria to appear

[10] U Teschner ldquoNew results about the bondage number of agraphrdquoDiscreteMathematics vol 171 no 1ndash3 pp 249ndash259 1997

[11] B L Hartnell and D F Rall ldquoA bound on the size of a graphwith given order and bondage numberrdquo Discrete Mathematicsvol 197-198 pp 409ndash413 1999 16th British CombinatorialConference (London 1997)

[12] B L Hartnell and D F Rall ldquoA characterization of trees inwhich no edge is essential to the domination numberrdquo ArsCombinatoria vol 33 pp 65ndash76 1992

[13] J Topp and P D Vestergaard ldquo120572119896- and 120574

119896-stable graphsrdquo

Discrete Mathematics vol 212 no 1-2 pp 149ndash160 2000[14] A Meir and J W Moon ldquoRelations between packing and

covering numbers of a treerdquo Pacific Journal of Mathematics vol61 no 1 pp 225ndash233 1975

[15] B L Hartnell L K Joslashrgensen P D Vestergaard and CA Whitehead ldquoEdge stability of the k-domination numberof treesrdquo Bulletin of the Institute of Combinatorics and ItsApplications vol 22 pp 31ndash40 1998

[16] F-T Hu and J-M Xu ldquoOn the complexity of the bondage andreinforcement problemsrdquo Journal of Complexity vol 28 no 2pp 192ndash201 2012

[17] B L Hartnell and D F Rall ldquoBounds on the bondage numberof a graphrdquo Discrete Mathematics vol 128 no 1ndash3 pp 173ndash1771994

[18] Y-L Wang ldquoOn the bondage number of a graphrdquo DiscreteMathematics vol 159 no 1ndash3 pp 291ndash294 1996

[19] G H Fricke TW Haynes S M Hedetniemi S T Hedetniemiand R C Laskar ldquoExcellent treesrdquo Bulletin of the Institute ofCombinatorics and Its Applications vol 34 pp 27ndash38 2002

[20] V Samodivkin ldquoThe bondage number of graphs good and badverticesrdquoDiscussiones Mathematicae GraphTheory vol 28 no3 pp 453ndash462 2008

[21] J E Dunbar T W Haynes U Teschner and L VolkmannldquoBondage insensitivity and reinforcementrdquo in Domination inGraphs vol 209 of Monographs and Textbooks in Pure andAppliedMathematics pp 471ndash489 Dekker NewYork NY USA1998

[22] H Liu and L Sun ldquoThe bondage and connectivity of a graphrdquoDiscrete Mathematics vol 263 no 1ndash3 pp 289ndash293 2003

[23] A-H Esfahanian and S L Hakimi ldquoOn computing a con-ditional edge-connectivity of a graphrdquo Information ProcessingLetters vol 27 no 4 pp 195ndash199 1988

[24] R C Brigham P Z Chinn and R D Dutton ldquoVertexdomination-critical graphsrdquoNetworks vol 18 no 3 pp 173ndash1791988

[25] V Samodivkin ldquoDomination with respect to nondegenerateproperties bondage numberrdquoThe Australasian Journal of Com-binatorics vol 45 pp 217ndash226 2009

[26] U Teschner ldquoA new upper bound for the bondage numberof graphs with small domination numberrdquo The AustralasianJournal of Combinatorics vol 12 pp 27ndash35 1995

32 International Journal of Combinatorics

[27] J Fulman D Hanson and G MacGillivray ldquoVertex dom-ination-critical graphsrdquoNetworks vol 25 no 2 pp 41ndash43 1995

[28] U Teschner ldquoA counterexample to a conjecture on the bondagenumber of a graphrdquo Discrete Mathematics vol 122 no 1ndash3 pp393ndash395 1993

[29] U Teschner ldquoThe bondage number of a graph G can be muchgreater than Δ(G)rdquo Ars Combinatoria vol 43 pp 81ndash87 1996

[30] L Volkmann ldquoOn graphs with equal domination and coveringnumbersrdquoDiscrete AppliedMathematics vol 51 no 1-2 pp 211ndash217 1994

[31] L A Sanchis ldquoMaximum number of edges in connected graphswith a given domination numberrdquoDiscreteMathematics vol 87no 1 pp 65ndash72 1991

[32] S Klavzar and N Seifter ldquoDominating Cartesian products ofcyclesrdquoDiscrete AppliedMathematics vol 59 no 2 pp 129ndash1361995

[33] H K Kim ldquoA note on Vizingrsquos conjecture about the bondagenumberrdquo Korean Journal of Mathematics vol 10 no 2 pp 39ndash42 2003

[34] M Y Sohn X Yuan and H S Jeong ldquoThe bondage number of1198623times119862

119899rdquo Journal of the KoreanMathematical Society vol 44 no

6 pp 1213ndash1231 2007[35] L Kang M Y Sohn and H K Kim ldquoBondage number of the

discrete torus 119862119899times 119862

4rdquo Discrete Mathematics vol 303 no 1ndash3

pp 80ndash86 2005[36] J Huang and J-M Xu ldquoThe bondage numbers and efficient

dominations of vertex-transitive graphsrdquoDiscrete Mathematicsvol 308 no 4 pp 571ndash582 2008

[37] C J Xiang X Yuan andM Y Sohn ldquoDomination and bondagenumber of119862

3times119862

119899rdquoArs Combinatoria vol 97 pp 299ndash310 2010

[38] M S Jacobson and L F Kinch ldquoOn the domination numberof products of graphs Irdquo Ars Combinatoria vol 18 pp 33ndash441984

[39] Y-Q Li ldquoA note on the bondage number of a graphrdquo ChineseQuarterly Journal of Mathematics vol 9 no 4 pp 1ndash3 1994

[40] F-T Hu and J-M Xu ldquoBondage number of mesh networksrdquoFrontiers ofMathematics inChina vol 7 no 5 pp 813ndash826 2012

[41] R Frucht and F Harary ldquoOn the corona of two graphsrdquoAequationes Mathematicae vol 4 pp 322ndash325 1970

[42] K Carlson and M Develin ldquoOn the bondage number of planarand directed graphsrdquoDiscreteMathematics vol 306 no 8-9 pp820ndash826 2006

[43] U Teschner ldquoOn the bondage number of block graphsrdquo ArsCombinatoria vol 46 pp 25ndash32 1997

[44] J Huang and J-M Xu ldquoNote on conjectures of bondagenumbers of planar graphsrdquo Applied Mathematical Sciences vol6 no 65ndash68 pp 3277ndash3287 2012

[45] L Kang and J Yuan ldquoBondage number of planar graphsrdquoDiscrete Mathematics vol 222 no 1ndash3 pp 191ndash198 2000

[46] M Fischermann D Rautenbach and L Volkmann ldquoRemarkson the bondage number of planar graphsrdquo Discrete Mathemat-ics vol 260 no 1ndash3 pp 57ndash67 2003

[47] J Hou G Liu and J Wu ldquoBondage number of planar graphswithout small cyclesrdquoUtilitasMathematica vol 84 pp 189ndash1992011

[48] J Huang and J-M Xu ldquoThe bondage numbers of graphs withsmall crossing numbersrdquo Discrete Mathematics vol 307 no 15pp 1881ndash1897 2007

[49] Q Ma S Zhang and J Wang ldquoBondage number of 1-planargraphrdquo Applied Mathematics vol 1 no 2 pp 101ndash103 2010

[50] J Hou and G Liu ldquoA bound on the bondage number of toroidalgraphsrdquoDiscreteMathematics Algorithms and Applications vol4 no 3 Article ID 1250046 5 pages 2012

[51] Y-C Cao J-M Xu and X-R Xu ldquoOn bondage number oftoroidal graphsrdquo Journal of University of Science and Technologyof China vol 39 no 3 pp 225ndash228 2009

[52] Y-C Cao J Huang and J-M Xu ldquoThe bondage number ofgraphs with crossing number less than fourrdquo Ars Combinatoriato appear

[53] H K Kim ldquoOn the bondage numbers of some graphsrdquo KoreanJournal of Mathematics vol 11 no 2 pp 11ndash15 2004

[54] A Gagarin and V Zverovich ldquoUpper bounds for the bondagenumber of graphs on topological surfacesrdquo Discrete Mathemat-ics 2012

[55] J Huang ldquoAn improved upper bound for the bondage numberof graphs on surfacesrdquoDiscrete Mathematics vol 312 no 18 pp2776ndash2781 2012

[56] A Gagarin and V Zverovich ldquoThe bondage number of graphson topological surfaces and Teschnerrsquos conjecturerdquo DiscreteMathematics vol 313 no 6 pp 796ndash808 2013

[57] V Samodivkin ldquoNote on the bondage number of graphs ontopological surfacesrdquo httparxivorgabs12086203

[58] E J Cockayne R M Dawes and S T Hedetniemi ldquoTotaldomination in graphsrdquoNetworks vol 10 no 3 pp 211ndash219 1980

[59] R Laskar J Pfaff S M Hedetniemi and S T HedetniemildquoOn the algorithmic complexity of total dominationrdquo Society forIndustrial and Applied Mathematics vol 5 no 3 pp 420ndash4251984

[60] J Pfaff R C Laskar and S T Hedetniemi ldquoNP-completeness oftotal and connected domination and irredundance for bipartitegraphsrdquo Tech Rep 428 Department of Mathematical SciencesClemson University 1983

[61] M A Henning ldquoA survey of selected recent results on totaldomination in graphsrdquoDiscrete Mathematics vol 309 no 1 pp32ndash63 2009

[62] V R Kulli and D K Patwari ldquoThe total bondage number ofa graphrdquo in Advances in Graph Theory pp 227ndash235 VishwaGulbarga India 1991

[63] F-T Hu Y Lu and J-M Xu ldquoThe total bondage number ofgrid graphsrdquo Discrete Applied Mathematics vol 160 no 16-17pp 2408ndash2418 2012

[64] J Cao M Shi and B Wu ldquoResearch on the total bondagenumber of a spectial networkrdquo in Proceedings of the 3rdInternational Joint Conference on Computational Sciences andOptimization (CSO rsquo10) pp 456ndash458 May 2010

[65] N Sridharan MD Elias and V S A Subramanian ldquoTotalbondage number of a graphrdquo AKCE International Journal ofGraphs and Combinatorics vol 4 no 2 pp 203ndash209 2007

[66] N Jafari Rad and J Raczek ldquoSome progress on total bondage ingraphsrdquo Graphs and Combinatorics 2011

[67] T W Haynes and P J Slater ldquoPaired-domination and thepaired-domatic numberrdquoCongressusNumerantium vol 109 pp65ndash72 1995

[68] T W Haynes and P J Slater ldquoPaired-domination in graphsrdquoNetworks vol 32 no 3 pp 199ndash206 1998

[69] X Hou ldquoA characterization of 2120574 120574119875-treesrdquoDiscrete Mathemat-

ics vol 308 no 15 pp 3420ndash3426 2008[70] K E Proffitt T W Haynes and P J Slater ldquoPaired-domination

in grid graphsrdquo Congressus Numerantium vol 150 pp 161ndash1722001

International Journal of Combinatorics 33

[71] H Qiao L KangM Cardei andD-Z Du ldquoPaired-dominationof treesrdquo Journal of Global Optimization vol 25 no 1 pp 43ndash542003

[72] E Shan L Kang and M A Henning ldquoA characterizationof trees with equal total domination and paired-dominationnumbersrdquo The Australasian Journal of Combinatorics vol 30pp 31ndash39 2004

[73] J Raczek ldquoPaired bondage in treesrdquo Discrete Mathematics vol308 no 23 pp 5570ndash5575 2008

[74] J A Telle and A Proskurowski ldquoAlgorithms for vertex parti-tioning problems on partial k-treesrdquo SIAM Journal on DiscreteMathematics vol 10 no 4 pp 529ndash550 1997

[75] G S Domke J H Hattingh M A Henning and L R MarkusldquoRestrained domination in treesrdquoDiscreteMathematics vol 211no 1ndash3 pp 1ndash9 2000

[76] G S Domke J H Hattingh M A Henning and L R MarkusldquoRestrained domination in graphs with minimum degree twordquoJournal of Combinatorial Mathematics and Combinatorial Com-puting vol 35 pp 239ndash254 2000

[77] G S Domke J H Hattingh S T Hedetniemi R C Laskarand L R Markus ldquoRestrained domination in graphsrdquo DiscreteMathematics vol 203 no 1ndash3 pp 61ndash69 1999

[78] J H Hattingh and E J Joubert ldquoAn upper bound for therestrained domination number of a graph with minimumdegree at least two in terms of order and minimum degreerdquoDiscrete Applied Mathematics vol 157 no 13 pp 2846ndash28582009

[79] M A Henning ldquoGraphs with large restrained dominationnumberrdquo Discrete Mathematics vol 197-198 pp 415ndash429 199916th British Combinatorial Conference (London 1997)

[80] J H Hattingh and A R Plummer ldquoRestrained bondage ingraphsrdquo Discrete Mathematics vol 308 no 23 pp 5446ndash54532008

[81] N Jafari Rad R Hasni J Raczek and L Volkmann ldquoTotalrestrained bondage in graphsrdquoActaMathematica Sinica vol 29no 6 pp 1033ndash1042 2013

[82] J Cyman and J Raczek ldquoOn the total restrained dominationnumber of a graphrdquoThe Australasian Journal of Combinatoricsvol 36 pp 91ndash100 2006

[83] M A Henning ldquoGraphs with large total domination numberrdquoJournal of Graph Theory vol 35 no 1 pp 21ndash45 2000

[84] D-X Ma X-G Chen and L Sun ldquoOn total restraineddomination in graphsrdquoCzechoslovakMathematical Journal vol55 no 1 pp 165ndash173 2005

[85] B Zelinka ldquoRemarks on restrained domination and totalrestrained domination in graphsrdquo Czechoslovak MathematicalJournal vol 55 no 2 pp 393ndash396 2005

[86] W Goddard and M A Henning ldquoIndependent domination ingraphs a survey and recent resultsrdquo Discrete Mathematics vol313 no 7 pp 839ndash854 2013

[87] J H Zhang H L Liu and L Sun ldquoIndependence bondagenumber and reinforcement number of some graphsrdquo Transac-tions of Beijing Institute of Technology vol 23 no 2 pp 140ndash1422003

[88] K D Dharmalingam Studies in Graph Theorymdashequitabledomination and bottleneck domination [PhD thesis] MaduraiKamaraj University Madurai India 2006

[89] GDeepakND Soner andAAlwardi ldquoThe equitable bondagenumber of a graphrdquo Research Journal of Pure Algebra vol 1 no9 pp 209ndash212 2011

[90] E Sampathkumar and H B Walikar ldquoThe connected domina-tion number of a graphrdquo Journal of Mathematical and PhysicalSciences vol 13 no 6 pp 607ndash613 1979

[91] K White M Farber and W Pulleyblank ldquoSteiner trees con-nected domination and strongly chordal graphsrdquoNetworks vol15 no 1 pp 109ndash124 1985

[92] M Cadei M X Cheng X Cheng and D-Z Du ldquoConnecteddomination in Ad Hoc wireless networksrdquo in Proceedings ofthe 6th International Conference on Computer Science andInformatics (CSampI rsquo02) 2002

[93] J F Fink and M S Jacobson ldquon-domination in graphsrdquo inGraph Theory with Applications to Algorithms and ComputerScience Wiley-Interscience Publications pp 283ndash300 WileyNew York NY USA 1985

[94] M Chellali O Favaron A Hansberg and L Volkmann ldquok-domination and k-independence in graphs a surveyrdquo Graphsand Combinatorics vol 28 no 1 pp 1ndash55 2012

[95] Y Lu X Hou J-M Xu andN Li ldquoTrees with uniqueminimump-dominating setsrdquo Utilitas Mathematica vol 86 pp 193ndash2052011

[96] Y Lu and J-M Xu ldquoThe p-bondage number of treesrdquo Graphsand Combinatorics vol 27 no 1 pp 129ndash141 2011

[97] Y Lu and J-M Xu ldquoThe 2-domination and 2-bondage numbersof grid graphs A manuscriptrdquo httparxivorgabs12044514

[98] M Krzywkowski ldquoNon-isolating 2-bondage in graphsrdquo TheJournal of the Mathematical Society of Japan vol 64 no 4 2012

[99] M A Henning O R Oellermann and H C Swart ldquoBoundson distance domination parametersrdquo Journal of CombinatoricsInformation amp System Sciences vol 16 no 1 pp 11ndash18 1991

[100] F Tian and J-M Xu ldquoBounds for distance domination numbersof graphsrdquo Journal of University of Science and Technology ofChina vol 34 no 5 pp 529ndash534 2004

[101] F Tian and J-M Xu ldquoDistance domination-critical graphsrdquoApplied Mathematics Letters vol 21 no 4 pp 416ndash420 2008

[102] F Tian and J-M Xu ldquoA note on distance domination numbersof graphsrdquo The Australasian Journal of Combinatorics vol 43pp 181ndash190 2009

[103] F Tian and J-M Xu ldquoAverage distances and distance domina-tion numbersrdquo Discrete Applied Mathematics vol 157 no 5 pp1113ndash1127 2009

[104] Z-L Liu F Tian and J-M Xu ldquoProbabilistic analysis of upperbounds for 2-connected distance k-dominating sets in graphsrdquoTheoretical Computer Science vol 410 no 38ndash40 pp 3804ndash3813 2009

[105] M A Henning O R Oellermann and H C Swart ldquoDistancedomination critical graphsrdquo Journal of Combinatorial Mathe-matics and Combinatorial Computing vol 44 pp 33ndash45 2003

[106] T J Bean M A Henning and H C Swart ldquoOn the integrityof distance domination in graphsrdquo The Australasian Journal ofCombinatorics vol 10 pp 29ndash43 1994

[107] M Fischermann and L Volkmann ldquoA remark on a conjecturefor the (kp)-domination numberrdquoUtilitasMathematica vol 67pp 223ndash227 2005

[108] T Korneffel D Meierling and L Volkmann ldquoA remark on the(22)-domination numberrdquo Discussiones Mathematicae GraphTheory vol 28 no 2 pp 361ndash366 2008

[109] Y Lu X Hou and J-M Xu ldquoOn the (22)-domination numberof treesrdquo Discussiones Mathematicae Graph Theory vol 30 no2 pp 185ndash199 2010

34 International Journal of Combinatorics

[110] H Li and J-M Xu ldquo(dm)-domination numbers of m-connected graphsrdquo Rapports de Recherche 1130 LRI UniversiteParis-Sud 1997

[111] S M Hedetniemi S T Hedetniemi and T V Wimer ldquoLineartime resource allocation algorithms for treesrdquo Tech Rep URI-014 Department of Mathematical Sciences Clemson Univer-sity 1987

[112] M Farber ldquoDomination independent domination and dualityin strongly chordal graphsrdquo Discrete Applied Mathematics vol7 no 2 pp 115ndash130 1984

[113] G S Domke and R C Laskar ldquoThe bondage and reinforcementnumbers of 120574

119891for some graphsrdquoDiscrete Mathematics vol 167-

168 pp 249ndash259 1997[114] G S Domke S T Hedetniemi and R C Laskar ldquoFractional

packings coverings and irredundance in graphsrdquo vol 66 pp227ndash238 1988

[115] E J Cockayne P A Dreyer Jr S M Hedetniemi andS T Hedetniemi ldquoRoman domination in graphsrdquo DiscreteMathematics vol 278 no 1ndash3 pp 11ndash22 2004

[116] I Stewart ldquoDefend the roman empirerdquo Scientific American vol281 pp 136ndash139 1999

[117] C S ReVelle and K E Rosing ldquoDefendens imperiumromanum a classical problem in military strategyrdquoThe Ameri-can Mathematical Monthly vol 107 no 7 pp 585ndash594 2000

[118] A Bahremandpour F-T Hu S M Sheikholeslami and J-MXu ldquoOn the Roman bondage number of a graphrdquo DiscreteMathematics Algorithms and Applications vol 5 no 1 ArticleID 1350001 15 pages 2013

[119] N Jafari Rad and L Volkmann ldquoRoman bondage in graphsrdquoDiscussiones Mathematicae Graph Theory vol 31 no 4 pp763ndash773 2011

[120] K Ebadi and L PushpaLatha ldquoRoman bondage and Romanreinforcement numbers of a graphrdquo International Journal ofContemporary Mathematical Sciences vol 5 pp 1487ndash14972010

[121] N Dehgardi S M Sheikholeslami and L Volkman ldquoOn theRoman k-bondage number of a graphrdquo AKCE InternationalJournal of Graphs and Combinatorics vol 8 pp 169ndash180 2011

[122] S Akbari and S Qajar ldquoA note on Roman bondage number ofgraphsrdquo Ars Combinatoria to Appear

[123] F-T Hu and J-M Xu ldquoRoman bondage numbers of somegraphsrdquo httparxivorgabs11093933

[124] N Jafari Rad and L Volkmann ldquoOn the Roman bondagenumber of planar graphsrdquo Graphs and Combinatorics vol 27no 4 pp 531ndash538 2011

[125] S Akbari M Khatirinejad and S Qajar ldquoA note on the romanbondage number of planar graphsrdquo Graphs and Combinatorics2012

[126] B Bresar M A Henning and D F Rall ldquoRainbow dominationin graphsrdquo Taiwanese Journal of Mathematics vol 12 no 1 pp213ndash225 2008

[127] B L Hartnell and D F Rall ldquoOn dominating the Cartesianproduct of a graph and 120572krdquo Discussiones Mathematicae GraphTheory vol 24 no 3 pp 389ndash402 2004

[128] N Dehgardi S M Sheikholeslami and L Volkman ldquoThe k-rainbow bondage number of a graphrdquo to appear in DiscreteApplied Mathematics

[129] J von Neumann and O Morgenstern Theory of Games andEconomic Behavior Princeton University Press Princeton NJUSA 1944

[130] C BergeTheorıe des Graphes et ses Applications 1958[131] O Ore Theory of Graphs vol 38 American Mathematical

Society Colloquium Publications 1962[132] E Shan and L Kang ldquoBondage number in oriented graphsrdquoArs

Combinatoria vol 84 pp 319ndash331 2007[133] J Huang and J-M Xu ldquoThe bondage numbers of extended

de Bruijn and Kautz digraphsrdquo Computers amp Mathematics withApplications vol 51 no 6-7 pp 1137ndash1147 2006

[134] J Huang and J-M Xu ldquoThe total domination and total bondagenumbers of extended de Bruijn and Kautz digraphsrdquoComputersamp Mathematics with Applications vol 53 no 8 pp 1206ndash12132007

[135] Y Shibata and Y Gonda ldquoExtension of de Bruijn graph andKautz graphrdquo Computers amp Mathematics with Applications vol30 no 9 pp 51ndash61 1995

[136] X Zhang J Liu and JMeng ldquoThebondage number in completet-partite digraphsrdquo Information Processing Letters vol 109 no17 pp 997ndash1000 2009

[137] D W Bange A E Barkauskas and P J Slater ldquoEfficient dom-inating sets in graphsrdquo in Applications of Discrete Mathematicspp 189ndash199 SIAM Philadelphia Pa USA 1988

[138] M Livingston and Q F Stout ldquoPerfect dominating setsrdquoCongressus Numerantium vol 79 pp 187ndash203 1990

[139] L Clark ldquoPerfect domination in random graphsrdquo Journal ofCombinatorial Mathematics and Combinatorial Computing vol14 pp 173ndash182 1993

[140] D W Bange A E Barkauskas L H Host and L H ClarkldquoEfficient domination of the orientations of a graphrdquo DiscreteMathematics vol 178 no 1ndash3 pp 1ndash14 1998

[141] A E Barkauskas and L H Host ldquoFinding efficient dominatingsets in oriented graphsrdquo Congressus Numerantium vol 98 pp27ndash32 1993

[142] I J Dejter and O Serra ldquoEfficient dominating sets in CayleygraphsrdquoDiscrete AppliedMathematics vol 129 no 2-3 pp 319ndash328 2003

[143] J Lee ldquoIndependent perfect domination sets in Cayley graphsrdquoJournal of Graph Theory vol 37 no 4 pp 213ndash219 2001

[144] WGu X Jia and J Shen ldquoIndependent perfect domination setsin meshes tori and treesrdquo Unpublished

[145] N Obradovic J Peters and G Ruzic ldquoEfficient domination incirculant graphswith two chord lengthsrdquo Information ProcessingLetters vol 102 no 6 pp 253ndash258 2007

[146] K Reji Kumar and G MacGillivray ldquoEfficient domination incirculant graphsrdquo Discrete Mathematics vol 313 no 6 pp 767ndash771 2013

[147] D Van Wieren M Livingston and Q F Stout ldquoPerfectdominating sets on cube-connected cyclesrdquo Congressus Numer-antium vol 97 pp 51ndash70 1993

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

International Journal of Combinatorics 17

Theorem 108 (Gagarin and Zverovich [56] 2013) Let 119866 be aconnected triangle-free graph of order 119899 If 119866 can be embeddedon the surface of its orientable or nonorientable genus of theEuler characteristic 120594 then

(a) 120594 ⩾ 1 implies 119887(119866) ⩽ 6(b) 120594 ⩽ 0 and 119899 gt minus8120594 imply 119887(119866) ⩽ 7(c) 120594 ⩽ minus1 and 119899 ⩽ minus8120594 imply 119887(119866) ⩽ 7minus4120594(1+radic2 minus 120594)

In [56] the authors also explicitly improved upperbounds in Theorem 103 and gave tight lower bounds forthe number of vertices of graphs 2-cell embeddable ontopological surfaces of a given genusThe interested reader isreferred to the original paper for details The following resultis interesting although its proof is easy

Theorem 109 (Samodivkin [57]) Let119866 be a graph with order119899 and girth 119892 If 119866 is embeddable on a surface whose Eulercharacteristic 120594 is as large as possible then

119887 (119866) ⩽ 3 +8

119892 minus 2minus

4120594119892

119899 (119892 minus 2) (39)

If 119866 is a planar graph with girth 119892 ⩾ 4 + 119894 for any 119894 isin0 1 2 then Theorem 109 leads to 119887(119866) ⩽ 6 minus 119894 which isthe result inTheorem 79 obtained by Fischermann et al [46]Also inmany cases the bound stated inTheorem 109 is betterthan those given byTheorems 103 and 105

8 Conditional Bondage Numbers

Since the concept of the bondage number is based upon thedomination all sorts of dominations which are variationsof the normal domination by adding restricted conditionsto the normal dominating set can develop a new ldquobondagenumberrdquo as long as a variation of domination number isgiven In this section we survey results on the bondagenumber under some restricted conditions

81 Total Bondage Numbers A dominating set 119878 of a graph119866 without isolated vertices is called total if the subgraphinduced by 119878 contains no isolated vertices The minimumcardinality of a total dominating set is called the totaldomination number of 119866 and denoted by 120574

119879(119866) It is clear

that 120574(119866) ⩽ 120574119879(119866) ⩽ 2120574(119866) for any graph 119866 without isolated

verticesThe total domination in graphs was introduced by Cock-

ayne et al [58] in 1980 Pfaff et al [59 60] in 1983 showedthat the problem determining total domination number forgeneral graphs is NP-complete even for bipartite graphs andchordal graphs Even now total domination in graphs hasbeen extensively studied in the literature In 2009 Henning[61] gave a survey of selected recent results on total domina-tion in graphs

The total bondage number of 119866 denoted by 119887119879(119866) is the

smallest cardinality of a subset 119861 sube 119864(119866) with the propertythat119866minus119861 contains no isolated vertices and 120574

119879(119866minus119861) gt 120574

119879(119866)

From definition 119887119879(119866)may not exist for some graphs for

example119866 = 1198701119899 We put 119887

119879(119866) = infin if 119887

119879(119866) does not exist

It is easy to see that 119887119879(119866) is finite for any connected graph 119866

other than 1198701 119870

2 119870

3 and 119870

1119899 In fact for such a graph 119866

since there is a path of length 3 in 119866 we can find 1198611sube 119864(119866)

such that 119866 minus 1198611is a spanning tree 119879 of 119866 containing a path

of length 3 So 120574119879(119879) = 120574

119879(119866minus119861

1) ⩾ 120574

119879(119866) For the tree119879 we

find 1198612sube 119864(119879) such that 120574

119879(119879 minus 119861

2) gt 120574

119879(119879) ⩾ 120574

119879(119866) Thus

we have 120574119879(119866minus119861

2minus119861

1) gt 120574

119879(119866) and so 119887

119879(119866) ⩽ |119861

1|+|119861

2| In

the following discussion we always assume that 119887119879(119866) exists

when a graph 119866 is mentionedIn 1991 Kulli and Patwari [62] first studied the total

bondage number of a graph and calculated the exact valuesof 119887

119879(119866) for some standard graphs

Theorem 110 (Kulli and Patwari [62] 1991) For 119899 ⩾ 4

119887119879(119862

119899) =

3 119894119891 119899 equiv 2 (mod 4) 2 119900119905ℎ119890119903119908119894119904119890

119887119879(119875

119899) =

2 119894119891 119899 equiv 2 (mod 4) 1 119900119905ℎ119890119903119908119894119904119890

119887119879(119870

119898119899) = 119898 119908119894119905ℎ 2 ⩽ 119898 ⩽ 119899

119887119879(119870

119899) =

4 119891119900119903 119899 = 4

2119899 minus 5 119891119900119903 119899 ⩾ 5

(40)

Recently Hu et al [63] have obtained some results on thetotal bondage number of the Cartesian product119875

119898times119875

119899of two

paths 119875119898and 119875

119899

Theorem 111 (Hu et al [63] 2012) For the Cartesian product119875119898times 119875

119899of two paths 119875

119898and 119875

119899

119887119879(119875

2times 119875

119899) =

1 119894119891 119899 equiv 0 (mod 3) 2 119894119891 119899 equiv 2 (mod 3) 3 119894119891 119899 equiv 1 (mod 3)

119887119879(119875

3times 119875

119899) = 1

119887119879(119875

4times 119875

119899)

= 1 119894119891 119899 equiv 1 (mod 5) = 2 119894119891 119899 equiv 4 (mod 5) ⩽ 3 119894119891 119899 equiv 2 (mod 5) ⩽ 4 119894119891 119899 equiv 0 3 (mod 5)

(41)

Generalized Petersen graphs are an important class ofcommonly used interconnection networks and have beenstudied recently By constructing a family of minimum totaldominating sets Cao et al [64] determined the total bondagenumber of the generalized Petersen graphs

FromTheorem 6 we know that 119887(119879) ⩽ 2 for a nontrivialtree 119879 But given any positive integer 119896 Sridharan et al [65]constructed a tree 119879 for which 119887

119879(119879) = 119896 Let119867

119896be the tree

obtained from a star1198701119896+1

by subdividing 119896 edges twice Thetree 119867

7is shown in Figure 6 It can be easily verified that

119887119879(119867

119896) = 119896 This fact can be stated as the following theorem

Theorem 112 (Sridharan et al [65] 2007) For any positiveinteger 119896 there exists a tree 119879 with 119887

119879(119879) = 119896

18 International Journal of Combinatorics

Figure 61198677

Combining Theorem 1 with Theorem 112 we have whatfollows

Corollary 113 (Sridharan et al [65] 2007) The bondagenumber and the total bondage number are unrelated even fortrees

However Sridharan et al [65] gave an upper bound onthe total bondage number of a tree in terms of its maximumdegree For any tree 119879 of order 119899 if 119879 =119870

1119899minus1 then 119887

119879(119879) =

minΔ(119879) (13)(119899 minus 1) Rad and Raczek [66] improved thisupper bound and gave a constructive characterization of acertain class of trees attaching the upper bound

Theorem 114 (Rad and Raczek [66] 2011) 119887119879(119879) ⩽ Δ(119879) minus 1

for any tree 119879 with maximum degree at least three

In general the decision problem for 119887119879(119866) is NP-complete

for any graph 119866 We state the decision problem for the totalbondage as follows

Problem 3 Consider the decision problemTotal Bondage ProblemInstance a graph 119866 and a positive integer 119887Question is 119887

119879(119866) ⩽ 119887

Theorem 115 (Hu and Xu [16] 2012) The total bondageproblem is NP-complete

Consequently in view of its computational hardnessit is significative to establish some sharp bounds on thetotal bondage number of a graph in terms of other graphicparameters In 1991 Kulli and Patwari [62] showed that119887119879(119866) ⩽ 2119899 minus 5 for any graph 119866 with order 119899 ⩾ 5 Sridharan

et al [65] improved this result as follows

Theorem 116 (Sridharan et al [65] 2007) For any graph 119866with order 119899 if 119899 ⩾ 5 then

119887119879(119866) ⩽

1

3(119899 minus 1) 119894119891 119866 119888119900119899119905119886119894119899119904 119899119900 119888119910119888119897119890119904

119899 minus 1 119894119891 119892 (119866) ⩾ 5

119899 minus 2 119894119891 119892 (119866) = 4

(42)

Rad and Raczek [66] also established some upperbounds on 119887

119879(119866) for a general graph 119866 In particular they

gave an upper bound on 119887119879(119866) in terms of the girth of

119866

Theorem 117 (Rad and Raczek [66] 2011) 119887119879(119866)⩽4Δ(119866) minus 5

for any graph 119866 with 119892(119866) ⩾ 4

82 Paired Bondage Numbers A dominating set 119878 of 119866 iscalled to be paired if the subgraph induced by 119878 containsa perfect matching The paired domination number of 119866denoted by 120574

119875(119866) is the minimum cardinality of a paired

dominating set of 119866 Clearly 120574119879(119866) ⩽ 120574

119875(119866) for every

connected graph 119866 with order at least two where 120574119879(119866) is

the total domination number of 119866 and 120574119875(119866) ⩽ 2120574(119866) for

any graph119866without isolated vertices Paired domination wasintroduced by Haynes and Slater [67 68] and further studiedin [69ndash72]

The paired bondage number of 119866 with 120575(119866) ⩾ 1 denotedby 119887P(119866) is the minimum cardinality among all sets of edges119861 sube 119864 such that 120575(119866 minus 119861) ⩾ 1 and 120574

119875(119866 minus 119861) gt 120574

119875(119866)

The concept of the paired bondage number was firstproposed by Raczek [73] in 2008The following observationsfollow immediately from the definition of the paired bondagenumber

Observation 2 Let 119866 be a graph with 120575(119866) ⩾ 1

(a) If 119867 is a subgraph of 119866 such that 120575(119867) ⩾ 1 then119887119875(119867) ⩽ 119887

119875(119866)

(b) If 119867 is a subgraph of 119866 such that 119887119875(119867) = 1 and 119896

is the number of edges removed to form119867 then 1 ⩽119887119875(119866) ⩽ 119896 + 1

(c) If 119909119910 isin 119864(119866) such that 119889119866(119909) 119889

119866(119910) gt 1 and 119909119910

belongs to each perfect matching of each minimumpaired dominating set of 119866 then 119887

119875(119866) = 1

Based on these observations 119887119875(119875

119899) ⩽ 2 In fact the

paired bondage number of a path has been determined

Theorem 118 (Raczek [73] 2008) For any positive integer 119896if 119899 ⩾ 2 then

119887119875(119875

119899) =

0 119894119891 119899 = 2 3 119900119903 5

1 119894119891 119899 = 4119896 4119896 + 3 119900119903 4119896 + 6

2 119900119905ℎ119890119903119908119894119904119890

(43)

For a cycle 119862119899of order 119899 ⩾ 3 since 120574

119875(119862

119899) = 120574

119875(119875

119899) from

Theorem 118 the following result holds immediately

Corollary 119 (Raczek [73] 2008) For any positive integer 119896if 119899 ⩾ 3 then

119887119875(119862

119899) =

0 119894119891 119899 = 3 119900119903 5

2 119894119891 119899 = 4119896 4119896 + 3 119900119903 4119896 + 6

3 119900119905ℎ119890119903119908119894119904119890

(44)

A wheel 119882119899 where 119899 ⩾ 4 is a graph with 119899 vertices

formed by connecting a single vertex to all vertices of a cycle119862119899minus1

Of course 120574119875(119882

119899) = 2

International Journal of Combinatorics 19

119896

Figure 7 A tree 119879 with 119887119875(119879) = 119896

Theorem 120 (Raczek [73] 2008) For positive integers 119899 and119898

119887119875(119870

119898119899) = 119898 119891119900119903 1 lt 119898 ⩽ 119899

119887119875(119882

119899) =

4 119894119891 119899 = 4

3 119894119891 119899 = 5

2 119900119905ℎ119890119903119908119894119904119890

(45)

Theorem 16 shows that if 119879 is a nontrivial tree then119887(119879) ⩽ 2 However no similar result exists for pairedbondage For any nonnegative integer 119896 let 119879

119896be a tree

obtained by subdividing all but one edge of a star 1198701119896+1

(asshown in Figure 7) It is easy to see that 119887

119875(119879

119896) = 119896 We

state this fact as the following theorem which is similar toTheorem 112

Theorem121 (Raczek [73] 2008) For any nonnegative integer119896 there exists a tree 119879 with 119887

119875(119879) = 119896

Consider the tree defined in Figure 5 for 119896 = 3 Then119887(119879

3) ⩽ 2 and 119887

119875(119879

3) = 3 On the other hand 119887(119875

4) = 2

and 119887119875(119875

4) = 1 Thus we have what follows which is similar

to Corollary 113

Corollary 122 The bondage number and the paired bondagenumber are unrelated even for trees

A constructive characterization of trees with 119887(119879) = 2is given by Hartnell and Rall in [12] Raczek [73] provideda constructive characterization of trees with 119887

119875(119879) = 0 In

order to state the characterization we define a labeling andthree simple operations on a tree119879 Let 119910 isin 119881(119879) and let ℓ(119910)be the label assigned to 119910

Operation T1 If ℓ(119910) = 119861 add a vertex 119909 and the

edge 119910119909 and let ℓ(119909) = 119860OperationT

2 If ℓ(119910) = 119862 add a path (119909

1 119909

2) and the

edge 1199101199091 and let ℓ(119909

1) = 119861 ℓ(119909

2) = 119860

Operation T3 If ℓ(119910) = 119861 add a path (119909

1 119909

2 119909

3)

and the edge 1199101199091 and let ℓ(119909

1) = 119862 ℓ(119909

2) = 119861 and

ℓ(1199093) = 119860

Let 1198752= (119906 V) with ℓ(119906) = 119860 and ℓ(V) = 119861 Let T be

the class of all trees obtained from the labeled 1198752by a finite

sequence of operationsT1 T

2 T

3

A tree 119879 in Figure 8 belongs to the familyTRaczek [73] obtained the following characterization of all

trees 119879 with 119887119875(119879) = 0

119860

119860

119860

119860

119860

119861 119862

119861 119862 119861

119861119860

Figure 8 A tree 119879 belong to the familyT

Theorem 123 (Raczek [73] 2008) For a tree 119879 119887119875(119879) = 0 if

and only if 119879 is inT

We state the decision problem for the paired bondage asfollows

Problem 4 Consider the decision problem

Paired Bondage ProblemInstance a graph 119866 and a positive integer 119887Question is 119887

119875(119866) ⩽ 119887

Conjecture 124 The paired bondage problem is NP-complete

83 Restrained Bondage Numbers A dominating set 119878 of 119866 iscalled to be restrained if the subgraph induced by 119878 containsno isolated vertices where 119878 = 119881(119866) 119878 The restraineddomination number of 119866 denoted by 120574

119877(119866) is the smallest

cardinality of a restrained dominating set of 119866 It is clear that120574119877(119866) exists and 120574(119866) ⩽ 120574

119877(119866) for any nonempty graph 119866

The concept of restrained domination was introducedby Telle and Proskurowski [74] in 1997 albeit indirectly asa vertex partitioning problem Concerning the study of therestrained domination numbers the reader is referred to [74ndash79]

In 2008 Hattingh and Plummer [80] defined therestrained bondage number 119887R(119866) of a nonempty graph 119866 tobe the minimum cardinality among all subsets 119861 sube 119864(119866) forwhich 120574

119877(119866 minus 119861) gt 120574

119877(119866)

For some simple graphs their restrained dominationnumbers can be easily determined and so restrained bondagenumbers have been also determined For example it is clearthat 120574

119877(119870

119899) = 1 Domke et al [77] showed that 120574

119877(119862

119899) =

119899 minus 2lfloor1198993rfloor for 119899 ⩾ 3 and 120574119877(119875

119899) = 119899 minus 2lfloor(119899 minus 1)3rfloor for 119899 ⩾ 1

Using these results Hattingh and Plummer [80] obtained therestricted bondage numbers for119870

119899 119862

119899 119875

119899 and119866 = 119870

11989911198992119899119905

as follows

Theorem 125 (Hattingh and Plummer [80] 2008) For 119899 ⩾ 3

119887R (119870119899) =

1 119894119891 119899 = 3

lceil119899

2rceil 119900119905ℎ119890119903119908119894119904119890

119887R (119862119899) =

1 119894119891 119899 equiv 0 (mod 3) 2 119900119905ℎ119890119903119908119894119904119890

119887R (119875119899) = 1 119899 ⩾ 4

20 International Journal of Combinatorics

119887R (119866)

=

lceil119898

2rceil 119894119891 119899

119898= 1 119886119899119889 119899

119898+1⩾ 2 (1 ⩽ 119898 lt 119905)

2119905 minus 2 119894119891 1198991= 119899

2= sdot sdot sdot = 119899

119905= 2 (119905 ⩾ 2)

2 119894119891 1198991= 2 119886119899119889 119899

2⩾ 3 (119905 = 2)

119905minus1

sum

119894=1

119899119894minus 1 119900119905ℎ119890119903119908119894119904119890

(46)

Theorem 126 (Hattingh and Plummer [80] 2008) For a tree119879 of order 119899 (⩾ 4) 119879 ≇ 119870

1119899minus1if and only if 119887R(119879) = 1

Theorem 126 shows that the restrained bondage numberof a tree can be computed in constant time However thedecision problem for 119887R(119866) is NP-complete even for bipartitegraphs

Problem 5 Consider the decision problem

Restrained Bondage Problem

Instance a graph 119866 and a positive integer 119887

Question is 119887R(119866) ⩽ 119887

Theorem 127 (Hattingh and Plummer [80] 2008) Therestrained bondage problem is NP-complete even for bipartitegraphs

Consequently in view of its computational hardnessit is significative to establish some sharp bounds on therestrained bondage number of a graph in terms of othergraphic parameters

Theorem 128 (Hattingh and Plummer [80] 2008) For anygraph 119866 with 120575(119866) ⩾ 2

119887R (119866) ⩽ min119909119910isin119864(119866)

119889119866(119909) + 119889

119866(119910) minus 2 (47)

Corollary 129 119887R(119866) ⩽ Δ(119866)+120575(119866)minus2 for any graph119866 with120575(119866) ⩾ 2

Notice that the bounds stated in Theorem 128 andCorollary 129 are sharp Indeed the class of cycles whoseorders are congruent to 1 2 (mod 3) have a restrained bondagenumber achieving these bounds (see Theorem 125)

Theorem 128 is an analogue of Theorem 14 A quitenatural problem is whether or not for restricted bondagenumber there are analogues of Theorem 16 Theorem 18Theorem 29 and so on Indeed we should consider all resultsfor the bondage number whether or not there are analoguesfor restricted bondage number

Theorem 130 (Hattingh and Plummer [80] 2008) For anygraph 119866 with 120574

119877(119866) = 2

119887R (119866) ⩽ Δ (119866) + 1 (48)

We now consider the relation between 119887119877(119866) and 119887(119866)

Theorem 131 (Hattingh and Plummer [80] 2008) If 120574119877(119866) =

120574(119866) for some graph 119866 then 119887R(119866) ⩽ 119887(119866)

Corollary 129 and Theorem 131 provide a possibility toattack Conjecture 73 by characterizing planar graphs with120574119877= 120574 and 119887R = 119887 when 3 ⩽ Δ ⩽ 6However we do not have 119887R(119866) = 119887(119866) for any graph

119866 even if 120574119877(119866) = 120574(119866) Observe that 120574

119877(119870

3) = 120574(119870

3) yet

119887119877(119870

3) = 1 and 119887(119870

3) = 2 We still may not claim that

119887119877(119866) = 119887(119866) even in the case that every 120574-set is a 120574

119877-set

The example 1198703again demonstrates this

Theorem 132 (Hattingh and Plummer [80] 2008) There isan infinite class of graphs inwhich each graph119866 satisfies 119887(119866) lt119887R(119866)

In fact such an infinite class of graphs can be obtained asfollows Let119867 be a connected graph BC(119867) a graph obtainedfrom 119867 by attaching ℓ (⩾ 2) vertices of degree 1 to eachvertex in 119867 and B = 119866 119866 = BC(119867) for some graph 119867such that 120575(119867) ⩾ 2 By Theorem 3 we can easily check that119887(119866) = 1 lt 2 ⩽ min120575(119867) ℓ = 119887R(119866) for any 119866 isin BMoreover there exists a graph 119866 isin B such that 119887R(119866) canbe much larger than 119887(119866) which is stated as the followingtheorem

Theorem 133 (Hattingh and Plummer [80] 2008) For eachpositive integer 119896 there is a graph119866 such that 119896 = 119887R(119866)minus119887(119866)

Combining Theorem 133 with Theorem 132 we have thefollowing corollary

Corollary 134 The bondage number and the restrainedbondage number are unrelated

Very recently Jafari Rad et al [81] have consideredthe total restrained bondage in graphs based on both totaldominating sets and restrained dominating sets and obtainedseveral properties exact values and bounds for the totalrestrained bondage number of a graph

A restrained dominating set 119878 of a graph 119866 without iso-lated vertices is called to be a total restrained dominating setif 119878 is a total dominating set The total restrained dominationnumber of119866 denoted by 120574TR (119866) is theminimum cardinalityof a total restrained dominating set of 119866 For referenceson this domination in graphs see [3 61 82ndash85] The totalrestrained bondage number 119887TR (119866) of a graph 119866 with noisolated vertex is the cardinality of a smallest set of edges 119861 sube119864(119866) for which119866minus119861 has no isolated vertex and 120574TR (119866minus119861) gt120574TR (119866) In the case that there is no such subset 119861 we define119887TR (119866) = infin

Ma et al [84] determined that 120574TR (119875119899) = 119899minus2lfloor(119899minus2)4rfloorfor 119899 ⩾ 3 120574TR (119862119899

) = 119899 minus 2lfloor1198994rfloor for 119899 ⩾ 3 120574TR (1198703) =

3 and 120574TR (119870119899) = 2 for 119899 = 3 and 120574TR (1198701119899

) = 1 + 119899

and 120574TR (119870119898119899) = 2 for 2 ⩽ 119898 ⩽ 119899 According to these

results Jafari Rad et al [81] determined the exact valuesof total restrained bondage numbers for the correspondinggraphs

International Journal of Combinatorics 21

Theorem 135 (Jafari Rad et al [81]) For an integer 119899

119887TR (119875119899) = infin 119894119891 2 ⩽ 119899 ⩽ 5

1 119894119891 119899 ⩾ 5

119887TR (119862119899) =

infin 119894119891 119899 = 3

1 119894119891 3 lt 119899 119899 equiv 0 1 (mod 4)2 119894119891 3 lt 119899 119899 equiv 2 3 (mod 4)

119887TR (119882119899) = 2 119891119900119903 119899 ⩾ 6

119887TR (119870119899) = 119899 minus 1 119891119900119903 119899 ⩾ 4

119887TR (119870119898119899) = 119898 minus 1 119891119900119903 2 ⩽ 119898 ⩽ 119899

(49)

119887TR (119870119899) = 119899minus1 shows the fact that for any integer 119899 ⩾ 3 there

exists a connected graph namely119870119899+1

such that 119887TR (119870119899+1) =

119899 that is the total restrained bondage number is unboundedfrom the above

A vertex is said to be a support vertex if it is adjacent toa vertex of degree one Let A be the family of all connectedgraphs such that119866 belongs toA if and only if every edge of119866is incident to a support vertex or 119866 is a cycle of length three

Proposition 136 (Cyman and Raczek [82] 2006) For aconnected graph119866 of order 119899 120574TR (119866) = 119899 if and only if119866 isin A

Hence if 119866 isin A then the removal of any subset of edgesof119866 cannot increase the total restrained domination numberof 119866 and thus 119887TR (119866) = infin When 119866 notin A a result similar toTheorem 6 is obtained

Theorem 137 (Jafari Rad et al [81]) For any tree 119879 notin A119887TR (119879) ⩽ 2 This bound is sharp

In the samepaper the authors provided a characterizationfor trees 119879 notin A with 119887TR (119879) = 1 or 2 The following result issimilar to Observation 1

Observation 3 (Jafari Rad et al [81]) If 119896 edges can beremoved from a graph 119866 to obtain a graph 119867 without anisolated vertex and with 119887TR (119867) = 119905 then 119887TR (119866) ⩽ 119896 + 119905

ApplyingTheorem 137 andObservation 3 Jafari Rad et alcharacterized all connected graphs 119866 with 119887TR (119866) = infin

Theorem 138 (Jafari Rad et al [81]) For a connected graph119866119887TR (119866) = infin if and only if 119866 isin A

84 Other Conditional Bondage Numbers A subset 119868 sube 119881(119866)is called an independent set if no two vertices in 119868 are adjacentin 119866 The maximum cardinality among all independent setsis called the independence number of 119866 denoted by 120572(119866)

A dominating set 119878 of a graph 119866 is called to be inde-pendent if 119878 is an independent set of 119866 The minimumcardinality among all independent dominating set is calledthe independence domination number of 119866 and denoted by120574119868(119866)

Since an independent dominating set is not only adominating set but also an independent set 120574(119866) ⩽ 120574

119868(119866) ⩽

120572(119866) for any graph 119866It is clear that a maximal independent set is certainly a

dominating set Thus an independent set is maximal if andonly if it is an independent dominating set and so 120574

119868(119866) is the

minimum cardinality among all maximal independent setsof 119866 This graph-theoretical invariant has been well studiedin the literature see for example Haynes et al [3] for earlyresults Goddard and Henning [86] for recent results onindependent domination in graphs

In 2003 Zhang et al [87] defined the independencebondage number 119887

119868(119866) of a nonempty graph 119866 to be the

minimum cardinality among all subsets 119861 sube 119864(119866) for which120574119868(119866 minus 119861) gt 120574

119868(119866) For some ordinary graphs their indepen-

dence domination numbers can be easily determined and soindependence bondage numbers have been also determinedClearly 119887

119868(119870

119899) = 1 if 119899 ⩾ 2

Theorem 139 (Zhang et al [87] 2003) For any integer 119899 ⩾ 4

119887119868(119862

119899) =

1 119894119891 119899 equiv 1 (mod 2) 2 119894119891 119899 equiv 0 (mod 2)

119887119868(119875

119899) =

1 119894119891 119899 equiv 0 (mod 2) 2 119894119891 119899 equiv 1 (mod 2)

119887119868(119870

119898119899) = 119899 119891119900119903 119898 ⩽ 119899

(50)

Apart from the above-mentioned results as far as we findno other results on the independence bondage number in thepresent literature we never knew much of any result on thisparameter for a tree

A subset 119878 of 119881(119866) is called an equitable dominatingset if for every 119910 isin 119881(119866 minus 119878) there exists a vertex 119909 isin 119878such that 119909119910 isin 119864(119866) and |119889

119866(119909) minus 119889

119866(119910)| ⩽ 1 proposed

by Dharmalingam [88] The minimum cardinality of such adominating set is called the equitable domination number andis denoted by 120574

119864(119866) Deepak et al [89] defined the equitable

bondage number 119887119864(119866) of a graph 119866 to be the cardinality of a

smallest set 119861 sube 119864(119866) of edges for which 120574119864(119866 minus 119861) gt 120574

119864(119866)

and determined 119887119864(119866) for some special graphs

Theorem 140 (Deepak et al [89] 2011) For any integer 119899 ⩾ 2

119887119864(119870

119899) = lceil

119899

2rceil

119887119864(119870

119898119899) = 119898 119891119900119903 0 ⩽ 119899 minus 119898 ⩽ 1

119887119864(119862

119899) =

3 119894119891 119899 equiv 1 (mod 3)2 119900119905ℎ119890119903119908119894119904119890

119887119864(119875

119899) =

2 119894119891 119899 equiv 1 (mod 3)1 119900119905ℎ119890119903119908119894119904119890

119887119864(119879) ⩽ 2 119891119900119903 119886119899119910 119899119900119899119905119903119894119907119894119886119897 119905119903119890119890 119879

119887119864(119866) ⩽ 119899 minus 1 119891119900119903 119886119899119910 119888119900119899119899119890119888119905119890119889 119892119903119886119901ℎ 119866 119900119891 119900119903119889119890119903 119899

(51)

22 International Journal of Combinatorics

As far as we know there are no more results on theequitable bondage number of graphs in the present literature

There are other variations of the normal domination byadding restricted conditions to the normal dominating setFor example the connected domination set of a graph119866 pro-posed by Sampathkumar and Walikar [90] is a dominatingset 119878 such that the subgraph induced by 119878 is connected Theproblem of finding a minimum connected domination set ina graph is equivalent to the problemof finding a spanning treewithmaximumnumber of vertices of degree one [91] and hassome important applications in wireless networks [92] thusit has received much research attention However for suchan important domination there is no result on its bondagenumber We believe that the topic is of significance for futureresearch

9 Generalized Bondage Numbers

There are various generalizations of the classical dominationsuch as 119901-domination distance domination fractional dom-ination Roman domination and rainbow domination Everysuch a generalization can lead to a corresponding bondageIn this section we introduce some of them

91 119901-Bondage Numbers In 1985 Fink and Jacobson [93]introduced the concept of 119901-domination Let 119901 be a positiveinteger A subset 119878 of 119881(119866) is a 119901-dominating set of 119866 if|119878 cap 119873

119866(119910)| ⩾ 119901 for every 119910 isin 119878 The 119901-domination number

120574119901(119866) is the minimum cardinality among all 119901-dominating

sets of 119866 Any 119901-dominating set of 119866 with cardinality 120574119901(119866)

is called a 120574119901-set of 119866 Note that a 120574

1-set is a classic minimum

dominating set Notice that every graph has a 119901-dominatingset since the vertex-set119881(119866) is such a setWe also note that a 1-dominating set is a dominating set and so 120574(119866) = 120574

1(119866) The

119901-domination number has received much research attentionsee a state-of-the-art survey article by Chellali et al [94]

It is clear from definition that every 119901-dominating setof a graph certainly contains all vertices of degree at most119901 minus 1 By this simple observation to avoid the trivial caseoccurrence we always assume that Δ(119866) ⩾ 119901 For 119901 ⩾ 2 Luet al [95] gave a constructive characterization of trees withunique minimum 119901-dominating sets

Recently Lu and Xu [96] have introduced the conceptto the 119901-bondage number of 119866 denoted by 119887

119901(119866) as the

minimum cardinality among all sets of edges 119861 sube 119864(119866) suchthat 120574

119901(119866 minus 119861) gt 120574

119901(119866) Clearly 119887

1(119866) = 119887(119866)

Lu and Xu [96] established a lower bound and an upperbound on 119887

119901(119879) for any integer 119901 ⩾ 2 and any tree 119879 with

Δ(119879) ⩾ 119901 and characterized all trees achieving the lowerbound and the upper bound respectively

Theorem 141 (Lu and Xu [96] 2011) For any integer 119901 ⩾ 2and any tree 119879 with Δ(119879) ⩾ 119901

1 ⩽ 119887119901(119879) ⩽ Δ (119879) minus 119901 + 1 (52)

Let 119878 be a given subset of 119881(119866) and for any 119909 isin 119878 let

119873119901(119909 119878 119866) = 119910 isin 119878 cap 119873

119866(119909)

1003816100381610038161003816119873119866(119910) cap 119878

1003816100381610038161003816 = 119901

(53)

Then all trees achieving the lower bound 1 can be character-ized as follows

Theorem 142 (Lu and Xu [96] 2011) Let 119879 be a tree withΔ(119879) ⩾ 119901 ⩾ 2 Then 119887

119901(119879) = 1 if and only if for any 120574

119901-set

119878 of 119879 there exists an edge 119909119910 isin (119878 119878) such that 119910 isin 119873119901(119909 119878 119879)

The notation 119878(119886 119887) denotes a double star obtained byadding an edge between the central vertices of two stars119870

1119886minus1

and1198701119887minus1

And the vertex with degree 119886 (resp 119887) in 119878(119886 119887) iscalled the 119871-central vertex (resp 119877-central vertex) of 119878(119886 119887)

To characterize all trees attaining the upper bound givenin Theorem 141 we define three types of operations on a tree119879 with Δ(119879) = Δ ⩾ 119901 + 1

Type 1 attach a pendant edge to a vertex 119910with 119889119879(119910) ⩽ 119901minus

2 in 119879

Type 2 attach a star 1198701Δminus1

to a vertex 119910 of 119879 by joining itscentral vertex to 119910 where 119910 in a 120574

119901-set of 119879 and

119889119879(119910) ⩽ Δ minus 1

Type 3 attach a double star 119878(119901 Δ minus 1) to a pendant vertex 119910of 119879 by coinciding its 119877-central vertex with 119910 wherethe unique neighbor of 119910 is in a 120574

119901-set of 119879

Let B = 119879 a tree obtained from 1198701Δ

or 119878(119901 Δ) by afinite sequence of operations of Types 1 2 and 3

Theorem 143 (Lu and Xu [96] 2011) A tree with the maxi-mum Δ ⩾ 119901 + 1 has 119901-bondage number Δ minus 119901 + 1 if and onlyif it belongs toB

Theorem 144 (Lu and Xu [97] 2012) Let 119866119898119899= 119875

119898times 119875

119899

Then

1198872(119866

2119899) = 1 119891119900119903 119899 ⩾ 2

1198872(119866

3119899) =

2 119894119891 119899 equiv 1 (mod 3) 1 119900119905ℎ119890119903119908119894119904119890

for 119899 ⩾ 2

1198872(119866

4119899) =

1 119894119891 119899 equiv 3 (mod 4) 2 119900119905ℎ119890119903119908119894119904119890

for 119899 ⩾ 7

(54)

Recently Krzywkowski [98] has proposed the conceptof the nonisolating 2-bondage of a graph The nonisolating2-bondage number of a graph 119866 denoted by 1198871015840

2(119866) is the

minimum cardinality among all sets of edges 119861 sube 119864(119866) suchthat 119866 minus 119861 has no isolated vertices and 120574

2(119866 minus 119861) gt 120574

2(119866) If

for every 119861 sube 119864(119866) either 1205742(119866 minus 119861) = 120574

2(119866) or 119866 minus 119861 has

isolated vertices then define 11988710158402(119866) = 0 and say that 119866 is a 2-

nonisolatedly strongly stable graph Krzywkowski presentedsome basic properties of nonisolating 2-bondage in graphsshowed that for every nonnegative integer 119896 there exists atree 119879 such 1198871015840

2(119879) = 119896 characterized all 2-non-isolatedly

strongly stable trees and determined 11988710158402(119866) for several special

graphs

International Journal of Combinatorics 23

Theorem 145 (Krzywkowski [98] 2012) For any positiveintegers 119899 and119898 with119898 ⩽ 119899

1198871015840

2(119870

119899) =

0 119894119891 119899 = 1 2 3

lfloor2119899

3rfloor 119900119905ℎ119890119903119908119894119904119890

1198871015840

2(119875

119899) =

0 119894119891 119899 = 1 2 3

1 119900119905ℎ119890119903119908119894119904119890

1198871015840

2(119862

119899) =

0 119894119891 119899 = 3

1 119894119891 119899 119894119904 119890V119890119899

2 119894119891 119899 119894119904 119900119889119889

1198871015840

2(119870

119898119899) =

3 119894119891 119898 = 119899 = 3

1 119894119891 119898 = 119899 = 4

2 119900119905ℎ119890119903119908119894119904119890

(55)

92 Distance Bondage Numbers A subset 119878 of vertices of agraph119866 is said to be a distance 119896-dominating set for119866 if everyvertex in 119866 not in 119878 is at distance at most 119896 from some vertexof 119878 The minimum cardinality of all distance 119896-dominatingsets is called the distance 119896-domination number of 119866 anddenoted by 120574

119896(119866) (do not confuse with the above-mentioned

120574119901(119866)) When 119896 = 1 a distance 1-dominating set is a normal

dominating set and so 1205741(119866) = 120574(119866) for any graph 119866 Thus

the distance 119896-domination is a generalization of the classicaldomination

The relation between 120574119896and 120572

119896(the 119896-independence

number defined in Section 22) for a tree obtained by Meirand Moon [14] who proved that 120574

119896(119879) = 120572

2119896(119879) for any tree

119879 (a special case for 119896 = 1 is stated in Proposition 10) Thefurther research results can be found in Henning et al [99]Tian and Xu [100ndash103] and Liu et al [104]

In 1998 Hartnell et al [15] defined the distance 119896-bondagenumber of 119866 denoted by 119887

119896(119866) to be the cardinality of a

smallest subset 119861 of edges of 119866 with the property that 120574119896(119866 minus

119861) gt 120574119896(119866) FromTheorem 6 it is clear that if119879 is a nontrivial

tree then 1 ⩽ 1198871(119879) ⩽ 2 Hartnell et al [15] generalized this

result to any integer 119896 ⩾ 1

Theorem 146 (Hartnell et al [15] 1998) For every nontrivialtree 119879 and positive integer 119896 1 ⩽ 119887

119896(119879) ⩽ 2

Hartnell et al [15] and Topp and Vestergaard [13] alsocharacterized the trees having distance 119896-bondage number 2In particular the class of trees for which 119887

1(119879) = 2 are just

those which have a unique maximum 2-independent set (seeTheorem 11)

Since when 119896 = 1 the distance 1-bondage number 1198871(119866)

is the classical bondage number 119887(119866) Theorem 12 gives theNP-hardness of deciding the distance 119896-bondage number ofgeneral graphs

Theorem 147 Given a nonempty undirected graph 119866 andpositive integers 119896 and 119887 with 119887 ⩽ 120576(119866) determining wetheror not 119887

119896(119866) ⩽ 119887 is NP-hard

For a vertex 119909 in119866 the open 119896-neighborhood119873119896(119909) of 119909 is

defined as119873119896(119909) = 119910 isin 119881(119866) 1 ⩽ 119889

119866(119909 119910) le 119896The closed

119896-neighborhood119873119896[119909] of 119909 in 119866 is defined as119873

119896(119909) cup 119909 Let

Δ119896(119866) = max 1003816100381610038161003816119873119896

(119909)1003816100381610038161003816 for any 119909 isin 119881 (119866) (56)

Clearly Δ1(119866) = Δ(119866) The 119896th power of a graph 119866 is the

graph119866119896 with vertex-set119881(119866119896

) = 119881(119866) and edge-set119864(119866119896

) =

119909119910 1 ⩽ 119889119866(119909 119910) ⩽ 119896 The following lemma holds directly

from the definition of 119866119896

Proposition 148 (Tian and Xu [101]) Δ(119866119896

) = Δ119896(119866) and

120574(119866119896

) = 120574119896(119866) for any graph 119866 and each 119896 ⩾ 1

A graph 119866 is 119896-distance domination-critical or 120574119896-critical

for short if 120574119896(119866 minus 119909) lt 120574

119896(119866) for every vertex 119909 in 119866

proposed by Henning et al [105]

Proposition 149 (Tian and Xu [101]) For each 119896 ⩾ 1 a graph119866 is 120574

119896-critical if and only if 119866119896 is 120574

119896-critical

By the above facts we suggest the following worthwhileproblem

Problem 6 Can we generalize the results on the bondage for119866 to 119866119896 In particular do the following propositions hold

(a) 119887(119866119896

) = 119887119896(119866) for any graph 119866 and each 119896 ⩾ 1

(b) 119887(119866119896

) ⩽ Δ119896(119866) if 119866 is not 120574

119896-critical

Let 119896 and 119901 be positive integers A subset 119878 of 119881(119866) isdefined to be a (119896 119901)-dominating set of 119866 if for any vertex119909 isin 119878 |119873

119896(119909) cap 119878| ⩾ 119901 The (119896 119901)-domination number of

119866 denoted by 120574119896119901(119866) is the minimum cardinality among

all (119896 119901)-dominating sets of 119866 Clearly for a graph 119866 a(1 1)-dominating set is a classical dominating set a (119896 1)-dominating set is a distance 119896-dominating set and a (1 119901)-dominating set is the above-mentioned 119901-dominating setThat is 120574

11(119866) = 120574(119866) 120574

1198961(119866) = 120574

119896(119866) and 120574

1119901(119866) = 120574

119901(119866)

The concept of (119896 119901)-domination in a graph 119866 is a gen-eralized domination which combines distance 119896-dominationand 119901-domination in 119866 So the investigation of (119896 119901)-domination of 119866 is more interesting and has received theattention of many researchers see for example [106ndash109]

More general Li and Xu [110] proposed the concept of the(ℓ 119908)-domination number of a119908-connected graph A subset119878 of 119881(119866) is defined to be an (ℓ 119908)-dominating set of 119866 iffor any vertex 119909 isin 119878 there are 119908 internally disjoint pathsof length at most ℓ from 119909 to some vertex in 119878 The (ℓ 119908)-domination number of119866 denoted by 120574

ℓ119908(119866) is theminimum

cardinality among all (ℓ 119908)-dominating sets of 119866 Clearly1205741198961(119866) = 120574

119896(119866) and 120574

1119901(119866) = 120574

119901(119866)

It is quite natural to propose the concept of bondage num-ber for (119896 119901)-domination or (ℓ 119908)-domination However asfar as we know none has proposed this concept until todayThis is a worthwhile topic for further research

24 International Journal of Combinatorics

93 Fractional Bondage Numbers If 120590 is a function mappingthe vertex-set 119881 into some set of real numbers then for anysubset 119878 sube 119881 let 120590(119878) = sum

119909isin119878120590(119909) Also let |120590| = 120590(119881)

A real-value function 120590 119881 rarr [0 1] is a dominatingfunction of a graph 119866 if for every 119909 isin 119881(119866) 120590(119873

119866[119909]) ⩾ 1

Thus if 119878 is a dominating set of 119866 120590 is a function where

120590 (119909) = 1 if 119909 isin 1198780 otherwise

(57)

then 120590 is a dominating function of119866 The fractional domina-tion number of 119866 denoted by 120574

119865(119866) is defined as follows

120574119865(119866) = min |120590| 120590 is a domination function of 119866

(58)

The 120574119865-bondage number of 119866 denoted by 119887

119865(119866) is

defined as the minimum cardinality of a subset 119861 sube 119864 whoseremoval results in 120574

119865(119866 minus 119861) gt 120574

119865(119866)

Hedetniemi et al [111] are the first to study fractionaldomination although Farber [112] introduced the idea indi-rectly The concept of the fractional domination numberwas proposed by Domke and Laskar [113] in 1997 Thefractional domination numbers for some ordinary graphs aredetermined

Proposition 150 (Domke et al [114] 1998 Domke and Laskar[113] 1997) (a) If 119866 is a 119896-regular graph with order 119899 then120574119865(119866) = 119899119896 + 1(b) 120574

119865(119862

119899) = 1198993 120574

119865(119875

119899) = lceil1198993rceil 120574

119865(119870

119899) = 1

(c) 120574119865(119866) = 1 if and only if Δ(119866) = 119899 minus 1

(d) 120574119865(119879) = 120574(119879) for any tree 119879

(e) 120574119865(119870

119899119898) = (119899(119898 + 1) + 119898(119899 + 1))(119899119898 minus 1) where

max119899119898 gt 1

The assertions (a)ndash(d) are due to Domke et al [114] andthe assertion (e) is due to Domke and Laskar [113] Accordingto these results Domke and Laskar [113] determined the 120574

119865-

bondage numbers for these graphs

Theorem 151 (Domke and Laskar [113] 1997) 119887119865(119870

119899) =

lceil1198992rceil 119887119865(119870

119899119898) = 1 wheremax119899119898 gt 1

119887119865(119862

119899) =

2 119894119891 119899 equiv 0 (mod 3) 1 119900119905ℎ119890119903119908119894119904119890

119887119865(119875

119899) =

2 119894119891 119899 equiv 1 (mod 3) 1 119900119905ℎ119890119903119908119894119904119890

(59)

It is easy to see that for any tree119879 119887119865(119879) ⩽ 2 In fact since

120574119865(119879) = 120574(119879) for any tree 119879 120574

119865(119879

1015840

) = 120574(1198791015840

) for any subgraph1198791015840 of 119879 It follows that

119887119865(119879) = min 119861 sube 119864 (119879) 120574

119865(119879 minus 119861) gt 120574

119865(119879)

= min 119861 sube 119864 (119879) 120574 (119879 minus 119861) gt 120574 (119879)

= 119887 (119879) ⩽ 2

(60)

Except for these no results on 119887119865(119866) have been known as

yet

94 Roman Bondage Numbers A Roman dominating func-tion on a graph 119866 is a labeling 119891 119881 rarr 0 1 2 such thatevery vertex with label 0 has at least one neighbor with label2 The weight of a Roman dominating function 119891 on 119866 is thevalue

119891 (119866) = sum

119906isin119881(119866)

119891 (119906) (61)

The minimum weight of a Roman dominating function ona graph 119866 is called the Roman domination number of 119866denoted by 120574RM (119866)

A Roman dominating function 119891 119881 rarr 0 1 2

can be represented by the ordered partition (1198810 119881

1 119881

2) (or

(119881119891

0 119881

119891

1 119881

119891

2) to refer to119891) of119881 where119881

119894= V isin 119881 | 119891(V) = 119894

In this representation its weight 119891(119866) = |1198811| + 2|119881

2| It is

clear that119881119891

1cup119881

119891

2is a dominating set of 119866 called the Roman

dominating set denoted by 119863119891

R = (1198811 1198812) Since 119881119891

1cup 119881

119891

2is

a dominating set when 119891 is a Roman dominating functionand since placing weight 2 at the vertices of a dominating setyields a Roman dominating function in [115] it is observedthat

120574 (119866) ⩽ 120574RM (119866) ⩽ 2120574 (119866) (62)

A graph 119866 is called to be Roman if 120574RM (119866) = 2120574(119866)The definition of the Roman domination function is

proposed implicitly by Stewart [116] and ReVelle and Rosing[117] Roman dominating numbers have been deeply studiedIn particular Bahremandpour et al [118] showed that theproblem determining the Roman domination number is NP-complete even for bipartite graphs

Let 119866 be a graph with maximum degree at least twoThe Roman bondage number 119887RM (119866) of 119866 is the minimumcardinality of all sets 1198641015840 sube 119864 for which 120574RM (119866 minus 119864

1015840

) gt

120574RM (119866) Since in study of Roman bondage number theassumption Δ(119866) ⩾ 2 is necessary we always assume thatwhen we discuss 119887RM (119866) all graphs involved satisfy Δ(119866) ⩾2 The Roman bondage number 119887RM (119866) is introduced byJafari Rad and Volkmann in [119]

Recently Bahremandpour et al [118] have shown that theproblem determining the Roman bondage number is NP-hard even for bipartite graphs

Problem 7 Consider the decision problem

Roman Bondage Problem

Instance a nonempty bipartite graph119866 and a positiveinteger 119896

Question is 119887RM (119866) ⩽ 119896

Theorem 152 (Bahremandpour et al [118] 2013) TheRomanbondage problem is NP-hard even for bipartite graphs

The exact value of 119887RM(119866) is known only for a few familyof graphs including the complete graphs cycles and paths

International Journal of Combinatorics 25

Theorem 153 (Jafari Rad and Volkmann [119] 2011) For anypositive integers 119899 and119898 with119898 ⩽ 119899

119887RM (119875119899) = 2 119894119891 119899 equiv 2 (mod 3) 1 119900119905ℎ119890119903119908119894119904119890

119887RM (119862119899) =

3 119894119891 119899 = 2 (mod 3) 2 119900119905ℎ119890119903119908119894119904119890

119887RM (119870119899) = lceil

119899

2rceil 119891119900119903 119899 ⩾ 3

119887RM (119870119898119899) =

1 119894119891 119898 = 1 119886119899119889 119899 = 1

4 119894119891 119898 = 119899 = 3

119898 119900119905ℎ119890119903119908119894119904119890

(63)

Lemma 154 (Cockayne et al [115] 2004) If 119866 is a graph oforder 119899 and contains vertices of degree 119899 minus 1 then 120574RM(119866) = 2

Using Lemma 154 the third conclusion in Theorem 153can be generalized to more general case which is similar toLemma 49

Theorem 155 (Jafari Rad and Volkmann [119] 2011) Let119866 bea graph with order 119899 ge 3 and 119905 the number of vertices of degree119899 minus 1 in 119866 If 119905 ⩾ 1 then 119887RM(119866) = lceil1199052rceil

Ebadi and PushpaLatha [120] conjectured that 119887RM(119866) ⩽119899 minus 1 for any graph 119866 of order 119899 ⩾ 3 Dehgardi et al[121] Akbari andQajar [122] independently showed that thisconjecture is true

Theorem 156 119887RM(119866) ⩽ 119899 minus 1 for any connected graph 119866 oforder 119899 ⩾ 3

For a complete 119905-partite graph we have the followingresult

Theorem 157 (Hu and Xu [123] 2011) Let 119866 = 11987011989811198982119898119905

bea complete 119905-partite graph with 119898

1= sdot sdot sdot = 119898

119894lt 119898

119894+1le sdot sdot sdot ⩽

119898119905 119905 ⩾ 2 and 119899 = sum119905

119895=1119898

119895 Then

119887RM (119866) =

lceil119894

2rceil 119894119891 119898

119894= 1 119886119899119889 119899 ⩾ 3

2 119894119891 119898119894= 2 119886119899119889 119894 = 1

119894 119894119891 119898119894= 2 119886119899119889 119894 ⩾ 2

4 119894119891 119898119894= 3 119886119899119889 119894 = 119905 = 2

119899 minus 1 119894119891 119898119894= 3 119886119899119889 119894 = 119905 ⩾ 3

119899 minus 119898119905

119894119891 119898119894⩾ 3 119886119899119889 119898

119905⩾ 4

(64)

Consider a complete 119905-partite graph 119870119905(3) for 119905 ⩾ 3

119887RM(119870119905(3)) = 119899 minus 1 by Theorem 157 where 119899 = 3119905

which shows that this upper bound of 119899 minus 1 on 119887RM(119866) inTheorem 156 is sharp This fact and 120574

119877(119870

119905(3)) = 4 give

a negative answer to the following two problems posed byDehgardi et al [121]Prove or Disprove for any connected graph 119866 of order 119899 ge 3(i) 119887RM(119866) = 119899 minus 1 if and only if 119866 cong 119870

3 (ii) 119887RM(119866) ⩽ 119899 minus

120574119877(119866) + 1Note that a complete 119905-partite graph 119870

119905(3) is (119899 minus 3)-

regular and 119887RM(119870119905(3)) = 119899 minus 1 for 119905 ⩾ 3 where 119899 = 3119905

Hu and Xu further determined that 119887119877(119866) = 119899 minus 2 for any

(119899 minus 3)-regular graph 119866 of order 119899 ⩽ 5 except 119870119905(3)

Theorem 158 (Hu and Xu [123] 2011) For any (119899minus3)-regulargraph 119866 of order 119899 other than119870

119905(3) 119887RM(119866) = 119899 minus 2 for 119899 ⩾ 5

For a tree 119879 with order 119899 ⩾ 3 Ebadi and PushpaLatha[120] and Jafari Rad and Volkmann [119] independentlyobtained an upper bound on 119887RM(119879)

Theorem 159 119887RM(119879) ⩽ 3 for any tree 119879 with order 119899 ⩾ 3

Theorem 160 (Bahremandpour et al [118] 2013) 119887RM(1198752 times119875119899) = 2 for 119899 ⩾ 2

Theorem 161 (Jafari Rad and Volkmann [119] 2011) Let119866 bea graph with order at least three

(a) If (119909 119910 119911) is a path of length 2 in 119866 then 119887RM(119866) ⩽119889119866(119909) + 119889

119866(119910) + 119889

119866(119911) minus 3 minus |119873

119866(119909) cap 119873

119866(119910)|

(b) If 119866 is connected then 119887RM(119866) ⩽ 120582(119866) + 2Δ(119866) minus 3

Theorem 161 (b) implies 119887RM(119866) ⩽ 120575(119866) + 2Δ(119866) minus 3 for aconnected graph 119866 Note that for a planar graph 119866 120575(119866) ⩽ 5moreover 120575(119866) ⩽ 3 if the girth at least 4 and 120575(119866) ⩽ 2 if thegirth at least 6 These two facts show that 119887RM(119866) ⩽ 2Δ(119866) +2 for a connected planar graph 119866 Jafari Rad and Volkmann[124] improved this bound

Theorem 162 (Jafari Rad and Volkmann [124] 2011) For anyconnected planar graph 119866 of order 119899 ⩾ 3 with girth 119892(119866)

119887RM (119866)

⩽

2Δ (119866)

Δ (119866) + 6

Δ (119866) + 5 119894119891 119866 119888119900119899119905119886119894119899119904 119899119900 V119890119903119905119894119888119890119904 119900119891 119889119890119892119903119890119890 119891119894V119890

Δ (119866) + 4 119894119891 119892 (119866) ⩾ 4

Δ (119866) + 3 119894119891 119892 (119866) ⩾ 5

Δ (119866) + 2 119894119891 119892 (119866) ⩾ 6

Δ (119866) + 1 119894119891 119892 (119866) ⩾ 8

(65)

According toTheorem 157 119887RM(119862119899) = 3 = Δ(119862

119899)+1 for a

cycle 119862119899of length 119899 ⩾ 8 with 119899 equiv 2 (mod 3) and therefore

the last upper bound inTheorem 162 is sharp at least for Δ =2

Combining the fact that every planar graph 119866 withminimum degree 5 contains an edge 119909119910 with 119889

119866(119909) = 5

and 119889119866(119910) isin 5 6 with Theorem 161 (a) Akbari et al [125]

obtained the following result

26 International Journal of Combinatorics

middot middot middot

middot middot middot

middot middot middot

middot middot middot

119866

Figure 9 The graph 119866 is constructed from 119866

Theorem 163 (Akbari et al [125] 2012) 119887RM(119866) ⩽ 15 forevery planar graph 119866

It remains open to show whether the bound inTheorem 163 is sharp or not Though finding a planargraph 119866 with 119887RM(119866) = 15 seems to be difficult Akbari etal [125] constructed an infinite family of planar graphs withRoman bondage number equal to 7 by proving the followingresult

Theorem 164 (Akbari et al [125] 2012) Let 119866 be a graph oforder 119899 and 119866 is the graph of order 5119899 obtained from 119866 byattaching the central vertex of a copy of a path 119875

5to each vertex

of119866 (see Figure 9) Then 120574RM(119866) = 4119899 and 119887RM(119866) = 120575(119866)+2

ByTheorem 164 infinitemany planar graphswith Romanbondage number 7 can be obtained by considering any planargraph 119866 with 120575(119866) = 5 (eg the icosahedron graph)

Conjecture 165 (Akbari et al [125] 2012) The Romanbondage number of every planar graph is at most 7

For general bounds the following observation is directlyobtained which is similar to Observation 1

Observation 4 Let 119866 be a graph of order 119899 with maximumdegree at least two Assume that 119867 is a spanning subgraphobtained from 119866 by removing 119896 edges If 120574RM(119867) = 120574RM(119866)then 119887

119877(119867) ⩽ 119887RM(119866) ⩽ 119887RM(119867) + 119896

Theorem 166 (Bahremandpour et al [118] 2013) For anyconnected graph 119866 of order 119899 if 119899 ⩾ 3 and 120574RM(119866) = 120574(119866) + 1then

119887RM (119866) ⩽ min 119887 (119866) 119899Δ (66)

where 119899Δis the number of vertices with maximum degree Δ in

119866

Observation 5 For any graph 119866 if 120574(119866) = 120574RM(119866) then119887RM(119866) ⩽ 119887(119866)

Theorem 167 (Bahremandpour et al [118] 2013) For everyRoman graph 119866

119887RM (119866) ⩾ 119887 (119866) (67)

The bound is sharp for cycles on 119899 vertices where 119899 equiv 0 (mod3)

The strict inequality in Theorem 167 can hold for exam-ple 119887(119862

3119896+2) = 2 lt 3 = 119887RM(1198623119896+2

) byTheorem 157A graph 119866 is called to be vertex Roman domination-

critical if 120574RM(119866 minus 119909) lt 120574RM(119866) for every vertex 119909 in 119866 If119866 has no isolated vertices then 120574RM(119866) ⩽ 2120574(119866) ⩽ 2120573(119866)If 120574RM(119866) = 2120573(119866) then 120574RM(119866) = 2120574(119866) and hence 119866 is aRoman graph In [30] Volkmann gave a lot of graphs with120574(119866) = 120573(119866) The following result is similar to Theorem 57

Theorem 168 (Bahremandpour et al [118] 2013) For anygraph 119866 if 120574RM(119866) = 2120573(119866) then

(a) 119887RM(119866) ⩾ 120575(119866)

(b) 119887RM(119866) ⩾ 120575(119866)+1 if119866 is a vertex Roman domination-critical graph

If 120574RM(119866) = 2 then obviously 119887RM(119866) ⩽ 120575(119866) So weassume 120574RM(119866) ⩾ 3 Dehgardi et al [121] proved 119887RM(119866) ⩽(120574RM(119866) minus 2)Δ(119866) + 1 and posed the following problem if 119866is a connected graph of order 119899 ⩾ 4 with 120574RM(119866) ⩾ 3 then

119887RM (119866) ⩽ (120574RM (119866) minus 2) Δ (119866) (68)

Theorem 161 (a) shows that the inequality (68) holds if120574RM(119866) ⩾ 5 Thus the bound in (68) is of interest only when120574RM(119866) is 3 or 4

Lemma 169 (Bahremandpour et al [118] 2013) For anynonempty graph 119866 of order 119899 ⩾ 3 120574RM(119866) = 3 if and onlyif Δ(119866) = 119899 minus 2

The following result shows that (68) holds for all graphs119866 of order 119899 ⩾ 4 with 120574RM(119866) = 3 or 4 which improvesTheorem 161 (b)

Theorem 170 (Bahremandpour et al [118] 2013) For anyconnected graph 119866 of order 119899 ⩾ 4

119887RM (119866) le Δ (119866) = 119899 minus 2 119894119891 120574RM (119866) = 3

Δ (119866) + 120575 (119866) minus 1 119894119891 120574RM (119866) = 4(69)

with the first equality if and only if 119866 cong 1198624

Recently Akbari and Qajar [122] have proved the follow-ing result

Theorem 171 (Akbari and Qajar [122]) For any connectedgraph 119866 of order 119899 ⩾ 3

119887RM (119866) ⩽ 119899 minus 120574RM (119866) + 5 (70)

International Journal of Combinatorics 27

We conclude this section with the following problems

Problem 8 Characterize all connected graphs 119866 of order 119899 ⩾3 for which 119887RM(119866) = 119899 minus 1

Problem 9 Prove or disprove if 119866 is a connected graph oforder 119899 ⩾ 3 then

119887RM (119866) ⩽ 119899 minus 120574RM (119866) + 3 (71)

Note that 120574119877(119870

119905(3)) = 4 and 119887RM(119870119905

(3)) = 119899minus1 where 119899 =3119905 for 119905 ⩾ 3 The upper bound on 119887RM(119866) given in Problem 9is sharp if Problem 9 has positive answer

95 Rainbow Bondage Numbers For a positive integer 119896 a119896-rainbow dominating function of a graph 119866 is a function 119891from the vertex-set 119881(119866) to the set of all subsets of the set1 2 119896 such that for any vertex 119909 in 119866 with 119891(119909) = 0 thecondition

⋃

119910isin119873119866(119909)

119891 (119910) = 1 2 119896 (72)

is fulfilledThe weight of a 119896-rainbow dominating function 119891on 119866 is the value

119891 (119866) = sum

119909isin119881(119866)

1003816100381610038161003816119891 (119909)1003816100381610038161003816 (73)

The 119896-rainbow domination number of a graph 119866 denoted by120574119903119896(119866) is the minimum weight of a 119896-rainbow dominating

function 119891 of 119866Note that 120574

1199031(119866) is the classical domination number 120574(119866)

The 119896-rainbow domination number is introduced by Bresaret al [126] Rainbow domination of a graph 119866 coincides withordinary domination of the Cartesian product of 119866 with thecomplete graph in particular 120574

119903119896(119866) = 120574(119866 times 119870

119896) for any

positive integer 119896 and any graph 119866 [126] Hartnell and Rall[127] proved that 120574(119879) lt 120574

1199032(119879) for any tree 119879 no graph has

120574 = 120574119903119896for 119896 ⩾ 3 for any 119896 ⩾ 2 and any graph 119866 of order 119899

min 119899 120574 (119866) + 119896 minus 2 ⩽ 120574119903119896(119866) ⩽ 119896120574 (119866) (74)

The introduction of rainbow domination is motivatedby the study of paired domination in Cartesian productsof graphs where certain upper bounds can be expressed interms of rainbow domination that is for any graph 119866 andany graph 119867 if 119881(119867) can be partitioned into 119896120574

119875-sets then

120574119875(119866 times 119867) ⩽ (1119896)120592(119867)120574

119903119896(119866) proved by Bresar et al [126]

Dehgardi et al [128] introduced the concept of 119896-rainbowbondage number of graphs Let 119866 be a graph with maximumdegree at least two The 119896-rainbow bondage number 119887

119903119896(119866)

of 119866 is the minimum cardinality of all sets 119861 sube 119864(119866) forwhich 120574

119903119896(119866 minus 119861) gt 120574

119903119896(119866) Since in the study of 119896-rainbow

bondage number the assumption Δ(119866) ⩾ 2 is necessarywe always assume that when we discuss 119887

119903119896(119866) all graphs

involved satisfy Δ(119866) ⩾ 2The 2-rainbow bondage number of some special families

of graphs has been determined

Theorem 172 (Dehgardi et al [128]) For any integer 119899 ⩾ 3

1198871199032(119875

119899) = 1 119891119900119903 119899 ⩾ 2

1198871199032(119862

119899) =

1 119894119891 119899 equiv 0 (mod 4) 2 119900119905ℎ119890119903119908119894119904119890

1198871199032(119870

119899119899) =

1 119894119891 119899 = 2

3 119894119891 119899 = 3

6 119894119891 119899 = 4

119898 119894119891 119899 ⩾ 5

(75)

Theorem 173 (Dehgardi et al [128]) For any connected graph119866 of order 119899 ⩾ 3 119887

1199032(119866) ⩽ 120582(119866) + 2Δ(119866) minus 3

Theorem 174 (Dehgardi et al [128]) For any tree 119879 of order119899 ⩾ 3 119887

1199032(119879) ⩽ 2

Theorem 175 (Dehgardi et al [128]) If 119866 is a planar graphwithmaximumdegree at least two then 119887

1199032(119866) ⩽ 15Moreover

if the girth 119892(119866) ⩾ 4 then 1198871199032(119866) ⩽ 11 and if 119892(119866) ⩾ 6 then

1198871199032(119866) ⩽ 9

Theorem 176 (Dehgardi et al [128]) For a complete bipartitegraph 119870

119901119902 if 2119896 + 1 ⩽ 119901 ⩽ 119902 then 119887

119903119896(119870

119901119902) = 119901

Theorem177 (Dehgardi et al [128]) For a complete graph119870119899

if 119899 ⩾ 119896 + 1 then 119887119903119896(119870

119899) = lceil119896119899(119896 + 1)rceil

The special case 119896 = 1 of Theorem 177 can be found inTheorem 1 The following is a simple observation similar toObservation 1

Observation 6 (Dehgardi et al [128]) Let 119866 be a graphwith maximum degree at least two 119867 a spanning subgraphobtained from 119866 by removing 119896 edges If 120574

119903119896(119867) = 120574

119903119896(119866)

then 119887119903119896(119867) ⩽ 119887

119903119896(119866) ⩽ 119887

119903119896(119867) + 119896

Theorem 178 (Dehgardi et al [128]) For every graph 119866 with120574119903119896(119866) = 119896120574(119866) 119887

119903119896(119866) ⩾ 119887(119866)

A graph 119866 is said to be vertex 119896-rainbow domination-critical if 120574

119903119896(119866 minus 119909) lt 120574

119903119896(119866) for every vertex 119909 in 119866 It is

clear that if 119866 has no isolated vertices then 120574119903119896(119866) ⩽ 119896120574(119866) ⩽

119896120573(119866) where 120573(119866) is the vertex-covering number of119866 Thusif 120574

119903119896(119866) = 119896120573(119866) then 120574

119903119896(119866) = 119896120574(119866)

Theorem 179 (Dehgardi et al [128]) For any graph 119866 if120574119903119896(119866) = 119896120573(119866) then

(a) 119887119903119896(119866) ⩾ 120575(119866)

(b) 119887119903119896(119866) ⩾ 120575(119866)+1 if119866 is vertex 119896-rainbow domination-

critical

As far as we know there are no more results on the 119896-rainbow bondage number of graphs in the literature

28 International Journal of Combinatorics

10 Results on Digraphs

Although domination has been extensively studied in undi-rected graphs it is natural to think of a dominating setas a one-way relationship between vertices of the graphIndeed among the earliest literatures on this subject vonNeumann and Morgenstern [129] used what is now calleddomination in digraphs to find solution (or kernels whichare independent dominating sets) for cooperative 119899-persongames Most likely the first formulation of domination byBerge [130] in 1958 was given in the context of digraphs andonly some years latter by Ore [131] in 1962 for undirectedgraphs Despite this history examination of domination andits variants in digraphs has been essentially overlooked (see[4] for an overview of the domination literature) Thus thereare few if any such results on domination for digraphs in theliterature

The bondage number and its related topics for undirectedgraph have become one of major areas both in theoreticaland applied researches However until recently Carlsonand Develin [42] Shan and Kang [132] and Huang andXu [133 134] studied the bondage number for digraphsindependently In this section we introduce their resultsfor general digraphs Results for some special digraphs suchas vertex-transitive digraphs are introduced in the nextsection

101 Upper Bounds for General Digraphs Let 119866 = (119881 119864)

be a digraph without loops and parallel edges A digraph 119866is called to be asymmetric if whenever (119909 119910) isin 119864(119866) then(119910 119909) notin 119864(119866) and to be symmetric if (119909 119910) isin 119864(119866) implies(119910 119909) isin 119864(119866) For a vertex 119909 of 119881(119866) the sets of out-neighbors and in-neighbors of 119909 are respectively defined as119873

+

119866(119909) = 119910 isin 119881(119866) (119909 119910) isin 119864(119866) and 119873minus

119866(119909) = 119909 isin

119881(119866) (119909 119910) isin 119864(119866) the out-degree and the in-degree of119909 are respectively defined as 119889+

119866(119909) = |119873

+

119866(119909)| and 119889minus

119866(119909) =

|119873+

119866(119909)| Denote themaximum and theminimum out-degree

(resp in-degree) of 119866 by Δ+(119866) and 120575+(119866) (resp Δminus(119866) and120575minus

(119866))The degree 119889119866(119909) of 119909 is defined as 119889+

119866(119909)+119889

minus

119866(119909) and

the maximum and the minimum degrees of119866 are denoted byΔ(119866) and 120575(119866) respectively that is Δ(119866) = max119889

119866(119909) 119909 isin

119881(119866) and 120575(119866) = min119889119866(119909) 119909 isin 119881(119866) Note that the

definitions here are different from the ones in the textbookon digraphs see for example Xu [1]

A subset 119878 of119881(119866) is called a dominating set if119881(119866) = 119878cup119873

+

119866(119878) where119873+

119866(119878) is the set of out-neighbors of 119878Then just

as for undirected graphs 120574(119866) is the minimum cardinalityof a dominating set and the bondage number 119887(119866) is thesmallest cardinality of a set 119861 of edges such that 120574(119866 minus 119861) gt120574(119866) if such a subset 119861 sube 119864(119866) exists Otherwise we put119887(119866) = infin

Some basic results for undirected graphs stated inSection 3 can be generalized to digraphs For exampleTheorem 18 is generalized by Carlson and Develin [42]Huang andXu [133] and Shan andKang [132] independentlyas follows

Theorem 180 Let 119866 be a digraph and (119909 119910) isin 119864(119866) Then119887(119866) ⩽ 119889

119866(119910) + 119889

minus

119866(119909) minus |119873

minus

119866(119909) cap 119873

minus

119866(119910)|

Corollary 181 For a digraph 119866 119887(119866) ⩽ 120575minus(119866) + Δ(119866)

Since 120575minus

(119866) ⩽ (12)Δ(119866) from Corollary 181Conjecture 53 is valid for digraphs

Corollary 182 For a digraph 119866 119887(119866) ⩽ (32)Δ(119866)

In the case of undirected graphs the bondage number of119866119899= 119870

119899times 119870

119899achieves this bound However it was shown

by Carlson and Develin in [42] that if we take the symmetricdigraph119866

119899 we haveΔ(119866

119899) = 4(119899minus1) 120574(119866

119899) = 119899 and 119887(119866

119899) ⩽

4(119899 minus 1) + 1 = Δ(119866119899) + 1 So this family of digraphs cannot

show that the bound in Corollary 182 is tightCorresponding to Conjecture 51 which is discredited

for undirected graphs and is valid for digraphs the sameconjecture for digraphs can be proposed as follows

Conjecture 183 (Carlson and Develin [42] 2006) 119887(119866) ⩽Δ(119866) + 1 for any digraph 119866

If Conjecture 183 is true then the following results showsthat this upper bound is tight since 119887(119870

119899times119870

119899) = Δ(119870

119899times119870

119899)+1

for a complete digraph 119870119899

In 2007 Shan and Kang [132] gave some tight upperbounds on the bondage numbers for some asymmetricdigraphs For example 119887(119879) ⩽ Δ(119879) for any asymmetricdirected tree 119879 119887(119866) ⩽ Δ(119866) for any asymmetric digraph 119866with order at least 4 and 120574(119866) ⩽ 2 For planar digraphs theyobtained the following results

Theorem 184 (Shan and Kang [132] 2007) Let 119866 be aasymmetric planar digraphThen 119887(119866) ⩽ Δ(119866)+2 and 119887(119866) ⩽Δ(119866) + 1 if Δ(119866) ⩾ 5 and 119889minus

119866(119909) ⩾ 3 for every vertex 119909 with

119889119866(119909) ⩾ 4

102 Results for Some Special Digraphs The exact values andbounds on 119887(119866) for some standard digraphs are determined

Theorem185 (Huang andXu [133] 2006) For a directed cycle119862119899and a directed path 119875

119899

119887 (119862119899) =

3 119894119891 119899 119894119904 119900119889119889

2 119894119891 119899 119894119904 119890V119890119899

119887 (119875119899) =

2 119894119891 119899 119894119904 119900119889119889

1 119894119891 119899 119894119904 119890V119890119899

(76)

For the de Bruijn digraph 119861(119889 119899) and the Kautz digraph119870(119889 119899)

119887 (119861 (119889 119899)) = 119889 119894119891 119899 119894119904 119900119889119889

119889 ⩽ 119887 (119861 (119889 119899)) ⩽ 2119889 119894119891 119899 119894119904 119890V119890119899

119887 (119870 (119889 119899)) = 119889 + 1

(77)

Like undirected graphs we can define the total domi-nation number and the total bondage number On the totalbondage numbers for some special digraphs the knownresults are as follows

International Journal of Combinatorics 29

Theorem 186 (Huang and Xu [134] 2007) For a directedcycle 119862

119899and a directed path 119875

119899 119887

119879(119875

119899) and 119887

119879(119862

119899) all do not

exist For a complete digraph 119870119899

119887119879(119870

119899) = infin 119894119891 119899 = 1 2

119887119879(119870

119899) = 3 119894119891 119899 = 3

119899 ⩽ 119887119879(119870

119899) ⩽ 2119899 minus 3 119894119891 119899 ⩾ 4

(78)

The extended de Bruijn digraph EB(119889 119899 1199021 119902

119901) and

the extended Kautz digraph EK(119889 119899 1199021 119902

119901) are intro-

duced by Shibata and Gonda [135] If 119901 = 1 then theyare the de Bruijn digraph B(119889 119899) and the Kautz digraphK(119889 119899) respectively Huang and Xu [134] determined theirtotal domination numbers In particular their total bondagenumbers for general cases are determined as follows

Theorem 187 (Huang andXu [134] 2007) If 119889 ⩾ 2 and 119902119894⩾ 2

for each 119894 = 1 2 119901 then

119887119879(EB(119889 119899 119902

1 119902

119901)) = 119889

119901

minus 1

119887119879(EK(119889 119899 119902

1 119902

119901)) = 119889

119901

(79)

In particular for the de Bruijn digraph B(119889 119899) and the Kautzdigraph K(119889 119899)

119887119879(B (119889 119899)) = 119889 minus 1 119887

119879(K (119889 119899)) = 119889 (80)

Zhang et al [136] determined the bondage number incomplete 119905-partite digraphs

Theorem 188 (Zhang et al [136] 2009) For a complete 119905-partite digraph 119870

11989911198992119899119905

where 1198991⩽ 119899

2⩽ sdot sdot sdot ⩽ 119899

119905

119887 (11987011989911198992119899119905

)

=

119898 119894119891 119899119898= 1 119899

119898+1⩾ 2 119891119900119903 119904119900119898119890

119898 (1 ⩽ 119898 lt 119905)

4119905 minus 3 119894119891 1198991= 119899

2= sdot sdot sdot = 119899

119905= 2

119905minus1

sum

119894=1

119899119894+ 2 (119905 minus 1) 119900119905ℎ119890119903119908119894119904119890

(81)

Since an undirected graph can be thought of as a sym-metric digraph any result for digraphs has an analogy forundirected graphs in general In view of this point studyingthe bondage number for digraphs is more significant thanfor undirected graphs Thus we should further study thebondage number of digraphs and try to generalize knownresults on the bondage number and related variants for undi-rected graphs to digraphs prove or disprove Conjecture 183In particular determine the exact values of 119887(B(119889 119899)) for aneven 119899 and 119887

119879(119870

119899) for 119899 ⩾ 4

11 Efficient Dominating Sets

A dominating set 119878 of a graph 119866 is called to be efficient if forevery vertex 119909 in119866 |119873

119866[119909]cap119878| = 1 if119866 is a undirected graph

or |119873minus

119866[119909] cap 119878| = 1 if 119866 is a directed graph The concept of an

efficient set was proposed by Bange et al [137] in 1988 In theliterature an efficient set is also called a perfect dominatingset (see Livingston and Stout [138] for an explanation)

From definition if 119878 is an efficient dominating set of agraph 119866 then 119878 is certainly an independent set and everyvertex not in 119878 is adjacent to exactly one vertex in 119878

It is also clear from definition that a dominating set 119878 isefficient if and only if N

119866[119878] = 119873

119866[119909] 119909 isin 119878 for the

undirected graph 119866 or N+

119866[119878] = 119873

+

119866[119909] 119909 isin 119878 for the

digraph119866 is a partition of119881(119866) where the induced subgraphby119873

119866[119909] or119873+

119866[119909] is a star or an out-star with the root 119909

The efficient domination has important applications inmany areas such as error-correcting codes and receivesmuch attention in the late years

The concept of efficient dominating sets is a measureof the efficiency of domination in graphs Unfortunately asshown in [137] not every graph has an efficient dominatingset andmoreover it is anNP-complete problem to determinewhether a given graph has an efficient dominating set Inaddition it was shown by Clark [139] in 1993 that for a widerange of 119901 almost every random undirected graph 119866 isin

G(120592 119901) has no efficient dominating sets This means thatundirected graphs possessing an efficient dominating set arerare However it is easy to show that every undirected graphhas an orientationwith an efficient dominating set (see Bangeet al [140])

In 1993 Barkauskas and Host [141] showed that deter-mining whether an arbitrary oriented graph has an efficientdominating set is NP-complete Even so the existence ofefficient dominating sets for some special classes of graphs hasbeen examined see for example Dejter and Serra [142] andLee [143] for Cayley graph Gu et al [144] formeshes and toriObradovic et al [145] Huang and Xu [36] Reji Kumar andMacGillivray [146] for circulant graphs Harary graphs andtori and Van Wieren et al [147] for cube-connected cycles

In this section we introduce results of the bondagenumber for some graphs with efficient dominating sets dueto Huang and Xu [36]

111 Results for General Graphs In this subsection we intro-duce some results on bondage numbers obtained by applyingefficient dominating sets We first state the two followinglemmas

Lemma 189 For any 119896-regular graph or digraph 119866 of order 119899120574(119866) ⩾ 119899(119896 + 1) with equality if and only if 119866 has an efficientdominating set In addition if 119866 has an efficient dominatingset then every efficient dominating set is certainly a 120574-set andvice versa

Let 119890 be an edge and 119878 a dominating set in 119866 We say that119890 supports 119878 if 119890 isin (119878 119878) where (119878 119878) = (119909 119910) isin 119864(119866) 119909 isin 119878 119910 isin 119878 Denote by 119904(119866) the minimum number of edgeswhich support all 120574-sets in 119866

Lemma 190 For any graph or digraph 119866 119887(119866) ⩾ 119904(119866) withequality if 119866 is regular and has an efficient dominating set

30 International Journal of Combinatorics

A graph 119866 is called to be vertex-transitive if its automor-phism group Aut(119866) acts transitively on its vertex-set 119881(119866)A vertex-transitive graph is regular Applying Lemmas 189and 190 Huang and Xu obtained some results on bondagenumbers for vertex-transitive graphs or digraphs

Theorem 191 (Huang and Xu [36] 2008) For any vertex-transitive graph or digraph 119866 of order 119899

119887 (119866) ⩾

lceil119899

2120574 (119866)rceil 119894119891 119866 119894119904 119906119899119889119894119903119890119888119905119890119889

lceil119899

120574 (119866)rceil 119894119891 119866 119894119904 119889119894119903119890119888119905119890119889

(82)

Theorem192 (Huang andXu [36] 2008) Let119866 be a 119896-regulargraph of order 119899 Then

119887(119866) ⩽ 119896 if119866 is undirected and 119899 ⩾ 120574(119866)(119896+1)minus119896+1119887(119866) ⩽ 119896+1+ℓ if119866 is directed and 119899 ⩾ 120574(119866)(119896+1)minusℓwith 0 ⩽ ℓ ⩽ 119896 minus 1

Next we introduce a better upper bound on 119887(119866) To thisaim we need the following concept which is a generalizationof the concept of the edge-covering of a graph 119866 For 1198811015840

sube

119881(119866) and 1198641015840 sube 119864(119866) we say that 1198641015840covers 1198811015840 and call 1198641015840an edge-covering for 1198811015840 if there exists an edge (119909 119910) isin 1198641015840 forany vertex 119909 isin 1198811015840 For 119910 isin 119881(119866) let 1205731015840[119910] be the minimumcardinality over all edge-coverings for119873minus

119866[119910]

Theorem 193 (Huang and Xu [36] 2008) If 119866 is a 119896-regulargraph with order 120574(119866)(119896 + 1) then 119887(119866) ⩽ 1205731015840[119910] for any 119910 isin119881(119866)

Theupper bound on 119887(119866) given inTheorem 193 is tight inview of 119887(119862

119899) for a cycle or a directed cycle 119862

119899(seeTheorems

1 and 185 resp)It is easy to see that for a 119896-regular graph 119866 lceil(119896 + 1)2rceil ⩽

1205731015840

[119910] ⩽ 119896 when 119866 is undirected and 1205731015840[119910] = 119896 + 1 when 119866 isdirected By this fact and Lemma 189 the following theoremis merely a simple combination of Theorems 191 and 193

Theorem 194 (Huang and Xu [36] 2008) Let 119866 be a vertex-transitive graph of degree 119896 If 119866 has an efficient dominatingset then

lceil119896 + 1

2rceil ⩽ 119887 (119866) ⩽ 119896 119894119891 119866 119894119904 119906119899119889119894119903119890119888119905119890119889

119887 (119866) = 119896 + 1 119894119891 119866 119894119904 119889119894119903119890119888119905119890119889

(83)

Theorem 195 (Huang and Xu [36] 2008) If 119866 is an undi-rected vertex-transitive cubic graph with order 4120574(119866) and girth119892(119866) ⩽ 5 then 119887(119866) = 2

The proof of Theorem 195 leads to a byproduct If 119892 =5 let (119906

1 119906

2 119906

3 119906

4 119906

5) be a cycle and let D

119894be the family

of efficient dominating sets containing 119906119894for each 119894 isin

1 2 3 4 5 ThenD119894capD

119895= 0 for 119894 = 1 2 5 Then 119866 has

at least 5119904 efficient dominating sets where 119904 = 119904(119866) But thereare only 4s (=ns120574) distinct efficient dominating sets in119866This

contradiction implies that an undirected vertex-transitivecubic graph with girth five has no efficient dominating setsBut a similar argument for 119892(119866) = 3 4 or 119892(119866) ⩾ 6 couldnot give any contradiction This is consistent with the resultthat CCC(119899) a vertex-transitive cubic graph with girth 119899 if3 ⩽ 119899 ⩽ 8 or girth 8 if 119899 ⩾ 9 has efficient dominating setsfor all 119899 ⩾ 3 except 119899 = 5 (see Theorem 199 in the followingsubsection)

112 Results for Cayley Graphs In this subsection we useTheorem 194 to determine the exact values or approximativevalues of bondage numbers for some special vertex-transitivegraphs by characterizing the existence of efficient dominatingsets in these graphs

Let Γ be a nontrivial finite group 119878 a nonempty subset ofΓ without the identity element of Γ A digraph 119866 is defined asfollows

119881 (119866) = Γ (119909 119910) isin 119864 (119866) lArrrArr 119909minus1

119910 isin 119878

for any 119909 119910 isin Γ(84)

is called a Cayley digraph of the group Γ with respect to119878 denoted by 119862

Γ(119878) If 119878minus1 = 119904minus1 119904 isin 119878 = 119878 then

119862Γ(119878) is symmetric and is called a Cayley undirected graph

a Cayley graph for short Cayley graphs or digraphs arecertainly vertex-transitive (see Xu [1])

A circulant graph 119866(119899 119878) of order 119899 is a Cayley graph119862119885119899

(119878) where 119885119899= 0 119899 minus 1 is the addition group of

order 119899 and 119878 is a nonempty subset of119885119899without the identity

element and hence is a vertex-transitive digraph of degree|119878| If 119878minus1 = 119878 then119866(119899 119878) is an undirected graph If 119878 = 1 119896where 2 ⩽ 119896 ⩽ 119899 minus 2 we write 119866(119899 1 119896) for 119866(119899 1 119896) or119866(119899 plusmn1 plusmn119896) and call it a double-loop circulant graph

Huang and Xu [36] showed that for directed 119866 =

119866(119899 1 119896) lceil1198993rceil ⩽ 120574(119866) ⩽ lceil1198992rceil and 119866 has an efficientdominating set if and only if 3 | 119899 and 119896 equiv 2 (mod 3) fordirected 119866 = 119866(119899 1 119896) and 119896 = 1198992 lceil1198995rceil ⩽ 120574(119866) ⩽ lceil1198993rceiland 119866 has an efficient dominating set if and only if 5 | 119899 and119896 equiv plusmn2 (mod 5) By Theorem 194 we obtain the bondagenumber of a double loop circulant graph if it has an efficientdominating set

Theorem 196 (Huang and Xu [36] 2008) Let 119866 be a double-loop circulant graph 119866(119899 1 119896) If 119866 is directed with 3 | 119899and 119896 equiv 2 (mod 3) or 119866 is undirected with 5 | 119899 and119896 equiv plusmn2 (mod 5) then 119887(119866) = 3

The 119898 times 119899 torus is the Cartesian product 119862119898times 119862

119899of

two cycles and is a Cayley graph 119862119885119898times119885119899

(119878) where 119878 =

(0 1) (1 0) for directed cycles and 119878 = (0 plusmn1) (plusmn1 0) forundirected cycles and hence is vertex-transitive Gu et al[144] showed that the undirected torus119862

119898times119862

119899has an efficient

dominating set if and only if both 119898 and 119899 are multiplesof 5 Huang and Xu [36] showed that the directed torus119862119898times 119862

119899has an efficient dominating set if and only if both

119898 and 119899 are multiples of 3 Moreover they found a necessarycondition for a dominating set containing the vertex (0 0)in 119862

119898times 119862

119899to be efficient and obtained the following

result

International Journal of Combinatorics 31

Theorem 197 (Huang and Xu [36] 2008) Let 119866 = 119862119898times 119862

119899

If 119866 is undirected and both119898 and 119899 are multiples of 5 or if 119866 isdirected and both119898 and 119899 are multiples of 3 then 119887(119866) = 3

The hypercube 119876119899

is the Cayley graph 119862Γ(119878)

where Γ = 1198852times sdot sdot sdot times 119885

2= (119885

2)119899 and 119878 =

100 sdot sdot sdot 0 010 sdot sdot sdot 0 00 sdot sdot sdot 01 Lee [143] showed that119876119899has an efficient dominating set if and only if 119899 = 2119898 minus 1

for a positive integer119898 ByTheorem 194 the following resultis obtained immediately

Theorem 198 (Huang and Xu [36] 2008) If 119899 = 2119898 minus 1 for apositive integer119898 then 2119898minus1

⩽ 119887(119876119899) ⩽ 2

119898

minus 1

Problem 10 Determine the exact value of 119887(119876119899) for 119899 = 2119898minus1

The 119899-dimensional cube-connected cycle denoted byCCC(119899) is constructed from the 119899-dimensional hypercube119876119899by replacing each vertex 119909 in 119876

119899with an undirected cycle

119862119899of length 119899 and linking the 119894th vertex of the 119862

119899to the 119894th

neighbor of 119909 It has been proved that CCC(119899) is a Cayleygraph and hence is a vertex-transitive graph with degree 3Van Wieren et al [147] proved that CCC(119899) has an efficientdominating set if and only if 119899 = 5 From Theorems 194 and195 the following result can be derived

Theorem 199 (Huang and Xu [36] 2008) Let 119866 = 119862119862119862(119899)be the 119899-dimensional cube-connected cycles with 119899 ⩾ 3 and119899 = 5 Then 120574(119866) = 1198992119899minus2 and 2 ⩽ 119887(119866) ⩽ 3 In addition119887(119862119862119862(3)) = 119887(119862119862119862(4)) = 2

Problem 11 Determine the exact value of 119887(CCC(119899)) for 119899 ⩾5

Acknowledgments

This work was supported by NNSF of China (Grants nos11071233 61272008) The author would like to express hisgratitude to the anonymous referees for their kind sugges-tions and comments on the original paper to ProfessorSheikholeslami and Professor Jafari Rad for sending thereferences [128] and [81] which resulted in this version

References

[1] J-M Xu Theory and Application of Graphs vol 10 of NetworkTheory and Applications Kluwer Academic Dordrecht TheNetherlands 2003

[2] S THedetniemi andR C Laskar ldquoBibliography on dominationin graphs and some basic definitions of domination parame-tersrdquo Discrete Mathematics vol 86 no 1ndash3 pp 257ndash277 1990

[3] TW Haynes S T Hedetniemi and P J Slater Fundamentals ofDomination in Graphs vol 208 ofMonographs and Textbooks inPure and Applied Mathematics Marcel Dekker New York NYUSA 1998

[4] TWHaynes S THedetniemi and P J Slater EdsDominationin Graphs vol 209 of Monographs and Textbooks in Pure andAppliedMathematicsMarcelDekker NewYorkNYUSA 1998

[5] M R Garey andD S JohnsonComputers and Intractability WH Freeman and Company San Francisco Calif USA 1979

[6] H B Walikar and B D Acharya ldquoDomination critical graphsrdquoNational Academy Science Letters vol 2 pp 70ndash72 1979

[7] D Bauer F Harary J Nieminen and C L Suffel ldquoDominationalteration sets in graphsrdquo Discrete Mathematics vol 47 no 2-3pp 153ndash161 1983

[8] J F Fink M S Jacobson L F Kinch and J Roberts ldquoThebondage number of a graphrdquo Discrete Mathematics vol 86 no1ndash3 pp 47ndash57 1990

[9] F-T Hu and J-M Xu ldquoThe bondage number of (119899 minus 3)-regulargraph G of order nrdquo Ars Combinatoria to appear

[10] U Teschner ldquoNew results about the bondage number of agraphrdquoDiscreteMathematics vol 171 no 1ndash3 pp 249ndash259 1997

[11] B L Hartnell and D F Rall ldquoA bound on the size of a graphwith given order and bondage numberrdquo Discrete Mathematicsvol 197-198 pp 409ndash413 1999 16th British CombinatorialConference (London 1997)

[12] B L Hartnell and D F Rall ldquoA characterization of trees inwhich no edge is essential to the domination numberrdquo ArsCombinatoria vol 33 pp 65ndash76 1992

[13] J Topp and P D Vestergaard ldquo120572119896- and 120574

119896-stable graphsrdquo

Discrete Mathematics vol 212 no 1-2 pp 149ndash160 2000[14] A Meir and J W Moon ldquoRelations between packing and

covering numbers of a treerdquo Pacific Journal of Mathematics vol61 no 1 pp 225ndash233 1975

[15] B L Hartnell L K Joslashrgensen P D Vestergaard and CA Whitehead ldquoEdge stability of the k-domination numberof treesrdquo Bulletin of the Institute of Combinatorics and ItsApplications vol 22 pp 31ndash40 1998

[16] F-T Hu and J-M Xu ldquoOn the complexity of the bondage andreinforcement problemsrdquo Journal of Complexity vol 28 no 2pp 192ndash201 2012

[17] B L Hartnell and D F Rall ldquoBounds on the bondage numberof a graphrdquo Discrete Mathematics vol 128 no 1ndash3 pp 173ndash1771994

[18] Y-L Wang ldquoOn the bondage number of a graphrdquo DiscreteMathematics vol 159 no 1ndash3 pp 291ndash294 1996

[19] G H Fricke TW Haynes S M Hedetniemi S T Hedetniemiand R C Laskar ldquoExcellent treesrdquo Bulletin of the Institute ofCombinatorics and Its Applications vol 34 pp 27ndash38 2002

[20] V Samodivkin ldquoThe bondage number of graphs good and badverticesrdquoDiscussiones Mathematicae GraphTheory vol 28 no3 pp 453ndash462 2008

[21] J E Dunbar T W Haynes U Teschner and L VolkmannldquoBondage insensitivity and reinforcementrdquo in Domination inGraphs vol 209 of Monographs and Textbooks in Pure andAppliedMathematics pp 471ndash489 Dekker NewYork NY USA1998

[22] H Liu and L Sun ldquoThe bondage and connectivity of a graphrdquoDiscrete Mathematics vol 263 no 1ndash3 pp 289ndash293 2003

[23] A-H Esfahanian and S L Hakimi ldquoOn computing a con-ditional edge-connectivity of a graphrdquo Information ProcessingLetters vol 27 no 4 pp 195ndash199 1988

[24] R C Brigham P Z Chinn and R D Dutton ldquoVertexdomination-critical graphsrdquoNetworks vol 18 no 3 pp 173ndash1791988

[25] V Samodivkin ldquoDomination with respect to nondegenerateproperties bondage numberrdquoThe Australasian Journal of Com-binatorics vol 45 pp 217ndash226 2009

[26] U Teschner ldquoA new upper bound for the bondage numberof graphs with small domination numberrdquo The AustralasianJournal of Combinatorics vol 12 pp 27ndash35 1995

32 International Journal of Combinatorics

[27] J Fulman D Hanson and G MacGillivray ldquoVertex dom-ination-critical graphsrdquoNetworks vol 25 no 2 pp 41ndash43 1995

[28] U Teschner ldquoA counterexample to a conjecture on the bondagenumber of a graphrdquo Discrete Mathematics vol 122 no 1ndash3 pp393ndash395 1993

[29] U Teschner ldquoThe bondage number of a graph G can be muchgreater than Δ(G)rdquo Ars Combinatoria vol 43 pp 81ndash87 1996

[30] L Volkmann ldquoOn graphs with equal domination and coveringnumbersrdquoDiscrete AppliedMathematics vol 51 no 1-2 pp 211ndash217 1994

[31] L A Sanchis ldquoMaximum number of edges in connected graphswith a given domination numberrdquoDiscreteMathematics vol 87no 1 pp 65ndash72 1991

[32] S Klavzar and N Seifter ldquoDominating Cartesian products ofcyclesrdquoDiscrete AppliedMathematics vol 59 no 2 pp 129ndash1361995

[33] H K Kim ldquoA note on Vizingrsquos conjecture about the bondagenumberrdquo Korean Journal of Mathematics vol 10 no 2 pp 39ndash42 2003

[34] M Y Sohn X Yuan and H S Jeong ldquoThe bondage number of1198623times119862

119899rdquo Journal of the KoreanMathematical Society vol 44 no

6 pp 1213ndash1231 2007[35] L Kang M Y Sohn and H K Kim ldquoBondage number of the

discrete torus 119862119899times 119862

4rdquo Discrete Mathematics vol 303 no 1ndash3

pp 80ndash86 2005[36] J Huang and J-M Xu ldquoThe bondage numbers and efficient

dominations of vertex-transitive graphsrdquoDiscrete Mathematicsvol 308 no 4 pp 571ndash582 2008

[37] C J Xiang X Yuan andM Y Sohn ldquoDomination and bondagenumber of119862

3times119862

119899rdquoArs Combinatoria vol 97 pp 299ndash310 2010

[38] M S Jacobson and L F Kinch ldquoOn the domination numberof products of graphs Irdquo Ars Combinatoria vol 18 pp 33ndash441984

[39] Y-Q Li ldquoA note on the bondage number of a graphrdquo ChineseQuarterly Journal of Mathematics vol 9 no 4 pp 1ndash3 1994

[40] F-T Hu and J-M Xu ldquoBondage number of mesh networksrdquoFrontiers ofMathematics inChina vol 7 no 5 pp 813ndash826 2012

[41] R Frucht and F Harary ldquoOn the corona of two graphsrdquoAequationes Mathematicae vol 4 pp 322ndash325 1970

[42] K Carlson and M Develin ldquoOn the bondage number of planarand directed graphsrdquoDiscreteMathematics vol 306 no 8-9 pp820ndash826 2006

[43] U Teschner ldquoOn the bondage number of block graphsrdquo ArsCombinatoria vol 46 pp 25ndash32 1997

[44] J Huang and J-M Xu ldquoNote on conjectures of bondagenumbers of planar graphsrdquo Applied Mathematical Sciences vol6 no 65ndash68 pp 3277ndash3287 2012

[45] L Kang and J Yuan ldquoBondage number of planar graphsrdquoDiscrete Mathematics vol 222 no 1ndash3 pp 191ndash198 2000

[46] M Fischermann D Rautenbach and L Volkmann ldquoRemarkson the bondage number of planar graphsrdquo Discrete Mathemat-ics vol 260 no 1ndash3 pp 57ndash67 2003

[47] J Hou G Liu and J Wu ldquoBondage number of planar graphswithout small cyclesrdquoUtilitasMathematica vol 84 pp 189ndash1992011

[48] J Huang and J-M Xu ldquoThe bondage numbers of graphs withsmall crossing numbersrdquo Discrete Mathematics vol 307 no 15pp 1881ndash1897 2007

[49] Q Ma S Zhang and J Wang ldquoBondage number of 1-planargraphrdquo Applied Mathematics vol 1 no 2 pp 101ndash103 2010

[50] J Hou and G Liu ldquoA bound on the bondage number of toroidalgraphsrdquoDiscreteMathematics Algorithms and Applications vol4 no 3 Article ID 1250046 5 pages 2012

[51] Y-C Cao J-M Xu and X-R Xu ldquoOn bondage number oftoroidal graphsrdquo Journal of University of Science and Technologyof China vol 39 no 3 pp 225ndash228 2009

[52] Y-C Cao J Huang and J-M Xu ldquoThe bondage number ofgraphs with crossing number less than fourrdquo Ars Combinatoriato appear

[53] H K Kim ldquoOn the bondage numbers of some graphsrdquo KoreanJournal of Mathematics vol 11 no 2 pp 11ndash15 2004

[54] A Gagarin and V Zverovich ldquoUpper bounds for the bondagenumber of graphs on topological surfacesrdquo Discrete Mathemat-ics 2012

[55] J Huang ldquoAn improved upper bound for the bondage numberof graphs on surfacesrdquoDiscrete Mathematics vol 312 no 18 pp2776ndash2781 2012

[56] A Gagarin and V Zverovich ldquoThe bondage number of graphson topological surfaces and Teschnerrsquos conjecturerdquo DiscreteMathematics vol 313 no 6 pp 796ndash808 2013

[57] V Samodivkin ldquoNote on the bondage number of graphs ontopological surfacesrdquo httparxivorgabs12086203

[58] E J Cockayne R M Dawes and S T Hedetniemi ldquoTotaldomination in graphsrdquoNetworks vol 10 no 3 pp 211ndash219 1980

[59] R Laskar J Pfaff S M Hedetniemi and S T HedetniemildquoOn the algorithmic complexity of total dominationrdquo Society forIndustrial and Applied Mathematics vol 5 no 3 pp 420ndash4251984

[60] J Pfaff R C Laskar and S T Hedetniemi ldquoNP-completeness oftotal and connected domination and irredundance for bipartitegraphsrdquo Tech Rep 428 Department of Mathematical SciencesClemson University 1983

[61] M A Henning ldquoA survey of selected recent results on totaldomination in graphsrdquoDiscrete Mathematics vol 309 no 1 pp32ndash63 2009

[62] V R Kulli and D K Patwari ldquoThe total bondage number ofa graphrdquo in Advances in Graph Theory pp 227ndash235 VishwaGulbarga India 1991

[63] F-T Hu Y Lu and J-M Xu ldquoThe total bondage number ofgrid graphsrdquo Discrete Applied Mathematics vol 160 no 16-17pp 2408ndash2418 2012

[64] J Cao M Shi and B Wu ldquoResearch on the total bondagenumber of a spectial networkrdquo in Proceedings of the 3rdInternational Joint Conference on Computational Sciences andOptimization (CSO rsquo10) pp 456ndash458 May 2010

[65] N Sridharan MD Elias and V S A Subramanian ldquoTotalbondage number of a graphrdquo AKCE International Journal ofGraphs and Combinatorics vol 4 no 2 pp 203ndash209 2007

[66] N Jafari Rad and J Raczek ldquoSome progress on total bondage ingraphsrdquo Graphs and Combinatorics 2011

[67] T W Haynes and P J Slater ldquoPaired-domination and thepaired-domatic numberrdquoCongressusNumerantium vol 109 pp65ndash72 1995

[68] T W Haynes and P J Slater ldquoPaired-domination in graphsrdquoNetworks vol 32 no 3 pp 199ndash206 1998

[69] X Hou ldquoA characterization of 2120574 120574119875-treesrdquoDiscrete Mathemat-

ics vol 308 no 15 pp 3420ndash3426 2008[70] K E Proffitt T W Haynes and P J Slater ldquoPaired-domination

in grid graphsrdquo Congressus Numerantium vol 150 pp 161ndash1722001

International Journal of Combinatorics 33

[71] H Qiao L KangM Cardei andD-Z Du ldquoPaired-dominationof treesrdquo Journal of Global Optimization vol 25 no 1 pp 43ndash542003

[72] E Shan L Kang and M A Henning ldquoA characterizationof trees with equal total domination and paired-dominationnumbersrdquo The Australasian Journal of Combinatorics vol 30pp 31ndash39 2004

[73] J Raczek ldquoPaired bondage in treesrdquo Discrete Mathematics vol308 no 23 pp 5570ndash5575 2008

[74] J A Telle and A Proskurowski ldquoAlgorithms for vertex parti-tioning problems on partial k-treesrdquo SIAM Journal on DiscreteMathematics vol 10 no 4 pp 529ndash550 1997

[75] G S Domke J H Hattingh M A Henning and L R MarkusldquoRestrained domination in treesrdquoDiscreteMathematics vol 211no 1ndash3 pp 1ndash9 2000

[76] G S Domke J H Hattingh M A Henning and L R MarkusldquoRestrained domination in graphs with minimum degree twordquoJournal of Combinatorial Mathematics and Combinatorial Com-puting vol 35 pp 239ndash254 2000

[77] G S Domke J H Hattingh S T Hedetniemi R C Laskarand L R Markus ldquoRestrained domination in graphsrdquo DiscreteMathematics vol 203 no 1ndash3 pp 61ndash69 1999

[78] J H Hattingh and E J Joubert ldquoAn upper bound for therestrained domination number of a graph with minimumdegree at least two in terms of order and minimum degreerdquoDiscrete Applied Mathematics vol 157 no 13 pp 2846ndash28582009

[79] M A Henning ldquoGraphs with large restrained dominationnumberrdquo Discrete Mathematics vol 197-198 pp 415ndash429 199916th British Combinatorial Conference (London 1997)

[80] J H Hattingh and A R Plummer ldquoRestrained bondage ingraphsrdquo Discrete Mathematics vol 308 no 23 pp 5446ndash54532008

[81] N Jafari Rad R Hasni J Raczek and L Volkmann ldquoTotalrestrained bondage in graphsrdquoActaMathematica Sinica vol 29no 6 pp 1033ndash1042 2013

[82] J Cyman and J Raczek ldquoOn the total restrained dominationnumber of a graphrdquoThe Australasian Journal of Combinatoricsvol 36 pp 91ndash100 2006

[83] M A Henning ldquoGraphs with large total domination numberrdquoJournal of Graph Theory vol 35 no 1 pp 21ndash45 2000

[84] D-X Ma X-G Chen and L Sun ldquoOn total restraineddomination in graphsrdquoCzechoslovakMathematical Journal vol55 no 1 pp 165ndash173 2005

[85] B Zelinka ldquoRemarks on restrained domination and totalrestrained domination in graphsrdquo Czechoslovak MathematicalJournal vol 55 no 2 pp 393ndash396 2005

[86] W Goddard and M A Henning ldquoIndependent domination ingraphs a survey and recent resultsrdquo Discrete Mathematics vol313 no 7 pp 839ndash854 2013

[87] J H Zhang H L Liu and L Sun ldquoIndependence bondagenumber and reinforcement number of some graphsrdquo Transac-tions of Beijing Institute of Technology vol 23 no 2 pp 140ndash1422003

[88] K D Dharmalingam Studies in Graph Theorymdashequitabledomination and bottleneck domination [PhD thesis] MaduraiKamaraj University Madurai India 2006

[89] GDeepakND Soner andAAlwardi ldquoThe equitable bondagenumber of a graphrdquo Research Journal of Pure Algebra vol 1 no9 pp 209ndash212 2011

[90] E Sampathkumar and H B Walikar ldquoThe connected domina-tion number of a graphrdquo Journal of Mathematical and PhysicalSciences vol 13 no 6 pp 607ndash613 1979

[91] K White M Farber and W Pulleyblank ldquoSteiner trees con-nected domination and strongly chordal graphsrdquoNetworks vol15 no 1 pp 109ndash124 1985

[92] M Cadei M X Cheng X Cheng and D-Z Du ldquoConnecteddomination in Ad Hoc wireless networksrdquo in Proceedings ofthe 6th International Conference on Computer Science andInformatics (CSampI rsquo02) 2002

[93] J F Fink and M S Jacobson ldquon-domination in graphsrdquo inGraph Theory with Applications to Algorithms and ComputerScience Wiley-Interscience Publications pp 283ndash300 WileyNew York NY USA 1985

[94] M Chellali O Favaron A Hansberg and L Volkmann ldquok-domination and k-independence in graphs a surveyrdquo Graphsand Combinatorics vol 28 no 1 pp 1ndash55 2012

[95] Y Lu X Hou J-M Xu andN Li ldquoTrees with uniqueminimump-dominating setsrdquo Utilitas Mathematica vol 86 pp 193ndash2052011

[96] Y Lu and J-M Xu ldquoThe p-bondage number of treesrdquo Graphsand Combinatorics vol 27 no 1 pp 129ndash141 2011

[97] Y Lu and J-M Xu ldquoThe 2-domination and 2-bondage numbersof grid graphs A manuscriptrdquo httparxivorgabs12044514

[98] M Krzywkowski ldquoNon-isolating 2-bondage in graphsrdquo TheJournal of the Mathematical Society of Japan vol 64 no 4 2012

[99] M A Henning O R Oellermann and H C Swart ldquoBoundson distance domination parametersrdquo Journal of CombinatoricsInformation amp System Sciences vol 16 no 1 pp 11ndash18 1991

[100] F Tian and J-M Xu ldquoBounds for distance domination numbersof graphsrdquo Journal of University of Science and Technology ofChina vol 34 no 5 pp 529ndash534 2004

[101] F Tian and J-M Xu ldquoDistance domination-critical graphsrdquoApplied Mathematics Letters vol 21 no 4 pp 416ndash420 2008

[102] F Tian and J-M Xu ldquoA note on distance domination numbersof graphsrdquo The Australasian Journal of Combinatorics vol 43pp 181ndash190 2009

[103] F Tian and J-M Xu ldquoAverage distances and distance domina-tion numbersrdquo Discrete Applied Mathematics vol 157 no 5 pp1113ndash1127 2009

[104] Z-L Liu F Tian and J-M Xu ldquoProbabilistic analysis of upperbounds for 2-connected distance k-dominating sets in graphsrdquoTheoretical Computer Science vol 410 no 38ndash40 pp 3804ndash3813 2009

[105] M A Henning O R Oellermann and H C Swart ldquoDistancedomination critical graphsrdquo Journal of Combinatorial Mathe-matics and Combinatorial Computing vol 44 pp 33ndash45 2003

[106] T J Bean M A Henning and H C Swart ldquoOn the integrityof distance domination in graphsrdquo The Australasian Journal ofCombinatorics vol 10 pp 29ndash43 1994

[107] M Fischermann and L Volkmann ldquoA remark on a conjecturefor the (kp)-domination numberrdquoUtilitasMathematica vol 67pp 223ndash227 2005

[108] T Korneffel D Meierling and L Volkmann ldquoA remark on the(22)-domination numberrdquo Discussiones Mathematicae GraphTheory vol 28 no 2 pp 361ndash366 2008

[109] Y Lu X Hou and J-M Xu ldquoOn the (22)-domination numberof treesrdquo Discussiones Mathematicae Graph Theory vol 30 no2 pp 185ndash199 2010

34 International Journal of Combinatorics

[110] H Li and J-M Xu ldquo(dm)-domination numbers of m-connected graphsrdquo Rapports de Recherche 1130 LRI UniversiteParis-Sud 1997

[111] S M Hedetniemi S T Hedetniemi and T V Wimer ldquoLineartime resource allocation algorithms for treesrdquo Tech Rep URI-014 Department of Mathematical Sciences Clemson Univer-sity 1987

[112] M Farber ldquoDomination independent domination and dualityin strongly chordal graphsrdquo Discrete Applied Mathematics vol7 no 2 pp 115ndash130 1984

[113] G S Domke and R C Laskar ldquoThe bondage and reinforcementnumbers of 120574

119891for some graphsrdquoDiscrete Mathematics vol 167-

168 pp 249ndash259 1997[114] G S Domke S T Hedetniemi and R C Laskar ldquoFractional

packings coverings and irredundance in graphsrdquo vol 66 pp227ndash238 1988

[115] E J Cockayne P A Dreyer Jr S M Hedetniemi andS T Hedetniemi ldquoRoman domination in graphsrdquo DiscreteMathematics vol 278 no 1ndash3 pp 11ndash22 2004

[116] I Stewart ldquoDefend the roman empirerdquo Scientific American vol281 pp 136ndash139 1999

[117] C S ReVelle and K E Rosing ldquoDefendens imperiumromanum a classical problem in military strategyrdquoThe Ameri-can Mathematical Monthly vol 107 no 7 pp 585ndash594 2000

[118] A Bahremandpour F-T Hu S M Sheikholeslami and J-MXu ldquoOn the Roman bondage number of a graphrdquo DiscreteMathematics Algorithms and Applications vol 5 no 1 ArticleID 1350001 15 pages 2013

[119] N Jafari Rad and L Volkmann ldquoRoman bondage in graphsrdquoDiscussiones Mathematicae Graph Theory vol 31 no 4 pp763ndash773 2011

[120] K Ebadi and L PushpaLatha ldquoRoman bondage and Romanreinforcement numbers of a graphrdquo International Journal ofContemporary Mathematical Sciences vol 5 pp 1487ndash14972010

[121] N Dehgardi S M Sheikholeslami and L Volkman ldquoOn theRoman k-bondage number of a graphrdquo AKCE InternationalJournal of Graphs and Combinatorics vol 8 pp 169ndash180 2011

[122] S Akbari and S Qajar ldquoA note on Roman bondage number ofgraphsrdquo Ars Combinatoria to Appear

[123] F-T Hu and J-M Xu ldquoRoman bondage numbers of somegraphsrdquo httparxivorgabs11093933

[124] N Jafari Rad and L Volkmann ldquoOn the Roman bondagenumber of planar graphsrdquo Graphs and Combinatorics vol 27no 4 pp 531ndash538 2011

[125] S Akbari M Khatirinejad and S Qajar ldquoA note on the romanbondage number of planar graphsrdquo Graphs and Combinatorics2012

[126] B Bresar M A Henning and D F Rall ldquoRainbow dominationin graphsrdquo Taiwanese Journal of Mathematics vol 12 no 1 pp213ndash225 2008

[127] B L Hartnell and D F Rall ldquoOn dominating the Cartesianproduct of a graph and 120572krdquo Discussiones Mathematicae GraphTheory vol 24 no 3 pp 389ndash402 2004

[128] N Dehgardi S M Sheikholeslami and L Volkman ldquoThe k-rainbow bondage number of a graphrdquo to appear in DiscreteApplied Mathematics

[129] J von Neumann and O Morgenstern Theory of Games andEconomic Behavior Princeton University Press Princeton NJUSA 1944

[130] C BergeTheorıe des Graphes et ses Applications 1958[131] O Ore Theory of Graphs vol 38 American Mathematical

Society Colloquium Publications 1962[132] E Shan and L Kang ldquoBondage number in oriented graphsrdquoArs

Combinatoria vol 84 pp 319ndash331 2007[133] J Huang and J-M Xu ldquoThe bondage numbers of extended

de Bruijn and Kautz digraphsrdquo Computers amp Mathematics withApplications vol 51 no 6-7 pp 1137ndash1147 2006

[134] J Huang and J-M Xu ldquoThe total domination and total bondagenumbers of extended de Bruijn and Kautz digraphsrdquoComputersamp Mathematics with Applications vol 53 no 8 pp 1206ndash12132007

[135] Y Shibata and Y Gonda ldquoExtension of de Bruijn graph andKautz graphrdquo Computers amp Mathematics with Applications vol30 no 9 pp 51ndash61 1995

[136] X Zhang J Liu and JMeng ldquoThebondage number in completet-partite digraphsrdquo Information Processing Letters vol 109 no17 pp 997ndash1000 2009

[137] D W Bange A E Barkauskas and P J Slater ldquoEfficient dom-inating sets in graphsrdquo in Applications of Discrete Mathematicspp 189ndash199 SIAM Philadelphia Pa USA 1988

[138] M Livingston and Q F Stout ldquoPerfect dominating setsrdquoCongressus Numerantium vol 79 pp 187ndash203 1990

[139] L Clark ldquoPerfect domination in random graphsrdquo Journal ofCombinatorial Mathematics and Combinatorial Computing vol14 pp 173ndash182 1993

[140] D W Bange A E Barkauskas L H Host and L H ClarkldquoEfficient domination of the orientations of a graphrdquo DiscreteMathematics vol 178 no 1ndash3 pp 1ndash14 1998

[141] A E Barkauskas and L H Host ldquoFinding efficient dominatingsets in oriented graphsrdquo Congressus Numerantium vol 98 pp27ndash32 1993

[142] I J Dejter and O Serra ldquoEfficient dominating sets in CayleygraphsrdquoDiscrete AppliedMathematics vol 129 no 2-3 pp 319ndash328 2003

[143] J Lee ldquoIndependent perfect domination sets in Cayley graphsrdquoJournal of Graph Theory vol 37 no 4 pp 213ndash219 2001

[144] WGu X Jia and J Shen ldquoIndependent perfect domination setsin meshes tori and treesrdquo Unpublished

[145] N Obradovic J Peters and G Ruzic ldquoEfficient domination incirculant graphswith two chord lengthsrdquo Information ProcessingLetters vol 102 no 6 pp 253ndash258 2007

[146] K Reji Kumar and G MacGillivray ldquoEfficient domination incirculant graphsrdquo Discrete Mathematics vol 313 no 6 pp 767ndash771 2013

[147] D Van Wieren M Livingston and Q F Stout ldquoPerfectdominating sets on cube-connected cyclesrdquo Congressus Numer-antium vol 97 pp 51ndash70 1993

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

18 International Journal of Combinatorics

Figure 61198677

Combining Theorem 1 with Theorem 112 we have whatfollows

Corollary 113 (Sridharan et al [65] 2007) The bondagenumber and the total bondage number are unrelated even fortrees

However Sridharan et al [65] gave an upper bound onthe total bondage number of a tree in terms of its maximumdegree For any tree 119879 of order 119899 if 119879 =119870

1119899minus1 then 119887

119879(119879) =

minΔ(119879) (13)(119899 minus 1) Rad and Raczek [66] improved thisupper bound and gave a constructive characterization of acertain class of trees attaching the upper bound

Theorem 114 (Rad and Raczek [66] 2011) 119887119879(119879) ⩽ Δ(119879) minus 1

for any tree 119879 with maximum degree at least three

In general the decision problem for 119887119879(119866) is NP-complete

for any graph 119866 We state the decision problem for the totalbondage as follows

Problem 3 Consider the decision problemTotal Bondage ProblemInstance a graph 119866 and a positive integer 119887Question is 119887

119879(119866) ⩽ 119887

Theorem 115 (Hu and Xu [16] 2012) The total bondageproblem is NP-complete

Consequently in view of its computational hardnessit is significative to establish some sharp bounds on thetotal bondage number of a graph in terms of other graphicparameters In 1991 Kulli and Patwari [62] showed that119887119879(119866) ⩽ 2119899 minus 5 for any graph 119866 with order 119899 ⩾ 5 Sridharan

et al [65] improved this result as follows

Theorem 116 (Sridharan et al [65] 2007) For any graph 119866with order 119899 if 119899 ⩾ 5 then

119887119879(119866) ⩽

1

3(119899 minus 1) 119894119891 119866 119888119900119899119905119886119894119899119904 119899119900 119888119910119888119897119890119904

119899 minus 1 119894119891 119892 (119866) ⩾ 5

119899 minus 2 119894119891 119892 (119866) = 4

(42)

Rad and Raczek [66] also established some upperbounds on 119887

119879(119866) for a general graph 119866 In particular they

gave an upper bound on 119887119879(119866) in terms of the girth of

119866

Theorem 117 (Rad and Raczek [66] 2011) 119887119879(119866)⩽4Δ(119866) minus 5

for any graph 119866 with 119892(119866) ⩾ 4

82 Paired Bondage Numbers A dominating set 119878 of 119866 iscalled to be paired if the subgraph induced by 119878 containsa perfect matching The paired domination number of 119866denoted by 120574

119875(119866) is the minimum cardinality of a paired

dominating set of 119866 Clearly 120574119879(119866) ⩽ 120574

119875(119866) for every

connected graph 119866 with order at least two where 120574119879(119866) is

the total domination number of 119866 and 120574119875(119866) ⩽ 2120574(119866) for

any graph119866without isolated vertices Paired domination wasintroduced by Haynes and Slater [67 68] and further studiedin [69ndash72]

The paired bondage number of 119866 with 120575(119866) ⩾ 1 denotedby 119887P(119866) is the minimum cardinality among all sets of edges119861 sube 119864 such that 120575(119866 minus 119861) ⩾ 1 and 120574

119875(119866 minus 119861) gt 120574

119875(119866)

The concept of the paired bondage number was firstproposed by Raczek [73] in 2008The following observationsfollow immediately from the definition of the paired bondagenumber

Observation 2 Let 119866 be a graph with 120575(119866) ⩾ 1

(a) If 119867 is a subgraph of 119866 such that 120575(119867) ⩾ 1 then119887119875(119867) ⩽ 119887

119875(119866)

(b) If 119867 is a subgraph of 119866 such that 119887119875(119867) = 1 and 119896

is the number of edges removed to form119867 then 1 ⩽119887119875(119866) ⩽ 119896 + 1

(c) If 119909119910 isin 119864(119866) such that 119889119866(119909) 119889

119866(119910) gt 1 and 119909119910

belongs to each perfect matching of each minimumpaired dominating set of 119866 then 119887

119875(119866) = 1

Based on these observations 119887119875(119875

119899) ⩽ 2 In fact the

paired bondage number of a path has been determined

Theorem 118 (Raczek [73] 2008) For any positive integer 119896if 119899 ⩾ 2 then

119887119875(119875

119899) =

0 119894119891 119899 = 2 3 119900119903 5

1 119894119891 119899 = 4119896 4119896 + 3 119900119903 4119896 + 6

2 119900119905ℎ119890119903119908119894119904119890

(43)

For a cycle 119862119899of order 119899 ⩾ 3 since 120574

119875(119862

119899) = 120574

119875(119875

119899) from

Theorem 118 the following result holds immediately

Corollary 119 (Raczek [73] 2008) For any positive integer 119896if 119899 ⩾ 3 then

119887119875(119862

119899) =

0 119894119891 119899 = 3 119900119903 5

2 119894119891 119899 = 4119896 4119896 + 3 119900119903 4119896 + 6

3 119900119905ℎ119890119903119908119894119904119890

(44)

A wheel 119882119899 where 119899 ⩾ 4 is a graph with 119899 vertices

formed by connecting a single vertex to all vertices of a cycle119862119899minus1

Of course 120574119875(119882

119899) = 2

International Journal of Combinatorics 19

119896

Figure 7 A tree 119879 with 119887119875(119879) = 119896

Theorem 120 (Raczek [73] 2008) For positive integers 119899 and119898

119887119875(119870

119898119899) = 119898 119891119900119903 1 lt 119898 ⩽ 119899

119887119875(119882

119899) =

4 119894119891 119899 = 4

3 119894119891 119899 = 5

2 119900119905ℎ119890119903119908119894119904119890

(45)

Theorem 16 shows that if 119879 is a nontrivial tree then119887(119879) ⩽ 2 However no similar result exists for pairedbondage For any nonnegative integer 119896 let 119879

119896be a tree

obtained by subdividing all but one edge of a star 1198701119896+1

(asshown in Figure 7) It is easy to see that 119887

119875(119879

119896) = 119896 We

state this fact as the following theorem which is similar toTheorem 112

Theorem121 (Raczek [73] 2008) For any nonnegative integer119896 there exists a tree 119879 with 119887

119875(119879) = 119896

Consider the tree defined in Figure 5 for 119896 = 3 Then119887(119879

3) ⩽ 2 and 119887

119875(119879

3) = 3 On the other hand 119887(119875

4) = 2

and 119887119875(119875

4) = 1 Thus we have what follows which is similar

to Corollary 113

Corollary 122 The bondage number and the paired bondagenumber are unrelated even for trees

A constructive characterization of trees with 119887(119879) = 2is given by Hartnell and Rall in [12] Raczek [73] provideda constructive characterization of trees with 119887

119875(119879) = 0 In

order to state the characterization we define a labeling andthree simple operations on a tree119879 Let 119910 isin 119881(119879) and let ℓ(119910)be the label assigned to 119910

Operation T1 If ℓ(119910) = 119861 add a vertex 119909 and the

edge 119910119909 and let ℓ(119909) = 119860OperationT

2 If ℓ(119910) = 119862 add a path (119909

1 119909

2) and the

edge 1199101199091 and let ℓ(119909

1) = 119861 ℓ(119909

2) = 119860

Operation T3 If ℓ(119910) = 119861 add a path (119909

1 119909

2 119909

3)

and the edge 1199101199091 and let ℓ(119909

1) = 119862 ℓ(119909

2) = 119861 and

ℓ(1199093) = 119860

Let 1198752= (119906 V) with ℓ(119906) = 119860 and ℓ(V) = 119861 Let T be

the class of all trees obtained from the labeled 1198752by a finite

sequence of operationsT1 T

2 T

3

A tree 119879 in Figure 8 belongs to the familyTRaczek [73] obtained the following characterization of all

trees 119879 with 119887119875(119879) = 0

119860

119860

119860

119860

119860

119861 119862

119861 119862 119861

119861119860

Figure 8 A tree 119879 belong to the familyT

Theorem 123 (Raczek [73] 2008) For a tree 119879 119887119875(119879) = 0 if

and only if 119879 is inT

We state the decision problem for the paired bondage asfollows

Problem 4 Consider the decision problem

Paired Bondage ProblemInstance a graph 119866 and a positive integer 119887Question is 119887

119875(119866) ⩽ 119887

Conjecture 124 The paired bondage problem is NP-complete

83 Restrained Bondage Numbers A dominating set 119878 of 119866 iscalled to be restrained if the subgraph induced by 119878 containsno isolated vertices where 119878 = 119881(119866) 119878 The restraineddomination number of 119866 denoted by 120574

119877(119866) is the smallest

cardinality of a restrained dominating set of 119866 It is clear that120574119877(119866) exists and 120574(119866) ⩽ 120574

119877(119866) for any nonempty graph 119866

The concept of restrained domination was introducedby Telle and Proskurowski [74] in 1997 albeit indirectly asa vertex partitioning problem Concerning the study of therestrained domination numbers the reader is referred to [74ndash79]

In 2008 Hattingh and Plummer [80] defined therestrained bondage number 119887R(119866) of a nonempty graph 119866 tobe the minimum cardinality among all subsets 119861 sube 119864(119866) forwhich 120574

119877(119866 minus 119861) gt 120574

119877(119866)

For some simple graphs their restrained dominationnumbers can be easily determined and so restrained bondagenumbers have been also determined For example it is clearthat 120574

119877(119870

119899) = 1 Domke et al [77] showed that 120574

119877(119862

119899) =

119899 minus 2lfloor1198993rfloor for 119899 ⩾ 3 and 120574119877(119875

119899) = 119899 minus 2lfloor(119899 minus 1)3rfloor for 119899 ⩾ 1

Using these results Hattingh and Plummer [80] obtained therestricted bondage numbers for119870

119899 119862

119899 119875

119899 and119866 = 119870

11989911198992119899119905

as follows

Theorem 125 (Hattingh and Plummer [80] 2008) For 119899 ⩾ 3

119887R (119870119899) =

1 119894119891 119899 = 3

lceil119899

2rceil 119900119905ℎ119890119903119908119894119904119890

119887R (119862119899) =

1 119894119891 119899 equiv 0 (mod 3) 2 119900119905ℎ119890119903119908119894119904119890

119887R (119875119899) = 1 119899 ⩾ 4

20 International Journal of Combinatorics

119887R (119866)

=

lceil119898

2rceil 119894119891 119899

119898= 1 119886119899119889 119899

119898+1⩾ 2 (1 ⩽ 119898 lt 119905)

2119905 minus 2 119894119891 1198991= 119899

2= sdot sdot sdot = 119899

119905= 2 (119905 ⩾ 2)

2 119894119891 1198991= 2 119886119899119889 119899

2⩾ 3 (119905 = 2)

119905minus1

sum

119894=1

119899119894minus 1 119900119905ℎ119890119903119908119894119904119890

(46)

Theorem 126 (Hattingh and Plummer [80] 2008) For a tree119879 of order 119899 (⩾ 4) 119879 ≇ 119870

1119899minus1if and only if 119887R(119879) = 1

Theorem 126 shows that the restrained bondage numberof a tree can be computed in constant time However thedecision problem for 119887R(119866) is NP-complete even for bipartitegraphs

Problem 5 Consider the decision problem

Restrained Bondage Problem

Instance a graph 119866 and a positive integer 119887

Question is 119887R(119866) ⩽ 119887

Theorem 127 (Hattingh and Plummer [80] 2008) Therestrained bondage problem is NP-complete even for bipartitegraphs

Consequently in view of its computational hardnessit is significative to establish some sharp bounds on therestrained bondage number of a graph in terms of othergraphic parameters

Theorem 128 (Hattingh and Plummer [80] 2008) For anygraph 119866 with 120575(119866) ⩾ 2

119887R (119866) ⩽ min119909119910isin119864(119866)

119889119866(119909) + 119889

119866(119910) minus 2 (47)

Corollary 129 119887R(119866) ⩽ Δ(119866)+120575(119866)minus2 for any graph119866 with120575(119866) ⩾ 2

Notice that the bounds stated in Theorem 128 andCorollary 129 are sharp Indeed the class of cycles whoseorders are congruent to 1 2 (mod 3) have a restrained bondagenumber achieving these bounds (see Theorem 125)

Theorem 128 is an analogue of Theorem 14 A quitenatural problem is whether or not for restricted bondagenumber there are analogues of Theorem 16 Theorem 18Theorem 29 and so on Indeed we should consider all resultsfor the bondage number whether or not there are analoguesfor restricted bondage number

Theorem 130 (Hattingh and Plummer [80] 2008) For anygraph 119866 with 120574

119877(119866) = 2

119887R (119866) ⩽ Δ (119866) + 1 (48)

We now consider the relation between 119887119877(119866) and 119887(119866)

Theorem 131 (Hattingh and Plummer [80] 2008) If 120574119877(119866) =

120574(119866) for some graph 119866 then 119887R(119866) ⩽ 119887(119866)

Corollary 129 and Theorem 131 provide a possibility toattack Conjecture 73 by characterizing planar graphs with120574119877= 120574 and 119887R = 119887 when 3 ⩽ Δ ⩽ 6However we do not have 119887R(119866) = 119887(119866) for any graph

119866 even if 120574119877(119866) = 120574(119866) Observe that 120574

119877(119870

3) = 120574(119870

3) yet

119887119877(119870

3) = 1 and 119887(119870

3) = 2 We still may not claim that

119887119877(119866) = 119887(119866) even in the case that every 120574-set is a 120574

119877-set

The example 1198703again demonstrates this

Theorem 132 (Hattingh and Plummer [80] 2008) There isan infinite class of graphs inwhich each graph119866 satisfies 119887(119866) lt119887R(119866)

In fact such an infinite class of graphs can be obtained asfollows Let119867 be a connected graph BC(119867) a graph obtainedfrom 119867 by attaching ℓ (⩾ 2) vertices of degree 1 to eachvertex in 119867 and B = 119866 119866 = BC(119867) for some graph 119867such that 120575(119867) ⩾ 2 By Theorem 3 we can easily check that119887(119866) = 1 lt 2 ⩽ min120575(119867) ℓ = 119887R(119866) for any 119866 isin BMoreover there exists a graph 119866 isin B such that 119887R(119866) canbe much larger than 119887(119866) which is stated as the followingtheorem

Theorem 133 (Hattingh and Plummer [80] 2008) For eachpositive integer 119896 there is a graph119866 such that 119896 = 119887R(119866)minus119887(119866)

Combining Theorem 133 with Theorem 132 we have thefollowing corollary

Corollary 134 The bondage number and the restrainedbondage number are unrelated

Very recently Jafari Rad et al [81] have consideredthe total restrained bondage in graphs based on both totaldominating sets and restrained dominating sets and obtainedseveral properties exact values and bounds for the totalrestrained bondage number of a graph

A restrained dominating set 119878 of a graph 119866 without iso-lated vertices is called to be a total restrained dominating setif 119878 is a total dominating set The total restrained dominationnumber of119866 denoted by 120574TR (119866) is theminimum cardinalityof a total restrained dominating set of 119866 For referenceson this domination in graphs see [3 61 82ndash85] The totalrestrained bondage number 119887TR (119866) of a graph 119866 with noisolated vertex is the cardinality of a smallest set of edges 119861 sube119864(119866) for which119866minus119861 has no isolated vertex and 120574TR (119866minus119861) gt120574TR (119866) In the case that there is no such subset 119861 we define119887TR (119866) = infin

Ma et al [84] determined that 120574TR (119875119899) = 119899minus2lfloor(119899minus2)4rfloorfor 119899 ⩾ 3 120574TR (119862119899

) = 119899 minus 2lfloor1198994rfloor for 119899 ⩾ 3 120574TR (1198703) =

3 and 120574TR (119870119899) = 2 for 119899 = 3 and 120574TR (1198701119899

) = 1 + 119899

and 120574TR (119870119898119899) = 2 for 2 ⩽ 119898 ⩽ 119899 According to these

results Jafari Rad et al [81] determined the exact valuesof total restrained bondage numbers for the correspondinggraphs

International Journal of Combinatorics 21

Theorem 135 (Jafari Rad et al [81]) For an integer 119899

119887TR (119875119899) = infin 119894119891 2 ⩽ 119899 ⩽ 5

1 119894119891 119899 ⩾ 5

119887TR (119862119899) =

infin 119894119891 119899 = 3

1 119894119891 3 lt 119899 119899 equiv 0 1 (mod 4)2 119894119891 3 lt 119899 119899 equiv 2 3 (mod 4)

119887TR (119882119899) = 2 119891119900119903 119899 ⩾ 6

119887TR (119870119899) = 119899 minus 1 119891119900119903 119899 ⩾ 4

119887TR (119870119898119899) = 119898 minus 1 119891119900119903 2 ⩽ 119898 ⩽ 119899

(49)

119887TR (119870119899) = 119899minus1 shows the fact that for any integer 119899 ⩾ 3 there

exists a connected graph namely119870119899+1

such that 119887TR (119870119899+1) =

119899 that is the total restrained bondage number is unboundedfrom the above

A vertex is said to be a support vertex if it is adjacent toa vertex of degree one Let A be the family of all connectedgraphs such that119866 belongs toA if and only if every edge of119866is incident to a support vertex or 119866 is a cycle of length three

Proposition 136 (Cyman and Raczek [82] 2006) For aconnected graph119866 of order 119899 120574TR (119866) = 119899 if and only if119866 isin A

Hence if 119866 isin A then the removal of any subset of edgesof119866 cannot increase the total restrained domination numberof 119866 and thus 119887TR (119866) = infin When 119866 notin A a result similar toTheorem 6 is obtained

Theorem 137 (Jafari Rad et al [81]) For any tree 119879 notin A119887TR (119879) ⩽ 2 This bound is sharp

In the samepaper the authors provided a characterizationfor trees 119879 notin A with 119887TR (119879) = 1 or 2 The following result issimilar to Observation 1

Observation 3 (Jafari Rad et al [81]) If 119896 edges can beremoved from a graph 119866 to obtain a graph 119867 without anisolated vertex and with 119887TR (119867) = 119905 then 119887TR (119866) ⩽ 119896 + 119905

ApplyingTheorem 137 andObservation 3 Jafari Rad et alcharacterized all connected graphs 119866 with 119887TR (119866) = infin

Theorem 138 (Jafari Rad et al [81]) For a connected graph119866119887TR (119866) = infin if and only if 119866 isin A

84 Other Conditional Bondage Numbers A subset 119868 sube 119881(119866)is called an independent set if no two vertices in 119868 are adjacentin 119866 The maximum cardinality among all independent setsis called the independence number of 119866 denoted by 120572(119866)

A dominating set 119878 of a graph 119866 is called to be inde-pendent if 119878 is an independent set of 119866 The minimumcardinality among all independent dominating set is calledthe independence domination number of 119866 and denoted by120574119868(119866)

Since an independent dominating set is not only adominating set but also an independent set 120574(119866) ⩽ 120574

119868(119866) ⩽

120572(119866) for any graph 119866It is clear that a maximal independent set is certainly a

dominating set Thus an independent set is maximal if andonly if it is an independent dominating set and so 120574

119868(119866) is the

minimum cardinality among all maximal independent setsof 119866 This graph-theoretical invariant has been well studiedin the literature see for example Haynes et al [3] for earlyresults Goddard and Henning [86] for recent results onindependent domination in graphs

In 2003 Zhang et al [87] defined the independencebondage number 119887

119868(119866) of a nonempty graph 119866 to be the

minimum cardinality among all subsets 119861 sube 119864(119866) for which120574119868(119866 minus 119861) gt 120574

119868(119866) For some ordinary graphs their indepen-

dence domination numbers can be easily determined and soindependence bondage numbers have been also determinedClearly 119887

119868(119870

119899) = 1 if 119899 ⩾ 2

Theorem 139 (Zhang et al [87] 2003) For any integer 119899 ⩾ 4

119887119868(119862

119899) =

1 119894119891 119899 equiv 1 (mod 2) 2 119894119891 119899 equiv 0 (mod 2)

119887119868(119875

119899) =

1 119894119891 119899 equiv 0 (mod 2) 2 119894119891 119899 equiv 1 (mod 2)

119887119868(119870

119898119899) = 119899 119891119900119903 119898 ⩽ 119899

(50)

Apart from the above-mentioned results as far as we findno other results on the independence bondage number in thepresent literature we never knew much of any result on thisparameter for a tree

A subset 119878 of 119881(119866) is called an equitable dominatingset if for every 119910 isin 119881(119866 minus 119878) there exists a vertex 119909 isin 119878such that 119909119910 isin 119864(119866) and |119889

119866(119909) minus 119889

119866(119910)| ⩽ 1 proposed

by Dharmalingam [88] The minimum cardinality of such adominating set is called the equitable domination number andis denoted by 120574

119864(119866) Deepak et al [89] defined the equitable

bondage number 119887119864(119866) of a graph 119866 to be the cardinality of a

smallest set 119861 sube 119864(119866) of edges for which 120574119864(119866 minus 119861) gt 120574

119864(119866)

and determined 119887119864(119866) for some special graphs

Theorem 140 (Deepak et al [89] 2011) For any integer 119899 ⩾ 2

119887119864(119870

119899) = lceil

119899

2rceil

119887119864(119870

119898119899) = 119898 119891119900119903 0 ⩽ 119899 minus 119898 ⩽ 1

119887119864(119862

119899) =

3 119894119891 119899 equiv 1 (mod 3)2 119900119905ℎ119890119903119908119894119904119890

119887119864(119875

119899) =

2 119894119891 119899 equiv 1 (mod 3)1 119900119905ℎ119890119903119908119894119904119890

119887119864(119879) ⩽ 2 119891119900119903 119886119899119910 119899119900119899119905119903119894119907119894119886119897 119905119903119890119890 119879

119887119864(119866) ⩽ 119899 minus 1 119891119900119903 119886119899119910 119888119900119899119899119890119888119905119890119889 119892119903119886119901ℎ 119866 119900119891 119900119903119889119890119903 119899

(51)

22 International Journal of Combinatorics

As far as we know there are no more results on theequitable bondage number of graphs in the present literature

There are other variations of the normal domination byadding restricted conditions to the normal dominating setFor example the connected domination set of a graph119866 pro-posed by Sampathkumar and Walikar [90] is a dominatingset 119878 such that the subgraph induced by 119878 is connected Theproblem of finding a minimum connected domination set ina graph is equivalent to the problemof finding a spanning treewithmaximumnumber of vertices of degree one [91] and hassome important applications in wireless networks [92] thusit has received much research attention However for suchan important domination there is no result on its bondagenumber We believe that the topic is of significance for futureresearch

9 Generalized Bondage Numbers

There are various generalizations of the classical dominationsuch as 119901-domination distance domination fractional dom-ination Roman domination and rainbow domination Everysuch a generalization can lead to a corresponding bondageIn this section we introduce some of them

91 119901-Bondage Numbers In 1985 Fink and Jacobson [93]introduced the concept of 119901-domination Let 119901 be a positiveinteger A subset 119878 of 119881(119866) is a 119901-dominating set of 119866 if|119878 cap 119873

119866(119910)| ⩾ 119901 for every 119910 isin 119878 The 119901-domination number

120574119901(119866) is the minimum cardinality among all 119901-dominating

sets of 119866 Any 119901-dominating set of 119866 with cardinality 120574119901(119866)

is called a 120574119901-set of 119866 Note that a 120574

1-set is a classic minimum

dominating set Notice that every graph has a 119901-dominatingset since the vertex-set119881(119866) is such a setWe also note that a 1-dominating set is a dominating set and so 120574(119866) = 120574

1(119866) The

119901-domination number has received much research attentionsee a state-of-the-art survey article by Chellali et al [94]

It is clear from definition that every 119901-dominating setof a graph certainly contains all vertices of degree at most119901 minus 1 By this simple observation to avoid the trivial caseoccurrence we always assume that Δ(119866) ⩾ 119901 For 119901 ⩾ 2 Luet al [95] gave a constructive characterization of trees withunique minimum 119901-dominating sets

Recently Lu and Xu [96] have introduced the conceptto the 119901-bondage number of 119866 denoted by 119887

119901(119866) as the

minimum cardinality among all sets of edges 119861 sube 119864(119866) suchthat 120574

119901(119866 minus 119861) gt 120574

119901(119866) Clearly 119887

1(119866) = 119887(119866)

Lu and Xu [96] established a lower bound and an upperbound on 119887

119901(119879) for any integer 119901 ⩾ 2 and any tree 119879 with

Δ(119879) ⩾ 119901 and characterized all trees achieving the lowerbound and the upper bound respectively

Theorem 141 (Lu and Xu [96] 2011) For any integer 119901 ⩾ 2and any tree 119879 with Δ(119879) ⩾ 119901

1 ⩽ 119887119901(119879) ⩽ Δ (119879) minus 119901 + 1 (52)

Let 119878 be a given subset of 119881(119866) and for any 119909 isin 119878 let

119873119901(119909 119878 119866) = 119910 isin 119878 cap 119873

119866(119909)

1003816100381610038161003816119873119866(119910) cap 119878

1003816100381610038161003816 = 119901

(53)

Then all trees achieving the lower bound 1 can be character-ized as follows

Theorem 142 (Lu and Xu [96] 2011) Let 119879 be a tree withΔ(119879) ⩾ 119901 ⩾ 2 Then 119887

119901(119879) = 1 if and only if for any 120574

119901-set

119878 of 119879 there exists an edge 119909119910 isin (119878 119878) such that 119910 isin 119873119901(119909 119878 119879)

The notation 119878(119886 119887) denotes a double star obtained byadding an edge between the central vertices of two stars119870

1119886minus1

and1198701119887minus1

And the vertex with degree 119886 (resp 119887) in 119878(119886 119887) iscalled the 119871-central vertex (resp 119877-central vertex) of 119878(119886 119887)

To characterize all trees attaining the upper bound givenin Theorem 141 we define three types of operations on a tree119879 with Δ(119879) = Δ ⩾ 119901 + 1

Type 1 attach a pendant edge to a vertex 119910with 119889119879(119910) ⩽ 119901minus

2 in 119879

Type 2 attach a star 1198701Δminus1

to a vertex 119910 of 119879 by joining itscentral vertex to 119910 where 119910 in a 120574

119901-set of 119879 and

119889119879(119910) ⩽ Δ minus 1

Type 3 attach a double star 119878(119901 Δ minus 1) to a pendant vertex 119910of 119879 by coinciding its 119877-central vertex with 119910 wherethe unique neighbor of 119910 is in a 120574

119901-set of 119879

Let B = 119879 a tree obtained from 1198701Δ

or 119878(119901 Δ) by afinite sequence of operations of Types 1 2 and 3

Theorem 143 (Lu and Xu [96] 2011) A tree with the maxi-mum Δ ⩾ 119901 + 1 has 119901-bondage number Δ minus 119901 + 1 if and onlyif it belongs toB

Theorem 144 (Lu and Xu [97] 2012) Let 119866119898119899= 119875

119898times 119875

119899

Then

1198872(119866

2119899) = 1 119891119900119903 119899 ⩾ 2

1198872(119866

3119899) =

2 119894119891 119899 equiv 1 (mod 3) 1 119900119905ℎ119890119903119908119894119904119890

for 119899 ⩾ 2

1198872(119866

4119899) =

1 119894119891 119899 equiv 3 (mod 4) 2 119900119905ℎ119890119903119908119894119904119890

for 119899 ⩾ 7

(54)

Recently Krzywkowski [98] has proposed the conceptof the nonisolating 2-bondage of a graph The nonisolating2-bondage number of a graph 119866 denoted by 1198871015840

2(119866) is the

minimum cardinality among all sets of edges 119861 sube 119864(119866) suchthat 119866 minus 119861 has no isolated vertices and 120574

2(119866 minus 119861) gt 120574

2(119866) If

for every 119861 sube 119864(119866) either 1205742(119866 minus 119861) = 120574

2(119866) or 119866 minus 119861 has

isolated vertices then define 11988710158402(119866) = 0 and say that 119866 is a 2-

nonisolatedly strongly stable graph Krzywkowski presentedsome basic properties of nonisolating 2-bondage in graphsshowed that for every nonnegative integer 119896 there exists atree 119879 such 1198871015840

2(119879) = 119896 characterized all 2-non-isolatedly

strongly stable trees and determined 11988710158402(119866) for several special

graphs

International Journal of Combinatorics 23

Theorem 145 (Krzywkowski [98] 2012) For any positiveintegers 119899 and119898 with119898 ⩽ 119899

1198871015840

2(119870

119899) =

0 119894119891 119899 = 1 2 3

lfloor2119899

3rfloor 119900119905ℎ119890119903119908119894119904119890

1198871015840

2(119875

119899) =

0 119894119891 119899 = 1 2 3

1 119900119905ℎ119890119903119908119894119904119890

1198871015840

2(119862

119899) =

0 119894119891 119899 = 3

1 119894119891 119899 119894119904 119890V119890119899

2 119894119891 119899 119894119904 119900119889119889

1198871015840

2(119870

119898119899) =

3 119894119891 119898 = 119899 = 3

1 119894119891 119898 = 119899 = 4

2 119900119905ℎ119890119903119908119894119904119890

(55)

92 Distance Bondage Numbers A subset 119878 of vertices of agraph119866 is said to be a distance 119896-dominating set for119866 if everyvertex in 119866 not in 119878 is at distance at most 119896 from some vertexof 119878 The minimum cardinality of all distance 119896-dominatingsets is called the distance 119896-domination number of 119866 anddenoted by 120574

119896(119866) (do not confuse with the above-mentioned

120574119901(119866)) When 119896 = 1 a distance 1-dominating set is a normal

dominating set and so 1205741(119866) = 120574(119866) for any graph 119866 Thus

the distance 119896-domination is a generalization of the classicaldomination

The relation between 120574119896and 120572

119896(the 119896-independence

number defined in Section 22) for a tree obtained by Meirand Moon [14] who proved that 120574

119896(119879) = 120572

2119896(119879) for any tree

119879 (a special case for 119896 = 1 is stated in Proposition 10) Thefurther research results can be found in Henning et al [99]Tian and Xu [100ndash103] and Liu et al [104]

In 1998 Hartnell et al [15] defined the distance 119896-bondagenumber of 119866 denoted by 119887

119896(119866) to be the cardinality of a

smallest subset 119861 of edges of 119866 with the property that 120574119896(119866 minus

119861) gt 120574119896(119866) FromTheorem 6 it is clear that if119879 is a nontrivial

tree then 1 ⩽ 1198871(119879) ⩽ 2 Hartnell et al [15] generalized this

result to any integer 119896 ⩾ 1

Theorem 146 (Hartnell et al [15] 1998) For every nontrivialtree 119879 and positive integer 119896 1 ⩽ 119887

119896(119879) ⩽ 2

Hartnell et al [15] and Topp and Vestergaard [13] alsocharacterized the trees having distance 119896-bondage number 2In particular the class of trees for which 119887

1(119879) = 2 are just

those which have a unique maximum 2-independent set (seeTheorem 11)

Since when 119896 = 1 the distance 1-bondage number 1198871(119866)

is the classical bondage number 119887(119866) Theorem 12 gives theNP-hardness of deciding the distance 119896-bondage number ofgeneral graphs

Theorem 147 Given a nonempty undirected graph 119866 andpositive integers 119896 and 119887 with 119887 ⩽ 120576(119866) determining wetheror not 119887

119896(119866) ⩽ 119887 is NP-hard

For a vertex 119909 in119866 the open 119896-neighborhood119873119896(119909) of 119909 is

defined as119873119896(119909) = 119910 isin 119881(119866) 1 ⩽ 119889

119866(119909 119910) le 119896The closed

119896-neighborhood119873119896[119909] of 119909 in 119866 is defined as119873

119896(119909) cup 119909 Let

Δ119896(119866) = max 1003816100381610038161003816119873119896

(119909)1003816100381610038161003816 for any 119909 isin 119881 (119866) (56)

Clearly Δ1(119866) = Δ(119866) The 119896th power of a graph 119866 is the

graph119866119896 with vertex-set119881(119866119896

) = 119881(119866) and edge-set119864(119866119896

) =

119909119910 1 ⩽ 119889119866(119909 119910) ⩽ 119896 The following lemma holds directly

from the definition of 119866119896

Proposition 148 (Tian and Xu [101]) Δ(119866119896

) = Δ119896(119866) and

120574(119866119896

) = 120574119896(119866) for any graph 119866 and each 119896 ⩾ 1

A graph 119866 is 119896-distance domination-critical or 120574119896-critical

for short if 120574119896(119866 minus 119909) lt 120574

119896(119866) for every vertex 119909 in 119866

proposed by Henning et al [105]

Proposition 149 (Tian and Xu [101]) For each 119896 ⩾ 1 a graph119866 is 120574

119896-critical if and only if 119866119896 is 120574

119896-critical

By the above facts we suggest the following worthwhileproblem

Problem 6 Can we generalize the results on the bondage for119866 to 119866119896 In particular do the following propositions hold

(a) 119887(119866119896

) = 119887119896(119866) for any graph 119866 and each 119896 ⩾ 1

(b) 119887(119866119896

) ⩽ Δ119896(119866) if 119866 is not 120574

119896-critical

Let 119896 and 119901 be positive integers A subset 119878 of 119881(119866) isdefined to be a (119896 119901)-dominating set of 119866 if for any vertex119909 isin 119878 |119873

119896(119909) cap 119878| ⩾ 119901 The (119896 119901)-domination number of

119866 denoted by 120574119896119901(119866) is the minimum cardinality among

all (119896 119901)-dominating sets of 119866 Clearly for a graph 119866 a(1 1)-dominating set is a classical dominating set a (119896 1)-dominating set is a distance 119896-dominating set and a (1 119901)-dominating set is the above-mentioned 119901-dominating setThat is 120574

11(119866) = 120574(119866) 120574

1198961(119866) = 120574

119896(119866) and 120574

1119901(119866) = 120574

119901(119866)

The concept of (119896 119901)-domination in a graph 119866 is a gen-eralized domination which combines distance 119896-dominationand 119901-domination in 119866 So the investigation of (119896 119901)-domination of 119866 is more interesting and has received theattention of many researchers see for example [106ndash109]

More general Li and Xu [110] proposed the concept of the(ℓ 119908)-domination number of a119908-connected graph A subset119878 of 119881(119866) is defined to be an (ℓ 119908)-dominating set of 119866 iffor any vertex 119909 isin 119878 there are 119908 internally disjoint pathsof length at most ℓ from 119909 to some vertex in 119878 The (ℓ 119908)-domination number of119866 denoted by 120574

ℓ119908(119866) is theminimum

cardinality among all (ℓ 119908)-dominating sets of 119866 Clearly1205741198961(119866) = 120574

119896(119866) and 120574

1119901(119866) = 120574

119901(119866)

It is quite natural to propose the concept of bondage num-ber for (119896 119901)-domination or (ℓ 119908)-domination However asfar as we know none has proposed this concept until todayThis is a worthwhile topic for further research

24 International Journal of Combinatorics

93 Fractional Bondage Numbers If 120590 is a function mappingthe vertex-set 119881 into some set of real numbers then for anysubset 119878 sube 119881 let 120590(119878) = sum

119909isin119878120590(119909) Also let |120590| = 120590(119881)

A real-value function 120590 119881 rarr [0 1] is a dominatingfunction of a graph 119866 if for every 119909 isin 119881(119866) 120590(119873

119866[119909]) ⩾ 1

Thus if 119878 is a dominating set of 119866 120590 is a function where

120590 (119909) = 1 if 119909 isin 1198780 otherwise

(57)

then 120590 is a dominating function of119866 The fractional domina-tion number of 119866 denoted by 120574

119865(119866) is defined as follows

120574119865(119866) = min |120590| 120590 is a domination function of 119866

(58)

The 120574119865-bondage number of 119866 denoted by 119887

119865(119866) is

defined as the minimum cardinality of a subset 119861 sube 119864 whoseremoval results in 120574

119865(119866 minus 119861) gt 120574

119865(119866)

Hedetniemi et al [111] are the first to study fractionaldomination although Farber [112] introduced the idea indi-rectly The concept of the fractional domination numberwas proposed by Domke and Laskar [113] in 1997 Thefractional domination numbers for some ordinary graphs aredetermined

Proposition 150 (Domke et al [114] 1998 Domke and Laskar[113] 1997) (a) If 119866 is a 119896-regular graph with order 119899 then120574119865(119866) = 119899119896 + 1(b) 120574

119865(119862

119899) = 1198993 120574

119865(119875

119899) = lceil1198993rceil 120574

119865(119870

119899) = 1

(c) 120574119865(119866) = 1 if and only if Δ(119866) = 119899 minus 1

(d) 120574119865(119879) = 120574(119879) for any tree 119879

(e) 120574119865(119870

119899119898) = (119899(119898 + 1) + 119898(119899 + 1))(119899119898 minus 1) where

max119899119898 gt 1

The assertions (a)ndash(d) are due to Domke et al [114] andthe assertion (e) is due to Domke and Laskar [113] Accordingto these results Domke and Laskar [113] determined the 120574

119865-

bondage numbers for these graphs

Theorem 151 (Domke and Laskar [113] 1997) 119887119865(119870

119899) =

lceil1198992rceil 119887119865(119870

119899119898) = 1 wheremax119899119898 gt 1

119887119865(119862

119899) =

2 119894119891 119899 equiv 0 (mod 3) 1 119900119905ℎ119890119903119908119894119904119890

119887119865(119875

119899) =

2 119894119891 119899 equiv 1 (mod 3) 1 119900119905ℎ119890119903119908119894119904119890

(59)

It is easy to see that for any tree119879 119887119865(119879) ⩽ 2 In fact since

120574119865(119879) = 120574(119879) for any tree 119879 120574

119865(119879

1015840

) = 120574(1198791015840

) for any subgraph1198791015840 of 119879 It follows that

119887119865(119879) = min 119861 sube 119864 (119879) 120574

119865(119879 minus 119861) gt 120574

119865(119879)

= min 119861 sube 119864 (119879) 120574 (119879 minus 119861) gt 120574 (119879)

= 119887 (119879) ⩽ 2

(60)

Except for these no results on 119887119865(119866) have been known as

yet

94 Roman Bondage Numbers A Roman dominating func-tion on a graph 119866 is a labeling 119891 119881 rarr 0 1 2 such thatevery vertex with label 0 has at least one neighbor with label2 The weight of a Roman dominating function 119891 on 119866 is thevalue

119891 (119866) = sum

119906isin119881(119866)

119891 (119906) (61)

The minimum weight of a Roman dominating function ona graph 119866 is called the Roman domination number of 119866denoted by 120574RM (119866)

A Roman dominating function 119891 119881 rarr 0 1 2

can be represented by the ordered partition (1198810 119881

1 119881

2) (or

(119881119891

0 119881

119891

1 119881

119891

2) to refer to119891) of119881 where119881

119894= V isin 119881 | 119891(V) = 119894

In this representation its weight 119891(119866) = |1198811| + 2|119881

2| It is

clear that119881119891

1cup119881

119891

2is a dominating set of 119866 called the Roman

dominating set denoted by 119863119891

R = (1198811 1198812) Since 119881119891

1cup 119881

119891

2is

a dominating set when 119891 is a Roman dominating functionand since placing weight 2 at the vertices of a dominating setyields a Roman dominating function in [115] it is observedthat

120574 (119866) ⩽ 120574RM (119866) ⩽ 2120574 (119866) (62)

A graph 119866 is called to be Roman if 120574RM (119866) = 2120574(119866)The definition of the Roman domination function is

proposed implicitly by Stewart [116] and ReVelle and Rosing[117] Roman dominating numbers have been deeply studiedIn particular Bahremandpour et al [118] showed that theproblem determining the Roman domination number is NP-complete even for bipartite graphs

Let 119866 be a graph with maximum degree at least twoThe Roman bondage number 119887RM (119866) of 119866 is the minimumcardinality of all sets 1198641015840 sube 119864 for which 120574RM (119866 minus 119864

1015840

) gt

120574RM (119866) Since in study of Roman bondage number theassumption Δ(119866) ⩾ 2 is necessary we always assume thatwhen we discuss 119887RM (119866) all graphs involved satisfy Δ(119866) ⩾2 The Roman bondage number 119887RM (119866) is introduced byJafari Rad and Volkmann in [119]

Recently Bahremandpour et al [118] have shown that theproblem determining the Roman bondage number is NP-hard even for bipartite graphs

Problem 7 Consider the decision problem

Roman Bondage Problem

Instance a nonempty bipartite graph119866 and a positiveinteger 119896

Question is 119887RM (119866) ⩽ 119896

Theorem 152 (Bahremandpour et al [118] 2013) TheRomanbondage problem is NP-hard even for bipartite graphs

The exact value of 119887RM(119866) is known only for a few familyof graphs including the complete graphs cycles and paths

International Journal of Combinatorics 25

Theorem 153 (Jafari Rad and Volkmann [119] 2011) For anypositive integers 119899 and119898 with119898 ⩽ 119899

119887RM (119875119899) = 2 119894119891 119899 equiv 2 (mod 3) 1 119900119905ℎ119890119903119908119894119904119890

119887RM (119862119899) =

3 119894119891 119899 = 2 (mod 3) 2 119900119905ℎ119890119903119908119894119904119890

119887RM (119870119899) = lceil

119899

2rceil 119891119900119903 119899 ⩾ 3

119887RM (119870119898119899) =

1 119894119891 119898 = 1 119886119899119889 119899 = 1

4 119894119891 119898 = 119899 = 3

119898 119900119905ℎ119890119903119908119894119904119890

(63)

Lemma 154 (Cockayne et al [115] 2004) If 119866 is a graph oforder 119899 and contains vertices of degree 119899 minus 1 then 120574RM(119866) = 2

Using Lemma 154 the third conclusion in Theorem 153can be generalized to more general case which is similar toLemma 49

Theorem 155 (Jafari Rad and Volkmann [119] 2011) Let119866 bea graph with order 119899 ge 3 and 119905 the number of vertices of degree119899 minus 1 in 119866 If 119905 ⩾ 1 then 119887RM(119866) = lceil1199052rceil

Ebadi and PushpaLatha [120] conjectured that 119887RM(119866) ⩽119899 minus 1 for any graph 119866 of order 119899 ⩾ 3 Dehgardi et al[121] Akbari andQajar [122] independently showed that thisconjecture is true

Theorem 156 119887RM(119866) ⩽ 119899 minus 1 for any connected graph 119866 oforder 119899 ⩾ 3

For a complete 119905-partite graph we have the followingresult

Theorem 157 (Hu and Xu [123] 2011) Let 119866 = 11987011989811198982119898119905

bea complete 119905-partite graph with 119898

1= sdot sdot sdot = 119898

119894lt 119898

119894+1le sdot sdot sdot ⩽

119898119905 119905 ⩾ 2 and 119899 = sum119905

119895=1119898

119895 Then

119887RM (119866) =

lceil119894

2rceil 119894119891 119898

119894= 1 119886119899119889 119899 ⩾ 3

2 119894119891 119898119894= 2 119886119899119889 119894 = 1

119894 119894119891 119898119894= 2 119886119899119889 119894 ⩾ 2

4 119894119891 119898119894= 3 119886119899119889 119894 = 119905 = 2

119899 minus 1 119894119891 119898119894= 3 119886119899119889 119894 = 119905 ⩾ 3

119899 minus 119898119905

119894119891 119898119894⩾ 3 119886119899119889 119898

119905⩾ 4

(64)

Consider a complete 119905-partite graph 119870119905(3) for 119905 ⩾ 3

119887RM(119870119905(3)) = 119899 minus 1 by Theorem 157 where 119899 = 3119905

which shows that this upper bound of 119899 minus 1 on 119887RM(119866) inTheorem 156 is sharp This fact and 120574

119877(119870

119905(3)) = 4 give

a negative answer to the following two problems posed byDehgardi et al [121]Prove or Disprove for any connected graph 119866 of order 119899 ge 3(i) 119887RM(119866) = 119899 minus 1 if and only if 119866 cong 119870

3 (ii) 119887RM(119866) ⩽ 119899 minus

120574119877(119866) + 1Note that a complete 119905-partite graph 119870

119905(3) is (119899 minus 3)-

regular and 119887RM(119870119905(3)) = 119899 minus 1 for 119905 ⩾ 3 where 119899 = 3119905

Hu and Xu further determined that 119887119877(119866) = 119899 minus 2 for any

(119899 minus 3)-regular graph 119866 of order 119899 ⩽ 5 except 119870119905(3)

Theorem 158 (Hu and Xu [123] 2011) For any (119899minus3)-regulargraph 119866 of order 119899 other than119870

119905(3) 119887RM(119866) = 119899 minus 2 for 119899 ⩾ 5

For a tree 119879 with order 119899 ⩾ 3 Ebadi and PushpaLatha[120] and Jafari Rad and Volkmann [119] independentlyobtained an upper bound on 119887RM(119879)

Theorem 159 119887RM(119879) ⩽ 3 for any tree 119879 with order 119899 ⩾ 3

Theorem 160 (Bahremandpour et al [118] 2013) 119887RM(1198752 times119875119899) = 2 for 119899 ⩾ 2

Theorem 161 (Jafari Rad and Volkmann [119] 2011) Let119866 bea graph with order at least three

(a) If (119909 119910 119911) is a path of length 2 in 119866 then 119887RM(119866) ⩽119889119866(119909) + 119889

119866(119910) + 119889

119866(119911) minus 3 minus |119873

119866(119909) cap 119873

119866(119910)|

(b) If 119866 is connected then 119887RM(119866) ⩽ 120582(119866) + 2Δ(119866) minus 3

Theorem 161 (b) implies 119887RM(119866) ⩽ 120575(119866) + 2Δ(119866) minus 3 for aconnected graph 119866 Note that for a planar graph 119866 120575(119866) ⩽ 5moreover 120575(119866) ⩽ 3 if the girth at least 4 and 120575(119866) ⩽ 2 if thegirth at least 6 These two facts show that 119887RM(119866) ⩽ 2Δ(119866) +2 for a connected planar graph 119866 Jafari Rad and Volkmann[124] improved this bound

Theorem 162 (Jafari Rad and Volkmann [124] 2011) For anyconnected planar graph 119866 of order 119899 ⩾ 3 with girth 119892(119866)

119887RM (119866)

⩽

2Δ (119866)

Δ (119866) + 6

Δ (119866) + 5 119894119891 119866 119888119900119899119905119886119894119899119904 119899119900 V119890119903119905119894119888119890119904 119900119891 119889119890119892119903119890119890 119891119894V119890

Δ (119866) + 4 119894119891 119892 (119866) ⩾ 4

Δ (119866) + 3 119894119891 119892 (119866) ⩾ 5

Δ (119866) + 2 119894119891 119892 (119866) ⩾ 6

Δ (119866) + 1 119894119891 119892 (119866) ⩾ 8

(65)

According toTheorem 157 119887RM(119862119899) = 3 = Δ(119862

119899)+1 for a

cycle 119862119899of length 119899 ⩾ 8 with 119899 equiv 2 (mod 3) and therefore

the last upper bound inTheorem 162 is sharp at least for Δ =2

Combining the fact that every planar graph 119866 withminimum degree 5 contains an edge 119909119910 with 119889

119866(119909) = 5

and 119889119866(119910) isin 5 6 with Theorem 161 (a) Akbari et al [125]

obtained the following result

26 International Journal of Combinatorics

middot middot middot

middot middot middot

middot middot middot

middot middot middot

119866

Figure 9 The graph 119866 is constructed from 119866

Theorem 163 (Akbari et al [125] 2012) 119887RM(119866) ⩽ 15 forevery planar graph 119866

It remains open to show whether the bound inTheorem 163 is sharp or not Though finding a planargraph 119866 with 119887RM(119866) = 15 seems to be difficult Akbari etal [125] constructed an infinite family of planar graphs withRoman bondage number equal to 7 by proving the followingresult

Theorem 164 (Akbari et al [125] 2012) Let 119866 be a graph oforder 119899 and 119866 is the graph of order 5119899 obtained from 119866 byattaching the central vertex of a copy of a path 119875

5to each vertex

of119866 (see Figure 9) Then 120574RM(119866) = 4119899 and 119887RM(119866) = 120575(119866)+2

ByTheorem 164 infinitemany planar graphswith Romanbondage number 7 can be obtained by considering any planargraph 119866 with 120575(119866) = 5 (eg the icosahedron graph)

Conjecture 165 (Akbari et al [125] 2012) The Romanbondage number of every planar graph is at most 7

For general bounds the following observation is directlyobtained which is similar to Observation 1

Observation 4 Let 119866 be a graph of order 119899 with maximumdegree at least two Assume that 119867 is a spanning subgraphobtained from 119866 by removing 119896 edges If 120574RM(119867) = 120574RM(119866)then 119887

119877(119867) ⩽ 119887RM(119866) ⩽ 119887RM(119867) + 119896

Theorem 166 (Bahremandpour et al [118] 2013) For anyconnected graph 119866 of order 119899 if 119899 ⩾ 3 and 120574RM(119866) = 120574(119866) + 1then

119887RM (119866) ⩽ min 119887 (119866) 119899Δ (66)

where 119899Δis the number of vertices with maximum degree Δ in

119866

Observation 5 For any graph 119866 if 120574(119866) = 120574RM(119866) then119887RM(119866) ⩽ 119887(119866)

Theorem 167 (Bahremandpour et al [118] 2013) For everyRoman graph 119866

119887RM (119866) ⩾ 119887 (119866) (67)

The bound is sharp for cycles on 119899 vertices where 119899 equiv 0 (mod3)

The strict inequality in Theorem 167 can hold for exam-ple 119887(119862

3119896+2) = 2 lt 3 = 119887RM(1198623119896+2

) byTheorem 157A graph 119866 is called to be vertex Roman domination-

critical if 120574RM(119866 minus 119909) lt 120574RM(119866) for every vertex 119909 in 119866 If119866 has no isolated vertices then 120574RM(119866) ⩽ 2120574(119866) ⩽ 2120573(119866)If 120574RM(119866) = 2120573(119866) then 120574RM(119866) = 2120574(119866) and hence 119866 is aRoman graph In [30] Volkmann gave a lot of graphs with120574(119866) = 120573(119866) The following result is similar to Theorem 57

Theorem 168 (Bahremandpour et al [118] 2013) For anygraph 119866 if 120574RM(119866) = 2120573(119866) then

(a) 119887RM(119866) ⩾ 120575(119866)

(b) 119887RM(119866) ⩾ 120575(119866)+1 if119866 is a vertex Roman domination-critical graph

If 120574RM(119866) = 2 then obviously 119887RM(119866) ⩽ 120575(119866) So weassume 120574RM(119866) ⩾ 3 Dehgardi et al [121] proved 119887RM(119866) ⩽(120574RM(119866) minus 2)Δ(119866) + 1 and posed the following problem if 119866is a connected graph of order 119899 ⩾ 4 with 120574RM(119866) ⩾ 3 then

119887RM (119866) ⩽ (120574RM (119866) minus 2) Δ (119866) (68)

Theorem 161 (a) shows that the inequality (68) holds if120574RM(119866) ⩾ 5 Thus the bound in (68) is of interest only when120574RM(119866) is 3 or 4

Lemma 169 (Bahremandpour et al [118] 2013) For anynonempty graph 119866 of order 119899 ⩾ 3 120574RM(119866) = 3 if and onlyif Δ(119866) = 119899 minus 2

The following result shows that (68) holds for all graphs119866 of order 119899 ⩾ 4 with 120574RM(119866) = 3 or 4 which improvesTheorem 161 (b)

Theorem 170 (Bahremandpour et al [118] 2013) For anyconnected graph 119866 of order 119899 ⩾ 4

119887RM (119866) le Δ (119866) = 119899 minus 2 119894119891 120574RM (119866) = 3

Δ (119866) + 120575 (119866) minus 1 119894119891 120574RM (119866) = 4(69)

with the first equality if and only if 119866 cong 1198624

Recently Akbari and Qajar [122] have proved the follow-ing result

Theorem 171 (Akbari and Qajar [122]) For any connectedgraph 119866 of order 119899 ⩾ 3

119887RM (119866) ⩽ 119899 minus 120574RM (119866) + 5 (70)

International Journal of Combinatorics 27

We conclude this section with the following problems

Problem 8 Characterize all connected graphs 119866 of order 119899 ⩾3 for which 119887RM(119866) = 119899 minus 1

Problem 9 Prove or disprove if 119866 is a connected graph oforder 119899 ⩾ 3 then

119887RM (119866) ⩽ 119899 minus 120574RM (119866) + 3 (71)

Note that 120574119877(119870

119905(3)) = 4 and 119887RM(119870119905

(3)) = 119899minus1 where 119899 =3119905 for 119905 ⩾ 3 The upper bound on 119887RM(119866) given in Problem 9is sharp if Problem 9 has positive answer

95 Rainbow Bondage Numbers For a positive integer 119896 a119896-rainbow dominating function of a graph 119866 is a function 119891from the vertex-set 119881(119866) to the set of all subsets of the set1 2 119896 such that for any vertex 119909 in 119866 with 119891(119909) = 0 thecondition

⋃

119910isin119873119866(119909)

119891 (119910) = 1 2 119896 (72)

is fulfilledThe weight of a 119896-rainbow dominating function 119891on 119866 is the value

119891 (119866) = sum

119909isin119881(119866)

1003816100381610038161003816119891 (119909)1003816100381610038161003816 (73)

The 119896-rainbow domination number of a graph 119866 denoted by120574119903119896(119866) is the minimum weight of a 119896-rainbow dominating

function 119891 of 119866Note that 120574

1199031(119866) is the classical domination number 120574(119866)

The 119896-rainbow domination number is introduced by Bresaret al [126] Rainbow domination of a graph 119866 coincides withordinary domination of the Cartesian product of 119866 with thecomplete graph in particular 120574

119903119896(119866) = 120574(119866 times 119870

119896) for any

positive integer 119896 and any graph 119866 [126] Hartnell and Rall[127] proved that 120574(119879) lt 120574

1199032(119879) for any tree 119879 no graph has

120574 = 120574119903119896for 119896 ⩾ 3 for any 119896 ⩾ 2 and any graph 119866 of order 119899

min 119899 120574 (119866) + 119896 minus 2 ⩽ 120574119903119896(119866) ⩽ 119896120574 (119866) (74)

The introduction of rainbow domination is motivatedby the study of paired domination in Cartesian productsof graphs where certain upper bounds can be expressed interms of rainbow domination that is for any graph 119866 andany graph 119867 if 119881(119867) can be partitioned into 119896120574

119875-sets then

120574119875(119866 times 119867) ⩽ (1119896)120592(119867)120574

119903119896(119866) proved by Bresar et al [126]

Dehgardi et al [128] introduced the concept of 119896-rainbowbondage number of graphs Let 119866 be a graph with maximumdegree at least two The 119896-rainbow bondage number 119887

119903119896(119866)

of 119866 is the minimum cardinality of all sets 119861 sube 119864(119866) forwhich 120574

119903119896(119866 minus 119861) gt 120574

119903119896(119866) Since in the study of 119896-rainbow

bondage number the assumption Δ(119866) ⩾ 2 is necessarywe always assume that when we discuss 119887

119903119896(119866) all graphs

involved satisfy Δ(119866) ⩾ 2The 2-rainbow bondage number of some special families

of graphs has been determined

Theorem 172 (Dehgardi et al [128]) For any integer 119899 ⩾ 3

1198871199032(119875

119899) = 1 119891119900119903 119899 ⩾ 2

1198871199032(119862

119899) =

1 119894119891 119899 equiv 0 (mod 4) 2 119900119905ℎ119890119903119908119894119904119890

1198871199032(119870

119899119899) =

1 119894119891 119899 = 2

3 119894119891 119899 = 3

6 119894119891 119899 = 4

119898 119894119891 119899 ⩾ 5

(75)

Theorem 173 (Dehgardi et al [128]) For any connected graph119866 of order 119899 ⩾ 3 119887

1199032(119866) ⩽ 120582(119866) + 2Δ(119866) minus 3

Theorem 174 (Dehgardi et al [128]) For any tree 119879 of order119899 ⩾ 3 119887

1199032(119879) ⩽ 2

Theorem 175 (Dehgardi et al [128]) If 119866 is a planar graphwithmaximumdegree at least two then 119887

1199032(119866) ⩽ 15Moreover

if the girth 119892(119866) ⩾ 4 then 1198871199032(119866) ⩽ 11 and if 119892(119866) ⩾ 6 then

1198871199032(119866) ⩽ 9

Theorem 176 (Dehgardi et al [128]) For a complete bipartitegraph 119870

119901119902 if 2119896 + 1 ⩽ 119901 ⩽ 119902 then 119887

119903119896(119870

119901119902) = 119901

Theorem177 (Dehgardi et al [128]) For a complete graph119870119899

if 119899 ⩾ 119896 + 1 then 119887119903119896(119870

119899) = lceil119896119899(119896 + 1)rceil

The special case 119896 = 1 of Theorem 177 can be found inTheorem 1 The following is a simple observation similar toObservation 1

Observation 6 (Dehgardi et al [128]) Let 119866 be a graphwith maximum degree at least two 119867 a spanning subgraphobtained from 119866 by removing 119896 edges If 120574

119903119896(119867) = 120574

119903119896(119866)

then 119887119903119896(119867) ⩽ 119887

119903119896(119866) ⩽ 119887

119903119896(119867) + 119896

Theorem 178 (Dehgardi et al [128]) For every graph 119866 with120574119903119896(119866) = 119896120574(119866) 119887

119903119896(119866) ⩾ 119887(119866)

A graph 119866 is said to be vertex 119896-rainbow domination-critical if 120574

119903119896(119866 minus 119909) lt 120574

119903119896(119866) for every vertex 119909 in 119866 It is

clear that if 119866 has no isolated vertices then 120574119903119896(119866) ⩽ 119896120574(119866) ⩽

119896120573(119866) where 120573(119866) is the vertex-covering number of119866 Thusif 120574

119903119896(119866) = 119896120573(119866) then 120574

119903119896(119866) = 119896120574(119866)

Theorem 179 (Dehgardi et al [128]) For any graph 119866 if120574119903119896(119866) = 119896120573(119866) then

(a) 119887119903119896(119866) ⩾ 120575(119866)

(b) 119887119903119896(119866) ⩾ 120575(119866)+1 if119866 is vertex 119896-rainbow domination-

critical

As far as we know there are no more results on the 119896-rainbow bondage number of graphs in the literature

28 International Journal of Combinatorics

10 Results on Digraphs

Although domination has been extensively studied in undi-rected graphs it is natural to think of a dominating setas a one-way relationship between vertices of the graphIndeed among the earliest literatures on this subject vonNeumann and Morgenstern [129] used what is now calleddomination in digraphs to find solution (or kernels whichare independent dominating sets) for cooperative 119899-persongames Most likely the first formulation of domination byBerge [130] in 1958 was given in the context of digraphs andonly some years latter by Ore [131] in 1962 for undirectedgraphs Despite this history examination of domination andits variants in digraphs has been essentially overlooked (see[4] for an overview of the domination literature) Thus thereare few if any such results on domination for digraphs in theliterature

The bondage number and its related topics for undirectedgraph have become one of major areas both in theoreticaland applied researches However until recently Carlsonand Develin [42] Shan and Kang [132] and Huang andXu [133 134] studied the bondage number for digraphsindependently In this section we introduce their resultsfor general digraphs Results for some special digraphs suchas vertex-transitive digraphs are introduced in the nextsection

101 Upper Bounds for General Digraphs Let 119866 = (119881 119864)

be a digraph without loops and parallel edges A digraph 119866is called to be asymmetric if whenever (119909 119910) isin 119864(119866) then(119910 119909) notin 119864(119866) and to be symmetric if (119909 119910) isin 119864(119866) implies(119910 119909) isin 119864(119866) For a vertex 119909 of 119881(119866) the sets of out-neighbors and in-neighbors of 119909 are respectively defined as119873

+

119866(119909) = 119910 isin 119881(119866) (119909 119910) isin 119864(119866) and 119873minus

119866(119909) = 119909 isin

119881(119866) (119909 119910) isin 119864(119866) the out-degree and the in-degree of119909 are respectively defined as 119889+

119866(119909) = |119873

+

119866(119909)| and 119889minus

119866(119909) =

|119873+

119866(119909)| Denote themaximum and theminimum out-degree

(resp in-degree) of 119866 by Δ+(119866) and 120575+(119866) (resp Δminus(119866) and120575minus

(119866))The degree 119889119866(119909) of 119909 is defined as 119889+

119866(119909)+119889

minus

119866(119909) and

the maximum and the minimum degrees of119866 are denoted byΔ(119866) and 120575(119866) respectively that is Δ(119866) = max119889

119866(119909) 119909 isin

119881(119866) and 120575(119866) = min119889119866(119909) 119909 isin 119881(119866) Note that the

definitions here are different from the ones in the textbookon digraphs see for example Xu [1]

A subset 119878 of119881(119866) is called a dominating set if119881(119866) = 119878cup119873

+

119866(119878) where119873+

119866(119878) is the set of out-neighbors of 119878Then just

as for undirected graphs 120574(119866) is the minimum cardinalityof a dominating set and the bondage number 119887(119866) is thesmallest cardinality of a set 119861 of edges such that 120574(119866 minus 119861) gt120574(119866) if such a subset 119861 sube 119864(119866) exists Otherwise we put119887(119866) = infin

Some basic results for undirected graphs stated inSection 3 can be generalized to digraphs For exampleTheorem 18 is generalized by Carlson and Develin [42]Huang andXu [133] and Shan andKang [132] independentlyas follows

Theorem 180 Let 119866 be a digraph and (119909 119910) isin 119864(119866) Then119887(119866) ⩽ 119889

119866(119910) + 119889

minus

119866(119909) minus |119873

minus

119866(119909) cap 119873

minus

119866(119910)|

Corollary 181 For a digraph 119866 119887(119866) ⩽ 120575minus(119866) + Δ(119866)

Since 120575minus

(119866) ⩽ (12)Δ(119866) from Corollary 181Conjecture 53 is valid for digraphs

Corollary 182 For a digraph 119866 119887(119866) ⩽ (32)Δ(119866)

In the case of undirected graphs the bondage number of119866119899= 119870

119899times 119870

119899achieves this bound However it was shown

by Carlson and Develin in [42] that if we take the symmetricdigraph119866

119899 we haveΔ(119866

119899) = 4(119899minus1) 120574(119866

119899) = 119899 and 119887(119866

119899) ⩽

4(119899 minus 1) + 1 = Δ(119866119899) + 1 So this family of digraphs cannot

show that the bound in Corollary 182 is tightCorresponding to Conjecture 51 which is discredited

for undirected graphs and is valid for digraphs the sameconjecture for digraphs can be proposed as follows

Conjecture 183 (Carlson and Develin [42] 2006) 119887(119866) ⩽Δ(119866) + 1 for any digraph 119866

If Conjecture 183 is true then the following results showsthat this upper bound is tight since 119887(119870

119899times119870

119899) = Δ(119870

119899times119870

119899)+1

for a complete digraph 119870119899

In 2007 Shan and Kang [132] gave some tight upperbounds on the bondage numbers for some asymmetricdigraphs For example 119887(119879) ⩽ Δ(119879) for any asymmetricdirected tree 119879 119887(119866) ⩽ Δ(119866) for any asymmetric digraph 119866with order at least 4 and 120574(119866) ⩽ 2 For planar digraphs theyobtained the following results

Theorem 184 (Shan and Kang [132] 2007) Let 119866 be aasymmetric planar digraphThen 119887(119866) ⩽ Δ(119866)+2 and 119887(119866) ⩽Δ(119866) + 1 if Δ(119866) ⩾ 5 and 119889minus

119866(119909) ⩾ 3 for every vertex 119909 with

119889119866(119909) ⩾ 4

102 Results for Some Special Digraphs The exact values andbounds on 119887(119866) for some standard digraphs are determined

Theorem185 (Huang andXu [133] 2006) For a directed cycle119862119899and a directed path 119875

119899

119887 (119862119899) =

3 119894119891 119899 119894119904 119900119889119889

2 119894119891 119899 119894119904 119890V119890119899

119887 (119875119899) =

2 119894119891 119899 119894119904 119900119889119889

1 119894119891 119899 119894119904 119890V119890119899

(76)

For the de Bruijn digraph 119861(119889 119899) and the Kautz digraph119870(119889 119899)

119887 (119861 (119889 119899)) = 119889 119894119891 119899 119894119904 119900119889119889

119889 ⩽ 119887 (119861 (119889 119899)) ⩽ 2119889 119894119891 119899 119894119904 119890V119890119899

119887 (119870 (119889 119899)) = 119889 + 1

(77)

Like undirected graphs we can define the total domi-nation number and the total bondage number On the totalbondage numbers for some special digraphs the knownresults are as follows

International Journal of Combinatorics 29

Theorem 186 (Huang and Xu [134] 2007) For a directedcycle 119862

119899and a directed path 119875

119899 119887

119879(119875

119899) and 119887

119879(119862

119899) all do not

exist For a complete digraph 119870119899

119887119879(119870

119899) = infin 119894119891 119899 = 1 2

119887119879(119870

119899) = 3 119894119891 119899 = 3

119899 ⩽ 119887119879(119870

119899) ⩽ 2119899 minus 3 119894119891 119899 ⩾ 4

(78)

The extended de Bruijn digraph EB(119889 119899 1199021 119902

119901) and

the extended Kautz digraph EK(119889 119899 1199021 119902

119901) are intro-

duced by Shibata and Gonda [135] If 119901 = 1 then theyare the de Bruijn digraph B(119889 119899) and the Kautz digraphK(119889 119899) respectively Huang and Xu [134] determined theirtotal domination numbers In particular their total bondagenumbers for general cases are determined as follows

Theorem 187 (Huang andXu [134] 2007) If 119889 ⩾ 2 and 119902119894⩾ 2

for each 119894 = 1 2 119901 then

119887119879(EB(119889 119899 119902

1 119902

119901)) = 119889

119901

minus 1

119887119879(EK(119889 119899 119902

1 119902

119901)) = 119889

119901

(79)

In particular for the de Bruijn digraph B(119889 119899) and the Kautzdigraph K(119889 119899)

119887119879(B (119889 119899)) = 119889 minus 1 119887

119879(K (119889 119899)) = 119889 (80)

Zhang et al [136] determined the bondage number incomplete 119905-partite digraphs

Theorem 188 (Zhang et al [136] 2009) For a complete 119905-partite digraph 119870

11989911198992119899119905

where 1198991⩽ 119899

2⩽ sdot sdot sdot ⩽ 119899

119905

119887 (11987011989911198992119899119905

)

=

119898 119894119891 119899119898= 1 119899

119898+1⩾ 2 119891119900119903 119904119900119898119890

119898 (1 ⩽ 119898 lt 119905)

4119905 minus 3 119894119891 1198991= 119899

2= sdot sdot sdot = 119899

119905= 2

119905minus1

sum

119894=1

119899119894+ 2 (119905 minus 1) 119900119905ℎ119890119903119908119894119904119890

(81)

Since an undirected graph can be thought of as a sym-metric digraph any result for digraphs has an analogy forundirected graphs in general In view of this point studyingthe bondage number for digraphs is more significant thanfor undirected graphs Thus we should further study thebondage number of digraphs and try to generalize knownresults on the bondage number and related variants for undi-rected graphs to digraphs prove or disprove Conjecture 183In particular determine the exact values of 119887(B(119889 119899)) for aneven 119899 and 119887

119879(119870

119899) for 119899 ⩾ 4

11 Efficient Dominating Sets

A dominating set 119878 of a graph 119866 is called to be efficient if forevery vertex 119909 in119866 |119873

119866[119909]cap119878| = 1 if119866 is a undirected graph

or |119873minus

119866[119909] cap 119878| = 1 if 119866 is a directed graph The concept of an

efficient set was proposed by Bange et al [137] in 1988 In theliterature an efficient set is also called a perfect dominatingset (see Livingston and Stout [138] for an explanation)

From definition if 119878 is an efficient dominating set of agraph 119866 then 119878 is certainly an independent set and everyvertex not in 119878 is adjacent to exactly one vertex in 119878

It is also clear from definition that a dominating set 119878 isefficient if and only if N

119866[119878] = 119873

119866[119909] 119909 isin 119878 for the

undirected graph 119866 or N+

119866[119878] = 119873

+

119866[119909] 119909 isin 119878 for the

digraph119866 is a partition of119881(119866) where the induced subgraphby119873

119866[119909] or119873+

119866[119909] is a star or an out-star with the root 119909

The efficient domination has important applications inmany areas such as error-correcting codes and receivesmuch attention in the late years

The concept of efficient dominating sets is a measureof the efficiency of domination in graphs Unfortunately asshown in [137] not every graph has an efficient dominatingset andmoreover it is anNP-complete problem to determinewhether a given graph has an efficient dominating set Inaddition it was shown by Clark [139] in 1993 that for a widerange of 119901 almost every random undirected graph 119866 isin

G(120592 119901) has no efficient dominating sets This means thatundirected graphs possessing an efficient dominating set arerare However it is easy to show that every undirected graphhas an orientationwith an efficient dominating set (see Bangeet al [140])

In 1993 Barkauskas and Host [141] showed that deter-mining whether an arbitrary oriented graph has an efficientdominating set is NP-complete Even so the existence ofefficient dominating sets for some special classes of graphs hasbeen examined see for example Dejter and Serra [142] andLee [143] for Cayley graph Gu et al [144] formeshes and toriObradovic et al [145] Huang and Xu [36] Reji Kumar andMacGillivray [146] for circulant graphs Harary graphs andtori and Van Wieren et al [147] for cube-connected cycles

In this section we introduce results of the bondagenumber for some graphs with efficient dominating sets dueto Huang and Xu [36]

111 Results for General Graphs In this subsection we intro-duce some results on bondage numbers obtained by applyingefficient dominating sets We first state the two followinglemmas

Lemma 189 For any 119896-regular graph or digraph 119866 of order 119899120574(119866) ⩾ 119899(119896 + 1) with equality if and only if 119866 has an efficientdominating set In addition if 119866 has an efficient dominatingset then every efficient dominating set is certainly a 120574-set andvice versa

Let 119890 be an edge and 119878 a dominating set in 119866 We say that119890 supports 119878 if 119890 isin (119878 119878) where (119878 119878) = (119909 119910) isin 119864(119866) 119909 isin 119878 119910 isin 119878 Denote by 119904(119866) the minimum number of edgeswhich support all 120574-sets in 119866

Lemma 190 For any graph or digraph 119866 119887(119866) ⩾ 119904(119866) withequality if 119866 is regular and has an efficient dominating set

30 International Journal of Combinatorics

A graph 119866 is called to be vertex-transitive if its automor-phism group Aut(119866) acts transitively on its vertex-set 119881(119866)A vertex-transitive graph is regular Applying Lemmas 189and 190 Huang and Xu obtained some results on bondagenumbers for vertex-transitive graphs or digraphs

Theorem 191 (Huang and Xu [36] 2008) For any vertex-transitive graph or digraph 119866 of order 119899

119887 (119866) ⩾

lceil119899

2120574 (119866)rceil 119894119891 119866 119894119904 119906119899119889119894119903119890119888119905119890119889

lceil119899

120574 (119866)rceil 119894119891 119866 119894119904 119889119894119903119890119888119905119890119889

(82)

Theorem192 (Huang andXu [36] 2008) Let119866 be a 119896-regulargraph of order 119899 Then

119887(119866) ⩽ 119896 if119866 is undirected and 119899 ⩾ 120574(119866)(119896+1)minus119896+1119887(119866) ⩽ 119896+1+ℓ if119866 is directed and 119899 ⩾ 120574(119866)(119896+1)minusℓwith 0 ⩽ ℓ ⩽ 119896 minus 1

Next we introduce a better upper bound on 119887(119866) To thisaim we need the following concept which is a generalizationof the concept of the edge-covering of a graph 119866 For 1198811015840

sube

119881(119866) and 1198641015840 sube 119864(119866) we say that 1198641015840covers 1198811015840 and call 1198641015840an edge-covering for 1198811015840 if there exists an edge (119909 119910) isin 1198641015840 forany vertex 119909 isin 1198811015840 For 119910 isin 119881(119866) let 1205731015840[119910] be the minimumcardinality over all edge-coverings for119873minus

119866[119910]

Theorem 193 (Huang and Xu [36] 2008) If 119866 is a 119896-regulargraph with order 120574(119866)(119896 + 1) then 119887(119866) ⩽ 1205731015840[119910] for any 119910 isin119881(119866)

Theupper bound on 119887(119866) given inTheorem 193 is tight inview of 119887(119862

119899) for a cycle or a directed cycle 119862

119899(seeTheorems

1 and 185 resp)It is easy to see that for a 119896-regular graph 119866 lceil(119896 + 1)2rceil ⩽

1205731015840

[119910] ⩽ 119896 when 119866 is undirected and 1205731015840[119910] = 119896 + 1 when 119866 isdirected By this fact and Lemma 189 the following theoremis merely a simple combination of Theorems 191 and 193

Theorem 194 (Huang and Xu [36] 2008) Let 119866 be a vertex-transitive graph of degree 119896 If 119866 has an efficient dominatingset then

lceil119896 + 1

2rceil ⩽ 119887 (119866) ⩽ 119896 119894119891 119866 119894119904 119906119899119889119894119903119890119888119905119890119889

119887 (119866) = 119896 + 1 119894119891 119866 119894119904 119889119894119903119890119888119905119890119889

(83)

Theorem 195 (Huang and Xu [36] 2008) If 119866 is an undi-rected vertex-transitive cubic graph with order 4120574(119866) and girth119892(119866) ⩽ 5 then 119887(119866) = 2

The proof of Theorem 195 leads to a byproduct If 119892 =5 let (119906

1 119906

2 119906

3 119906

4 119906

5) be a cycle and let D

119894be the family

of efficient dominating sets containing 119906119894for each 119894 isin

1 2 3 4 5 ThenD119894capD

119895= 0 for 119894 = 1 2 5 Then 119866 has

at least 5119904 efficient dominating sets where 119904 = 119904(119866) But thereare only 4s (=ns120574) distinct efficient dominating sets in119866This

contradiction implies that an undirected vertex-transitivecubic graph with girth five has no efficient dominating setsBut a similar argument for 119892(119866) = 3 4 or 119892(119866) ⩾ 6 couldnot give any contradiction This is consistent with the resultthat CCC(119899) a vertex-transitive cubic graph with girth 119899 if3 ⩽ 119899 ⩽ 8 or girth 8 if 119899 ⩾ 9 has efficient dominating setsfor all 119899 ⩾ 3 except 119899 = 5 (see Theorem 199 in the followingsubsection)

112 Results for Cayley Graphs In this subsection we useTheorem 194 to determine the exact values or approximativevalues of bondage numbers for some special vertex-transitivegraphs by characterizing the existence of efficient dominatingsets in these graphs

Let Γ be a nontrivial finite group 119878 a nonempty subset ofΓ without the identity element of Γ A digraph 119866 is defined asfollows

119881 (119866) = Γ (119909 119910) isin 119864 (119866) lArrrArr 119909minus1

119910 isin 119878

for any 119909 119910 isin Γ(84)

is called a Cayley digraph of the group Γ with respect to119878 denoted by 119862

Γ(119878) If 119878minus1 = 119904minus1 119904 isin 119878 = 119878 then

119862Γ(119878) is symmetric and is called a Cayley undirected graph

a Cayley graph for short Cayley graphs or digraphs arecertainly vertex-transitive (see Xu [1])

A circulant graph 119866(119899 119878) of order 119899 is a Cayley graph119862119885119899

(119878) where 119885119899= 0 119899 minus 1 is the addition group of

order 119899 and 119878 is a nonempty subset of119885119899without the identity

element and hence is a vertex-transitive digraph of degree|119878| If 119878minus1 = 119878 then119866(119899 119878) is an undirected graph If 119878 = 1 119896where 2 ⩽ 119896 ⩽ 119899 minus 2 we write 119866(119899 1 119896) for 119866(119899 1 119896) or119866(119899 plusmn1 plusmn119896) and call it a double-loop circulant graph

Huang and Xu [36] showed that for directed 119866 =

119866(119899 1 119896) lceil1198993rceil ⩽ 120574(119866) ⩽ lceil1198992rceil and 119866 has an efficientdominating set if and only if 3 | 119899 and 119896 equiv 2 (mod 3) fordirected 119866 = 119866(119899 1 119896) and 119896 = 1198992 lceil1198995rceil ⩽ 120574(119866) ⩽ lceil1198993rceiland 119866 has an efficient dominating set if and only if 5 | 119899 and119896 equiv plusmn2 (mod 5) By Theorem 194 we obtain the bondagenumber of a double loop circulant graph if it has an efficientdominating set

Theorem 196 (Huang and Xu [36] 2008) Let 119866 be a double-loop circulant graph 119866(119899 1 119896) If 119866 is directed with 3 | 119899and 119896 equiv 2 (mod 3) or 119866 is undirected with 5 | 119899 and119896 equiv plusmn2 (mod 5) then 119887(119866) = 3

The 119898 times 119899 torus is the Cartesian product 119862119898times 119862

119899of

two cycles and is a Cayley graph 119862119885119898times119885119899

(119878) where 119878 =

(0 1) (1 0) for directed cycles and 119878 = (0 plusmn1) (plusmn1 0) forundirected cycles and hence is vertex-transitive Gu et al[144] showed that the undirected torus119862

119898times119862

119899has an efficient

dominating set if and only if both 119898 and 119899 are multiplesof 5 Huang and Xu [36] showed that the directed torus119862119898times 119862

119899has an efficient dominating set if and only if both

119898 and 119899 are multiples of 3 Moreover they found a necessarycondition for a dominating set containing the vertex (0 0)in 119862

119898times 119862

119899to be efficient and obtained the following

result

International Journal of Combinatorics 31

Theorem 197 (Huang and Xu [36] 2008) Let 119866 = 119862119898times 119862

119899

If 119866 is undirected and both119898 and 119899 are multiples of 5 or if 119866 isdirected and both119898 and 119899 are multiples of 3 then 119887(119866) = 3

The hypercube 119876119899

is the Cayley graph 119862Γ(119878)

where Γ = 1198852times sdot sdot sdot times 119885

2= (119885

2)119899 and 119878 =

100 sdot sdot sdot 0 010 sdot sdot sdot 0 00 sdot sdot sdot 01 Lee [143] showed that119876119899has an efficient dominating set if and only if 119899 = 2119898 minus 1

for a positive integer119898 ByTheorem 194 the following resultis obtained immediately

Theorem 198 (Huang and Xu [36] 2008) If 119899 = 2119898 minus 1 for apositive integer119898 then 2119898minus1

⩽ 119887(119876119899) ⩽ 2

119898

minus 1

Problem 10 Determine the exact value of 119887(119876119899) for 119899 = 2119898minus1

The 119899-dimensional cube-connected cycle denoted byCCC(119899) is constructed from the 119899-dimensional hypercube119876119899by replacing each vertex 119909 in 119876

119899with an undirected cycle

119862119899of length 119899 and linking the 119894th vertex of the 119862

119899to the 119894th

neighbor of 119909 It has been proved that CCC(119899) is a Cayleygraph and hence is a vertex-transitive graph with degree 3Van Wieren et al [147] proved that CCC(119899) has an efficientdominating set if and only if 119899 = 5 From Theorems 194 and195 the following result can be derived

Theorem 199 (Huang and Xu [36] 2008) Let 119866 = 119862119862119862(119899)be the 119899-dimensional cube-connected cycles with 119899 ⩾ 3 and119899 = 5 Then 120574(119866) = 1198992119899minus2 and 2 ⩽ 119887(119866) ⩽ 3 In addition119887(119862119862119862(3)) = 119887(119862119862119862(4)) = 2

Problem 11 Determine the exact value of 119887(CCC(119899)) for 119899 ⩾5

Acknowledgments

This work was supported by NNSF of China (Grants nos11071233 61272008) The author would like to express hisgratitude to the anonymous referees for their kind sugges-tions and comments on the original paper to ProfessorSheikholeslami and Professor Jafari Rad for sending thereferences [128] and [81] which resulted in this version

References

[1] J-M Xu Theory and Application of Graphs vol 10 of NetworkTheory and Applications Kluwer Academic Dordrecht TheNetherlands 2003

[2] S THedetniemi andR C Laskar ldquoBibliography on dominationin graphs and some basic definitions of domination parame-tersrdquo Discrete Mathematics vol 86 no 1ndash3 pp 257ndash277 1990

[3] TW Haynes S T Hedetniemi and P J Slater Fundamentals ofDomination in Graphs vol 208 ofMonographs and Textbooks inPure and Applied Mathematics Marcel Dekker New York NYUSA 1998

[4] TWHaynes S THedetniemi and P J Slater EdsDominationin Graphs vol 209 of Monographs and Textbooks in Pure andAppliedMathematicsMarcelDekker NewYorkNYUSA 1998

[5] M R Garey andD S JohnsonComputers and Intractability WH Freeman and Company San Francisco Calif USA 1979

[6] H B Walikar and B D Acharya ldquoDomination critical graphsrdquoNational Academy Science Letters vol 2 pp 70ndash72 1979

[7] D Bauer F Harary J Nieminen and C L Suffel ldquoDominationalteration sets in graphsrdquo Discrete Mathematics vol 47 no 2-3pp 153ndash161 1983

[8] J F Fink M S Jacobson L F Kinch and J Roberts ldquoThebondage number of a graphrdquo Discrete Mathematics vol 86 no1ndash3 pp 47ndash57 1990

[9] F-T Hu and J-M Xu ldquoThe bondage number of (119899 minus 3)-regulargraph G of order nrdquo Ars Combinatoria to appear

[10] U Teschner ldquoNew results about the bondage number of agraphrdquoDiscreteMathematics vol 171 no 1ndash3 pp 249ndash259 1997

[11] B L Hartnell and D F Rall ldquoA bound on the size of a graphwith given order and bondage numberrdquo Discrete Mathematicsvol 197-198 pp 409ndash413 1999 16th British CombinatorialConference (London 1997)

[12] B L Hartnell and D F Rall ldquoA characterization of trees inwhich no edge is essential to the domination numberrdquo ArsCombinatoria vol 33 pp 65ndash76 1992

[13] J Topp and P D Vestergaard ldquo120572119896- and 120574

119896-stable graphsrdquo

Discrete Mathematics vol 212 no 1-2 pp 149ndash160 2000[14] A Meir and J W Moon ldquoRelations between packing and

covering numbers of a treerdquo Pacific Journal of Mathematics vol61 no 1 pp 225ndash233 1975

[15] B L Hartnell L K Joslashrgensen P D Vestergaard and CA Whitehead ldquoEdge stability of the k-domination numberof treesrdquo Bulletin of the Institute of Combinatorics and ItsApplications vol 22 pp 31ndash40 1998

[16] F-T Hu and J-M Xu ldquoOn the complexity of the bondage andreinforcement problemsrdquo Journal of Complexity vol 28 no 2pp 192ndash201 2012

[17] B L Hartnell and D F Rall ldquoBounds on the bondage numberof a graphrdquo Discrete Mathematics vol 128 no 1ndash3 pp 173ndash1771994

[18] Y-L Wang ldquoOn the bondage number of a graphrdquo DiscreteMathematics vol 159 no 1ndash3 pp 291ndash294 1996

[19] G H Fricke TW Haynes S M Hedetniemi S T Hedetniemiand R C Laskar ldquoExcellent treesrdquo Bulletin of the Institute ofCombinatorics and Its Applications vol 34 pp 27ndash38 2002

[20] V Samodivkin ldquoThe bondage number of graphs good and badverticesrdquoDiscussiones Mathematicae GraphTheory vol 28 no3 pp 453ndash462 2008

[21] J E Dunbar T W Haynes U Teschner and L VolkmannldquoBondage insensitivity and reinforcementrdquo in Domination inGraphs vol 209 of Monographs and Textbooks in Pure andAppliedMathematics pp 471ndash489 Dekker NewYork NY USA1998

[22] H Liu and L Sun ldquoThe bondage and connectivity of a graphrdquoDiscrete Mathematics vol 263 no 1ndash3 pp 289ndash293 2003

[23] A-H Esfahanian and S L Hakimi ldquoOn computing a con-ditional edge-connectivity of a graphrdquo Information ProcessingLetters vol 27 no 4 pp 195ndash199 1988

[24] R C Brigham P Z Chinn and R D Dutton ldquoVertexdomination-critical graphsrdquoNetworks vol 18 no 3 pp 173ndash1791988

[25] V Samodivkin ldquoDomination with respect to nondegenerateproperties bondage numberrdquoThe Australasian Journal of Com-binatorics vol 45 pp 217ndash226 2009

[26] U Teschner ldquoA new upper bound for the bondage numberof graphs with small domination numberrdquo The AustralasianJournal of Combinatorics vol 12 pp 27ndash35 1995

32 International Journal of Combinatorics

[27] J Fulman D Hanson and G MacGillivray ldquoVertex dom-ination-critical graphsrdquoNetworks vol 25 no 2 pp 41ndash43 1995

[28] U Teschner ldquoA counterexample to a conjecture on the bondagenumber of a graphrdquo Discrete Mathematics vol 122 no 1ndash3 pp393ndash395 1993

[29] U Teschner ldquoThe bondage number of a graph G can be muchgreater than Δ(G)rdquo Ars Combinatoria vol 43 pp 81ndash87 1996

[30] L Volkmann ldquoOn graphs with equal domination and coveringnumbersrdquoDiscrete AppliedMathematics vol 51 no 1-2 pp 211ndash217 1994

[31] L A Sanchis ldquoMaximum number of edges in connected graphswith a given domination numberrdquoDiscreteMathematics vol 87no 1 pp 65ndash72 1991

[32] S Klavzar and N Seifter ldquoDominating Cartesian products ofcyclesrdquoDiscrete AppliedMathematics vol 59 no 2 pp 129ndash1361995

[33] H K Kim ldquoA note on Vizingrsquos conjecture about the bondagenumberrdquo Korean Journal of Mathematics vol 10 no 2 pp 39ndash42 2003

[34] M Y Sohn X Yuan and H S Jeong ldquoThe bondage number of1198623times119862

119899rdquo Journal of the KoreanMathematical Society vol 44 no

6 pp 1213ndash1231 2007[35] L Kang M Y Sohn and H K Kim ldquoBondage number of the

discrete torus 119862119899times 119862

4rdquo Discrete Mathematics vol 303 no 1ndash3

pp 80ndash86 2005[36] J Huang and J-M Xu ldquoThe bondage numbers and efficient

dominations of vertex-transitive graphsrdquoDiscrete Mathematicsvol 308 no 4 pp 571ndash582 2008

[37] C J Xiang X Yuan andM Y Sohn ldquoDomination and bondagenumber of119862

3times119862

119899rdquoArs Combinatoria vol 97 pp 299ndash310 2010

[38] M S Jacobson and L F Kinch ldquoOn the domination numberof products of graphs Irdquo Ars Combinatoria vol 18 pp 33ndash441984

[39] Y-Q Li ldquoA note on the bondage number of a graphrdquo ChineseQuarterly Journal of Mathematics vol 9 no 4 pp 1ndash3 1994

[40] F-T Hu and J-M Xu ldquoBondage number of mesh networksrdquoFrontiers ofMathematics inChina vol 7 no 5 pp 813ndash826 2012

[41] R Frucht and F Harary ldquoOn the corona of two graphsrdquoAequationes Mathematicae vol 4 pp 322ndash325 1970

[42] K Carlson and M Develin ldquoOn the bondage number of planarand directed graphsrdquoDiscreteMathematics vol 306 no 8-9 pp820ndash826 2006

[43] U Teschner ldquoOn the bondage number of block graphsrdquo ArsCombinatoria vol 46 pp 25ndash32 1997

[44] J Huang and J-M Xu ldquoNote on conjectures of bondagenumbers of planar graphsrdquo Applied Mathematical Sciences vol6 no 65ndash68 pp 3277ndash3287 2012

[45] L Kang and J Yuan ldquoBondage number of planar graphsrdquoDiscrete Mathematics vol 222 no 1ndash3 pp 191ndash198 2000

[46] M Fischermann D Rautenbach and L Volkmann ldquoRemarkson the bondage number of planar graphsrdquo Discrete Mathemat-ics vol 260 no 1ndash3 pp 57ndash67 2003

[47] J Hou G Liu and J Wu ldquoBondage number of planar graphswithout small cyclesrdquoUtilitasMathematica vol 84 pp 189ndash1992011

[48] J Huang and J-M Xu ldquoThe bondage numbers of graphs withsmall crossing numbersrdquo Discrete Mathematics vol 307 no 15pp 1881ndash1897 2007

[49] Q Ma S Zhang and J Wang ldquoBondage number of 1-planargraphrdquo Applied Mathematics vol 1 no 2 pp 101ndash103 2010

[50] J Hou and G Liu ldquoA bound on the bondage number of toroidalgraphsrdquoDiscreteMathematics Algorithms and Applications vol4 no 3 Article ID 1250046 5 pages 2012

[51] Y-C Cao J-M Xu and X-R Xu ldquoOn bondage number oftoroidal graphsrdquo Journal of University of Science and Technologyof China vol 39 no 3 pp 225ndash228 2009

[52] Y-C Cao J Huang and J-M Xu ldquoThe bondage number ofgraphs with crossing number less than fourrdquo Ars Combinatoriato appear

[53] H K Kim ldquoOn the bondage numbers of some graphsrdquo KoreanJournal of Mathematics vol 11 no 2 pp 11ndash15 2004

[54] A Gagarin and V Zverovich ldquoUpper bounds for the bondagenumber of graphs on topological surfacesrdquo Discrete Mathemat-ics 2012

[55] J Huang ldquoAn improved upper bound for the bondage numberof graphs on surfacesrdquoDiscrete Mathematics vol 312 no 18 pp2776ndash2781 2012

[56] A Gagarin and V Zverovich ldquoThe bondage number of graphson topological surfaces and Teschnerrsquos conjecturerdquo DiscreteMathematics vol 313 no 6 pp 796ndash808 2013

[57] V Samodivkin ldquoNote on the bondage number of graphs ontopological surfacesrdquo httparxivorgabs12086203

[58] E J Cockayne R M Dawes and S T Hedetniemi ldquoTotaldomination in graphsrdquoNetworks vol 10 no 3 pp 211ndash219 1980

[59] R Laskar J Pfaff S M Hedetniemi and S T HedetniemildquoOn the algorithmic complexity of total dominationrdquo Society forIndustrial and Applied Mathematics vol 5 no 3 pp 420ndash4251984

[60] J Pfaff R C Laskar and S T Hedetniemi ldquoNP-completeness oftotal and connected domination and irredundance for bipartitegraphsrdquo Tech Rep 428 Department of Mathematical SciencesClemson University 1983

[61] M A Henning ldquoA survey of selected recent results on totaldomination in graphsrdquoDiscrete Mathematics vol 309 no 1 pp32ndash63 2009

[62] V R Kulli and D K Patwari ldquoThe total bondage number ofa graphrdquo in Advances in Graph Theory pp 227ndash235 VishwaGulbarga India 1991

[63] F-T Hu Y Lu and J-M Xu ldquoThe total bondage number ofgrid graphsrdquo Discrete Applied Mathematics vol 160 no 16-17pp 2408ndash2418 2012

[64] J Cao M Shi and B Wu ldquoResearch on the total bondagenumber of a spectial networkrdquo in Proceedings of the 3rdInternational Joint Conference on Computational Sciences andOptimization (CSO rsquo10) pp 456ndash458 May 2010

[65] N Sridharan MD Elias and V S A Subramanian ldquoTotalbondage number of a graphrdquo AKCE International Journal ofGraphs and Combinatorics vol 4 no 2 pp 203ndash209 2007

[66] N Jafari Rad and J Raczek ldquoSome progress on total bondage ingraphsrdquo Graphs and Combinatorics 2011

[67] T W Haynes and P J Slater ldquoPaired-domination and thepaired-domatic numberrdquoCongressusNumerantium vol 109 pp65ndash72 1995

[68] T W Haynes and P J Slater ldquoPaired-domination in graphsrdquoNetworks vol 32 no 3 pp 199ndash206 1998

[69] X Hou ldquoA characterization of 2120574 120574119875-treesrdquoDiscrete Mathemat-

ics vol 308 no 15 pp 3420ndash3426 2008[70] K E Proffitt T W Haynes and P J Slater ldquoPaired-domination

in grid graphsrdquo Congressus Numerantium vol 150 pp 161ndash1722001

International Journal of Combinatorics 33

[71] H Qiao L KangM Cardei andD-Z Du ldquoPaired-dominationof treesrdquo Journal of Global Optimization vol 25 no 1 pp 43ndash542003

[72] E Shan L Kang and M A Henning ldquoA characterizationof trees with equal total domination and paired-dominationnumbersrdquo The Australasian Journal of Combinatorics vol 30pp 31ndash39 2004

[73] J Raczek ldquoPaired bondage in treesrdquo Discrete Mathematics vol308 no 23 pp 5570ndash5575 2008

[74] J A Telle and A Proskurowski ldquoAlgorithms for vertex parti-tioning problems on partial k-treesrdquo SIAM Journal on DiscreteMathematics vol 10 no 4 pp 529ndash550 1997

[75] G S Domke J H Hattingh M A Henning and L R MarkusldquoRestrained domination in treesrdquoDiscreteMathematics vol 211no 1ndash3 pp 1ndash9 2000

[76] G S Domke J H Hattingh M A Henning and L R MarkusldquoRestrained domination in graphs with minimum degree twordquoJournal of Combinatorial Mathematics and Combinatorial Com-puting vol 35 pp 239ndash254 2000

[77] G S Domke J H Hattingh S T Hedetniemi R C Laskarand L R Markus ldquoRestrained domination in graphsrdquo DiscreteMathematics vol 203 no 1ndash3 pp 61ndash69 1999

[78] J H Hattingh and E J Joubert ldquoAn upper bound for therestrained domination number of a graph with minimumdegree at least two in terms of order and minimum degreerdquoDiscrete Applied Mathematics vol 157 no 13 pp 2846ndash28582009

[79] M A Henning ldquoGraphs with large restrained dominationnumberrdquo Discrete Mathematics vol 197-198 pp 415ndash429 199916th British Combinatorial Conference (London 1997)

[80] J H Hattingh and A R Plummer ldquoRestrained bondage ingraphsrdquo Discrete Mathematics vol 308 no 23 pp 5446ndash54532008

[81] N Jafari Rad R Hasni J Raczek and L Volkmann ldquoTotalrestrained bondage in graphsrdquoActaMathematica Sinica vol 29no 6 pp 1033ndash1042 2013

[82] J Cyman and J Raczek ldquoOn the total restrained dominationnumber of a graphrdquoThe Australasian Journal of Combinatoricsvol 36 pp 91ndash100 2006

[83] M A Henning ldquoGraphs with large total domination numberrdquoJournal of Graph Theory vol 35 no 1 pp 21ndash45 2000

[84] D-X Ma X-G Chen and L Sun ldquoOn total restraineddomination in graphsrdquoCzechoslovakMathematical Journal vol55 no 1 pp 165ndash173 2005

[85] B Zelinka ldquoRemarks on restrained domination and totalrestrained domination in graphsrdquo Czechoslovak MathematicalJournal vol 55 no 2 pp 393ndash396 2005

[86] W Goddard and M A Henning ldquoIndependent domination ingraphs a survey and recent resultsrdquo Discrete Mathematics vol313 no 7 pp 839ndash854 2013

[87] J H Zhang H L Liu and L Sun ldquoIndependence bondagenumber and reinforcement number of some graphsrdquo Transac-tions of Beijing Institute of Technology vol 23 no 2 pp 140ndash1422003

[88] K D Dharmalingam Studies in Graph Theorymdashequitabledomination and bottleneck domination [PhD thesis] MaduraiKamaraj University Madurai India 2006

[89] GDeepakND Soner andAAlwardi ldquoThe equitable bondagenumber of a graphrdquo Research Journal of Pure Algebra vol 1 no9 pp 209ndash212 2011

[90] E Sampathkumar and H B Walikar ldquoThe connected domina-tion number of a graphrdquo Journal of Mathematical and PhysicalSciences vol 13 no 6 pp 607ndash613 1979

[91] K White M Farber and W Pulleyblank ldquoSteiner trees con-nected domination and strongly chordal graphsrdquoNetworks vol15 no 1 pp 109ndash124 1985

[92] M Cadei M X Cheng X Cheng and D-Z Du ldquoConnecteddomination in Ad Hoc wireless networksrdquo in Proceedings ofthe 6th International Conference on Computer Science andInformatics (CSampI rsquo02) 2002

[93] J F Fink and M S Jacobson ldquon-domination in graphsrdquo inGraph Theory with Applications to Algorithms and ComputerScience Wiley-Interscience Publications pp 283ndash300 WileyNew York NY USA 1985

[94] M Chellali O Favaron A Hansberg and L Volkmann ldquok-domination and k-independence in graphs a surveyrdquo Graphsand Combinatorics vol 28 no 1 pp 1ndash55 2012

[95] Y Lu X Hou J-M Xu andN Li ldquoTrees with uniqueminimump-dominating setsrdquo Utilitas Mathematica vol 86 pp 193ndash2052011

[96] Y Lu and J-M Xu ldquoThe p-bondage number of treesrdquo Graphsand Combinatorics vol 27 no 1 pp 129ndash141 2011

[97] Y Lu and J-M Xu ldquoThe 2-domination and 2-bondage numbersof grid graphs A manuscriptrdquo httparxivorgabs12044514

[98] M Krzywkowski ldquoNon-isolating 2-bondage in graphsrdquo TheJournal of the Mathematical Society of Japan vol 64 no 4 2012

[99] M A Henning O R Oellermann and H C Swart ldquoBoundson distance domination parametersrdquo Journal of CombinatoricsInformation amp System Sciences vol 16 no 1 pp 11ndash18 1991

[100] F Tian and J-M Xu ldquoBounds for distance domination numbersof graphsrdquo Journal of University of Science and Technology ofChina vol 34 no 5 pp 529ndash534 2004

[101] F Tian and J-M Xu ldquoDistance domination-critical graphsrdquoApplied Mathematics Letters vol 21 no 4 pp 416ndash420 2008

[102] F Tian and J-M Xu ldquoA note on distance domination numbersof graphsrdquo The Australasian Journal of Combinatorics vol 43pp 181ndash190 2009

[103] F Tian and J-M Xu ldquoAverage distances and distance domina-tion numbersrdquo Discrete Applied Mathematics vol 157 no 5 pp1113ndash1127 2009

[104] Z-L Liu F Tian and J-M Xu ldquoProbabilistic analysis of upperbounds for 2-connected distance k-dominating sets in graphsrdquoTheoretical Computer Science vol 410 no 38ndash40 pp 3804ndash3813 2009

[105] M A Henning O R Oellermann and H C Swart ldquoDistancedomination critical graphsrdquo Journal of Combinatorial Mathe-matics and Combinatorial Computing vol 44 pp 33ndash45 2003

[106] T J Bean M A Henning and H C Swart ldquoOn the integrityof distance domination in graphsrdquo The Australasian Journal ofCombinatorics vol 10 pp 29ndash43 1994

[107] M Fischermann and L Volkmann ldquoA remark on a conjecturefor the (kp)-domination numberrdquoUtilitasMathematica vol 67pp 223ndash227 2005

[108] T Korneffel D Meierling and L Volkmann ldquoA remark on the(22)-domination numberrdquo Discussiones Mathematicae GraphTheory vol 28 no 2 pp 361ndash366 2008

[109] Y Lu X Hou and J-M Xu ldquoOn the (22)-domination numberof treesrdquo Discussiones Mathematicae Graph Theory vol 30 no2 pp 185ndash199 2010

34 International Journal of Combinatorics

[110] H Li and J-M Xu ldquo(dm)-domination numbers of m-connected graphsrdquo Rapports de Recherche 1130 LRI UniversiteParis-Sud 1997

[111] S M Hedetniemi S T Hedetniemi and T V Wimer ldquoLineartime resource allocation algorithms for treesrdquo Tech Rep URI-014 Department of Mathematical Sciences Clemson Univer-sity 1987

[112] M Farber ldquoDomination independent domination and dualityin strongly chordal graphsrdquo Discrete Applied Mathematics vol7 no 2 pp 115ndash130 1984

[113] G S Domke and R C Laskar ldquoThe bondage and reinforcementnumbers of 120574

119891for some graphsrdquoDiscrete Mathematics vol 167-

168 pp 249ndash259 1997[114] G S Domke S T Hedetniemi and R C Laskar ldquoFractional

packings coverings and irredundance in graphsrdquo vol 66 pp227ndash238 1988

[115] E J Cockayne P A Dreyer Jr S M Hedetniemi andS T Hedetniemi ldquoRoman domination in graphsrdquo DiscreteMathematics vol 278 no 1ndash3 pp 11ndash22 2004

[116] I Stewart ldquoDefend the roman empirerdquo Scientific American vol281 pp 136ndash139 1999

[117] C S ReVelle and K E Rosing ldquoDefendens imperiumromanum a classical problem in military strategyrdquoThe Ameri-can Mathematical Monthly vol 107 no 7 pp 585ndash594 2000

[118] A Bahremandpour F-T Hu S M Sheikholeslami and J-MXu ldquoOn the Roman bondage number of a graphrdquo DiscreteMathematics Algorithms and Applications vol 5 no 1 ArticleID 1350001 15 pages 2013

[119] N Jafari Rad and L Volkmann ldquoRoman bondage in graphsrdquoDiscussiones Mathematicae Graph Theory vol 31 no 4 pp763ndash773 2011

[120] K Ebadi and L PushpaLatha ldquoRoman bondage and Romanreinforcement numbers of a graphrdquo International Journal ofContemporary Mathematical Sciences vol 5 pp 1487ndash14972010

[121] N Dehgardi S M Sheikholeslami and L Volkman ldquoOn theRoman k-bondage number of a graphrdquo AKCE InternationalJournal of Graphs and Combinatorics vol 8 pp 169ndash180 2011

[122] S Akbari and S Qajar ldquoA note on Roman bondage number ofgraphsrdquo Ars Combinatoria to Appear

[123] F-T Hu and J-M Xu ldquoRoman bondage numbers of somegraphsrdquo httparxivorgabs11093933

[124] N Jafari Rad and L Volkmann ldquoOn the Roman bondagenumber of planar graphsrdquo Graphs and Combinatorics vol 27no 4 pp 531ndash538 2011

[125] S Akbari M Khatirinejad and S Qajar ldquoA note on the romanbondage number of planar graphsrdquo Graphs and Combinatorics2012

[126] B Bresar M A Henning and D F Rall ldquoRainbow dominationin graphsrdquo Taiwanese Journal of Mathematics vol 12 no 1 pp213ndash225 2008

[127] B L Hartnell and D F Rall ldquoOn dominating the Cartesianproduct of a graph and 120572krdquo Discussiones Mathematicae GraphTheory vol 24 no 3 pp 389ndash402 2004

[128] N Dehgardi S M Sheikholeslami and L Volkman ldquoThe k-rainbow bondage number of a graphrdquo to appear in DiscreteApplied Mathematics

[129] J von Neumann and O Morgenstern Theory of Games andEconomic Behavior Princeton University Press Princeton NJUSA 1944

[130] C BergeTheorıe des Graphes et ses Applications 1958[131] O Ore Theory of Graphs vol 38 American Mathematical

Society Colloquium Publications 1962[132] E Shan and L Kang ldquoBondage number in oriented graphsrdquoArs

Combinatoria vol 84 pp 319ndash331 2007[133] J Huang and J-M Xu ldquoThe bondage numbers of extended

de Bruijn and Kautz digraphsrdquo Computers amp Mathematics withApplications vol 51 no 6-7 pp 1137ndash1147 2006

[134] J Huang and J-M Xu ldquoThe total domination and total bondagenumbers of extended de Bruijn and Kautz digraphsrdquoComputersamp Mathematics with Applications vol 53 no 8 pp 1206ndash12132007

[135] Y Shibata and Y Gonda ldquoExtension of de Bruijn graph andKautz graphrdquo Computers amp Mathematics with Applications vol30 no 9 pp 51ndash61 1995

[136] X Zhang J Liu and JMeng ldquoThebondage number in completet-partite digraphsrdquo Information Processing Letters vol 109 no17 pp 997ndash1000 2009

[137] D W Bange A E Barkauskas and P J Slater ldquoEfficient dom-inating sets in graphsrdquo in Applications of Discrete Mathematicspp 189ndash199 SIAM Philadelphia Pa USA 1988

[138] M Livingston and Q F Stout ldquoPerfect dominating setsrdquoCongressus Numerantium vol 79 pp 187ndash203 1990

[139] L Clark ldquoPerfect domination in random graphsrdquo Journal ofCombinatorial Mathematics and Combinatorial Computing vol14 pp 173ndash182 1993

[140] D W Bange A E Barkauskas L H Host and L H ClarkldquoEfficient domination of the orientations of a graphrdquo DiscreteMathematics vol 178 no 1ndash3 pp 1ndash14 1998

[141] A E Barkauskas and L H Host ldquoFinding efficient dominatingsets in oriented graphsrdquo Congressus Numerantium vol 98 pp27ndash32 1993

[142] I J Dejter and O Serra ldquoEfficient dominating sets in CayleygraphsrdquoDiscrete AppliedMathematics vol 129 no 2-3 pp 319ndash328 2003

[143] J Lee ldquoIndependent perfect domination sets in Cayley graphsrdquoJournal of Graph Theory vol 37 no 4 pp 213ndash219 2001

[144] WGu X Jia and J Shen ldquoIndependent perfect domination setsin meshes tori and treesrdquo Unpublished

[145] N Obradovic J Peters and G Ruzic ldquoEfficient domination incirculant graphswith two chord lengthsrdquo Information ProcessingLetters vol 102 no 6 pp 253ndash258 2007

[146] K Reji Kumar and G MacGillivray ldquoEfficient domination incirculant graphsrdquo Discrete Mathematics vol 313 no 6 pp 767ndash771 2013

[147] D Van Wieren M Livingston and Q F Stout ldquoPerfectdominating sets on cube-connected cyclesrdquo Congressus Numer-antium vol 97 pp 51ndash70 1993

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

International Journal of Combinatorics 19

119896

Figure 7 A tree 119879 with 119887119875(119879) = 119896

Theorem 120 (Raczek [73] 2008) For positive integers 119899 and119898

119887119875(119870

119898119899) = 119898 119891119900119903 1 lt 119898 ⩽ 119899

119887119875(119882

119899) =

4 119894119891 119899 = 4

3 119894119891 119899 = 5

2 119900119905ℎ119890119903119908119894119904119890

(45)

Theorem 16 shows that if 119879 is a nontrivial tree then119887(119879) ⩽ 2 However no similar result exists for pairedbondage For any nonnegative integer 119896 let 119879

119896be a tree

obtained by subdividing all but one edge of a star 1198701119896+1

(asshown in Figure 7) It is easy to see that 119887

119875(119879

119896) = 119896 We

state this fact as the following theorem which is similar toTheorem 112

Theorem121 (Raczek [73] 2008) For any nonnegative integer119896 there exists a tree 119879 with 119887

119875(119879) = 119896

Consider the tree defined in Figure 5 for 119896 = 3 Then119887(119879

3) ⩽ 2 and 119887

119875(119879

3) = 3 On the other hand 119887(119875

4) = 2

and 119887119875(119875

4) = 1 Thus we have what follows which is similar

to Corollary 113

Corollary 122 The bondage number and the paired bondagenumber are unrelated even for trees

A constructive characterization of trees with 119887(119879) = 2is given by Hartnell and Rall in [12] Raczek [73] provideda constructive characterization of trees with 119887

119875(119879) = 0 In

order to state the characterization we define a labeling andthree simple operations on a tree119879 Let 119910 isin 119881(119879) and let ℓ(119910)be the label assigned to 119910

Operation T1 If ℓ(119910) = 119861 add a vertex 119909 and the

edge 119910119909 and let ℓ(119909) = 119860OperationT

2 If ℓ(119910) = 119862 add a path (119909

1 119909

2) and the

edge 1199101199091 and let ℓ(119909

1) = 119861 ℓ(119909

2) = 119860

Operation T3 If ℓ(119910) = 119861 add a path (119909

1 119909

2 119909

3)

and the edge 1199101199091 and let ℓ(119909

1) = 119862 ℓ(119909

2) = 119861 and

ℓ(1199093) = 119860

Let 1198752= (119906 V) with ℓ(119906) = 119860 and ℓ(V) = 119861 Let T be

the class of all trees obtained from the labeled 1198752by a finite

sequence of operationsT1 T

2 T

3

A tree 119879 in Figure 8 belongs to the familyTRaczek [73] obtained the following characterization of all

trees 119879 with 119887119875(119879) = 0

119860

119860

119860

119860

119860

119861 119862

119861 119862 119861

119861119860

Figure 8 A tree 119879 belong to the familyT

Theorem 123 (Raczek [73] 2008) For a tree 119879 119887119875(119879) = 0 if

and only if 119879 is inT

We state the decision problem for the paired bondage asfollows

Problem 4 Consider the decision problem

Paired Bondage ProblemInstance a graph 119866 and a positive integer 119887Question is 119887

119875(119866) ⩽ 119887

Conjecture 124 The paired bondage problem is NP-complete

83 Restrained Bondage Numbers A dominating set 119878 of 119866 iscalled to be restrained if the subgraph induced by 119878 containsno isolated vertices where 119878 = 119881(119866) 119878 The restraineddomination number of 119866 denoted by 120574

119877(119866) is the smallest

cardinality of a restrained dominating set of 119866 It is clear that120574119877(119866) exists and 120574(119866) ⩽ 120574

119877(119866) for any nonempty graph 119866

The concept of restrained domination was introducedby Telle and Proskurowski [74] in 1997 albeit indirectly asa vertex partitioning problem Concerning the study of therestrained domination numbers the reader is referred to [74ndash79]

In 2008 Hattingh and Plummer [80] defined therestrained bondage number 119887R(119866) of a nonempty graph 119866 tobe the minimum cardinality among all subsets 119861 sube 119864(119866) forwhich 120574

119877(119866 minus 119861) gt 120574

119877(119866)

For some simple graphs their restrained dominationnumbers can be easily determined and so restrained bondagenumbers have been also determined For example it is clearthat 120574

119877(119870

119899) = 1 Domke et al [77] showed that 120574

119877(119862

119899) =

119899 minus 2lfloor1198993rfloor for 119899 ⩾ 3 and 120574119877(119875

119899) = 119899 minus 2lfloor(119899 minus 1)3rfloor for 119899 ⩾ 1

Using these results Hattingh and Plummer [80] obtained therestricted bondage numbers for119870

119899 119862

119899 119875

119899 and119866 = 119870

11989911198992119899119905

as follows

Theorem 125 (Hattingh and Plummer [80] 2008) For 119899 ⩾ 3

119887R (119870119899) =

1 119894119891 119899 = 3

lceil119899

2rceil 119900119905ℎ119890119903119908119894119904119890

119887R (119862119899) =

1 119894119891 119899 equiv 0 (mod 3) 2 119900119905ℎ119890119903119908119894119904119890

119887R (119875119899) = 1 119899 ⩾ 4

20 International Journal of Combinatorics

119887R (119866)

=

lceil119898

2rceil 119894119891 119899

119898= 1 119886119899119889 119899

119898+1⩾ 2 (1 ⩽ 119898 lt 119905)

2119905 minus 2 119894119891 1198991= 119899

2= sdot sdot sdot = 119899

119905= 2 (119905 ⩾ 2)

2 119894119891 1198991= 2 119886119899119889 119899

2⩾ 3 (119905 = 2)

119905minus1

sum

119894=1

119899119894minus 1 119900119905ℎ119890119903119908119894119904119890

(46)

Theorem 126 (Hattingh and Plummer [80] 2008) For a tree119879 of order 119899 (⩾ 4) 119879 ≇ 119870

1119899minus1if and only if 119887R(119879) = 1

Theorem 126 shows that the restrained bondage numberof a tree can be computed in constant time However thedecision problem for 119887R(119866) is NP-complete even for bipartitegraphs

Problem 5 Consider the decision problem

Restrained Bondage Problem

Instance a graph 119866 and a positive integer 119887

Question is 119887R(119866) ⩽ 119887

Theorem 127 (Hattingh and Plummer [80] 2008) Therestrained bondage problem is NP-complete even for bipartitegraphs

Consequently in view of its computational hardnessit is significative to establish some sharp bounds on therestrained bondage number of a graph in terms of othergraphic parameters

Theorem 128 (Hattingh and Plummer [80] 2008) For anygraph 119866 with 120575(119866) ⩾ 2

119887R (119866) ⩽ min119909119910isin119864(119866)

119889119866(119909) + 119889

119866(119910) minus 2 (47)

Corollary 129 119887R(119866) ⩽ Δ(119866)+120575(119866)minus2 for any graph119866 with120575(119866) ⩾ 2

Notice that the bounds stated in Theorem 128 andCorollary 129 are sharp Indeed the class of cycles whoseorders are congruent to 1 2 (mod 3) have a restrained bondagenumber achieving these bounds (see Theorem 125)

Theorem 128 is an analogue of Theorem 14 A quitenatural problem is whether or not for restricted bondagenumber there are analogues of Theorem 16 Theorem 18Theorem 29 and so on Indeed we should consider all resultsfor the bondage number whether or not there are analoguesfor restricted bondage number

Theorem 130 (Hattingh and Plummer [80] 2008) For anygraph 119866 with 120574

119877(119866) = 2

119887R (119866) ⩽ Δ (119866) + 1 (48)

We now consider the relation between 119887119877(119866) and 119887(119866)

Theorem 131 (Hattingh and Plummer [80] 2008) If 120574119877(119866) =

120574(119866) for some graph 119866 then 119887R(119866) ⩽ 119887(119866)

Corollary 129 and Theorem 131 provide a possibility toattack Conjecture 73 by characterizing planar graphs with120574119877= 120574 and 119887R = 119887 when 3 ⩽ Δ ⩽ 6However we do not have 119887R(119866) = 119887(119866) for any graph

119866 even if 120574119877(119866) = 120574(119866) Observe that 120574

119877(119870

3) = 120574(119870

3) yet

119887119877(119870

3) = 1 and 119887(119870

3) = 2 We still may not claim that

119887119877(119866) = 119887(119866) even in the case that every 120574-set is a 120574

119877-set

The example 1198703again demonstrates this

Theorem 132 (Hattingh and Plummer [80] 2008) There isan infinite class of graphs inwhich each graph119866 satisfies 119887(119866) lt119887R(119866)

In fact such an infinite class of graphs can be obtained asfollows Let119867 be a connected graph BC(119867) a graph obtainedfrom 119867 by attaching ℓ (⩾ 2) vertices of degree 1 to eachvertex in 119867 and B = 119866 119866 = BC(119867) for some graph 119867such that 120575(119867) ⩾ 2 By Theorem 3 we can easily check that119887(119866) = 1 lt 2 ⩽ min120575(119867) ℓ = 119887R(119866) for any 119866 isin BMoreover there exists a graph 119866 isin B such that 119887R(119866) canbe much larger than 119887(119866) which is stated as the followingtheorem

Theorem 133 (Hattingh and Plummer [80] 2008) For eachpositive integer 119896 there is a graph119866 such that 119896 = 119887R(119866)minus119887(119866)

Combining Theorem 133 with Theorem 132 we have thefollowing corollary

Corollary 134 The bondage number and the restrainedbondage number are unrelated

Very recently Jafari Rad et al [81] have consideredthe total restrained bondage in graphs based on both totaldominating sets and restrained dominating sets and obtainedseveral properties exact values and bounds for the totalrestrained bondage number of a graph

A restrained dominating set 119878 of a graph 119866 without iso-lated vertices is called to be a total restrained dominating setif 119878 is a total dominating set The total restrained dominationnumber of119866 denoted by 120574TR (119866) is theminimum cardinalityof a total restrained dominating set of 119866 For referenceson this domination in graphs see [3 61 82ndash85] The totalrestrained bondage number 119887TR (119866) of a graph 119866 with noisolated vertex is the cardinality of a smallest set of edges 119861 sube119864(119866) for which119866minus119861 has no isolated vertex and 120574TR (119866minus119861) gt120574TR (119866) In the case that there is no such subset 119861 we define119887TR (119866) = infin

Ma et al [84] determined that 120574TR (119875119899) = 119899minus2lfloor(119899minus2)4rfloorfor 119899 ⩾ 3 120574TR (119862119899

) = 119899 minus 2lfloor1198994rfloor for 119899 ⩾ 3 120574TR (1198703) =

3 and 120574TR (119870119899) = 2 for 119899 = 3 and 120574TR (1198701119899

) = 1 + 119899

and 120574TR (119870119898119899) = 2 for 2 ⩽ 119898 ⩽ 119899 According to these

results Jafari Rad et al [81] determined the exact valuesof total restrained bondage numbers for the correspondinggraphs

International Journal of Combinatorics 21

Theorem 135 (Jafari Rad et al [81]) For an integer 119899

119887TR (119875119899) = infin 119894119891 2 ⩽ 119899 ⩽ 5

1 119894119891 119899 ⩾ 5

119887TR (119862119899) =

infin 119894119891 119899 = 3

1 119894119891 3 lt 119899 119899 equiv 0 1 (mod 4)2 119894119891 3 lt 119899 119899 equiv 2 3 (mod 4)

119887TR (119882119899) = 2 119891119900119903 119899 ⩾ 6

119887TR (119870119899) = 119899 minus 1 119891119900119903 119899 ⩾ 4

119887TR (119870119898119899) = 119898 minus 1 119891119900119903 2 ⩽ 119898 ⩽ 119899

(49)

119887TR (119870119899) = 119899minus1 shows the fact that for any integer 119899 ⩾ 3 there

exists a connected graph namely119870119899+1

such that 119887TR (119870119899+1) =

119899 that is the total restrained bondage number is unboundedfrom the above

A vertex is said to be a support vertex if it is adjacent toa vertex of degree one Let A be the family of all connectedgraphs such that119866 belongs toA if and only if every edge of119866is incident to a support vertex or 119866 is a cycle of length three

Proposition 136 (Cyman and Raczek [82] 2006) For aconnected graph119866 of order 119899 120574TR (119866) = 119899 if and only if119866 isin A

Hence if 119866 isin A then the removal of any subset of edgesof119866 cannot increase the total restrained domination numberof 119866 and thus 119887TR (119866) = infin When 119866 notin A a result similar toTheorem 6 is obtained

Theorem 137 (Jafari Rad et al [81]) For any tree 119879 notin A119887TR (119879) ⩽ 2 This bound is sharp

In the samepaper the authors provided a characterizationfor trees 119879 notin A with 119887TR (119879) = 1 or 2 The following result issimilar to Observation 1

Observation 3 (Jafari Rad et al [81]) If 119896 edges can beremoved from a graph 119866 to obtain a graph 119867 without anisolated vertex and with 119887TR (119867) = 119905 then 119887TR (119866) ⩽ 119896 + 119905

ApplyingTheorem 137 andObservation 3 Jafari Rad et alcharacterized all connected graphs 119866 with 119887TR (119866) = infin

Theorem 138 (Jafari Rad et al [81]) For a connected graph119866119887TR (119866) = infin if and only if 119866 isin A

84 Other Conditional Bondage Numbers A subset 119868 sube 119881(119866)is called an independent set if no two vertices in 119868 are adjacentin 119866 The maximum cardinality among all independent setsis called the independence number of 119866 denoted by 120572(119866)

A dominating set 119878 of a graph 119866 is called to be inde-pendent if 119878 is an independent set of 119866 The minimumcardinality among all independent dominating set is calledthe independence domination number of 119866 and denoted by120574119868(119866)

Since an independent dominating set is not only adominating set but also an independent set 120574(119866) ⩽ 120574

119868(119866) ⩽

120572(119866) for any graph 119866It is clear that a maximal independent set is certainly a

dominating set Thus an independent set is maximal if andonly if it is an independent dominating set and so 120574

119868(119866) is the

minimum cardinality among all maximal independent setsof 119866 This graph-theoretical invariant has been well studiedin the literature see for example Haynes et al [3] for earlyresults Goddard and Henning [86] for recent results onindependent domination in graphs

In 2003 Zhang et al [87] defined the independencebondage number 119887

119868(119866) of a nonempty graph 119866 to be the

minimum cardinality among all subsets 119861 sube 119864(119866) for which120574119868(119866 minus 119861) gt 120574

119868(119866) For some ordinary graphs their indepen-

dence domination numbers can be easily determined and soindependence bondage numbers have been also determinedClearly 119887

119868(119870

119899) = 1 if 119899 ⩾ 2

Theorem 139 (Zhang et al [87] 2003) For any integer 119899 ⩾ 4

119887119868(119862

119899) =

1 119894119891 119899 equiv 1 (mod 2) 2 119894119891 119899 equiv 0 (mod 2)

119887119868(119875

119899) =

1 119894119891 119899 equiv 0 (mod 2) 2 119894119891 119899 equiv 1 (mod 2)

119887119868(119870

119898119899) = 119899 119891119900119903 119898 ⩽ 119899

(50)

Apart from the above-mentioned results as far as we findno other results on the independence bondage number in thepresent literature we never knew much of any result on thisparameter for a tree

A subset 119878 of 119881(119866) is called an equitable dominatingset if for every 119910 isin 119881(119866 minus 119878) there exists a vertex 119909 isin 119878such that 119909119910 isin 119864(119866) and |119889

119866(119909) minus 119889

119866(119910)| ⩽ 1 proposed

by Dharmalingam [88] The minimum cardinality of such adominating set is called the equitable domination number andis denoted by 120574

119864(119866) Deepak et al [89] defined the equitable

bondage number 119887119864(119866) of a graph 119866 to be the cardinality of a

smallest set 119861 sube 119864(119866) of edges for which 120574119864(119866 minus 119861) gt 120574

119864(119866)

and determined 119887119864(119866) for some special graphs

Theorem 140 (Deepak et al [89] 2011) For any integer 119899 ⩾ 2

119887119864(119870

119899) = lceil

119899

2rceil

119887119864(119870

119898119899) = 119898 119891119900119903 0 ⩽ 119899 minus 119898 ⩽ 1

119887119864(119862

119899) =

3 119894119891 119899 equiv 1 (mod 3)2 119900119905ℎ119890119903119908119894119904119890

119887119864(119875

119899) =

2 119894119891 119899 equiv 1 (mod 3)1 119900119905ℎ119890119903119908119894119904119890

119887119864(119879) ⩽ 2 119891119900119903 119886119899119910 119899119900119899119905119903119894119907119894119886119897 119905119903119890119890 119879

119887119864(119866) ⩽ 119899 minus 1 119891119900119903 119886119899119910 119888119900119899119899119890119888119905119890119889 119892119903119886119901ℎ 119866 119900119891 119900119903119889119890119903 119899

(51)

22 International Journal of Combinatorics

As far as we know there are no more results on theequitable bondage number of graphs in the present literature

There are other variations of the normal domination byadding restricted conditions to the normal dominating setFor example the connected domination set of a graph119866 pro-posed by Sampathkumar and Walikar [90] is a dominatingset 119878 such that the subgraph induced by 119878 is connected Theproblem of finding a minimum connected domination set ina graph is equivalent to the problemof finding a spanning treewithmaximumnumber of vertices of degree one [91] and hassome important applications in wireless networks [92] thusit has received much research attention However for suchan important domination there is no result on its bondagenumber We believe that the topic is of significance for futureresearch

9 Generalized Bondage Numbers

There are various generalizations of the classical dominationsuch as 119901-domination distance domination fractional dom-ination Roman domination and rainbow domination Everysuch a generalization can lead to a corresponding bondageIn this section we introduce some of them

91 119901-Bondage Numbers In 1985 Fink and Jacobson [93]introduced the concept of 119901-domination Let 119901 be a positiveinteger A subset 119878 of 119881(119866) is a 119901-dominating set of 119866 if|119878 cap 119873

119866(119910)| ⩾ 119901 for every 119910 isin 119878 The 119901-domination number

120574119901(119866) is the minimum cardinality among all 119901-dominating

sets of 119866 Any 119901-dominating set of 119866 with cardinality 120574119901(119866)

is called a 120574119901-set of 119866 Note that a 120574

1-set is a classic minimum

dominating set Notice that every graph has a 119901-dominatingset since the vertex-set119881(119866) is such a setWe also note that a 1-dominating set is a dominating set and so 120574(119866) = 120574

1(119866) The

119901-domination number has received much research attentionsee a state-of-the-art survey article by Chellali et al [94]

It is clear from definition that every 119901-dominating setof a graph certainly contains all vertices of degree at most119901 minus 1 By this simple observation to avoid the trivial caseoccurrence we always assume that Δ(119866) ⩾ 119901 For 119901 ⩾ 2 Luet al [95] gave a constructive characterization of trees withunique minimum 119901-dominating sets

Recently Lu and Xu [96] have introduced the conceptto the 119901-bondage number of 119866 denoted by 119887

119901(119866) as the

minimum cardinality among all sets of edges 119861 sube 119864(119866) suchthat 120574

119901(119866 minus 119861) gt 120574

119901(119866) Clearly 119887

1(119866) = 119887(119866)

Lu and Xu [96] established a lower bound and an upperbound on 119887

119901(119879) for any integer 119901 ⩾ 2 and any tree 119879 with

Δ(119879) ⩾ 119901 and characterized all trees achieving the lowerbound and the upper bound respectively

Theorem 141 (Lu and Xu [96] 2011) For any integer 119901 ⩾ 2and any tree 119879 with Δ(119879) ⩾ 119901

1 ⩽ 119887119901(119879) ⩽ Δ (119879) minus 119901 + 1 (52)

Let 119878 be a given subset of 119881(119866) and for any 119909 isin 119878 let

119873119901(119909 119878 119866) = 119910 isin 119878 cap 119873

119866(119909)

1003816100381610038161003816119873119866(119910) cap 119878

1003816100381610038161003816 = 119901

(53)

Then all trees achieving the lower bound 1 can be character-ized as follows

Theorem 142 (Lu and Xu [96] 2011) Let 119879 be a tree withΔ(119879) ⩾ 119901 ⩾ 2 Then 119887

119901(119879) = 1 if and only if for any 120574

119901-set

119878 of 119879 there exists an edge 119909119910 isin (119878 119878) such that 119910 isin 119873119901(119909 119878 119879)

The notation 119878(119886 119887) denotes a double star obtained byadding an edge between the central vertices of two stars119870

1119886minus1

and1198701119887minus1

And the vertex with degree 119886 (resp 119887) in 119878(119886 119887) iscalled the 119871-central vertex (resp 119877-central vertex) of 119878(119886 119887)

To characterize all trees attaining the upper bound givenin Theorem 141 we define three types of operations on a tree119879 with Δ(119879) = Δ ⩾ 119901 + 1

Type 1 attach a pendant edge to a vertex 119910with 119889119879(119910) ⩽ 119901minus

2 in 119879

Type 2 attach a star 1198701Δminus1

to a vertex 119910 of 119879 by joining itscentral vertex to 119910 where 119910 in a 120574

119901-set of 119879 and

119889119879(119910) ⩽ Δ minus 1

Type 3 attach a double star 119878(119901 Δ minus 1) to a pendant vertex 119910of 119879 by coinciding its 119877-central vertex with 119910 wherethe unique neighbor of 119910 is in a 120574

119901-set of 119879

Let B = 119879 a tree obtained from 1198701Δ

or 119878(119901 Δ) by afinite sequence of operations of Types 1 2 and 3

Theorem 143 (Lu and Xu [96] 2011) A tree with the maxi-mum Δ ⩾ 119901 + 1 has 119901-bondage number Δ minus 119901 + 1 if and onlyif it belongs toB

Theorem 144 (Lu and Xu [97] 2012) Let 119866119898119899= 119875

119898times 119875

119899

Then

1198872(119866

2119899) = 1 119891119900119903 119899 ⩾ 2

1198872(119866

3119899) =

2 119894119891 119899 equiv 1 (mod 3) 1 119900119905ℎ119890119903119908119894119904119890

for 119899 ⩾ 2

1198872(119866

4119899) =

1 119894119891 119899 equiv 3 (mod 4) 2 119900119905ℎ119890119903119908119894119904119890

for 119899 ⩾ 7

(54)

Recently Krzywkowski [98] has proposed the conceptof the nonisolating 2-bondage of a graph The nonisolating2-bondage number of a graph 119866 denoted by 1198871015840

2(119866) is the

minimum cardinality among all sets of edges 119861 sube 119864(119866) suchthat 119866 minus 119861 has no isolated vertices and 120574

2(119866 minus 119861) gt 120574

2(119866) If

for every 119861 sube 119864(119866) either 1205742(119866 minus 119861) = 120574

2(119866) or 119866 minus 119861 has

isolated vertices then define 11988710158402(119866) = 0 and say that 119866 is a 2-

nonisolatedly strongly stable graph Krzywkowski presentedsome basic properties of nonisolating 2-bondage in graphsshowed that for every nonnegative integer 119896 there exists atree 119879 such 1198871015840

2(119879) = 119896 characterized all 2-non-isolatedly

strongly stable trees and determined 11988710158402(119866) for several special

graphs

International Journal of Combinatorics 23

Theorem 145 (Krzywkowski [98] 2012) For any positiveintegers 119899 and119898 with119898 ⩽ 119899

1198871015840

2(119870

119899) =

0 119894119891 119899 = 1 2 3

lfloor2119899

3rfloor 119900119905ℎ119890119903119908119894119904119890

1198871015840

2(119875

119899) =

0 119894119891 119899 = 1 2 3

1 119900119905ℎ119890119903119908119894119904119890

1198871015840

2(119862

119899) =

0 119894119891 119899 = 3

1 119894119891 119899 119894119904 119890V119890119899

2 119894119891 119899 119894119904 119900119889119889

1198871015840

2(119870

119898119899) =

3 119894119891 119898 = 119899 = 3

1 119894119891 119898 = 119899 = 4

2 119900119905ℎ119890119903119908119894119904119890

(55)

92 Distance Bondage Numbers A subset 119878 of vertices of agraph119866 is said to be a distance 119896-dominating set for119866 if everyvertex in 119866 not in 119878 is at distance at most 119896 from some vertexof 119878 The minimum cardinality of all distance 119896-dominatingsets is called the distance 119896-domination number of 119866 anddenoted by 120574

119896(119866) (do not confuse with the above-mentioned

120574119901(119866)) When 119896 = 1 a distance 1-dominating set is a normal

dominating set and so 1205741(119866) = 120574(119866) for any graph 119866 Thus

the distance 119896-domination is a generalization of the classicaldomination

The relation between 120574119896and 120572

119896(the 119896-independence

number defined in Section 22) for a tree obtained by Meirand Moon [14] who proved that 120574

119896(119879) = 120572

2119896(119879) for any tree

119879 (a special case for 119896 = 1 is stated in Proposition 10) Thefurther research results can be found in Henning et al [99]Tian and Xu [100ndash103] and Liu et al [104]

In 1998 Hartnell et al [15] defined the distance 119896-bondagenumber of 119866 denoted by 119887

119896(119866) to be the cardinality of a

smallest subset 119861 of edges of 119866 with the property that 120574119896(119866 minus

119861) gt 120574119896(119866) FromTheorem 6 it is clear that if119879 is a nontrivial

tree then 1 ⩽ 1198871(119879) ⩽ 2 Hartnell et al [15] generalized this

result to any integer 119896 ⩾ 1

Theorem 146 (Hartnell et al [15] 1998) For every nontrivialtree 119879 and positive integer 119896 1 ⩽ 119887

119896(119879) ⩽ 2

Hartnell et al [15] and Topp and Vestergaard [13] alsocharacterized the trees having distance 119896-bondage number 2In particular the class of trees for which 119887

1(119879) = 2 are just

those which have a unique maximum 2-independent set (seeTheorem 11)

Since when 119896 = 1 the distance 1-bondage number 1198871(119866)

is the classical bondage number 119887(119866) Theorem 12 gives theNP-hardness of deciding the distance 119896-bondage number ofgeneral graphs

Theorem 147 Given a nonempty undirected graph 119866 andpositive integers 119896 and 119887 with 119887 ⩽ 120576(119866) determining wetheror not 119887

119896(119866) ⩽ 119887 is NP-hard

For a vertex 119909 in119866 the open 119896-neighborhood119873119896(119909) of 119909 is

defined as119873119896(119909) = 119910 isin 119881(119866) 1 ⩽ 119889

119866(119909 119910) le 119896The closed

119896-neighborhood119873119896[119909] of 119909 in 119866 is defined as119873

119896(119909) cup 119909 Let

Δ119896(119866) = max 1003816100381610038161003816119873119896

(119909)1003816100381610038161003816 for any 119909 isin 119881 (119866) (56)

Clearly Δ1(119866) = Δ(119866) The 119896th power of a graph 119866 is the

graph119866119896 with vertex-set119881(119866119896

) = 119881(119866) and edge-set119864(119866119896

) =

119909119910 1 ⩽ 119889119866(119909 119910) ⩽ 119896 The following lemma holds directly

from the definition of 119866119896

Proposition 148 (Tian and Xu [101]) Δ(119866119896

) = Δ119896(119866) and

120574(119866119896

) = 120574119896(119866) for any graph 119866 and each 119896 ⩾ 1

A graph 119866 is 119896-distance domination-critical or 120574119896-critical

for short if 120574119896(119866 minus 119909) lt 120574

119896(119866) for every vertex 119909 in 119866

proposed by Henning et al [105]

Proposition 149 (Tian and Xu [101]) For each 119896 ⩾ 1 a graph119866 is 120574

119896-critical if and only if 119866119896 is 120574

119896-critical

By the above facts we suggest the following worthwhileproblem

Problem 6 Can we generalize the results on the bondage for119866 to 119866119896 In particular do the following propositions hold

(a) 119887(119866119896

) = 119887119896(119866) for any graph 119866 and each 119896 ⩾ 1

(b) 119887(119866119896

) ⩽ Δ119896(119866) if 119866 is not 120574

119896-critical

Let 119896 and 119901 be positive integers A subset 119878 of 119881(119866) isdefined to be a (119896 119901)-dominating set of 119866 if for any vertex119909 isin 119878 |119873

119896(119909) cap 119878| ⩾ 119901 The (119896 119901)-domination number of

119866 denoted by 120574119896119901(119866) is the minimum cardinality among

all (119896 119901)-dominating sets of 119866 Clearly for a graph 119866 a(1 1)-dominating set is a classical dominating set a (119896 1)-dominating set is a distance 119896-dominating set and a (1 119901)-dominating set is the above-mentioned 119901-dominating setThat is 120574

11(119866) = 120574(119866) 120574

1198961(119866) = 120574

119896(119866) and 120574

1119901(119866) = 120574

119901(119866)

The concept of (119896 119901)-domination in a graph 119866 is a gen-eralized domination which combines distance 119896-dominationand 119901-domination in 119866 So the investigation of (119896 119901)-domination of 119866 is more interesting and has received theattention of many researchers see for example [106ndash109]

More general Li and Xu [110] proposed the concept of the(ℓ 119908)-domination number of a119908-connected graph A subset119878 of 119881(119866) is defined to be an (ℓ 119908)-dominating set of 119866 iffor any vertex 119909 isin 119878 there are 119908 internally disjoint pathsof length at most ℓ from 119909 to some vertex in 119878 The (ℓ 119908)-domination number of119866 denoted by 120574

ℓ119908(119866) is theminimum

cardinality among all (ℓ 119908)-dominating sets of 119866 Clearly1205741198961(119866) = 120574

119896(119866) and 120574

1119901(119866) = 120574

119901(119866)

It is quite natural to propose the concept of bondage num-ber for (119896 119901)-domination or (ℓ 119908)-domination However asfar as we know none has proposed this concept until todayThis is a worthwhile topic for further research

24 International Journal of Combinatorics

93 Fractional Bondage Numbers If 120590 is a function mappingthe vertex-set 119881 into some set of real numbers then for anysubset 119878 sube 119881 let 120590(119878) = sum

119909isin119878120590(119909) Also let |120590| = 120590(119881)

A real-value function 120590 119881 rarr [0 1] is a dominatingfunction of a graph 119866 if for every 119909 isin 119881(119866) 120590(119873

119866[119909]) ⩾ 1

Thus if 119878 is a dominating set of 119866 120590 is a function where

120590 (119909) = 1 if 119909 isin 1198780 otherwise

(57)

then 120590 is a dominating function of119866 The fractional domina-tion number of 119866 denoted by 120574

119865(119866) is defined as follows

120574119865(119866) = min |120590| 120590 is a domination function of 119866

(58)

The 120574119865-bondage number of 119866 denoted by 119887

119865(119866) is

defined as the minimum cardinality of a subset 119861 sube 119864 whoseremoval results in 120574

119865(119866 minus 119861) gt 120574

119865(119866)

Hedetniemi et al [111] are the first to study fractionaldomination although Farber [112] introduced the idea indi-rectly The concept of the fractional domination numberwas proposed by Domke and Laskar [113] in 1997 Thefractional domination numbers for some ordinary graphs aredetermined

Proposition 150 (Domke et al [114] 1998 Domke and Laskar[113] 1997) (a) If 119866 is a 119896-regular graph with order 119899 then120574119865(119866) = 119899119896 + 1(b) 120574

119865(119862

119899) = 1198993 120574

119865(119875

119899) = lceil1198993rceil 120574

119865(119870

119899) = 1

(c) 120574119865(119866) = 1 if and only if Δ(119866) = 119899 minus 1

(d) 120574119865(119879) = 120574(119879) for any tree 119879

(e) 120574119865(119870

119899119898) = (119899(119898 + 1) + 119898(119899 + 1))(119899119898 minus 1) where

max119899119898 gt 1

The assertions (a)ndash(d) are due to Domke et al [114] andthe assertion (e) is due to Domke and Laskar [113] Accordingto these results Domke and Laskar [113] determined the 120574

119865-

bondage numbers for these graphs

Theorem 151 (Domke and Laskar [113] 1997) 119887119865(119870

119899) =

lceil1198992rceil 119887119865(119870

119899119898) = 1 wheremax119899119898 gt 1

119887119865(119862

119899) =

2 119894119891 119899 equiv 0 (mod 3) 1 119900119905ℎ119890119903119908119894119904119890

119887119865(119875

119899) =

2 119894119891 119899 equiv 1 (mod 3) 1 119900119905ℎ119890119903119908119894119904119890

(59)

It is easy to see that for any tree119879 119887119865(119879) ⩽ 2 In fact since

120574119865(119879) = 120574(119879) for any tree 119879 120574

119865(119879

1015840

) = 120574(1198791015840

) for any subgraph1198791015840 of 119879 It follows that

119887119865(119879) = min 119861 sube 119864 (119879) 120574

119865(119879 minus 119861) gt 120574

119865(119879)

= min 119861 sube 119864 (119879) 120574 (119879 minus 119861) gt 120574 (119879)

= 119887 (119879) ⩽ 2

(60)

Except for these no results on 119887119865(119866) have been known as

yet

94 Roman Bondage Numbers A Roman dominating func-tion on a graph 119866 is a labeling 119891 119881 rarr 0 1 2 such thatevery vertex with label 0 has at least one neighbor with label2 The weight of a Roman dominating function 119891 on 119866 is thevalue

119891 (119866) = sum

119906isin119881(119866)

119891 (119906) (61)

The minimum weight of a Roman dominating function ona graph 119866 is called the Roman domination number of 119866denoted by 120574RM (119866)

A Roman dominating function 119891 119881 rarr 0 1 2

can be represented by the ordered partition (1198810 119881

1 119881

2) (or

(119881119891

0 119881

119891

1 119881

119891

2) to refer to119891) of119881 where119881

119894= V isin 119881 | 119891(V) = 119894

In this representation its weight 119891(119866) = |1198811| + 2|119881

2| It is

clear that119881119891

1cup119881

119891

2is a dominating set of 119866 called the Roman

dominating set denoted by 119863119891

R = (1198811 1198812) Since 119881119891

1cup 119881

119891

2is

a dominating set when 119891 is a Roman dominating functionand since placing weight 2 at the vertices of a dominating setyields a Roman dominating function in [115] it is observedthat

120574 (119866) ⩽ 120574RM (119866) ⩽ 2120574 (119866) (62)

A graph 119866 is called to be Roman if 120574RM (119866) = 2120574(119866)The definition of the Roman domination function is

proposed implicitly by Stewart [116] and ReVelle and Rosing[117] Roman dominating numbers have been deeply studiedIn particular Bahremandpour et al [118] showed that theproblem determining the Roman domination number is NP-complete even for bipartite graphs

Let 119866 be a graph with maximum degree at least twoThe Roman bondage number 119887RM (119866) of 119866 is the minimumcardinality of all sets 1198641015840 sube 119864 for which 120574RM (119866 minus 119864

1015840

) gt

120574RM (119866) Since in study of Roman bondage number theassumption Δ(119866) ⩾ 2 is necessary we always assume thatwhen we discuss 119887RM (119866) all graphs involved satisfy Δ(119866) ⩾2 The Roman bondage number 119887RM (119866) is introduced byJafari Rad and Volkmann in [119]

Recently Bahremandpour et al [118] have shown that theproblem determining the Roman bondage number is NP-hard even for bipartite graphs

Problem 7 Consider the decision problem

Roman Bondage Problem

Instance a nonempty bipartite graph119866 and a positiveinteger 119896

Question is 119887RM (119866) ⩽ 119896

Theorem 152 (Bahremandpour et al [118] 2013) TheRomanbondage problem is NP-hard even for bipartite graphs

The exact value of 119887RM(119866) is known only for a few familyof graphs including the complete graphs cycles and paths

International Journal of Combinatorics 25

Theorem 153 (Jafari Rad and Volkmann [119] 2011) For anypositive integers 119899 and119898 with119898 ⩽ 119899

119887RM (119875119899) = 2 119894119891 119899 equiv 2 (mod 3) 1 119900119905ℎ119890119903119908119894119904119890

119887RM (119862119899) =

3 119894119891 119899 = 2 (mod 3) 2 119900119905ℎ119890119903119908119894119904119890

119887RM (119870119899) = lceil

119899

2rceil 119891119900119903 119899 ⩾ 3

119887RM (119870119898119899) =

1 119894119891 119898 = 1 119886119899119889 119899 = 1

4 119894119891 119898 = 119899 = 3

119898 119900119905ℎ119890119903119908119894119904119890

(63)

Lemma 154 (Cockayne et al [115] 2004) If 119866 is a graph oforder 119899 and contains vertices of degree 119899 minus 1 then 120574RM(119866) = 2

Using Lemma 154 the third conclusion in Theorem 153can be generalized to more general case which is similar toLemma 49

Theorem 155 (Jafari Rad and Volkmann [119] 2011) Let119866 bea graph with order 119899 ge 3 and 119905 the number of vertices of degree119899 minus 1 in 119866 If 119905 ⩾ 1 then 119887RM(119866) = lceil1199052rceil

Ebadi and PushpaLatha [120] conjectured that 119887RM(119866) ⩽119899 minus 1 for any graph 119866 of order 119899 ⩾ 3 Dehgardi et al[121] Akbari andQajar [122] independently showed that thisconjecture is true

Theorem 156 119887RM(119866) ⩽ 119899 minus 1 for any connected graph 119866 oforder 119899 ⩾ 3

For a complete 119905-partite graph we have the followingresult

Theorem 157 (Hu and Xu [123] 2011) Let 119866 = 11987011989811198982119898119905

bea complete 119905-partite graph with 119898

1= sdot sdot sdot = 119898

119894lt 119898

119894+1le sdot sdot sdot ⩽

119898119905 119905 ⩾ 2 and 119899 = sum119905

119895=1119898

119895 Then

119887RM (119866) =

lceil119894

2rceil 119894119891 119898

119894= 1 119886119899119889 119899 ⩾ 3

2 119894119891 119898119894= 2 119886119899119889 119894 = 1

119894 119894119891 119898119894= 2 119886119899119889 119894 ⩾ 2

4 119894119891 119898119894= 3 119886119899119889 119894 = 119905 = 2

119899 minus 1 119894119891 119898119894= 3 119886119899119889 119894 = 119905 ⩾ 3

119899 minus 119898119905

119894119891 119898119894⩾ 3 119886119899119889 119898

119905⩾ 4

(64)

Consider a complete 119905-partite graph 119870119905(3) for 119905 ⩾ 3

119887RM(119870119905(3)) = 119899 minus 1 by Theorem 157 where 119899 = 3119905

which shows that this upper bound of 119899 minus 1 on 119887RM(119866) inTheorem 156 is sharp This fact and 120574

119877(119870

119905(3)) = 4 give

a negative answer to the following two problems posed byDehgardi et al [121]Prove or Disprove for any connected graph 119866 of order 119899 ge 3(i) 119887RM(119866) = 119899 minus 1 if and only if 119866 cong 119870

3 (ii) 119887RM(119866) ⩽ 119899 minus

120574119877(119866) + 1Note that a complete 119905-partite graph 119870

119905(3) is (119899 minus 3)-

regular and 119887RM(119870119905(3)) = 119899 minus 1 for 119905 ⩾ 3 where 119899 = 3119905

Hu and Xu further determined that 119887119877(119866) = 119899 minus 2 for any

(119899 minus 3)-regular graph 119866 of order 119899 ⩽ 5 except 119870119905(3)

Theorem 158 (Hu and Xu [123] 2011) For any (119899minus3)-regulargraph 119866 of order 119899 other than119870

119905(3) 119887RM(119866) = 119899 minus 2 for 119899 ⩾ 5

For a tree 119879 with order 119899 ⩾ 3 Ebadi and PushpaLatha[120] and Jafari Rad and Volkmann [119] independentlyobtained an upper bound on 119887RM(119879)

Theorem 159 119887RM(119879) ⩽ 3 for any tree 119879 with order 119899 ⩾ 3

Theorem 160 (Bahremandpour et al [118] 2013) 119887RM(1198752 times119875119899) = 2 for 119899 ⩾ 2

Theorem 161 (Jafari Rad and Volkmann [119] 2011) Let119866 bea graph with order at least three

(a) If (119909 119910 119911) is a path of length 2 in 119866 then 119887RM(119866) ⩽119889119866(119909) + 119889

119866(119910) + 119889

119866(119911) minus 3 minus |119873

119866(119909) cap 119873

119866(119910)|

(b) If 119866 is connected then 119887RM(119866) ⩽ 120582(119866) + 2Δ(119866) minus 3

Theorem 161 (b) implies 119887RM(119866) ⩽ 120575(119866) + 2Δ(119866) minus 3 for aconnected graph 119866 Note that for a planar graph 119866 120575(119866) ⩽ 5moreover 120575(119866) ⩽ 3 if the girth at least 4 and 120575(119866) ⩽ 2 if thegirth at least 6 These two facts show that 119887RM(119866) ⩽ 2Δ(119866) +2 for a connected planar graph 119866 Jafari Rad and Volkmann[124] improved this bound

Theorem 162 (Jafari Rad and Volkmann [124] 2011) For anyconnected planar graph 119866 of order 119899 ⩾ 3 with girth 119892(119866)

119887RM (119866)

⩽

2Δ (119866)

Δ (119866) + 6

Δ (119866) + 5 119894119891 119866 119888119900119899119905119886119894119899119904 119899119900 V119890119903119905119894119888119890119904 119900119891 119889119890119892119903119890119890 119891119894V119890

Δ (119866) + 4 119894119891 119892 (119866) ⩾ 4

Δ (119866) + 3 119894119891 119892 (119866) ⩾ 5

Δ (119866) + 2 119894119891 119892 (119866) ⩾ 6

Δ (119866) + 1 119894119891 119892 (119866) ⩾ 8

(65)

According toTheorem 157 119887RM(119862119899) = 3 = Δ(119862

119899)+1 for a

cycle 119862119899of length 119899 ⩾ 8 with 119899 equiv 2 (mod 3) and therefore

the last upper bound inTheorem 162 is sharp at least for Δ =2

Combining the fact that every planar graph 119866 withminimum degree 5 contains an edge 119909119910 with 119889

119866(119909) = 5

and 119889119866(119910) isin 5 6 with Theorem 161 (a) Akbari et al [125]

obtained the following result

26 International Journal of Combinatorics

middot middot middot

middot middot middot

middot middot middot

middot middot middot

119866

Figure 9 The graph 119866 is constructed from 119866

Theorem 163 (Akbari et al [125] 2012) 119887RM(119866) ⩽ 15 forevery planar graph 119866

It remains open to show whether the bound inTheorem 163 is sharp or not Though finding a planargraph 119866 with 119887RM(119866) = 15 seems to be difficult Akbari etal [125] constructed an infinite family of planar graphs withRoman bondage number equal to 7 by proving the followingresult

Theorem 164 (Akbari et al [125] 2012) Let 119866 be a graph oforder 119899 and 119866 is the graph of order 5119899 obtained from 119866 byattaching the central vertex of a copy of a path 119875

5to each vertex

of119866 (see Figure 9) Then 120574RM(119866) = 4119899 and 119887RM(119866) = 120575(119866)+2

ByTheorem 164 infinitemany planar graphswith Romanbondage number 7 can be obtained by considering any planargraph 119866 with 120575(119866) = 5 (eg the icosahedron graph)

Conjecture 165 (Akbari et al [125] 2012) The Romanbondage number of every planar graph is at most 7

For general bounds the following observation is directlyobtained which is similar to Observation 1

Observation 4 Let 119866 be a graph of order 119899 with maximumdegree at least two Assume that 119867 is a spanning subgraphobtained from 119866 by removing 119896 edges If 120574RM(119867) = 120574RM(119866)then 119887

119877(119867) ⩽ 119887RM(119866) ⩽ 119887RM(119867) + 119896

Theorem 166 (Bahremandpour et al [118] 2013) For anyconnected graph 119866 of order 119899 if 119899 ⩾ 3 and 120574RM(119866) = 120574(119866) + 1then

119887RM (119866) ⩽ min 119887 (119866) 119899Δ (66)

where 119899Δis the number of vertices with maximum degree Δ in

119866

Observation 5 For any graph 119866 if 120574(119866) = 120574RM(119866) then119887RM(119866) ⩽ 119887(119866)

Theorem 167 (Bahremandpour et al [118] 2013) For everyRoman graph 119866

119887RM (119866) ⩾ 119887 (119866) (67)

The bound is sharp for cycles on 119899 vertices where 119899 equiv 0 (mod3)

The strict inequality in Theorem 167 can hold for exam-ple 119887(119862

3119896+2) = 2 lt 3 = 119887RM(1198623119896+2

) byTheorem 157A graph 119866 is called to be vertex Roman domination-

critical if 120574RM(119866 minus 119909) lt 120574RM(119866) for every vertex 119909 in 119866 If119866 has no isolated vertices then 120574RM(119866) ⩽ 2120574(119866) ⩽ 2120573(119866)If 120574RM(119866) = 2120573(119866) then 120574RM(119866) = 2120574(119866) and hence 119866 is aRoman graph In [30] Volkmann gave a lot of graphs with120574(119866) = 120573(119866) The following result is similar to Theorem 57

Theorem 168 (Bahremandpour et al [118] 2013) For anygraph 119866 if 120574RM(119866) = 2120573(119866) then

(a) 119887RM(119866) ⩾ 120575(119866)

(b) 119887RM(119866) ⩾ 120575(119866)+1 if119866 is a vertex Roman domination-critical graph

If 120574RM(119866) = 2 then obviously 119887RM(119866) ⩽ 120575(119866) So weassume 120574RM(119866) ⩾ 3 Dehgardi et al [121] proved 119887RM(119866) ⩽(120574RM(119866) minus 2)Δ(119866) + 1 and posed the following problem if 119866is a connected graph of order 119899 ⩾ 4 with 120574RM(119866) ⩾ 3 then

119887RM (119866) ⩽ (120574RM (119866) minus 2) Δ (119866) (68)

Theorem 161 (a) shows that the inequality (68) holds if120574RM(119866) ⩾ 5 Thus the bound in (68) is of interest only when120574RM(119866) is 3 or 4

Lemma 169 (Bahremandpour et al [118] 2013) For anynonempty graph 119866 of order 119899 ⩾ 3 120574RM(119866) = 3 if and onlyif Δ(119866) = 119899 minus 2

The following result shows that (68) holds for all graphs119866 of order 119899 ⩾ 4 with 120574RM(119866) = 3 or 4 which improvesTheorem 161 (b)

Theorem 170 (Bahremandpour et al [118] 2013) For anyconnected graph 119866 of order 119899 ⩾ 4

119887RM (119866) le Δ (119866) = 119899 minus 2 119894119891 120574RM (119866) = 3

Δ (119866) + 120575 (119866) minus 1 119894119891 120574RM (119866) = 4(69)

with the first equality if and only if 119866 cong 1198624

Recently Akbari and Qajar [122] have proved the follow-ing result

Theorem 171 (Akbari and Qajar [122]) For any connectedgraph 119866 of order 119899 ⩾ 3

119887RM (119866) ⩽ 119899 minus 120574RM (119866) + 5 (70)

International Journal of Combinatorics 27

We conclude this section with the following problems

Problem 8 Characterize all connected graphs 119866 of order 119899 ⩾3 for which 119887RM(119866) = 119899 minus 1

Problem 9 Prove or disprove if 119866 is a connected graph oforder 119899 ⩾ 3 then

119887RM (119866) ⩽ 119899 minus 120574RM (119866) + 3 (71)

Note that 120574119877(119870

119905(3)) = 4 and 119887RM(119870119905

(3)) = 119899minus1 where 119899 =3119905 for 119905 ⩾ 3 The upper bound on 119887RM(119866) given in Problem 9is sharp if Problem 9 has positive answer

95 Rainbow Bondage Numbers For a positive integer 119896 a119896-rainbow dominating function of a graph 119866 is a function 119891from the vertex-set 119881(119866) to the set of all subsets of the set1 2 119896 such that for any vertex 119909 in 119866 with 119891(119909) = 0 thecondition

⋃

119910isin119873119866(119909)

119891 (119910) = 1 2 119896 (72)

is fulfilledThe weight of a 119896-rainbow dominating function 119891on 119866 is the value

119891 (119866) = sum

119909isin119881(119866)

1003816100381610038161003816119891 (119909)1003816100381610038161003816 (73)

The 119896-rainbow domination number of a graph 119866 denoted by120574119903119896(119866) is the minimum weight of a 119896-rainbow dominating

function 119891 of 119866Note that 120574

1199031(119866) is the classical domination number 120574(119866)

The 119896-rainbow domination number is introduced by Bresaret al [126] Rainbow domination of a graph 119866 coincides withordinary domination of the Cartesian product of 119866 with thecomplete graph in particular 120574

119903119896(119866) = 120574(119866 times 119870

119896) for any

positive integer 119896 and any graph 119866 [126] Hartnell and Rall[127] proved that 120574(119879) lt 120574

1199032(119879) for any tree 119879 no graph has

120574 = 120574119903119896for 119896 ⩾ 3 for any 119896 ⩾ 2 and any graph 119866 of order 119899

min 119899 120574 (119866) + 119896 minus 2 ⩽ 120574119903119896(119866) ⩽ 119896120574 (119866) (74)

The introduction of rainbow domination is motivatedby the study of paired domination in Cartesian productsof graphs where certain upper bounds can be expressed interms of rainbow domination that is for any graph 119866 andany graph 119867 if 119881(119867) can be partitioned into 119896120574

119875-sets then

120574119875(119866 times 119867) ⩽ (1119896)120592(119867)120574

119903119896(119866) proved by Bresar et al [126]

Dehgardi et al [128] introduced the concept of 119896-rainbowbondage number of graphs Let 119866 be a graph with maximumdegree at least two The 119896-rainbow bondage number 119887

119903119896(119866)

of 119866 is the minimum cardinality of all sets 119861 sube 119864(119866) forwhich 120574

119903119896(119866 minus 119861) gt 120574

119903119896(119866) Since in the study of 119896-rainbow

bondage number the assumption Δ(119866) ⩾ 2 is necessarywe always assume that when we discuss 119887

119903119896(119866) all graphs

involved satisfy Δ(119866) ⩾ 2The 2-rainbow bondage number of some special families

of graphs has been determined

Theorem 172 (Dehgardi et al [128]) For any integer 119899 ⩾ 3

1198871199032(119875

119899) = 1 119891119900119903 119899 ⩾ 2

1198871199032(119862

119899) =

1 119894119891 119899 equiv 0 (mod 4) 2 119900119905ℎ119890119903119908119894119904119890

1198871199032(119870

119899119899) =

1 119894119891 119899 = 2

3 119894119891 119899 = 3

6 119894119891 119899 = 4

119898 119894119891 119899 ⩾ 5

(75)

Theorem 173 (Dehgardi et al [128]) For any connected graph119866 of order 119899 ⩾ 3 119887

1199032(119866) ⩽ 120582(119866) + 2Δ(119866) minus 3

Theorem 174 (Dehgardi et al [128]) For any tree 119879 of order119899 ⩾ 3 119887

1199032(119879) ⩽ 2

Theorem 175 (Dehgardi et al [128]) If 119866 is a planar graphwithmaximumdegree at least two then 119887

1199032(119866) ⩽ 15Moreover

if the girth 119892(119866) ⩾ 4 then 1198871199032(119866) ⩽ 11 and if 119892(119866) ⩾ 6 then

1198871199032(119866) ⩽ 9

Theorem 176 (Dehgardi et al [128]) For a complete bipartitegraph 119870

119901119902 if 2119896 + 1 ⩽ 119901 ⩽ 119902 then 119887

119903119896(119870

119901119902) = 119901

Theorem177 (Dehgardi et al [128]) For a complete graph119870119899

if 119899 ⩾ 119896 + 1 then 119887119903119896(119870

119899) = lceil119896119899(119896 + 1)rceil

The special case 119896 = 1 of Theorem 177 can be found inTheorem 1 The following is a simple observation similar toObservation 1

Observation 6 (Dehgardi et al [128]) Let 119866 be a graphwith maximum degree at least two 119867 a spanning subgraphobtained from 119866 by removing 119896 edges If 120574

119903119896(119867) = 120574

119903119896(119866)

then 119887119903119896(119867) ⩽ 119887

119903119896(119866) ⩽ 119887

119903119896(119867) + 119896

Theorem 178 (Dehgardi et al [128]) For every graph 119866 with120574119903119896(119866) = 119896120574(119866) 119887

119903119896(119866) ⩾ 119887(119866)

A graph 119866 is said to be vertex 119896-rainbow domination-critical if 120574

119903119896(119866 minus 119909) lt 120574

119903119896(119866) for every vertex 119909 in 119866 It is

clear that if 119866 has no isolated vertices then 120574119903119896(119866) ⩽ 119896120574(119866) ⩽

119896120573(119866) where 120573(119866) is the vertex-covering number of119866 Thusif 120574

119903119896(119866) = 119896120573(119866) then 120574

119903119896(119866) = 119896120574(119866)

Theorem 179 (Dehgardi et al [128]) For any graph 119866 if120574119903119896(119866) = 119896120573(119866) then

(a) 119887119903119896(119866) ⩾ 120575(119866)

(b) 119887119903119896(119866) ⩾ 120575(119866)+1 if119866 is vertex 119896-rainbow domination-

critical

As far as we know there are no more results on the 119896-rainbow bondage number of graphs in the literature

28 International Journal of Combinatorics

10 Results on Digraphs

Although domination has been extensively studied in undi-rected graphs it is natural to think of a dominating setas a one-way relationship between vertices of the graphIndeed among the earliest literatures on this subject vonNeumann and Morgenstern [129] used what is now calleddomination in digraphs to find solution (or kernels whichare independent dominating sets) for cooperative 119899-persongames Most likely the first formulation of domination byBerge [130] in 1958 was given in the context of digraphs andonly some years latter by Ore [131] in 1962 for undirectedgraphs Despite this history examination of domination andits variants in digraphs has been essentially overlooked (see[4] for an overview of the domination literature) Thus thereare few if any such results on domination for digraphs in theliterature

The bondage number and its related topics for undirectedgraph have become one of major areas both in theoreticaland applied researches However until recently Carlsonand Develin [42] Shan and Kang [132] and Huang andXu [133 134] studied the bondage number for digraphsindependently In this section we introduce their resultsfor general digraphs Results for some special digraphs suchas vertex-transitive digraphs are introduced in the nextsection

101 Upper Bounds for General Digraphs Let 119866 = (119881 119864)

be a digraph without loops and parallel edges A digraph 119866is called to be asymmetric if whenever (119909 119910) isin 119864(119866) then(119910 119909) notin 119864(119866) and to be symmetric if (119909 119910) isin 119864(119866) implies(119910 119909) isin 119864(119866) For a vertex 119909 of 119881(119866) the sets of out-neighbors and in-neighbors of 119909 are respectively defined as119873

+

119866(119909) = 119910 isin 119881(119866) (119909 119910) isin 119864(119866) and 119873minus

119866(119909) = 119909 isin

119881(119866) (119909 119910) isin 119864(119866) the out-degree and the in-degree of119909 are respectively defined as 119889+

119866(119909) = |119873

+

119866(119909)| and 119889minus

119866(119909) =

|119873+

119866(119909)| Denote themaximum and theminimum out-degree

(resp in-degree) of 119866 by Δ+(119866) and 120575+(119866) (resp Δminus(119866) and120575minus

(119866))The degree 119889119866(119909) of 119909 is defined as 119889+

119866(119909)+119889

minus

119866(119909) and

the maximum and the minimum degrees of119866 are denoted byΔ(119866) and 120575(119866) respectively that is Δ(119866) = max119889

119866(119909) 119909 isin

119881(119866) and 120575(119866) = min119889119866(119909) 119909 isin 119881(119866) Note that the

definitions here are different from the ones in the textbookon digraphs see for example Xu [1]

A subset 119878 of119881(119866) is called a dominating set if119881(119866) = 119878cup119873

+

119866(119878) where119873+

119866(119878) is the set of out-neighbors of 119878Then just

as for undirected graphs 120574(119866) is the minimum cardinalityof a dominating set and the bondage number 119887(119866) is thesmallest cardinality of a set 119861 of edges such that 120574(119866 minus 119861) gt120574(119866) if such a subset 119861 sube 119864(119866) exists Otherwise we put119887(119866) = infin

Some basic results for undirected graphs stated inSection 3 can be generalized to digraphs For exampleTheorem 18 is generalized by Carlson and Develin [42]Huang andXu [133] and Shan andKang [132] independentlyas follows

Theorem 180 Let 119866 be a digraph and (119909 119910) isin 119864(119866) Then119887(119866) ⩽ 119889

119866(119910) + 119889

minus

119866(119909) minus |119873

minus

119866(119909) cap 119873

minus

119866(119910)|

Corollary 181 For a digraph 119866 119887(119866) ⩽ 120575minus(119866) + Δ(119866)

Since 120575minus

(119866) ⩽ (12)Δ(119866) from Corollary 181Conjecture 53 is valid for digraphs

Corollary 182 For a digraph 119866 119887(119866) ⩽ (32)Δ(119866)

In the case of undirected graphs the bondage number of119866119899= 119870

119899times 119870

119899achieves this bound However it was shown

by Carlson and Develin in [42] that if we take the symmetricdigraph119866

119899 we haveΔ(119866

119899) = 4(119899minus1) 120574(119866

119899) = 119899 and 119887(119866

119899) ⩽

4(119899 minus 1) + 1 = Δ(119866119899) + 1 So this family of digraphs cannot

show that the bound in Corollary 182 is tightCorresponding to Conjecture 51 which is discredited

for undirected graphs and is valid for digraphs the sameconjecture for digraphs can be proposed as follows

Conjecture 183 (Carlson and Develin [42] 2006) 119887(119866) ⩽Δ(119866) + 1 for any digraph 119866

If Conjecture 183 is true then the following results showsthat this upper bound is tight since 119887(119870

119899times119870

119899) = Δ(119870

119899times119870

119899)+1

for a complete digraph 119870119899

In 2007 Shan and Kang [132] gave some tight upperbounds on the bondage numbers for some asymmetricdigraphs For example 119887(119879) ⩽ Δ(119879) for any asymmetricdirected tree 119879 119887(119866) ⩽ Δ(119866) for any asymmetric digraph 119866with order at least 4 and 120574(119866) ⩽ 2 For planar digraphs theyobtained the following results

Theorem 184 (Shan and Kang [132] 2007) Let 119866 be aasymmetric planar digraphThen 119887(119866) ⩽ Δ(119866)+2 and 119887(119866) ⩽Δ(119866) + 1 if Δ(119866) ⩾ 5 and 119889minus

119866(119909) ⩾ 3 for every vertex 119909 with

119889119866(119909) ⩾ 4

102 Results for Some Special Digraphs The exact values andbounds on 119887(119866) for some standard digraphs are determined

Theorem185 (Huang andXu [133] 2006) For a directed cycle119862119899and a directed path 119875

119899

119887 (119862119899) =

3 119894119891 119899 119894119904 119900119889119889

2 119894119891 119899 119894119904 119890V119890119899

119887 (119875119899) =

2 119894119891 119899 119894119904 119900119889119889

1 119894119891 119899 119894119904 119890V119890119899

(76)

For the de Bruijn digraph 119861(119889 119899) and the Kautz digraph119870(119889 119899)

119887 (119861 (119889 119899)) = 119889 119894119891 119899 119894119904 119900119889119889

119889 ⩽ 119887 (119861 (119889 119899)) ⩽ 2119889 119894119891 119899 119894119904 119890V119890119899

119887 (119870 (119889 119899)) = 119889 + 1

(77)

Like undirected graphs we can define the total domi-nation number and the total bondage number On the totalbondage numbers for some special digraphs the knownresults are as follows

International Journal of Combinatorics 29

Theorem 186 (Huang and Xu [134] 2007) For a directedcycle 119862

119899and a directed path 119875

119899 119887

119879(119875

119899) and 119887

119879(119862

119899) all do not

exist For a complete digraph 119870119899

119887119879(119870

119899) = infin 119894119891 119899 = 1 2

119887119879(119870

119899) = 3 119894119891 119899 = 3

119899 ⩽ 119887119879(119870

119899) ⩽ 2119899 minus 3 119894119891 119899 ⩾ 4

(78)

The extended de Bruijn digraph EB(119889 119899 1199021 119902

119901) and

the extended Kautz digraph EK(119889 119899 1199021 119902

119901) are intro-

duced by Shibata and Gonda [135] If 119901 = 1 then theyare the de Bruijn digraph B(119889 119899) and the Kautz digraphK(119889 119899) respectively Huang and Xu [134] determined theirtotal domination numbers In particular their total bondagenumbers for general cases are determined as follows

Theorem 187 (Huang andXu [134] 2007) If 119889 ⩾ 2 and 119902119894⩾ 2

for each 119894 = 1 2 119901 then

119887119879(EB(119889 119899 119902

1 119902

119901)) = 119889

119901

minus 1

119887119879(EK(119889 119899 119902

1 119902

119901)) = 119889

119901

(79)

In particular for the de Bruijn digraph B(119889 119899) and the Kautzdigraph K(119889 119899)

119887119879(B (119889 119899)) = 119889 minus 1 119887

119879(K (119889 119899)) = 119889 (80)

Zhang et al [136] determined the bondage number incomplete 119905-partite digraphs

Theorem 188 (Zhang et al [136] 2009) For a complete 119905-partite digraph 119870

11989911198992119899119905

where 1198991⩽ 119899

2⩽ sdot sdot sdot ⩽ 119899

119905

119887 (11987011989911198992119899119905

)

=

119898 119894119891 119899119898= 1 119899

119898+1⩾ 2 119891119900119903 119904119900119898119890

119898 (1 ⩽ 119898 lt 119905)

4119905 minus 3 119894119891 1198991= 119899

2= sdot sdot sdot = 119899

119905= 2

119905minus1

sum

119894=1

119899119894+ 2 (119905 minus 1) 119900119905ℎ119890119903119908119894119904119890

(81)

Since an undirected graph can be thought of as a sym-metric digraph any result for digraphs has an analogy forundirected graphs in general In view of this point studyingthe bondage number for digraphs is more significant thanfor undirected graphs Thus we should further study thebondage number of digraphs and try to generalize knownresults on the bondage number and related variants for undi-rected graphs to digraphs prove or disprove Conjecture 183In particular determine the exact values of 119887(B(119889 119899)) for aneven 119899 and 119887

119879(119870

119899) for 119899 ⩾ 4

11 Efficient Dominating Sets

A dominating set 119878 of a graph 119866 is called to be efficient if forevery vertex 119909 in119866 |119873

119866[119909]cap119878| = 1 if119866 is a undirected graph

or |119873minus

119866[119909] cap 119878| = 1 if 119866 is a directed graph The concept of an

efficient set was proposed by Bange et al [137] in 1988 In theliterature an efficient set is also called a perfect dominatingset (see Livingston and Stout [138] for an explanation)

From definition if 119878 is an efficient dominating set of agraph 119866 then 119878 is certainly an independent set and everyvertex not in 119878 is adjacent to exactly one vertex in 119878

It is also clear from definition that a dominating set 119878 isefficient if and only if N

119866[119878] = 119873

119866[119909] 119909 isin 119878 for the

undirected graph 119866 or N+

119866[119878] = 119873

+

119866[119909] 119909 isin 119878 for the

digraph119866 is a partition of119881(119866) where the induced subgraphby119873

119866[119909] or119873+

119866[119909] is a star or an out-star with the root 119909

The efficient domination has important applications inmany areas such as error-correcting codes and receivesmuch attention in the late years

The concept of efficient dominating sets is a measureof the efficiency of domination in graphs Unfortunately asshown in [137] not every graph has an efficient dominatingset andmoreover it is anNP-complete problem to determinewhether a given graph has an efficient dominating set Inaddition it was shown by Clark [139] in 1993 that for a widerange of 119901 almost every random undirected graph 119866 isin

G(120592 119901) has no efficient dominating sets This means thatundirected graphs possessing an efficient dominating set arerare However it is easy to show that every undirected graphhas an orientationwith an efficient dominating set (see Bangeet al [140])

In 1993 Barkauskas and Host [141] showed that deter-mining whether an arbitrary oriented graph has an efficientdominating set is NP-complete Even so the existence ofefficient dominating sets for some special classes of graphs hasbeen examined see for example Dejter and Serra [142] andLee [143] for Cayley graph Gu et al [144] formeshes and toriObradovic et al [145] Huang and Xu [36] Reji Kumar andMacGillivray [146] for circulant graphs Harary graphs andtori and Van Wieren et al [147] for cube-connected cycles

In this section we introduce results of the bondagenumber for some graphs with efficient dominating sets dueto Huang and Xu [36]

111 Results for General Graphs In this subsection we intro-duce some results on bondage numbers obtained by applyingefficient dominating sets We first state the two followinglemmas

Lemma 189 For any 119896-regular graph or digraph 119866 of order 119899120574(119866) ⩾ 119899(119896 + 1) with equality if and only if 119866 has an efficientdominating set In addition if 119866 has an efficient dominatingset then every efficient dominating set is certainly a 120574-set andvice versa

Let 119890 be an edge and 119878 a dominating set in 119866 We say that119890 supports 119878 if 119890 isin (119878 119878) where (119878 119878) = (119909 119910) isin 119864(119866) 119909 isin 119878 119910 isin 119878 Denote by 119904(119866) the minimum number of edgeswhich support all 120574-sets in 119866

Lemma 190 For any graph or digraph 119866 119887(119866) ⩾ 119904(119866) withequality if 119866 is regular and has an efficient dominating set

30 International Journal of Combinatorics

A graph 119866 is called to be vertex-transitive if its automor-phism group Aut(119866) acts transitively on its vertex-set 119881(119866)A vertex-transitive graph is regular Applying Lemmas 189and 190 Huang and Xu obtained some results on bondagenumbers for vertex-transitive graphs or digraphs

Theorem 191 (Huang and Xu [36] 2008) For any vertex-transitive graph or digraph 119866 of order 119899

119887 (119866) ⩾

lceil119899

2120574 (119866)rceil 119894119891 119866 119894119904 119906119899119889119894119903119890119888119905119890119889

lceil119899

120574 (119866)rceil 119894119891 119866 119894119904 119889119894119903119890119888119905119890119889

(82)

Theorem192 (Huang andXu [36] 2008) Let119866 be a 119896-regulargraph of order 119899 Then

119887(119866) ⩽ 119896 if119866 is undirected and 119899 ⩾ 120574(119866)(119896+1)minus119896+1119887(119866) ⩽ 119896+1+ℓ if119866 is directed and 119899 ⩾ 120574(119866)(119896+1)minusℓwith 0 ⩽ ℓ ⩽ 119896 minus 1

Next we introduce a better upper bound on 119887(119866) To thisaim we need the following concept which is a generalizationof the concept of the edge-covering of a graph 119866 For 1198811015840

sube

119881(119866) and 1198641015840 sube 119864(119866) we say that 1198641015840covers 1198811015840 and call 1198641015840an edge-covering for 1198811015840 if there exists an edge (119909 119910) isin 1198641015840 forany vertex 119909 isin 1198811015840 For 119910 isin 119881(119866) let 1205731015840[119910] be the minimumcardinality over all edge-coverings for119873minus

119866[119910]

Theorem 193 (Huang and Xu [36] 2008) If 119866 is a 119896-regulargraph with order 120574(119866)(119896 + 1) then 119887(119866) ⩽ 1205731015840[119910] for any 119910 isin119881(119866)

Theupper bound on 119887(119866) given inTheorem 193 is tight inview of 119887(119862

119899) for a cycle or a directed cycle 119862

119899(seeTheorems

1 and 185 resp)It is easy to see that for a 119896-regular graph 119866 lceil(119896 + 1)2rceil ⩽

1205731015840

[119910] ⩽ 119896 when 119866 is undirected and 1205731015840[119910] = 119896 + 1 when 119866 isdirected By this fact and Lemma 189 the following theoremis merely a simple combination of Theorems 191 and 193

Theorem 194 (Huang and Xu [36] 2008) Let 119866 be a vertex-transitive graph of degree 119896 If 119866 has an efficient dominatingset then

lceil119896 + 1

2rceil ⩽ 119887 (119866) ⩽ 119896 119894119891 119866 119894119904 119906119899119889119894119903119890119888119905119890119889

119887 (119866) = 119896 + 1 119894119891 119866 119894119904 119889119894119903119890119888119905119890119889

(83)

Theorem 195 (Huang and Xu [36] 2008) If 119866 is an undi-rected vertex-transitive cubic graph with order 4120574(119866) and girth119892(119866) ⩽ 5 then 119887(119866) = 2

The proof of Theorem 195 leads to a byproduct If 119892 =5 let (119906

1 119906

2 119906

3 119906

4 119906

5) be a cycle and let D

119894be the family

of efficient dominating sets containing 119906119894for each 119894 isin

1 2 3 4 5 ThenD119894capD

119895= 0 for 119894 = 1 2 5 Then 119866 has

at least 5119904 efficient dominating sets where 119904 = 119904(119866) But thereare only 4s (=ns120574) distinct efficient dominating sets in119866This

contradiction implies that an undirected vertex-transitivecubic graph with girth five has no efficient dominating setsBut a similar argument for 119892(119866) = 3 4 or 119892(119866) ⩾ 6 couldnot give any contradiction This is consistent with the resultthat CCC(119899) a vertex-transitive cubic graph with girth 119899 if3 ⩽ 119899 ⩽ 8 or girth 8 if 119899 ⩾ 9 has efficient dominating setsfor all 119899 ⩾ 3 except 119899 = 5 (see Theorem 199 in the followingsubsection)

112 Results for Cayley Graphs In this subsection we useTheorem 194 to determine the exact values or approximativevalues of bondage numbers for some special vertex-transitivegraphs by characterizing the existence of efficient dominatingsets in these graphs

Let Γ be a nontrivial finite group 119878 a nonempty subset ofΓ without the identity element of Γ A digraph 119866 is defined asfollows

119881 (119866) = Γ (119909 119910) isin 119864 (119866) lArrrArr 119909minus1

119910 isin 119878

for any 119909 119910 isin Γ(84)

is called a Cayley digraph of the group Γ with respect to119878 denoted by 119862

Γ(119878) If 119878minus1 = 119904minus1 119904 isin 119878 = 119878 then

119862Γ(119878) is symmetric and is called a Cayley undirected graph

a Cayley graph for short Cayley graphs or digraphs arecertainly vertex-transitive (see Xu [1])

A circulant graph 119866(119899 119878) of order 119899 is a Cayley graph119862119885119899

(119878) where 119885119899= 0 119899 minus 1 is the addition group of

order 119899 and 119878 is a nonempty subset of119885119899without the identity

element and hence is a vertex-transitive digraph of degree|119878| If 119878minus1 = 119878 then119866(119899 119878) is an undirected graph If 119878 = 1 119896where 2 ⩽ 119896 ⩽ 119899 minus 2 we write 119866(119899 1 119896) for 119866(119899 1 119896) or119866(119899 plusmn1 plusmn119896) and call it a double-loop circulant graph

Huang and Xu [36] showed that for directed 119866 =

119866(119899 1 119896) lceil1198993rceil ⩽ 120574(119866) ⩽ lceil1198992rceil and 119866 has an efficientdominating set if and only if 3 | 119899 and 119896 equiv 2 (mod 3) fordirected 119866 = 119866(119899 1 119896) and 119896 = 1198992 lceil1198995rceil ⩽ 120574(119866) ⩽ lceil1198993rceiland 119866 has an efficient dominating set if and only if 5 | 119899 and119896 equiv plusmn2 (mod 5) By Theorem 194 we obtain the bondagenumber of a double loop circulant graph if it has an efficientdominating set

Theorem 196 (Huang and Xu [36] 2008) Let 119866 be a double-loop circulant graph 119866(119899 1 119896) If 119866 is directed with 3 | 119899and 119896 equiv 2 (mod 3) or 119866 is undirected with 5 | 119899 and119896 equiv plusmn2 (mod 5) then 119887(119866) = 3

The 119898 times 119899 torus is the Cartesian product 119862119898times 119862

119899of

two cycles and is a Cayley graph 119862119885119898times119885119899

(119878) where 119878 =

(0 1) (1 0) for directed cycles and 119878 = (0 plusmn1) (plusmn1 0) forundirected cycles and hence is vertex-transitive Gu et al[144] showed that the undirected torus119862

119898times119862

119899has an efficient

dominating set if and only if both 119898 and 119899 are multiplesof 5 Huang and Xu [36] showed that the directed torus119862119898times 119862

119899has an efficient dominating set if and only if both

119898 and 119899 are multiples of 3 Moreover they found a necessarycondition for a dominating set containing the vertex (0 0)in 119862

119898times 119862

119899to be efficient and obtained the following

result

International Journal of Combinatorics 31

Theorem 197 (Huang and Xu [36] 2008) Let 119866 = 119862119898times 119862

119899

If 119866 is undirected and both119898 and 119899 are multiples of 5 or if 119866 isdirected and both119898 and 119899 are multiples of 3 then 119887(119866) = 3

The hypercube 119876119899

is the Cayley graph 119862Γ(119878)

where Γ = 1198852times sdot sdot sdot times 119885

2= (119885

2)119899 and 119878 =

100 sdot sdot sdot 0 010 sdot sdot sdot 0 00 sdot sdot sdot 01 Lee [143] showed that119876119899has an efficient dominating set if and only if 119899 = 2119898 minus 1

for a positive integer119898 ByTheorem 194 the following resultis obtained immediately

Theorem 198 (Huang and Xu [36] 2008) If 119899 = 2119898 minus 1 for apositive integer119898 then 2119898minus1

⩽ 119887(119876119899) ⩽ 2

119898

minus 1

Problem 10 Determine the exact value of 119887(119876119899) for 119899 = 2119898minus1

The 119899-dimensional cube-connected cycle denoted byCCC(119899) is constructed from the 119899-dimensional hypercube119876119899by replacing each vertex 119909 in 119876

119899with an undirected cycle

119862119899of length 119899 and linking the 119894th vertex of the 119862

119899to the 119894th

neighbor of 119909 It has been proved that CCC(119899) is a Cayleygraph and hence is a vertex-transitive graph with degree 3Van Wieren et al [147] proved that CCC(119899) has an efficientdominating set if and only if 119899 = 5 From Theorems 194 and195 the following result can be derived

Theorem 199 (Huang and Xu [36] 2008) Let 119866 = 119862119862119862(119899)be the 119899-dimensional cube-connected cycles with 119899 ⩾ 3 and119899 = 5 Then 120574(119866) = 1198992119899minus2 and 2 ⩽ 119887(119866) ⩽ 3 In addition119887(119862119862119862(3)) = 119887(119862119862119862(4)) = 2

Problem 11 Determine the exact value of 119887(CCC(119899)) for 119899 ⩾5

Acknowledgments

This work was supported by NNSF of China (Grants nos11071233 61272008) The author would like to express hisgratitude to the anonymous referees for their kind sugges-tions and comments on the original paper to ProfessorSheikholeslami and Professor Jafari Rad for sending thereferences [128] and [81] which resulted in this version

References

[1] J-M Xu Theory and Application of Graphs vol 10 of NetworkTheory and Applications Kluwer Academic Dordrecht TheNetherlands 2003

[2] S THedetniemi andR C Laskar ldquoBibliography on dominationin graphs and some basic definitions of domination parame-tersrdquo Discrete Mathematics vol 86 no 1ndash3 pp 257ndash277 1990

[3] TW Haynes S T Hedetniemi and P J Slater Fundamentals ofDomination in Graphs vol 208 ofMonographs and Textbooks inPure and Applied Mathematics Marcel Dekker New York NYUSA 1998

[4] TWHaynes S THedetniemi and P J Slater EdsDominationin Graphs vol 209 of Monographs and Textbooks in Pure andAppliedMathematicsMarcelDekker NewYorkNYUSA 1998

[5] M R Garey andD S JohnsonComputers and Intractability WH Freeman and Company San Francisco Calif USA 1979

[6] H B Walikar and B D Acharya ldquoDomination critical graphsrdquoNational Academy Science Letters vol 2 pp 70ndash72 1979

[7] D Bauer F Harary J Nieminen and C L Suffel ldquoDominationalteration sets in graphsrdquo Discrete Mathematics vol 47 no 2-3pp 153ndash161 1983

[8] J F Fink M S Jacobson L F Kinch and J Roberts ldquoThebondage number of a graphrdquo Discrete Mathematics vol 86 no1ndash3 pp 47ndash57 1990

[9] F-T Hu and J-M Xu ldquoThe bondage number of (119899 minus 3)-regulargraph G of order nrdquo Ars Combinatoria to appear

[10] U Teschner ldquoNew results about the bondage number of agraphrdquoDiscreteMathematics vol 171 no 1ndash3 pp 249ndash259 1997

[11] B L Hartnell and D F Rall ldquoA bound on the size of a graphwith given order and bondage numberrdquo Discrete Mathematicsvol 197-198 pp 409ndash413 1999 16th British CombinatorialConference (London 1997)

[12] B L Hartnell and D F Rall ldquoA characterization of trees inwhich no edge is essential to the domination numberrdquo ArsCombinatoria vol 33 pp 65ndash76 1992

[13] J Topp and P D Vestergaard ldquo120572119896- and 120574

119896-stable graphsrdquo

Discrete Mathematics vol 212 no 1-2 pp 149ndash160 2000[14] A Meir and J W Moon ldquoRelations between packing and

covering numbers of a treerdquo Pacific Journal of Mathematics vol61 no 1 pp 225ndash233 1975

[15] B L Hartnell L K Joslashrgensen P D Vestergaard and CA Whitehead ldquoEdge stability of the k-domination numberof treesrdquo Bulletin of the Institute of Combinatorics and ItsApplications vol 22 pp 31ndash40 1998

[16] F-T Hu and J-M Xu ldquoOn the complexity of the bondage andreinforcement problemsrdquo Journal of Complexity vol 28 no 2pp 192ndash201 2012

[17] B L Hartnell and D F Rall ldquoBounds on the bondage numberof a graphrdquo Discrete Mathematics vol 128 no 1ndash3 pp 173ndash1771994

[18] Y-L Wang ldquoOn the bondage number of a graphrdquo DiscreteMathematics vol 159 no 1ndash3 pp 291ndash294 1996

[19] G H Fricke TW Haynes S M Hedetniemi S T Hedetniemiand R C Laskar ldquoExcellent treesrdquo Bulletin of the Institute ofCombinatorics and Its Applications vol 34 pp 27ndash38 2002

[20] V Samodivkin ldquoThe bondage number of graphs good and badverticesrdquoDiscussiones Mathematicae GraphTheory vol 28 no3 pp 453ndash462 2008

[21] J E Dunbar T W Haynes U Teschner and L VolkmannldquoBondage insensitivity and reinforcementrdquo in Domination inGraphs vol 209 of Monographs and Textbooks in Pure andAppliedMathematics pp 471ndash489 Dekker NewYork NY USA1998

[22] H Liu and L Sun ldquoThe bondage and connectivity of a graphrdquoDiscrete Mathematics vol 263 no 1ndash3 pp 289ndash293 2003

[23] A-H Esfahanian and S L Hakimi ldquoOn computing a con-ditional edge-connectivity of a graphrdquo Information ProcessingLetters vol 27 no 4 pp 195ndash199 1988

[24] R C Brigham P Z Chinn and R D Dutton ldquoVertexdomination-critical graphsrdquoNetworks vol 18 no 3 pp 173ndash1791988

[25] V Samodivkin ldquoDomination with respect to nondegenerateproperties bondage numberrdquoThe Australasian Journal of Com-binatorics vol 45 pp 217ndash226 2009

[26] U Teschner ldquoA new upper bound for the bondage numberof graphs with small domination numberrdquo The AustralasianJournal of Combinatorics vol 12 pp 27ndash35 1995

32 International Journal of Combinatorics

[27] J Fulman D Hanson and G MacGillivray ldquoVertex dom-ination-critical graphsrdquoNetworks vol 25 no 2 pp 41ndash43 1995

[28] U Teschner ldquoA counterexample to a conjecture on the bondagenumber of a graphrdquo Discrete Mathematics vol 122 no 1ndash3 pp393ndash395 1993

[29] U Teschner ldquoThe bondage number of a graph G can be muchgreater than Δ(G)rdquo Ars Combinatoria vol 43 pp 81ndash87 1996

[30] L Volkmann ldquoOn graphs with equal domination and coveringnumbersrdquoDiscrete AppliedMathematics vol 51 no 1-2 pp 211ndash217 1994

[31] L A Sanchis ldquoMaximum number of edges in connected graphswith a given domination numberrdquoDiscreteMathematics vol 87no 1 pp 65ndash72 1991

[32] S Klavzar and N Seifter ldquoDominating Cartesian products ofcyclesrdquoDiscrete AppliedMathematics vol 59 no 2 pp 129ndash1361995

[33] H K Kim ldquoA note on Vizingrsquos conjecture about the bondagenumberrdquo Korean Journal of Mathematics vol 10 no 2 pp 39ndash42 2003

[34] M Y Sohn X Yuan and H S Jeong ldquoThe bondage number of1198623times119862

119899rdquo Journal of the KoreanMathematical Society vol 44 no

6 pp 1213ndash1231 2007[35] L Kang M Y Sohn and H K Kim ldquoBondage number of the

discrete torus 119862119899times 119862

4rdquo Discrete Mathematics vol 303 no 1ndash3

pp 80ndash86 2005[36] J Huang and J-M Xu ldquoThe bondage numbers and efficient

dominations of vertex-transitive graphsrdquoDiscrete Mathematicsvol 308 no 4 pp 571ndash582 2008

[37] C J Xiang X Yuan andM Y Sohn ldquoDomination and bondagenumber of119862

3times119862

119899rdquoArs Combinatoria vol 97 pp 299ndash310 2010

[38] M S Jacobson and L F Kinch ldquoOn the domination numberof products of graphs Irdquo Ars Combinatoria vol 18 pp 33ndash441984

[39] Y-Q Li ldquoA note on the bondage number of a graphrdquo ChineseQuarterly Journal of Mathematics vol 9 no 4 pp 1ndash3 1994

[40] F-T Hu and J-M Xu ldquoBondage number of mesh networksrdquoFrontiers ofMathematics inChina vol 7 no 5 pp 813ndash826 2012

[41] R Frucht and F Harary ldquoOn the corona of two graphsrdquoAequationes Mathematicae vol 4 pp 322ndash325 1970

[42] K Carlson and M Develin ldquoOn the bondage number of planarand directed graphsrdquoDiscreteMathematics vol 306 no 8-9 pp820ndash826 2006

[43] U Teschner ldquoOn the bondage number of block graphsrdquo ArsCombinatoria vol 46 pp 25ndash32 1997

[44] J Huang and J-M Xu ldquoNote on conjectures of bondagenumbers of planar graphsrdquo Applied Mathematical Sciences vol6 no 65ndash68 pp 3277ndash3287 2012

[45] L Kang and J Yuan ldquoBondage number of planar graphsrdquoDiscrete Mathematics vol 222 no 1ndash3 pp 191ndash198 2000

[46] M Fischermann D Rautenbach and L Volkmann ldquoRemarkson the bondage number of planar graphsrdquo Discrete Mathemat-ics vol 260 no 1ndash3 pp 57ndash67 2003

[47] J Hou G Liu and J Wu ldquoBondage number of planar graphswithout small cyclesrdquoUtilitasMathematica vol 84 pp 189ndash1992011

[48] J Huang and J-M Xu ldquoThe bondage numbers of graphs withsmall crossing numbersrdquo Discrete Mathematics vol 307 no 15pp 1881ndash1897 2007

[49] Q Ma S Zhang and J Wang ldquoBondage number of 1-planargraphrdquo Applied Mathematics vol 1 no 2 pp 101ndash103 2010

[50] J Hou and G Liu ldquoA bound on the bondage number of toroidalgraphsrdquoDiscreteMathematics Algorithms and Applications vol4 no 3 Article ID 1250046 5 pages 2012

[51] Y-C Cao J-M Xu and X-R Xu ldquoOn bondage number oftoroidal graphsrdquo Journal of University of Science and Technologyof China vol 39 no 3 pp 225ndash228 2009

[52] Y-C Cao J Huang and J-M Xu ldquoThe bondage number ofgraphs with crossing number less than fourrdquo Ars Combinatoriato appear

[53] H K Kim ldquoOn the bondage numbers of some graphsrdquo KoreanJournal of Mathematics vol 11 no 2 pp 11ndash15 2004

[54] A Gagarin and V Zverovich ldquoUpper bounds for the bondagenumber of graphs on topological surfacesrdquo Discrete Mathemat-ics 2012

[55] J Huang ldquoAn improved upper bound for the bondage numberof graphs on surfacesrdquoDiscrete Mathematics vol 312 no 18 pp2776ndash2781 2012

[56] A Gagarin and V Zverovich ldquoThe bondage number of graphson topological surfaces and Teschnerrsquos conjecturerdquo DiscreteMathematics vol 313 no 6 pp 796ndash808 2013

[57] V Samodivkin ldquoNote on the bondage number of graphs ontopological surfacesrdquo httparxivorgabs12086203

[58] E J Cockayne R M Dawes and S T Hedetniemi ldquoTotaldomination in graphsrdquoNetworks vol 10 no 3 pp 211ndash219 1980

[59] R Laskar J Pfaff S M Hedetniemi and S T HedetniemildquoOn the algorithmic complexity of total dominationrdquo Society forIndustrial and Applied Mathematics vol 5 no 3 pp 420ndash4251984

[60] J Pfaff R C Laskar and S T Hedetniemi ldquoNP-completeness oftotal and connected domination and irredundance for bipartitegraphsrdquo Tech Rep 428 Department of Mathematical SciencesClemson University 1983

[61] M A Henning ldquoA survey of selected recent results on totaldomination in graphsrdquoDiscrete Mathematics vol 309 no 1 pp32ndash63 2009

[62] V R Kulli and D K Patwari ldquoThe total bondage number ofa graphrdquo in Advances in Graph Theory pp 227ndash235 VishwaGulbarga India 1991

[63] F-T Hu Y Lu and J-M Xu ldquoThe total bondage number ofgrid graphsrdquo Discrete Applied Mathematics vol 160 no 16-17pp 2408ndash2418 2012

[64] J Cao M Shi and B Wu ldquoResearch on the total bondagenumber of a spectial networkrdquo in Proceedings of the 3rdInternational Joint Conference on Computational Sciences andOptimization (CSO rsquo10) pp 456ndash458 May 2010

[65] N Sridharan MD Elias and V S A Subramanian ldquoTotalbondage number of a graphrdquo AKCE International Journal ofGraphs and Combinatorics vol 4 no 2 pp 203ndash209 2007

[66] N Jafari Rad and J Raczek ldquoSome progress on total bondage ingraphsrdquo Graphs and Combinatorics 2011

[67] T W Haynes and P J Slater ldquoPaired-domination and thepaired-domatic numberrdquoCongressusNumerantium vol 109 pp65ndash72 1995

[68] T W Haynes and P J Slater ldquoPaired-domination in graphsrdquoNetworks vol 32 no 3 pp 199ndash206 1998

[69] X Hou ldquoA characterization of 2120574 120574119875-treesrdquoDiscrete Mathemat-

ics vol 308 no 15 pp 3420ndash3426 2008[70] K E Proffitt T W Haynes and P J Slater ldquoPaired-domination

in grid graphsrdquo Congressus Numerantium vol 150 pp 161ndash1722001

International Journal of Combinatorics 33

[71] H Qiao L KangM Cardei andD-Z Du ldquoPaired-dominationof treesrdquo Journal of Global Optimization vol 25 no 1 pp 43ndash542003

[72] E Shan L Kang and M A Henning ldquoA characterizationof trees with equal total domination and paired-dominationnumbersrdquo The Australasian Journal of Combinatorics vol 30pp 31ndash39 2004

[73] J Raczek ldquoPaired bondage in treesrdquo Discrete Mathematics vol308 no 23 pp 5570ndash5575 2008

[74] J A Telle and A Proskurowski ldquoAlgorithms for vertex parti-tioning problems on partial k-treesrdquo SIAM Journal on DiscreteMathematics vol 10 no 4 pp 529ndash550 1997

[75] G S Domke J H Hattingh M A Henning and L R MarkusldquoRestrained domination in treesrdquoDiscreteMathematics vol 211no 1ndash3 pp 1ndash9 2000

[76] G S Domke J H Hattingh M A Henning and L R MarkusldquoRestrained domination in graphs with minimum degree twordquoJournal of Combinatorial Mathematics and Combinatorial Com-puting vol 35 pp 239ndash254 2000

[77] G S Domke J H Hattingh S T Hedetniemi R C Laskarand L R Markus ldquoRestrained domination in graphsrdquo DiscreteMathematics vol 203 no 1ndash3 pp 61ndash69 1999

[78] J H Hattingh and E J Joubert ldquoAn upper bound for therestrained domination number of a graph with minimumdegree at least two in terms of order and minimum degreerdquoDiscrete Applied Mathematics vol 157 no 13 pp 2846ndash28582009

[79] M A Henning ldquoGraphs with large restrained dominationnumberrdquo Discrete Mathematics vol 197-198 pp 415ndash429 199916th British Combinatorial Conference (London 1997)

[80] J H Hattingh and A R Plummer ldquoRestrained bondage ingraphsrdquo Discrete Mathematics vol 308 no 23 pp 5446ndash54532008

[81] N Jafari Rad R Hasni J Raczek and L Volkmann ldquoTotalrestrained bondage in graphsrdquoActaMathematica Sinica vol 29no 6 pp 1033ndash1042 2013

[82] J Cyman and J Raczek ldquoOn the total restrained dominationnumber of a graphrdquoThe Australasian Journal of Combinatoricsvol 36 pp 91ndash100 2006

[83] M A Henning ldquoGraphs with large total domination numberrdquoJournal of Graph Theory vol 35 no 1 pp 21ndash45 2000

[84] D-X Ma X-G Chen and L Sun ldquoOn total restraineddomination in graphsrdquoCzechoslovakMathematical Journal vol55 no 1 pp 165ndash173 2005

[85] B Zelinka ldquoRemarks on restrained domination and totalrestrained domination in graphsrdquo Czechoslovak MathematicalJournal vol 55 no 2 pp 393ndash396 2005

[86] W Goddard and M A Henning ldquoIndependent domination ingraphs a survey and recent resultsrdquo Discrete Mathematics vol313 no 7 pp 839ndash854 2013

[87] J H Zhang H L Liu and L Sun ldquoIndependence bondagenumber and reinforcement number of some graphsrdquo Transac-tions of Beijing Institute of Technology vol 23 no 2 pp 140ndash1422003

[88] K D Dharmalingam Studies in Graph Theorymdashequitabledomination and bottleneck domination [PhD thesis] MaduraiKamaraj University Madurai India 2006

[89] GDeepakND Soner andAAlwardi ldquoThe equitable bondagenumber of a graphrdquo Research Journal of Pure Algebra vol 1 no9 pp 209ndash212 2011

[90] E Sampathkumar and H B Walikar ldquoThe connected domina-tion number of a graphrdquo Journal of Mathematical and PhysicalSciences vol 13 no 6 pp 607ndash613 1979

[91] K White M Farber and W Pulleyblank ldquoSteiner trees con-nected domination and strongly chordal graphsrdquoNetworks vol15 no 1 pp 109ndash124 1985

[92] M Cadei M X Cheng X Cheng and D-Z Du ldquoConnecteddomination in Ad Hoc wireless networksrdquo in Proceedings ofthe 6th International Conference on Computer Science andInformatics (CSampI rsquo02) 2002

[93] J F Fink and M S Jacobson ldquon-domination in graphsrdquo inGraph Theory with Applications to Algorithms and ComputerScience Wiley-Interscience Publications pp 283ndash300 WileyNew York NY USA 1985

[94] M Chellali O Favaron A Hansberg and L Volkmann ldquok-domination and k-independence in graphs a surveyrdquo Graphsand Combinatorics vol 28 no 1 pp 1ndash55 2012

[95] Y Lu X Hou J-M Xu andN Li ldquoTrees with uniqueminimump-dominating setsrdquo Utilitas Mathematica vol 86 pp 193ndash2052011

[96] Y Lu and J-M Xu ldquoThe p-bondage number of treesrdquo Graphsand Combinatorics vol 27 no 1 pp 129ndash141 2011

[97] Y Lu and J-M Xu ldquoThe 2-domination and 2-bondage numbersof grid graphs A manuscriptrdquo httparxivorgabs12044514

[98] M Krzywkowski ldquoNon-isolating 2-bondage in graphsrdquo TheJournal of the Mathematical Society of Japan vol 64 no 4 2012

[99] M A Henning O R Oellermann and H C Swart ldquoBoundson distance domination parametersrdquo Journal of CombinatoricsInformation amp System Sciences vol 16 no 1 pp 11ndash18 1991

[100] F Tian and J-M Xu ldquoBounds for distance domination numbersof graphsrdquo Journal of University of Science and Technology ofChina vol 34 no 5 pp 529ndash534 2004

[101] F Tian and J-M Xu ldquoDistance domination-critical graphsrdquoApplied Mathematics Letters vol 21 no 4 pp 416ndash420 2008

[102] F Tian and J-M Xu ldquoA note on distance domination numbersof graphsrdquo The Australasian Journal of Combinatorics vol 43pp 181ndash190 2009

[103] F Tian and J-M Xu ldquoAverage distances and distance domina-tion numbersrdquo Discrete Applied Mathematics vol 157 no 5 pp1113ndash1127 2009

[104] Z-L Liu F Tian and J-M Xu ldquoProbabilistic analysis of upperbounds for 2-connected distance k-dominating sets in graphsrdquoTheoretical Computer Science vol 410 no 38ndash40 pp 3804ndash3813 2009

[105] M A Henning O R Oellermann and H C Swart ldquoDistancedomination critical graphsrdquo Journal of Combinatorial Mathe-matics and Combinatorial Computing vol 44 pp 33ndash45 2003

[106] T J Bean M A Henning and H C Swart ldquoOn the integrityof distance domination in graphsrdquo The Australasian Journal ofCombinatorics vol 10 pp 29ndash43 1994

[107] M Fischermann and L Volkmann ldquoA remark on a conjecturefor the (kp)-domination numberrdquoUtilitasMathematica vol 67pp 223ndash227 2005

[108] T Korneffel D Meierling and L Volkmann ldquoA remark on the(22)-domination numberrdquo Discussiones Mathematicae GraphTheory vol 28 no 2 pp 361ndash366 2008

[109] Y Lu X Hou and J-M Xu ldquoOn the (22)-domination numberof treesrdquo Discussiones Mathematicae Graph Theory vol 30 no2 pp 185ndash199 2010

34 International Journal of Combinatorics

[110] H Li and J-M Xu ldquo(dm)-domination numbers of m-connected graphsrdquo Rapports de Recherche 1130 LRI UniversiteParis-Sud 1997

[111] S M Hedetniemi S T Hedetniemi and T V Wimer ldquoLineartime resource allocation algorithms for treesrdquo Tech Rep URI-014 Department of Mathematical Sciences Clemson Univer-sity 1987

[112] M Farber ldquoDomination independent domination and dualityin strongly chordal graphsrdquo Discrete Applied Mathematics vol7 no 2 pp 115ndash130 1984

[113] G S Domke and R C Laskar ldquoThe bondage and reinforcementnumbers of 120574

119891for some graphsrdquoDiscrete Mathematics vol 167-

168 pp 249ndash259 1997[114] G S Domke S T Hedetniemi and R C Laskar ldquoFractional

packings coverings and irredundance in graphsrdquo vol 66 pp227ndash238 1988

[115] E J Cockayne P A Dreyer Jr S M Hedetniemi andS T Hedetniemi ldquoRoman domination in graphsrdquo DiscreteMathematics vol 278 no 1ndash3 pp 11ndash22 2004

[116] I Stewart ldquoDefend the roman empirerdquo Scientific American vol281 pp 136ndash139 1999

[117] C S ReVelle and K E Rosing ldquoDefendens imperiumromanum a classical problem in military strategyrdquoThe Ameri-can Mathematical Monthly vol 107 no 7 pp 585ndash594 2000

[118] A Bahremandpour F-T Hu S M Sheikholeslami and J-MXu ldquoOn the Roman bondage number of a graphrdquo DiscreteMathematics Algorithms and Applications vol 5 no 1 ArticleID 1350001 15 pages 2013

[119] N Jafari Rad and L Volkmann ldquoRoman bondage in graphsrdquoDiscussiones Mathematicae Graph Theory vol 31 no 4 pp763ndash773 2011

[120] K Ebadi and L PushpaLatha ldquoRoman bondage and Romanreinforcement numbers of a graphrdquo International Journal ofContemporary Mathematical Sciences vol 5 pp 1487ndash14972010

[121] N Dehgardi S M Sheikholeslami and L Volkman ldquoOn theRoman k-bondage number of a graphrdquo AKCE InternationalJournal of Graphs and Combinatorics vol 8 pp 169ndash180 2011

[122] S Akbari and S Qajar ldquoA note on Roman bondage number ofgraphsrdquo Ars Combinatoria to Appear

[123] F-T Hu and J-M Xu ldquoRoman bondage numbers of somegraphsrdquo httparxivorgabs11093933

[124] N Jafari Rad and L Volkmann ldquoOn the Roman bondagenumber of planar graphsrdquo Graphs and Combinatorics vol 27no 4 pp 531ndash538 2011

[125] S Akbari M Khatirinejad and S Qajar ldquoA note on the romanbondage number of planar graphsrdquo Graphs and Combinatorics2012

[126] B Bresar M A Henning and D F Rall ldquoRainbow dominationin graphsrdquo Taiwanese Journal of Mathematics vol 12 no 1 pp213ndash225 2008

[127] B L Hartnell and D F Rall ldquoOn dominating the Cartesianproduct of a graph and 120572krdquo Discussiones Mathematicae GraphTheory vol 24 no 3 pp 389ndash402 2004

[128] N Dehgardi S M Sheikholeslami and L Volkman ldquoThe k-rainbow bondage number of a graphrdquo to appear in DiscreteApplied Mathematics

[129] J von Neumann and O Morgenstern Theory of Games andEconomic Behavior Princeton University Press Princeton NJUSA 1944

[130] C BergeTheorıe des Graphes et ses Applications 1958[131] O Ore Theory of Graphs vol 38 American Mathematical

Society Colloquium Publications 1962[132] E Shan and L Kang ldquoBondage number in oriented graphsrdquoArs

Combinatoria vol 84 pp 319ndash331 2007[133] J Huang and J-M Xu ldquoThe bondage numbers of extended

de Bruijn and Kautz digraphsrdquo Computers amp Mathematics withApplications vol 51 no 6-7 pp 1137ndash1147 2006

[134] J Huang and J-M Xu ldquoThe total domination and total bondagenumbers of extended de Bruijn and Kautz digraphsrdquoComputersamp Mathematics with Applications vol 53 no 8 pp 1206ndash12132007

[135] Y Shibata and Y Gonda ldquoExtension of de Bruijn graph andKautz graphrdquo Computers amp Mathematics with Applications vol30 no 9 pp 51ndash61 1995

[136] X Zhang J Liu and JMeng ldquoThebondage number in completet-partite digraphsrdquo Information Processing Letters vol 109 no17 pp 997ndash1000 2009

[137] D W Bange A E Barkauskas and P J Slater ldquoEfficient dom-inating sets in graphsrdquo in Applications of Discrete Mathematicspp 189ndash199 SIAM Philadelphia Pa USA 1988

[138] M Livingston and Q F Stout ldquoPerfect dominating setsrdquoCongressus Numerantium vol 79 pp 187ndash203 1990

[139] L Clark ldquoPerfect domination in random graphsrdquo Journal ofCombinatorial Mathematics and Combinatorial Computing vol14 pp 173ndash182 1993

[140] D W Bange A E Barkauskas L H Host and L H ClarkldquoEfficient domination of the orientations of a graphrdquo DiscreteMathematics vol 178 no 1ndash3 pp 1ndash14 1998

[141] A E Barkauskas and L H Host ldquoFinding efficient dominatingsets in oriented graphsrdquo Congressus Numerantium vol 98 pp27ndash32 1993

[142] I J Dejter and O Serra ldquoEfficient dominating sets in CayleygraphsrdquoDiscrete AppliedMathematics vol 129 no 2-3 pp 319ndash328 2003

[143] J Lee ldquoIndependent perfect domination sets in Cayley graphsrdquoJournal of Graph Theory vol 37 no 4 pp 213ndash219 2001

[144] WGu X Jia and J Shen ldquoIndependent perfect domination setsin meshes tori and treesrdquo Unpublished

[145] N Obradovic J Peters and G Ruzic ldquoEfficient domination incirculant graphswith two chord lengthsrdquo Information ProcessingLetters vol 102 no 6 pp 253ndash258 2007

[146] K Reji Kumar and G MacGillivray ldquoEfficient domination incirculant graphsrdquo Discrete Mathematics vol 313 no 6 pp 767ndash771 2013

[147] D Van Wieren M Livingston and Q F Stout ldquoPerfectdominating sets on cube-connected cyclesrdquo Congressus Numer-antium vol 97 pp 51ndash70 1993

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

20 International Journal of Combinatorics

119887R (119866)

=

lceil119898

2rceil 119894119891 119899

119898= 1 119886119899119889 119899

119898+1⩾ 2 (1 ⩽ 119898 lt 119905)

2119905 minus 2 119894119891 1198991= 119899

2= sdot sdot sdot = 119899

119905= 2 (119905 ⩾ 2)

2 119894119891 1198991= 2 119886119899119889 119899

2⩾ 3 (119905 = 2)

119905minus1

sum

119894=1

119899119894minus 1 119900119905ℎ119890119903119908119894119904119890

(46)

Theorem 126 (Hattingh and Plummer [80] 2008) For a tree119879 of order 119899 (⩾ 4) 119879 ≇ 119870

1119899minus1if and only if 119887R(119879) = 1

Theorem 126 shows that the restrained bondage numberof a tree can be computed in constant time However thedecision problem for 119887R(119866) is NP-complete even for bipartitegraphs

Problem 5 Consider the decision problem

Restrained Bondage Problem

Instance a graph 119866 and a positive integer 119887

Question is 119887R(119866) ⩽ 119887

Theorem 127 (Hattingh and Plummer [80] 2008) Therestrained bondage problem is NP-complete even for bipartitegraphs

Consequently in view of its computational hardnessit is significative to establish some sharp bounds on therestrained bondage number of a graph in terms of othergraphic parameters

Theorem 128 (Hattingh and Plummer [80] 2008) For anygraph 119866 with 120575(119866) ⩾ 2

119887R (119866) ⩽ min119909119910isin119864(119866)

119889119866(119909) + 119889

119866(119910) minus 2 (47)

Corollary 129 119887R(119866) ⩽ Δ(119866)+120575(119866)minus2 for any graph119866 with120575(119866) ⩾ 2

Notice that the bounds stated in Theorem 128 andCorollary 129 are sharp Indeed the class of cycles whoseorders are congruent to 1 2 (mod 3) have a restrained bondagenumber achieving these bounds (see Theorem 125)

Theorem 128 is an analogue of Theorem 14 A quitenatural problem is whether or not for restricted bondagenumber there are analogues of Theorem 16 Theorem 18Theorem 29 and so on Indeed we should consider all resultsfor the bondage number whether or not there are analoguesfor restricted bondage number

Theorem 130 (Hattingh and Plummer [80] 2008) For anygraph 119866 with 120574

119877(119866) = 2

119887R (119866) ⩽ Δ (119866) + 1 (48)

We now consider the relation between 119887119877(119866) and 119887(119866)

Theorem 131 (Hattingh and Plummer [80] 2008) If 120574119877(119866) =

120574(119866) for some graph 119866 then 119887R(119866) ⩽ 119887(119866)

Corollary 129 and Theorem 131 provide a possibility toattack Conjecture 73 by characterizing planar graphs with120574119877= 120574 and 119887R = 119887 when 3 ⩽ Δ ⩽ 6However we do not have 119887R(119866) = 119887(119866) for any graph

119866 even if 120574119877(119866) = 120574(119866) Observe that 120574

119877(119870

3) = 120574(119870

3) yet

119887119877(119870

3) = 1 and 119887(119870

3) = 2 We still may not claim that

119887119877(119866) = 119887(119866) even in the case that every 120574-set is a 120574

119877-set

The example 1198703again demonstrates this

Theorem 132 (Hattingh and Plummer [80] 2008) There isan infinite class of graphs inwhich each graph119866 satisfies 119887(119866) lt119887R(119866)

In fact such an infinite class of graphs can be obtained asfollows Let119867 be a connected graph BC(119867) a graph obtainedfrom 119867 by attaching ℓ (⩾ 2) vertices of degree 1 to eachvertex in 119867 and B = 119866 119866 = BC(119867) for some graph 119867such that 120575(119867) ⩾ 2 By Theorem 3 we can easily check that119887(119866) = 1 lt 2 ⩽ min120575(119867) ℓ = 119887R(119866) for any 119866 isin BMoreover there exists a graph 119866 isin B such that 119887R(119866) canbe much larger than 119887(119866) which is stated as the followingtheorem

Theorem 133 (Hattingh and Plummer [80] 2008) For eachpositive integer 119896 there is a graph119866 such that 119896 = 119887R(119866)minus119887(119866)

Combining Theorem 133 with Theorem 132 we have thefollowing corollary

Corollary 134 The bondage number and the restrainedbondage number are unrelated

Very recently Jafari Rad et al [81] have consideredthe total restrained bondage in graphs based on both totaldominating sets and restrained dominating sets and obtainedseveral properties exact values and bounds for the totalrestrained bondage number of a graph

A restrained dominating set 119878 of a graph 119866 without iso-lated vertices is called to be a total restrained dominating setif 119878 is a total dominating set The total restrained dominationnumber of119866 denoted by 120574TR (119866) is theminimum cardinalityof a total restrained dominating set of 119866 For referenceson this domination in graphs see [3 61 82ndash85] The totalrestrained bondage number 119887TR (119866) of a graph 119866 with noisolated vertex is the cardinality of a smallest set of edges 119861 sube119864(119866) for which119866minus119861 has no isolated vertex and 120574TR (119866minus119861) gt120574TR (119866) In the case that there is no such subset 119861 we define119887TR (119866) = infin

Ma et al [84] determined that 120574TR (119875119899) = 119899minus2lfloor(119899minus2)4rfloorfor 119899 ⩾ 3 120574TR (119862119899

) = 119899 minus 2lfloor1198994rfloor for 119899 ⩾ 3 120574TR (1198703) =

3 and 120574TR (119870119899) = 2 for 119899 = 3 and 120574TR (1198701119899

) = 1 + 119899

and 120574TR (119870119898119899) = 2 for 2 ⩽ 119898 ⩽ 119899 According to these

results Jafari Rad et al [81] determined the exact valuesof total restrained bondage numbers for the correspondinggraphs

International Journal of Combinatorics 21

Theorem 135 (Jafari Rad et al [81]) For an integer 119899

119887TR (119875119899) = infin 119894119891 2 ⩽ 119899 ⩽ 5

1 119894119891 119899 ⩾ 5

119887TR (119862119899) =

infin 119894119891 119899 = 3

1 119894119891 3 lt 119899 119899 equiv 0 1 (mod 4)2 119894119891 3 lt 119899 119899 equiv 2 3 (mod 4)

119887TR (119882119899) = 2 119891119900119903 119899 ⩾ 6

119887TR (119870119899) = 119899 minus 1 119891119900119903 119899 ⩾ 4

119887TR (119870119898119899) = 119898 minus 1 119891119900119903 2 ⩽ 119898 ⩽ 119899

(49)

119887TR (119870119899) = 119899minus1 shows the fact that for any integer 119899 ⩾ 3 there

exists a connected graph namely119870119899+1

such that 119887TR (119870119899+1) =

119899 that is the total restrained bondage number is unboundedfrom the above

A vertex is said to be a support vertex if it is adjacent toa vertex of degree one Let A be the family of all connectedgraphs such that119866 belongs toA if and only if every edge of119866is incident to a support vertex or 119866 is a cycle of length three

Proposition 136 (Cyman and Raczek [82] 2006) For aconnected graph119866 of order 119899 120574TR (119866) = 119899 if and only if119866 isin A

Hence if 119866 isin A then the removal of any subset of edgesof119866 cannot increase the total restrained domination numberof 119866 and thus 119887TR (119866) = infin When 119866 notin A a result similar toTheorem 6 is obtained

Theorem 137 (Jafari Rad et al [81]) For any tree 119879 notin A119887TR (119879) ⩽ 2 This bound is sharp

In the samepaper the authors provided a characterizationfor trees 119879 notin A with 119887TR (119879) = 1 or 2 The following result issimilar to Observation 1

Observation 3 (Jafari Rad et al [81]) If 119896 edges can beremoved from a graph 119866 to obtain a graph 119867 without anisolated vertex and with 119887TR (119867) = 119905 then 119887TR (119866) ⩽ 119896 + 119905

ApplyingTheorem 137 andObservation 3 Jafari Rad et alcharacterized all connected graphs 119866 with 119887TR (119866) = infin

Theorem 138 (Jafari Rad et al [81]) For a connected graph119866119887TR (119866) = infin if and only if 119866 isin A

84 Other Conditional Bondage Numbers A subset 119868 sube 119881(119866)is called an independent set if no two vertices in 119868 are adjacentin 119866 The maximum cardinality among all independent setsis called the independence number of 119866 denoted by 120572(119866)

A dominating set 119878 of a graph 119866 is called to be inde-pendent if 119878 is an independent set of 119866 The minimumcardinality among all independent dominating set is calledthe independence domination number of 119866 and denoted by120574119868(119866)

Since an independent dominating set is not only adominating set but also an independent set 120574(119866) ⩽ 120574

119868(119866) ⩽

120572(119866) for any graph 119866It is clear that a maximal independent set is certainly a

dominating set Thus an independent set is maximal if andonly if it is an independent dominating set and so 120574

119868(119866) is the

minimum cardinality among all maximal independent setsof 119866 This graph-theoretical invariant has been well studiedin the literature see for example Haynes et al [3] for earlyresults Goddard and Henning [86] for recent results onindependent domination in graphs

In 2003 Zhang et al [87] defined the independencebondage number 119887

119868(119866) of a nonempty graph 119866 to be the

minimum cardinality among all subsets 119861 sube 119864(119866) for which120574119868(119866 minus 119861) gt 120574

119868(119866) For some ordinary graphs their indepen-

dence domination numbers can be easily determined and soindependence bondage numbers have been also determinedClearly 119887

119868(119870

119899) = 1 if 119899 ⩾ 2

Theorem 139 (Zhang et al [87] 2003) For any integer 119899 ⩾ 4

119887119868(119862

119899) =

1 119894119891 119899 equiv 1 (mod 2) 2 119894119891 119899 equiv 0 (mod 2)

119887119868(119875

119899) =

1 119894119891 119899 equiv 0 (mod 2) 2 119894119891 119899 equiv 1 (mod 2)

119887119868(119870

119898119899) = 119899 119891119900119903 119898 ⩽ 119899

(50)

Apart from the above-mentioned results as far as we findno other results on the independence bondage number in thepresent literature we never knew much of any result on thisparameter for a tree

A subset 119878 of 119881(119866) is called an equitable dominatingset if for every 119910 isin 119881(119866 minus 119878) there exists a vertex 119909 isin 119878such that 119909119910 isin 119864(119866) and |119889

119866(119909) minus 119889

119866(119910)| ⩽ 1 proposed

by Dharmalingam [88] The minimum cardinality of such adominating set is called the equitable domination number andis denoted by 120574

119864(119866) Deepak et al [89] defined the equitable

bondage number 119887119864(119866) of a graph 119866 to be the cardinality of a

smallest set 119861 sube 119864(119866) of edges for which 120574119864(119866 minus 119861) gt 120574

119864(119866)

and determined 119887119864(119866) for some special graphs

Theorem 140 (Deepak et al [89] 2011) For any integer 119899 ⩾ 2

119887119864(119870

119899) = lceil

119899

2rceil

119887119864(119870

119898119899) = 119898 119891119900119903 0 ⩽ 119899 minus 119898 ⩽ 1

119887119864(119862

119899) =

3 119894119891 119899 equiv 1 (mod 3)2 119900119905ℎ119890119903119908119894119904119890

119887119864(119875

119899) =

2 119894119891 119899 equiv 1 (mod 3)1 119900119905ℎ119890119903119908119894119904119890

119887119864(119879) ⩽ 2 119891119900119903 119886119899119910 119899119900119899119905119903119894119907119894119886119897 119905119903119890119890 119879

119887119864(119866) ⩽ 119899 minus 1 119891119900119903 119886119899119910 119888119900119899119899119890119888119905119890119889 119892119903119886119901ℎ 119866 119900119891 119900119903119889119890119903 119899

(51)

22 International Journal of Combinatorics

As far as we know there are no more results on theequitable bondage number of graphs in the present literature

There are other variations of the normal domination byadding restricted conditions to the normal dominating setFor example the connected domination set of a graph119866 pro-posed by Sampathkumar and Walikar [90] is a dominatingset 119878 such that the subgraph induced by 119878 is connected Theproblem of finding a minimum connected domination set ina graph is equivalent to the problemof finding a spanning treewithmaximumnumber of vertices of degree one [91] and hassome important applications in wireless networks [92] thusit has received much research attention However for suchan important domination there is no result on its bondagenumber We believe that the topic is of significance for futureresearch

9 Generalized Bondage Numbers

There are various generalizations of the classical dominationsuch as 119901-domination distance domination fractional dom-ination Roman domination and rainbow domination Everysuch a generalization can lead to a corresponding bondageIn this section we introduce some of them

91 119901-Bondage Numbers In 1985 Fink and Jacobson [93]introduced the concept of 119901-domination Let 119901 be a positiveinteger A subset 119878 of 119881(119866) is a 119901-dominating set of 119866 if|119878 cap 119873

119866(119910)| ⩾ 119901 for every 119910 isin 119878 The 119901-domination number

120574119901(119866) is the minimum cardinality among all 119901-dominating

sets of 119866 Any 119901-dominating set of 119866 with cardinality 120574119901(119866)

is called a 120574119901-set of 119866 Note that a 120574

1-set is a classic minimum

dominating set Notice that every graph has a 119901-dominatingset since the vertex-set119881(119866) is such a setWe also note that a 1-dominating set is a dominating set and so 120574(119866) = 120574

1(119866) The

119901-domination number has received much research attentionsee a state-of-the-art survey article by Chellali et al [94]

It is clear from definition that every 119901-dominating setof a graph certainly contains all vertices of degree at most119901 minus 1 By this simple observation to avoid the trivial caseoccurrence we always assume that Δ(119866) ⩾ 119901 For 119901 ⩾ 2 Luet al [95] gave a constructive characterization of trees withunique minimum 119901-dominating sets

Recently Lu and Xu [96] have introduced the conceptto the 119901-bondage number of 119866 denoted by 119887

119901(119866) as the

minimum cardinality among all sets of edges 119861 sube 119864(119866) suchthat 120574

119901(119866 minus 119861) gt 120574

119901(119866) Clearly 119887

1(119866) = 119887(119866)

Lu and Xu [96] established a lower bound and an upperbound on 119887

119901(119879) for any integer 119901 ⩾ 2 and any tree 119879 with

Δ(119879) ⩾ 119901 and characterized all trees achieving the lowerbound and the upper bound respectively

Theorem 141 (Lu and Xu [96] 2011) For any integer 119901 ⩾ 2and any tree 119879 with Δ(119879) ⩾ 119901

1 ⩽ 119887119901(119879) ⩽ Δ (119879) minus 119901 + 1 (52)

Let 119878 be a given subset of 119881(119866) and for any 119909 isin 119878 let

119873119901(119909 119878 119866) = 119910 isin 119878 cap 119873

119866(119909)

1003816100381610038161003816119873119866(119910) cap 119878

1003816100381610038161003816 = 119901

(53)

Then all trees achieving the lower bound 1 can be character-ized as follows

Theorem 142 (Lu and Xu [96] 2011) Let 119879 be a tree withΔ(119879) ⩾ 119901 ⩾ 2 Then 119887

119901(119879) = 1 if and only if for any 120574

119901-set

119878 of 119879 there exists an edge 119909119910 isin (119878 119878) such that 119910 isin 119873119901(119909 119878 119879)

The notation 119878(119886 119887) denotes a double star obtained byadding an edge between the central vertices of two stars119870

1119886minus1

and1198701119887minus1

And the vertex with degree 119886 (resp 119887) in 119878(119886 119887) iscalled the 119871-central vertex (resp 119877-central vertex) of 119878(119886 119887)

To characterize all trees attaining the upper bound givenin Theorem 141 we define three types of operations on a tree119879 with Δ(119879) = Δ ⩾ 119901 + 1

Type 1 attach a pendant edge to a vertex 119910with 119889119879(119910) ⩽ 119901minus

2 in 119879

Type 2 attach a star 1198701Δminus1

to a vertex 119910 of 119879 by joining itscentral vertex to 119910 where 119910 in a 120574

119901-set of 119879 and

119889119879(119910) ⩽ Δ minus 1

Type 3 attach a double star 119878(119901 Δ minus 1) to a pendant vertex 119910of 119879 by coinciding its 119877-central vertex with 119910 wherethe unique neighbor of 119910 is in a 120574

119901-set of 119879

Let B = 119879 a tree obtained from 1198701Δ

or 119878(119901 Δ) by afinite sequence of operations of Types 1 2 and 3

Theorem 143 (Lu and Xu [96] 2011) A tree with the maxi-mum Δ ⩾ 119901 + 1 has 119901-bondage number Δ minus 119901 + 1 if and onlyif it belongs toB

Theorem 144 (Lu and Xu [97] 2012) Let 119866119898119899= 119875

119898times 119875

119899

Then

1198872(119866

2119899) = 1 119891119900119903 119899 ⩾ 2

1198872(119866

3119899) =

2 119894119891 119899 equiv 1 (mod 3) 1 119900119905ℎ119890119903119908119894119904119890

for 119899 ⩾ 2

1198872(119866

4119899) =

1 119894119891 119899 equiv 3 (mod 4) 2 119900119905ℎ119890119903119908119894119904119890

for 119899 ⩾ 7

(54)

Recently Krzywkowski [98] has proposed the conceptof the nonisolating 2-bondage of a graph The nonisolating2-bondage number of a graph 119866 denoted by 1198871015840

2(119866) is the

minimum cardinality among all sets of edges 119861 sube 119864(119866) suchthat 119866 minus 119861 has no isolated vertices and 120574

2(119866 minus 119861) gt 120574

2(119866) If

for every 119861 sube 119864(119866) either 1205742(119866 minus 119861) = 120574

2(119866) or 119866 minus 119861 has

isolated vertices then define 11988710158402(119866) = 0 and say that 119866 is a 2-

nonisolatedly strongly stable graph Krzywkowski presentedsome basic properties of nonisolating 2-bondage in graphsshowed that for every nonnegative integer 119896 there exists atree 119879 such 1198871015840

2(119879) = 119896 characterized all 2-non-isolatedly

strongly stable trees and determined 11988710158402(119866) for several special

graphs

International Journal of Combinatorics 23

Theorem 145 (Krzywkowski [98] 2012) For any positiveintegers 119899 and119898 with119898 ⩽ 119899

1198871015840

2(119870

119899) =

0 119894119891 119899 = 1 2 3

lfloor2119899

3rfloor 119900119905ℎ119890119903119908119894119904119890

1198871015840

2(119875

119899) =

0 119894119891 119899 = 1 2 3

1 119900119905ℎ119890119903119908119894119904119890

1198871015840

2(119862

119899) =

0 119894119891 119899 = 3

1 119894119891 119899 119894119904 119890V119890119899

2 119894119891 119899 119894119904 119900119889119889

1198871015840

2(119870

119898119899) =

3 119894119891 119898 = 119899 = 3

1 119894119891 119898 = 119899 = 4

2 119900119905ℎ119890119903119908119894119904119890

(55)

92 Distance Bondage Numbers A subset 119878 of vertices of agraph119866 is said to be a distance 119896-dominating set for119866 if everyvertex in 119866 not in 119878 is at distance at most 119896 from some vertexof 119878 The minimum cardinality of all distance 119896-dominatingsets is called the distance 119896-domination number of 119866 anddenoted by 120574

119896(119866) (do not confuse with the above-mentioned

120574119901(119866)) When 119896 = 1 a distance 1-dominating set is a normal

dominating set and so 1205741(119866) = 120574(119866) for any graph 119866 Thus

the distance 119896-domination is a generalization of the classicaldomination

The relation between 120574119896and 120572

119896(the 119896-independence

number defined in Section 22) for a tree obtained by Meirand Moon [14] who proved that 120574

119896(119879) = 120572

2119896(119879) for any tree

119879 (a special case for 119896 = 1 is stated in Proposition 10) Thefurther research results can be found in Henning et al [99]Tian and Xu [100ndash103] and Liu et al [104]

In 1998 Hartnell et al [15] defined the distance 119896-bondagenumber of 119866 denoted by 119887

119896(119866) to be the cardinality of a

smallest subset 119861 of edges of 119866 with the property that 120574119896(119866 minus

119861) gt 120574119896(119866) FromTheorem 6 it is clear that if119879 is a nontrivial

tree then 1 ⩽ 1198871(119879) ⩽ 2 Hartnell et al [15] generalized this

result to any integer 119896 ⩾ 1

Theorem 146 (Hartnell et al [15] 1998) For every nontrivialtree 119879 and positive integer 119896 1 ⩽ 119887

119896(119879) ⩽ 2

Hartnell et al [15] and Topp and Vestergaard [13] alsocharacterized the trees having distance 119896-bondage number 2In particular the class of trees for which 119887

1(119879) = 2 are just

those which have a unique maximum 2-independent set (seeTheorem 11)

Since when 119896 = 1 the distance 1-bondage number 1198871(119866)

is the classical bondage number 119887(119866) Theorem 12 gives theNP-hardness of deciding the distance 119896-bondage number ofgeneral graphs

Theorem 147 Given a nonempty undirected graph 119866 andpositive integers 119896 and 119887 with 119887 ⩽ 120576(119866) determining wetheror not 119887

119896(119866) ⩽ 119887 is NP-hard

For a vertex 119909 in119866 the open 119896-neighborhood119873119896(119909) of 119909 is

defined as119873119896(119909) = 119910 isin 119881(119866) 1 ⩽ 119889

119866(119909 119910) le 119896The closed

119896-neighborhood119873119896[119909] of 119909 in 119866 is defined as119873

119896(119909) cup 119909 Let

Δ119896(119866) = max 1003816100381610038161003816119873119896

(119909)1003816100381610038161003816 for any 119909 isin 119881 (119866) (56)

Clearly Δ1(119866) = Δ(119866) The 119896th power of a graph 119866 is the

graph119866119896 with vertex-set119881(119866119896

) = 119881(119866) and edge-set119864(119866119896

) =

119909119910 1 ⩽ 119889119866(119909 119910) ⩽ 119896 The following lemma holds directly

from the definition of 119866119896

Proposition 148 (Tian and Xu [101]) Δ(119866119896

) = Δ119896(119866) and

120574(119866119896

) = 120574119896(119866) for any graph 119866 and each 119896 ⩾ 1

A graph 119866 is 119896-distance domination-critical or 120574119896-critical

for short if 120574119896(119866 minus 119909) lt 120574

119896(119866) for every vertex 119909 in 119866

proposed by Henning et al [105]

Proposition 149 (Tian and Xu [101]) For each 119896 ⩾ 1 a graph119866 is 120574

119896-critical if and only if 119866119896 is 120574

119896-critical

By the above facts we suggest the following worthwhileproblem

Problem 6 Can we generalize the results on the bondage for119866 to 119866119896 In particular do the following propositions hold

(a) 119887(119866119896

) = 119887119896(119866) for any graph 119866 and each 119896 ⩾ 1

(b) 119887(119866119896

) ⩽ Δ119896(119866) if 119866 is not 120574

119896-critical

Let 119896 and 119901 be positive integers A subset 119878 of 119881(119866) isdefined to be a (119896 119901)-dominating set of 119866 if for any vertex119909 isin 119878 |119873

119896(119909) cap 119878| ⩾ 119901 The (119896 119901)-domination number of

119866 denoted by 120574119896119901(119866) is the minimum cardinality among

all (119896 119901)-dominating sets of 119866 Clearly for a graph 119866 a(1 1)-dominating set is a classical dominating set a (119896 1)-dominating set is a distance 119896-dominating set and a (1 119901)-dominating set is the above-mentioned 119901-dominating setThat is 120574

11(119866) = 120574(119866) 120574

1198961(119866) = 120574

119896(119866) and 120574

1119901(119866) = 120574

119901(119866)

The concept of (119896 119901)-domination in a graph 119866 is a gen-eralized domination which combines distance 119896-dominationand 119901-domination in 119866 So the investigation of (119896 119901)-domination of 119866 is more interesting and has received theattention of many researchers see for example [106ndash109]

More general Li and Xu [110] proposed the concept of the(ℓ 119908)-domination number of a119908-connected graph A subset119878 of 119881(119866) is defined to be an (ℓ 119908)-dominating set of 119866 iffor any vertex 119909 isin 119878 there are 119908 internally disjoint pathsof length at most ℓ from 119909 to some vertex in 119878 The (ℓ 119908)-domination number of119866 denoted by 120574

ℓ119908(119866) is theminimum

cardinality among all (ℓ 119908)-dominating sets of 119866 Clearly1205741198961(119866) = 120574

119896(119866) and 120574

1119901(119866) = 120574

119901(119866)

It is quite natural to propose the concept of bondage num-ber for (119896 119901)-domination or (ℓ 119908)-domination However asfar as we know none has proposed this concept until todayThis is a worthwhile topic for further research

24 International Journal of Combinatorics

93 Fractional Bondage Numbers If 120590 is a function mappingthe vertex-set 119881 into some set of real numbers then for anysubset 119878 sube 119881 let 120590(119878) = sum

119909isin119878120590(119909) Also let |120590| = 120590(119881)

A real-value function 120590 119881 rarr [0 1] is a dominatingfunction of a graph 119866 if for every 119909 isin 119881(119866) 120590(119873

119866[119909]) ⩾ 1

Thus if 119878 is a dominating set of 119866 120590 is a function where

120590 (119909) = 1 if 119909 isin 1198780 otherwise

(57)

then 120590 is a dominating function of119866 The fractional domina-tion number of 119866 denoted by 120574

119865(119866) is defined as follows

120574119865(119866) = min |120590| 120590 is a domination function of 119866

(58)

The 120574119865-bondage number of 119866 denoted by 119887

119865(119866) is

defined as the minimum cardinality of a subset 119861 sube 119864 whoseremoval results in 120574

119865(119866 minus 119861) gt 120574

119865(119866)

Hedetniemi et al [111] are the first to study fractionaldomination although Farber [112] introduced the idea indi-rectly The concept of the fractional domination numberwas proposed by Domke and Laskar [113] in 1997 Thefractional domination numbers for some ordinary graphs aredetermined

Proposition 150 (Domke et al [114] 1998 Domke and Laskar[113] 1997) (a) If 119866 is a 119896-regular graph with order 119899 then120574119865(119866) = 119899119896 + 1(b) 120574

119865(119862

119899) = 1198993 120574

119865(119875

119899) = lceil1198993rceil 120574

119865(119870

119899) = 1

(c) 120574119865(119866) = 1 if and only if Δ(119866) = 119899 minus 1

(d) 120574119865(119879) = 120574(119879) for any tree 119879

(e) 120574119865(119870

119899119898) = (119899(119898 + 1) + 119898(119899 + 1))(119899119898 minus 1) where

max119899119898 gt 1

The assertions (a)ndash(d) are due to Domke et al [114] andthe assertion (e) is due to Domke and Laskar [113] Accordingto these results Domke and Laskar [113] determined the 120574

119865-

bondage numbers for these graphs

Theorem 151 (Domke and Laskar [113] 1997) 119887119865(119870

119899) =

lceil1198992rceil 119887119865(119870

119899119898) = 1 wheremax119899119898 gt 1

119887119865(119862

119899) =

2 119894119891 119899 equiv 0 (mod 3) 1 119900119905ℎ119890119903119908119894119904119890

119887119865(119875

119899) =

2 119894119891 119899 equiv 1 (mod 3) 1 119900119905ℎ119890119903119908119894119904119890

(59)

It is easy to see that for any tree119879 119887119865(119879) ⩽ 2 In fact since

120574119865(119879) = 120574(119879) for any tree 119879 120574

119865(119879

1015840

) = 120574(1198791015840

) for any subgraph1198791015840 of 119879 It follows that

119887119865(119879) = min 119861 sube 119864 (119879) 120574

119865(119879 minus 119861) gt 120574

119865(119879)

= min 119861 sube 119864 (119879) 120574 (119879 minus 119861) gt 120574 (119879)

= 119887 (119879) ⩽ 2

(60)

Except for these no results on 119887119865(119866) have been known as

yet

94 Roman Bondage Numbers A Roman dominating func-tion on a graph 119866 is a labeling 119891 119881 rarr 0 1 2 such thatevery vertex with label 0 has at least one neighbor with label2 The weight of a Roman dominating function 119891 on 119866 is thevalue

119891 (119866) = sum

119906isin119881(119866)

119891 (119906) (61)

The minimum weight of a Roman dominating function ona graph 119866 is called the Roman domination number of 119866denoted by 120574RM (119866)

A Roman dominating function 119891 119881 rarr 0 1 2

can be represented by the ordered partition (1198810 119881

1 119881

2) (or

(119881119891

0 119881

119891

1 119881

119891

2) to refer to119891) of119881 where119881

119894= V isin 119881 | 119891(V) = 119894

In this representation its weight 119891(119866) = |1198811| + 2|119881

2| It is

clear that119881119891

1cup119881

119891

2is a dominating set of 119866 called the Roman

dominating set denoted by 119863119891

R = (1198811 1198812) Since 119881119891

1cup 119881

119891

2is

a dominating set when 119891 is a Roman dominating functionand since placing weight 2 at the vertices of a dominating setyields a Roman dominating function in [115] it is observedthat

120574 (119866) ⩽ 120574RM (119866) ⩽ 2120574 (119866) (62)

A graph 119866 is called to be Roman if 120574RM (119866) = 2120574(119866)The definition of the Roman domination function is

proposed implicitly by Stewart [116] and ReVelle and Rosing[117] Roman dominating numbers have been deeply studiedIn particular Bahremandpour et al [118] showed that theproblem determining the Roman domination number is NP-complete even for bipartite graphs

Let 119866 be a graph with maximum degree at least twoThe Roman bondage number 119887RM (119866) of 119866 is the minimumcardinality of all sets 1198641015840 sube 119864 for which 120574RM (119866 minus 119864

1015840

) gt

120574RM (119866) Since in study of Roman bondage number theassumption Δ(119866) ⩾ 2 is necessary we always assume thatwhen we discuss 119887RM (119866) all graphs involved satisfy Δ(119866) ⩾2 The Roman bondage number 119887RM (119866) is introduced byJafari Rad and Volkmann in [119]

Recently Bahremandpour et al [118] have shown that theproblem determining the Roman bondage number is NP-hard even for bipartite graphs

Problem 7 Consider the decision problem

Roman Bondage Problem

Instance a nonempty bipartite graph119866 and a positiveinteger 119896

Question is 119887RM (119866) ⩽ 119896

Theorem 152 (Bahremandpour et al [118] 2013) TheRomanbondage problem is NP-hard even for bipartite graphs

The exact value of 119887RM(119866) is known only for a few familyof graphs including the complete graphs cycles and paths

International Journal of Combinatorics 25

Theorem 153 (Jafari Rad and Volkmann [119] 2011) For anypositive integers 119899 and119898 with119898 ⩽ 119899

119887RM (119875119899) = 2 119894119891 119899 equiv 2 (mod 3) 1 119900119905ℎ119890119903119908119894119904119890

119887RM (119862119899) =

3 119894119891 119899 = 2 (mod 3) 2 119900119905ℎ119890119903119908119894119904119890

119887RM (119870119899) = lceil

119899

2rceil 119891119900119903 119899 ⩾ 3

119887RM (119870119898119899) =

1 119894119891 119898 = 1 119886119899119889 119899 = 1

4 119894119891 119898 = 119899 = 3

119898 119900119905ℎ119890119903119908119894119904119890

(63)

Lemma 154 (Cockayne et al [115] 2004) If 119866 is a graph oforder 119899 and contains vertices of degree 119899 minus 1 then 120574RM(119866) = 2

Using Lemma 154 the third conclusion in Theorem 153can be generalized to more general case which is similar toLemma 49

Theorem 155 (Jafari Rad and Volkmann [119] 2011) Let119866 bea graph with order 119899 ge 3 and 119905 the number of vertices of degree119899 minus 1 in 119866 If 119905 ⩾ 1 then 119887RM(119866) = lceil1199052rceil

Ebadi and PushpaLatha [120] conjectured that 119887RM(119866) ⩽119899 minus 1 for any graph 119866 of order 119899 ⩾ 3 Dehgardi et al[121] Akbari andQajar [122] independently showed that thisconjecture is true

Theorem 156 119887RM(119866) ⩽ 119899 minus 1 for any connected graph 119866 oforder 119899 ⩾ 3

For a complete 119905-partite graph we have the followingresult

Theorem 157 (Hu and Xu [123] 2011) Let 119866 = 11987011989811198982119898119905

bea complete 119905-partite graph with 119898

1= sdot sdot sdot = 119898

119894lt 119898

119894+1le sdot sdot sdot ⩽

119898119905 119905 ⩾ 2 and 119899 = sum119905

119895=1119898

119895 Then

119887RM (119866) =

lceil119894

2rceil 119894119891 119898

119894= 1 119886119899119889 119899 ⩾ 3

2 119894119891 119898119894= 2 119886119899119889 119894 = 1

119894 119894119891 119898119894= 2 119886119899119889 119894 ⩾ 2

4 119894119891 119898119894= 3 119886119899119889 119894 = 119905 = 2

119899 minus 1 119894119891 119898119894= 3 119886119899119889 119894 = 119905 ⩾ 3

119899 minus 119898119905

119894119891 119898119894⩾ 3 119886119899119889 119898

119905⩾ 4

(64)

Consider a complete 119905-partite graph 119870119905(3) for 119905 ⩾ 3

119887RM(119870119905(3)) = 119899 minus 1 by Theorem 157 where 119899 = 3119905

which shows that this upper bound of 119899 minus 1 on 119887RM(119866) inTheorem 156 is sharp This fact and 120574

119877(119870

119905(3)) = 4 give

a negative answer to the following two problems posed byDehgardi et al [121]Prove or Disprove for any connected graph 119866 of order 119899 ge 3(i) 119887RM(119866) = 119899 minus 1 if and only if 119866 cong 119870

3 (ii) 119887RM(119866) ⩽ 119899 minus

120574119877(119866) + 1Note that a complete 119905-partite graph 119870

119905(3) is (119899 minus 3)-

regular and 119887RM(119870119905(3)) = 119899 minus 1 for 119905 ⩾ 3 where 119899 = 3119905

Hu and Xu further determined that 119887119877(119866) = 119899 minus 2 for any

(119899 minus 3)-regular graph 119866 of order 119899 ⩽ 5 except 119870119905(3)

Theorem 158 (Hu and Xu [123] 2011) For any (119899minus3)-regulargraph 119866 of order 119899 other than119870

119905(3) 119887RM(119866) = 119899 minus 2 for 119899 ⩾ 5

For a tree 119879 with order 119899 ⩾ 3 Ebadi and PushpaLatha[120] and Jafari Rad and Volkmann [119] independentlyobtained an upper bound on 119887RM(119879)

Theorem 159 119887RM(119879) ⩽ 3 for any tree 119879 with order 119899 ⩾ 3

Theorem 160 (Bahremandpour et al [118] 2013) 119887RM(1198752 times119875119899) = 2 for 119899 ⩾ 2

Theorem 161 (Jafari Rad and Volkmann [119] 2011) Let119866 bea graph with order at least three

(a) If (119909 119910 119911) is a path of length 2 in 119866 then 119887RM(119866) ⩽119889119866(119909) + 119889

119866(119910) + 119889

119866(119911) minus 3 minus |119873

119866(119909) cap 119873

119866(119910)|

(b) If 119866 is connected then 119887RM(119866) ⩽ 120582(119866) + 2Δ(119866) minus 3

Theorem 161 (b) implies 119887RM(119866) ⩽ 120575(119866) + 2Δ(119866) minus 3 for aconnected graph 119866 Note that for a planar graph 119866 120575(119866) ⩽ 5moreover 120575(119866) ⩽ 3 if the girth at least 4 and 120575(119866) ⩽ 2 if thegirth at least 6 These two facts show that 119887RM(119866) ⩽ 2Δ(119866) +2 for a connected planar graph 119866 Jafari Rad and Volkmann[124] improved this bound

Theorem 162 (Jafari Rad and Volkmann [124] 2011) For anyconnected planar graph 119866 of order 119899 ⩾ 3 with girth 119892(119866)

119887RM (119866)

⩽

2Δ (119866)

Δ (119866) + 6

Δ (119866) + 5 119894119891 119866 119888119900119899119905119886119894119899119904 119899119900 V119890119903119905119894119888119890119904 119900119891 119889119890119892119903119890119890 119891119894V119890

Δ (119866) + 4 119894119891 119892 (119866) ⩾ 4

Δ (119866) + 3 119894119891 119892 (119866) ⩾ 5

Δ (119866) + 2 119894119891 119892 (119866) ⩾ 6

Δ (119866) + 1 119894119891 119892 (119866) ⩾ 8

(65)

According toTheorem 157 119887RM(119862119899) = 3 = Δ(119862

119899)+1 for a

cycle 119862119899of length 119899 ⩾ 8 with 119899 equiv 2 (mod 3) and therefore

the last upper bound inTheorem 162 is sharp at least for Δ =2

Combining the fact that every planar graph 119866 withminimum degree 5 contains an edge 119909119910 with 119889

119866(119909) = 5

and 119889119866(119910) isin 5 6 with Theorem 161 (a) Akbari et al [125]

obtained the following result

26 International Journal of Combinatorics

middot middot middot

middot middot middot

middot middot middot

middot middot middot

119866

Figure 9 The graph 119866 is constructed from 119866

Theorem 163 (Akbari et al [125] 2012) 119887RM(119866) ⩽ 15 forevery planar graph 119866

It remains open to show whether the bound inTheorem 163 is sharp or not Though finding a planargraph 119866 with 119887RM(119866) = 15 seems to be difficult Akbari etal [125] constructed an infinite family of planar graphs withRoman bondage number equal to 7 by proving the followingresult

Theorem 164 (Akbari et al [125] 2012) Let 119866 be a graph oforder 119899 and 119866 is the graph of order 5119899 obtained from 119866 byattaching the central vertex of a copy of a path 119875

5to each vertex

of119866 (see Figure 9) Then 120574RM(119866) = 4119899 and 119887RM(119866) = 120575(119866)+2

ByTheorem 164 infinitemany planar graphswith Romanbondage number 7 can be obtained by considering any planargraph 119866 with 120575(119866) = 5 (eg the icosahedron graph)

Conjecture 165 (Akbari et al [125] 2012) The Romanbondage number of every planar graph is at most 7

For general bounds the following observation is directlyobtained which is similar to Observation 1

Observation 4 Let 119866 be a graph of order 119899 with maximumdegree at least two Assume that 119867 is a spanning subgraphobtained from 119866 by removing 119896 edges If 120574RM(119867) = 120574RM(119866)then 119887

119877(119867) ⩽ 119887RM(119866) ⩽ 119887RM(119867) + 119896

Theorem 166 (Bahremandpour et al [118] 2013) For anyconnected graph 119866 of order 119899 if 119899 ⩾ 3 and 120574RM(119866) = 120574(119866) + 1then

119887RM (119866) ⩽ min 119887 (119866) 119899Δ (66)

where 119899Δis the number of vertices with maximum degree Δ in

119866

Observation 5 For any graph 119866 if 120574(119866) = 120574RM(119866) then119887RM(119866) ⩽ 119887(119866)

Theorem 167 (Bahremandpour et al [118] 2013) For everyRoman graph 119866

119887RM (119866) ⩾ 119887 (119866) (67)

The bound is sharp for cycles on 119899 vertices where 119899 equiv 0 (mod3)

The strict inequality in Theorem 167 can hold for exam-ple 119887(119862

3119896+2) = 2 lt 3 = 119887RM(1198623119896+2

) byTheorem 157A graph 119866 is called to be vertex Roman domination-

critical if 120574RM(119866 minus 119909) lt 120574RM(119866) for every vertex 119909 in 119866 If119866 has no isolated vertices then 120574RM(119866) ⩽ 2120574(119866) ⩽ 2120573(119866)If 120574RM(119866) = 2120573(119866) then 120574RM(119866) = 2120574(119866) and hence 119866 is aRoman graph In [30] Volkmann gave a lot of graphs with120574(119866) = 120573(119866) The following result is similar to Theorem 57

Theorem 168 (Bahremandpour et al [118] 2013) For anygraph 119866 if 120574RM(119866) = 2120573(119866) then

(a) 119887RM(119866) ⩾ 120575(119866)

(b) 119887RM(119866) ⩾ 120575(119866)+1 if119866 is a vertex Roman domination-critical graph

If 120574RM(119866) = 2 then obviously 119887RM(119866) ⩽ 120575(119866) So weassume 120574RM(119866) ⩾ 3 Dehgardi et al [121] proved 119887RM(119866) ⩽(120574RM(119866) minus 2)Δ(119866) + 1 and posed the following problem if 119866is a connected graph of order 119899 ⩾ 4 with 120574RM(119866) ⩾ 3 then

119887RM (119866) ⩽ (120574RM (119866) minus 2) Δ (119866) (68)

Theorem 161 (a) shows that the inequality (68) holds if120574RM(119866) ⩾ 5 Thus the bound in (68) is of interest only when120574RM(119866) is 3 or 4

Lemma 169 (Bahremandpour et al [118] 2013) For anynonempty graph 119866 of order 119899 ⩾ 3 120574RM(119866) = 3 if and onlyif Δ(119866) = 119899 minus 2

The following result shows that (68) holds for all graphs119866 of order 119899 ⩾ 4 with 120574RM(119866) = 3 or 4 which improvesTheorem 161 (b)

Theorem 170 (Bahremandpour et al [118] 2013) For anyconnected graph 119866 of order 119899 ⩾ 4

119887RM (119866) le Δ (119866) = 119899 minus 2 119894119891 120574RM (119866) = 3

Δ (119866) + 120575 (119866) minus 1 119894119891 120574RM (119866) = 4(69)

with the first equality if and only if 119866 cong 1198624

Recently Akbari and Qajar [122] have proved the follow-ing result

Theorem 171 (Akbari and Qajar [122]) For any connectedgraph 119866 of order 119899 ⩾ 3

119887RM (119866) ⩽ 119899 minus 120574RM (119866) + 5 (70)

International Journal of Combinatorics 27

We conclude this section with the following problems

Problem 8 Characterize all connected graphs 119866 of order 119899 ⩾3 for which 119887RM(119866) = 119899 minus 1

Problem 9 Prove or disprove if 119866 is a connected graph oforder 119899 ⩾ 3 then

119887RM (119866) ⩽ 119899 minus 120574RM (119866) + 3 (71)

Note that 120574119877(119870

119905(3)) = 4 and 119887RM(119870119905

(3)) = 119899minus1 where 119899 =3119905 for 119905 ⩾ 3 The upper bound on 119887RM(119866) given in Problem 9is sharp if Problem 9 has positive answer

95 Rainbow Bondage Numbers For a positive integer 119896 a119896-rainbow dominating function of a graph 119866 is a function 119891from the vertex-set 119881(119866) to the set of all subsets of the set1 2 119896 such that for any vertex 119909 in 119866 with 119891(119909) = 0 thecondition

⋃

119910isin119873119866(119909)

119891 (119910) = 1 2 119896 (72)

is fulfilledThe weight of a 119896-rainbow dominating function 119891on 119866 is the value

119891 (119866) = sum

119909isin119881(119866)

1003816100381610038161003816119891 (119909)1003816100381610038161003816 (73)

The 119896-rainbow domination number of a graph 119866 denoted by120574119903119896(119866) is the minimum weight of a 119896-rainbow dominating

function 119891 of 119866Note that 120574

1199031(119866) is the classical domination number 120574(119866)

The 119896-rainbow domination number is introduced by Bresaret al [126] Rainbow domination of a graph 119866 coincides withordinary domination of the Cartesian product of 119866 with thecomplete graph in particular 120574

119903119896(119866) = 120574(119866 times 119870

119896) for any

positive integer 119896 and any graph 119866 [126] Hartnell and Rall[127] proved that 120574(119879) lt 120574

1199032(119879) for any tree 119879 no graph has

120574 = 120574119903119896for 119896 ⩾ 3 for any 119896 ⩾ 2 and any graph 119866 of order 119899

min 119899 120574 (119866) + 119896 minus 2 ⩽ 120574119903119896(119866) ⩽ 119896120574 (119866) (74)

The introduction of rainbow domination is motivatedby the study of paired domination in Cartesian productsof graphs where certain upper bounds can be expressed interms of rainbow domination that is for any graph 119866 andany graph 119867 if 119881(119867) can be partitioned into 119896120574

119875-sets then

120574119875(119866 times 119867) ⩽ (1119896)120592(119867)120574

119903119896(119866) proved by Bresar et al [126]

Dehgardi et al [128] introduced the concept of 119896-rainbowbondage number of graphs Let 119866 be a graph with maximumdegree at least two The 119896-rainbow bondage number 119887

119903119896(119866)

of 119866 is the minimum cardinality of all sets 119861 sube 119864(119866) forwhich 120574

119903119896(119866 minus 119861) gt 120574

119903119896(119866) Since in the study of 119896-rainbow

bondage number the assumption Δ(119866) ⩾ 2 is necessarywe always assume that when we discuss 119887

119903119896(119866) all graphs

involved satisfy Δ(119866) ⩾ 2The 2-rainbow bondage number of some special families

of graphs has been determined

Theorem 172 (Dehgardi et al [128]) For any integer 119899 ⩾ 3

1198871199032(119875

119899) = 1 119891119900119903 119899 ⩾ 2

1198871199032(119862

119899) =

1 119894119891 119899 equiv 0 (mod 4) 2 119900119905ℎ119890119903119908119894119904119890

1198871199032(119870

119899119899) =

1 119894119891 119899 = 2

3 119894119891 119899 = 3

6 119894119891 119899 = 4

119898 119894119891 119899 ⩾ 5

(75)

Theorem 173 (Dehgardi et al [128]) For any connected graph119866 of order 119899 ⩾ 3 119887

1199032(119866) ⩽ 120582(119866) + 2Δ(119866) minus 3

Theorem 174 (Dehgardi et al [128]) For any tree 119879 of order119899 ⩾ 3 119887

1199032(119879) ⩽ 2

Theorem 175 (Dehgardi et al [128]) If 119866 is a planar graphwithmaximumdegree at least two then 119887

1199032(119866) ⩽ 15Moreover

if the girth 119892(119866) ⩾ 4 then 1198871199032(119866) ⩽ 11 and if 119892(119866) ⩾ 6 then

1198871199032(119866) ⩽ 9

Theorem 176 (Dehgardi et al [128]) For a complete bipartitegraph 119870

119901119902 if 2119896 + 1 ⩽ 119901 ⩽ 119902 then 119887

119903119896(119870

119901119902) = 119901

Theorem177 (Dehgardi et al [128]) For a complete graph119870119899

if 119899 ⩾ 119896 + 1 then 119887119903119896(119870

119899) = lceil119896119899(119896 + 1)rceil

The special case 119896 = 1 of Theorem 177 can be found inTheorem 1 The following is a simple observation similar toObservation 1

Observation 6 (Dehgardi et al [128]) Let 119866 be a graphwith maximum degree at least two 119867 a spanning subgraphobtained from 119866 by removing 119896 edges If 120574

119903119896(119867) = 120574

119903119896(119866)

then 119887119903119896(119867) ⩽ 119887

119903119896(119866) ⩽ 119887

119903119896(119867) + 119896

Theorem 178 (Dehgardi et al [128]) For every graph 119866 with120574119903119896(119866) = 119896120574(119866) 119887

119903119896(119866) ⩾ 119887(119866)

A graph 119866 is said to be vertex 119896-rainbow domination-critical if 120574

119903119896(119866 minus 119909) lt 120574

119903119896(119866) for every vertex 119909 in 119866 It is

clear that if 119866 has no isolated vertices then 120574119903119896(119866) ⩽ 119896120574(119866) ⩽

119896120573(119866) where 120573(119866) is the vertex-covering number of119866 Thusif 120574

119903119896(119866) = 119896120573(119866) then 120574

119903119896(119866) = 119896120574(119866)

Theorem 179 (Dehgardi et al [128]) For any graph 119866 if120574119903119896(119866) = 119896120573(119866) then

(a) 119887119903119896(119866) ⩾ 120575(119866)

(b) 119887119903119896(119866) ⩾ 120575(119866)+1 if119866 is vertex 119896-rainbow domination-

critical

As far as we know there are no more results on the 119896-rainbow bondage number of graphs in the literature

28 International Journal of Combinatorics

10 Results on Digraphs

Although domination has been extensively studied in undi-rected graphs it is natural to think of a dominating setas a one-way relationship between vertices of the graphIndeed among the earliest literatures on this subject vonNeumann and Morgenstern [129] used what is now calleddomination in digraphs to find solution (or kernels whichare independent dominating sets) for cooperative 119899-persongames Most likely the first formulation of domination byBerge [130] in 1958 was given in the context of digraphs andonly some years latter by Ore [131] in 1962 for undirectedgraphs Despite this history examination of domination andits variants in digraphs has been essentially overlooked (see[4] for an overview of the domination literature) Thus thereare few if any such results on domination for digraphs in theliterature

The bondage number and its related topics for undirectedgraph have become one of major areas both in theoreticaland applied researches However until recently Carlsonand Develin [42] Shan and Kang [132] and Huang andXu [133 134] studied the bondage number for digraphsindependently In this section we introduce their resultsfor general digraphs Results for some special digraphs suchas vertex-transitive digraphs are introduced in the nextsection

101 Upper Bounds for General Digraphs Let 119866 = (119881 119864)

be a digraph without loops and parallel edges A digraph 119866is called to be asymmetric if whenever (119909 119910) isin 119864(119866) then(119910 119909) notin 119864(119866) and to be symmetric if (119909 119910) isin 119864(119866) implies(119910 119909) isin 119864(119866) For a vertex 119909 of 119881(119866) the sets of out-neighbors and in-neighbors of 119909 are respectively defined as119873

+

119866(119909) = 119910 isin 119881(119866) (119909 119910) isin 119864(119866) and 119873minus

119866(119909) = 119909 isin

119881(119866) (119909 119910) isin 119864(119866) the out-degree and the in-degree of119909 are respectively defined as 119889+

119866(119909) = |119873

+

119866(119909)| and 119889minus

119866(119909) =

|119873+

119866(119909)| Denote themaximum and theminimum out-degree

(resp in-degree) of 119866 by Δ+(119866) and 120575+(119866) (resp Δminus(119866) and120575minus

(119866))The degree 119889119866(119909) of 119909 is defined as 119889+

119866(119909)+119889

minus

119866(119909) and

the maximum and the minimum degrees of119866 are denoted byΔ(119866) and 120575(119866) respectively that is Δ(119866) = max119889

119866(119909) 119909 isin

119881(119866) and 120575(119866) = min119889119866(119909) 119909 isin 119881(119866) Note that the

definitions here are different from the ones in the textbookon digraphs see for example Xu [1]

A subset 119878 of119881(119866) is called a dominating set if119881(119866) = 119878cup119873

+

119866(119878) where119873+

119866(119878) is the set of out-neighbors of 119878Then just

as for undirected graphs 120574(119866) is the minimum cardinalityof a dominating set and the bondage number 119887(119866) is thesmallest cardinality of a set 119861 of edges such that 120574(119866 minus 119861) gt120574(119866) if such a subset 119861 sube 119864(119866) exists Otherwise we put119887(119866) = infin

Some basic results for undirected graphs stated inSection 3 can be generalized to digraphs For exampleTheorem 18 is generalized by Carlson and Develin [42]Huang andXu [133] and Shan andKang [132] independentlyas follows

Theorem 180 Let 119866 be a digraph and (119909 119910) isin 119864(119866) Then119887(119866) ⩽ 119889

119866(119910) + 119889

minus

119866(119909) minus |119873

minus

119866(119909) cap 119873

minus

119866(119910)|

Corollary 181 For a digraph 119866 119887(119866) ⩽ 120575minus(119866) + Δ(119866)

Since 120575minus

(119866) ⩽ (12)Δ(119866) from Corollary 181Conjecture 53 is valid for digraphs

Corollary 182 For a digraph 119866 119887(119866) ⩽ (32)Δ(119866)

In the case of undirected graphs the bondage number of119866119899= 119870

119899times 119870

119899achieves this bound However it was shown

by Carlson and Develin in [42] that if we take the symmetricdigraph119866

119899 we haveΔ(119866

119899) = 4(119899minus1) 120574(119866

119899) = 119899 and 119887(119866

119899) ⩽

4(119899 minus 1) + 1 = Δ(119866119899) + 1 So this family of digraphs cannot

show that the bound in Corollary 182 is tightCorresponding to Conjecture 51 which is discredited

for undirected graphs and is valid for digraphs the sameconjecture for digraphs can be proposed as follows

Conjecture 183 (Carlson and Develin [42] 2006) 119887(119866) ⩽Δ(119866) + 1 for any digraph 119866

If Conjecture 183 is true then the following results showsthat this upper bound is tight since 119887(119870

119899times119870

119899) = Δ(119870

119899times119870

119899)+1

for a complete digraph 119870119899

In 2007 Shan and Kang [132] gave some tight upperbounds on the bondage numbers for some asymmetricdigraphs For example 119887(119879) ⩽ Δ(119879) for any asymmetricdirected tree 119879 119887(119866) ⩽ Δ(119866) for any asymmetric digraph 119866with order at least 4 and 120574(119866) ⩽ 2 For planar digraphs theyobtained the following results

Theorem 184 (Shan and Kang [132] 2007) Let 119866 be aasymmetric planar digraphThen 119887(119866) ⩽ Δ(119866)+2 and 119887(119866) ⩽Δ(119866) + 1 if Δ(119866) ⩾ 5 and 119889minus

119866(119909) ⩾ 3 for every vertex 119909 with

119889119866(119909) ⩾ 4

102 Results for Some Special Digraphs The exact values andbounds on 119887(119866) for some standard digraphs are determined

Theorem185 (Huang andXu [133] 2006) For a directed cycle119862119899and a directed path 119875

119899

119887 (119862119899) =

3 119894119891 119899 119894119904 119900119889119889

2 119894119891 119899 119894119904 119890V119890119899

119887 (119875119899) =

2 119894119891 119899 119894119904 119900119889119889

1 119894119891 119899 119894119904 119890V119890119899

(76)

For the de Bruijn digraph 119861(119889 119899) and the Kautz digraph119870(119889 119899)

119887 (119861 (119889 119899)) = 119889 119894119891 119899 119894119904 119900119889119889

119889 ⩽ 119887 (119861 (119889 119899)) ⩽ 2119889 119894119891 119899 119894119904 119890V119890119899

119887 (119870 (119889 119899)) = 119889 + 1

(77)

Like undirected graphs we can define the total domi-nation number and the total bondage number On the totalbondage numbers for some special digraphs the knownresults are as follows

International Journal of Combinatorics 29

Theorem 186 (Huang and Xu [134] 2007) For a directedcycle 119862

119899and a directed path 119875

119899 119887

119879(119875

119899) and 119887

119879(119862

119899) all do not

exist For a complete digraph 119870119899

119887119879(119870

119899) = infin 119894119891 119899 = 1 2

119887119879(119870

119899) = 3 119894119891 119899 = 3

119899 ⩽ 119887119879(119870

119899) ⩽ 2119899 minus 3 119894119891 119899 ⩾ 4

(78)

The extended de Bruijn digraph EB(119889 119899 1199021 119902

119901) and

the extended Kautz digraph EK(119889 119899 1199021 119902

119901) are intro-

duced by Shibata and Gonda [135] If 119901 = 1 then theyare the de Bruijn digraph B(119889 119899) and the Kautz digraphK(119889 119899) respectively Huang and Xu [134] determined theirtotal domination numbers In particular their total bondagenumbers for general cases are determined as follows

Theorem 187 (Huang andXu [134] 2007) If 119889 ⩾ 2 and 119902119894⩾ 2

for each 119894 = 1 2 119901 then

119887119879(EB(119889 119899 119902

1 119902

119901)) = 119889

119901

minus 1

119887119879(EK(119889 119899 119902

1 119902

119901)) = 119889

119901

(79)

In particular for the de Bruijn digraph B(119889 119899) and the Kautzdigraph K(119889 119899)

119887119879(B (119889 119899)) = 119889 minus 1 119887

119879(K (119889 119899)) = 119889 (80)

Zhang et al [136] determined the bondage number incomplete 119905-partite digraphs

Theorem 188 (Zhang et al [136] 2009) For a complete 119905-partite digraph 119870

11989911198992119899119905

where 1198991⩽ 119899

2⩽ sdot sdot sdot ⩽ 119899

119905

119887 (11987011989911198992119899119905

)

=

119898 119894119891 119899119898= 1 119899

119898+1⩾ 2 119891119900119903 119904119900119898119890

119898 (1 ⩽ 119898 lt 119905)

4119905 minus 3 119894119891 1198991= 119899

2= sdot sdot sdot = 119899

119905= 2

119905minus1

sum

119894=1

119899119894+ 2 (119905 minus 1) 119900119905ℎ119890119903119908119894119904119890

(81)

Since an undirected graph can be thought of as a sym-metric digraph any result for digraphs has an analogy forundirected graphs in general In view of this point studyingthe bondage number for digraphs is more significant thanfor undirected graphs Thus we should further study thebondage number of digraphs and try to generalize knownresults on the bondage number and related variants for undi-rected graphs to digraphs prove or disprove Conjecture 183In particular determine the exact values of 119887(B(119889 119899)) for aneven 119899 and 119887

119879(119870

119899) for 119899 ⩾ 4

11 Efficient Dominating Sets

A dominating set 119878 of a graph 119866 is called to be efficient if forevery vertex 119909 in119866 |119873

119866[119909]cap119878| = 1 if119866 is a undirected graph

or |119873minus

119866[119909] cap 119878| = 1 if 119866 is a directed graph The concept of an

efficient set was proposed by Bange et al [137] in 1988 In theliterature an efficient set is also called a perfect dominatingset (see Livingston and Stout [138] for an explanation)

From definition if 119878 is an efficient dominating set of agraph 119866 then 119878 is certainly an independent set and everyvertex not in 119878 is adjacent to exactly one vertex in 119878

It is also clear from definition that a dominating set 119878 isefficient if and only if N

119866[119878] = 119873

119866[119909] 119909 isin 119878 for the

undirected graph 119866 or N+

119866[119878] = 119873

+

119866[119909] 119909 isin 119878 for the

digraph119866 is a partition of119881(119866) where the induced subgraphby119873

119866[119909] or119873+

119866[119909] is a star or an out-star with the root 119909

The efficient domination has important applications inmany areas such as error-correcting codes and receivesmuch attention in the late years

The concept of efficient dominating sets is a measureof the efficiency of domination in graphs Unfortunately asshown in [137] not every graph has an efficient dominatingset andmoreover it is anNP-complete problem to determinewhether a given graph has an efficient dominating set Inaddition it was shown by Clark [139] in 1993 that for a widerange of 119901 almost every random undirected graph 119866 isin

G(120592 119901) has no efficient dominating sets This means thatundirected graphs possessing an efficient dominating set arerare However it is easy to show that every undirected graphhas an orientationwith an efficient dominating set (see Bangeet al [140])

In 1993 Barkauskas and Host [141] showed that deter-mining whether an arbitrary oriented graph has an efficientdominating set is NP-complete Even so the existence ofefficient dominating sets for some special classes of graphs hasbeen examined see for example Dejter and Serra [142] andLee [143] for Cayley graph Gu et al [144] formeshes and toriObradovic et al [145] Huang and Xu [36] Reji Kumar andMacGillivray [146] for circulant graphs Harary graphs andtori and Van Wieren et al [147] for cube-connected cycles

In this section we introduce results of the bondagenumber for some graphs with efficient dominating sets dueto Huang and Xu [36]

111 Results for General Graphs In this subsection we intro-duce some results on bondage numbers obtained by applyingefficient dominating sets We first state the two followinglemmas

Lemma 189 For any 119896-regular graph or digraph 119866 of order 119899120574(119866) ⩾ 119899(119896 + 1) with equality if and only if 119866 has an efficientdominating set In addition if 119866 has an efficient dominatingset then every efficient dominating set is certainly a 120574-set andvice versa

Let 119890 be an edge and 119878 a dominating set in 119866 We say that119890 supports 119878 if 119890 isin (119878 119878) where (119878 119878) = (119909 119910) isin 119864(119866) 119909 isin 119878 119910 isin 119878 Denote by 119904(119866) the minimum number of edgeswhich support all 120574-sets in 119866

Lemma 190 For any graph or digraph 119866 119887(119866) ⩾ 119904(119866) withequality if 119866 is regular and has an efficient dominating set

30 International Journal of Combinatorics

A graph 119866 is called to be vertex-transitive if its automor-phism group Aut(119866) acts transitively on its vertex-set 119881(119866)A vertex-transitive graph is regular Applying Lemmas 189and 190 Huang and Xu obtained some results on bondagenumbers for vertex-transitive graphs or digraphs

Theorem 191 (Huang and Xu [36] 2008) For any vertex-transitive graph or digraph 119866 of order 119899

119887 (119866) ⩾

lceil119899

2120574 (119866)rceil 119894119891 119866 119894119904 119906119899119889119894119903119890119888119905119890119889

lceil119899

120574 (119866)rceil 119894119891 119866 119894119904 119889119894119903119890119888119905119890119889

(82)

Theorem192 (Huang andXu [36] 2008) Let119866 be a 119896-regulargraph of order 119899 Then

119887(119866) ⩽ 119896 if119866 is undirected and 119899 ⩾ 120574(119866)(119896+1)minus119896+1119887(119866) ⩽ 119896+1+ℓ if119866 is directed and 119899 ⩾ 120574(119866)(119896+1)minusℓwith 0 ⩽ ℓ ⩽ 119896 minus 1

Next we introduce a better upper bound on 119887(119866) To thisaim we need the following concept which is a generalizationof the concept of the edge-covering of a graph 119866 For 1198811015840

sube

119881(119866) and 1198641015840 sube 119864(119866) we say that 1198641015840covers 1198811015840 and call 1198641015840an edge-covering for 1198811015840 if there exists an edge (119909 119910) isin 1198641015840 forany vertex 119909 isin 1198811015840 For 119910 isin 119881(119866) let 1205731015840[119910] be the minimumcardinality over all edge-coverings for119873minus

119866[119910]

Theorem 193 (Huang and Xu [36] 2008) If 119866 is a 119896-regulargraph with order 120574(119866)(119896 + 1) then 119887(119866) ⩽ 1205731015840[119910] for any 119910 isin119881(119866)

Theupper bound on 119887(119866) given inTheorem 193 is tight inview of 119887(119862

119899) for a cycle or a directed cycle 119862

119899(seeTheorems

1 and 185 resp)It is easy to see that for a 119896-regular graph 119866 lceil(119896 + 1)2rceil ⩽

1205731015840

[119910] ⩽ 119896 when 119866 is undirected and 1205731015840[119910] = 119896 + 1 when 119866 isdirected By this fact and Lemma 189 the following theoremis merely a simple combination of Theorems 191 and 193

Theorem 194 (Huang and Xu [36] 2008) Let 119866 be a vertex-transitive graph of degree 119896 If 119866 has an efficient dominatingset then

lceil119896 + 1

2rceil ⩽ 119887 (119866) ⩽ 119896 119894119891 119866 119894119904 119906119899119889119894119903119890119888119905119890119889

119887 (119866) = 119896 + 1 119894119891 119866 119894119904 119889119894119903119890119888119905119890119889

(83)

Theorem 195 (Huang and Xu [36] 2008) If 119866 is an undi-rected vertex-transitive cubic graph with order 4120574(119866) and girth119892(119866) ⩽ 5 then 119887(119866) = 2

The proof of Theorem 195 leads to a byproduct If 119892 =5 let (119906

1 119906

2 119906

3 119906

4 119906

5) be a cycle and let D

119894be the family

of efficient dominating sets containing 119906119894for each 119894 isin

1 2 3 4 5 ThenD119894capD

119895= 0 for 119894 = 1 2 5 Then 119866 has

at least 5119904 efficient dominating sets where 119904 = 119904(119866) But thereare only 4s (=ns120574) distinct efficient dominating sets in119866This

contradiction implies that an undirected vertex-transitivecubic graph with girth five has no efficient dominating setsBut a similar argument for 119892(119866) = 3 4 or 119892(119866) ⩾ 6 couldnot give any contradiction This is consistent with the resultthat CCC(119899) a vertex-transitive cubic graph with girth 119899 if3 ⩽ 119899 ⩽ 8 or girth 8 if 119899 ⩾ 9 has efficient dominating setsfor all 119899 ⩾ 3 except 119899 = 5 (see Theorem 199 in the followingsubsection)

112 Results for Cayley Graphs In this subsection we useTheorem 194 to determine the exact values or approximativevalues of bondage numbers for some special vertex-transitivegraphs by characterizing the existence of efficient dominatingsets in these graphs

Let Γ be a nontrivial finite group 119878 a nonempty subset ofΓ without the identity element of Γ A digraph 119866 is defined asfollows

119881 (119866) = Γ (119909 119910) isin 119864 (119866) lArrrArr 119909minus1

119910 isin 119878

for any 119909 119910 isin Γ(84)

is called a Cayley digraph of the group Γ with respect to119878 denoted by 119862

Γ(119878) If 119878minus1 = 119904minus1 119904 isin 119878 = 119878 then

119862Γ(119878) is symmetric and is called a Cayley undirected graph

a Cayley graph for short Cayley graphs or digraphs arecertainly vertex-transitive (see Xu [1])

A circulant graph 119866(119899 119878) of order 119899 is a Cayley graph119862119885119899

(119878) where 119885119899= 0 119899 minus 1 is the addition group of

order 119899 and 119878 is a nonempty subset of119885119899without the identity

element and hence is a vertex-transitive digraph of degree|119878| If 119878minus1 = 119878 then119866(119899 119878) is an undirected graph If 119878 = 1 119896where 2 ⩽ 119896 ⩽ 119899 minus 2 we write 119866(119899 1 119896) for 119866(119899 1 119896) or119866(119899 plusmn1 plusmn119896) and call it a double-loop circulant graph

Huang and Xu [36] showed that for directed 119866 =

119866(119899 1 119896) lceil1198993rceil ⩽ 120574(119866) ⩽ lceil1198992rceil and 119866 has an efficientdominating set if and only if 3 | 119899 and 119896 equiv 2 (mod 3) fordirected 119866 = 119866(119899 1 119896) and 119896 = 1198992 lceil1198995rceil ⩽ 120574(119866) ⩽ lceil1198993rceiland 119866 has an efficient dominating set if and only if 5 | 119899 and119896 equiv plusmn2 (mod 5) By Theorem 194 we obtain the bondagenumber of a double loop circulant graph if it has an efficientdominating set

Theorem 196 (Huang and Xu [36] 2008) Let 119866 be a double-loop circulant graph 119866(119899 1 119896) If 119866 is directed with 3 | 119899and 119896 equiv 2 (mod 3) or 119866 is undirected with 5 | 119899 and119896 equiv plusmn2 (mod 5) then 119887(119866) = 3

The 119898 times 119899 torus is the Cartesian product 119862119898times 119862

119899of

two cycles and is a Cayley graph 119862119885119898times119885119899

(119878) where 119878 =

(0 1) (1 0) for directed cycles and 119878 = (0 plusmn1) (plusmn1 0) forundirected cycles and hence is vertex-transitive Gu et al[144] showed that the undirected torus119862

119898times119862

119899has an efficient

dominating set if and only if both 119898 and 119899 are multiplesof 5 Huang and Xu [36] showed that the directed torus119862119898times 119862

119899has an efficient dominating set if and only if both

119898 and 119899 are multiples of 3 Moreover they found a necessarycondition for a dominating set containing the vertex (0 0)in 119862

119898times 119862

119899to be efficient and obtained the following

result

International Journal of Combinatorics 31

Theorem 197 (Huang and Xu [36] 2008) Let 119866 = 119862119898times 119862

119899

If 119866 is undirected and both119898 and 119899 are multiples of 5 or if 119866 isdirected and both119898 and 119899 are multiples of 3 then 119887(119866) = 3

The hypercube 119876119899

is the Cayley graph 119862Γ(119878)

where Γ = 1198852times sdot sdot sdot times 119885

2= (119885

2)119899 and 119878 =

100 sdot sdot sdot 0 010 sdot sdot sdot 0 00 sdot sdot sdot 01 Lee [143] showed that119876119899has an efficient dominating set if and only if 119899 = 2119898 minus 1

for a positive integer119898 ByTheorem 194 the following resultis obtained immediately

Theorem 198 (Huang and Xu [36] 2008) If 119899 = 2119898 minus 1 for apositive integer119898 then 2119898minus1

⩽ 119887(119876119899) ⩽ 2

119898

minus 1

Problem 10 Determine the exact value of 119887(119876119899) for 119899 = 2119898minus1

The 119899-dimensional cube-connected cycle denoted byCCC(119899) is constructed from the 119899-dimensional hypercube119876119899by replacing each vertex 119909 in 119876

119899with an undirected cycle

119862119899of length 119899 and linking the 119894th vertex of the 119862

119899to the 119894th

neighbor of 119909 It has been proved that CCC(119899) is a Cayleygraph and hence is a vertex-transitive graph with degree 3Van Wieren et al [147] proved that CCC(119899) has an efficientdominating set if and only if 119899 = 5 From Theorems 194 and195 the following result can be derived

Theorem 199 (Huang and Xu [36] 2008) Let 119866 = 119862119862119862(119899)be the 119899-dimensional cube-connected cycles with 119899 ⩾ 3 and119899 = 5 Then 120574(119866) = 1198992119899minus2 and 2 ⩽ 119887(119866) ⩽ 3 In addition119887(119862119862119862(3)) = 119887(119862119862119862(4)) = 2

Problem 11 Determine the exact value of 119887(CCC(119899)) for 119899 ⩾5

Acknowledgments

This work was supported by NNSF of China (Grants nos11071233 61272008) The author would like to express hisgratitude to the anonymous referees for their kind sugges-tions and comments on the original paper to ProfessorSheikholeslami and Professor Jafari Rad for sending thereferences [128] and [81] which resulted in this version

References

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[2] S THedetniemi andR C Laskar ldquoBibliography on dominationin graphs and some basic definitions of domination parame-tersrdquo Discrete Mathematics vol 86 no 1ndash3 pp 257ndash277 1990

[3] TW Haynes S T Hedetniemi and P J Slater Fundamentals ofDomination in Graphs vol 208 ofMonographs and Textbooks inPure and Applied Mathematics Marcel Dekker New York NYUSA 1998

[4] TWHaynes S THedetniemi and P J Slater EdsDominationin Graphs vol 209 of Monographs and Textbooks in Pure andAppliedMathematicsMarcelDekker NewYorkNYUSA 1998

[5] M R Garey andD S JohnsonComputers and Intractability WH Freeman and Company San Francisco Calif USA 1979

[6] H B Walikar and B D Acharya ldquoDomination critical graphsrdquoNational Academy Science Letters vol 2 pp 70ndash72 1979

[7] D Bauer F Harary J Nieminen and C L Suffel ldquoDominationalteration sets in graphsrdquo Discrete Mathematics vol 47 no 2-3pp 153ndash161 1983

[8] J F Fink M S Jacobson L F Kinch and J Roberts ldquoThebondage number of a graphrdquo Discrete Mathematics vol 86 no1ndash3 pp 47ndash57 1990

[9] F-T Hu and J-M Xu ldquoThe bondage number of (119899 minus 3)-regulargraph G of order nrdquo Ars Combinatoria to appear

[10] U Teschner ldquoNew results about the bondage number of agraphrdquoDiscreteMathematics vol 171 no 1ndash3 pp 249ndash259 1997

[11] B L Hartnell and D F Rall ldquoA bound on the size of a graphwith given order and bondage numberrdquo Discrete Mathematicsvol 197-198 pp 409ndash413 1999 16th British CombinatorialConference (London 1997)

[12] B L Hartnell and D F Rall ldquoA characterization of trees inwhich no edge is essential to the domination numberrdquo ArsCombinatoria vol 33 pp 65ndash76 1992

[13] J Topp and P D Vestergaard ldquo120572119896- and 120574

119896-stable graphsrdquo

Discrete Mathematics vol 212 no 1-2 pp 149ndash160 2000[14] A Meir and J W Moon ldquoRelations between packing and

covering numbers of a treerdquo Pacific Journal of Mathematics vol61 no 1 pp 225ndash233 1975

[15] B L Hartnell L K Joslashrgensen P D Vestergaard and CA Whitehead ldquoEdge stability of the k-domination numberof treesrdquo Bulletin of the Institute of Combinatorics and ItsApplications vol 22 pp 31ndash40 1998

[16] F-T Hu and J-M Xu ldquoOn the complexity of the bondage andreinforcement problemsrdquo Journal of Complexity vol 28 no 2pp 192ndash201 2012

[17] B L Hartnell and D F Rall ldquoBounds on the bondage numberof a graphrdquo Discrete Mathematics vol 128 no 1ndash3 pp 173ndash1771994

[18] Y-L Wang ldquoOn the bondage number of a graphrdquo DiscreteMathematics vol 159 no 1ndash3 pp 291ndash294 1996

[19] G H Fricke TW Haynes S M Hedetniemi S T Hedetniemiand R C Laskar ldquoExcellent treesrdquo Bulletin of the Institute ofCombinatorics and Its Applications vol 34 pp 27ndash38 2002

[20] V Samodivkin ldquoThe bondage number of graphs good and badverticesrdquoDiscussiones Mathematicae GraphTheory vol 28 no3 pp 453ndash462 2008

[21] J E Dunbar T W Haynes U Teschner and L VolkmannldquoBondage insensitivity and reinforcementrdquo in Domination inGraphs vol 209 of Monographs and Textbooks in Pure andAppliedMathematics pp 471ndash489 Dekker NewYork NY USA1998

[22] H Liu and L Sun ldquoThe bondage and connectivity of a graphrdquoDiscrete Mathematics vol 263 no 1ndash3 pp 289ndash293 2003

[23] A-H Esfahanian and S L Hakimi ldquoOn computing a con-ditional edge-connectivity of a graphrdquo Information ProcessingLetters vol 27 no 4 pp 195ndash199 1988

[24] R C Brigham P Z Chinn and R D Dutton ldquoVertexdomination-critical graphsrdquoNetworks vol 18 no 3 pp 173ndash1791988

[25] V Samodivkin ldquoDomination with respect to nondegenerateproperties bondage numberrdquoThe Australasian Journal of Com-binatorics vol 45 pp 217ndash226 2009

[26] U Teschner ldquoA new upper bound for the bondage numberof graphs with small domination numberrdquo The AustralasianJournal of Combinatorics vol 12 pp 27ndash35 1995

32 International Journal of Combinatorics

[27] J Fulman D Hanson and G MacGillivray ldquoVertex dom-ination-critical graphsrdquoNetworks vol 25 no 2 pp 41ndash43 1995

[28] U Teschner ldquoA counterexample to a conjecture on the bondagenumber of a graphrdquo Discrete Mathematics vol 122 no 1ndash3 pp393ndash395 1993

[29] U Teschner ldquoThe bondage number of a graph G can be muchgreater than Δ(G)rdquo Ars Combinatoria vol 43 pp 81ndash87 1996

[30] L Volkmann ldquoOn graphs with equal domination and coveringnumbersrdquoDiscrete AppliedMathematics vol 51 no 1-2 pp 211ndash217 1994

[31] L A Sanchis ldquoMaximum number of edges in connected graphswith a given domination numberrdquoDiscreteMathematics vol 87no 1 pp 65ndash72 1991

[32] S Klavzar and N Seifter ldquoDominating Cartesian products ofcyclesrdquoDiscrete AppliedMathematics vol 59 no 2 pp 129ndash1361995

[33] H K Kim ldquoA note on Vizingrsquos conjecture about the bondagenumberrdquo Korean Journal of Mathematics vol 10 no 2 pp 39ndash42 2003

[34] M Y Sohn X Yuan and H S Jeong ldquoThe bondage number of1198623times119862

119899rdquo Journal of the KoreanMathematical Society vol 44 no

6 pp 1213ndash1231 2007[35] L Kang M Y Sohn and H K Kim ldquoBondage number of the

discrete torus 119862119899times 119862

4rdquo Discrete Mathematics vol 303 no 1ndash3

pp 80ndash86 2005[36] J Huang and J-M Xu ldquoThe bondage numbers and efficient

dominations of vertex-transitive graphsrdquoDiscrete Mathematicsvol 308 no 4 pp 571ndash582 2008

[37] C J Xiang X Yuan andM Y Sohn ldquoDomination and bondagenumber of119862

3times119862

119899rdquoArs Combinatoria vol 97 pp 299ndash310 2010

[38] M S Jacobson and L F Kinch ldquoOn the domination numberof products of graphs Irdquo Ars Combinatoria vol 18 pp 33ndash441984

[39] Y-Q Li ldquoA note on the bondage number of a graphrdquo ChineseQuarterly Journal of Mathematics vol 9 no 4 pp 1ndash3 1994

[40] F-T Hu and J-M Xu ldquoBondage number of mesh networksrdquoFrontiers ofMathematics inChina vol 7 no 5 pp 813ndash826 2012

[41] R Frucht and F Harary ldquoOn the corona of two graphsrdquoAequationes Mathematicae vol 4 pp 322ndash325 1970

[42] K Carlson and M Develin ldquoOn the bondage number of planarand directed graphsrdquoDiscreteMathematics vol 306 no 8-9 pp820ndash826 2006

[43] U Teschner ldquoOn the bondage number of block graphsrdquo ArsCombinatoria vol 46 pp 25ndash32 1997

[44] J Huang and J-M Xu ldquoNote on conjectures of bondagenumbers of planar graphsrdquo Applied Mathematical Sciences vol6 no 65ndash68 pp 3277ndash3287 2012

[45] L Kang and J Yuan ldquoBondage number of planar graphsrdquoDiscrete Mathematics vol 222 no 1ndash3 pp 191ndash198 2000

[46] M Fischermann D Rautenbach and L Volkmann ldquoRemarkson the bondage number of planar graphsrdquo Discrete Mathemat-ics vol 260 no 1ndash3 pp 57ndash67 2003

[47] J Hou G Liu and J Wu ldquoBondage number of planar graphswithout small cyclesrdquoUtilitasMathematica vol 84 pp 189ndash1992011

[48] J Huang and J-M Xu ldquoThe bondage numbers of graphs withsmall crossing numbersrdquo Discrete Mathematics vol 307 no 15pp 1881ndash1897 2007

[49] Q Ma S Zhang and J Wang ldquoBondage number of 1-planargraphrdquo Applied Mathematics vol 1 no 2 pp 101ndash103 2010

[50] J Hou and G Liu ldquoA bound on the bondage number of toroidalgraphsrdquoDiscreteMathematics Algorithms and Applications vol4 no 3 Article ID 1250046 5 pages 2012

[51] Y-C Cao J-M Xu and X-R Xu ldquoOn bondage number oftoroidal graphsrdquo Journal of University of Science and Technologyof China vol 39 no 3 pp 225ndash228 2009

[52] Y-C Cao J Huang and J-M Xu ldquoThe bondage number ofgraphs with crossing number less than fourrdquo Ars Combinatoriato appear

[53] H K Kim ldquoOn the bondage numbers of some graphsrdquo KoreanJournal of Mathematics vol 11 no 2 pp 11ndash15 2004

[54] A Gagarin and V Zverovich ldquoUpper bounds for the bondagenumber of graphs on topological surfacesrdquo Discrete Mathemat-ics 2012

[55] J Huang ldquoAn improved upper bound for the bondage numberof graphs on surfacesrdquoDiscrete Mathematics vol 312 no 18 pp2776ndash2781 2012

[56] A Gagarin and V Zverovich ldquoThe bondage number of graphson topological surfaces and Teschnerrsquos conjecturerdquo DiscreteMathematics vol 313 no 6 pp 796ndash808 2013

[57] V Samodivkin ldquoNote on the bondage number of graphs ontopological surfacesrdquo httparxivorgabs12086203

[58] E J Cockayne R M Dawes and S T Hedetniemi ldquoTotaldomination in graphsrdquoNetworks vol 10 no 3 pp 211ndash219 1980

[59] R Laskar J Pfaff S M Hedetniemi and S T HedetniemildquoOn the algorithmic complexity of total dominationrdquo Society forIndustrial and Applied Mathematics vol 5 no 3 pp 420ndash4251984

[60] J Pfaff R C Laskar and S T Hedetniemi ldquoNP-completeness oftotal and connected domination and irredundance for bipartitegraphsrdquo Tech Rep 428 Department of Mathematical SciencesClemson University 1983

[61] M A Henning ldquoA survey of selected recent results on totaldomination in graphsrdquoDiscrete Mathematics vol 309 no 1 pp32ndash63 2009

[62] V R Kulli and D K Patwari ldquoThe total bondage number ofa graphrdquo in Advances in Graph Theory pp 227ndash235 VishwaGulbarga India 1991

[63] F-T Hu Y Lu and J-M Xu ldquoThe total bondage number ofgrid graphsrdquo Discrete Applied Mathematics vol 160 no 16-17pp 2408ndash2418 2012

[64] J Cao M Shi and B Wu ldquoResearch on the total bondagenumber of a spectial networkrdquo in Proceedings of the 3rdInternational Joint Conference on Computational Sciences andOptimization (CSO rsquo10) pp 456ndash458 May 2010

[65] N Sridharan MD Elias and V S A Subramanian ldquoTotalbondage number of a graphrdquo AKCE International Journal ofGraphs and Combinatorics vol 4 no 2 pp 203ndash209 2007

[66] N Jafari Rad and J Raczek ldquoSome progress on total bondage ingraphsrdquo Graphs and Combinatorics 2011

[67] T W Haynes and P J Slater ldquoPaired-domination and thepaired-domatic numberrdquoCongressusNumerantium vol 109 pp65ndash72 1995

[68] T W Haynes and P J Slater ldquoPaired-domination in graphsrdquoNetworks vol 32 no 3 pp 199ndash206 1998

[69] X Hou ldquoA characterization of 2120574 120574119875-treesrdquoDiscrete Mathemat-

ics vol 308 no 15 pp 3420ndash3426 2008[70] K E Proffitt T W Haynes and P J Slater ldquoPaired-domination

in grid graphsrdquo Congressus Numerantium vol 150 pp 161ndash1722001

International Journal of Combinatorics 33

[71] H Qiao L KangM Cardei andD-Z Du ldquoPaired-dominationof treesrdquo Journal of Global Optimization vol 25 no 1 pp 43ndash542003

[72] E Shan L Kang and M A Henning ldquoA characterizationof trees with equal total domination and paired-dominationnumbersrdquo The Australasian Journal of Combinatorics vol 30pp 31ndash39 2004

[73] J Raczek ldquoPaired bondage in treesrdquo Discrete Mathematics vol308 no 23 pp 5570ndash5575 2008

[74] J A Telle and A Proskurowski ldquoAlgorithms for vertex parti-tioning problems on partial k-treesrdquo SIAM Journal on DiscreteMathematics vol 10 no 4 pp 529ndash550 1997

[75] G S Domke J H Hattingh M A Henning and L R MarkusldquoRestrained domination in treesrdquoDiscreteMathematics vol 211no 1ndash3 pp 1ndash9 2000

[76] G S Domke J H Hattingh M A Henning and L R MarkusldquoRestrained domination in graphs with minimum degree twordquoJournal of Combinatorial Mathematics and Combinatorial Com-puting vol 35 pp 239ndash254 2000

[77] G S Domke J H Hattingh S T Hedetniemi R C Laskarand L R Markus ldquoRestrained domination in graphsrdquo DiscreteMathematics vol 203 no 1ndash3 pp 61ndash69 1999

[78] J H Hattingh and E J Joubert ldquoAn upper bound for therestrained domination number of a graph with minimumdegree at least two in terms of order and minimum degreerdquoDiscrete Applied Mathematics vol 157 no 13 pp 2846ndash28582009

[79] M A Henning ldquoGraphs with large restrained dominationnumberrdquo Discrete Mathematics vol 197-198 pp 415ndash429 199916th British Combinatorial Conference (London 1997)

[80] J H Hattingh and A R Plummer ldquoRestrained bondage ingraphsrdquo Discrete Mathematics vol 308 no 23 pp 5446ndash54532008

[81] N Jafari Rad R Hasni J Raczek and L Volkmann ldquoTotalrestrained bondage in graphsrdquoActaMathematica Sinica vol 29no 6 pp 1033ndash1042 2013

[82] J Cyman and J Raczek ldquoOn the total restrained dominationnumber of a graphrdquoThe Australasian Journal of Combinatoricsvol 36 pp 91ndash100 2006

[83] M A Henning ldquoGraphs with large total domination numberrdquoJournal of Graph Theory vol 35 no 1 pp 21ndash45 2000

[84] D-X Ma X-G Chen and L Sun ldquoOn total restraineddomination in graphsrdquoCzechoslovakMathematical Journal vol55 no 1 pp 165ndash173 2005

[85] B Zelinka ldquoRemarks on restrained domination and totalrestrained domination in graphsrdquo Czechoslovak MathematicalJournal vol 55 no 2 pp 393ndash396 2005

[86] W Goddard and M A Henning ldquoIndependent domination ingraphs a survey and recent resultsrdquo Discrete Mathematics vol313 no 7 pp 839ndash854 2013

[87] J H Zhang H L Liu and L Sun ldquoIndependence bondagenumber and reinforcement number of some graphsrdquo Transac-tions of Beijing Institute of Technology vol 23 no 2 pp 140ndash1422003

[88] K D Dharmalingam Studies in Graph Theorymdashequitabledomination and bottleneck domination [PhD thesis] MaduraiKamaraj University Madurai India 2006

[89] GDeepakND Soner andAAlwardi ldquoThe equitable bondagenumber of a graphrdquo Research Journal of Pure Algebra vol 1 no9 pp 209ndash212 2011

[90] E Sampathkumar and H B Walikar ldquoThe connected domina-tion number of a graphrdquo Journal of Mathematical and PhysicalSciences vol 13 no 6 pp 607ndash613 1979

[91] K White M Farber and W Pulleyblank ldquoSteiner trees con-nected domination and strongly chordal graphsrdquoNetworks vol15 no 1 pp 109ndash124 1985

[92] M Cadei M X Cheng X Cheng and D-Z Du ldquoConnecteddomination in Ad Hoc wireless networksrdquo in Proceedings ofthe 6th International Conference on Computer Science andInformatics (CSampI rsquo02) 2002

[93] J F Fink and M S Jacobson ldquon-domination in graphsrdquo inGraph Theory with Applications to Algorithms and ComputerScience Wiley-Interscience Publications pp 283ndash300 WileyNew York NY USA 1985

[94] M Chellali O Favaron A Hansberg and L Volkmann ldquok-domination and k-independence in graphs a surveyrdquo Graphsand Combinatorics vol 28 no 1 pp 1ndash55 2012

[95] Y Lu X Hou J-M Xu andN Li ldquoTrees with uniqueminimump-dominating setsrdquo Utilitas Mathematica vol 86 pp 193ndash2052011

[96] Y Lu and J-M Xu ldquoThe p-bondage number of treesrdquo Graphsand Combinatorics vol 27 no 1 pp 129ndash141 2011

[97] Y Lu and J-M Xu ldquoThe 2-domination and 2-bondage numbersof grid graphs A manuscriptrdquo httparxivorgabs12044514

[98] M Krzywkowski ldquoNon-isolating 2-bondage in graphsrdquo TheJournal of the Mathematical Society of Japan vol 64 no 4 2012

[99] M A Henning O R Oellermann and H C Swart ldquoBoundson distance domination parametersrdquo Journal of CombinatoricsInformation amp System Sciences vol 16 no 1 pp 11ndash18 1991

[100] F Tian and J-M Xu ldquoBounds for distance domination numbersof graphsrdquo Journal of University of Science and Technology ofChina vol 34 no 5 pp 529ndash534 2004

[101] F Tian and J-M Xu ldquoDistance domination-critical graphsrdquoApplied Mathematics Letters vol 21 no 4 pp 416ndash420 2008

[102] F Tian and J-M Xu ldquoA note on distance domination numbersof graphsrdquo The Australasian Journal of Combinatorics vol 43pp 181ndash190 2009

[103] F Tian and J-M Xu ldquoAverage distances and distance domina-tion numbersrdquo Discrete Applied Mathematics vol 157 no 5 pp1113ndash1127 2009

[104] Z-L Liu F Tian and J-M Xu ldquoProbabilistic analysis of upperbounds for 2-connected distance k-dominating sets in graphsrdquoTheoretical Computer Science vol 410 no 38ndash40 pp 3804ndash3813 2009

[105] M A Henning O R Oellermann and H C Swart ldquoDistancedomination critical graphsrdquo Journal of Combinatorial Mathe-matics and Combinatorial Computing vol 44 pp 33ndash45 2003

[106] T J Bean M A Henning and H C Swart ldquoOn the integrityof distance domination in graphsrdquo The Australasian Journal ofCombinatorics vol 10 pp 29ndash43 1994

[107] M Fischermann and L Volkmann ldquoA remark on a conjecturefor the (kp)-domination numberrdquoUtilitasMathematica vol 67pp 223ndash227 2005

[108] T Korneffel D Meierling and L Volkmann ldquoA remark on the(22)-domination numberrdquo Discussiones Mathematicae GraphTheory vol 28 no 2 pp 361ndash366 2008

[109] Y Lu X Hou and J-M Xu ldquoOn the (22)-domination numberof treesrdquo Discussiones Mathematicae Graph Theory vol 30 no2 pp 185ndash199 2010

34 International Journal of Combinatorics

[110] H Li and J-M Xu ldquo(dm)-domination numbers of m-connected graphsrdquo Rapports de Recherche 1130 LRI UniversiteParis-Sud 1997

[111] S M Hedetniemi S T Hedetniemi and T V Wimer ldquoLineartime resource allocation algorithms for treesrdquo Tech Rep URI-014 Department of Mathematical Sciences Clemson Univer-sity 1987

[112] M Farber ldquoDomination independent domination and dualityin strongly chordal graphsrdquo Discrete Applied Mathematics vol7 no 2 pp 115ndash130 1984

[113] G S Domke and R C Laskar ldquoThe bondage and reinforcementnumbers of 120574

119891for some graphsrdquoDiscrete Mathematics vol 167-

168 pp 249ndash259 1997[114] G S Domke S T Hedetniemi and R C Laskar ldquoFractional

packings coverings and irredundance in graphsrdquo vol 66 pp227ndash238 1988

[115] E J Cockayne P A Dreyer Jr S M Hedetniemi andS T Hedetniemi ldquoRoman domination in graphsrdquo DiscreteMathematics vol 278 no 1ndash3 pp 11ndash22 2004

[116] I Stewart ldquoDefend the roman empirerdquo Scientific American vol281 pp 136ndash139 1999

[117] C S ReVelle and K E Rosing ldquoDefendens imperiumromanum a classical problem in military strategyrdquoThe Ameri-can Mathematical Monthly vol 107 no 7 pp 585ndash594 2000

[118] A Bahremandpour F-T Hu S M Sheikholeslami and J-MXu ldquoOn the Roman bondage number of a graphrdquo DiscreteMathematics Algorithms and Applications vol 5 no 1 ArticleID 1350001 15 pages 2013

[119] N Jafari Rad and L Volkmann ldquoRoman bondage in graphsrdquoDiscussiones Mathematicae Graph Theory vol 31 no 4 pp763ndash773 2011

[120] K Ebadi and L PushpaLatha ldquoRoman bondage and Romanreinforcement numbers of a graphrdquo International Journal ofContemporary Mathematical Sciences vol 5 pp 1487ndash14972010

[121] N Dehgardi S M Sheikholeslami and L Volkman ldquoOn theRoman k-bondage number of a graphrdquo AKCE InternationalJournal of Graphs and Combinatorics vol 8 pp 169ndash180 2011

[122] S Akbari and S Qajar ldquoA note on Roman bondage number ofgraphsrdquo Ars Combinatoria to Appear

[123] F-T Hu and J-M Xu ldquoRoman bondage numbers of somegraphsrdquo httparxivorgabs11093933

[124] N Jafari Rad and L Volkmann ldquoOn the Roman bondagenumber of planar graphsrdquo Graphs and Combinatorics vol 27no 4 pp 531ndash538 2011

[125] S Akbari M Khatirinejad and S Qajar ldquoA note on the romanbondage number of planar graphsrdquo Graphs and Combinatorics2012

[126] B Bresar M A Henning and D F Rall ldquoRainbow dominationin graphsrdquo Taiwanese Journal of Mathematics vol 12 no 1 pp213ndash225 2008

[127] B L Hartnell and D F Rall ldquoOn dominating the Cartesianproduct of a graph and 120572krdquo Discussiones Mathematicae GraphTheory vol 24 no 3 pp 389ndash402 2004

[128] N Dehgardi S M Sheikholeslami and L Volkman ldquoThe k-rainbow bondage number of a graphrdquo to appear in DiscreteApplied Mathematics

[129] J von Neumann and O Morgenstern Theory of Games andEconomic Behavior Princeton University Press Princeton NJUSA 1944

[130] C BergeTheorıe des Graphes et ses Applications 1958[131] O Ore Theory of Graphs vol 38 American Mathematical

Society Colloquium Publications 1962[132] E Shan and L Kang ldquoBondage number in oriented graphsrdquoArs

Combinatoria vol 84 pp 319ndash331 2007[133] J Huang and J-M Xu ldquoThe bondage numbers of extended

de Bruijn and Kautz digraphsrdquo Computers amp Mathematics withApplications vol 51 no 6-7 pp 1137ndash1147 2006

[134] J Huang and J-M Xu ldquoThe total domination and total bondagenumbers of extended de Bruijn and Kautz digraphsrdquoComputersamp Mathematics with Applications vol 53 no 8 pp 1206ndash12132007

[135] Y Shibata and Y Gonda ldquoExtension of de Bruijn graph andKautz graphrdquo Computers amp Mathematics with Applications vol30 no 9 pp 51ndash61 1995

[136] X Zhang J Liu and JMeng ldquoThebondage number in completet-partite digraphsrdquo Information Processing Letters vol 109 no17 pp 997ndash1000 2009

[137] D W Bange A E Barkauskas and P J Slater ldquoEfficient dom-inating sets in graphsrdquo in Applications of Discrete Mathematicspp 189ndash199 SIAM Philadelphia Pa USA 1988

[138] M Livingston and Q F Stout ldquoPerfect dominating setsrdquoCongressus Numerantium vol 79 pp 187ndash203 1990

[139] L Clark ldquoPerfect domination in random graphsrdquo Journal ofCombinatorial Mathematics and Combinatorial Computing vol14 pp 173ndash182 1993

[140] D W Bange A E Barkauskas L H Host and L H ClarkldquoEfficient domination of the orientations of a graphrdquo DiscreteMathematics vol 178 no 1ndash3 pp 1ndash14 1998

[141] A E Barkauskas and L H Host ldquoFinding efficient dominatingsets in oriented graphsrdquo Congressus Numerantium vol 98 pp27ndash32 1993

[142] I J Dejter and O Serra ldquoEfficient dominating sets in CayleygraphsrdquoDiscrete AppliedMathematics vol 129 no 2-3 pp 319ndash328 2003

[143] J Lee ldquoIndependent perfect domination sets in Cayley graphsrdquoJournal of Graph Theory vol 37 no 4 pp 213ndash219 2001

[144] WGu X Jia and J Shen ldquoIndependent perfect domination setsin meshes tori and treesrdquo Unpublished

[145] N Obradovic J Peters and G Ruzic ldquoEfficient domination incirculant graphswith two chord lengthsrdquo Information ProcessingLetters vol 102 no 6 pp 253ndash258 2007

[146] K Reji Kumar and G MacGillivray ldquoEfficient domination incirculant graphsrdquo Discrete Mathematics vol 313 no 6 pp 767ndash771 2013

[147] D Van Wieren M Livingston and Q F Stout ldquoPerfectdominating sets on cube-connected cyclesrdquo Congressus Numer-antium vol 97 pp 51ndash70 1993

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

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Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

International Journal of Combinatorics 21

Theorem 135 (Jafari Rad et al [81]) For an integer 119899

119887TR (119875119899) = infin 119894119891 2 ⩽ 119899 ⩽ 5

1 119894119891 119899 ⩾ 5

119887TR (119862119899) =

infin 119894119891 119899 = 3

1 119894119891 3 lt 119899 119899 equiv 0 1 (mod 4)2 119894119891 3 lt 119899 119899 equiv 2 3 (mod 4)

119887TR (119882119899) = 2 119891119900119903 119899 ⩾ 6

119887TR (119870119899) = 119899 minus 1 119891119900119903 119899 ⩾ 4

119887TR (119870119898119899) = 119898 minus 1 119891119900119903 2 ⩽ 119898 ⩽ 119899

(49)

119887TR (119870119899) = 119899minus1 shows the fact that for any integer 119899 ⩾ 3 there

exists a connected graph namely119870119899+1

such that 119887TR (119870119899+1) =

119899 that is the total restrained bondage number is unboundedfrom the above

A vertex is said to be a support vertex if it is adjacent toa vertex of degree one Let A be the family of all connectedgraphs such that119866 belongs toA if and only if every edge of119866is incident to a support vertex or 119866 is a cycle of length three

Proposition 136 (Cyman and Raczek [82] 2006) For aconnected graph119866 of order 119899 120574TR (119866) = 119899 if and only if119866 isin A

Hence if 119866 isin A then the removal of any subset of edgesof119866 cannot increase the total restrained domination numberof 119866 and thus 119887TR (119866) = infin When 119866 notin A a result similar toTheorem 6 is obtained

Theorem 137 (Jafari Rad et al [81]) For any tree 119879 notin A119887TR (119879) ⩽ 2 This bound is sharp

In the samepaper the authors provided a characterizationfor trees 119879 notin A with 119887TR (119879) = 1 or 2 The following result issimilar to Observation 1

Observation 3 (Jafari Rad et al [81]) If 119896 edges can beremoved from a graph 119866 to obtain a graph 119867 without anisolated vertex and with 119887TR (119867) = 119905 then 119887TR (119866) ⩽ 119896 + 119905

ApplyingTheorem 137 andObservation 3 Jafari Rad et alcharacterized all connected graphs 119866 with 119887TR (119866) = infin

Theorem 138 (Jafari Rad et al [81]) For a connected graph119866119887TR (119866) = infin if and only if 119866 isin A

84 Other Conditional Bondage Numbers A subset 119868 sube 119881(119866)is called an independent set if no two vertices in 119868 are adjacentin 119866 The maximum cardinality among all independent setsis called the independence number of 119866 denoted by 120572(119866)

A dominating set 119878 of a graph 119866 is called to be inde-pendent if 119878 is an independent set of 119866 The minimumcardinality among all independent dominating set is calledthe independence domination number of 119866 and denoted by120574119868(119866)

Since an independent dominating set is not only adominating set but also an independent set 120574(119866) ⩽ 120574

119868(119866) ⩽

120572(119866) for any graph 119866It is clear that a maximal independent set is certainly a

dominating set Thus an independent set is maximal if andonly if it is an independent dominating set and so 120574

119868(119866) is the

minimum cardinality among all maximal independent setsof 119866 This graph-theoretical invariant has been well studiedin the literature see for example Haynes et al [3] for earlyresults Goddard and Henning [86] for recent results onindependent domination in graphs

In 2003 Zhang et al [87] defined the independencebondage number 119887

119868(119866) of a nonempty graph 119866 to be the

minimum cardinality among all subsets 119861 sube 119864(119866) for which120574119868(119866 minus 119861) gt 120574

119868(119866) For some ordinary graphs their indepen-

dence domination numbers can be easily determined and soindependence bondage numbers have been also determinedClearly 119887

119868(119870

119899) = 1 if 119899 ⩾ 2

Theorem 139 (Zhang et al [87] 2003) For any integer 119899 ⩾ 4

119887119868(119862

119899) =

1 119894119891 119899 equiv 1 (mod 2) 2 119894119891 119899 equiv 0 (mod 2)

119887119868(119875

119899) =

1 119894119891 119899 equiv 0 (mod 2) 2 119894119891 119899 equiv 1 (mod 2)

119887119868(119870

119898119899) = 119899 119891119900119903 119898 ⩽ 119899

(50)

Apart from the above-mentioned results as far as we findno other results on the independence bondage number in thepresent literature we never knew much of any result on thisparameter for a tree

A subset 119878 of 119881(119866) is called an equitable dominatingset if for every 119910 isin 119881(119866 minus 119878) there exists a vertex 119909 isin 119878such that 119909119910 isin 119864(119866) and |119889

119866(119909) minus 119889

119866(119910)| ⩽ 1 proposed

by Dharmalingam [88] The minimum cardinality of such adominating set is called the equitable domination number andis denoted by 120574

119864(119866) Deepak et al [89] defined the equitable

bondage number 119887119864(119866) of a graph 119866 to be the cardinality of a

smallest set 119861 sube 119864(119866) of edges for which 120574119864(119866 minus 119861) gt 120574

119864(119866)

and determined 119887119864(119866) for some special graphs

Theorem 140 (Deepak et al [89] 2011) For any integer 119899 ⩾ 2

119887119864(119870

119899) = lceil

119899

2rceil

119887119864(119870

119898119899) = 119898 119891119900119903 0 ⩽ 119899 minus 119898 ⩽ 1

119887119864(119862

119899) =

3 119894119891 119899 equiv 1 (mod 3)2 119900119905ℎ119890119903119908119894119904119890

119887119864(119875

119899) =

2 119894119891 119899 equiv 1 (mod 3)1 119900119905ℎ119890119903119908119894119904119890

119887119864(119879) ⩽ 2 119891119900119903 119886119899119910 119899119900119899119905119903119894119907119894119886119897 119905119903119890119890 119879

119887119864(119866) ⩽ 119899 minus 1 119891119900119903 119886119899119910 119888119900119899119899119890119888119905119890119889 119892119903119886119901ℎ 119866 119900119891 119900119903119889119890119903 119899

(51)

22 International Journal of Combinatorics

As far as we know there are no more results on theequitable bondage number of graphs in the present literature

There are other variations of the normal domination byadding restricted conditions to the normal dominating setFor example the connected domination set of a graph119866 pro-posed by Sampathkumar and Walikar [90] is a dominatingset 119878 such that the subgraph induced by 119878 is connected Theproblem of finding a minimum connected domination set ina graph is equivalent to the problemof finding a spanning treewithmaximumnumber of vertices of degree one [91] and hassome important applications in wireless networks [92] thusit has received much research attention However for suchan important domination there is no result on its bondagenumber We believe that the topic is of significance for futureresearch

9 Generalized Bondage Numbers

There are various generalizations of the classical dominationsuch as 119901-domination distance domination fractional dom-ination Roman domination and rainbow domination Everysuch a generalization can lead to a corresponding bondageIn this section we introduce some of them

91 119901-Bondage Numbers In 1985 Fink and Jacobson [93]introduced the concept of 119901-domination Let 119901 be a positiveinteger A subset 119878 of 119881(119866) is a 119901-dominating set of 119866 if|119878 cap 119873

119866(119910)| ⩾ 119901 for every 119910 isin 119878 The 119901-domination number

120574119901(119866) is the minimum cardinality among all 119901-dominating

sets of 119866 Any 119901-dominating set of 119866 with cardinality 120574119901(119866)

is called a 120574119901-set of 119866 Note that a 120574

1-set is a classic minimum

dominating set Notice that every graph has a 119901-dominatingset since the vertex-set119881(119866) is such a setWe also note that a 1-dominating set is a dominating set and so 120574(119866) = 120574

1(119866) The

119901-domination number has received much research attentionsee a state-of-the-art survey article by Chellali et al [94]

It is clear from definition that every 119901-dominating setof a graph certainly contains all vertices of degree at most119901 minus 1 By this simple observation to avoid the trivial caseoccurrence we always assume that Δ(119866) ⩾ 119901 For 119901 ⩾ 2 Luet al [95] gave a constructive characterization of trees withunique minimum 119901-dominating sets

Recently Lu and Xu [96] have introduced the conceptto the 119901-bondage number of 119866 denoted by 119887

119901(119866) as the

minimum cardinality among all sets of edges 119861 sube 119864(119866) suchthat 120574

119901(119866 minus 119861) gt 120574

119901(119866) Clearly 119887

1(119866) = 119887(119866)

Lu and Xu [96] established a lower bound and an upperbound on 119887

119901(119879) for any integer 119901 ⩾ 2 and any tree 119879 with

Δ(119879) ⩾ 119901 and characterized all trees achieving the lowerbound and the upper bound respectively

Theorem 141 (Lu and Xu [96] 2011) For any integer 119901 ⩾ 2and any tree 119879 with Δ(119879) ⩾ 119901

1 ⩽ 119887119901(119879) ⩽ Δ (119879) minus 119901 + 1 (52)

Let 119878 be a given subset of 119881(119866) and for any 119909 isin 119878 let

119873119901(119909 119878 119866) = 119910 isin 119878 cap 119873

119866(119909)

1003816100381610038161003816119873119866(119910) cap 119878

1003816100381610038161003816 = 119901

(53)

Then all trees achieving the lower bound 1 can be character-ized as follows

Theorem 142 (Lu and Xu [96] 2011) Let 119879 be a tree withΔ(119879) ⩾ 119901 ⩾ 2 Then 119887

119901(119879) = 1 if and only if for any 120574

119901-set

119878 of 119879 there exists an edge 119909119910 isin (119878 119878) such that 119910 isin 119873119901(119909 119878 119879)

The notation 119878(119886 119887) denotes a double star obtained byadding an edge between the central vertices of two stars119870

1119886minus1

and1198701119887minus1

And the vertex with degree 119886 (resp 119887) in 119878(119886 119887) iscalled the 119871-central vertex (resp 119877-central vertex) of 119878(119886 119887)

To characterize all trees attaining the upper bound givenin Theorem 141 we define three types of operations on a tree119879 with Δ(119879) = Δ ⩾ 119901 + 1

Type 1 attach a pendant edge to a vertex 119910with 119889119879(119910) ⩽ 119901minus

2 in 119879

Type 2 attach a star 1198701Δminus1

to a vertex 119910 of 119879 by joining itscentral vertex to 119910 where 119910 in a 120574

119901-set of 119879 and

119889119879(119910) ⩽ Δ minus 1

Type 3 attach a double star 119878(119901 Δ minus 1) to a pendant vertex 119910of 119879 by coinciding its 119877-central vertex with 119910 wherethe unique neighbor of 119910 is in a 120574

119901-set of 119879

Let B = 119879 a tree obtained from 1198701Δ

or 119878(119901 Δ) by afinite sequence of operations of Types 1 2 and 3

Theorem 143 (Lu and Xu [96] 2011) A tree with the maxi-mum Δ ⩾ 119901 + 1 has 119901-bondage number Δ minus 119901 + 1 if and onlyif it belongs toB

Theorem 144 (Lu and Xu [97] 2012) Let 119866119898119899= 119875

119898times 119875

119899

Then

1198872(119866

2119899) = 1 119891119900119903 119899 ⩾ 2

1198872(119866

3119899) =

2 119894119891 119899 equiv 1 (mod 3) 1 119900119905ℎ119890119903119908119894119904119890

for 119899 ⩾ 2

1198872(119866

4119899) =

1 119894119891 119899 equiv 3 (mod 4) 2 119900119905ℎ119890119903119908119894119904119890

for 119899 ⩾ 7

(54)

Recently Krzywkowski [98] has proposed the conceptof the nonisolating 2-bondage of a graph The nonisolating2-bondage number of a graph 119866 denoted by 1198871015840

2(119866) is the

minimum cardinality among all sets of edges 119861 sube 119864(119866) suchthat 119866 minus 119861 has no isolated vertices and 120574

2(119866 minus 119861) gt 120574

2(119866) If

for every 119861 sube 119864(119866) either 1205742(119866 minus 119861) = 120574

2(119866) or 119866 minus 119861 has

isolated vertices then define 11988710158402(119866) = 0 and say that 119866 is a 2-

nonisolatedly strongly stable graph Krzywkowski presentedsome basic properties of nonisolating 2-bondage in graphsshowed that for every nonnegative integer 119896 there exists atree 119879 such 1198871015840

2(119879) = 119896 characterized all 2-non-isolatedly

strongly stable trees and determined 11988710158402(119866) for several special

graphs

International Journal of Combinatorics 23

Theorem 145 (Krzywkowski [98] 2012) For any positiveintegers 119899 and119898 with119898 ⩽ 119899

1198871015840

2(119870

119899) =

0 119894119891 119899 = 1 2 3

lfloor2119899

3rfloor 119900119905ℎ119890119903119908119894119904119890

1198871015840

2(119875

119899) =

0 119894119891 119899 = 1 2 3

1 119900119905ℎ119890119903119908119894119904119890

1198871015840

2(119862

119899) =

0 119894119891 119899 = 3

1 119894119891 119899 119894119904 119890V119890119899

2 119894119891 119899 119894119904 119900119889119889

1198871015840

2(119870

119898119899) =

3 119894119891 119898 = 119899 = 3

1 119894119891 119898 = 119899 = 4

2 119900119905ℎ119890119903119908119894119904119890

(55)

92 Distance Bondage Numbers A subset 119878 of vertices of agraph119866 is said to be a distance 119896-dominating set for119866 if everyvertex in 119866 not in 119878 is at distance at most 119896 from some vertexof 119878 The minimum cardinality of all distance 119896-dominatingsets is called the distance 119896-domination number of 119866 anddenoted by 120574

119896(119866) (do not confuse with the above-mentioned

120574119901(119866)) When 119896 = 1 a distance 1-dominating set is a normal

dominating set and so 1205741(119866) = 120574(119866) for any graph 119866 Thus

the distance 119896-domination is a generalization of the classicaldomination

The relation between 120574119896and 120572

119896(the 119896-independence

number defined in Section 22) for a tree obtained by Meirand Moon [14] who proved that 120574

119896(119879) = 120572

2119896(119879) for any tree

119879 (a special case for 119896 = 1 is stated in Proposition 10) Thefurther research results can be found in Henning et al [99]Tian and Xu [100ndash103] and Liu et al [104]

In 1998 Hartnell et al [15] defined the distance 119896-bondagenumber of 119866 denoted by 119887

119896(119866) to be the cardinality of a

smallest subset 119861 of edges of 119866 with the property that 120574119896(119866 minus

119861) gt 120574119896(119866) FromTheorem 6 it is clear that if119879 is a nontrivial

tree then 1 ⩽ 1198871(119879) ⩽ 2 Hartnell et al [15] generalized this

result to any integer 119896 ⩾ 1

Theorem 146 (Hartnell et al [15] 1998) For every nontrivialtree 119879 and positive integer 119896 1 ⩽ 119887

119896(119879) ⩽ 2

Hartnell et al [15] and Topp and Vestergaard [13] alsocharacterized the trees having distance 119896-bondage number 2In particular the class of trees for which 119887

1(119879) = 2 are just

those which have a unique maximum 2-independent set (seeTheorem 11)

Since when 119896 = 1 the distance 1-bondage number 1198871(119866)

is the classical bondage number 119887(119866) Theorem 12 gives theNP-hardness of deciding the distance 119896-bondage number ofgeneral graphs

Theorem 147 Given a nonempty undirected graph 119866 andpositive integers 119896 and 119887 with 119887 ⩽ 120576(119866) determining wetheror not 119887

119896(119866) ⩽ 119887 is NP-hard

For a vertex 119909 in119866 the open 119896-neighborhood119873119896(119909) of 119909 is

defined as119873119896(119909) = 119910 isin 119881(119866) 1 ⩽ 119889

119866(119909 119910) le 119896The closed

119896-neighborhood119873119896[119909] of 119909 in 119866 is defined as119873

119896(119909) cup 119909 Let

Δ119896(119866) = max 1003816100381610038161003816119873119896

(119909)1003816100381610038161003816 for any 119909 isin 119881 (119866) (56)

Clearly Δ1(119866) = Δ(119866) The 119896th power of a graph 119866 is the

graph119866119896 with vertex-set119881(119866119896

) = 119881(119866) and edge-set119864(119866119896

) =

119909119910 1 ⩽ 119889119866(119909 119910) ⩽ 119896 The following lemma holds directly

from the definition of 119866119896

Proposition 148 (Tian and Xu [101]) Δ(119866119896

) = Δ119896(119866) and

120574(119866119896

) = 120574119896(119866) for any graph 119866 and each 119896 ⩾ 1

A graph 119866 is 119896-distance domination-critical or 120574119896-critical

for short if 120574119896(119866 minus 119909) lt 120574

119896(119866) for every vertex 119909 in 119866

proposed by Henning et al [105]

Proposition 149 (Tian and Xu [101]) For each 119896 ⩾ 1 a graph119866 is 120574

119896-critical if and only if 119866119896 is 120574

119896-critical

By the above facts we suggest the following worthwhileproblem

Problem 6 Can we generalize the results on the bondage for119866 to 119866119896 In particular do the following propositions hold

(a) 119887(119866119896

) = 119887119896(119866) for any graph 119866 and each 119896 ⩾ 1

(b) 119887(119866119896

) ⩽ Δ119896(119866) if 119866 is not 120574

119896-critical

Let 119896 and 119901 be positive integers A subset 119878 of 119881(119866) isdefined to be a (119896 119901)-dominating set of 119866 if for any vertex119909 isin 119878 |119873

119896(119909) cap 119878| ⩾ 119901 The (119896 119901)-domination number of

119866 denoted by 120574119896119901(119866) is the minimum cardinality among

all (119896 119901)-dominating sets of 119866 Clearly for a graph 119866 a(1 1)-dominating set is a classical dominating set a (119896 1)-dominating set is a distance 119896-dominating set and a (1 119901)-dominating set is the above-mentioned 119901-dominating setThat is 120574

11(119866) = 120574(119866) 120574

1198961(119866) = 120574

119896(119866) and 120574

1119901(119866) = 120574

119901(119866)

The concept of (119896 119901)-domination in a graph 119866 is a gen-eralized domination which combines distance 119896-dominationand 119901-domination in 119866 So the investigation of (119896 119901)-domination of 119866 is more interesting and has received theattention of many researchers see for example [106ndash109]

More general Li and Xu [110] proposed the concept of the(ℓ 119908)-domination number of a119908-connected graph A subset119878 of 119881(119866) is defined to be an (ℓ 119908)-dominating set of 119866 iffor any vertex 119909 isin 119878 there are 119908 internally disjoint pathsof length at most ℓ from 119909 to some vertex in 119878 The (ℓ 119908)-domination number of119866 denoted by 120574

ℓ119908(119866) is theminimum

cardinality among all (ℓ 119908)-dominating sets of 119866 Clearly1205741198961(119866) = 120574

119896(119866) and 120574

1119901(119866) = 120574

119901(119866)

It is quite natural to propose the concept of bondage num-ber for (119896 119901)-domination or (ℓ 119908)-domination However asfar as we know none has proposed this concept until todayThis is a worthwhile topic for further research

24 International Journal of Combinatorics

93 Fractional Bondage Numbers If 120590 is a function mappingthe vertex-set 119881 into some set of real numbers then for anysubset 119878 sube 119881 let 120590(119878) = sum

119909isin119878120590(119909) Also let |120590| = 120590(119881)

A real-value function 120590 119881 rarr [0 1] is a dominatingfunction of a graph 119866 if for every 119909 isin 119881(119866) 120590(119873

119866[119909]) ⩾ 1

Thus if 119878 is a dominating set of 119866 120590 is a function where

120590 (119909) = 1 if 119909 isin 1198780 otherwise

(57)

then 120590 is a dominating function of119866 The fractional domina-tion number of 119866 denoted by 120574

119865(119866) is defined as follows

120574119865(119866) = min |120590| 120590 is a domination function of 119866

(58)

The 120574119865-bondage number of 119866 denoted by 119887

119865(119866) is

defined as the minimum cardinality of a subset 119861 sube 119864 whoseremoval results in 120574

119865(119866 minus 119861) gt 120574

119865(119866)

Hedetniemi et al [111] are the first to study fractionaldomination although Farber [112] introduced the idea indi-rectly The concept of the fractional domination numberwas proposed by Domke and Laskar [113] in 1997 Thefractional domination numbers for some ordinary graphs aredetermined

Proposition 150 (Domke et al [114] 1998 Domke and Laskar[113] 1997) (a) If 119866 is a 119896-regular graph with order 119899 then120574119865(119866) = 119899119896 + 1(b) 120574

119865(119862

119899) = 1198993 120574

119865(119875

119899) = lceil1198993rceil 120574

119865(119870

119899) = 1

(c) 120574119865(119866) = 1 if and only if Δ(119866) = 119899 minus 1

(d) 120574119865(119879) = 120574(119879) for any tree 119879

(e) 120574119865(119870

119899119898) = (119899(119898 + 1) + 119898(119899 + 1))(119899119898 minus 1) where

max119899119898 gt 1

The assertions (a)ndash(d) are due to Domke et al [114] andthe assertion (e) is due to Domke and Laskar [113] Accordingto these results Domke and Laskar [113] determined the 120574

119865-

bondage numbers for these graphs

Theorem 151 (Domke and Laskar [113] 1997) 119887119865(119870

119899) =

lceil1198992rceil 119887119865(119870

119899119898) = 1 wheremax119899119898 gt 1

119887119865(119862

119899) =

2 119894119891 119899 equiv 0 (mod 3) 1 119900119905ℎ119890119903119908119894119904119890

119887119865(119875

119899) =

2 119894119891 119899 equiv 1 (mod 3) 1 119900119905ℎ119890119903119908119894119904119890

(59)

It is easy to see that for any tree119879 119887119865(119879) ⩽ 2 In fact since

120574119865(119879) = 120574(119879) for any tree 119879 120574

119865(119879

1015840

) = 120574(1198791015840

) for any subgraph1198791015840 of 119879 It follows that

119887119865(119879) = min 119861 sube 119864 (119879) 120574

119865(119879 minus 119861) gt 120574

119865(119879)

= min 119861 sube 119864 (119879) 120574 (119879 minus 119861) gt 120574 (119879)

= 119887 (119879) ⩽ 2

(60)

Except for these no results on 119887119865(119866) have been known as

yet

94 Roman Bondage Numbers A Roman dominating func-tion on a graph 119866 is a labeling 119891 119881 rarr 0 1 2 such thatevery vertex with label 0 has at least one neighbor with label2 The weight of a Roman dominating function 119891 on 119866 is thevalue

119891 (119866) = sum

119906isin119881(119866)

119891 (119906) (61)

The minimum weight of a Roman dominating function ona graph 119866 is called the Roman domination number of 119866denoted by 120574RM (119866)

A Roman dominating function 119891 119881 rarr 0 1 2

can be represented by the ordered partition (1198810 119881

1 119881

2) (or

(119881119891

0 119881

119891

1 119881

119891

2) to refer to119891) of119881 where119881

119894= V isin 119881 | 119891(V) = 119894

In this representation its weight 119891(119866) = |1198811| + 2|119881

2| It is

clear that119881119891

1cup119881

119891

2is a dominating set of 119866 called the Roman

dominating set denoted by 119863119891

R = (1198811 1198812) Since 119881119891

1cup 119881

119891

2is

a dominating set when 119891 is a Roman dominating functionand since placing weight 2 at the vertices of a dominating setyields a Roman dominating function in [115] it is observedthat

120574 (119866) ⩽ 120574RM (119866) ⩽ 2120574 (119866) (62)

A graph 119866 is called to be Roman if 120574RM (119866) = 2120574(119866)The definition of the Roman domination function is

proposed implicitly by Stewart [116] and ReVelle and Rosing[117] Roman dominating numbers have been deeply studiedIn particular Bahremandpour et al [118] showed that theproblem determining the Roman domination number is NP-complete even for bipartite graphs

Let 119866 be a graph with maximum degree at least twoThe Roman bondage number 119887RM (119866) of 119866 is the minimumcardinality of all sets 1198641015840 sube 119864 for which 120574RM (119866 minus 119864

1015840

) gt

120574RM (119866) Since in study of Roman bondage number theassumption Δ(119866) ⩾ 2 is necessary we always assume thatwhen we discuss 119887RM (119866) all graphs involved satisfy Δ(119866) ⩾2 The Roman bondage number 119887RM (119866) is introduced byJafari Rad and Volkmann in [119]

Recently Bahremandpour et al [118] have shown that theproblem determining the Roman bondage number is NP-hard even for bipartite graphs

Problem 7 Consider the decision problem

Roman Bondage Problem

Instance a nonempty bipartite graph119866 and a positiveinteger 119896

Question is 119887RM (119866) ⩽ 119896

Theorem 152 (Bahremandpour et al [118] 2013) TheRomanbondage problem is NP-hard even for bipartite graphs

The exact value of 119887RM(119866) is known only for a few familyof graphs including the complete graphs cycles and paths

International Journal of Combinatorics 25

Theorem 153 (Jafari Rad and Volkmann [119] 2011) For anypositive integers 119899 and119898 with119898 ⩽ 119899

119887RM (119875119899) = 2 119894119891 119899 equiv 2 (mod 3) 1 119900119905ℎ119890119903119908119894119904119890

119887RM (119862119899) =

3 119894119891 119899 = 2 (mod 3) 2 119900119905ℎ119890119903119908119894119904119890

119887RM (119870119899) = lceil

119899

2rceil 119891119900119903 119899 ⩾ 3

119887RM (119870119898119899) =

1 119894119891 119898 = 1 119886119899119889 119899 = 1

4 119894119891 119898 = 119899 = 3

119898 119900119905ℎ119890119903119908119894119904119890

(63)

Lemma 154 (Cockayne et al [115] 2004) If 119866 is a graph oforder 119899 and contains vertices of degree 119899 minus 1 then 120574RM(119866) = 2

Using Lemma 154 the third conclusion in Theorem 153can be generalized to more general case which is similar toLemma 49

Theorem 155 (Jafari Rad and Volkmann [119] 2011) Let119866 bea graph with order 119899 ge 3 and 119905 the number of vertices of degree119899 minus 1 in 119866 If 119905 ⩾ 1 then 119887RM(119866) = lceil1199052rceil

Ebadi and PushpaLatha [120] conjectured that 119887RM(119866) ⩽119899 minus 1 for any graph 119866 of order 119899 ⩾ 3 Dehgardi et al[121] Akbari andQajar [122] independently showed that thisconjecture is true

Theorem 156 119887RM(119866) ⩽ 119899 minus 1 for any connected graph 119866 oforder 119899 ⩾ 3

For a complete 119905-partite graph we have the followingresult

Theorem 157 (Hu and Xu [123] 2011) Let 119866 = 11987011989811198982119898119905

bea complete 119905-partite graph with 119898

1= sdot sdot sdot = 119898

119894lt 119898

119894+1le sdot sdot sdot ⩽

119898119905 119905 ⩾ 2 and 119899 = sum119905

119895=1119898

119895 Then

119887RM (119866) =

lceil119894

2rceil 119894119891 119898

119894= 1 119886119899119889 119899 ⩾ 3

2 119894119891 119898119894= 2 119886119899119889 119894 = 1

119894 119894119891 119898119894= 2 119886119899119889 119894 ⩾ 2

4 119894119891 119898119894= 3 119886119899119889 119894 = 119905 = 2

119899 minus 1 119894119891 119898119894= 3 119886119899119889 119894 = 119905 ⩾ 3

119899 minus 119898119905

119894119891 119898119894⩾ 3 119886119899119889 119898

119905⩾ 4

(64)

Consider a complete 119905-partite graph 119870119905(3) for 119905 ⩾ 3

119887RM(119870119905(3)) = 119899 minus 1 by Theorem 157 where 119899 = 3119905

which shows that this upper bound of 119899 minus 1 on 119887RM(119866) inTheorem 156 is sharp This fact and 120574

119877(119870

119905(3)) = 4 give

a negative answer to the following two problems posed byDehgardi et al [121]Prove or Disprove for any connected graph 119866 of order 119899 ge 3(i) 119887RM(119866) = 119899 minus 1 if and only if 119866 cong 119870

3 (ii) 119887RM(119866) ⩽ 119899 minus

120574119877(119866) + 1Note that a complete 119905-partite graph 119870

119905(3) is (119899 minus 3)-

regular and 119887RM(119870119905(3)) = 119899 minus 1 for 119905 ⩾ 3 where 119899 = 3119905

Hu and Xu further determined that 119887119877(119866) = 119899 minus 2 for any

(119899 minus 3)-regular graph 119866 of order 119899 ⩽ 5 except 119870119905(3)

Theorem 158 (Hu and Xu [123] 2011) For any (119899minus3)-regulargraph 119866 of order 119899 other than119870

119905(3) 119887RM(119866) = 119899 minus 2 for 119899 ⩾ 5

For a tree 119879 with order 119899 ⩾ 3 Ebadi and PushpaLatha[120] and Jafari Rad and Volkmann [119] independentlyobtained an upper bound on 119887RM(119879)

Theorem 159 119887RM(119879) ⩽ 3 for any tree 119879 with order 119899 ⩾ 3

Theorem 160 (Bahremandpour et al [118] 2013) 119887RM(1198752 times119875119899) = 2 for 119899 ⩾ 2

Theorem 161 (Jafari Rad and Volkmann [119] 2011) Let119866 bea graph with order at least three

(a) If (119909 119910 119911) is a path of length 2 in 119866 then 119887RM(119866) ⩽119889119866(119909) + 119889

119866(119910) + 119889

119866(119911) minus 3 minus |119873

119866(119909) cap 119873

119866(119910)|

(b) If 119866 is connected then 119887RM(119866) ⩽ 120582(119866) + 2Δ(119866) minus 3

Theorem 161 (b) implies 119887RM(119866) ⩽ 120575(119866) + 2Δ(119866) minus 3 for aconnected graph 119866 Note that for a planar graph 119866 120575(119866) ⩽ 5moreover 120575(119866) ⩽ 3 if the girth at least 4 and 120575(119866) ⩽ 2 if thegirth at least 6 These two facts show that 119887RM(119866) ⩽ 2Δ(119866) +2 for a connected planar graph 119866 Jafari Rad and Volkmann[124] improved this bound

Theorem 162 (Jafari Rad and Volkmann [124] 2011) For anyconnected planar graph 119866 of order 119899 ⩾ 3 with girth 119892(119866)

119887RM (119866)

⩽

2Δ (119866)

Δ (119866) + 6

Δ (119866) + 5 119894119891 119866 119888119900119899119905119886119894119899119904 119899119900 V119890119903119905119894119888119890119904 119900119891 119889119890119892119903119890119890 119891119894V119890

Δ (119866) + 4 119894119891 119892 (119866) ⩾ 4

Δ (119866) + 3 119894119891 119892 (119866) ⩾ 5

Δ (119866) + 2 119894119891 119892 (119866) ⩾ 6

Δ (119866) + 1 119894119891 119892 (119866) ⩾ 8

(65)

According toTheorem 157 119887RM(119862119899) = 3 = Δ(119862

119899)+1 for a

cycle 119862119899of length 119899 ⩾ 8 with 119899 equiv 2 (mod 3) and therefore

the last upper bound inTheorem 162 is sharp at least for Δ =2

Combining the fact that every planar graph 119866 withminimum degree 5 contains an edge 119909119910 with 119889

119866(119909) = 5

and 119889119866(119910) isin 5 6 with Theorem 161 (a) Akbari et al [125]

obtained the following result

26 International Journal of Combinatorics

middot middot middot

middot middot middot

middot middot middot

middot middot middot

119866

Figure 9 The graph 119866 is constructed from 119866

Theorem 163 (Akbari et al [125] 2012) 119887RM(119866) ⩽ 15 forevery planar graph 119866

It remains open to show whether the bound inTheorem 163 is sharp or not Though finding a planargraph 119866 with 119887RM(119866) = 15 seems to be difficult Akbari etal [125] constructed an infinite family of planar graphs withRoman bondage number equal to 7 by proving the followingresult

Theorem 164 (Akbari et al [125] 2012) Let 119866 be a graph oforder 119899 and 119866 is the graph of order 5119899 obtained from 119866 byattaching the central vertex of a copy of a path 119875

5to each vertex

of119866 (see Figure 9) Then 120574RM(119866) = 4119899 and 119887RM(119866) = 120575(119866)+2

ByTheorem 164 infinitemany planar graphswith Romanbondage number 7 can be obtained by considering any planargraph 119866 with 120575(119866) = 5 (eg the icosahedron graph)

Conjecture 165 (Akbari et al [125] 2012) The Romanbondage number of every planar graph is at most 7

For general bounds the following observation is directlyobtained which is similar to Observation 1

Observation 4 Let 119866 be a graph of order 119899 with maximumdegree at least two Assume that 119867 is a spanning subgraphobtained from 119866 by removing 119896 edges If 120574RM(119867) = 120574RM(119866)then 119887

119877(119867) ⩽ 119887RM(119866) ⩽ 119887RM(119867) + 119896

Theorem 166 (Bahremandpour et al [118] 2013) For anyconnected graph 119866 of order 119899 if 119899 ⩾ 3 and 120574RM(119866) = 120574(119866) + 1then

119887RM (119866) ⩽ min 119887 (119866) 119899Δ (66)

where 119899Δis the number of vertices with maximum degree Δ in

119866

Observation 5 For any graph 119866 if 120574(119866) = 120574RM(119866) then119887RM(119866) ⩽ 119887(119866)

Theorem 167 (Bahremandpour et al [118] 2013) For everyRoman graph 119866

119887RM (119866) ⩾ 119887 (119866) (67)

The bound is sharp for cycles on 119899 vertices where 119899 equiv 0 (mod3)

The strict inequality in Theorem 167 can hold for exam-ple 119887(119862

3119896+2) = 2 lt 3 = 119887RM(1198623119896+2

) byTheorem 157A graph 119866 is called to be vertex Roman domination-

critical if 120574RM(119866 minus 119909) lt 120574RM(119866) for every vertex 119909 in 119866 If119866 has no isolated vertices then 120574RM(119866) ⩽ 2120574(119866) ⩽ 2120573(119866)If 120574RM(119866) = 2120573(119866) then 120574RM(119866) = 2120574(119866) and hence 119866 is aRoman graph In [30] Volkmann gave a lot of graphs with120574(119866) = 120573(119866) The following result is similar to Theorem 57

Theorem 168 (Bahremandpour et al [118] 2013) For anygraph 119866 if 120574RM(119866) = 2120573(119866) then

(a) 119887RM(119866) ⩾ 120575(119866)

(b) 119887RM(119866) ⩾ 120575(119866)+1 if119866 is a vertex Roman domination-critical graph

If 120574RM(119866) = 2 then obviously 119887RM(119866) ⩽ 120575(119866) So weassume 120574RM(119866) ⩾ 3 Dehgardi et al [121] proved 119887RM(119866) ⩽(120574RM(119866) minus 2)Δ(119866) + 1 and posed the following problem if 119866is a connected graph of order 119899 ⩾ 4 with 120574RM(119866) ⩾ 3 then

119887RM (119866) ⩽ (120574RM (119866) minus 2) Δ (119866) (68)

Theorem 161 (a) shows that the inequality (68) holds if120574RM(119866) ⩾ 5 Thus the bound in (68) is of interest only when120574RM(119866) is 3 or 4

Lemma 169 (Bahremandpour et al [118] 2013) For anynonempty graph 119866 of order 119899 ⩾ 3 120574RM(119866) = 3 if and onlyif Δ(119866) = 119899 minus 2

The following result shows that (68) holds for all graphs119866 of order 119899 ⩾ 4 with 120574RM(119866) = 3 or 4 which improvesTheorem 161 (b)

Theorem 170 (Bahremandpour et al [118] 2013) For anyconnected graph 119866 of order 119899 ⩾ 4

119887RM (119866) le Δ (119866) = 119899 minus 2 119894119891 120574RM (119866) = 3

Δ (119866) + 120575 (119866) minus 1 119894119891 120574RM (119866) = 4(69)

with the first equality if and only if 119866 cong 1198624

Recently Akbari and Qajar [122] have proved the follow-ing result

Theorem 171 (Akbari and Qajar [122]) For any connectedgraph 119866 of order 119899 ⩾ 3

119887RM (119866) ⩽ 119899 minus 120574RM (119866) + 5 (70)

International Journal of Combinatorics 27

We conclude this section with the following problems

Problem 8 Characterize all connected graphs 119866 of order 119899 ⩾3 for which 119887RM(119866) = 119899 minus 1

Problem 9 Prove or disprove if 119866 is a connected graph oforder 119899 ⩾ 3 then

119887RM (119866) ⩽ 119899 minus 120574RM (119866) + 3 (71)

Note that 120574119877(119870

119905(3)) = 4 and 119887RM(119870119905

(3)) = 119899minus1 where 119899 =3119905 for 119905 ⩾ 3 The upper bound on 119887RM(119866) given in Problem 9is sharp if Problem 9 has positive answer

95 Rainbow Bondage Numbers For a positive integer 119896 a119896-rainbow dominating function of a graph 119866 is a function 119891from the vertex-set 119881(119866) to the set of all subsets of the set1 2 119896 such that for any vertex 119909 in 119866 with 119891(119909) = 0 thecondition

⋃

119910isin119873119866(119909)

119891 (119910) = 1 2 119896 (72)

is fulfilledThe weight of a 119896-rainbow dominating function 119891on 119866 is the value

119891 (119866) = sum

119909isin119881(119866)

1003816100381610038161003816119891 (119909)1003816100381610038161003816 (73)

The 119896-rainbow domination number of a graph 119866 denoted by120574119903119896(119866) is the minimum weight of a 119896-rainbow dominating

function 119891 of 119866Note that 120574

1199031(119866) is the classical domination number 120574(119866)

The 119896-rainbow domination number is introduced by Bresaret al [126] Rainbow domination of a graph 119866 coincides withordinary domination of the Cartesian product of 119866 with thecomplete graph in particular 120574

119903119896(119866) = 120574(119866 times 119870

119896) for any

positive integer 119896 and any graph 119866 [126] Hartnell and Rall[127] proved that 120574(119879) lt 120574

1199032(119879) for any tree 119879 no graph has

120574 = 120574119903119896for 119896 ⩾ 3 for any 119896 ⩾ 2 and any graph 119866 of order 119899

min 119899 120574 (119866) + 119896 minus 2 ⩽ 120574119903119896(119866) ⩽ 119896120574 (119866) (74)

The introduction of rainbow domination is motivatedby the study of paired domination in Cartesian productsof graphs where certain upper bounds can be expressed interms of rainbow domination that is for any graph 119866 andany graph 119867 if 119881(119867) can be partitioned into 119896120574

119875-sets then

120574119875(119866 times 119867) ⩽ (1119896)120592(119867)120574

119903119896(119866) proved by Bresar et al [126]

Dehgardi et al [128] introduced the concept of 119896-rainbowbondage number of graphs Let 119866 be a graph with maximumdegree at least two The 119896-rainbow bondage number 119887

119903119896(119866)

of 119866 is the minimum cardinality of all sets 119861 sube 119864(119866) forwhich 120574

119903119896(119866 minus 119861) gt 120574

119903119896(119866) Since in the study of 119896-rainbow

bondage number the assumption Δ(119866) ⩾ 2 is necessarywe always assume that when we discuss 119887

119903119896(119866) all graphs

involved satisfy Δ(119866) ⩾ 2The 2-rainbow bondage number of some special families

of graphs has been determined

Theorem 172 (Dehgardi et al [128]) For any integer 119899 ⩾ 3

1198871199032(119875

119899) = 1 119891119900119903 119899 ⩾ 2

1198871199032(119862

119899) =

1 119894119891 119899 equiv 0 (mod 4) 2 119900119905ℎ119890119903119908119894119904119890

1198871199032(119870

119899119899) =

1 119894119891 119899 = 2

3 119894119891 119899 = 3

6 119894119891 119899 = 4

119898 119894119891 119899 ⩾ 5

(75)

Theorem 173 (Dehgardi et al [128]) For any connected graph119866 of order 119899 ⩾ 3 119887

1199032(119866) ⩽ 120582(119866) + 2Δ(119866) minus 3

Theorem 174 (Dehgardi et al [128]) For any tree 119879 of order119899 ⩾ 3 119887

1199032(119879) ⩽ 2

Theorem 175 (Dehgardi et al [128]) If 119866 is a planar graphwithmaximumdegree at least two then 119887

1199032(119866) ⩽ 15Moreover

if the girth 119892(119866) ⩾ 4 then 1198871199032(119866) ⩽ 11 and if 119892(119866) ⩾ 6 then

1198871199032(119866) ⩽ 9

Theorem 176 (Dehgardi et al [128]) For a complete bipartitegraph 119870

119901119902 if 2119896 + 1 ⩽ 119901 ⩽ 119902 then 119887

119903119896(119870

119901119902) = 119901

Theorem177 (Dehgardi et al [128]) For a complete graph119870119899

if 119899 ⩾ 119896 + 1 then 119887119903119896(119870

119899) = lceil119896119899(119896 + 1)rceil

The special case 119896 = 1 of Theorem 177 can be found inTheorem 1 The following is a simple observation similar toObservation 1

Observation 6 (Dehgardi et al [128]) Let 119866 be a graphwith maximum degree at least two 119867 a spanning subgraphobtained from 119866 by removing 119896 edges If 120574

119903119896(119867) = 120574

119903119896(119866)

then 119887119903119896(119867) ⩽ 119887

119903119896(119866) ⩽ 119887

119903119896(119867) + 119896

Theorem 178 (Dehgardi et al [128]) For every graph 119866 with120574119903119896(119866) = 119896120574(119866) 119887

119903119896(119866) ⩾ 119887(119866)

A graph 119866 is said to be vertex 119896-rainbow domination-critical if 120574

119903119896(119866 minus 119909) lt 120574

119903119896(119866) for every vertex 119909 in 119866 It is

clear that if 119866 has no isolated vertices then 120574119903119896(119866) ⩽ 119896120574(119866) ⩽

119896120573(119866) where 120573(119866) is the vertex-covering number of119866 Thusif 120574

119903119896(119866) = 119896120573(119866) then 120574

119903119896(119866) = 119896120574(119866)

Theorem 179 (Dehgardi et al [128]) For any graph 119866 if120574119903119896(119866) = 119896120573(119866) then

(a) 119887119903119896(119866) ⩾ 120575(119866)

(b) 119887119903119896(119866) ⩾ 120575(119866)+1 if119866 is vertex 119896-rainbow domination-

critical

As far as we know there are no more results on the 119896-rainbow bondage number of graphs in the literature

28 International Journal of Combinatorics

10 Results on Digraphs

Although domination has been extensively studied in undi-rected graphs it is natural to think of a dominating setas a one-way relationship between vertices of the graphIndeed among the earliest literatures on this subject vonNeumann and Morgenstern [129] used what is now calleddomination in digraphs to find solution (or kernels whichare independent dominating sets) for cooperative 119899-persongames Most likely the first formulation of domination byBerge [130] in 1958 was given in the context of digraphs andonly some years latter by Ore [131] in 1962 for undirectedgraphs Despite this history examination of domination andits variants in digraphs has been essentially overlooked (see[4] for an overview of the domination literature) Thus thereare few if any such results on domination for digraphs in theliterature

The bondage number and its related topics for undirectedgraph have become one of major areas both in theoreticaland applied researches However until recently Carlsonand Develin [42] Shan and Kang [132] and Huang andXu [133 134] studied the bondage number for digraphsindependently In this section we introduce their resultsfor general digraphs Results for some special digraphs suchas vertex-transitive digraphs are introduced in the nextsection

101 Upper Bounds for General Digraphs Let 119866 = (119881 119864)

be a digraph without loops and parallel edges A digraph 119866is called to be asymmetric if whenever (119909 119910) isin 119864(119866) then(119910 119909) notin 119864(119866) and to be symmetric if (119909 119910) isin 119864(119866) implies(119910 119909) isin 119864(119866) For a vertex 119909 of 119881(119866) the sets of out-neighbors and in-neighbors of 119909 are respectively defined as119873

+

119866(119909) = 119910 isin 119881(119866) (119909 119910) isin 119864(119866) and 119873minus

119866(119909) = 119909 isin

119881(119866) (119909 119910) isin 119864(119866) the out-degree and the in-degree of119909 are respectively defined as 119889+

119866(119909) = |119873

+

119866(119909)| and 119889minus

119866(119909) =

|119873+

119866(119909)| Denote themaximum and theminimum out-degree

(resp in-degree) of 119866 by Δ+(119866) and 120575+(119866) (resp Δminus(119866) and120575minus

(119866))The degree 119889119866(119909) of 119909 is defined as 119889+

119866(119909)+119889

minus

119866(119909) and

the maximum and the minimum degrees of119866 are denoted byΔ(119866) and 120575(119866) respectively that is Δ(119866) = max119889

119866(119909) 119909 isin

119881(119866) and 120575(119866) = min119889119866(119909) 119909 isin 119881(119866) Note that the

definitions here are different from the ones in the textbookon digraphs see for example Xu [1]

A subset 119878 of119881(119866) is called a dominating set if119881(119866) = 119878cup119873

+

119866(119878) where119873+

119866(119878) is the set of out-neighbors of 119878Then just

as for undirected graphs 120574(119866) is the minimum cardinalityof a dominating set and the bondage number 119887(119866) is thesmallest cardinality of a set 119861 of edges such that 120574(119866 minus 119861) gt120574(119866) if such a subset 119861 sube 119864(119866) exists Otherwise we put119887(119866) = infin

Some basic results for undirected graphs stated inSection 3 can be generalized to digraphs For exampleTheorem 18 is generalized by Carlson and Develin [42]Huang andXu [133] and Shan andKang [132] independentlyas follows

Theorem 180 Let 119866 be a digraph and (119909 119910) isin 119864(119866) Then119887(119866) ⩽ 119889

119866(119910) + 119889

minus

119866(119909) minus |119873

minus

119866(119909) cap 119873

minus

119866(119910)|

Corollary 181 For a digraph 119866 119887(119866) ⩽ 120575minus(119866) + Δ(119866)

Since 120575minus

(119866) ⩽ (12)Δ(119866) from Corollary 181Conjecture 53 is valid for digraphs

Corollary 182 For a digraph 119866 119887(119866) ⩽ (32)Δ(119866)

In the case of undirected graphs the bondage number of119866119899= 119870

119899times 119870

119899achieves this bound However it was shown

by Carlson and Develin in [42] that if we take the symmetricdigraph119866

119899 we haveΔ(119866

119899) = 4(119899minus1) 120574(119866

119899) = 119899 and 119887(119866

119899) ⩽

4(119899 minus 1) + 1 = Δ(119866119899) + 1 So this family of digraphs cannot

show that the bound in Corollary 182 is tightCorresponding to Conjecture 51 which is discredited

for undirected graphs and is valid for digraphs the sameconjecture for digraphs can be proposed as follows

Conjecture 183 (Carlson and Develin [42] 2006) 119887(119866) ⩽Δ(119866) + 1 for any digraph 119866

If Conjecture 183 is true then the following results showsthat this upper bound is tight since 119887(119870

119899times119870

119899) = Δ(119870

119899times119870

119899)+1

for a complete digraph 119870119899

In 2007 Shan and Kang [132] gave some tight upperbounds on the bondage numbers for some asymmetricdigraphs For example 119887(119879) ⩽ Δ(119879) for any asymmetricdirected tree 119879 119887(119866) ⩽ Δ(119866) for any asymmetric digraph 119866with order at least 4 and 120574(119866) ⩽ 2 For planar digraphs theyobtained the following results

Theorem 184 (Shan and Kang [132] 2007) Let 119866 be aasymmetric planar digraphThen 119887(119866) ⩽ Δ(119866)+2 and 119887(119866) ⩽Δ(119866) + 1 if Δ(119866) ⩾ 5 and 119889minus

119866(119909) ⩾ 3 for every vertex 119909 with

119889119866(119909) ⩾ 4

102 Results for Some Special Digraphs The exact values andbounds on 119887(119866) for some standard digraphs are determined

Theorem185 (Huang andXu [133] 2006) For a directed cycle119862119899and a directed path 119875

119899

119887 (119862119899) =

3 119894119891 119899 119894119904 119900119889119889

2 119894119891 119899 119894119904 119890V119890119899

119887 (119875119899) =

2 119894119891 119899 119894119904 119900119889119889

1 119894119891 119899 119894119904 119890V119890119899

(76)

For the de Bruijn digraph 119861(119889 119899) and the Kautz digraph119870(119889 119899)

119887 (119861 (119889 119899)) = 119889 119894119891 119899 119894119904 119900119889119889

119889 ⩽ 119887 (119861 (119889 119899)) ⩽ 2119889 119894119891 119899 119894119904 119890V119890119899

119887 (119870 (119889 119899)) = 119889 + 1

(77)

Like undirected graphs we can define the total domi-nation number and the total bondage number On the totalbondage numbers for some special digraphs the knownresults are as follows

International Journal of Combinatorics 29

Theorem 186 (Huang and Xu [134] 2007) For a directedcycle 119862

119899and a directed path 119875

119899 119887

119879(119875

119899) and 119887

119879(119862

119899) all do not

exist For a complete digraph 119870119899

119887119879(119870

119899) = infin 119894119891 119899 = 1 2

119887119879(119870

119899) = 3 119894119891 119899 = 3

119899 ⩽ 119887119879(119870

119899) ⩽ 2119899 minus 3 119894119891 119899 ⩾ 4

(78)

The extended de Bruijn digraph EB(119889 119899 1199021 119902

119901) and

the extended Kautz digraph EK(119889 119899 1199021 119902

119901) are intro-

duced by Shibata and Gonda [135] If 119901 = 1 then theyare the de Bruijn digraph B(119889 119899) and the Kautz digraphK(119889 119899) respectively Huang and Xu [134] determined theirtotal domination numbers In particular their total bondagenumbers for general cases are determined as follows

Theorem 187 (Huang andXu [134] 2007) If 119889 ⩾ 2 and 119902119894⩾ 2

for each 119894 = 1 2 119901 then

119887119879(EB(119889 119899 119902

1 119902

119901)) = 119889

119901

minus 1

119887119879(EK(119889 119899 119902

1 119902

119901)) = 119889

119901

(79)

In particular for the de Bruijn digraph B(119889 119899) and the Kautzdigraph K(119889 119899)

119887119879(B (119889 119899)) = 119889 minus 1 119887

119879(K (119889 119899)) = 119889 (80)

Zhang et al [136] determined the bondage number incomplete 119905-partite digraphs

Theorem 188 (Zhang et al [136] 2009) For a complete 119905-partite digraph 119870

11989911198992119899119905

where 1198991⩽ 119899

2⩽ sdot sdot sdot ⩽ 119899

119905

119887 (11987011989911198992119899119905

)

=

119898 119894119891 119899119898= 1 119899

119898+1⩾ 2 119891119900119903 119904119900119898119890

119898 (1 ⩽ 119898 lt 119905)

4119905 minus 3 119894119891 1198991= 119899

2= sdot sdot sdot = 119899

119905= 2

119905minus1

sum

119894=1

119899119894+ 2 (119905 minus 1) 119900119905ℎ119890119903119908119894119904119890

(81)

Since an undirected graph can be thought of as a sym-metric digraph any result for digraphs has an analogy forundirected graphs in general In view of this point studyingthe bondage number for digraphs is more significant thanfor undirected graphs Thus we should further study thebondage number of digraphs and try to generalize knownresults on the bondage number and related variants for undi-rected graphs to digraphs prove or disprove Conjecture 183In particular determine the exact values of 119887(B(119889 119899)) for aneven 119899 and 119887

119879(119870

119899) for 119899 ⩾ 4

11 Efficient Dominating Sets

A dominating set 119878 of a graph 119866 is called to be efficient if forevery vertex 119909 in119866 |119873

119866[119909]cap119878| = 1 if119866 is a undirected graph

or |119873minus

119866[119909] cap 119878| = 1 if 119866 is a directed graph The concept of an

efficient set was proposed by Bange et al [137] in 1988 In theliterature an efficient set is also called a perfect dominatingset (see Livingston and Stout [138] for an explanation)

From definition if 119878 is an efficient dominating set of agraph 119866 then 119878 is certainly an independent set and everyvertex not in 119878 is adjacent to exactly one vertex in 119878

It is also clear from definition that a dominating set 119878 isefficient if and only if N

119866[119878] = 119873

119866[119909] 119909 isin 119878 for the

undirected graph 119866 or N+

119866[119878] = 119873

+

119866[119909] 119909 isin 119878 for the

digraph119866 is a partition of119881(119866) where the induced subgraphby119873

119866[119909] or119873+

119866[119909] is a star or an out-star with the root 119909

The efficient domination has important applications inmany areas such as error-correcting codes and receivesmuch attention in the late years

The concept of efficient dominating sets is a measureof the efficiency of domination in graphs Unfortunately asshown in [137] not every graph has an efficient dominatingset andmoreover it is anNP-complete problem to determinewhether a given graph has an efficient dominating set Inaddition it was shown by Clark [139] in 1993 that for a widerange of 119901 almost every random undirected graph 119866 isin

G(120592 119901) has no efficient dominating sets This means thatundirected graphs possessing an efficient dominating set arerare However it is easy to show that every undirected graphhas an orientationwith an efficient dominating set (see Bangeet al [140])

In 1993 Barkauskas and Host [141] showed that deter-mining whether an arbitrary oriented graph has an efficientdominating set is NP-complete Even so the existence ofefficient dominating sets for some special classes of graphs hasbeen examined see for example Dejter and Serra [142] andLee [143] for Cayley graph Gu et al [144] formeshes and toriObradovic et al [145] Huang and Xu [36] Reji Kumar andMacGillivray [146] for circulant graphs Harary graphs andtori and Van Wieren et al [147] for cube-connected cycles

In this section we introduce results of the bondagenumber for some graphs with efficient dominating sets dueto Huang and Xu [36]

111 Results for General Graphs In this subsection we intro-duce some results on bondage numbers obtained by applyingefficient dominating sets We first state the two followinglemmas

Lemma 189 For any 119896-regular graph or digraph 119866 of order 119899120574(119866) ⩾ 119899(119896 + 1) with equality if and only if 119866 has an efficientdominating set In addition if 119866 has an efficient dominatingset then every efficient dominating set is certainly a 120574-set andvice versa

Let 119890 be an edge and 119878 a dominating set in 119866 We say that119890 supports 119878 if 119890 isin (119878 119878) where (119878 119878) = (119909 119910) isin 119864(119866) 119909 isin 119878 119910 isin 119878 Denote by 119904(119866) the minimum number of edgeswhich support all 120574-sets in 119866

Lemma 190 For any graph or digraph 119866 119887(119866) ⩾ 119904(119866) withequality if 119866 is regular and has an efficient dominating set

30 International Journal of Combinatorics

A graph 119866 is called to be vertex-transitive if its automor-phism group Aut(119866) acts transitively on its vertex-set 119881(119866)A vertex-transitive graph is regular Applying Lemmas 189and 190 Huang and Xu obtained some results on bondagenumbers for vertex-transitive graphs or digraphs

Theorem 191 (Huang and Xu [36] 2008) For any vertex-transitive graph or digraph 119866 of order 119899

119887 (119866) ⩾

lceil119899

2120574 (119866)rceil 119894119891 119866 119894119904 119906119899119889119894119903119890119888119905119890119889

lceil119899

120574 (119866)rceil 119894119891 119866 119894119904 119889119894119903119890119888119905119890119889

(82)

Theorem192 (Huang andXu [36] 2008) Let119866 be a 119896-regulargraph of order 119899 Then

119887(119866) ⩽ 119896 if119866 is undirected and 119899 ⩾ 120574(119866)(119896+1)minus119896+1119887(119866) ⩽ 119896+1+ℓ if119866 is directed and 119899 ⩾ 120574(119866)(119896+1)minusℓwith 0 ⩽ ℓ ⩽ 119896 minus 1

Next we introduce a better upper bound on 119887(119866) To thisaim we need the following concept which is a generalizationof the concept of the edge-covering of a graph 119866 For 1198811015840

sube

119881(119866) and 1198641015840 sube 119864(119866) we say that 1198641015840covers 1198811015840 and call 1198641015840an edge-covering for 1198811015840 if there exists an edge (119909 119910) isin 1198641015840 forany vertex 119909 isin 1198811015840 For 119910 isin 119881(119866) let 1205731015840[119910] be the minimumcardinality over all edge-coverings for119873minus

119866[119910]

Theorem 193 (Huang and Xu [36] 2008) If 119866 is a 119896-regulargraph with order 120574(119866)(119896 + 1) then 119887(119866) ⩽ 1205731015840[119910] for any 119910 isin119881(119866)

Theupper bound on 119887(119866) given inTheorem 193 is tight inview of 119887(119862

119899) for a cycle or a directed cycle 119862

119899(seeTheorems

1 and 185 resp)It is easy to see that for a 119896-regular graph 119866 lceil(119896 + 1)2rceil ⩽

1205731015840

[119910] ⩽ 119896 when 119866 is undirected and 1205731015840[119910] = 119896 + 1 when 119866 isdirected By this fact and Lemma 189 the following theoremis merely a simple combination of Theorems 191 and 193

Theorem 194 (Huang and Xu [36] 2008) Let 119866 be a vertex-transitive graph of degree 119896 If 119866 has an efficient dominatingset then

lceil119896 + 1

2rceil ⩽ 119887 (119866) ⩽ 119896 119894119891 119866 119894119904 119906119899119889119894119903119890119888119905119890119889

119887 (119866) = 119896 + 1 119894119891 119866 119894119904 119889119894119903119890119888119905119890119889

(83)

Theorem 195 (Huang and Xu [36] 2008) If 119866 is an undi-rected vertex-transitive cubic graph with order 4120574(119866) and girth119892(119866) ⩽ 5 then 119887(119866) = 2

The proof of Theorem 195 leads to a byproduct If 119892 =5 let (119906

1 119906

2 119906

3 119906

4 119906

5) be a cycle and let D

119894be the family

of efficient dominating sets containing 119906119894for each 119894 isin

1 2 3 4 5 ThenD119894capD

119895= 0 for 119894 = 1 2 5 Then 119866 has

at least 5119904 efficient dominating sets where 119904 = 119904(119866) But thereare only 4s (=ns120574) distinct efficient dominating sets in119866This

contradiction implies that an undirected vertex-transitivecubic graph with girth five has no efficient dominating setsBut a similar argument for 119892(119866) = 3 4 or 119892(119866) ⩾ 6 couldnot give any contradiction This is consistent with the resultthat CCC(119899) a vertex-transitive cubic graph with girth 119899 if3 ⩽ 119899 ⩽ 8 or girth 8 if 119899 ⩾ 9 has efficient dominating setsfor all 119899 ⩾ 3 except 119899 = 5 (see Theorem 199 in the followingsubsection)

112 Results for Cayley Graphs In this subsection we useTheorem 194 to determine the exact values or approximativevalues of bondage numbers for some special vertex-transitivegraphs by characterizing the existence of efficient dominatingsets in these graphs

Let Γ be a nontrivial finite group 119878 a nonempty subset ofΓ without the identity element of Γ A digraph 119866 is defined asfollows

119881 (119866) = Γ (119909 119910) isin 119864 (119866) lArrrArr 119909minus1

119910 isin 119878

for any 119909 119910 isin Γ(84)

is called a Cayley digraph of the group Γ with respect to119878 denoted by 119862

Γ(119878) If 119878minus1 = 119904minus1 119904 isin 119878 = 119878 then

119862Γ(119878) is symmetric and is called a Cayley undirected graph

a Cayley graph for short Cayley graphs or digraphs arecertainly vertex-transitive (see Xu [1])

A circulant graph 119866(119899 119878) of order 119899 is a Cayley graph119862119885119899

(119878) where 119885119899= 0 119899 minus 1 is the addition group of

order 119899 and 119878 is a nonempty subset of119885119899without the identity

element and hence is a vertex-transitive digraph of degree|119878| If 119878minus1 = 119878 then119866(119899 119878) is an undirected graph If 119878 = 1 119896where 2 ⩽ 119896 ⩽ 119899 minus 2 we write 119866(119899 1 119896) for 119866(119899 1 119896) or119866(119899 plusmn1 plusmn119896) and call it a double-loop circulant graph

Huang and Xu [36] showed that for directed 119866 =

119866(119899 1 119896) lceil1198993rceil ⩽ 120574(119866) ⩽ lceil1198992rceil and 119866 has an efficientdominating set if and only if 3 | 119899 and 119896 equiv 2 (mod 3) fordirected 119866 = 119866(119899 1 119896) and 119896 = 1198992 lceil1198995rceil ⩽ 120574(119866) ⩽ lceil1198993rceiland 119866 has an efficient dominating set if and only if 5 | 119899 and119896 equiv plusmn2 (mod 5) By Theorem 194 we obtain the bondagenumber of a double loop circulant graph if it has an efficientdominating set

Theorem 196 (Huang and Xu [36] 2008) Let 119866 be a double-loop circulant graph 119866(119899 1 119896) If 119866 is directed with 3 | 119899and 119896 equiv 2 (mod 3) or 119866 is undirected with 5 | 119899 and119896 equiv plusmn2 (mod 5) then 119887(119866) = 3

The 119898 times 119899 torus is the Cartesian product 119862119898times 119862

119899of

two cycles and is a Cayley graph 119862119885119898times119885119899

(119878) where 119878 =

(0 1) (1 0) for directed cycles and 119878 = (0 plusmn1) (plusmn1 0) forundirected cycles and hence is vertex-transitive Gu et al[144] showed that the undirected torus119862

119898times119862

119899has an efficient

dominating set if and only if both 119898 and 119899 are multiplesof 5 Huang and Xu [36] showed that the directed torus119862119898times 119862

119899has an efficient dominating set if and only if both

119898 and 119899 are multiples of 3 Moreover they found a necessarycondition for a dominating set containing the vertex (0 0)in 119862

119898times 119862

119899to be efficient and obtained the following

result

International Journal of Combinatorics 31

Theorem 197 (Huang and Xu [36] 2008) Let 119866 = 119862119898times 119862

119899

If 119866 is undirected and both119898 and 119899 are multiples of 5 or if 119866 isdirected and both119898 and 119899 are multiples of 3 then 119887(119866) = 3

The hypercube 119876119899

is the Cayley graph 119862Γ(119878)

where Γ = 1198852times sdot sdot sdot times 119885

2= (119885

2)119899 and 119878 =

100 sdot sdot sdot 0 010 sdot sdot sdot 0 00 sdot sdot sdot 01 Lee [143] showed that119876119899has an efficient dominating set if and only if 119899 = 2119898 minus 1

for a positive integer119898 ByTheorem 194 the following resultis obtained immediately

Theorem 198 (Huang and Xu [36] 2008) If 119899 = 2119898 minus 1 for apositive integer119898 then 2119898minus1

⩽ 119887(119876119899) ⩽ 2

119898

minus 1

Problem 10 Determine the exact value of 119887(119876119899) for 119899 = 2119898minus1

The 119899-dimensional cube-connected cycle denoted byCCC(119899) is constructed from the 119899-dimensional hypercube119876119899by replacing each vertex 119909 in 119876

119899with an undirected cycle

119862119899of length 119899 and linking the 119894th vertex of the 119862

119899to the 119894th

neighbor of 119909 It has been proved that CCC(119899) is a Cayleygraph and hence is a vertex-transitive graph with degree 3Van Wieren et al [147] proved that CCC(119899) has an efficientdominating set if and only if 119899 = 5 From Theorems 194 and195 the following result can be derived

Theorem 199 (Huang and Xu [36] 2008) Let 119866 = 119862119862119862(119899)be the 119899-dimensional cube-connected cycles with 119899 ⩾ 3 and119899 = 5 Then 120574(119866) = 1198992119899minus2 and 2 ⩽ 119887(119866) ⩽ 3 In addition119887(119862119862119862(3)) = 119887(119862119862119862(4)) = 2

Problem 11 Determine the exact value of 119887(CCC(119899)) for 119899 ⩾5

Acknowledgments

This work was supported by NNSF of China (Grants nos11071233 61272008) The author would like to express hisgratitude to the anonymous referees for their kind sugges-tions and comments on the original paper to ProfessorSheikholeslami and Professor Jafari Rad for sending thereferences [128] and [81] which resulted in this version

References

[1] J-M Xu Theory and Application of Graphs vol 10 of NetworkTheory and Applications Kluwer Academic Dordrecht TheNetherlands 2003

[2] S THedetniemi andR C Laskar ldquoBibliography on dominationin graphs and some basic definitions of domination parame-tersrdquo Discrete Mathematics vol 86 no 1ndash3 pp 257ndash277 1990

[3] TW Haynes S T Hedetniemi and P J Slater Fundamentals ofDomination in Graphs vol 208 ofMonographs and Textbooks inPure and Applied Mathematics Marcel Dekker New York NYUSA 1998

[4] TWHaynes S THedetniemi and P J Slater EdsDominationin Graphs vol 209 of Monographs and Textbooks in Pure andAppliedMathematicsMarcelDekker NewYorkNYUSA 1998

[5] M R Garey andD S JohnsonComputers and Intractability WH Freeman and Company San Francisco Calif USA 1979

[6] H B Walikar and B D Acharya ldquoDomination critical graphsrdquoNational Academy Science Letters vol 2 pp 70ndash72 1979

[7] D Bauer F Harary J Nieminen and C L Suffel ldquoDominationalteration sets in graphsrdquo Discrete Mathematics vol 47 no 2-3pp 153ndash161 1983

[8] J F Fink M S Jacobson L F Kinch and J Roberts ldquoThebondage number of a graphrdquo Discrete Mathematics vol 86 no1ndash3 pp 47ndash57 1990

[9] F-T Hu and J-M Xu ldquoThe bondage number of (119899 minus 3)-regulargraph G of order nrdquo Ars Combinatoria to appear

[10] U Teschner ldquoNew results about the bondage number of agraphrdquoDiscreteMathematics vol 171 no 1ndash3 pp 249ndash259 1997

[11] B L Hartnell and D F Rall ldquoA bound on the size of a graphwith given order and bondage numberrdquo Discrete Mathematicsvol 197-198 pp 409ndash413 1999 16th British CombinatorialConference (London 1997)

[12] B L Hartnell and D F Rall ldquoA characterization of trees inwhich no edge is essential to the domination numberrdquo ArsCombinatoria vol 33 pp 65ndash76 1992

[13] J Topp and P D Vestergaard ldquo120572119896- and 120574

119896-stable graphsrdquo

Discrete Mathematics vol 212 no 1-2 pp 149ndash160 2000[14] A Meir and J W Moon ldquoRelations between packing and

covering numbers of a treerdquo Pacific Journal of Mathematics vol61 no 1 pp 225ndash233 1975

[15] B L Hartnell L K Joslashrgensen P D Vestergaard and CA Whitehead ldquoEdge stability of the k-domination numberof treesrdquo Bulletin of the Institute of Combinatorics and ItsApplications vol 22 pp 31ndash40 1998

[16] F-T Hu and J-M Xu ldquoOn the complexity of the bondage andreinforcement problemsrdquo Journal of Complexity vol 28 no 2pp 192ndash201 2012

[17] B L Hartnell and D F Rall ldquoBounds on the bondage numberof a graphrdquo Discrete Mathematics vol 128 no 1ndash3 pp 173ndash1771994

[18] Y-L Wang ldquoOn the bondage number of a graphrdquo DiscreteMathematics vol 159 no 1ndash3 pp 291ndash294 1996

[19] G H Fricke TW Haynes S M Hedetniemi S T Hedetniemiand R C Laskar ldquoExcellent treesrdquo Bulletin of the Institute ofCombinatorics and Its Applications vol 34 pp 27ndash38 2002

[20] V Samodivkin ldquoThe bondage number of graphs good and badverticesrdquoDiscussiones Mathematicae GraphTheory vol 28 no3 pp 453ndash462 2008

[21] J E Dunbar T W Haynes U Teschner and L VolkmannldquoBondage insensitivity and reinforcementrdquo in Domination inGraphs vol 209 of Monographs and Textbooks in Pure andAppliedMathematics pp 471ndash489 Dekker NewYork NY USA1998

[22] H Liu and L Sun ldquoThe bondage and connectivity of a graphrdquoDiscrete Mathematics vol 263 no 1ndash3 pp 289ndash293 2003

[23] A-H Esfahanian and S L Hakimi ldquoOn computing a con-ditional edge-connectivity of a graphrdquo Information ProcessingLetters vol 27 no 4 pp 195ndash199 1988

[24] R C Brigham P Z Chinn and R D Dutton ldquoVertexdomination-critical graphsrdquoNetworks vol 18 no 3 pp 173ndash1791988

[25] V Samodivkin ldquoDomination with respect to nondegenerateproperties bondage numberrdquoThe Australasian Journal of Com-binatorics vol 45 pp 217ndash226 2009

[26] U Teschner ldquoA new upper bound for the bondage numberof graphs with small domination numberrdquo The AustralasianJournal of Combinatorics vol 12 pp 27ndash35 1995

32 International Journal of Combinatorics

[27] J Fulman D Hanson and G MacGillivray ldquoVertex dom-ination-critical graphsrdquoNetworks vol 25 no 2 pp 41ndash43 1995

[28] U Teschner ldquoA counterexample to a conjecture on the bondagenumber of a graphrdquo Discrete Mathematics vol 122 no 1ndash3 pp393ndash395 1993

[29] U Teschner ldquoThe bondage number of a graph G can be muchgreater than Δ(G)rdquo Ars Combinatoria vol 43 pp 81ndash87 1996

[30] L Volkmann ldquoOn graphs with equal domination and coveringnumbersrdquoDiscrete AppliedMathematics vol 51 no 1-2 pp 211ndash217 1994

[31] L A Sanchis ldquoMaximum number of edges in connected graphswith a given domination numberrdquoDiscreteMathematics vol 87no 1 pp 65ndash72 1991

[32] S Klavzar and N Seifter ldquoDominating Cartesian products ofcyclesrdquoDiscrete AppliedMathematics vol 59 no 2 pp 129ndash1361995

[33] H K Kim ldquoA note on Vizingrsquos conjecture about the bondagenumberrdquo Korean Journal of Mathematics vol 10 no 2 pp 39ndash42 2003

[34] M Y Sohn X Yuan and H S Jeong ldquoThe bondage number of1198623times119862

119899rdquo Journal of the KoreanMathematical Society vol 44 no

6 pp 1213ndash1231 2007[35] L Kang M Y Sohn and H K Kim ldquoBondage number of the

discrete torus 119862119899times 119862

4rdquo Discrete Mathematics vol 303 no 1ndash3

pp 80ndash86 2005[36] J Huang and J-M Xu ldquoThe bondage numbers and efficient

dominations of vertex-transitive graphsrdquoDiscrete Mathematicsvol 308 no 4 pp 571ndash582 2008

[37] C J Xiang X Yuan andM Y Sohn ldquoDomination and bondagenumber of119862

3times119862

119899rdquoArs Combinatoria vol 97 pp 299ndash310 2010

[38] M S Jacobson and L F Kinch ldquoOn the domination numberof products of graphs Irdquo Ars Combinatoria vol 18 pp 33ndash441984

[39] Y-Q Li ldquoA note on the bondage number of a graphrdquo ChineseQuarterly Journal of Mathematics vol 9 no 4 pp 1ndash3 1994

[40] F-T Hu and J-M Xu ldquoBondage number of mesh networksrdquoFrontiers ofMathematics inChina vol 7 no 5 pp 813ndash826 2012

[41] R Frucht and F Harary ldquoOn the corona of two graphsrdquoAequationes Mathematicae vol 4 pp 322ndash325 1970

[42] K Carlson and M Develin ldquoOn the bondage number of planarand directed graphsrdquoDiscreteMathematics vol 306 no 8-9 pp820ndash826 2006

[43] U Teschner ldquoOn the bondage number of block graphsrdquo ArsCombinatoria vol 46 pp 25ndash32 1997

[44] J Huang and J-M Xu ldquoNote on conjectures of bondagenumbers of planar graphsrdquo Applied Mathematical Sciences vol6 no 65ndash68 pp 3277ndash3287 2012

[45] L Kang and J Yuan ldquoBondage number of planar graphsrdquoDiscrete Mathematics vol 222 no 1ndash3 pp 191ndash198 2000

[46] M Fischermann D Rautenbach and L Volkmann ldquoRemarkson the bondage number of planar graphsrdquo Discrete Mathemat-ics vol 260 no 1ndash3 pp 57ndash67 2003

[47] J Hou G Liu and J Wu ldquoBondage number of planar graphswithout small cyclesrdquoUtilitasMathematica vol 84 pp 189ndash1992011

[48] J Huang and J-M Xu ldquoThe bondage numbers of graphs withsmall crossing numbersrdquo Discrete Mathematics vol 307 no 15pp 1881ndash1897 2007

[49] Q Ma S Zhang and J Wang ldquoBondage number of 1-planargraphrdquo Applied Mathematics vol 1 no 2 pp 101ndash103 2010

[50] J Hou and G Liu ldquoA bound on the bondage number of toroidalgraphsrdquoDiscreteMathematics Algorithms and Applications vol4 no 3 Article ID 1250046 5 pages 2012

[51] Y-C Cao J-M Xu and X-R Xu ldquoOn bondage number oftoroidal graphsrdquo Journal of University of Science and Technologyof China vol 39 no 3 pp 225ndash228 2009

[52] Y-C Cao J Huang and J-M Xu ldquoThe bondage number ofgraphs with crossing number less than fourrdquo Ars Combinatoriato appear

[53] H K Kim ldquoOn the bondage numbers of some graphsrdquo KoreanJournal of Mathematics vol 11 no 2 pp 11ndash15 2004

[54] A Gagarin and V Zverovich ldquoUpper bounds for the bondagenumber of graphs on topological surfacesrdquo Discrete Mathemat-ics 2012

[55] J Huang ldquoAn improved upper bound for the bondage numberof graphs on surfacesrdquoDiscrete Mathematics vol 312 no 18 pp2776ndash2781 2012

[56] A Gagarin and V Zverovich ldquoThe bondage number of graphson topological surfaces and Teschnerrsquos conjecturerdquo DiscreteMathematics vol 313 no 6 pp 796ndash808 2013

[57] V Samodivkin ldquoNote on the bondage number of graphs ontopological surfacesrdquo httparxivorgabs12086203

[58] E J Cockayne R M Dawes and S T Hedetniemi ldquoTotaldomination in graphsrdquoNetworks vol 10 no 3 pp 211ndash219 1980

[59] R Laskar J Pfaff S M Hedetniemi and S T HedetniemildquoOn the algorithmic complexity of total dominationrdquo Society forIndustrial and Applied Mathematics vol 5 no 3 pp 420ndash4251984

[60] J Pfaff R C Laskar and S T Hedetniemi ldquoNP-completeness oftotal and connected domination and irredundance for bipartitegraphsrdquo Tech Rep 428 Department of Mathematical SciencesClemson University 1983

[61] M A Henning ldquoA survey of selected recent results on totaldomination in graphsrdquoDiscrete Mathematics vol 309 no 1 pp32ndash63 2009

[62] V R Kulli and D K Patwari ldquoThe total bondage number ofa graphrdquo in Advances in Graph Theory pp 227ndash235 VishwaGulbarga India 1991

[63] F-T Hu Y Lu and J-M Xu ldquoThe total bondage number ofgrid graphsrdquo Discrete Applied Mathematics vol 160 no 16-17pp 2408ndash2418 2012

[64] J Cao M Shi and B Wu ldquoResearch on the total bondagenumber of a spectial networkrdquo in Proceedings of the 3rdInternational Joint Conference on Computational Sciences andOptimization (CSO rsquo10) pp 456ndash458 May 2010

[65] N Sridharan MD Elias and V S A Subramanian ldquoTotalbondage number of a graphrdquo AKCE International Journal ofGraphs and Combinatorics vol 4 no 2 pp 203ndash209 2007

[66] N Jafari Rad and J Raczek ldquoSome progress on total bondage ingraphsrdquo Graphs and Combinatorics 2011

[67] T W Haynes and P J Slater ldquoPaired-domination and thepaired-domatic numberrdquoCongressusNumerantium vol 109 pp65ndash72 1995

[68] T W Haynes and P J Slater ldquoPaired-domination in graphsrdquoNetworks vol 32 no 3 pp 199ndash206 1998

[69] X Hou ldquoA characterization of 2120574 120574119875-treesrdquoDiscrete Mathemat-

ics vol 308 no 15 pp 3420ndash3426 2008[70] K E Proffitt T W Haynes and P J Slater ldquoPaired-domination

in grid graphsrdquo Congressus Numerantium vol 150 pp 161ndash1722001

International Journal of Combinatorics 33

[71] H Qiao L KangM Cardei andD-Z Du ldquoPaired-dominationof treesrdquo Journal of Global Optimization vol 25 no 1 pp 43ndash542003

[72] E Shan L Kang and M A Henning ldquoA characterizationof trees with equal total domination and paired-dominationnumbersrdquo The Australasian Journal of Combinatorics vol 30pp 31ndash39 2004

[73] J Raczek ldquoPaired bondage in treesrdquo Discrete Mathematics vol308 no 23 pp 5570ndash5575 2008

[74] J A Telle and A Proskurowski ldquoAlgorithms for vertex parti-tioning problems on partial k-treesrdquo SIAM Journal on DiscreteMathematics vol 10 no 4 pp 529ndash550 1997

[75] G S Domke J H Hattingh M A Henning and L R MarkusldquoRestrained domination in treesrdquoDiscreteMathematics vol 211no 1ndash3 pp 1ndash9 2000

[76] G S Domke J H Hattingh M A Henning and L R MarkusldquoRestrained domination in graphs with minimum degree twordquoJournal of Combinatorial Mathematics and Combinatorial Com-puting vol 35 pp 239ndash254 2000

[77] G S Domke J H Hattingh S T Hedetniemi R C Laskarand L R Markus ldquoRestrained domination in graphsrdquo DiscreteMathematics vol 203 no 1ndash3 pp 61ndash69 1999

[78] J H Hattingh and E J Joubert ldquoAn upper bound for therestrained domination number of a graph with minimumdegree at least two in terms of order and minimum degreerdquoDiscrete Applied Mathematics vol 157 no 13 pp 2846ndash28582009

[79] M A Henning ldquoGraphs with large restrained dominationnumberrdquo Discrete Mathematics vol 197-198 pp 415ndash429 199916th British Combinatorial Conference (London 1997)

[80] J H Hattingh and A R Plummer ldquoRestrained bondage ingraphsrdquo Discrete Mathematics vol 308 no 23 pp 5446ndash54532008

[81] N Jafari Rad R Hasni J Raczek and L Volkmann ldquoTotalrestrained bondage in graphsrdquoActaMathematica Sinica vol 29no 6 pp 1033ndash1042 2013

[82] J Cyman and J Raczek ldquoOn the total restrained dominationnumber of a graphrdquoThe Australasian Journal of Combinatoricsvol 36 pp 91ndash100 2006

[83] M A Henning ldquoGraphs with large total domination numberrdquoJournal of Graph Theory vol 35 no 1 pp 21ndash45 2000

[84] D-X Ma X-G Chen and L Sun ldquoOn total restraineddomination in graphsrdquoCzechoslovakMathematical Journal vol55 no 1 pp 165ndash173 2005

[85] B Zelinka ldquoRemarks on restrained domination and totalrestrained domination in graphsrdquo Czechoslovak MathematicalJournal vol 55 no 2 pp 393ndash396 2005

[86] W Goddard and M A Henning ldquoIndependent domination ingraphs a survey and recent resultsrdquo Discrete Mathematics vol313 no 7 pp 839ndash854 2013

[87] J H Zhang H L Liu and L Sun ldquoIndependence bondagenumber and reinforcement number of some graphsrdquo Transac-tions of Beijing Institute of Technology vol 23 no 2 pp 140ndash1422003

[88] K D Dharmalingam Studies in Graph Theorymdashequitabledomination and bottleneck domination [PhD thesis] MaduraiKamaraj University Madurai India 2006

[89] GDeepakND Soner andAAlwardi ldquoThe equitable bondagenumber of a graphrdquo Research Journal of Pure Algebra vol 1 no9 pp 209ndash212 2011

[90] E Sampathkumar and H B Walikar ldquoThe connected domina-tion number of a graphrdquo Journal of Mathematical and PhysicalSciences vol 13 no 6 pp 607ndash613 1979

[91] K White M Farber and W Pulleyblank ldquoSteiner trees con-nected domination and strongly chordal graphsrdquoNetworks vol15 no 1 pp 109ndash124 1985

[92] M Cadei M X Cheng X Cheng and D-Z Du ldquoConnecteddomination in Ad Hoc wireless networksrdquo in Proceedings ofthe 6th International Conference on Computer Science andInformatics (CSampI rsquo02) 2002

[93] J F Fink and M S Jacobson ldquon-domination in graphsrdquo inGraph Theory with Applications to Algorithms and ComputerScience Wiley-Interscience Publications pp 283ndash300 WileyNew York NY USA 1985

[94] M Chellali O Favaron A Hansberg and L Volkmann ldquok-domination and k-independence in graphs a surveyrdquo Graphsand Combinatorics vol 28 no 1 pp 1ndash55 2012

[95] Y Lu X Hou J-M Xu andN Li ldquoTrees with uniqueminimump-dominating setsrdquo Utilitas Mathematica vol 86 pp 193ndash2052011

[96] Y Lu and J-M Xu ldquoThe p-bondage number of treesrdquo Graphsand Combinatorics vol 27 no 1 pp 129ndash141 2011

[97] Y Lu and J-M Xu ldquoThe 2-domination and 2-bondage numbersof grid graphs A manuscriptrdquo httparxivorgabs12044514

[98] M Krzywkowski ldquoNon-isolating 2-bondage in graphsrdquo TheJournal of the Mathematical Society of Japan vol 64 no 4 2012

[99] M A Henning O R Oellermann and H C Swart ldquoBoundson distance domination parametersrdquo Journal of CombinatoricsInformation amp System Sciences vol 16 no 1 pp 11ndash18 1991

[100] F Tian and J-M Xu ldquoBounds for distance domination numbersof graphsrdquo Journal of University of Science and Technology ofChina vol 34 no 5 pp 529ndash534 2004

[101] F Tian and J-M Xu ldquoDistance domination-critical graphsrdquoApplied Mathematics Letters vol 21 no 4 pp 416ndash420 2008

[102] F Tian and J-M Xu ldquoA note on distance domination numbersof graphsrdquo The Australasian Journal of Combinatorics vol 43pp 181ndash190 2009

[103] F Tian and J-M Xu ldquoAverage distances and distance domina-tion numbersrdquo Discrete Applied Mathematics vol 157 no 5 pp1113ndash1127 2009

[104] Z-L Liu F Tian and J-M Xu ldquoProbabilistic analysis of upperbounds for 2-connected distance k-dominating sets in graphsrdquoTheoretical Computer Science vol 410 no 38ndash40 pp 3804ndash3813 2009

[105] M A Henning O R Oellermann and H C Swart ldquoDistancedomination critical graphsrdquo Journal of Combinatorial Mathe-matics and Combinatorial Computing vol 44 pp 33ndash45 2003

[106] T J Bean M A Henning and H C Swart ldquoOn the integrityof distance domination in graphsrdquo The Australasian Journal ofCombinatorics vol 10 pp 29ndash43 1994

[107] M Fischermann and L Volkmann ldquoA remark on a conjecturefor the (kp)-domination numberrdquoUtilitasMathematica vol 67pp 223ndash227 2005

[108] T Korneffel D Meierling and L Volkmann ldquoA remark on the(22)-domination numberrdquo Discussiones Mathematicae GraphTheory vol 28 no 2 pp 361ndash366 2008

[109] Y Lu X Hou and J-M Xu ldquoOn the (22)-domination numberof treesrdquo Discussiones Mathematicae Graph Theory vol 30 no2 pp 185ndash199 2010

34 International Journal of Combinatorics

[110] H Li and J-M Xu ldquo(dm)-domination numbers of m-connected graphsrdquo Rapports de Recherche 1130 LRI UniversiteParis-Sud 1997

[111] S M Hedetniemi S T Hedetniemi and T V Wimer ldquoLineartime resource allocation algorithms for treesrdquo Tech Rep URI-014 Department of Mathematical Sciences Clemson Univer-sity 1987

[112] M Farber ldquoDomination independent domination and dualityin strongly chordal graphsrdquo Discrete Applied Mathematics vol7 no 2 pp 115ndash130 1984

[113] G S Domke and R C Laskar ldquoThe bondage and reinforcementnumbers of 120574

119891for some graphsrdquoDiscrete Mathematics vol 167-

168 pp 249ndash259 1997[114] G S Domke S T Hedetniemi and R C Laskar ldquoFractional

packings coverings and irredundance in graphsrdquo vol 66 pp227ndash238 1988

[115] E J Cockayne P A Dreyer Jr S M Hedetniemi andS T Hedetniemi ldquoRoman domination in graphsrdquo DiscreteMathematics vol 278 no 1ndash3 pp 11ndash22 2004

[116] I Stewart ldquoDefend the roman empirerdquo Scientific American vol281 pp 136ndash139 1999

[117] C S ReVelle and K E Rosing ldquoDefendens imperiumromanum a classical problem in military strategyrdquoThe Ameri-can Mathematical Monthly vol 107 no 7 pp 585ndash594 2000

[118] A Bahremandpour F-T Hu S M Sheikholeslami and J-MXu ldquoOn the Roman bondage number of a graphrdquo DiscreteMathematics Algorithms and Applications vol 5 no 1 ArticleID 1350001 15 pages 2013

[119] N Jafari Rad and L Volkmann ldquoRoman bondage in graphsrdquoDiscussiones Mathematicae Graph Theory vol 31 no 4 pp763ndash773 2011

[120] K Ebadi and L PushpaLatha ldquoRoman bondage and Romanreinforcement numbers of a graphrdquo International Journal ofContemporary Mathematical Sciences vol 5 pp 1487ndash14972010

[121] N Dehgardi S M Sheikholeslami and L Volkman ldquoOn theRoman k-bondage number of a graphrdquo AKCE InternationalJournal of Graphs and Combinatorics vol 8 pp 169ndash180 2011

[122] S Akbari and S Qajar ldquoA note on Roman bondage number ofgraphsrdquo Ars Combinatoria to Appear

[123] F-T Hu and J-M Xu ldquoRoman bondage numbers of somegraphsrdquo httparxivorgabs11093933

[124] N Jafari Rad and L Volkmann ldquoOn the Roman bondagenumber of planar graphsrdquo Graphs and Combinatorics vol 27no 4 pp 531ndash538 2011

[125] S Akbari M Khatirinejad and S Qajar ldquoA note on the romanbondage number of planar graphsrdquo Graphs and Combinatorics2012

[126] B Bresar M A Henning and D F Rall ldquoRainbow dominationin graphsrdquo Taiwanese Journal of Mathematics vol 12 no 1 pp213ndash225 2008

[127] B L Hartnell and D F Rall ldquoOn dominating the Cartesianproduct of a graph and 120572krdquo Discussiones Mathematicae GraphTheory vol 24 no 3 pp 389ndash402 2004

[128] N Dehgardi S M Sheikholeslami and L Volkman ldquoThe k-rainbow bondage number of a graphrdquo to appear in DiscreteApplied Mathematics

[129] J von Neumann and O Morgenstern Theory of Games andEconomic Behavior Princeton University Press Princeton NJUSA 1944

[130] C BergeTheorıe des Graphes et ses Applications 1958[131] O Ore Theory of Graphs vol 38 American Mathematical

Society Colloquium Publications 1962[132] E Shan and L Kang ldquoBondage number in oriented graphsrdquoArs

Combinatoria vol 84 pp 319ndash331 2007[133] J Huang and J-M Xu ldquoThe bondage numbers of extended

de Bruijn and Kautz digraphsrdquo Computers amp Mathematics withApplications vol 51 no 6-7 pp 1137ndash1147 2006

[134] J Huang and J-M Xu ldquoThe total domination and total bondagenumbers of extended de Bruijn and Kautz digraphsrdquoComputersamp Mathematics with Applications vol 53 no 8 pp 1206ndash12132007

[135] Y Shibata and Y Gonda ldquoExtension of de Bruijn graph andKautz graphrdquo Computers amp Mathematics with Applications vol30 no 9 pp 51ndash61 1995

[136] X Zhang J Liu and JMeng ldquoThebondage number in completet-partite digraphsrdquo Information Processing Letters vol 109 no17 pp 997ndash1000 2009

[137] D W Bange A E Barkauskas and P J Slater ldquoEfficient dom-inating sets in graphsrdquo in Applications of Discrete Mathematicspp 189ndash199 SIAM Philadelphia Pa USA 1988

[138] M Livingston and Q F Stout ldquoPerfect dominating setsrdquoCongressus Numerantium vol 79 pp 187ndash203 1990

[139] L Clark ldquoPerfect domination in random graphsrdquo Journal ofCombinatorial Mathematics and Combinatorial Computing vol14 pp 173ndash182 1993

[140] D W Bange A E Barkauskas L H Host and L H ClarkldquoEfficient domination of the orientations of a graphrdquo DiscreteMathematics vol 178 no 1ndash3 pp 1ndash14 1998

[141] A E Barkauskas and L H Host ldquoFinding efficient dominatingsets in oriented graphsrdquo Congressus Numerantium vol 98 pp27ndash32 1993

[142] I J Dejter and O Serra ldquoEfficient dominating sets in CayleygraphsrdquoDiscrete AppliedMathematics vol 129 no 2-3 pp 319ndash328 2003

[143] J Lee ldquoIndependent perfect domination sets in Cayley graphsrdquoJournal of Graph Theory vol 37 no 4 pp 213ndash219 2001

[144] WGu X Jia and J Shen ldquoIndependent perfect domination setsin meshes tori and treesrdquo Unpublished

[145] N Obradovic J Peters and G Ruzic ldquoEfficient domination incirculant graphswith two chord lengthsrdquo Information ProcessingLetters vol 102 no 6 pp 253ndash258 2007

[146] K Reji Kumar and G MacGillivray ldquoEfficient domination incirculant graphsrdquo Discrete Mathematics vol 313 no 6 pp 767ndash771 2013

[147] D Van Wieren M Livingston and Q F Stout ldquoPerfectdominating sets on cube-connected cyclesrdquo Congressus Numer-antium vol 97 pp 51ndash70 1993

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

22 International Journal of Combinatorics

As far as we know there are no more results on theequitable bondage number of graphs in the present literature

There are other variations of the normal domination byadding restricted conditions to the normal dominating setFor example the connected domination set of a graph119866 pro-posed by Sampathkumar and Walikar [90] is a dominatingset 119878 such that the subgraph induced by 119878 is connected Theproblem of finding a minimum connected domination set ina graph is equivalent to the problemof finding a spanning treewithmaximumnumber of vertices of degree one [91] and hassome important applications in wireless networks [92] thusit has received much research attention However for suchan important domination there is no result on its bondagenumber We believe that the topic is of significance for futureresearch

9 Generalized Bondage Numbers

There are various generalizations of the classical dominationsuch as 119901-domination distance domination fractional dom-ination Roman domination and rainbow domination Everysuch a generalization can lead to a corresponding bondageIn this section we introduce some of them

91 119901-Bondage Numbers In 1985 Fink and Jacobson [93]introduced the concept of 119901-domination Let 119901 be a positiveinteger A subset 119878 of 119881(119866) is a 119901-dominating set of 119866 if|119878 cap 119873

119866(119910)| ⩾ 119901 for every 119910 isin 119878 The 119901-domination number

120574119901(119866) is the minimum cardinality among all 119901-dominating

sets of 119866 Any 119901-dominating set of 119866 with cardinality 120574119901(119866)

is called a 120574119901-set of 119866 Note that a 120574

1-set is a classic minimum

dominating set Notice that every graph has a 119901-dominatingset since the vertex-set119881(119866) is such a setWe also note that a 1-dominating set is a dominating set and so 120574(119866) = 120574

1(119866) The

119901-domination number has received much research attentionsee a state-of-the-art survey article by Chellali et al [94]

It is clear from definition that every 119901-dominating setof a graph certainly contains all vertices of degree at most119901 minus 1 By this simple observation to avoid the trivial caseoccurrence we always assume that Δ(119866) ⩾ 119901 For 119901 ⩾ 2 Luet al [95] gave a constructive characterization of trees withunique minimum 119901-dominating sets

Recently Lu and Xu [96] have introduced the conceptto the 119901-bondage number of 119866 denoted by 119887

119901(119866) as the

minimum cardinality among all sets of edges 119861 sube 119864(119866) suchthat 120574

119901(119866 minus 119861) gt 120574

119901(119866) Clearly 119887

1(119866) = 119887(119866)

Lu and Xu [96] established a lower bound and an upperbound on 119887

119901(119879) for any integer 119901 ⩾ 2 and any tree 119879 with

Δ(119879) ⩾ 119901 and characterized all trees achieving the lowerbound and the upper bound respectively

Theorem 141 (Lu and Xu [96] 2011) For any integer 119901 ⩾ 2and any tree 119879 with Δ(119879) ⩾ 119901

1 ⩽ 119887119901(119879) ⩽ Δ (119879) minus 119901 + 1 (52)

Let 119878 be a given subset of 119881(119866) and for any 119909 isin 119878 let

119873119901(119909 119878 119866) = 119910 isin 119878 cap 119873

119866(119909)

1003816100381610038161003816119873119866(119910) cap 119878

1003816100381610038161003816 = 119901

(53)

Then all trees achieving the lower bound 1 can be character-ized as follows

Theorem 142 (Lu and Xu [96] 2011) Let 119879 be a tree withΔ(119879) ⩾ 119901 ⩾ 2 Then 119887

119901(119879) = 1 if and only if for any 120574

119901-set

119878 of 119879 there exists an edge 119909119910 isin (119878 119878) such that 119910 isin 119873119901(119909 119878 119879)

The notation 119878(119886 119887) denotes a double star obtained byadding an edge between the central vertices of two stars119870

1119886minus1

and1198701119887minus1

And the vertex with degree 119886 (resp 119887) in 119878(119886 119887) iscalled the 119871-central vertex (resp 119877-central vertex) of 119878(119886 119887)

To characterize all trees attaining the upper bound givenin Theorem 141 we define three types of operations on a tree119879 with Δ(119879) = Δ ⩾ 119901 + 1

Type 1 attach a pendant edge to a vertex 119910with 119889119879(119910) ⩽ 119901minus

2 in 119879

Type 2 attach a star 1198701Δminus1

to a vertex 119910 of 119879 by joining itscentral vertex to 119910 where 119910 in a 120574

119901-set of 119879 and

119889119879(119910) ⩽ Δ minus 1

Type 3 attach a double star 119878(119901 Δ minus 1) to a pendant vertex 119910of 119879 by coinciding its 119877-central vertex with 119910 wherethe unique neighbor of 119910 is in a 120574

119901-set of 119879

Let B = 119879 a tree obtained from 1198701Δ

or 119878(119901 Δ) by afinite sequence of operations of Types 1 2 and 3

Theorem 143 (Lu and Xu [96] 2011) A tree with the maxi-mum Δ ⩾ 119901 + 1 has 119901-bondage number Δ minus 119901 + 1 if and onlyif it belongs toB

Theorem 144 (Lu and Xu [97] 2012) Let 119866119898119899= 119875

119898times 119875

119899

Then

1198872(119866

2119899) = 1 119891119900119903 119899 ⩾ 2

1198872(119866

3119899) =

2 119894119891 119899 equiv 1 (mod 3) 1 119900119905ℎ119890119903119908119894119904119890

for 119899 ⩾ 2

1198872(119866

4119899) =

1 119894119891 119899 equiv 3 (mod 4) 2 119900119905ℎ119890119903119908119894119904119890

for 119899 ⩾ 7

(54)

Recently Krzywkowski [98] has proposed the conceptof the nonisolating 2-bondage of a graph The nonisolating2-bondage number of a graph 119866 denoted by 1198871015840

2(119866) is the

minimum cardinality among all sets of edges 119861 sube 119864(119866) suchthat 119866 minus 119861 has no isolated vertices and 120574

2(119866 minus 119861) gt 120574

2(119866) If

for every 119861 sube 119864(119866) either 1205742(119866 minus 119861) = 120574

2(119866) or 119866 minus 119861 has

isolated vertices then define 11988710158402(119866) = 0 and say that 119866 is a 2-

nonisolatedly strongly stable graph Krzywkowski presentedsome basic properties of nonisolating 2-bondage in graphsshowed that for every nonnegative integer 119896 there exists atree 119879 such 1198871015840

2(119879) = 119896 characterized all 2-non-isolatedly

strongly stable trees and determined 11988710158402(119866) for several special

graphs

International Journal of Combinatorics 23

Theorem 145 (Krzywkowski [98] 2012) For any positiveintegers 119899 and119898 with119898 ⩽ 119899

1198871015840

2(119870

119899) =

0 119894119891 119899 = 1 2 3

lfloor2119899

3rfloor 119900119905ℎ119890119903119908119894119904119890

1198871015840

2(119875

119899) =

0 119894119891 119899 = 1 2 3

1 119900119905ℎ119890119903119908119894119904119890

1198871015840

2(119862

119899) =

0 119894119891 119899 = 3

1 119894119891 119899 119894119904 119890V119890119899

2 119894119891 119899 119894119904 119900119889119889

1198871015840

2(119870

119898119899) =

3 119894119891 119898 = 119899 = 3

1 119894119891 119898 = 119899 = 4

2 119900119905ℎ119890119903119908119894119904119890

(55)

92 Distance Bondage Numbers A subset 119878 of vertices of agraph119866 is said to be a distance 119896-dominating set for119866 if everyvertex in 119866 not in 119878 is at distance at most 119896 from some vertexof 119878 The minimum cardinality of all distance 119896-dominatingsets is called the distance 119896-domination number of 119866 anddenoted by 120574

119896(119866) (do not confuse with the above-mentioned

120574119901(119866)) When 119896 = 1 a distance 1-dominating set is a normal

dominating set and so 1205741(119866) = 120574(119866) for any graph 119866 Thus

the distance 119896-domination is a generalization of the classicaldomination

The relation between 120574119896and 120572

119896(the 119896-independence

number defined in Section 22) for a tree obtained by Meirand Moon [14] who proved that 120574

119896(119879) = 120572

2119896(119879) for any tree

119879 (a special case for 119896 = 1 is stated in Proposition 10) Thefurther research results can be found in Henning et al [99]Tian and Xu [100ndash103] and Liu et al [104]

In 1998 Hartnell et al [15] defined the distance 119896-bondagenumber of 119866 denoted by 119887

119896(119866) to be the cardinality of a

smallest subset 119861 of edges of 119866 with the property that 120574119896(119866 minus

119861) gt 120574119896(119866) FromTheorem 6 it is clear that if119879 is a nontrivial

tree then 1 ⩽ 1198871(119879) ⩽ 2 Hartnell et al [15] generalized this

result to any integer 119896 ⩾ 1

Theorem 146 (Hartnell et al [15] 1998) For every nontrivialtree 119879 and positive integer 119896 1 ⩽ 119887

119896(119879) ⩽ 2

Hartnell et al [15] and Topp and Vestergaard [13] alsocharacterized the trees having distance 119896-bondage number 2In particular the class of trees for which 119887

1(119879) = 2 are just

those which have a unique maximum 2-independent set (seeTheorem 11)

Since when 119896 = 1 the distance 1-bondage number 1198871(119866)

is the classical bondage number 119887(119866) Theorem 12 gives theNP-hardness of deciding the distance 119896-bondage number ofgeneral graphs

Theorem 147 Given a nonempty undirected graph 119866 andpositive integers 119896 and 119887 with 119887 ⩽ 120576(119866) determining wetheror not 119887

119896(119866) ⩽ 119887 is NP-hard

For a vertex 119909 in119866 the open 119896-neighborhood119873119896(119909) of 119909 is

defined as119873119896(119909) = 119910 isin 119881(119866) 1 ⩽ 119889

119866(119909 119910) le 119896The closed

119896-neighborhood119873119896[119909] of 119909 in 119866 is defined as119873

119896(119909) cup 119909 Let

Δ119896(119866) = max 1003816100381610038161003816119873119896

(119909)1003816100381610038161003816 for any 119909 isin 119881 (119866) (56)

Clearly Δ1(119866) = Δ(119866) The 119896th power of a graph 119866 is the

graph119866119896 with vertex-set119881(119866119896

) = 119881(119866) and edge-set119864(119866119896

) =

119909119910 1 ⩽ 119889119866(119909 119910) ⩽ 119896 The following lemma holds directly

from the definition of 119866119896

Proposition 148 (Tian and Xu [101]) Δ(119866119896

) = Δ119896(119866) and

120574(119866119896

) = 120574119896(119866) for any graph 119866 and each 119896 ⩾ 1

A graph 119866 is 119896-distance domination-critical or 120574119896-critical

for short if 120574119896(119866 minus 119909) lt 120574

119896(119866) for every vertex 119909 in 119866

proposed by Henning et al [105]

Proposition 149 (Tian and Xu [101]) For each 119896 ⩾ 1 a graph119866 is 120574

119896-critical if and only if 119866119896 is 120574

119896-critical

By the above facts we suggest the following worthwhileproblem

Problem 6 Can we generalize the results on the bondage for119866 to 119866119896 In particular do the following propositions hold

(a) 119887(119866119896

) = 119887119896(119866) for any graph 119866 and each 119896 ⩾ 1

(b) 119887(119866119896

) ⩽ Δ119896(119866) if 119866 is not 120574

119896-critical

Let 119896 and 119901 be positive integers A subset 119878 of 119881(119866) isdefined to be a (119896 119901)-dominating set of 119866 if for any vertex119909 isin 119878 |119873

119896(119909) cap 119878| ⩾ 119901 The (119896 119901)-domination number of

119866 denoted by 120574119896119901(119866) is the minimum cardinality among

all (119896 119901)-dominating sets of 119866 Clearly for a graph 119866 a(1 1)-dominating set is a classical dominating set a (119896 1)-dominating set is a distance 119896-dominating set and a (1 119901)-dominating set is the above-mentioned 119901-dominating setThat is 120574

11(119866) = 120574(119866) 120574

1198961(119866) = 120574

119896(119866) and 120574

1119901(119866) = 120574

119901(119866)

The concept of (119896 119901)-domination in a graph 119866 is a gen-eralized domination which combines distance 119896-dominationand 119901-domination in 119866 So the investigation of (119896 119901)-domination of 119866 is more interesting and has received theattention of many researchers see for example [106ndash109]

More general Li and Xu [110] proposed the concept of the(ℓ 119908)-domination number of a119908-connected graph A subset119878 of 119881(119866) is defined to be an (ℓ 119908)-dominating set of 119866 iffor any vertex 119909 isin 119878 there are 119908 internally disjoint pathsof length at most ℓ from 119909 to some vertex in 119878 The (ℓ 119908)-domination number of119866 denoted by 120574

ℓ119908(119866) is theminimum

cardinality among all (ℓ 119908)-dominating sets of 119866 Clearly1205741198961(119866) = 120574

119896(119866) and 120574

1119901(119866) = 120574

119901(119866)

It is quite natural to propose the concept of bondage num-ber for (119896 119901)-domination or (ℓ 119908)-domination However asfar as we know none has proposed this concept until todayThis is a worthwhile topic for further research

24 International Journal of Combinatorics

93 Fractional Bondage Numbers If 120590 is a function mappingthe vertex-set 119881 into some set of real numbers then for anysubset 119878 sube 119881 let 120590(119878) = sum

119909isin119878120590(119909) Also let |120590| = 120590(119881)

A real-value function 120590 119881 rarr [0 1] is a dominatingfunction of a graph 119866 if for every 119909 isin 119881(119866) 120590(119873

119866[119909]) ⩾ 1

Thus if 119878 is a dominating set of 119866 120590 is a function where

120590 (119909) = 1 if 119909 isin 1198780 otherwise

(57)

then 120590 is a dominating function of119866 The fractional domina-tion number of 119866 denoted by 120574

119865(119866) is defined as follows

120574119865(119866) = min |120590| 120590 is a domination function of 119866

(58)

The 120574119865-bondage number of 119866 denoted by 119887

119865(119866) is

defined as the minimum cardinality of a subset 119861 sube 119864 whoseremoval results in 120574

119865(119866 minus 119861) gt 120574

119865(119866)

Hedetniemi et al [111] are the first to study fractionaldomination although Farber [112] introduced the idea indi-rectly The concept of the fractional domination numberwas proposed by Domke and Laskar [113] in 1997 Thefractional domination numbers for some ordinary graphs aredetermined

Proposition 150 (Domke et al [114] 1998 Domke and Laskar[113] 1997) (a) If 119866 is a 119896-regular graph with order 119899 then120574119865(119866) = 119899119896 + 1(b) 120574

119865(119862

119899) = 1198993 120574

119865(119875

119899) = lceil1198993rceil 120574

119865(119870

119899) = 1

(c) 120574119865(119866) = 1 if and only if Δ(119866) = 119899 minus 1

(d) 120574119865(119879) = 120574(119879) for any tree 119879

(e) 120574119865(119870

119899119898) = (119899(119898 + 1) + 119898(119899 + 1))(119899119898 minus 1) where

max119899119898 gt 1

The assertions (a)ndash(d) are due to Domke et al [114] andthe assertion (e) is due to Domke and Laskar [113] Accordingto these results Domke and Laskar [113] determined the 120574

119865-

bondage numbers for these graphs

Theorem 151 (Domke and Laskar [113] 1997) 119887119865(119870

119899) =

lceil1198992rceil 119887119865(119870

119899119898) = 1 wheremax119899119898 gt 1

119887119865(119862

119899) =

2 119894119891 119899 equiv 0 (mod 3) 1 119900119905ℎ119890119903119908119894119904119890

119887119865(119875

119899) =

2 119894119891 119899 equiv 1 (mod 3) 1 119900119905ℎ119890119903119908119894119904119890

(59)

It is easy to see that for any tree119879 119887119865(119879) ⩽ 2 In fact since

120574119865(119879) = 120574(119879) for any tree 119879 120574

119865(119879

1015840

) = 120574(1198791015840

) for any subgraph1198791015840 of 119879 It follows that

119887119865(119879) = min 119861 sube 119864 (119879) 120574

119865(119879 minus 119861) gt 120574

119865(119879)

= min 119861 sube 119864 (119879) 120574 (119879 minus 119861) gt 120574 (119879)

= 119887 (119879) ⩽ 2

(60)

Except for these no results on 119887119865(119866) have been known as

yet

94 Roman Bondage Numbers A Roman dominating func-tion on a graph 119866 is a labeling 119891 119881 rarr 0 1 2 such thatevery vertex with label 0 has at least one neighbor with label2 The weight of a Roman dominating function 119891 on 119866 is thevalue

119891 (119866) = sum

119906isin119881(119866)

119891 (119906) (61)

The minimum weight of a Roman dominating function ona graph 119866 is called the Roman domination number of 119866denoted by 120574RM (119866)

A Roman dominating function 119891 119881 rarr 0 1 2

can be represented by the ordered partition (1198810 119881

1 119881

2) (or

(119881119891

0 119881

119891

1 119881

119891

2) to refer to119891) of119881 where119881

119894= V isin 119881 | 119891(V) = 119894

In this representation its weight 119891(119866) = |1198811| + 2|119881

2| It is

clear that119881119891

1cup119881

119891

2is a dominating set of 119866 called the Roman

dominating set denoted by 119863119891

R = (1198811 1198812) Since 119881119891

1cup 119881

119891

2is

a dominating set when 119891 is a Roman dominating functionand since placing weight 2 at the vertices of a dominating setyields a Roman dominating function in [115] it is observedthat

120574 (119866) ⩽ 120574RM (119866) ⩽ 2120574 (119866) (62)

A graph 119866 is called to be Roman if 120574RM (119866) = 2120574(119866)The definition of the Roman domination function is

proposed implicitly by Stewart [116] and ReVelle and Rosing[117] Roman dominating numbers have been deeply studiedIn particular Bahremandpour et al [118] showed that theproblem determining the Roman domination number is NP-complete even for bipartite graphs

Let 119866 be a graph with maximum degree at least twoThe Roman bondage number 119887RM (119866) of 119866 is the minimumcardinality of all sets 1198641015840 sube 119864 for which 120574RM (119866 minus 119864

1015840

) gt

120574RM (119866) Since in study of Roman bondage number theassumption Δ(119866) ⩾ 2 is necessary we always assume thatwhen we discuss 119887RM (119866) all graphs involved satisfy Δ(119866) ⩾2 The Roman bondage number 119887RM (119866) is introduced byJafari Rad and Volkmann in [119]

Recently Bahremandpour et al [118] have shown that theproblem determining the Roman bondage number is NP-hard even for bipartite graphs

Problem 7 Consider the decision problem

Roman Bondage Problem

Instance a nonempty bipartite graph119866 and a positiveinteger 119896

Question is 119887RM (119866) ⩽ 119896

Theorem 152 (Bahremandpour et al [118] 2013) TheRomanbondage problem is NP-hard even for bipartite graphs

The exact value of 119887RM(119866) is known only for a few familyof graphs including the complete graphs cycles and paths

International Journal of Combinatorics 25

Theorem 153 (Jafari Rad and Volkmann [119] 2011) For anypositive integers 119899 and119898 with119898 ⩽ 119899

119887RM (119875119899) = 2 119894119891 119899 equiv 2 (mod 3) 1 119900119905ℎ119890119903119908119894119904119890

119887RM (119862119899) =

3 119894119891 119899 = 2 (mod 3) 2 119900119905ℎ119890119903119908119894119904119890

119887RM (119870119899) = lceil

119899

2rceil 119891119900119903 119899 ⩾ 3

119887RM (119870119898119899) =

1 119894119891 119898 = 1 119886119899119889 119899 = 1

4 119894119891 119898 = 119899 = 3

119898 119900119905ℎ119890119903119908119894119904119890

(63)

Lemma 154 (Cockayne et al [115] 2004) If 119866 is a graph oforder 119899 and contains vertices of degree 119899 minus 1 then 120574RM(119866) = 2

Using Lemma 154 the third conclusion in Theorem 153can be generalized to more general case which is similar toLemma 49

Theorem 155 (Jafari Rad and Volkmann [119] 2011) Let119866 bea graph with order 119899 ge 3 and 119905 the number of vertices of degree119899 minus 1 in 119866 If 119905 ⩾ 1 then 119887RM(119866) = lceil1199052rceil

Ebadi and PushpaLatha [120] conjectured that 119887RM(119866) ⩽119899 minus 1 for any graph 119866 of order 119899 ⩾ 3 Dehgardi et al[121] Akbari andQajar [122] independently showed that thisconjecture is true

Theorem 156 119887RM(119866) ⩽ 119899 minus 1 for any connected graph 119866 oforder 119899 ⩾ 3

For a complete 119905-partite graph we have the followingresult

Theorem 157 (Hu and Xu [123] 2011) Let 119866 = 11987011989811198982119898119905

bea complete 119905-partite graph with 119898

1= sdot sdot sdot = 119898

119894lt 119898

119894+1le sdot sdot sdot ⩽

119898119905 119905 ⩾ 2 and 119899 = sum119905

119895=1119898

119895 Then

119887RM (119866) =

lceil119894

2rceil 119894119891 119898

119894= 1 119886119899119889 119899 ⩾ 3

2 119894119891 119898119894= 2 119886119899119889 119894 = 1

119894 119894119891 119898119894= 2 119886119899119889 119894 ⩾ 2

4 119894119891 119898119894= 3 119886119899119889 119894 = 119905 = 2

119899 minus 1 119894119891 119898119894= 3 119886119899119889 119894 = 119905 ⩾ 3

119899 minus 119898119905

119894119891 119898119894⩾ 3 119886119899119889 119898

119905⩾ 4

(64)

Consider a complete 119905-partite graph 119870119905(3) for 119905 ⩾ 3

119887RM(119870119905(3)) = 119899 minus 1 by Theorem 157 where 119899 = 3119905

which shows that this upper bound of 119899 minus 1 on 119887RM(119866) inTheorem 156 is sharp This fact and 120574

119877(119870

119905(3)) = 4 give

a negative answer to the following two problems posed byDehgardi et al [121]Prove or Disprove for any connected graph 119866 of order 119899 ge 3(i) 119887RM(119866) = 119899 minus 1 if and only if 119866 cong 119870

3 (ii) 119887RM(119866) ⩽ 119899 minus

120574119877(119866) + 1Note that a complete 119905-partite graph 119870

119905(3) is (119899 minus 3)-

regular and 119887RM(119870119905(3)) = 119899 minus 1 for 119905 ⩾ 3 where 119899 = 3119905

Hu and Xu further determined that 119887119877(119866) = 119899 minus 2 for any

(119899 minus 3)-regular graph 119866 of order 119899 ⩽ 5 except 119870119905(3)

Theorem 158 (Hu and Xu [123] 2011) For any (119899minus3)-regulargraph 119866 of order 119899 other than119870

119905(3) 119887RM(119866) = 119899 minus 2 for 119899 ⩾ 5

For a tree 119879 with order 119899 ⩾ 3 Ebadi and PushpaLatha[120] and Jafari Rad and Volkmann [119] independentlyobtained an upper bound on 119887RM(119879)

Theorem 159 119887RM(119879) ⩽ 3 for any tree 119879 with order 119899 ⩾ 3

Theorem 160 (Bahremandpour et al [118] 2013) 119887RM(1198752 times119875119899) = 2 for 119899 ⩾ 2

Theorem 161 (Jafari Rad and Volkmann [119] 2011) Let119866 bea graph with order at least three

(a) If (119909 119910 119911) is a path of length 2 in 119866 then 119887RM(119866) ⩽119889119866(119909) + 119889

119866(119910) + 119889

119866(119911) minus 3 minus |119873

119866(119909) cap 119873

119866(119910)|

(b) If 119866 is connected then 119887RM(119866) ⩽ 120582(119866) + 2Δ(119866) minus 3

Theorem 161 (b) implies 119887RM(119866) ⩽ 120575(119866) + 2Δ(119866) minus 3 for aconnected graph 119866 Note that for a planar graph 119866 120575(119866) ⩽ 5moreover 120575(119866) ⩽ 3 if the girth at least 4 and 120575(119866) ⩽ 2 if thegirth at least 6 These two facts show that 119887RM(119866) ⩽ 2Δ(119866) +2 for a connected planar graph 119866 Jafari Rad and Volkmann[124] improved this bound

Theorem 162 (Jafari Rad and Volkmann [124] 2011) For anyconnected planar graph 119866 of order 119899 ⩾ 3 with girth 119892(119866)

119887RM (119866)

⩽

2Δ (119866)

Δ (119866) + 6

Δ (119866) + 5 119894119891 119866 119888119900119899119905119886119894119899119904 119899119900 V119890119903119905119894119888119890119904 119900119891 119889119890119892119903119890119890 119891119894V119890

Δ (119866) + 4 119894119891 119892 (119866) ⩾ 4

Δ (119866) + 3 119894119891 119892 (119866) ⩾ 5

Δ (119866) + 2 119894119891 119892 (119866) ⩾ 6

Δ (119866) + 1 119894119891 119892 (119866) ⩾ 8

(65)

According toTheorem 157 119887RM(119862119899) = 3 = Δ(119862

119899)+1 for a

cycle 119862119899of length 119899 ⩾ 8 with 119899 equiv 2 (mod 3) and therefore

the last upper bound inTheorem 162 is sharp at least for Δ =2

Combining the fact that every planar graph 119866 withminimum degree 5 contains an edge 119909119910 with 119889

119866(119909) = 5

and 119889119866(119910) isin 5 6 with Theorem 161 (a) Akbari et al [125]

obtained the following result

26 International Journal of Combinatorics

middot middot middot

middot middot middot

middot middot middot

middot middot middot

119866

Figure 9 The graph 119866 is constructed from 119866

Theorem 163 (Akbari et al [125] 2012) 119887RM(119866) ⩽ 15 forevery planar graph 119866

It remains open to show whether the bound inTheorem 163 is sharp or not Though finding a planargraph 119866 with 119887RM(119866) = 15 seems to be difficult Akbari etal [125] constructed an infinite family of planar graphs withRoman bondage number equal to 7 by proving the followingresult

Theorem 164 (Akbari et al [125] 2012) Let 119866 be a graph oforder 119899 and 119866 is the graph of order 5119899 obtained from 119866 byattaching the central vertex of a copy of a path 119875

5to each vertex

of119866 (see Figure 9) Then 120574RM(119866) = 4119899 and 119887RM(119866) = 120575(119866)+2

ByTheorem 164 infinitemany planar graphswith Romanbondage number 7 can be obtained by considering any planargraph 119866 with 120575(119866) = 5 (eg the icosahedron graph)

Conjecture 165 (Akbari et al [125] 2012) The Romanbondage number of every planar graph is at most 7

For general bounds the following observation is directlyobtained which is similar to Observation 1

Observation 4 Let 119866 be a graph of order 119899 with maximumdegree at least two Assume that 119867 is a spanning subgraphobtained from 119866 by removing 119896 edges If 120574RM(119867) = 120574RM(119866)then 119887

119877(119867) ⩽ 119887RM(119866) ⩽ 119887RM(119867) + 119896

Theorem 166 (Bahremandpour et al [118] 2013) For anyconnected graph 119866 of order 119899 if 119899 ⩾ 3 and 120574RM(119866) = 120574(119866) + 1then

119887RM (119866) ⩽ min 119887 (119866) 119899Δ (66)

where 119899Δis the number of vertices with maximum degree Δ in

119866

Observation 5 For any graph 119866 if 120574(119866) = 120574RM(119866) then119887RM(119866) ⩽ 119887(119866)

Theorem 167 (Bahremandpour et al [118] 2013) For everyRoman graph 119866

119887RM (119866) ⩾ 119887 (119866) (67)

The bound is sharp for cycles on 119899 vertices where 119899 equiv 0 (mod3)

The strict inequality in Theorem 167 can hold for exam-ple 119887(119862

3119896+2) = 2 lt 3 = 119887RM(1198623119896+2

) byTheorem 157A graph 119866 is called to be vertex Roman domination-

critical if 120574RM(119866 minus 119909) lt 120574RM(119866) for every vertex 119909 in 119866 If119866 has no isolated vertices then 120574RM(119866) ⩽ 2120574(119866) ⩽ 2120573(119866)If 120574RM(119866) = 2120573(119866) then 120574RM(119866) = 2120574(119866) and hence 119866 is aRoman graph In [30] Volkmann gave a lot of graphs with120574(119866) = 120573(119866) The following result is similar to Theorem 57

Theorem 168 (Bahremandpour et al [118] 2013) For anygraph 119866 if 120574RM(119866) = 2120573(119866) then

(a) 119887RM(119866) ⩾ 120575(119866)

(b) 119887RM(119866) ⩾ 120575(119866)+1 if119866 is a vertex Roman domination-critical graph

If 120574RM(119866) = 2 then obviously 119887RM(119866) ⩽ 120575(119866) So weassume 120574RM(119866) ⩾ 3 Dehgardi et al [121] proved 119887RM(119866) ⩽(120574RM(119866) minus 2)Δ(119866) + 1 and posed the following problem if 119866is a connected graph of order 119899 ⩾ 4 with 120574RM(119866) ⩾ 3 then

119887RM (119866) ⩽ (120574RM (119866) minus 2) Δ (119866) (68)

Theorem 161 (a) shows that the inequality (68) holds if120574RM(119866) ⩾ 5 Thus the bound in (68) is of interest only when120574RM(119866) is 3 or 4

Lemma 169 (Bahremandpour et al [118] 2013) For anynonempty graph 119866 of order 119899 ⩾ 3 120574RM(119866) = 3 if and onlyif Δ(119866) = 119899 minus 2

The following result shows that (68) holds for all graphs119866 of order 119899 ⩾ 4 with 120574RM(119866) = 3 or 4 which improvesTheorem 161 (b)

Theorem 170 (Bahremandpour et al [118] 2013) For anyconnected graph 119866 of order 119899 ⩾ 4

119887RM (119866) le Δ (119866) = 119899 minus 2 119894119891 120574RM (119866) = 3

Δ (119866) + 120575 (119866) minus 1 119894119891 120574RM (119866) = 4(69)

with the first equality if and only if 119866 cong 1198624

Recently Akbari and Qajar [122] have proved the follow-ing result

Theorem 171 (Akbari and Qajar [122]) For any connectedgraph 119866 of order 119899 ⩾ 3

119887RM (119866) ⩽ 119899 minus 120574RM (119866) + 5 (70)

International Journal of Combinatorics 27

We conclude this section with the following problems

Problem 8 Characterize all connected graphs 119866 of order 119899 ⩾3 for which 119887RM(119866) = 119899 minus 1

Problem 9 Prove or disprove if 119866 is a connected graph oforder 119899 ⩾ 3 then

119887RM (119866) ⩽ 119899 minus 120574RM (119866) + 3 (71)

Note that 120574119877(119870

119905(3)) = 4 and 119887RM(119870119905

(3)) = 119899minus1 where 119899 =3119905 for 119905 ⩾ 3 The upper bound on 119887RM(119866) given in Problem 9is sharp if Problem 9 has positive answer

95 Rainbow Bondage Numbers For a positive integer 119896 a119896-rainbow dominating function of a graph 119866 is a function 119891from the vertex-set 119881(119866) to the set of all subsets of the set1 2 119896 such that for any vertex 119909 in 119866 with 119891(119909) = 0 thecondition

⋃

119910isin119873119866(119909)

119891 (119910) = 1 2 119896 (72)

is fulfilledThe weight of a 119896-rainbow dominating function 119891on 119866 is the value

119891 (119866) = sum

119909isin119881(119866)

1003816100381610038161003816119891 (119909)1003816100381610038161003816 (73)

The 119896-rainbow domination number of a graph 119866 denoted by120574119903119896(119866) is the minimum weight of a 119896-rainbow dominating

function 119891 of 119866Note that 120574

1199031(119866) is the classical domination number 120574(119866)

The 119896-rainbow domination number is introduced by Bresaret al [126] Rainbow domination of a graph 119866 coincides withordinary domination of the Cartesian product of 119866 with thecomplete graph in particular 120574

119903119896(119866) = 120574(119866 times 119870

119896) for any

positive integer 119896 and any graph 119866 [126] Hartnell and Rall[127] proved that 120574(119879) lt 120574

1199032(119879) for any tree 119879 no graph has

120574 = 120574119903119896for 119896 ⩾ 3 for any 119896 ⩾ 2 and any graph 119866 of order 119899

min 119899 120574 (119866) + 119896 minus 2 ⩽ 120574119903119896(119866) ⩽ 119896120574 (119866) (74)

The introduction of rainbow domination is motivatedby the study of paired domination in Cartesian productsof graphs where certain upper bounds can be expressed interms of rainbow domination that is for any graph 119866 andany graph 119867 if 119881(119867) can be partitioned into 119896120574

119875-sets then

120574119875(119866 times 119867) ⩽ (1119896)120592(119867)120574

119903119896(119866) proved by Bresar et al [126]

Dehgardi et al [128] introduced the concept of 119896-rainbowbondage number of graphs Let 119866 be a graph with maximumdegree at least two The 119896-rainbow bondage number 119887

119903119896(119866)

of 119866 is the minimum cardinality of all sets 119861 sube 119864(119866) forwhich 120574

119903119896(119866 minus 119861) gt 120574

119903119896(119866) Since in the study of 119896-rainbow

bondage number the assumption Δ(119866) ⩾ 2 is necessarywe always assume that when we discuss 119887

119903119896(119866) all graphs

involved satisfy Δ(119866) ⩾ 2The 2-rainbow bondage number of some special families

of graphs has been determined

Theorem 172 (Dehgardi et al [128]) For any integer 119899 ⩾ 3

1198871199032(119875

119899) = 1 119891119900119903 119899 ⩾ 2

1198871199032(119862

119899) =

1 119894119891 119899 equiv 0 (mod 4) 2 119900119905ℎ119890119903119908119894119904119890

1198871199032(119870

119899119899) =

1 119894119891 119899 = 2

3 119894119891 119899 = 3

6 119894119891 119899 = 4

119898 119894119891 119899 ⩾ 5

(75)

Theorem 173 (Dehgardi et al [128]) For any connected graph119866 of order 119899 ⩾ 3 119887

1199032(119866) ⩽ 120582(119866) + 2Δ(119866) minus 3

Theorem 174 (Dehgardi et al [128]) For any tree 119879 of order119899 ⩾ 3 119887

1199032(119879) ⩽ 2

Theorem 175 (Dehgardi et al [128]) If 119866 is a planar graphwithmaximumdegree at least two then 119887

1199032(119866) ⩽ 15Moreover

if the girth 119892(119866) ⩾ 4 then 1198871199032(119866) ⩽ 11 and if 119892(119866) ⩾ 6 then

1198871199032(119866) ⩽ 9

Theorem 176 (Dehgardi et al [128]) For a complete bipartitegraph 119870

119901119902 if 2119896 + 1 ⩽ 119901 ⩽ 119902 then 119887

119903119896(119870

119901119902) = 119901

Theorem177 (Dehgardi et al [128]) For a complete graph119870119899

if 119899 ⩾ 119896 + 1 then 119887119903119896(119870

119899) = lceil119896119899(119896 + 1)rceil

The special case 119896 = 1 of Theorem 177 can be found inTheorem 1 The following is a simple observation similar toObservation 1

Observation 6 (Dehgardi et al [128]) Let 119866 be a graphwith maximum degree at least two 119867 a spanning subgraphobtained from 119866 by removing 119896 edges If 120574

119903119896(119867) = 120574

119903119896(119866)

then 119887119903119896(119867) ⩽ 119887

119903119896(119866) ⩽ 119887

119903119896(119867) + 119896

Theorem 178 (Dehgardi et al [128]) For every graph 119866 with120574119903119896(119866) = 119896120574(119866) 119887

119903119896(119866) ⩾ 119887(119866)

A graph 119866 is said to be vertex 119896-rainbow domination-critical if 120574

119903119896(119866 minus 119909) lt 120574

119903119896(119866) for every vertex 119909 in 119866 It is

clear that if 119866 has no isolated vertices then 120574119903119896(119866) ⩽ 119896120574(119866) ⩽

119896120573(119866) where 120573(119866) is the vertex-covering number of119866 Thusif 120574

119903119896(119866) = 119896120573(119866) then 120574

119903119896(119866) = 119896120574(119866)

Theorem 179 (Dehgardi et al [128]) For any graph 119866 if120574119903119896(119866) = 119896120573(119866) then

(a) 119887119903119896(119866) ⩾ 120575(119866)

(b) 119887119903119896(119866) ⩾ 120575(119866)+1 if119866 is vertex 119896-rainbow domination-

critical

As far as we know there are no more results on the 119896-rainbow bondage number of graphs in the literature

28 International Journal of Combinatorics

10 Results on Digraphs

Although domination has been extensively studied in undi-rected graphs it is natural to think of a dominating setas a one-way relationship between vertices of the graphIndeed among the earliest literatures on this subject vonNeumann and Morgenstern [129] used what is now calleddomination in digraphs to find solution (or kernels whichare independent dominating sets) for cooperative 119899-persongames Most likely the first formulation of domination byBerge [130] in 1958 was given in the context of digraphs andonly some years latter by Ore [131] in 1962 for undirectedgraphs Despite this history examination of domination andits variants in digraphs has been essentially overlooked (see[4] for an overview of the domination literature) Thus thereare few if any such results on domination for digraphs in theliterature

The bondage number and its related topics for undirectedgraph have become one of major areas both in theoreticaland applied researches However until recently Carlsonand Develin [42] Shan and Kang [132] and Huang andXu [133 134] studied the bondage number for digraphsindependently In this section we introduce their resultsfor general digraphs Results for some special digraphs suchas vertex-transitive digraphs are introduced in the nextsection

101 Upper Bounds for General Digraphs Let 119866 = (119881 119864)

be a digraph without loops and parallel edges A digraph 119866is called to be asymmetric if whenever (119909 119910) isin 119864(119866) then(119910 119909) notin 119864(119866) and to be symmetric if (119909 119910) isin 119864(119866) implies(119910 119909) isin 119864(119866) For a vertex 119909 of 119881(119866) the sets of out-neighbors and in-neighbors of 119909 are respectively defined as119873

+

119866(119909) = 119910 isin 119881(119866) (119909 119910) isin 119864(119866) and 119873minus

119866(119909) = 119909 isin

119881(119866) (119909 119910) isin 119864(119866) the out-degree and the in-degree of119909 are respectively defined as 119889+

119866(119909) = |119873

+

119866(119909)| and 119889minus

119866(119909) =

|119873+

119866(119909)| Denote themaximum and theminimum out-degree

(resp in-degree) of 119866 by Δ+(119866) and 120575+(119866) (resp Δminus(119866) and120575minus

(119866))The degree 119889119866(119909) of 119909 is defined as 119889+

119866(119909)+119889

minus

119866(119909) and

the maximum and the minimum degrees of119866 are denoted byΔ(119866) and 120575(119866) respectively that is Δ(119866) = max119889

119866(119909) 119909 isin

119881(119866) and 120575(119866) = min119889119866(119909) 119909 isin 119881(119866) Note that the

definitions here are different from the ones in the textbookon digraphs see for example Xu [1]

A subset 119878 of119881(119866) is called a dominating set if119881(119866) = 119878cup119873

+

119866(119878) where119873+

119866(119878) is the set of out-neighbors of 119878Then just

as for undirected graphs 120574(119866) is the minimum cardinalityof a dominating set and the bondage number 119887(119866) is thesmallest cardinality of a set 119861 of edges such that 120574(119866 minus 119861) gt120574(119866) if such a subset 119861 sube 119864(119866) exists Otherwise we put119887(119866) = infin

Some basic results for undirected graphs stated inSection 3 can be generalized to digraphs For exampleTheorem 18 is generalized by Carlson and Develin [42]Huang andXu [133] and Shan andKang [132] independentlyas follows

Theorem 180 Let 119866 be a digraph and (119909 119910) isin 119864(119866) Then119887(119866) ⩽ 119889

119866(119910) + 119889

minus

119866(119909) minus |119873

minus

119866(119909) cap 119873

minus

119866(119910)|

Corollary 181 For a digraph 119866 119887(119866) ⩽ 120575minus(119866) + Δ(119866)

Since 120575minus

(119866) ⩽ (12)Δ(119866) from Corollary 181Conjecture 53 is valid for digraphs

Corollary 182 For a digraph 119866 119887(119866) ⩽ (32)Δ(119866)

In the case of undirected graphs the bondage number of119866119899= 119870

119899times 119870

119899achieves this bound However it was shown

by Carlson and Develin in [42] that if we take the symmetricdigraph119866

119899 we haveΔ(119866

119899) = 4(119899minus1) 120574(119866

119899) = 119899 and 119887(119866

119899) ⩽

4(119899 minus 1) + 1 = Δ(119866119899) + 1 So this family of digraphs cannot

show that the bound in Corollary 182 is tightCorresponding to Conjecture 51 which is discredited

for undirected graphs and is valid for digraphs the sameconjecture for digraphs can be proposed as follows

Conjecture 183 (Carlson and Develin [42] 2006) 119887(119866) ⩽Δ(119866) + 1 for any digraph 119866

If Conjecture 183 is true then the following results showsthat this upper bound is tight since 119887(119870

119899times119870

119899) = Δ(119870

119899times119870

119899)+1

for a complete digraph 119870119899

In 2007 Shan and Kang [132] gave some tight upperbounds on the bondage numbers for some asymmetricdigraphs For example 119887(119879) ⩽ Δ(119879) for any asymmetricdirected tree 119879 119887(119866) ⩽ Δ(119866) for any asymmetric digraph 119866with order at least 4 and 120574(119866) ⩽ 2 For planar digraphs theyobtained the following results

Theorem 184 (Shan and Kang [132] 2007) Let 119866 be aasymmetric planar digraphThen 119887(119866) ⩽ Δ(119866)+2 and 119887(119866) ⩽Δ(119866) + 1 if Δ(119866) ⩾ 5 and 119889minus

119866(119909) ⩾ 3 for every vertex 119909 with

119889119866(119909) ⩾ 4

102 Results for Some Special Digraphs The exact values andbounds on 119887(119866) for some standard digraphs are determined

Theorem185 (Huang andXu [133] 2006) For a directed cycle119862119899and a directed path 119875

119899

119887 (119862119899) =

3 119894119891 119899 119894119904 119900119889119889

2 119894119891 119899 119894119904 119890V119890119899

119887 (119875119899) =

2 119894119891 119899 119894119904 119900119889119889

1 119894119891 119899 119894119904 119890V119890119899

(76)

For the de Bruijn digraph 119861(119889 119899) and the Kautz digraph119870(119889 119899)

119887 (119861 (119889 119899)) = 119889 119894119891 119899 119894119904 119900119889119889

119889 ⩽ 119887 (119861 (119889 119899)) ⩽ 2119889 119894119891 119899 119894119904 119890V119890119899

119887 (119870 (119889 119899)) = 119889 + 1

(77)

Like undirected graphs we can define the total domi-nation number and the total bondage number On the totalbondage numbers for some special digraphs the knownresults are as follows

International Journal of Combinatorics 29

Theorem 186 (Huang and Xu [134] 2007) For a directedcycle 119862

119899and a directed path 119875

119899 119887

119879(119875

119899) and 119887

119879(119862

119899) all do not

exist For a complete digraph 119870119899

119887119879(119870

119899) = infin 119894119891 119899 = 1 2

119887119879(119870

119899) = 3 119894119891 119899 = 3

119899 ⩽ 119887119879(119870

119899) ⩽ 2119899 minus 3 119894119891 119899 ⩾ 4

(78)

The extended de Bruijn digraph EB(119889 119899 1199021 119902

119901) and

the extended Kautz digraph EK(119889 119899 1199021 119902

119901) are intro-

duced by Shibata and Gonda [135] If 119901 = 1 then theyare the de Bruijn digraph B(119889 119899) and the Kautz digraphK(119889 119899) respectively Huang and Xu [134] determined theirtotal domination numbers In particular their total bondagenumbers for general cases are determined as follows

Theorem 187 (Huang andXu [134] 2007) If 119889 ⩾ 2 and 119902119894⩾ 2

for each 119894 = 1 2 119901 then

119887119879(EB(119889 119899 119902

1 119902

119901)) = 119889

119901

minus 1

119887119879(EK(119889 119899 119902

1 119902

119901)) = 119889

119901

(79)

In particular for the de Bruijn digraph B(119889 119899) and the Kautzdigraph K(119889 119899)

119887119879(B (119889 119899)) = 119889 minus 1 119887

119879(K (119889 119899)) = 119889 (80)

Zhang et al [136] determined the bondage number incomplete 119905-partite digraphs

Theorem 188 (Zhang et al [136] 2009) For a complete 119905-partite digraph 119870

11989911198992119899119905

where 1198991⩽ 119899

2⩽ sdot sdot sdot ⩽ 119899

119905

119887 (11987011989911198992119899119905

)

=

119898 119894119891 119899119898= 1 119899

119898+1⩾ 2 119891119900119903 119904119900119898119890

119898 (1 ⩽ 119898 lt 119905)

4119905 minus 3 119894119891 1198991= 119899

2= sdot sdot sdot = 119899

119905= 2

119905minus1

sum

119894=1

119899119894+ 2 (119905 minus 1) 119900119905ℎ119890119903119908119894119904119890

(81)

Since an undirected graph can be thought of as a sym-metric digraph any result for digraphs has an analogy forundirected graphs in general In view of this point studyingthe bondage number for digraphs is more significant thanfor undirected graphs Thus we should further study thebondage number of digraphs and try to generalize knownresults on the bondage number and related variants for undi-rected graphs to digraphs prove or disprove Conjecture 183In particular determine the exact values of 119887(B(119889 119899)) for aneven 119899 and 119887

119879(119870

119899) for 119899 ⩾ 4

11 Efficient Dominating Sets

A dominating set 119878 of a graph 119866 is called to be efficient if forevery vertex 119909 in119866 |119873

119866[119909]cap119878| = 1 if119866 is a undirected graph

or |119873minus

119866[119909] cap 119878| = 1 if 119866 is a directed graph The concept of an

efficient set was proposed by Bange et al [137] in 1988 In theliterature an efficient set is also called a perfect dominatingset (see Livingston and Stout [138] for an explanation)

From definition if 119878 is an efficient dominating set of agraph 119866 then 119878 is certainly an independent set and everyvertex not in 119878 is adjacent to exactly one vertex in 119878

It is also clear from definition that a dominating set 119878 isefficient if and only if N

119866[119878] = 119873

119866[119909] 119909 isin 119878 for the

undirected graph 119866 or N+

119866[119878] = 119873

+

119866[119909] 119909 isin 119878 for the

digraph119866 is a partition of119881(119866) where the induced subgraphby119873

119866[119909] or119873+

119866[119909] is a star or an out-star with the root 119909

The efficient domination has important applications inmany areas such as error-correcting codes and receivesmuch attention in the late years

The concept of efficient dominating sets is a measureof the efficiency of domination in graphs Unfortunately asshown in [137] not every graph has an efficient dominatingset andmoreover it is anNP-complete problem to determinewhether a given graph has an efficient dominating set Inaddition it was shown by Clark [139] in 1993 that for a widerange of 119901 almost every random undirected graph 119866 isin

G(120592 119901) has no efficient dominating sets This means thatundirected graphs possessing an efficient dominating set arerare However it is easy to show that every undirected graphhas an orientationwith an efficient dominating set (see Bangeet al [140])

In 1993 Barkauskas and Host [141] showed that deter-mining whether an arbitrary oriented graph has an efficientdominating set is NP-complete Even so the existence ofefficient dominating sets for some special classes of graphs hasbeen examined see for example Dejter and Serra [142] andLee [143] for Cayley graph Gu et al [144] formeshes and toriObradovic et al [145] Huang and Xu [36] Reji Kumar andMacGillivray [146] for circulant graphs Harary graphs andtori and Van Wieren et al [147] for cube-connected cycles

In this section we introduce results of the bondagenumber for some graphs with efficient dominating sets dueto Huang and Xu [36]

111 Results for General Graphs In this subsection we intro-duce some results on bondage numbers obtained by applyingefficient dominating sets We first state the two followinglemmas

Lemma 189 For any 119896-regular graph or digraph 119866 of order 119899120574(119866) ⩾ 119899(119896 + 1) with equality if and only if 119866 has an efficientdominating set In addition if 119866 has an efficient dominatingset then every efficient dominating set is certainly a 120574-set andvice versa

Let 119890 be an edge and 119878 a dominating set in 119866 We say that119890 supports 119878 if 119890 isin (119878 119878) where (119878 119878) = (119909 119910) isin 119864(119866) 119909 isin 119878 119910 isin 119878 Denote by 119904(119866) the minimum number of edgeswhich support all 120574-sets in 119866

Lemma 190 For any graph or digraph 119866 119887(119866) ⩾ 119904(119866) withequality if 119866 is regular and has an efficient dominating set

30 International Journal of Combinatorics

A graph 119866 is called to be vertex-transitive if its automor-phism group Aut(119866) acts transitively on its vertex-set 119881(119866)A vertex-transitive graph is regular Applying Lemmas 189and 190 Huang and Xu obtained some results on bondagenumbers for vertex-transitive graphs or digraphs

Theorem 191 (Huang and Xu [36] 2008) For any vertex-transitive graph or digraph 119866 of order 119899

119887 (119866) ⩾

lceil119899

2120574 (119866)rceil 119894119891 119866 119894119904 119906119899119889119894119903119890119888119905119890119889

lceil119899

120574 (119866)rceil 119894119891 119866 119894119904 119889119894119903119890119888119905119890119889

(82)

Theorem192 (Huang andXu [36] 2008) Let119866 be a 119896-regulargraph of order 119899 Then

119887(119866) ⩽ 119896 if119866 is undirected and 119899 ⩾ 120574(119866)(119896+1)minus119896+1119887(119866) ⩽ 119896+1+ℓ if119866 is directed and 119899 ⩾ 120574(119866)(119896+1)minusℓwith 0 ⩽ ℓ ⩽ 119896 minus 1

Next we introduce a better upper bound on 119887(119866) To thisaim we need the following concept which is a generalizationof the concept of the edge-covering of a graph 119866 For 1198811015840

sube

119881(119866) and 1198641015840 sube 119864(119866) we say that 1198641015840covers 1198811015840 and call 1198641015840an edge-covering for 1198811015840 if there exists an edge (119909 119910) isin 1198641015840 forany vertex 119909 isin 1198811015840 For 119910 isin 119881(119866) let 1205731015840[119910] be the minimumcardinality over all edge-coverings for119873minus

119866[119910]

Theorem 193 (Huang and Xu [36] 2008) If 119866 is a 119896-regulargraph with order 120574(119866)(119896 + 1) then 119887(119866) ⩽ 1205731015840[119910] for any 119910 isin119881(119866)

Theupper bound on 119887(119866) given inTheorem 193 is tight inview of 119887(119862

119899) for a cycle or a directed cycle 119862

119899(seeTheorems

1 and 185 resp)It is easy to see that for a 119896-regular graph 119866 lceil(119896 + 1)2rceil ⩽

1205731015840

[119910] ⩽ 119896 when 119866 is undirected and 1205731015840[119910] = 119896 + 1 when 119866 isdirected By this fact and Lemma 189 the following theoremis merely a simple combination of Theorems 191 and 193

Theorem 194 (Huang and Xu [36] 2008) Let 119866 be a vertex-transitive graph of degree 119896 If 119866 has an efficient dominatingset then

lceil119896 + 1

2rceil ⩽ 119887 (119866) ⩽ 119896 119894119891 119866 119894119904 119906119899119889119894119903119890119888119905119890119889

119887 (119866) = 119896 + 1 119894119891 119866 119894119904 119889119894119903119890119888119905119890119889

(83)

Theorem 195 (Huang and Xu [36] 2008) If 119866 is an undi-rected vertex-transitive cubic graph with order 4120574(119866) and girth119892(119866) ⩽ 5 then 119887(119866) = 2

The proof of Theorem 195 leads to a byproduct If 119892 =5 let (119906

1 119906

2 119906

3 119906

4 119906

5) be a cycle and let D

119894be the family

of efficient dominating sets containing 119906119894for each 119894 isin

1 2 3 4 5 ThenD119894capD

119895= 0 for 119894 = 1 2 5 Then 119866 has

at least 5119904 efficient dominating sets where 119904 = 119904(119866) But thereare only 4s (=ns120574) distinct efficient dominating sets in119866This

contradiction implies that an undirected vertex-transitivecubic graph with girth five has no efficient dominating setsBut a similar argument for 119892(119866) = 3 4 or 119892(119866) ⩾ 6 couldnot give any contradiction This is consistent with the resultthat CCC(119899) a vertex-transitive cubic graph with girth 119899 if3 ⩽ 119899 ⩽ 8 or girth 8 if 119899 ⩾ 9 has efficient dominating setsfor all 119899 ⩾ 3 except 119899 = 5 (see Theorem 199 in the followingsubsection)

112 Results for Cayley Graphs In this subsection we useTheorem 194 to determine the exact values or approximativevalues of bondage numbers for some special vertex-transitivegraphs by characterizing the existence of efficient dominatingsets in these graphs

Let Γ be a nontrivial finite group 119878 a nonempty subset ofΓ without the identity element of Γ A digraph 119866 is defined asfollows

119881 (119866) = Γ (119909 119910) isin 119864 (119866) lArrrArr 119909minus1

119910 isin 119878

for any 119909 119910 isin Γ(84)

is called a Cayley digraph of the group Γ with respect to119878 denoted by 119862

Γ(119878) If 119878minus1 = 119904minus1 119904 isin 119878 = 119878 then

119862Γ(119878) is symmetric and is called a Cayley undirected graph

a Cayley graph for short Cayley graphs or digraphs arecertainly vertex-transitive (see Xu [1])

A circulant graph 119866(119899 119878) of order 119899 is a Cayley graph119862119885119899

(119878) where 119885119899= 0 119899 minus 1 is the addition group of

order 119899 and 119878 is a nonempty subset of119885119899without the identity

element and hence is a vertex-transitive digraph of degree|119878| If 119878minus1 = 119878 then119866(119899 119878) is an undirected graph If 119878 = 1 119896where 2 ⩽ 119896 ⩽ 119899 minus 2 we write 119866(119899 1 119896) for 119866(119899 1 119896) or119866(119899 plusmn1 plusmn119896) and call it a double-loop circulant graph

Huang and Xu [36] showed that for directed 119866 =

119866(119899 1 119896) lceil1198993rceil ⩽ 120574(119866) ⩽ lceil1198992rceil and 119866 has an efficientdominating set if and only if 3 | 119899 and 119896 equiv 2 (mod 3) fordirected 119866 = 119866(119899 1 119896) and 119896 = 1198992 lceil1198995rceil ⩽ 120574(119866) ⩽ lceil1198993rceiland 119866 has an efficient dominating set if and only if 5 | 119899 and119896 equiv plusmn2 (mod 5) By Theorem 194 we obtain the bondagenumber of a double loop circulant graph if it has an efficientdominating set

Theorem 196 (Huang and Xu [36] 2008) Let 119866 be a double-loop circulant graph 119866(119899 1 119896) If 119866 is directed with 3 | 119899and 119896 equiv 2 (mod 3) or 119866 is undirected with 5 | 119899 and119896 equiv plusmn2 (mod 5) then 119887(119866) = 3

The 119898 times 119899 torus is the Cartesian product 119862119898times 119862

119899of

two cycles and is a Cayley graph 119862119885119898times119885119899

(119878) where 119878 =

(0 1) (1 0) for directed cycles and 119878 = (0 plusmn1) (plusmn1 0) forundirected cycles and hence is vertex-transitive Gu et al[144] showed that the undirected torus119862

119898times119862

119899has an efficient

dominating set if and only if both 119898 and 119899 are multiplesof 5 Huang and Xu [36] showed that the directed torus119862119898times 119862

119899has an efficient dominating set if and only if both

119898 and 119899 are multiples of 3 Moreover they found a necessarycondition for a dominating set containing the vertex (0 0)in 119862

119898times 119862

119899to be efficient and obtained the following

result

International Journal of Combinatorics 31

Theorem 197 (Huang and Xu [36] 2008) Let 119866 = 119862119898times 119862

119899

If 119866 is undirected and both119898 and 119899 are multiples of 5 or if 119866 isdirected and both119898 and 119899 are multiples of 3 then 119887(119866) = 3

The hypercube 119876119899

is the Cayley graph 119862Γ(119878)

where Γ = 1198852times sdot sdot sdot times 119885

2= (119885

2)119899 and 119878 =

100 sdot sdot sdot 0 010 sdot sdot sdot 0 00 sdot sdot sdot 01 Lee [143] showed that119876119899has an efficient dominating set if and only if 119899 = 2119898 minus 1

for a positive integer119898 ByTheorem 194 the following resultis obtained immediately

Theorem 198 (Huang and Xu [36] 2008) If 119899 = 2119898 minus 1 for apositive integer119898 then 2119898minus1

⩽ 119887(119876119899) ⩽ 2

119898

minus 1

Problem 10 Determine the exact value of 119887(119876119899) for 119899 = 2119898minus1

The 119899-dimensional cube-connected cycle denoted byCCC(119899) is constructed from the 119899-dimensional hypercube119876119899by replacing each vertex 119909 in 119876

119899with an undirected cycle

119862119899of length 119899 and linking the 119894th vertex of the 119862

119899to the 119894th

neighbor of 119909 It has been proved that CCC(119899) is a Cayleygraph and hence is a vertex-transitive graph with degree 3Van Wieren et al [147] proved that CCC(119899) has an efficientdominating set if and only if 119899 = 5 From Theorems 194 and195 the following result can be derived

Theorem 199 (Huang and Xu [36] 2008) Let 119866 = 119862119862119862(119899)be the 119899-dimensional cube-connected cycles with 119899 ⩾ 3 and119899 = 5 Then 120574(119866) = 1198992119899minus2 and 2 ⩽ 119887(119866) ⩽ 3 In addition119887(119862119862119862(3)) = 119887(119862119862119862(4)) = 2

Problem 11 Determine the exact value of 119887(CCC(119899)) for 119899 ⩾5

Acknowledgments

This work was supported by NNSF of China (Grants nos11071233 61272008) The author would like to express hisgratitude to the anonymous referees for their kind sugges-tions and comments on the original paper to ProfessorSheikholeslami and Professor Jafari Rad for sending thereferences [128] and [81] which resulted in this version

References

[1] J-M Xu Theory and Application of Graphs vol 10 of NetworkTheory and Applications Kluwer Academic Dordrecht TheNetherlands 2003

[2] S THedetniemi andR C Laskar ldquoBibliography on dominationin graphs and some basic definitions of domination parame-tersrdquo Discrete Mathematics vol 86 no 1ndash3 pp 257ndash277 1990

[3] TW Haynes S T Hedetniemi and P J Slater Fundamentals ofDomination in Graphs vol 208 ofMonographs and Textbooks inPure and Applied Mathematics Marcel Dekker New York NYUSA 1998

[4] TWHaynes S THedetniemi and P J Slater EdsDominationin Graphs vol 209 of Monographs and Textbooks in Pure andAppliedMathematicsMarcelDekker NewYorkNYUSA 1998

[5] M R Garey andD S JohnsonComputers and Intractability WH Freeman and Company San Francisco Calif USA 1979

[6] H B Walikar and B D Acharya ldquoDomination critical graphsrdquoNational Academy Science Letters vol 2 pp 70ndash72 1979

[7] D Bauer F Harary J Nieminen and C L Suffel ldquoDominationalteration sets in graphsrdquo Discrete Mathematics vol 47 no 2-3pp 153ndash161 1983

[8] J F Fink M S Jacobson L F Kinch and J Roberts ldquoThebondage number of a graphrdquo Discrete Mathematics vol 86 no1ndash3 pp 47ndash57 1990

[9] F-T Hu and J-M Xu ldquoThe bondage number of (119899 minus 3)-regulargraph G of order nrdquo Ars Combinatoria to appear

[10] U Teschner ldquoNew results about the bondage number of agraphrdquoDiscreteMathematics vol 171 no 1ndash3 pp 249ndash259 1997

[11] B L Hartnell and D F Rall ldquoA bound on the size of a graphwith given order and bondage numberrdquo Discrete Mathematicsvol 197-198 pp 409ndash413 1999 16th British CombinatorialConference (London 1997)

[12] B L Hartnell and D F Rall ldquoA characterization of trees inwhich no edge is essential to the domination numberrdquo ArsCombinatoria vol 33 pp 65ndash76 1992

[13] J Topp and P D Vestergaard ldquo120572119896- and 120574

119896-stable graphsrdquo

Discrete Mathematics vol 212 no 1-2 pp 149ndash160 2000[14] A Meir and J W Moon ldquoRelations between packing and

covering numbers of a treerdquo Pacific Journal of Mathematics vol61 no 1 pp 225ndash233 1975

[15] B L Hartnell L K Joslashrgensen P D Vestergaard and CA Whitehead ldquoEdge stability of the k-domination numberof treesrdquo Bulletin of the Institute of Combinatorics and ItsApplications vol 22 pp 31ndash40 1998

[16] F-T Hu and J-M Xu ldquoOn the complexity of the bondage andreinforcement problemsrdquo Journal of Complexity vol 28 no 2pp 192ndash201 2012

[17] B L Hartnell and D F Rall ldquoBounds on the bondage numberof a graphrdquo Discrete Mathematics vol 128 no 1ndash3 pp 173ndash1771994

[18] Y-L Wang ldquoOn the bondage number of a graphrdquo DiscreteMathematics vol 159 no 1ndash3 pp 291ndash294 1996

[19] G H Fricke TW Haynes S M Hedetniemi S T Hedetniemiand R C Laskar ldquoExcellent treesrdquo Bulletin of the Institute ofCombinatorics and Its Applications vol 34 pp 27ndash38 2002

[20] V Samodivkin ldquoThe bondage number of graphs good and badverticesrdquoDiscussiones Mathematicae GraphTheory vol 28 no3 pp 453ndash462 2008

[21] J E Dunbar T W Haynes U Teschner and L VolkmannldquoBondage insensitivity and reinforcementrdquo in Domination inGraphs vol 209 of Monographs and Textbooks in Pure andAppliedMathematics pp 471ndash489 Dekker NewYork NY USA1998

[22] H Liu and L Sun ldquoThe bondage and connectivity of a graphrdquoDiscrete Mathematics vol 263 no 1ndash3 pp 289ndash293 2003

[23] A-H Esfahanian and S L Hakimi ldquoOn computing a con-ditional edge-connectivity of a graphrdquo Information ProcessingLetters vol 27 no 4 pp 195ndash199 1988

[24] R C Brigham P Z Chinn and R D Dutton ldquoVertexdomination-critical graphsrdquoNetworks vol 18 no 3 pp 173ndash1791988

[25] V Samodivkin ldquoDomination with respect to nondegenerateproperties bondage numberrdquoThe Australasian Journal of Com-binatorics vol 45 pp 217ndash226 2009

[26] U Teschner ldquoA new upper bound for the bondage numberof graphs with small domination numberrdquo The AustralasianJournal of Combinatorics vol 12 pp 27ndash35 1995

32 International Journal of Combinatorics

[27] J Fulman D Hanson and G MacGillivray ldquoVertex dom-ination-critical graphsrdquoNetworks vol 25 no 2 pp 41ndash43 1995

[28] U Teschner ldquoA counterexample to a conjecture on the bondagenumber of a graphrdquo Discrete Mathematics vol 122 no 1ndash3 pp393ndash395 1993

[29] U Teschner ldquoThe bondage number of a graph G can be muchgreater than Δ(G)rdquo Ars Combinatoria vol 43 pp 81ndash87 1996

[30] L Volkmann ldquoOn graphs with equal domination and coveringnumbersrdquoDiscrete AppliedMathematics vol 51 no 1-2 pp 211ndash217 1994

[31] L A Sanchis ldquoMaximum number of edges in connected graphswith a given domination numberrdquoDiscreteMathematics vol 87no 1 pp 65ndash72 1991

[32] S Klavzar and N Seifter ldquoDominating Cartesian products ofcyclesrdquoDiscrete AppliedMathematics vol 59 no 2 pp 129ndash1361995

[33] H K Kim ldquoA note on Vizingrsquos conjecture about the bondagenumberrdquo Korean Journal of Mathematics vol 10 no 2 pp 39ndash42 2003

[34] M Y Sohn X Yuan and H S Jeong ldquoThe bondage number of1198623times119862

119899rdquo Journal of the KoreanMathematical Society vol 44 no

6 pp 1213ndash1231 2007[35] L Kang M Y Sohn and H K Kim ldquoBondage number of the

discrete torus 119862119899times 119862

4rdquo Discrete Mathematics vol 303 no 1ndash3

pp 80ndash86 2005[36] J Huang and J-M Xu ldquoThe bondage numbers and efficient

dominations of vertex-transitive graphsrdquoDiscrete Mathematicsvol 308 no 4 pp 571ndash582 2008

[37] C J Xiang X Yuan andM Y Sohn ldquoDomination and bondagenumber of119862

3times119862

119899rdquoArs Combinatoria vol 97 pp 299ndash310 2010

[38] M S Jacobson and L F Kinch ldquoOn the domination numberof products of graphs Irdquo Ars Combinatoria vol 18 pp 33ndash441984

[39] Y-Q Li ldquoA note on the bondage number of a graphrdquo ChineseQuarterly Journal of Mathematics vol 9 no 4 pp 1ndash3 1994

[40] F-T Hu and J-M Xu ldquoBondage number of mesh networksrdquoFrontiers ofMathematics inChina vol 7 no 5 pp 813ndash826 2012

[41] R Frucht and F Harary ldquoOn the corona of two graphsrdquoAequationes Mathematicae vol 4 pp 322ndash325 1970

[42] K Carlson and M Develin ldquoOn the bondage number of planarand directed graphsrdquoDiscreteMathematics vol 306 no 8-9 pp820ndash826 2006

[43] U Teschner ldquoOn the bondage number of block graphsrdquo ArsCombinatoria vol 46 pp 25ndash32 1997

[44] J Huang and J-M Xu ldquoNote on conjectures of bondagenumbers of planar graphsrdquo Applied Mathematical Sciences vol6 no 65ndash68 pp 3277ndash3287 2012

[45] L Kang and J Yuan ldquoBondage number of planar graphsrdquoDiscrete Mathematics vol 222 no 1ndash3 pp 191ndash198 2000

[46] M Fischermann D Rautenbach and L Volkmann ldquoRemarkson the bondage number of planar graphsrdquo Discrete Mathemat-ics vol 260 no 1ndash3 pp 57ndash67 2003

[47] J Hou G Liu and J Wu ldquoBondage number of planar graphswithout small cyclesrdquoUtilitasMathematica vol 84 pp 189ndash1992011

[48] J Huang and J-M Xu ldquoThe bondage numbers of graphs withsmall crossing numbersrdquo Discrete Mathematics vol 307 no 15pp 1881ndash1897 2007

[49] Q Ma S Zhang and J Wang ldquoBondage number of 1-planargraphrdquo Applied Mathematics vol 1 no 2 pp 101ndash103 2010

[50] J Hou and G Liu ldquoA bound on the bondage number of toroidalgraphsrdquoDiscreteMathematics Algorithms and Applications vol4 no 3 Article ID 1250046 5 pages 2012

[51] Y-C Cao J-M Xu and X-R Xu ldquoOn bondage number oftoroidal graphsrdquo Journal of University of Science and Technologyof China vol 39 no 3 pp 225ndash228 2009

[52] Y-C Cao J Huang and J-M Xu ldquoThe bondage number ofgraphs with crossing number less than fourrdquo Ars Combinatoriato appear

[53] H K Kim ldquoOn the bondage numbers of some graphsrdquo KoreanJournal of Mathematics vol 11 no 2 pp 11ndash15 2004

[54] A Gagarin and V Zverovich ldquoUpper bounds for the bondagenumber of graphs on topological surfacesrdquo Discrete Mathemat-ics 2012

[55] J Huang ldquoAn improved upper bound for the bondage numberof graphs on surfacesrdquoDiscrete Mathematics vol 312 no 18 pp2776ndash2781 2012

[56] A Gagarin and V Zverovich ldquoThe bondage number of graphson topological surfaces and Teschnerrsquos conjecturerdquo DiscreteMathematics vol 313 no 6 pp 796ndash808 2013

[57] V Samodivkin ldquoNote on the bondage number of graphs ontopological surfacesrdquo httparxivorgabs12086203

[58] E J Cockayne R M Dawes and S T Hedetniemi ldquoTotaldomination in graphsrdquoNetworks vol 10 no 3 pp 211ndash219 1980

[59] R Laskar J Pfaff S M Hedetniemi and S T HedetniemildquoOn the algorithmic complexity of total dominationrdquo Society forIndustrial and Applied Mathematics vol 5 no 3 pp 420ndash4251984

[60] J Pfaff R C Laskar and S T Hedetniemi ldquoNP-completeness oftotal and connected domination and irredundance for bipartitegraphsrdquo Tech Rep 428 Department of Mathematical SciencesClemson University 1983

[61] M A Henning ldquoA survey of selected recent results on totaldomination in graphsrdquoDiscrete Mathematics vol 309 no 1 pp32ndash63 2009

[62] V R Kulli and D K Patwari ldquoThe total bondage number ofa graphrdquo in Advances in Graph Theory pp 227ndash235 VishwaGulbarga India 1991

[63] F-T Hu Y Lu and J-M Xu ldquoThe total bondage number ofgrid graphsrdquo Discrete Applied Mathematics vol 160 no 16-17pp 2408ndash2418 2012

[64] J Cao M Shi and B Wu ldquoResearch on the total bondagenumber of a spectial networkrdquo in Proceedings of the 3rdInternational Joint Conference on Computational Sciences andOptimization (CSO rsquo10) pp 456ndash458 May 2010

[65] N Sridharan MD Elias and V S A Subramanian ldquoTotalbondage number of a graphrdquo AKCE International Journal ofGraphs and Combinatorics vol 4 no 2 pp 203ndash209 2007

[66] N Jafari Rad and J Raczek ldquoSome progress on total bondage ingraphsrdquo Graphs and Combinatorics 2011

[67] T W Haynes and P J Slater ldquoPaired-domination and thepaired-domatic numberrdquoCongressusNumerantium vol 109 pp65ndash72 1995

[68] T W Haynes and P J Slater ldquoPaired-domination in graphsrdquoNetworks vol 32 no 3 pp 199ndash206 1998

[69] X Hou ldquoA characterization of 2120574 120574119875-treesrdquoDiscrete Mathemat-

ics vol 308 no 15 pp 3420ndash3426 2008[70] K E Proffitt T W Haynes and P J Slater ldquoPaired-domination

in grid graphsrdquo Congressus Numerantium vol 150 pp 161ndash1722001

International Journal of Combinatorics 33

[71] H Qiao L KangM Cardei andD-Z Du ldquoPaired-dominationof treesrdquo Journal of Global Optimization vol 25 no 1 pp 43ndash542003

[72] E Shan L Kang and M A Henning ldquoA characterizationof trees with equal total domination and paired-dominationnumbersrdquo The Australasian Journal of Combinatorics vol 30pp 31ndash39 2004

[73] J Raczek ldquoPaired bondage in treesrdquo Discrete Mathematics vol308 no 23 pp 5570ndash5575 2008

[74] J A Telle and A Proskurowski ldquoAlgorithms for vertex parti-tioning problems on partial k-treesrdquo SIAM Journal on DiscreteMathematics vol 10 no 4 pp 529ndash550 1997

[75] G S Domke J H Hattingh M A Henning and L R MarkusldquoRestrained domination in treesrdquoDiscreteMathematics vol 211no 1ndash3 pp 1ndash9 2000

[76] G S Domke J H Hattingh M A Henning and L R MarkusldquoRestrained domination in graphs with minimum degree twordquoJournal of Combinatorial Mathematics and Combinatorial Com-puting vol 35 pp 239ndash254 2000

[77] G S Domke J H Hattingh S T Hedetniemi R C Laskarand L R Markus ldquoRestrained domination in graphsrdquo DiscreteMathematics vol 203 no 1ndash3 pp 61ndash69 1999

[78] J H Hattingh and E J Joubert ldquoAn upper bound for therestrained domination number of a graph with minimumdegree at least two in terms of order and minimum degreerdquoDiscrete Applied Mathematics vol 157 no 13 pp 2846ndash28582009

[79] M A Henning ldquoGraphs with large restrained dominationnumberrdquo Discrete Mathematics vol 197-198 pp 415ndash429 199916th British Combinatorial Conference (London 1997)

[80] J H Hattingh and A R Plummer ldquoRestrained bondage ingraphsrdquo Discrete Mathematics vol 308 no 23 pp 5446ndash54532008

[81] N Jafari Rad R Hasni J Raczek and L Volkmann ldquoTotalrestrained bondage in graphsrdquoActaMathematica Sinica vol 29no 6 pp 1033ndash1042 2013

[82] J Cyman and J Raczek ldquoOn the total restrained dominationnumber of a graphrdquoThe Australasian Journal of Combinatoricsvol 36 pp 91ndash100 2006

[83] M A Henning ldquoGraphs with large total domination numberrdquoJournal of Graph Theory vol 35 no 1 pp 21ndash45 2000

[84] D-X Ma X-G Chen and L Sun ldquoOn total restraineddomination in graphsrdquoCzechoslovakMathematical Journal vol55 no 1 pp 165ndash173 2005

[85] B Zelinka ldquoRemarks on restrained domination and totalrestrained domination in graphsrdquo Czechoslovak MathematicalJournal vol 55 no 2 pp 393ndash396 2005

[86] W Goddard and M A Henning ldquoIndependent domination ingraphs a survey and recent resultsrdquo Discrete Mathematics vol313 no 7 pp 839ndash854 2013

[87] J H Zhang H L Liu and L Sun ldquoIndependence bondagenumber and reinforcement number of some graphsrdquo Transac-tions of Beijing Institute of Technology vol 23 no 2 pp 140ndash1422003

[88] K D Dharmalingam Studies in Graph Theorymdashequitabledomination and bottleneck domination [PhD thesis] MaduraiKamaraj University Madurai India 2006

[89] GDeepakND Soner andAAlwardi ldquoThe equitable bondagenumber of a graphrdquo Research Journal of Pure Algebra vol 1 no9 pp 209ndash212 2011

[90] E Sampathkumar and H B Walikar ldquoThe connected domina-tion number of a graphrdquo Journal of Mathematical and PhysicalSciences vol 13 no 6 pp 607ndash613 1979

[91] K White M Farber and W Pulleyblank ldquoSteiner trees con-nected domination and strongly chordal graphsrdquoNetworks vol15 no 1 pp 109ndash124 1985

[92] M Cadei M X Cheng X Cheng and D-Z Du ldquoConnecteddomination in Ad Hoc wireless networksrdquo in Proceedings ofthe 6th International Conference on Computer Science andInformatics (CSampI rsquo02) 2002

[93] J F Fink and M S Jacobson ldquon-domination in graphsrdquo inGraph Theory with Applications to Algorithms and ComputerScience Wiley-Interscience Publications pp 283ndash300 WileyNew York NY USA 1985

[94] M Chellali O Favaron A Hansberg and L Volkmann ldquok-domination and k-independence in graphs a surveyrdquo Graphsand Combinatorics vol 28 no 1 pp 1ndash55 2012

[95] Y Lu X Hou J-M Xu andN Li ldquoTrees with uniqueminimump-dominating setsrdquo Utilitas Mathematica vol 86 pp 193ndash2052011

[96] Y Lu and J-M Xu ldquoThe p-bondage number of treesrdquo Graphsand Combinatorics vol 27 no 1 pp 129ndash141 2011

[97] Y Lu and J-M Xu ldquoThe 2-domination and 2-bondage numbersof grid graphs A manuscriptrdquo httparxivorgabs12044514

[98] M Krzywkowski ldquoNon-isolating 2-bondage in graphsrdquo TheJournal of the Mathematical Society of Japan vol 64 no 4 2012

[99] M A Henning O R Oellermann and H C Swart ldquoBoundson distance domination parametersrdquo Journal of CombinatoricsInformation amp System Sciences vol 16 no 1 pp 11ndash18 1991

[100] F Tian and J-M Xu ldquoBounds for distance domination numbersof graphsrdquo Journal of University of Science and Technology ofChina vol 34 no 5 pp 529ndash534 2004

[101] F Tian and J-M Xu ldquoDistance domination-critical graphsrdquoApplied Mathematics Letters vol 21 no 4 pp 416ndash420 2008

[102] F Tian and J-M Xu ldquoA note on distance domination numbersof graphsrdquo The Australasian Journal of Combinatorics vol 43pp 181ndash190 2009

[103] F Tian and J-M Xu ldquoAverage distances and distance domina-tion numbersrdquo Discrete Applied Mathematics vol 157 no 5 pp1113ndash1127 2009

[104] Z-L Liu F Tian and J-M Xu ldquoProbabilistic analysis of upperbounds for 2-connected distance k-dominating sets in graphsrdquoTheoretical Computer Science vol 410 no 38ndash40 pp 3804ndash3813 2009

[105] M A Henning O R Oellermann and H C Swart ldquoDistancedomination critical graphsrdquo Journal of Combinatorial Mathe-matics and Combinatorial Computing vol 44 pp 33ndash45 2003

[106] T J Bean M A Henning and H C Swart ldquoOn the integrityof distance domination in graphsrdquo The Australasian Journal ofCombinatorics vol 10 pp 29ndash43 1994

[107] M Fischermann and L Volkmann ldquoA remark on a conjecturefor the (kp)-domination numberrdquoUtilitasMathematica vol 67pp 223ndash227 2005

[108] T Korneffel D Meierling and L Volkmann ldquoA remark on the(22)-domination numberrdquo Discussiones Mathematicae GraphTheory vol 28 no 2 pp 361ndash366 2008

[109] Y Lu X Hou and J-M Xu ldquoOn the (22)-domination numberof treesrdquo Discussiones Mathematicae Graph Theory vol 30 no2 pp 185ndash199 2010

34 International Journal of Combinatorics

[110] H Li and J-M Xu ldquo(dm)-domination numbers of m-connected graphsrdquo Rapports de Recherche 1130 LRI UniversiteParis-Sud 1997

[111] S M Hedetniemi S T Hedetniemi and T V Wimer ldquoLineartime resource allocation algorithms for treesrdquo Tech Rep URI-014 Department of Mathematical Sciences Clemson Univer-sity 1987

[112] M Farber ldquoDomination independent domination and dualityin strongly chordal graphsrdquo Discrete Applied Mathematics vol7 no 2 pp 115ndash130 1984

[113] G S Domke and R C Laskar ldquoThe bondage and reinforcementnumbers of 120574

119891for some graphsrdquoDiscrete Mathematics vol 167-

168 pp 249ndash259 1997[114] G S Domke S T Hedetniemi and R C Laskar ldquoFractional

packings coverings and irredundance in graphsrdquo vol 66 pp227ndash238 1988

[115] E J Cockayne P A Dreyer Jr S M Hedetniemi andS T Hedetniemi ldquoRoman domination in graphsrdquo DiscreteMathematics vol 278 no 1ndash3 pp 11ndash22 2004

[116] I Stewart ldquoDefend the roman empirerdquo Scientific American vol281 pp 136ndash139 1999

[117] C S ReVelle and K E Rosing ldquoDefendens imperiumromanum a classical problem in military strategyrdquoThe Ameri-can Mathematical Monthly vol 107 no 7 pp 585ndash594 2000

[118] A Bahremandpour F-T Hu S M Sheikholeslami and J-MXu ldquoOn the Roman bondage number of a graphrdquo DiscreteMathematics Algorithms and Applications vol 5 no 1 ArticleID 1350001 15 pages 2013

[119] N Jafari Rad and L Volkmann ldquoRoman bondage in graphsrdquoDiscussiones Mathematicae Graph Theory vol 31 no 4 pp763ndash773 2011

[120] K Ebadi and L PushpaLatha ldquoRoman bondage and Romanreinforcement numbers of a graphrdquo International Journal ofContemporary Mathematical Sciences vol 5 pp 1487ndash14972010

[121] N Dehgardi S M Sheikholeslami and L Volkman ldquoOn theRoman k-bondage number of a graphrdquo AKCE InternationalJournal of Graphs and Combinatorics vol 8 pp 169ndash180 2011

[122] S Akbari and S Qajar ldquoA note on Roman bondage number ofgraphsrdquo Ars Combinatoria to Appear

[123] F-T Hu and J-M Xu ldquoRoman bondage numbers of somegraphsrdquo httparxivorgabs11093933

[124] N Jafari Rad and L Volkmann ldquoOn the Roman bondagenumber of planar graphsrdquo Graphs and Combinatorics vol 27no 4 pp 531ndash538 2011

[125] S Akbari M Khatirinejad and S Qajar ldquoA note on the romanbondage number of planar graphsrdquo Graphs and Combinatorics2012

[126] B Bresar M A Henning and D F Rall ldquoRainbow dominationin graphsrdquo Taiwanese Journal of Mathematics vol 12 no 1 pp213ndash225 2008

[127] B L Hartnell and D F Rall ldquoOn dominating the Cartesianproduct of a graph and 120572krdquo Discussiones Mathematicae GraphTheory vol 24 no 3 pp 389ndash402 2004

[128] N Dehgardi S M Sheikholeslami and L Volkman ldquoThe k-rainbow bondage number of a graphrdquo to appear in DiscreteApplied Mathematics

[129] J von Neumann and O Morgenstern Theory of Games andEconomic Behavior Princeton University Press Princeton NJUSA 1944

[130] C BergeTheorıe des Graphes et ses Applications 1958[131] O Ore Theory of Graphs vol 38 American Mathematical

Society Colloquium Publications 1962[132] E Shan and L Kang ldquoBondage number in oriented graphsrdquoArs

Combinatoria vol 84 pp 319ndash331 2007[133] J Huang and J-M Xu ldquoThe bondage numbers of extended

de Bruijn and Kautz digraphsrdquo Computers amp Mathematics withApplications vol 51 no 6-7 pp 1137ndash1147 2006

[134] J Huang and J-M Xu ldquoThe total domination and total bondagenumbers of extended de Bruijn and Kautz digraphsrdquoComputersamp Mathematics with Applications vol 53 no 8 pp 1206ndash12132007

[135] Y Shibata and Y Gonda ldquoExtension of de Bruijn graph andKautz graphrdquo Computers amp Mathematics with Applications vol30 no 9 pp 51ndash61 1995

[136] X Zhang J Liu and JMeng ldquoThebondage number in completet-partite digraphsrdquo Information Processing Letters vol 109 no17 pp 997ndash1000 2009

[137] D W Bange A E Barkauskas and P J Slater ldquoEfficient dom-inating sets in graphsrdquo in Applications of Discrete Mathematicspp 189ndash199 SIAM Philadelphia Pa USA 1988

[138] M Livingston and Q F Stout ldquoPerfect dominating setsrdquoCongressus Numerantium vol 79 pp 187ndash203 1990

[139] L Clark ldquoPerfect domination in random graphsrdquo Journal ofCombinatorial Mathematics and Combinatorial Computing vol14 pp 173ndash182 1993

[140] D W Bange A E Barkauskas L H Host and L H ClarkldquoEfficient domination of the orientations of a graphrdquo DiscreteMathematics vol 178 no 1ndash3 pp 1ndash14 1998

[141] A E Barkauskas and L H Host ldquoFinding efficient dominatingsets in oriented graphsrdquo Congressus Numerantium vol 98 pp27ndash32 1993

[142] I J Dejter and O Serra ldquoEfficient dominating sets in CayleygraphsrdquoDiscrete AppliedMathematics vol 129 no 2-3 pp 319ndash328 2003

[143] J Lee ldquoIndependent perfect domination sets in Cayley graphsrdquoJournal of Graph Theory vol 37 no 4 pp 213ndash219 2001

[144] WGu X Jia and J Shen ldquoIndependent perfect domination setsin meshes tori and treesrdquo Unpublished

[145] N Obradovic J Peters and G Ruzic ldquoEfficient domination incirculant graphswith two chord lengthsrdquo Information ProcessingLetters vol 102 no 6 pp 253ndash258 2007

[146] K Reji Kumar and G MacGillivray ldquoEfficient domination incirculant graphsrdquo Discrete Mathematics vol 313 no 6 pp 767ndash771 2013

[147] D Van Wieren M Livingston and Q F Stout ldquoPerfectdominating sets on cube-connected cyclesrdquo Congressus Numer-antium vol 97 pp 51ndash70 1993

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

International Journal of Combinatorics 23

Theorem 145 (Krzywkowski [98] 2012) For any positiveintegers 119899 and119898 with119898 ⩽ 119899

1198871015840

2(119870

119899) =

0 119894119891 119899 = 1 2 3

lfloor2119899

3rfloor 119900119905ℎ119890119903119908119894119904119890

1198871015840

2(119875

119899) =

0 119894119891 119899 = 1 2 3

1 119900119905ℎ119890119903119908119894119904119890

1198871015840

2(119862

119899) =

0 119894119891 119899 = 3

1 119894119891 119899 119894119904 119890V119890119899

2 119894119891 119899 119894119904 119900119889119889

1198871015840

2(119870

119898119899) =

3 119894119891 119898 = 119899 = 3

1 119894119891 119898 = 119899 = 4

2 119900119905ℎ119890119903119908119894119904119890

(55)

92 Distance Bondage Numbers A subset 119878 of vertices of agraph119866 is said to be a distance 119896-dominating set for119866 if everyvertex in 119866 not in 119878 is at distance at most 119896 from some vertexof 119878 The minimum cardinality of all distance 119896-dominatingsets is called the distance 119896-domination number of 119866 anddenoted by 120574

119896(119866) (do not confuse with the above-mentioned

120574119901(119866)) When 119896 = 1 a distance 1-dominating set is a normal

dominating set and so 1205741(119866) = 120574(119866) for any graph 119866 Thus

the distance 119896-domination is a generalization of the classicaldomination

The relation between 120574119896and 120572

119896(the 119896-independence

number defined in Section 22) for a tree obtained by Meirand Moon [14] who proved that 120574

119896(119879) = 120572

2119896(119879) for any tree

119879 (a special case for 119896 = 1 is stated in Proposition 10) Thefurther research results can be found in Henning et al [99]Tian and Xu [100ndash103] and Liu et al [104]

In 1998 Hartnell et al [15] defined the distance 119896-bondagenumber of 119866 denoted by 119887

119896(119866) to be the cardinality of a

smallest subset 119861 of edges of 119866 with the property that 120574119896(119866 minus

119861) gt 120574119896(119866) FromTheorem 6 it is clear that if119879 is a nontrivial

tree then 1 ⩽ 1198871(119879) ⩽ 2 Hartnell et al [15] generalized this

result to any integer 119896 ⩾ 1

Theorem 146 (Hartnell et al [15] 1998) For every nontrivialtree 119879 and positive integer 119896 1 ⩽ 119887

119896(119879) ⩽ 2

Hartnell et al [15] and Topp and Vestergaard [13] alsocharacterized the trees having distance 119896-bondage number 2In particular the class of trees for which 119887

1(119879) = 2 are just

those which have a unique maximum 2-independent set (seeTheorem 11)

Since when 119896 = 1 the distance 1-bondage number 1198871(119866)

is the classical bondage number 119887(119866) Theorem 12 gives theNP-hardness of deciding the distance 119896-bondage number ofgeneral graphs

Theorem 147 Given a nonempty undirected graph 119866 andpositive integers 119896 and 119887 with 119887 ⩽ 120576(119866) determining wetheror not 119887

119896(119866) ⩽ 119887 is NP-hard

For a vertex 119909 in119866 the open 119896-neighborhood119873119896(119909) of 119909 is

defined as119873119896(119909) = 119910 isin 119881(119866) 1 ⩽ 119889

119866(119909 119910) le 119896The closed

119896-neighborhood119873119896[119909] of 119909 in 119866 is defined as119873

119896(119909) cup 119909 Let

Δ119896(119866) = max 1003816100381610038161003816119873119896

(119909)1003816100381610038161003816 for any 119909 isin 119881 (119866) (56)

Clearly Δ1(119866) = Δ(119866) The 119896th power of a graph 119866 is the

graph119866119896 with vertex-set119881(119866119896

) = 119881(119866) and edge-set119864(119866119896

) =

119909119910 1 ⩽ 119889119866(119909 119910) ⩽ 119896 The following lemma holds directly

from the definition of 119866119896

Proposition 148 (Tian and Xu [101]) Δ(119866119896

) = Δ119896(119866) and

120574(119866119896

) = 120574119896(119866) for any graph 119866 and each 119896 ⩾ 1

A graph 119866 is 119896-distance domination-critical or 120574119896-critical

for short if 120574119896(119866 minus 119909) lt 120574

119896(119866) for every vertex 119909 in 119866

proposed by Henning et al [105]

Proposition 149 (Tian and Xu [101]) For each 119896 ⩾ 1 a graph119866 is 120574

119896-critical if and only if 119866119896 is 120574

119896-critical

By the above facts we suggest the following worthwhileproblem

Problem 6 Can we generalize the results on the bondage for119866 to 119866119896 In particular do the following propositions hold

(a) 119887(119866119896

) = 119887119896(119866) for any graph 119866 and each 119896 ⩾ 1

(b) 119887(119866119896

) ⩽ Δ119896(119866) if 119866 is not 120574

119896-critical

Let 119896 and 119901 be positive integers A subset 119878 of 119881(119866) isdefined to be a (119896 119901)-dominating set of 119866 if for any vertex119909 isin 119878 |119873

119896(119909) cap 119878| ⩾ 119901 The (119896 119901)-domination number of

119866 denoted by 120574119896119901(119866) is the minimum cardinality among

all (119896 119901)-dominating sets of 119866 Clearly for a graph 119866 a(1 1)-dominating set is a classical dominating set a (119896 1)-dominating set is a distance 119896-dominating set and a (1 119901)-dominating set is the above-mentioned 119901-dominating setThat is 120574

11(119866) = 120574(119866) 120574

1198961(119866) = 120574

119896(119866) and 120574

1119901(119866) = 120574

119901(119866)

The concept of (119896 119901)-domination in a graph 119866 is a gen-eralized domination which combines distance 119896-dominationand 119901-domination in 119866 So the investigation of (119896 119901)-domination of 119866 is more interesting and has received theattention of many researchers see for example [106ndash109]

More general Li and Xu [110] proposed the concept of the(ℓ 119908)-domination number of a119908-connected graph A subset119878 of 119881(119866) is defined to be an (ℓ 119908)-dominating set of 119866 iffor any vertex 119909 isin 119878 there are 119908 internally disjoint pathsof length at most ℓ from 119909 to some vertex in 119878 The (ℓ 119908)-domination number of119866 denoted by 120574

ℓ119908(119866) is theminimum

cardinality among all (ℓ 119908)-dominating sets of 119866 Clearly1205741198961(119866) = 120574

119896(119866) and 120574

1119901(119866) = 120574

119901(119866)

It is quite natural to propose the concept of bondage num-ber for (119896 119901)-domination or (ℓ 119908)-domination However asfar as we know none has proposed this concept until todayThis is a worthwhile topic for further research

24 International Journal of Combinatorics

93 Fractional Bondage Numbers If 120590 is a function mappingthe vertex-set 119881 into some set of real numbers then for anysubset 119878 sube 119881 let 120590(119878) = sum

119909isin119878120590(119909) Also let |120590| = 120590(119881)

A real-value function 120590 119881 rarr [0 1] is a dominatingfunction of a graph 119866 if for every 119909 isin 119881(119866) 120590(119873

119866[119909]) ⩾ 1

Thus if 119878 is a dominating set of 119866 120590 is a function where

120590 (119909) = 1 if 119909 isin 1198780 otherwise

(57)

then 120590 is a dominating function of119866 The fractional domina-tion number of 119866 denoted by 120574

119865(119866) is defined as follows

120574119865(119866) = min |120590| 120590 is a domination function of 119866

(58)

The 120574119865-bondage number of 119866 denoted by 119887

119865(119866) is

defined as the minimum cardinality of a subset 119861 sube 119864 whoseremoval results in 120574

119865(119866 minus 119861) gt 120574

119865(119866)

Hedetniemi et al [111] are the first to study fractionaldomination although Farber [112] introduced the idea indi-rectly The concept of the fractional domination numberwas proposed by Domke and Laskar [113] in 1997 Thefractional domination numbers for some ordinary graphs aredetermined

Proposition 150 (Domke et al [114] 1998 Domke and Laskar[113] 1997) (a) If 119866 is a 119896-regular graph with order 119899 then120574119865(119866) = 119899119896 + 1(b) 120574

119865(119862

119899) = 1198993 120574

119865(119875

119899) = lceil1198993rceil 120574

119865(119870

119899) = 1

(c) 120574119865(119866) = 1 if and only if Δ(119866) = 119899 minus 1

(d) 120574119865(119879) = 120574(119879) for any tree 119879

(e) 120574119865(119870

119899119898) = (119899(119898 + 1) + 119898(119899 + 1))(119899119898 minus 1) where

max119899119898 gt 1

The assertions (a)ndash(d) are due to Domke et al [114] andthe assertion (e) is due to Domke and Laskar [113] Accordingto these results Domke and Laskar [113] determined the 120574

119865-

bondage numbers for these graphs

Theorem 151 (Domke and Laskar [113] 1997) 119887119865(119870

119899) =

lceil1198992rceil 119887119865(119870

119899119898) = 1 wheremax119899119898 gt 1

119887119865(119862

119899) =

2 119894119891 119899 equiv 0 (mod 3) 1 119900119905ℎ119890119903119908119894119904119890

119887119865(119875

119899) =

2 119894119891 119899 equiv 1 (mod 3) 1 119900119905ℎ119890119903119908119894119904119890

(59)

It is easy to see that for any tree119879 119887119865(119879) ⩽ 2 In fact since

120574119865(119879) = 120574(119879) for any tree 119879 120574

119865(119879

1015840

) = 120574(1198791015840

) for any subgraph1198791015840 of 119879 It follows that

119887119865(119879) = min 119861 sube 119864 (119879) 120574

119865(119879 minus 119861) gt 120574

119865(119879)

= min 119861 sube 119864 (119879) 120574 (119879 minus 119861) gt 120574 (119879)

= 119887 (119879) ⩽ 2

(60)

Except for these no results on 119887119865(119866) have been known as

yet

94 Roman Bondage Numbers A Roman dominating func-tion on a graph 119866 is a labeling 119891 119881 rarr 0 1 2 such thatevery vertex with label 0 has at least one neighbor with label2 The weight of a Roman dominating function 119891 on 119866 is thevalue

119891 (119866) = sum

119906isin119881(119866)

119891 (119906) (61)

The minimum weight of a Roman dominating function ona graph 119866 is called the Roman domination number of 119866denoted by 120574RM (119866)

A Roman dominating function 119891 119881 rarr 0 1 2

can be represented by the ordered partition (1198810 119881

1 119881

2) (or

(119881119891

0 119881

119891

1 119881

119891

2) to refer to119891) of119881 where119881

119894= V isin 119881 | 119891(V) = 119894

In this representation its weight 119891(119866) = |1198811| + 2|119881

2| It is

clear that119881119891

1cup119881

119891

2is a dominating set of 119866 called the Roman

dominating set denoted by 119863119891

R = (1198811 1198812) Since 119881119891

1cup 119881

119891

2is

a dominating set when 119891 is a Roman dominating functionand since placing weight 2 at the vertices of a dominating setyields a Roman dominating function in [115] it is observedthat

120574 (119866) ⩽ 120574RM (119866) ⩽ 2120574 (119866) (62)

A graph 119866 is called to be Roman if 120574RM (119866) = 2120574(119866)The definition of the Roman domination function is

proposed implicitly by Stewart [116] and ReVelle and Rosing[117] Roman dominating numbers have been deeply studiedIn particular Bahremandpour et al [118] showed that theproblem determining the Roman domination number is NP-complete even for bipartite graphs

Let 119866 be a graph with maximum degree at least twoThe Roman bondage number 119887RM (119866) of 119866 is the minimumcardinality of all sets 1198641015840 sube 119864 for which 120574RM (119866 minus 119864

1015840

) gt

120574RM (119866) Since in study of Roman bondage number theassumption Δ(119866) ⩾ 2 is necessary we always assume thatwhen we discuss 119887RM (119866) all graphs involved satisfy Δ(119866) ⩾2 The Roman bondage number 119887RM (119866) is introduced byJafari Rad and Volkmann in [119]

Recently Bahremandpour et al [118] have shown that theproblem determining the Roman bondage number is NP-hard even for bipartite graphs

Problem 7 Consider the decision problem

Roman Bondage Problem

Instance a nonempty bipartite graph119866 and a positiveinteger 119896

Question is 119887RM (119866) ⩽ 119896

Theorem 152 (Bahremandpour et al [118] 2013) TheRomanbondage problem is NP-hard even for bipartite graphs

The exact value of 119887RM(119866) is known only for a few familyof graphs including the complete graphs cycles and paths

International Journal of Combinatorics 25

Theorem 153 (Jafari Rad and Volkmann [119] 2011) For anypositive integers 119899 and119898 with119898 ⩽ 119899

119887RM (119875119899) = 2 119894119891 119899 equiv 2 (mod 3) 1 119900119905ℎ119890119903119908119894119904119890

119887RM (119862119899) =

3 119894119891 119899 = 2 (mod 3) 2 119900119905ℎ119890119903119908119894119904119890

119887RM (119870119899) = lceil

119899

2rceil 119891119900119903 119899 ⩾ 3

119887RM (119870119898119899) =

1 119894119891 119898 = 1 119886119899119889 119899 = 1

4 119894119891 119898 = 119899 = 3

119898 119900119905ℎ119890119903119908119894119904119890

(63)

Lemma 154 (Cockayne et al [115] 2004) If 119866 is a graph oforder 119899 and contains vertices of degree 119899 minus 1 then 120574RM(119866) = 2

Using Lemma 154 the third conclusion in Theorem 153can be generalized to more general case which is similar toLemma 49

Theorem 155 (Jafari Rad and Volkmann [119] 2011) Let119866 bea graph with order 119899 ge 3 and 119905 the number of vertices of degree119899 minus 1 in 119866 If 119905 ⩾ 1 then 119887RM(119866) = lceil1199052rceil

Ebadi and PushpaLatha [120] conjectured that 119887RM(119866) ⩽119899 minus 1 for any graph 119866 of order 119899 ⩾ 3 Dehgardi et al[121] Akbari andQajar [122] independently showed that thisconjecture is true

Theorem 156 119887RM(119866) ⩽ 119899 minus 1 for any connected graph 119866 oforder 119899 ⩾ 3

For a complete 119905-partite graph we have the followingresult

Theorem 157 (Hu and Xu [123] 2011) Let 119866 = 11987011989811198982119898119905

bea complete 119905-partite graph with 119898

1= sdot sdot sdot = 119898

119894lt 119898

119894+1le sdot sdot sdot ⩽

119898119905 119905 ⩾ 2 and 119899 = sum119905

119895=1119898

119895 Then

119887RM (119866) =

lceil119894

2rceil 119894119891 119898

119894= 1 119886119899119889 119899 ⩾ 3

2 119894119891 119898119894= 2 119886119899119889 119894 = 1

119894 119894119891 119898119894= 2 119886119899119889 119894 ⩾ 2

4 119894119891 119898119894= 3 119886119899119889 119894 = 119905 = 2

119899 minus 1 119894119891 119898119894= 3 119886119899119889 119894 = 119905 ⩾ 3

119899 minus 119898119905

119894119891 119898119894⩾ 3 119886119899119889 119898

119905⩾ 4

(64)

Consider a complete 119905-partite graph 119870119905(3) for 119905 ⩾ 3

119887RM(119870119905(3)) = 119899 minus 1 by Theorem 157 where 119899 = 3119905

which shows that this upper bound of 119899 minus 1 on 119887RM(119866) inTheorem 156 is sharp This fact and 120574

119877(119870

119905(3)) = 4 give

a negative answer to the following two problems posed byDehgardi et al [121]Prove or Disprove for any connected graph 119866 of order 119899 ge 3(i) 119887RM(119866) = 119899 minus 1 if and only if 119866 cong 119870

3 (ii) 119887RM(119866) ⩽ 119899 minus

120574119877(119866) + 1Note that a complete 119905-partite graph 119870

119905(3) is (119899 minus 3)-

regular and 119887RM(119870119905(3)) = 119899 minus 1 for 119905 ⩾ 3 where 119899 = 3119905

Hu and Xu further determined that 119887119877(119866) = 119899 minus 2 for any

(119899 minus 3)-regular graph 119866 of order 119899 ⩽ 5 except 119870119905(3)

Theorem 158 (Hu and Xu [123] 2011) For any (119899minus3)-regulargraph 119866 of order 119899 other than119870

119905(3) 119887RM(119866) = 119899 minus 2 for 119899 ⩾ 5

For a tree 119879 with order 119899 ⩾ 3 Ebadi and PushpaLatha[120] and Jafari Rad and Volkmann [119] independentlyobtained an upper bound on 119887RM(119879)

Theorem 159 119887RM(119879) ⩽ 3 for any tree 119879 with order 119899 ⩾ 3

Theorem 160 (Bahremandpour et al [118] 2013) 119887RM(1198752 times119875119899) = 2 for 119899 ⩾ 2

Theorem 161 (Jafari Rad and Volkmann [119] 2011) Let119866 bea graph with order at least three

(a) If (119909 119910 119911) is a path of length 2 in 119866 then 119887RM(119866) ⩽119889119866(119909) + 119889

119866(119910) + 119889

119866(119911) minus 3 minus |119873

119866(119909) cap 119873

119866(119910)|

(b) If 119866 is connected then 119887RM(119866) ⩽ 120582(119866) + 2Δ(119866) minus 3

Theorem 161 (b) implies 119887RM(119866) ⩽ 120575(119866) + 2Δ(119866) minus 3 for aconnected graph 119866 Note that for a planar graph 119866 120575(119866) ⩽ 5moreover 120575(119866) ⩽ 3 if the girth at least 4 and 120575(119866) ⩽ 2 if thegirth at least 6 These two facts show that 119887RM(119866) ⩽ 2Δ(119866) +2 for a connected planar graph 119866 Jafari Rad and Volkmann[124] improved this bound

Theorem 162 (Jafari Rad and Volkmann [124] 2011) For anyconnected planar graph 119866 of order 119899 ⩾ 3 with girth 119892(119866)

119887RM (119866)

⩽

2Δ (119866)

Δ (119866) + 6

Δ (119866) + 5 119894119891 119866 119888119900119899119905119886119894119899119904 119899119900 V119890119903119905119894119888119890119904 119900119891 119889119890119892119903119890119890 119891119894V119890

Δ (119866) + 4 119894119891 119892 (119866) ⩾ 4

Δ (119866) + 3 119894119891 119892 (119866) ⩾ 5

Δ (119866) + 2 119894119891 119892 (119866) ⩾ 6

Δ (119866) + 1 119894119891 119892 (119866) ⩾ 8

(65)

According toTheorem 157 119887RM(119862119899) = 3 = Δ(119862

119899)+1 for a

cycle 119862119899of length 119899 ⩾ 8 with 119899 equiv 2 (mod 3) and therefore

the last upper bound inTheorem 162 is sharp at least for Δ =2

Combining the fact that every planar graph 119866 withminimum degree 5 contains an edge 119909119910 with 119889

119866(119909) = 5

and 119889119866(119910) isin 5 6 with Theorem 161 (a) Akbari et al [125]

obtained the following result

26 International Journal of Combinatorics

middot middot middot

middot middot middot

middot middot middot

middot middot middot

119866

Figure 9 The graph 119866 is constructed from 119866

Theorem 163 (Akbari et al [125] 2012) 119887RM(119866) ⩽ 15 forevery planar graph 119866

It remains open to show whether the bound inTheorem 163 is sharp or not Though finding a planargraph 119866 with 119887RM(119866) = 15 seems to be difficult Akbari etal [125] constructed an infinite family of planar graphs withRoman bondage number equal to 7 by proving the followingresult

Theorem 164 (Akbari et al [125] 2012) Let 119866 be a graph oforder 119899 and 119866 is the graph of order 5119899 obtained from 119866 byattaching the central vertex of a copy of a path 119875

5to each vertex

of119866 (see Figure 9) Then 120574RM(119866) = 4119899 and 119887RM(119866) = 120575(119866)+2

ByTheorem 164 infinitemany planar graphswith Romanbondage number 7 can be obtained by considering any planargraph 119866 with 120575(119866) = 5 (eg the icosahedron graph)

Conjecture 165 (Akbari et al [125] 2012) The Romanbondage number of every planar graph is at most 7

For general bounds the following observation is directlyobtained which is similar to Observation 1

Observation 4 Let 119866 be a graph of order 119899 with maximumdegree at least two Assume that 119867 is a spanning subgraphobtained from 119866 by removing 119896 edges If 120574RM(119867) = 120574RM(119866)then 119887

119877(119867) ⩽ 119887RM(119866) ⩽ 119887RM(119867) + 119896

Theorem 166 (Bahremandpour et al [118] 2013) For anyconnected graph 119866 of order 119899 if 119899 ⩾ 3 and 120574RM(119866) = 120574(119866) + 1then

119887RM (119866) ⩽ min 119887 (119866) 119899Δ (66)

where 119899Δis the number of vertices with maximum degree Δ in

119866

Observation 5 For any graph 119866 if 120574(119866) = 120574RM(119866) then119887RM(119866) ⩽ 119887(119866)

Theorem 167 (Bahremandpour et al [118] 2013) For everyRoman graph 119866

119887RM (119866) ⩾ 119887 (119866) (67)

The bound is sharp for cycles on 119899 vertices where 119899 equiv 0 (mod3)

The strict inequality in Theorem 167 can hold for exam-ple 119887(119862

3119896+2) = 2 lt 3 = 119887RM(1198623119896+2

) byTheorem 157A graph 119866 is called to be vertex Roman domination-

critical if 120574RM(119866 minus 119909) lt 120574RM(119866) for every vertex 119909 in 119866 If119866 has no isolated vertices then 120574RM(119866) ⩽ 2120574(119866) ⩽ 2120573(119866)If 120574RM(119866) = 2120573(119866) then 120574RM(119866) = 2120574(119866) and hence 119866 is aRoman graph In [30] Volkmann gave a lot of graphs with120574(119866) = 120573(119866) The following result is similar to Theorem 57

Theorem 168 (Bahremandpour et al [118] 2013) For anygraph 119866 if 120574RM(119866) = 2120573(119866) then

(a) 119887RM(119866) ⩾ 120575(119866)

(b) 119887RM(119866) ⩾ 120575(119866)+1 if119866 is a vertex Roman domination-critical graph

If 120574RM(119866) = 2 then obviously 119887RM(119866) ⩽ 120575(119866) So weassume 120574RM(119866) ⩾ 3 Dehgardi et al [121] proved 119887RM(119866) ⩽(120574RM(119866) minus 2)Δ(119866) + 1 and posed the following problem if 119866is a connected graph of order 119899 ⩾ 4 with 120574RM(119866) ⩾ 3 then

119887RM (119866) ⩽ (120574RM (119866) minus 2) Δ (119866) (68)

Theorem 161 (a) shows that the inequality (68) holds if120574RM(119866) ⩾ 5 Thus the bound in (68) is of interest only when120574RM(119866) is 3 or 4

Lemma 169 (Bahremandpour et al [118] 2013) For anynonempty graph 119866 of order 119899 ⩾ 3 120574RM(119866) = 3 if and onlyif Δ(119866) = 119899 minus 2

The following result shows that (68) holds for all graphs119866 of order 119899 ⩾ 4 with 120574RM(119866) = 3 or 4 which improvesTheorem 161 (b)

Theorem 170 (Bahremandpour et al [118] 2013) For anyconnected graph 119866 of order 119899 ⩾ 4

119887RM (119866) le Δ (119866) = 119899 minus 2 119894119891 120574RM (119866) = 3

Δ (119866) + 120575 (119866) minus 1 119894119891 120574RM (119866) = 4(69)

with the first equality if and only if 119866 cong 1198624

Recently Akbari and Qajar [122] have proved the follow-ing result

Theorem 171 (Akbari and Qajar [122]) For any connectedgraph 119866 of order 119899 ⩾ 3

119887RM (119866) ⩽ 119899 minus 120574RM (119866) + 5 (70)

International Journal of Combinatorics 27

We conclude this section with the following problems

Problem 8 Characterize all connected graphs 119866 of order 119899 ⩾3 for which 119887RM(119866) = 119899 minus 1

Problem 9 Prove or disprove if 119866 is a connected graph oforder 119899 ⩾ 3 then

119887RM (119866) ⩽ 119899 minus 120574RM (119866) + 3 (71)

Note that 120574119877(119870

119905(3)) = 4 and 119887RM(119870119905

(3)) = 119899minus1 where 119899 =3119905 for 119905 ⩾ 3 The upper bound on 119887RM(119866) given in Problem 9is sharp if Problem 9 has positive answer

95 Rainbow Bondage Numbers For a positive integer 119896 a119896-rainbow dominating function of a graph 119866 is a function 119891from the vertex-set 119881(119866) to the set of all subsets of the set1 2 119896 such that for any vertex 119909 in 119866 with 119891(119909) = 0 thecondition

⋃

119910isin119873119866(119909)

119891 (119910) = 1 2 119896 (72)

is fulfilledThe weight of a 119896-rainbow dominating function 119891on 119866 is the value

119891 (119866) = sum

119909isin119881(119866)

1003816100381610038161003816119891 (119909)1003816100381610038161003816 (73)

The 119896-rainbow domination number of a graph 119866 denoted by120574119903119896(119866) is the minimum weight of a 119896-rainbow dominating

function 119891 of 119866Note that 120574

1199031(119866) is the classical domination number 120574(119866)

The 119896-rainbow domination number is introduced by Bresaret al [126] Rainbow domination of a graph 119866 coincides withordinary domination of the Cartesian product of 119866 with thecomplete graph in particular 120574

119903119896(119866) = 120574(119866 times 119870

119896) for any

positive integer 119896 and any graph 119866 [126] Hartnell and Rall[127] proved that 120574(119879) lt 120574

1199032(119879) for any tree 119879 no graph has

120574 = 120574119903119896for 119896 ⩾ 3 for any 119896 ⩾ 2 and any graph 119866 of order 119899

min 119899 120574 (119866) + 119896 minus 2 ⩽ 120574119903119896(119866) ⩽ 119896120574 (119866) (74)

The introduction of rainbow domination is motivatedby the study of paired domination in Cartesian productsof graphs where certain upper bounds can be expressed interms of rainbow domination that is for any graph 119866 andany graph 119867 if 119881(119867) can be partitioned into 119896120574

119875-sets then

120574119875(119866 times 119867) ⩽ (1119896)120592(119867)120574

119903119896(119866) proved by Bresar et al [126]

Dehgardi et al [128] introduced the concept of 119896-rainbowbondage number of graphs Let 119866 be a graph with maximumdegree at least two The 119896-rainbow bondage number 119887

119903119896(119866)

of 119866 is the minimum cardinality of all sets 119861 sube 119864(119866) forwhich 120574

119903119896(119866 minus 119861) gt 120574

119903119896(119866) Since in the study of 119896-rainbow

bondage number the assumption Δ(119866) ⩾ 2 is necessarywe always assume that when we discuss 119887

119903119896(119866) all graphs

involved satisfy Δ(119866) ⩾ 2The 2-rainbow bondage number of some special families

of graphs has been determined

Theorem 172 (Dehgardi et al [128]) For any integer 119899 ⩾ 3

1198871199032(119875

119899) = 1 119891119900119903 119899 ⩾ 2

1198871199032(119862

119899) =

1 119894119891 119899 equiv 0 (mod 4) 2 119900119905ℎ119890119903119908119894119904119890

1198871199032(119870

119899119899) =

1 119894119891 119899 = 2

3 119894119891 119899 = 3

6 119894119891 119899 = 4

119898 119894119891 119899 ⩾ 5

(75)

Theorem 173 (Dehgardi et al [128]) For any connected graph119866 of order 119899 ⩾ 3 119887

1199032(119866) ⩽ 120582(119866) + 2Δ(119866) minus 3

Theorem 174 (Dehgardi et al [128]) For any tree 119879 of order119899 ⩾ 3 119887

1199032(119879) ⩽ 2

Theorem 175 (Dehgardi et al [128]) If 119866 is a planar graphwithmaximumdegree at least two then 119887

1199032(119866) ⩽ 15Moreover

if the girth 119892(119866) ⩾ 4 then 1198871199032(119866) ⩽ 11 and if 119892(119866) ⩾ 6 then

1198871199032(119866) ⩽ 9

Theorem 176 (Dehgardi et al [128]) For a complete bipartitegraph 119870

119901119902 if 2119896 + 1 ⩽ 119901 ⩽ 119902 then 119887

119903119896(119870

119901119902) = 119901

Theorem177 (Dehgardi et al [128]) For a complete graph119870119899

if 119899 ⩾ 119896 + 1 then 119887119903119896(119870

119899) = lceil119896119899(119896 + 1)rceil

The special case 119896 = 1 of Theorem 177 can be found inTheorem 1 The following is a simple observation similar toObservation 1

Observation 6 (Dehgardi et al [128]) Let 119866 be a graphwith maximum degree at least two 119867 a spanning subgraphobtained from 119866 by removing 119896 edges If 120574

119903119896(119867) = 120574

119903119896(119866)

then 119887119903119896(119867) ⩽ 119887

119903119896(119866) ⩽ 119887

119903119896(119867) + 119896

Theorem 178 (Dehgardi et al [128]) For every graph 119866 with120574119903119896(119866) = 119896120574(119866) 119887

119903119896(119866) ⩾ 119887(119866)

A graph 119866 is said to be vertex 119896-rainbow domination-critical if 120574

119903119896(119866 minus 119909) lt 120574

119903119896(119866) for every vertex 119909 in 119866 It is

clear that if 119866 has no isolated vertices then 120574119903119896(119866) ⩽ 119896120574(119866) ⩽

119896120573(119866) where 120573(119866) is the vertex-covering number of119866 Thusif 120574

119903119896(119866) = 119896120573(119866) then 120574

119903119896(119866) = 119896120574(119866)

Theorem 179 (Dehgardi et al [128]) For any graph 119866 if120574119903119896(119866) = 119896120573(119866) then

(a) 119887119903119896(119866) ⩾ 120575(119866)

(b) 119887119903119896(119866) ⩾ 120575(119866)+1 if119866 is vertex 119896-rainbow domination-

critical

As far as we know there are no more results on the 119896-rainbow bondage number of graphs in the literature

28 International Journal of Combinatorics

10 Results on Digraphs

Although domination has been extensively studied in undi-rected graphs it is natural to think of a dominating setas a one-way relationship between vertices of the graphIndeed among the earliest literatures on this subject vonNeumann and Morgenstern [129] used what is now calleddomination in digraphs to find solution (or kernels whichare independent dominating sets) for cooperative 119899-persongames Most likely the first formulation of domination byBerge [130] in 1958 was given in the context of digraphs andonly some years latter by Ore [131] in 1962 for undirectedgraphs Despite this history examination of domination andits variants in digraphs has been essentially overlooked (see[4] for an overview of the domination literature) Thus thereare few if any such results on domination for digraphs in theliterature

The bondage number and its related topics for undirectedgraph have become one of major areas both in theoreticaland applied researches However until recently Carlsonand Develin [42] Shan and Kang [132] and Huang andXu [133 134] studied the bondage number for digraphsindependently In this section we introduce their resultsfor general digraphs Results for some special digraphs suchas vertex-transitive digraphs are introduced in the nextsection

101 Upper Bounds for General Digraphs Let 119866 = (119881 119864)

be a digraph without loops and parallel edges A digraph 119866is called to be asymmetric if whenever (119909 119910) isin 119864(119866) then(119910 119909) notin 119864(119866) and to be symmetric if (119909 119910) isin 119864(119866) implies(119910 119909) isin 119864(119866) For a vertex 119909 of 119881(119866) the sets of out-neighbors and in-neighbors of 119909 are respectively defined as119873

+

119866(119909) = 119910 isin 119881(119866) (119909 119910) isin 119864(119866) and 119873minus

119866(119909) = 119909 isin

119881(119866) (119909 119910) isin 119864(119866) the out-degree and the in-degree of119909 are respectively defined as 119889+

119866(119909) = |119873

+

119866(119909)| and 119889minus

119866(119909) =

|119873+

119866(119909)| Denote themaximum and theminimum out-degree

(resp in-degree) of 119866 by Δ+(119866) and 120575+(119866) (resp Δminus(119866) and120575minus

(119866))The degree 119889119866(119909) of 119909 is defined as 119889+

119866(119909)+119889

minus

119866(119909) and

the maximum and the minimum degrees of119866 are denoted byΔ(119866) and 120575(119866) respectively that is Δ(119866) = max119889

119866(119909) 119909 isin

119881(119866) and 120575(119866) = min119889119866(119909) 119909 isin 119881(119866) Note that the

definitions here are different from the ones in the textbookon digraphs see for example Xu [1]

A subset 119878 of119881(119866) is called a dominating set if119881(119866) = 119878cup119873

+

119866(119878) where119873+

119866(119878) is the set of out-neighbors of 119878Then just

as for undirected graphs 120574(119866) is the minimum cardinalityof a dominating set and the bondage number 119887(119866) is thesmallest cardinality of a set 119861 of edges such that 120574(119866 minus 119861) gt120574(119866) if such a subset 119861 sube 119864(119866) exists Otherwise we put119887(119866) = infin

Some basic results for undirected graphs stated inSection 3 can be generalized to digraphs For exampleTheorem 18 is generalized by Carlson and Develin [42]Huang andXu [133] and Shan andKang [132] independentlyas follows

Theorem 180 Let 119866 be a digraph and (119909 119910) isin 119864(119866) Then119887(119866) ⩽ 119889

119866(119910) + 119889

minus

119866(119909) minus |119873

minus

119866(119909) cap 119873

minus

119866(119910)|

Corollary 181 For a digraph 119866 119887(119866) ⩽ 120575minus(119866) + Δ(119866)

Since 120575minus

(119866) ⩽ (12)Δ(119866) from Corollary 181Conjecture 53 is valid for digraphs

Corollary 182 For a digraph 119866 119887(119866) ⩽ (32)Δ(119866)

In the case of undirected graphs the bondage number of119866119899= 119870

119899times 119870

119899achieves this bound However it was shown

by Carlson and Develin in [42] that if we take the symmetricdigraph119866

119899 we haveΔ(119866

119899) = 4(119899minus1) 120574(119866

119899) = 119899 and 119887(119866

119899) ⩽

4(119899 minus 1) + 1 = Δ(119866119899) + 1 So this family of digraphs cannot

show that the bound in Corollary 182 is tightCorresponding to Conjecture 51 which is discredited

for undirected graphs and is valid for digraphs the sameconjecture for digraphs can be proposed as follows

Conjecture 183 (Carlson and Develin [42] 2006) 119887(119866) ⩽Δ(119866) + 1 for any digraph 119866

If Conjecture 183 is true then the following results showsthat this upper bound is tight since 119887(119870

119899times119870

119899) = Δ(119870

119899times119870

119899)+1

for a complete digraph 119870119899

In 2007 Shan and Kang [132] gave some tight upperbounds on the bondage numbers for some asymmetricdigraphs For example 119887(119879) ⩽ Δ(119879) for any asymmetricdirected tree 119879 119887(119866) ⩽ Δ(119866) for any asymmetric digraph 119866with order at least 4 and 120574(119866) ⩽ 2 For planar digraphs theyobtained the following results

Theorem 184 (Shan and Kang [132] 2007) Let 119866 be aasymmetric planar digraphThen 119887(119866) ⩽ Δ(119866)+2 and 119887(119866) ⩽Δ(119866) + 1 if Δ(119866) ⩾ 5 and 119889minus

119866(119909) ⩾ 3 for every vertex 119909 with

119889119866(119909) ⩾ 4

102 Results for Some Special Digraphs The exact values andbounds on 119887(119866) for some standard digraphs are determined

Theorem185 (Huang andXu [133] 2006) For a directed cycle119862119899and a directed path 119875

119899

119887 (119862119899) =

3 119894119891 119899 119894119904 119900119889119889

2 119894119891 119899 119894119904 119890V119890119899

119887 (119875119899) =

2 119894119891 119899 119894119904 119900119889119889

1 119894119891 119899 119894119904 119890V119890119899

(76)

For the de Bruijn digraph 119861(119889 119899) and the Kautz digraph119870(119889 119899)

119887 (119861 (119889 119899)) = 119889 119894119891 119899 119894119904 119900119889119889

119889 ⩽ 119887 (119861 (119889 119899)) ⩽ 2119889 119894119891 119899 119894119904 119890V119890119899

119887 (119870 (119889 119899)) = 119889 + 1

(77)

Like undirected graphs we can define the total domi-nation number and the total bondage number On the totalbondage numbers for some special digraphs the knownresults are as follows

International Journal of Combinatorics 29

Theorem 186 (Huang and Xu [134] 2007) For a directedcycle 119862

119899and a directed path 119875

119899 119887

119879(119875

119899) and 119887

119879(119862

119899) all do not

exist For a complete digraph 119870119899

119887119879(119870

119899) = infin 119894119891 119899 = 1 2

119887119879(119870

119899) = 3 119894119891 119899 = 3

119899 ⩽ 119887119879(119870

119899) ⩽ 2119899 minus 3 119894119891 119899 ⩾ 4

(78)

The extended de Bruijn digraph EB(119889 119899 1199021 119902

119901) and

the extended Kautz digraph EK(119889 119899 1199021 119902

119901) are intro-

duced by Shibata and Gonda [135] If 119901 = 1 then theyare the de Bruijn digraph B(119889 119899) and the Kautz digraphK(119889 119899) respectively Huang and Xu [134] determined theirtotal domination numbers In particular their total bondagenumbers for general cases are determined as follows

Theorem 187 (Huang andXu [134] 2007) If 119889 ⩾ 2 and 119902119894⩾ 2

for each 119894 = 1 2 119901 then

119887119879(EB(119889 119899 119902

1 119902

119901)) = 119889

119901

minus 1

119887119879(EK(119889 119899 119902

1 119902

119901)) = 119889

119901

(79)

In particular for the de Bruijn digraph B(119889 119899) and the Kautzdigraph K(119889 119899)

119887119879(B (119889 119899)) = 119889 minus 1 119887

119879(K (119889 119899)) = 119889 (80)

Zhang et al [136] determined the bondage number incomplete 119905-partite digraphs

Theorem 188 (Zhang et al [136] 2009) For a complete 119905-partite digraph 119870

11989911198992119899119905

where 1198991⩽ 119899

2⩽ sdot sdot sdot ⩽ 119899

119905

119887 (11987011989911198992119899119905

)

=

119898 119894119891 119899119898= 1 119899

119898+1⩾ 2 119891119900119903 119904119900119898119890

119898 (1 ⩽ 119898 lt 119905)

4119905 minus 3 119894119891 1198991= 119899

2= sdot sdot sdot = 119899

119905= 2

119905minus1

sum

119894=1

119899119894+ 2 (119905 minus 1) 119900119905ℎ119890119903119908119894119904119890

(81)

Since an undirected graph can be thought of as a sym-metric digraph any result for digraphs has an analogy forundirected graphs in general In view of this point studyingthe bondage number for digraphs is more significant thanfor undirected graphs Thus we should further study thebondage number of digraphs and try to generalize knownresults on the bondage number and related variants for undi-rected graphs to digraphs prove or disprove Conjecture 183In particular determine the exact values of 119887(B(119889 119899)) for aneven 119899 and 119887

119879(119870

119899) for 119899 ⩾ 4

11 Efficient Dominating Sets

A dominating set 119878 of a graph 119866 is called to be efficient if forevery vertex 119909 in119866 |119873

119866[119909]cap119878| = 1 if119866 is a undirected graph

or |119873minus

119866[119909] cap 119878| = 1 if 119866 is a directed graph The concept of an

efficient set was proposed by Bange et al [137] in 1988 In theliterature an efficient set is also called a perfect dominatingset (see Livingston and Stout [138] for an explanation)

From definition if 119878 is an efficient dominating set of agraph 119866 then 119878 is certainly an independent set and everyvertex not in 119878 is adjacent to exactly one vertex in 119878

It is also clear from definition that a dominating set 119878 isefficient if and only if N

119866[119878] = 119873

119866[119909] 119909 isin 119878 for the

undirected graph 119866 or N+

119866[119878] = 119873

+

119866[119909] 119909 isin 119878 for the

digraph119866 is a partition of119881(119866) where the induced subgraphby119873

119866[119909] or119873+

119866[119909] is a star or an out-star with the root 119909

The efficient domination has important applications inmany areas such as error-correcting codes and receivesmuch attention in the late years

The concept of efficient dominating sets is a measureof the efficiency of domination in graphs Unfortunately asshown in [137] not every graph has an efficient dominatingset andmoreover it is anNP-complete problem to determinewhether a given graph has an efficient dominating set Inaddition it was shown by Clark [139] in 1993 that for a widerange of 119901 almost every random undirected graph 119866 isin

G(120592 119901) has no efficient dominating sets This means thatundirected graphs possessing an efficient dominating set arerare However it is easy to show that every undirected graphhas an orientationwith an efficient dominating set (see Bangeet al [140])

In 1993 Barkauskas and Host [141] showed that deter-mining whether an arbitrary oriented graph has an efficientdominating set is NP-complete Even so the existence ofefficient dominating sets for some special classes of graphs hasbeen examined see for example Dejter and Serra [142] andLee [143] for Cayley graph Gu et al [144] formeshes and toriObradovic et al [145] Huang and Xu [36] Reji Kumar andMacGillivray [146] for circulant graphs Harary graphs andtori and Van Wieren et al [147] for cube-connected cycles

In this section we introduce results of the bondagenumber for some graphs with efficient dominating sets dueto Huang and Xu [36]

111 Results for General Graphs In this subsection we intro-duce some results on bondage numbers obtained by applyingefficient dominating sets We first state the two followinglemmas

Lemma 189 For any 119896-regular graph or digraph 119866 of order 119899120574(119866) ⩾ 119899(119896 + 1) with equality if and only if 119866 has an efficientdominating set In addition if 119866 has an efficient dominatingset then every efficient dominating set is certainly a 120574-set andvice versa

Let 119890 be an edge and 119878 a dominating set in 119866 We say that119890 supports 119878 if 119890 isin (119878 119878) where (119878 119878) = (119909 119910) isin 119864(119866) 119909 isin 119878 119910 isin 119878 Denote by 119904(119866) the minimum number of edgeswhich support all 120574-sets in 119866

Lemma 190 For any graph or digraph 119866 119887(119866) ⩾ 119904(119866) withequality if 119866 is regular and has an efficient dominating set

30 International Journal of Combinatorics

A graph 119866 is called to be vertex-transitive if its automor-phism group Aut(119866) acts transitively on its vertex-set 119881(119866)A vertex-transitive graph is regular Applying Lemmas 189and 190 Huang and Xu obtained some results on bondagenumbers for vertex-transitive graphs or digraphs

Theorem 191 (Huang and Xu [36] 2008) For any vertex-transitive graph or digraph 119866 of order 119899

119887 (119866) ⩾

lceil119899

2120574 (119866)rceil 119894119891 119866 119894119904 119906119899119889119894119903119890119888119905119890119889

lceil119899

120574 (119866)rceil 119894119891 119866 119894119904 119889119894119903119890119888119905119890119889

(82)

Theorem192 (Huang andXu [36] 2008) Let119866 be a 119896-regulargraph of order 119899 Then

119887(119866) ⩽ 119896 if119866 is undirected and 119899 ⩾ 120574(119866)(119896+1)minus119896+1119887(119866) ⩽ 119896+1+ℓ if119866 is directed and 119899 ⩾ 120574(119866)(119896+1)minusℓwith 0 ⩽ ℓ ⩽ 119896 minus 1

Next we introduce a better upper bound on 119887(119866) To thisaim we need the following concept which is a generalizationof the concept of the edge-covering of a graph 119866 For 1198811015840

sube

119881(119866) and 1198641015840 sube 119864(119866) we say that 1198641015840covers 1198811015840 and call 1198641015840an edge-covering for 1198811015840 if there exists an edge (119909 119910) isin 1198641015840 forany vertex 119909 isin 1198811015840 For 119910 isin 119881(119866) let 1205731015840[119910] be the minimumcardinality over all edge-coverings for119873minus

119866[119910]

Theorem 193 (Huang and Xu [36] 2008) If 119866 is a 119896-regulargraph with order 120574(119866)(119896 + 1) then 119887(119866) ⩽ 1205731015840[119910] for any 119910 isin119881(119866)

Theupper bound on 119887(119866) given inTheorem 193 is tight inview of 119887(119862

119899) for a cycle or a directed cycle 119862

119899(seeTheorems

1 and 185 resp)It is easy to see that for a 119896-regular graph 119866 lceil(119896 + 1)2rceil ⩽

1205731015840

[119910] ⩽ 119896 when 119866 is undirected and 1205731015840[119910] = 119896 + 1 when 119866 isdirected By this fact and Lemma 189 the following theoremis merely a simple combination of Theorems 191 and 193

Theorem 194 (Huang and Xu [36] 2008) Let 119866 be a vertex-transitive graph of degree 119896 If 119866 has an efficient dominatingset then

lceil119896 + 1

2rceil ⩽ 119887 (119866) ⩽ 119896 119894119891 119866 119894119904 119906119899119889119894119903119890119888119905119890119889

119887 (119866) = 119896 + 1 119894119891 119866 119894119904 119889119894119903119890119888119905119890119889

(83)

Theorem 195 (Huang and Xu [36] 2008) If 119866 is an undi-rected vertex-transitive cubic graph with order 4120574(119866) and girth119892(119866) ⩽ 5 then 119887(119866) = 2

The proof of Theorem 195 leads to a byproduct If 119892 =5 let (119906

1 119906

2 119906

3 119906

4 119906

5) be a cycle and let D

119894be the family

of efficient dominating sets containing 119906119894for each 119894 isin

1 2 3 4 5 ThenD119894capD

119895= 0 for 119894 = 1 2 5 Then 119866 has

at least 5119904 efficient dominating sets where 119904 = 119904(119866) But thereare only 4s (=ns120574) distinct efficient dominating sets in119866This

contradiction implies that an undirected vertex-transitivecubic graph with girth five has no efficient dominating setsBut a similar argument for 119892(119866) = 3 4 or 119892(119866) ⩾ 6 couldnot give any contradiction This is consistent with the resultthat CCC(119899) a vertex-transitive cubic graph with girth 119899 if3 ⩽ 119899 ⩽ 8 or girth 8 if 119899 ⩾ 9 has efficient dominating setsfor all 119899 ⩾ 3 except 119899 = 5 (see Theorem 199 in the followingsubsection)

112 Results for Cayley Graphs In this subsection we useTheorem 194 to determine the exact values or approximativevalues of bondage numbers for some special vertex-transitivegraphs by characterizing the existence of efficient dominatingsets in these graphs

Let Γ be a nontrivial finite group 119878 a nonempty subset ofΓ without the identity element of Γ A digraph 119866 is defined asfollows

119881 (119866) = Γ (119909 119910) isin 119864 (119866) lArrrArr 119909minus1

119910 isin 119878

for any 119909 119910 isin Γ(84)

is called a Cayley digraph of the group Γ with respect to119878 denoted by 119862

Γ(119878) If 119878minus1 = 119904minus1 119904 isin 119878 = 119878 then

119862Γ(119878) is symmetric and is called a Cayley undirected graph

a Cayley graph for short Cayley graphs or digraphs arecertainly vertex-transitive (see Xu [1])

A circulant graph 119866(119899 119878) of order 119899 is a Cayley graph119862119885119899

(119878) where 119885119899= 0 119899 minus 1 is the addition group of

order 119899 and 119878 is a nonempty subset of119885119899without the identity

element and hence is a vertex-transitive digraph of degree|119878| If 119878minus1 = 119878 then119866(119899 119878) is an undirected graph If 119878 = 1 119896where 2 ⩽ 119896 ⩽ 119899 minus 2 we write 119866(119899 1 119896) for 119866(119899 1 119896) or119866(119899 plusmn1 plusmn119896) and call it a double-loop circulant graph

Huang and Xu [36] showed that for directed 119866 =

119866(119899 1 119896) lceil1198993rceil ⩽ 120574(119866) ⩽ lceil1198992rceil and 119866 has an efficientdominating set if and only if 3 | 119899 and 119896 equiv 2 (mod 3) fordirected 119866 = 119866(119899 1 119896) and 119896 = 1198992 lceil1198995rceil ⩽ 120574(119866) ⩽ lceil1198993rceiland 119866 has an efficient dominating set if and only if 5 | 119899 and119896 equiv plusmn2 (mod 5) By Theorem 194 we obtain the bondagenumber of a double loop circulant graph if it has an efficientdominating set

Theorem 196 (Huang and Xu [36] 2008) Let 119866 be a double-loop circulant graph 119866(119899 1 119896) If 119866 is directed with 3 | 119899and 119896 equiv 2 (mod 3) or 119866 is undirected with 5 | 119899 and119896 equiv plusmn2 (mod 5) then 119887(119866) = 3

The 119898 times 119899 torus is the Cartesian product 119862119898times 119862

119899of

two cycles and is a Cayley graph 119862119885119898times119885119899

(119878) where 119878 =

(0 1) (1 0) for directed cycles and 119878 = (0 plusmn1) (plusmn1 0) forundirected cycles and hence is vertex-transitive Gu et al[144] showed that the undirected torus119862

119898times119862

119899has an efficient

dominating set if and only if both 119898 and 119899 are multiplesof 5 Huang and Xu [36] showed that the directed torus119862119898times 119862

119899has an efficient dominating set if and only if both

119898 and 119899 are multiples of 3 Moreover they found a necessarycondition for a dominating set containing the vertex (0 0)in 119862

119898times 119862

119899to be efficient and obtained the following

result

International Journal of Combinatorics 31

Theorem 197 (Huang and Xu [36] 2008) Let 119866 = 119862119898times 119862

119899

If 119866 is undirected and both119898 and 119899 are multiples of 5 or if 119866 isdirected and both119898 and 119899 are multiples of 3 then 119887(119866) = 3

The hypercube 119876119899

is the Cayley graph 119862Γ(119878)

where Γ = 1198852times sdot sdot sdot times 119885

2= (119885

2)119899 and 119878 =

100 sdot sdot sdot 0 010 sdot sdot sdot 0 00 sdot sdot sdot 01 Lee [143] showed that119876119899has an efficient dominating set if and only if 119899 = 2119898 minus 1

for a positive integer119898 ByTheorem 194 the following resultis obtained immediately

Theorem 198 (Huang and Xu [36] 2008) If 119899 = 2119898 minus 1 for apositive integer119898 then 2119898minus1

⩽ 119887(119876119899) ⩽ 2

119898

minus 1

Problem 10 Determine the exact value of 119887(119876119899) for 119899 = 2119898minus1

The 119899-dimensional cube-connected cycle denoted byCCC(119899) is constructed from the 119899-dimensional hypercube119876119899by replacing each vertex 119909 in 119876

119899with an undirected cycle

119862119899of length 119899 and linking the 119894th vertex of the 119862

119899to the 119894th

neighbor of 119909 It has been proved that CCC(119899) is a Cayleygraph and hence is a vertex-transitive graph with degree 3Van Wieren et al [147] proved that CCC(119899) has an efficientdominating set if and only if 119899 = 5 From Theorems 194 and195 the following result can be derived

Theorem 199 (Huang and Xu [36] 2008) Let 119866 = 119862119862119862(119899)be the 119899-dimensional cube-connected cycles with 119899 ⩾ 3 and119899 = 5 Then 120574(119866) = 1198992119899minus2 and 2 ⩽ 119887(119866) ⩽ 3 In addition119887(119862119862119862(3)) = 119887(119862119862119862(4)) = 2

Problem 11 Determine the exact value of 119887(CCC(119899)) for 119899 ⩾5

Acknowledgments

This work was supported by NNSF of China (Grants nos11071233 61272008) The author would like to express hisgratitude to the anonymous referees for their kind sugges-tions and comments on the original paper to ProfessorSheikholeslami and Professor Jafari Rad for sending thereferences [128] and [81] which resulted in this version

References

[1] J-M Xu Theory and Application of Graphs vol 10 of NetworkTheory and Applications Kluwer Academic Dordrecht TheNetherlands 2003

[2] S THedetniemi andR C Laskar ldquoBibliography on dominationin graphs and some basic definitions of domination parame-tersrdquo Discrete Mathematics vol 86 no 1ndash3 pp 257ndash277 1990

[3] TW Haynes S T Hedetniemi and P J Slater Fundamentals ofDomination in Graphs vol 208 ofMonographs and Textbooks inPure and Applied Mathematics Marcel Dekker New York NYUSA 1998

[4] TWHaynes S THedetniemi and P J Slater EdsDominationin Graphs vol 209 of Monographs and Textbooks in Pure andAppliedMathematicsMarcelDekker NewYorkNYUSA 1998

[5] M R Garey andD S JohnsonComputers and Intractability WH Freeman and Company San Francisco Calif USA 1979

[6] H B Walikar and B D Acharya ldquoDomination critical graphsrdquoNational Academy Science Letters vol 2 pp 70ndash72 1979

[7] D Bauer F Harary J Nieminen and C L Suffel ldquoDominationalteration sets in graphsrdquo Discrete Mathematics vol 47 no 2-3pp 153ndash161 1983

[8] J F Fink M S Jacobson L F Kinch and J Roberts ldquoThebondage number of a graphrdquo Discrete Mathematics vol 86 no1ndash3 pp 47ndash57 1990

[9] F-T Hu and J-M Xu ldquoThe bondage number of (119899 minus 3)-regulargraph G of order nrdquo Ars Combinatoria to appear

[10] U Teschner ldquoNew results about the bondage number of agraphrdquoDiscreteMathematics vol 171 no 1ndash3 pp 249ndash259 1997

[11] B L Hartnell and D F Rall ldquoA bound on the size of a graphwith given order and bondage numberrdquo Discrete Mathematicsvol 197-198 pp 409ndash413 1999 16th British CombinatorialConference (London 1997)

[12] B L Hartnell and D F Rall ldquoA characterization of trees inwhich no edge is essential to the domination numberrdquo ArsCombinatoria vol 33 pp 65ndash76 1992

[13] J Topp and P D Vestergaard ldquo120572119896- and 120574

119896-stable graphsrdquo

Discrete Mathematics vol 212 no 1-2 pp 149ndash160 2000[14] A Meir and J W Moon ldquoRelations between packing and

covering numbers of a treerdquo Pacific Journal of Mathematics vol61 no 1 pp 225ndash233 1975

[15] B L Hartnell L K Joslashrgensen P D Vestergaard and CA Whitehead ldquoEdge stability of the k-domination numberof treesrdquo Bulletin of the Institute of Combinatorics and ItsApplications vol 22 pp 31ndash40 1998

[16] F-T Hu and J-M Xu ldquoOn the complexity of the bondage andreinforcement problemsrdquo Journal of Complexity vol 28 no 2pp 192ndash201 2012

[17] B L Hartnell and D F Rall ldquoBounds on the bondage numberof a graphrdquo Discrete Mathematics vol 128 no 1ndash3 pp 173ndash1771994

[18] Y-L Wang ldquoOn the bondage number of a graphrdquo DiscreteMathematics vol 159 no 1ndash3 pp 291ndash294 1996

[19] G H Fricke TW Haynes S M Hedetniemi S T Hedetniemiand R C Laskar ldquoExcellent treesrdquo Bulletin of the Institute ofCombinatorics and Its Applications vol 34 pp 27ndash38 2002

[20] V Samodivkin ldquoThe bondage number of graphs good and badverticesrdquoDiscussiones Mathematicae GraphTheory vol 28 no3 pp 453ndash462 2008

[21] J E Dunbar T W Haynes U Teschner and L VolkmannldquoBondage insensitivity and reinforcementrdquo in Domination inGraphs vol 209 of Monographs and Textbooks in Pure andAppliedMathematics pp 471ndash489 Dekker NewYork NY USA1998

[22] H Liu and L Sun ldquoThe bondage and connectivity of a graphrdquoDiscrete Mathematics vol 263 no 1ndash3 pp 289ndash293 2003

[23] A-H Esfahanian and S L Hakimi ldquoOn computing a con-ditional edge-connectivity of a graphrdquo Information ProcessingLetters vol 27 no 4 pp 195ndash199 1988

[24] R C Brigham P Z Chinn and R D Dutton ldquoVertexdomination-critical graphsrdquoNetworks vol 18 no 3 pp 173ndash1791988

[25] V Samodivkin ldquoDomination with respect to nondegenerateproperties bondage numberrdquoThe Australasian Journal of Com-binatorics vol 45 pp 217ndash226 2009

[26] U Teschner ldquoA new upper bound for the bondage numberof graphs with small domination numberrdquo The AustralasianJournal of Combinatorics vol 12 pp 27ndash35 1995

32 International Journal of Combinatorics

[27] J Fulman D Hanson and G MacGillivray ldquoVertex dom-ination-critical graphsrdquoNetworks vol 25 no 2 pp 41ndash43 1995

[28] U Teschner ldquoA counterexample to a conjecture on the bondagenumber of a graphrdquo Discrete Mathematics vol 122 no 1ndash3 pp393ndash395 1993

[29] U Teschner ldquoThe bondage number of a graph G can be muchgreater than Δ(G)rdquo Ars Combinatoria vol 43 pp 81ndash87 1996

[30] L Volkmann ldquoOn graphs with equal domination and coveringnumbersrdquoDiscrete AppliedMathematics vol 51 no 1-2 pp 211ndash217 1994

[31] L A Sanchis ldquoMaximum number of edges in connected graphswith a given domination numberrdquoDiscreteMathematics vol 87no 1 pp 65ndash72 1991

[32] S Klavzar and N Seifter ldquoDominating Cartesian products ofcyclesrdquoDiscrete AppliedMathematics vol 59 no 2 pp 129ndash1361995

[33] H K Kim ldquoA note on Vizingrsquos conjecture about the bondagenumberrdquo Korean Journal of Mathematics vol 10 no 2 pp 39ndash42 2003

[34] M Y Sohn X Yuan and H S Jeong ldquoThe bondage number of1198623times119862

119899rdquo Journal of the KoreanMathematical Society vol 44 no

6 pp 1213ndash1231 2007[35] L Kang M Y Sohn and H K Kim ldquoBondage number of the

discrete torus 119862119899times 119862

4rdquo Discrete Mathematics vol 303 no 1ndash3

pp 80ndash86 2005[36] J Huang and J-M Xu ldquoThe bondage numbers and efficient

dominations of vertex-transitive graphsrdquoDiscrete Mathematicsvol 308 no 4 pp 571ndash582 2008

[37] C J Xiang X Yuan andM Y Sohn ldquoDomination and bondagenumber of119862

3times119862

119899rdquoArs Combinatoria vol 97 pp 299ndash310 2010

[38] M S Jacobson and L F Kinch ldquoOn the domination numberof products of graphs Irdquo Ars Combinatoria vol 18 pp 33ndash441984

[39] Y-Q Li ldquoA note on the bondage number of a graphrdquo ChineseQuarterly Journal of Mathematics vol 9 no 4 pp 1ndash3 1994

[40] F-T Hu and J-M Xu ldquoBondage number of mesh networksrdquoFrontiers ofMathematics inChina vol 7 no 5 pp 813ndash826 2012

[41] R Frucht and F Harary ldquoOn the corona of two graphsrdquoAequationes Mathematicae vol 4 pp 322ndash325 1970

[42] K Carlson and M Develin ldquoOn the bondage number of planarand directed graphsrdquoDiscreteMathematics vol 306 no 8-9 pp820ndash826 2006

[43] U Teschner ldquoOn the bondage number of block graphsrdquo ArsCombinatoria vol 46 pp 25ndash32 1997

[44] J Huang and J-M Xu ldquoNote on conjectures of bondagenumbers of planar graphsrdquo Applied Mathematical Sciences vol6 no 65ndash68 pp 3277ndash3287 2012

[45] L Kang and J Yuan ldquoBondage number of planar graphsrdquoDiscrete Mathematics vol 222 no 1ndash3 pp 191ndash198 2000

[46] M Fischermann D Rautenbach and L Volkmann ldquoRemarkson the bondage number of planar graphsrdquo Discrete Mathemat-ics vol 260 no 1ndash3 pp 57ndash67 2003

[47] J Hou G Liu and J Wu ldquoBondage number of planar graphswithout small cyclesrdquoUtilitasMathematica vol 84 pp 189ndash1992011

[48] J Huang and J-M Xu ldquoThe bondage numbers of graphs withsmall crossing numbersrdquo Discrete Mathematics vol 307 no 15pp 1881ndash1897 2007

[49] Q Ma S Zhang and J Wang ldquoBondage number of 1-planargraphrdquo Applied Mathematics vol 1 no 2 pp 101ndash103 2010

[50] J Hou and G Liu ldquoA bound on the bondage number of toroidalgraphsrdquoDiscreteMathematics Algorithms and Applications vol4 no 3 Article ID 1250046 5 pages 2012

[51] Y-C Cao J-M Xu and X-R Xu ldquoOn bondage number oftoroidal graphsrdquo Journal of University of Science and Technologyof China vol 39 no 3 pp 225ndash228 2009

[52] Y-C Cao J Huang and J-M Xu ldquoThe bondage number ofgraphs with crossing number less than fourrdquo Ars Combinatoriato appear

[53] H K Kim ldquoOn the bondage numbers of some graphsrdquo KoreanJournal of Mathematics vol 11 no 2 pp 11ndash15 2004

[54] A Gagarin and V Zverovich ldquoUpper bounds for the bondagenumber of graphs on topological surfacesrdquo Discrete Mathemat-ics 2012

[55] J Huang ldquoAn improved upper bound for the bondage numberof graphs on surfacesrdquoDiscrete Mathematics vol 312 no 18 pp2776ndash2781 2012

[56] A Gagarin and V Zverovich ldquoThe bondage number of graphson topological surfaces and Teschnerrsquos conjecturerdquo DiscreteMathematics vol 313 no 6 pp 796ndash808 2013

[57] V Samodivkin ldquoNote on the bondage number of graphs ontopological surfacesrdquo httparxivorgabs12086203

[58] E J Cockayne R M Dawes and S T Hedetniemi ldquoTotaldomination in graphsrdquoNetworks vol 10 no 3 pp 211ndash219 1980

[59] R Laskar J Pfaff S M Hedetniemi and S T HedetniemildquoOn the algorithmic complexity of total dominationrdquo Society forIndustrial and Applied Mathematics vol 5 no 3 pp 420ndash4251984

[60] J Pfaff R C Laskar and S T Hedetniemi ldquoNP-completeness oftotal and connected domination and irredundance for bipartitegraphsrdquo Tech Rep 428 Department of Mathematical SciencesClemson University 1983

[61] M A Henning ldquoA survey of selected recent results on totaldomination in graphsrdquoDiscrete Mathematics vol 309 no 1 pp32ndash63 2009

[62] V R Kulli and D K Patwari ldquoThe total bondage number ofa graphrdquo in Advances in Graph Theory pp 227ndash235 VishwaGulbarga India 1991

[63] F-T Hu Y Lu and J-M Xu ldquoThe total bondage number ofgrid graphsrdquo Discrete Applied Mathematics vol 160 no 16-17pp 2408ndash2418 2012

[64] J Cao M Shi and B Wu ldquoResearch on the total bondagenumber of a spectial networkrdquo in Proceedings of the 3rdInternational Joint Conference on Computational Sciences andOptimization (CSO rsquo10) pp 456ndash458 May 2010

[65] N Sridharan MD Elias and V S A Subramanian ldquoTotalbondage number of a graphrdquo AKCE International Journal ofGraphs and Combinatorics vol 4 no 2 pp 203ndash209 2007

[66] N Jafari Rad and J Raczek ldquoSome progress on total bondage ingraphsrdquo Graphs and Combinatorics 2011

[67] T W Haynes and P J Slater ldquoPaired-domination and thepaired-domatic numberrdquoCongressusNumerantium vol 109 pp65ndash72 1995

[68] T W Haynes and P J Slater ldquoPaired-domination in graphsrdquoNetworks vol 32 no 3 pp 199ndash206 1998

[69] X Hou ldquoA characterization of 2120574 120574119875-treesrdquoDiscrete Mathemat-

ics vol 308 no 15 pp 3420ndash3426 2008[70] K E Proffitt T W Haynes and P J Slater ldquoPaired-domination

in grid graphsrdquo Congressus Numerantium vol 150 pp 161ndash1722001

International Journal of Combinatorics 33

[71] H Qiao L KangM Cardei andD-Z Du ldquoPaired-dominationof treesrdquo Journal of Global Optimization vol 25 no 1 pp 43ndash542003

[72] E Shan L Kang and M A Henning ldquoA characterizationof trees with equal total domination and paired-dominationnumbersrdquo The Australasian Journal of Combinatorics vol 30pp 31ndash39 2004

[73] J Raczek ldquoPaired bondage in treesrdquo Discrete Mathematics vol308 no 23 pp 5570ndash5575 2008

[74] J A Telle and A Proskurowski ldquoAlgorithms for vertex parti-tioning problems on partial k-treesrdquo SIAM Journal on DiscreteMathematics vol 10 no 4 pp 529ndash550 1997

[75] G S Domke J H Hattingh M A Henning and L R MarkusldquoRestrained domination in treesrdquoDiscreteMathematics vol 211no 1ndash3 pp 1ndash9 2000

[76] G S Domke J H Hattingh M A Henning and L R MarkusldquoRestrained domination in graphs with minimum degree twordquoJournal of Combinatorial Mathematics and Combinatorial Com-puting vol 35 pp 239ndash254 2000

[77] G S Domke J H Hattingh S T Hedetniemi R C Laskarand L R Markus ldquoRestrained domination in graphsrdquo DiscreteMathematics vol 203 no 1ndash3 pp 61ndash69 1999

[78] J H Hattingh and E J Joubert ldquoAn upper bound for therestrained domination number of a graph with minimumdegree at least two in terms of order and minimum degreerdquoDiscrete Applied Mathematics vol 157 no 13 pp 2846ndash28582009

[79] M A Henning ldquoGraphs with large restrained dominationnumberrdquo Discrete Mathematics vol 197-198 pp 415ndash429 199916th British Combinatorial Conference (London 1997)

[80] J H Hattingh and A R Plummer ldquoRestrained bondage ingraphsrdquo Discrete Mathematics vol 308 no 23 pp 5446ndash54532008

[81] N Jafari Rad R Hasni J Raczek and L Volkmann ldquoTotalrestrained bondage in graphsrdquoActaMathematica Sinica vol 29no 6 pp 1033ndash1042 2013

[82] J Cyman and J Raczek ldquoOn the total restrained dominationnumber of a graphrdquoThe Australasian Journal of Combinatoricsvol 36 pp 91ndash100 2006

[83] M A Henning ldquoGraphs with large total domination numberrdquoJournal of Graph Theory vol 35 no 1 pp 21ndash45 2000

[84] D-X Ma X-G Chen and L Sun ldquoOn total restraineddomination in graphsrdquoCzechoslovakMathematical Journal vol55 no 1 pp 165ndash173 2005

[85] B Zelinka ldquoRemarks on restrained domination and totalrestrained domination in graphsrdquo Czechoslovak MathematicalJournal vol 55 no 2 pp 393ndash396 2005

[86] W Goddard and M A Henning ldquoIndependent domination ingraphs a survey and recent resultsrdquo Discrete Mathematics vol313 no 7 pp 839ndash854 2013

[87] J H Zhang H L Liu and L Sun ldquoIndependence bondagenumber and reinforcement number of some graphsrdquo Transac-tions of Beijing Institute of Technology vol 23 no 2 pp 140ndash1422003

[88] K D Dharmalingam Studies in Graph Theorymdashequitabledomination and bottleneck domination [PhD thesis] MaduraiKamaraj University Madurai India 2006

[89] GDeepakND Soner andAAlwardi ldquoThe equitable bondagenumber of a graphrdquo Research Journal of Pure Algebra vol 1 no9 pp 209ndash212 2011

[90] E Sampathkumar and H B Walikar ldquoThe connected domina-tion number of a graphrdquo Journal of Mathematical and PhysicalSciences vol 13 no 6 pp 607ndash613 1979

[91] K White M Farber and W Pulleyblank ldquoSteiner trees con-nected domination and strongly chordal graphsrdquoNetworks vol15 no 1 pp 109ndash124 1985

[92] M Cadei M X Cheng X Cheng and D-Z Du ldquoConnecteddomination in Ad Hoc wireless networksrdquo in Proceedings ofthe 6th International Conference on Computer Science andInformatics (CSampI rsquo02) 2002

[93] J F Fink and M S Jacobson ldquon-domination in graphsrdquo inGraph Theory with Applications to Algorithms and ComputerScience Wiley-Interscience Publications pp 283ndash300 WileyNew York NY USA 1985

[94] M Chellali O Favaron A Hansberg and L Volkmann ldquok-domination and k-independence in graphs a surveyrdquo Graphsand Combinatorics vol 28 no 1 pp 1ndash55 2012

[95] Y Lu X Hou J-M Xu andN Li ldquoTrees with uniqueminimump-dominating setsrdquo Utilitas Mathematica vol 86 pp 193ndash2052011

[96] Y Lu and J-M Xu ldquoThe p-bondage number of treesrdquo Graphsand Combinatorics vol 27 no 1 pp 129ndash141 2011

[97] Y Lu and J-M Xu ldquoThe 2-domination and 2-bondage numbersof grid graphs A manuscriptrdquo httparxivorgabs12044514

[98] M Krzywkowski ldquoNon-isolating 2-bondage in graphsrdquo TheJournal of the Mathematical Society of Japan vol 64 no 4 2012

[99] M A Henning O R Oellermann and H C Swart ldquoBoundson distance domination parametersrdquo Journal of CombinatoricsInformation amp System Sciences vol 16 no 1 pp 11ndash18 1991

[100] F Tian and J-M Xu ldquoBounds for distance domination numbersof graphsrdquo Journal of University of Science and Technology ofChina vol 34 no 5 pp 529ndash534 2004

[101] F Tian and J-M Xu ldquoDistance domination-critical graphsrdquoApplied Mathematics Letters vol 21 no 4 pp 416ndash420 2008

[102] F Tian and J-M Xu ldquoA note on distance domination numbersof graphsrdquo The Australasian Journal of Combinatorics vol 43pp 181ndash190 2009

[103] F Tian and J-M Xu ldquoAverage distances and distance domina-tion numbersrdquo Discrete Applied Mathematics vol 157 no 5 pp1113ndash1127 2009

[104] Z-L Liu F Tian and J-M Xu ldquoProbabilistic analysis of upperbounds for 2-connected distance k-dominating sets in graphsrdquoTheoretical Computer Science vol 410 no 38ndash40 pp 3804ndash3813 2009

[105] M A Henning O R Oellermann and H C Swart ldquoDistancedomination critical graphsrdquo Journal of Combinatorial Mathe-matics and Combinatorial Computing vol 44 pp 33ndash45 2003

[106] T J Bean M A Henning and H C Swart ldquoOn the integrityof distance domination in graphsrdquo The Australasian Journal ofCombinatorics vol 10 pp 29ndash43 1994

[107] M Fischermann and L Volkmann ldquoA remark on a conjecturefor the (kp)-domination numberrdquoUtilitasMathematica vol 67pp 223ndash227 2005

[108] T Korneffel D Meierling and L Volkmann ldquoA remark on the(22)-domination numberrdquo Discussiones Mathematicae GraphTheory vol 28 no 2 pp 361ndash366 2008

[109] Y Lu X Hou and J-M Xu ldquoOn the (22)-domination numberof treesrdquo Discussiones Mathematicae Graph Theory vol 30 no2 pp 185ndash199 2010

34 International Journal of Combinatorics

[110] H Li and J-M Xu ldquo(dm)-domination numbers of m-connected graphsrdquo Rapports de Recherche 1130 LRI UniversiteParis-Sud 1997

[111] S M Hedetniemi S T Hedetniemi and T V Wimer ldquoLineartime resource allocation algorithms for treesrdquo Tech Rep URI-014 Department of Mathematical Sciences Clemson Univer-sity 1987

[112] M Farber ldquoDomination independent domination and dualityin strongly chordal graphsrdquo Discrete Applied Mathematics vol7 no 2 pp 115ndash130 1984

[113] G S Domke and R C Laskar ldquoThe bondage and reinforcementnumbers of 120574

119891for some graphsrdquoDiscrete Mathematics vol 167-

168 pp 249ndash259 1997[114] G S Domke S T Hedetniemi and R C Laskar ldquoFractional

packings coverings and irredundance in graphsrdquo vol 66 pp227ndash238 1988

[115] E J Cockayne P A Dreyer Jr S M Hedetniemi andS T Hedetniemi ldquoRoman domination in graphsrdquo DiscreteMathematics vol 278 no 1ndash3 pp 11ndash22 2004

[116] I Stewart ldquoDefend the roman empirerdquo Scientific American vol281 pp 136ndash139 1999

[117] C S ReVelle and K E Rosing ldquoDefendens imperiumromanum a classical problem in military strategyrdquoThe Ameri-can Mathematical Monthly vol 107 no 7 pp 585ndash594 2000

[118] A Bahremandpour F-T Hu S M Sheikholeslami and J-MXu ldquoOn the Roman bondage number of a graphrdquo DiscreteMathematics Algorithms and Applications vol 5 no 1 ArticleID 1350001 15 pages 2013

[119] N Jafari Rad and L Volkmann ldquoRoman bondage in graphsrdquoDiscussiones Mathematicae Graph Theory vol 31 no 4 pp763ndash773 2011

[120] K Ebadi and L PushpaLatha ldquoRoman bondage and Romanreinforcement numbers of a graphrdquo International Journal ofContemporary Mathematical Sciences vol 5 pp 1487ndash14972010

[121] N Dehgardi S M Sheikholeslami and L Volkman ldquoOn theRoman k-bondage number of a graphrdquo AKCE InternationalJournal of Graphs and Combinatorics vol 8 pp 169ndash180 2011

[122] S Akbari and S Qajar ldquoA note on Roman bondage number ofgraphsrdquo Ars Combinatoria to Appear

[123] F-T Hu and J-M Xu ldquoRoman bondage numbers of somegraphsrdquo httparxivorgabs11093933

[124] N Jafari Rad and L Volkmann ldquoOn the Roman bondagenumber of planar graphsrdquo Graphs and Combinatorics vol 27no 4 pp 531ndash538 2011

[125] S Akbari M Khatirinejad and S Qajar ldquoA note on the romanbondage number of planar graphsrdquo Graphs and Combinatorics2012

[126] B Bresar M A Henning and D F Rall ldquoRainbow dominationin graphsrdquo Taiwanese Journal of Mathematics vol 12 no 1 pp213ndash225 2008

[127] B L Hartnell and D F Rall ldquoOn dominating the Cartesianproduct of a graph and 120572krdquo Discussiones Mathematicae GraphTheory vol 24 no 3 pp 389ndash402 2004

[128] N Dehgardi S M Sheikholeslami and L Volkman ldquoThe k-rainbow bondage number of a graphrdquo to appear in DiscreteApplied Mathematics

[129] J von Neumann and O Morgenstern Theory of Games andEconomic Behavior Princeton University Press Princeton NJUSA 1944

[130] C BergeTheorıe des Graphes et ses Applications 1958[131] O Ore Theory of Graphs vol 38 American Mathematical

Society Colloquium Publications 1962[132] E Shan and L Kang ldquoBondage number in oriented graphsrdquoArs

Combinatoria vol 84 pp 319ndash331 2007[133] J Huang and J-M Xu ldquoThe bondage numbers of extended

de Bruijn and Kautz digraphsrdquo Computers amp Mathematics withApplications vol 51 no 6-7 pp 1137ndash1147 2006

[134] J Huang and J-M Xu ldquoThe total domination and total bondagenumbers of extended de Bruijn and Kautz digraphsrdquoComputersamp Mathematics with Applications vol 53 no 8 pp 1206ndash12132007

[135] Y Shibata and Y Gonda ldquoExtension of de Bruijn graph andKautz graphrdquo Computers amp Mathematics with Applications vol30 no 9 pp 51ndash61 1995

[136] X Zhang J Liu and JMeng ldquoThebondage number in completet-partite digraphsrdquo Information Processing Letters vol 109 no17 pp 997ndash1000 2009

[137] D W Bange A E Barkauskas and P J Slater ldquoEfficient dom-inating sets in graphsrdquo in Applications of Discrete Mathematicspp 189ndash199 SIAM Philadelphia Pa USA 1988

[138] M Livingston and Q F Stout ldquoPerfect dominating setsrdquoCongressus Numerantium vol 79 pp 187ndash203 1990

[139] L Clark ldquoPerfect domination in random graphsrdquo Journal ofCombinatorial Mathematics and Combinatorial Computing vol14 pp 173ndash182 1993

[140] D W Bange A E Barkauskas L H Host and L H ClarkldquoEfficient domination of the orientations of a graphrdquo DiscreteMathematics vol 178 no 1ndash3 pp 1ndash14 1998

[141] A E Barkauskas and L H Host ldquoFinding efficient dominatingsets in oriented graphsrdquo Congressus Numerantium vol 98 pp27ndash32 1993

[142] I J Dejter and O Serra ldquoEfficient dominating sets in CayleygraphsrdquoDiscrete AppliedMathematics vol 129 no 2-3 pp 319ndash328 2003

[143] J Lee ldquoIndependent perfect domination sets in Cayley graphsrdquoJournal of Graph Theory vol 37 no 4 pp 213ndash219 2001

[144] WGu X Jia and J Shen ldquoIndependent perfect domination setsin meshes tori and treesrdquo Unpublished

[145] N Obradovic J Peters and G Ruzic ldquoEfficient domination incirculant graphswith two chord lengthsrdquo Information ProcessingLetters vol 102 no 6 pp 253ndash258 2007

[146] K Reji Kumar and G MacGillivray ldquoEfficient domination incirculant graphsrdquo Discrete Mathematics vol 313 no 6 pp 767ndash771 2013

[147] D Van Wieren M Livingston and Q F Stout ldquoPerfectdominating sets on cube-connected cyclesrdquo Congressus Numer-antium vol 97 pp 51ndash70 1993

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Operations ResearchAdvances in

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

24 International Journal of Combinatorics

93 Fractional Bondage Numbers If 120590 is a function mappingthe vertex-set 119881 into some set of real numbers then for anysubset 119878 sube 119881 let 120590(119878) = sum

119909isin119878120590(119909) Also let |120590| = 120590(119881)

A real-value function 120590 119881 rarr [0 1] is a dominatingfunction of a graph 119866 if for every 119909 isin 119881(119866) 120590(119873

119866[119909]) ⩾ 1

Thus if 119878 is a dominating set of 119866 120590 is a function where

120590 (119909) = 1 if 119909 isin 1198780 otherwise

(57)

then 120590 is a dominating function of119866 The fractional domina-tion number of 119866 denoted by 120574

119865(119866) is defined as follows

120574119865(119866) = min |120590| 120590 is a domination function of 119866

(58)

The 120574119865-bondage number of 119866 denoted by 119887

119865(119866) is

defined as the minimum cardinality of a subset 119861 sube 119864 whoseremoval results in 120574

119865(119866 minus 119861) gt 120574

119865(119866)

Hedetniemi et al [111] are the first to study fractionaldomination although Farber [112] introduced the idea indi-rectly The concept of the fractional domination numberwas proposed by Domke and Laskar [113] in 1997 Thefractional domination numbers for some ordinary graphs aredetermined

Proposition 150 (Domke et al [114] 1998 Domke and Laskar[113] 1997) (a) If 119866 is a 119896-regular graph with order 119899 then120574119865(119866) = 119899119896 + 1(b) 120574

119865(119862

119899) = 1198993 120574

119865(119875

119899) = lceil1198993rceil 120574

119865(119870

119899) = 1

(c) 120574119865(119866) = 1 if and only if Δ(119866) = 119899 minus 1

(d) 120574119865(119879) = 120574(119879) for any tree 119879

(e) 120574119865(119870

119899119898) = (119899(119898 + 1) + 119898(119899 + 1))(119899119898 minus 1) where

max119899119898 gt 1

The assertions (a)ndash(d) are due to Domke et al [114] andthe assertion (e) is due to Domke and Laskar [113] Accordingto these results Domke and Laskar [113] determined the 120574

119865-

bondage numbers for these graphs

Theorem 151 (Domke and Laskar [113] 1997) 119887119865(119870

119899) =

lceil1198992rceil 119887119865(119870

119899119898) = 1 wheremax119899119898 gt 1

119887119865(119862

119899) =

2 119894119891 119899 equiv 0 (mod 3) 1 119900119905ℎ119890119903119908119894119904119890

119887119865(119875

119899) =

2 119894119891 119899 equiv 1 (mod 3) 1 119900119905ℎ119890119903119908119894119904119890

(59)

It is easy to see that for any tree119879 119887119865(119879) ⩽ 2 In fact since

120574119865(119879) = 120574(119879) for any tree 119879 120574

119865(119879

1015840

) = 120574(1198791015840

) for any subgraph1198791015840 of 119879 It follows that

119887119865(119879) = min 119861 sube 119864 (119879) 120574

119865(119879 minus 119861) gt 120574

119865(119879)

= min 119861 sube 119864 (119879) 120574 (119879 minus 119861) gt 120574 (119879)

= 119887 (119879) ⩽ 2

(60)

Except for these no results on 119887119865(119866) have been known as

yet

94 Roman Bondage Numbers A Roman dominating func-tion on a graph 119866 is a labeling 119891 119881 rarr 0 1 2 such thatevery vertex with label 0 has at least one neighbor with label2 The weight of a Roman dominating function 119891 on 119866 is thevalue

119891 (119866) = sum

119906isin119881(119866)

119891 (119906) (61)

The minimum weight of a Roman dominating function ona graph 119866 is called the Roman domination number of 119866denoted by 120574RM (119866)

A Roman dominating function 119891 119881 rarr 0 1 2

can be represented by the ordered partition (1198810 119881

1 119881

2) (or

(119881119891

0 119881

119891

1 119881

119891

2) to refer to119891) of119881 where119881

119894= V isin 119881 | 119891(V) = 119894

In this representation its weight 119891(119866) = |1198811| + 2|119881

2| It is

clear that119881119891

1cup119881

119891

2is a dominating set of 119866 called the Roman

dominating set denoted by 119863119891

R = (1198811 1198812) Since 119881119891

1cup 119881

119891

2is

a dominating set when 119891 is a Roman dominating functionand since placing weight 2 at the vertices of a dominating setyields a Roman dominating function in [115] it is observedthat

120574 (119866) ⩽ 120574RM (119866) ⩽ 2120574 (119866) (62)

A graph 119866 is called to be Roman if 120574RM (119866) = 2120574(119866)The definition of the Roman domination function is

proposed implicitly by Stewart [116] and ReVelle and Rosing[117] Roman dominating numbers have been deeply studiedIn particular Bahremandpour et al [118] showed that theproblem determining the Roman domination number is NP-complete even for bipartite graphs

Let 119866 be a graph with maximum degree at least twoThe Roman bondage number 119887RM (119866) of 119866 is the minimumcardinality of all sets 1198641015840 sube 119864 for which 120574RM (119866 minus 119864

1015840

) gt

120574RM (119866) Since in study of Roman bondage number theassumption Δ(119866) ⩾ 2 is necessary we always assume thatwhen we discuss 119887RM (119866) all graphs involved satisfy Δ(119866) ⩾2 The Roman bondage number 119887RM (119866) is introduced byJafari Rad and Volkmann in [119]

Recently Bahremandpour et al [118] have shown that theproblem determining the Roman bondage number is NP-hard even for bipartite graphs

Problem 7 Consider the decision problem

Roman Bondage Problem

Instance a nonempty bipartite graph119866 and a positiveinteger 119896

Question is 119887RM (119866) ⩽ 119896

Theorem 152 (Bahremandpour et al [118] 2013) TheRomanbondage problem is NP-hard even for bipartite graphs

The exact value of 119887RM(119866) is known only for a few familyof graphs including the complete graphs cycles and paths

International Journal of Combinatorics 25

Theorem 153 (Jafari Rad and Volkmann [119] 2011) For anypositive integers 119899 and119898 with119898 ⩽ 119899

119887RM (119875119899) = 2 119894119891 119899 equiv 2 (mod 3) 1 119900119905ℎ119890119903119908119894119904119890

119887RM (119862119899) =

3 119894119891 119899 = 2 (mod 3) 2 119900119905ℎ119890119903119908119894119904119890

119887RM (119870119899) = lceil

119899

2rceil 119891119900119903 119899 ⩾ 3

119887RM (119870119898119899) =

1 119894119891 119898 = 1 119886119899119889 119899 = 1

4 119894119891 119898 = 119899 = 3

119898 119900119905ℎ119890119903119908119894119904119890

(63)

Lemma 154 (Cockayne et al [115] 2004) If 119866 is a graph oforder 119899 and contains vertices of degree 119899 minus 1 then 120574RM(119866) = 2

Using Lemma 154 the third conclusion in Theorem 153can be generalized to more general case which is similar toLemma 49

Theorem 155 (Jafari Rad and Volkmann [119] 2011) Let119866 bea graph with order 119899 ge 3 and 119905 the number of vertices of degree119899 minus 1 in 119866 If 119905 ⩾ 1 then 119887RM(119866) = lceil1199052rceil

Ebadi and PushpaLatha [120] conjectured that 119887RM(119866) ⩽119899 minus 1 for any graph 119866 of order 119899 ⩾ 3 Dehgardi et al[121] Akbari andQajar [122] independently showed that thisconjecture is true

Theorem 156 119887RM(119866) ⩽ 119899 minus 1 for any connected graph 119866 oforder 119899 ⩾ 3

For a complete 119905-partite graph we have the followingresult

Theorem 157 (Hu and Xu [123] 2011) Let 119866 = 11987011989811198982119898119905

bea complete 119905-partite graph with 119898

1= sdot sdot sdot = 119898

119894lt 119898

119894+1le sdot sdot sdot ⩽

119898119905 119905 ⩾ 2 and 119899 = sum119905

119895=1119898

119895 Then

119887RM (119866) =

lceil119894

2rceil 119894119891 119898

119894= 1 119886119899119889 119899 ⩾ 3

2 119894119891 119898119894= 2 119886119899119889 119894 = 1

119894 119894119891 119898119894= 2 119886119899119889 119894 ⩾ 2

4 119894119891 119898119894= 3 119886119899119889 119894 = 119905 = 2

119899 minus 1 119894119891 119898119894= 3 119886119899119889 119894 = 119905 ⩾ 3

119899 minus 119898119905

119894119891 119898119894⩾ 3 119886119899119889 119898

119905⩾ 4

(64)

Consider a complete 119905-partite graph 119870119905(3) for 119905 ⩾ 3

119887RM(119870119905(3)) = 119899 minus 1 by Theorem 157 where 119899 = 3119905

which shows that this upper bound of 119899 minus 1 on 119887RM(119866) inTheorem 156 is sharp This fact and 120574

119877(119870

119905(3)) = 4 give

a negative answer to the following two problems posed byDehgardi et al [121]Prove or Disprove for any connected graph 119866 of order 119899 ge 3(i) 119887RM(119866) = 119899 minus 1 if and only if 119866 cong 119870

3 (ii) 119887RM(119866) ⩽ 119899 minus

120574119877(119866) + 1Note that a complete 119905-partite graph 119870

119905(3) is (119899 minus 3)-

regular and 119887RM(119870119905(3)) = 119899 minus 1 for 119905 ⩾ 3 where 119899 = 3119905

Hu and Xu further determined that 119887119877(119866) = 119899 minus 2 for any

(119899 minus 3)-regular graph 119866 of order 119899 ⩽ 5 except 119870119905(3)

Theorem 158 (Hu and Xu [123] 2011) For any (119899minus3)-regulargraph 119866 of order 119899 other than119870

119905(3) 119887RM(119866) = 119899 minus 2 for 119899 ⩾ 5

For a tree 119879 with order 119899 ⩾ 3 Ebadi and PushpaLatha[120] and Jafari Rad and Volkmann [119] independentlyobtained an upper bound on 119887RM(119879)

Theorem 159 119887RM(119879) ⩽ 3 for any tree 119879 with order 119899 ⩾ 3

Theorem 160 (Bahremandpour et al [118] 2013) 119887RM(1198752 times119875119899) = 2 for 119899 ⩾ 2

Theorem 161 (Jafari Rad and Volkmann [119] 2011) Let119866 bea graph with order at least three

(a) If (119909 119910 119911) is a path of length 2 in 119866 then 119887RM(119866) ⩽119889119866(119909) + 119889

119866(119910) + 119889

119866(119911) minus 3 minus |119873

119866(119909) cap 119873

119866(119910)|

(b) If 119866 is connected then 119887RM(119866) ⩽ 120582(119866) + 2Δ(119866) minus 3

Theorem 161 (b) implies 119887RM(119866) ⩽ 120575(119866) + 2Δ(119866) minus 3 for aconnected graph 119866 Note that for a planar graph 119866 120575(119866) ⩽ 5moreover 120575(119866) ⩽ 3 if the girth at least 4 and 120575(119866) ⩽ 2 if thegirth at least 6 These two facts show that 119887RM(119866) ⩽ 2Δ(119866) +2 for a connected planar graph 119866 Jafari Rad and Volkmann[124] improved this bound

Theorem 162 (Jafari Rad and Volkmann [124] 2011) For anyconnected planar graph 119866 of order 119899 ⩾ 3 with girth 119892(119866)

119887RM (119866)

⩽

2Δ (119866)

Δ (119866) + 6

Δ (119866) + 5 119894119891 119866 119888119900119899119905119886119894119899119904 119899119900 V119890119903119905119894119888119890119904 119900119891 119889119890119892119903119890119890 119891119894V119890

Δ (119866) + 4 119894119891 119892 (119866) ⩾ 4

Δ (119866) + 3 119894119891 119892 (119866) ⩾ 5

Δ (119866) + 2 119894119891 119892 (119866) ⩾ 6

Δ (119866) + 1 119894119891 119892 (119866) ⩾ 8

(65)

According toTheorem 157 119887RM(119862119899) = 3 = Δ(119862

119899)+1 for a

cycle 119862119899of length 119899 ⩾ 8 with 119899 equiv 2 (mod 3) and therefore

the last upper bound inTheorem 162 is sharp at least for Δ =2

Combining the fact that every planar graph 119866 withminimum degree 5 contains an edge 119909119910 with 119889

119866(119909) = 5

and 119889119866(119910) isin 5 6 with Theorem 161 (a) Akbari et al [125]

obtained the following result

26 International Journal of Combinatorics

middot middot middot

middot middot middot

middot middot middot

middot middot middot

119866

Figure 9 The graph 119866 is constructed from 119866

Theorem 163 (Akbari et al [125] 2012) 119887RM(119866) ⩽ 15 forevery planar graph 119866

It remains open to show whether the bound inTheorem 163 is sharp or not Though finding a planargraph 119866 with 119887RM(119866) = 15 seems to be difficult Akbari etal [125] constructed an infinite family of planar graphs withRoman bondage number equal to 7 by proving the followingresult

Theorem 164 (Akbari et al [125] 2012) Let 119866 be a graph oforder 119899 and 119866 is the graph of order 5119899 obtained from 119866 byattaching the central vertex of a copy of a path 119875

5to each vertex

of119866 (see Figure 9) Then 120574RM(119866) = 4119899 and 119887RM(119866) = 120575(119866)+2

ByTheorem 164 infinitemany planar graphswith Romanbondage number 7 can be obtained by considering any planargraph 119866 with 120575(119866) = 5 (eg the icosahedron graph)

Conjecture 165 (Akbari et al [125] 2012) The Romanbondage number of every planar graph is at most 7

For general bounds the following observation is directlyobtained which is similar to Observation 1

Observation 4 Let 119866 be a graph of order 119899 with maximumdegree at least two Assume that 119867 is a spanning subgraphobtained from 119866 by removing 119896 edges If 120574RM(119867) = 120574RM(119866)then 119887

119877(119867) ⩽ 119887RM(119866) ⩽ 119887RM(119867) + 119896

Theorem 166 (Bahremandpour et al [118] 2013) For anyconnected graph 119866 of order 119899 if 119899 ⩾ 3 and 120574RM(119866) = 120574(119866) + 1then

119887RM (119866) ⩽ min 119887 (119866) 119899Δ (66)

where 119899Δis the number of vertices with maximum degree Δ in

119866

Observation 5 For any graph 119866 if 120574(119866) = 120574RM(119866) then119887RM(119866) ⩽ 119887(119866)

Theorem 167 (Bahremandpour et al [118] 2013) For everyRoman graph 119866

119887RM (119866) ⩾ 119887 (119866) (67)

The bound is sharp for cycles on 119899 vertices where 119899 equiv 0 (mod3)

The strict inequality in Theorem 167 can hold for exam-ple 119887(119862

3119896+2) = 2 lt 3 = 119887RM(1198623119896+2

) byTheorem 157A graph 119866 is called to be vertex Roman domination-

critical if 120574RM(119866 minus 119909) lt 120574RM(119866) for every vertex 119909 in 119866 If119866 has no isolated vertices then 120574RM(119866) ⩽ 2120574(119866) ⩽ 2120573(119866)If 120574RM(119866) = 2120573(119866) then 120574RM(119866) = 2120574(119866) and hence 119866 is aRoman graph In [30] Volkmann gave a lot of graphs with120574(119866) = 120573(119866) The following result is similar to Theorem 57

Theorem 168 (Bahremandpour et al [118] 2013) For anygraph 119866 if 120574RM(119866) = 2120573(119866) then

(a) 119887RM(119866) ⩾ 120575(119866)

(b) 119887RM(119866) ⩾ 120575(119866)+1 if119866 is a vertex Roman domination-critical graph

If 120574RM(119866) = 2 then obviously 119887RM(119866) ⩽ 120575(119866) So weassume 120574RM(119866) ⩾ 3 Dehgardi et al [121] proved 119887RM(119866) ⩽(120574RM(119866) minus 2)Δ(119866) + 1 and posed the following problem if 119866is a connected graph of order 119899 ⩾ 4 with 120574RM(119866) ⩾ 3 then

119887RM (119866) ⩽ (120574RM (119866) minus 2) Δ (119866) (68)

Theorem 161 (a) shows that the inequality (68) holds if120574RM(119866) ⩾ 5 Thus the bound in (68) is of interest only when120574RM(119866) is 3 or 4

Lemma 169 (Bahremandpour et al [118] 2013) For anynonempty graph 119866 of order 119899 ⩾ 3 120574RM(119866) = 3 if and onlyif Δ(119866) = 119899 minus 2

The following result shows that (68) holds for all graphs119866 of order 119899 ⩾ 4 with 120574RM(119866) = 3 or 4 which improvesTheorem 161 (b)

Theorem 170 (Bahremandpour et al [118] 2013) For anyconnected graph 119866 of order 119899 ⩾ 4

119887RM (119866) le Δ (119866) = 119899 minus 2 119894119891 120574RM (119866) = 3

Δ (119866) + 120575 (119866) minus 1 119894119891 120574RM (119866) = 4(69)

with the first equality if and only if 119866 cong 1198624

Recently Akbari and Qajar [122] have proved the follow-ing result

Theorem 171 (Akbari and Qajar [122]) For any connectedgraph 119866 of order 119899 ⩾ 3

119887RM (119866) ⩽ 119899 minus 120574RM (119866) + 5 (70)

International Journal of Combinatorics 27

We conclude this section with the following problems

Problem 8 Characterize all connected graphs 119866 of order 119899 ⩾3 for which 119887RM(119866) = 119899 minus 1

Problem 9 Prove or disprove if 119866 is a connected graph oforder 119899 ⩾ 3 then

119887RM (119866) ⩽ 119899 minus 120574RM (119866) + 3 (71)

Note that 120574119877(119870

119905(3)) = 4 and 119887RM(119870119905

(3)) = 119899minus1 where 119899 =3119905 for 119905 ⩾ 3 The upper bound on 119887RM(119866) given in Problem 9is sharp if Problem 9 has positive answer

95 Rainbow Bondage Numbers For a positive integer 119896 a119896-rainbow dominating function of a graph 119866 is a function 119891from the vertex-set 119881(119866) to the set of all subsets of the set1 2 119896 such that for any vertex 119909 in 119866 with 119891(119909) = 0 thecondition

⋃

119910isin119873119866(119909)

119891 (119910) = 1 2 119896 (72)

is fulfilledThe weight of a 119896-rainbow dominating function 119891on 119866 is the value

119891 (119866) = sum

119909isin119881(119866)

1003816100381610038161003816119891 (119909)1003816100381610038161003816 (73)

The 119896-rainbow domination number of a graph 119866 denoted by120574119903119896(119866) is the minimum weight of a 119896-rainbow dominating

function 119891 of 119866Note that 120574

1199031(119866) is the classical domination number 120574(119866)

The 119896-rainbow domination number is introduced by Bresaret al [126] Rainbow domination of a graph 119866 coincides withordinary domination of the Cartesian product of 119866 with thecomplete graph in particular 120574

119903119896(119866) = 120574(119866 times 119870

119896) for any

positive integer 119896 and any graph 119866 [126] Hartnell and Rall[127] proved that 120574(119879) lt 120574

1199032(119879) for any tree 119879 no graph has

120574 = 120574119903119896for 119896 ⩾ 3 for any 119896 ⩾ 2 and any graph 119866 of order 119899

min 119899 120574 (119866) + 119896 minus 2 ⩽ 120574119903119896(119866) ⩽ 119896120574 (119866) (74)

The introduction of rainbow domination is motivatedby the study of paired domination in Cartesian productsof graphs where certain upper bounds can be expressed interms of rainbow domination that is for any graph 119866 andany graph 119867 if 119881(119867) can be partitioned into 119896120574

119875-sets then

120574119875(119866 times 119867) ⩽ (1119896)120592(119867)120574

119903119896(119866) proved by Bresar et al [126]

Dehgardi et al [128] introduced the concept of 119896-rainbowbondage number of graphs Let 119866 be a graph with maximumdegree at least two The 119896-rainbow bondage number 119887

119903119896(119866)

of 119866 is the minimum cardinality of all sets 119861 sube 119864(119866) forwhich 120574

119903119896(119866 minus 119861) gt 120574

119903119896(119866) Since in the study of 119896-rainbow

bondage number the assumption Δ(119866) ⩾ 2 is necessarywe always assume that when we discuss 119887

119903119896(119866) all graphs

involved satisfy Δ(119866) ⩾ 2The 2-rainbow bondage number of some special families

of graphs has been determined

Theorem 172 (Dehgardi et al [128]) For any integer 119899 ⩾ 3

1198871199032(119875

119899) = 1 119891119900119903 119899 ⩾ 2

1198871199032(119862

119899) =

1 119894119891 119899 equiv 0 (mod 4) 2 119900119905ℎ119890119903119908119894119904119890

1198871199032(119870

119899119899) =

1 119894119891 119899 = 2

3 119894119891 119899 = 3

6 119894119891 119899 = 4

119898 119894119891 119899 ⩾ 5

(75)

Theorem 173 (Dehgardi et al [128]) For any connected graph119866 of order 119899 ⩾ 3 119887

1199032(119866) ⩽ 120582(119866) + 2Δ(119866) minus 3

Theorem 174 (Dehgardi et al [128]) For any tree 119879 of order119899 ⩾ 3 119887

1199032(119879) ⩽ 2

Theorem 175 (Dehgardi et al [128]) If 119866 is a planar graphwithmaximumdegree at least two then 119887

1199032(119866) ⩽ 15Moreover

if the girth 119892(119866) ⩾ 4 then 1198871199032(119866) ⩽ 11 and if 119892(119866) ⩾ 6 then

1198871199032(119866) ⩽ 9

Theorem 176 (Dehgardi et al [128]) For a complete bipartitegraph 119870

119901119902 if 2119896 + 1 ⩽ 119901 ⩽ 119902 then 119887

119903119896(119870

119901119902) = 119901

Theorem177 (Dehgardi et al [128]) For a complete graph119870119899

if 119899 ⩾ 119896 + 1 then 119887119903119896(119870

119899) = lceil119896119899(119896 + 1)rceil

The special case 119896 = 1 of Theorem 177 can be found inTheorem 1 The following is a simple observation similar toObservation 1

Observation 6 (Dehgardi et al [128]) Let 119866 be a graphwith maximum degree at least two 119867 a spanning subgraphobtained from 119866 by removing 119896 edges If 120574

119903119896(119867) = 120574

119903119896(119866)

then 119887119903119896(119867) ⩽ 119887

119903119896(119866) ⩽ 119887

119903119896(119867) + 119896

Theorem 178 (Dehgardi et al [128]) For every graph 119866 with120574119903119896(119866) = 119896120574(119866) 119887

119903119896(119866) ⩾ 119887(119866)

A graph 119866 is said to be vertex 119896-rainbow domination-critical if 120574

119903119896(119866 minus 119909) lt 120574

119903119896(119866) for every vertex 119909 in 119866 It is

clear that if 119866 has no isolated vertices then 120574119903119896(119866) ⩽ 119896120574(119866) ⩽

119896120573(119866) where 120573(119866) is the vertex-covering number of119866 Thusif 120574

119903119896(119866) = 119896120573(119866) then 120574

119903119896(119866) = 119896120574(119866)

Theorem 179 (Dehgardi et al [128]) For any graph 119866 if120574119903119896(119866) = 119896120573(119866) then

(a) 119887119903119896(119866) ⩾ 120575(119866)

(b) 119887119903119896(119866) ⩾ 120575(119866)+1 if119866 is vertex 119896-rainbow domination-

critical

As far as we know there are no more results on the 119896-rainbow bondage number of graphs in the literature

28 International Journal of Combinatorics

10 Results on Digraphs

Although domination has been extensively studied in undi-rected graphs it is natural to think of a dominating setas a one-way relationship between vertices of the graphIndeed among the earliest literatures on this subject vonNeumann and Morgenstern [129] used what is now calleddomination in digraphs to find solution (or kernels whichare independent dominating sets) for cooperative 119899-persongames Most likely the first formulation of domination byBerge [130] in 1958 was given in the context of digraphs andonly some years latter by Ore [131] in 1962 for undirectedgraphs Despite this history examination of domination andits variants in digraphs has been essentially overlooked (see[4] for an overview of the domination literature) Thus thereare few if any such results on domination for digraphs in theliterature

The bondage number and its related topics for undirectedgraph have become one of major areas both in theoreticaland applied researches However until recently Carlsonand Develin [42] Shan and Kang [132] and Huang andXu [133 134] studied the bondage number for digraphsindependently In this section we introduce their resultsfor general digraphs Results for some special digraphs suchas vertex-transitive digraphs are introduced in the nextsection

101 Upper Bounds for General Digraphs Let 119866 = (119881 119864)

be a digraph without loops and parallel edges A digraph 119866is called to be asymmetric if whenever (119909 119910) isin 119864(119866) then(119910 119909) notin 119864(119866) and to be symmetric if (119909 119910) isin 119864(119866) implies(119910 119909) isin 119864(119866) For a vertex 119909 of 119881(119866) the sets of out-neighbors and in-neighbors of 119909 are respectively defined as119873

+

119866(119909) = 119910 isin 119881(119866) (119909 119910) isin 119864(119866) and 119873minus

119866(119909) = 119909 isin

119881(119866) (119909 119910) isin 119864(119866) the out-degree and the in-degree of119909 are respectively defined as 119889+

119866(119909) = |119873

+

119866(119909)| and 119889minus

119866(119909) =

|119873+

119866(119909)| Denote themaximum and theminimum out-degree

(resp in-degree) of 119866 by Δ+(119866) and 120575+(119866) (resp Δminus(119866) and120575minus

(119866))The degree 119889119866(119909) of 119909 is defined as 119889+

119866(119909)+119889

minus

119866(119909) and

the maximum and the minimum degrees of119866 are denoted byΔ(119866) and 120575(119866) respectively that is Δ(119866) = max119889

119866(119909) 119909 isin

119881(119866) and 120575(119866) = min119889119866(119909) 119909 isin 119881(119866) Note that the

definitions here are different from the ones in the textbookon digraphs see for example Xu [1]

A subset 119878 of119881(119866) is called a dominating set if119881(119866) = 119878cup119873

+

119866(119878) where119873+

119866(119878) is the set of out-neighbors of 119878Then just

as for undirected graphs 120574(119866) is the minimum cardinalityof a dominating set and the bondage number 119887(119866) is thesmallest cardinality of a set 119861 of edges such that 120574(119866 minus 119861) gt120574(119866) if such a subset 119861 sube 119864(119866) exists Otherwise we put119887(119866) = infin

Some basic results for undirected graphs stated inSection 3 can be generalized to digraphs For exampleTheorem 18 is generalized by Carlson and Develin [42]Huang andXu [133] and Shan andKang [132] independentlyas follows

Theorem 180 Let 119866 be a digraph and (119909 119910) isin 119864(119866) Then119887(119866) ⩽ 119889

119866(119910) + 119889

minus

119866(119909) minus |119873

minus

119866(119909) cap 119873

minus

119866(119910)|

Corollary 181 For a digraph 119866 119887(119866) ⩽ 120575minus(119866) + Δ(119866)

Since 120575minus

(119866) ⩽ (12)Δ(119866) from Corollary 181Conjecture 53 is valid for digraphs

Corollary 182 For a digraph 119866 119887(119866) ⩽ (32)Δ(119866)

In the case of undirected graphs the bondage number of119866119899= 119870

119899times 119870

119899achieves this bound However it was shown

by Carlson and Develin in [42] that if we take the symmetricdigraph119866

119899 we haveΔ(119866

119899) = 4(119899minus1) 120574(119866

119899) = 119899 and 119887(119866

119899) ⩽

4(119899 minus 1) + 1 = Δ(119866119899) + 1 So this family of digraphs cannot

show that the bound in Corollary 182 is tightCorresponding to Conjecture 51 which is discredited

for undirected graphs and is valid for digraphs the sameconjecture for digraphs can be proposed as follows

Conjecture 183 (Carlson and Develin [42] 2006) 119887(119866) ⩽Δ(119866) + 1 for any digraph 119866

If Conjecture 183 is true then the following results showsthat this upper bound is tight since 119887(119870

119899times119870

119899) = Δ(119870

119899times119870

119899)+1

for a complete digraph 119870119899

In 2007 Shan and Kang [132] gave some tight upperbounds on the bondage numbers for some asymmetricdigraphs For example 119887(119879) ⩽ Δ(119879) for any asymmetricdirected tree 119879 119887(119866) ⩽ Δ(119866) for any asymmetric digraph 119866with order at least 4 and 120574(119866) ⩽ 2 For planar digraphs theyobtained the following results

Theorem 184 (Shan and Kang [132] 2007) Let 119866 be aasymmetric planar digraphThen 119887(119866) ⩽ Δ(119866)+2 and 119887(119866) ⩽Δ(119866) + 1 if Δ(119866) ⩾ 5 and 119889minus

119866(119909) ⩾ 3 for every vertex 119909 with

119889119866(119909) ⩾ 4

102 Results for Some Special Digraphs The exact values andbounds on 119887(119866) for some standard digraphs are determined

Theorem185 (Huang andXu [133] 2006) For a directed cycle119862119899and a directed path 119875

119899

119887 (119862119899) =

3 119894119891 119899 119894119904 119900119889119889

2 119894119891 119899 119894119904 119890V119890119899

119887 (119875119899) =

2 119894119891 119899 119894119904 119900119889119889

1 119894119891 119899 119894119904 119890V119890119899

(76)

For the de Bruijn digraph 119861(119889 119899) and the Kautz digraph119870(119889 119899)

119887 (119861 (119889 119899)) = 119889 119894119891 119899 119894119904 119900119889119889

119889 ⩽ 119887 (119861 (119889 119899)) ⩽ 2119889 119894119891 119899 119894119904 119890V119890119899

119887 (119870 (119889 119899)) = 119889 + 1

(77)

Like undirected graphs we can define the total domi-nation number and the total bondage number On the totalbondage numbers for some special digraphs the knownresults are as follows

International Journal of Combinatorics 29

Theorem 186 (Huang and Xu [134] 2007) For a directedcycle 119862

119899and a directed path 119875

119899 119887

119879(119875

119899) and 119887

119879(119862

119899) all do not

exist For a complete digraph 119870119899

119887119879(119870

119899) = infin 119894119891 119899 = 1 2

119887119879(119870

119899) = 3 119894119891 119899 = 3

119899 ⩽ 119887119879(119870

119899) ⩽ 2119899 minus 3 119894119891 119899 ⩾ 4

(78)

The extended de Bruijn digraph EB(119889 119899 1199021 119902

119901) and

the extended Kautz digraph EK(119889 119899 1199021 119902

119901) are intro-

duced by Shibata and Gonda [135] If 119901 = 1 then theyare the de Bruijn digraph B(119889 119899) and the Kautz digraphK(119889 119899) respectively Huang and Xu [134] determined theirtotal domination numbers In particular their total bondagenumbers for general cases are determined as follows

Theorem 187 (Huang andXu [134] 2007) If 119889 ⩾ 2 and 119902119894⩾ 2

for each 119894 = 1 2 119901 then

119887119879(EB(119889 119899 119902

1 119902

119901)) = 119889

119901

minus 1

119887119879(EK(119889 119899 119902

1 119902

119901)) = 119889

119901

(79)

In particular for the de Bruijn digraph B(119889 119899) and the Kautzdigraph K(119889 119899)

119887119879(B (119889 119899)) = 119889 minus 1 119887

119879(K (119889 119899)) = 119889 (80)

Zhang et al [136] determined the bondage number incomplete 119905-partite digraphs

Theorem 188 (Zhang et al [136] 2009) For a complete 119905-partite digraph 119870

11989911198992119899119905

where 1198991⩽ 119899

2⩽ sdot sdot sdot ⩽ 119899

119905

119887 (11987011989911198992119899119905

)

=

119898 119894119891 119899119898= 1 119899

119898+1⩾ 2 119891119900119903 119904119900119898119890

119898 (1 ⩽ 119898 lt 119905)

4119905 minus 3 119894119891 1198991= 119899

2= sdot sdot sdot = 119899

119905= 2

119905minus1

sum

119894=1

119899119894+ 2 (119905 minus 1) 119900119905ℎ119890119903119908119894119904119890

(81)

Since an undirected graph can be thought of as a sym-metric digraph any result for digraphs has an analogy forundirected graphs in general In view of this point studyingthe bondage number for digraphs is more significant thanfor undirected graphs Thus we should further study thebondage number of digraphs and try to generalize knownresults on the bondage number and related variants for undi-rected graphs to digraphs prove or disprove Conjecture 183In particular determine the exact values of 119887(B(119889 119899)) for aneven 119899 and 119887

119879(119870

119899) for 119899 ⩾ 4

11 Efficient Dominating Sets

A dominating set 119878 of a graph 119866 is called to be efficient if forevery vertex 119909 in119866 |119873

119866[119909]cap119878| = 1 if119866 is a undirected graph

or |119873minus

119866[119909] cap 119878| = 1 if 119866 is a directed graph The concept of an

efficient set was proposed by Bange et al [137] in 1988 In theliterature an efficient set is also called a perfect dominatingset (see Livingston and Stout [138] for an explanation)

From definition if 119878 is an efficient dominating set of agraph 119866 then 119878 is certainly an independent set and everyvertex not in 119878 is adjacent to exactly one vertex in 119878

It is also clear from definition that a dominating set 119878 isefficient if and only if N

119866[119878] = 119873

119866[119909] 119909 isin 119878 for the

undirected graph 119866 or N+

119866[119878] = 119873

+

119866[119909] 119909 isin 119878 for the

digraph119866 is a partition of119881(119866) where the induced subgraphby119873

119866[119909] or119873+

119866[119909] is a star or an out-star with the root 119909

The efficient domination has important applications inmany areas such as error-correcting codes and receivesmuch attention in the late years

The concept of efficient dominating sets is a measureof the efficiency of domination in graphs Unfortunately asshown in [137] not every graph has an efficient dominatingset andmoreover it is anNP-complete problem to determinewhether a given graph has an efficient dominating set Inaddition it was shown by Clark [139] in 1993 that for a widerange of 119901 almost every random undirected graph 119866 isin

G(120592 119901) has no efficient dominating sets This means thatundirected graphs possessing an efficient dominating set arerare However it is easy to show that every undirected graphhas an orientationwith an efficient dominating set (see Bangeet al [140])

In 1993 Barkauskas and Host [141] showed that deter-mining whether an arbitrary oriented graph has an efficientdominating set is NP-complete Even so the existence ofefficient dominating sets for some special classes of graphs hasbeen examined see for example Dejter and Serra [142] andLee [143] for Cayley graph Gu et al [144] formeshes and toriObradovic et al [145] Huang and Xu [36] Reji Kumar andMacGillivray [146] for circulant graphs Harary graphs andtori and Van Wieren et al [147] for cube-connected cycles

In this section we introduce results of the bondagenumber for some graphs with efficient dominating sets dueto Huang and Xu [36]

111 Results for General Graphs In this subsection we intro-duce some results on bondage numbers obtained by applyingefficient dominating sets We first state the two followinglemmas

Lemma 189 For any 119896-regular graph or digraph 119866 of order 119899120574(119866) ⩾ 119899(119896 + 1) with equality if and only if 119866 has an efficientdominating set In addition if 119866 has an efficient dominatingset then every efficient dominating set is certainly a 120574-set andvice versa

Let 119890 be an edge and 119878 a dominating set in 119866 We say that119890 supports 119878 if 119890 isin (119878 119878) where (119878 119878) = (119909 119910) isin 119864(119866) 119909 isin 119878 119910 isin 119878 Denote by 119904(119866) the minimum number of edgeswhich support all 120574-sets in 119866

Lemma 190 For any graph or digraph 119866 119887(119866) ⩾ 119904(119866) withequality if 119866 is regular and has an efficient dominating set

30 International Journal of Combinatorics

A graph 119866 is called to be vertex-transitive if its automor-phism group Aut(119866) acts transitively on its vertex-set 119881(119866)A vertex-transitive graph is regular Applying Lemmas 189and 190 Huang and Xu obtained some results on bondagenumbers for vertex-transitive graphs or digraphs

Theorem 191 (Huang and Xu [36] 2008) For any vertex-transitive graph or digraph 119866 of order 119899

119887 (119866) ⩾

lceil119899

2120574 (119866)rceil 119894119891 119866 119894119904 119906119899119889119894119903119890119888119905119890119889

lceil119899

120574 (119866)rceil 119894119891 119866 119894119904 119889119894119903119890119888119905119890119889

(82)

Theorem192 (Huang andXu [36] 2008) Let119866 be a 119896-regulargraph of order 119899 Then

119887(119866) ⩽ 119896 if119866 is undirected and 119899 ⩾ 120574(119866)(119896+1)minus119896+1119887(119866) ⩽ 119896+1+ℓ if119866 is directed and 119899 ⩾ 120574(119866)(119896+1)minusℓwith 0 ⩽ ℓ ⩽ 119896 minus 1

Next we introduce a better upper bound on 119887(119866) To thisaim we need the following concept which is a generalizationof the concept of the edge-covering of a graph 119866 For 1198811015840

sube

119881(119866) and 1198641015840 sube 119864(119866) we say that 1198641015840covers 1198811015840 and call 1198641015840an edge-covering for 1198811015840 if there exists an edge (119909 119910) isin 1198641015840 forany vertex 119909 isin 1198811015840 For 119910 isin 119881(119866) let 1205731015840[119910] be the minimumcardinality over all edge-coverings for119873minus

119866[119910]

Theorem 193 (Huang and Xu [36] 2008) If 119866 is a 119896-regulargraph with order 120574(119866)(119896 + 1) then 119887(119866) ⩽ 1205731015840[119910] for any 119910 isin119881(119866)

Theupper bound on 119887(119866) given inTheorem 193 is tight inview of 119887(119862

119899) for a cycle or a directed cycle 119862

119899(seeTheorems

1 and 185 resp)It is easy to see that for a 119896-regular graph 119866 lceil(119896 + 1)2rceil ⩽

1205731015840

[119910] ⩽ 119896 when 119866 is undirected and 1205731015840[119910] = 119896 + 1 when 119866 isdirected By this fact and Lemma 189 the following theoremis merely a simple combination of Theorems 191 and 193

Theorem 194 (Huang and Xu [36] 2008) Let 119866 be a vertex-transitive graph of degree 119896 If 119866 has an efficient dominatingset then

lceil119896 + 1

2rceil ⩽ 119887 (119866) ⩽ 119896 119894119891 119866 119894119904 119906119899119889119894119903119890119888119905119890119889

119887 (119866) = 119896 + 1 119894119891 119866 119894119904 119889119894119903119890119888119905119890119889

(83)

Theorem 195 (Huang and Xu [36] 2008) If 119866 is an undi-rected vertex-transitive cubic graph with order 4120574(119866) and girth119892(119866) ⩽ 5 then 119887(119866) = 2

The proof of Theorem 195 leads to a byproduct If 119892 =5 let (119906

1 119906

2 119906

3 119906

4 119906

5) be a cycle and let D

119894be the family

of efficient dominating sets containing 119906119894for each 119894 isin

1 2 3 4 5 ThenD119894capD

119895= 0 for 119894 = 1 2 5 Then 119866 has

at least 5119904 efficient dominating sets where 119904 = 119904(119866) But thereare only 4s (=ns120574) distinct efficient dominating sets in119866This

contradiction implies that an undirected vertex-transitivecubic graph with girth five has no efficient dominating setsBut a similar argument for 119892(119866) = 3 4 or 119892(119866) ⩾ 6 couldnot give any contradiction This is consistent with the resultthat CCC(119899) a vertex-transitive cubic graph with girth 119899 if3 ⩽ 119899 ⩽ 8 or girth 8 if 119899 ⩾ 9 has efficient dominating setsfor all 119899 ⩾ 3 except 119899 = 5 (see Theorem 199 in the followingsubsection)

112 Results for Cayley Graphs In this subsection we useTheorem 194 to determine the exact values or approximativevalues of bondage numbers for some special vertex-transitivegraphs by characterizing the existence of efficient dominatingsets in these graphs

Let Γ be a nontrivial finite group 119878 a nonempty subset ofΓ without the identity element of Γ A digraph 119866 is defined asfollows

119881 (119866) = Γ (119909 119910) isin 119864 (119866) lArrrArr 119909minus1

119910 isin 119878

for any 119909 119910 isin Γ(84)

is called a Cayley digraph of the group Γ with respect to119878 denoted by 119862

Γ(119878) If 119878minus1 = 119904minus1 119904 isin 119878 = 119878 then

119862Γ(119878) is symmetric and is called a Cayley undirected graph

a Cayley graph for short Cayley graphs or digraphs arecertainly vertex-transitive (see Xu [1])

A circulant graph 119866(119899 119878) of order 119899 is a Cayley graph119862119885119899

(119878) where 119885119899= 0 119899 minus 1 is the addition group of

order 119899 and 119878 is a nonempty subset of119885119899without the identity

element and hence is a vertex-transitive digraph of degree|119878| If 119878minus1 = 119878 then119866(119899 119878) is an undirected graph If 119878 = 1 119896where 2 ⩽ 119896 ⩽ 119899 minus 2 we write 119866(119899 1 119896) for 119866(119899 1 119896) or119866(119899 plusmn1 plusmn119896) and call it a double-loop circulant graph

Huang and Xu [36] showed that for directed 119866 =

119866(119899 1 119896) lceil1198993rceil ⩽ 120574(119866) ⩽ lceil1198992rceil and 119866 has an efficientdominating set if and only if 3 | 119899 and 119896 equiv 2 (mod 3) fordirected 119866 = 119866(119899 1 119896) and 119896 = 1198992 lceil1198995rceil ⩽ 120574(119866) ⩽ lceil1198993rceiland 119866 has an efficient dominating set if and only if 5 | 119899 and119896 equiv plusmn2 (mod 5) By Theorem 194 we obtain the bondagenumber of a double loop circulant graph if it has an efficientdominating set

Theorem 196 (Huang and Xu [36] 2008) Let 119866 be a double-loop circulant graph 119866(119899 1 119896) If 119866 is directed with 3 | 119899and 119896 equiv 2 (mod 3) or 119866 is undirected with 5 | 119899 and119896 equiv plusmn2 (mod 5) then 119887(119866) = 3

The 119898 times 119899 torus is the Cartesian product 119862119898times 119862

119899of

two cycles and is a Cayley graph 119862119885119898times119885119899

(119878) where 119878 =

(0 1) (1 0) for directed cycles and 119878 = (0 plusmn1) (plusmn1 0) forundirected cycles and hence is vertex-transitive Gu et al[144] showed that the undirected torus119862

119898times119862

119899has an efficient

dominating set if and only if both 119898 and 119899 are multiplesof 5 Huang and Xu [36] showed that the directed torus119862119898times 119862

119899has an efficient dominating set if and only if both

119898 and 119899 are multiples of 3 Moreover they found a necessarycondition for a dominating set containing the vertex (0 0)in 119862

119898times 119862

119899to be efficient and obtained the following

result

International Journal of Combinatorics 31

Theorem 197 (Huang and Xu [36] 2008) Let 119866 = 119862119898times 119862

119899

If 119866 is undirected and both119898 and 119899 are multiples of 5 or if 119866 isdirected and both119898 and 119899 are multiples of 3 then 119887(119866) = 3

The hypercube 119876119899

is the Cayley graph 119862Γ(119878)

where Γ = 1198852times sdot sdot sdot times 119885

2= (119885

2)119899 and 119878 =

100 sdot sdot sdot 0 010 sdot sdot sdot 0 00 sdot sdot sdot 01 Lee [143] showed that119876119899has an efficient dominating set if and only if 119899 = 2119898 minus 1

for a positive integer119898 ByTheorem 194 the following resultis obtained immediately

Theorem 198 (Huang and Xu [36] 2008) If 119899 = 2119898 minus 1 for apositive integer119898 then 2119898minus1

⩽ 119887(119876119899) ⩽ 2

119898

minus 1

Problem 10 Determine the exact value of 119887(119876119899) for 119899 = 2119898minus1

The 119899-dimensional cube-connected cycle denoted byCCC(119899) is constructed from the 119899-dimensional hypercube119876119899by replacing each vertex 119909 in 119876

119899with an undirected cycle

119862119899of length 119899 and linking the 119894th vertex of the 119862

119899to the 119894th

neighbor of 119909 It has been proved that CCC(119899) is a Cayleygraph and hence is a vertex-transitive graph with degree 3Van Wieren et al [147] proved that CCC(119899) has an efficientdominating set if and only if 119899 = 5 From Theorems 194 and195 the following result can be derived

Theorem 199 (Huang and Xu [36] 2008) Let 119866 = 119862119862119862(119899)be the 119899-dimensional cube-connected cycles with 119899 ⩾ 3 and119899 = 5 Then 120574(119866) = 1198992119899minus2 and 2 ⩽ 119887(119866) ⩽ 3 In addition119887(119862119862119862(3)) = 119887(119862119862119862(4)) = 2

Problem 11 Determine the exact value of 119887(CCC(119899)) for 119899 ⩾5

Acknowledgments

This work was supported by NNSF of China (Grants nos11071233 61272008) The author would like to express hisgratitude to the anonymous referees for their kind sugges-tions and comments on the original paper to ProfessorSheikholeslami and Professor Jafari Rad for sending thereferences [128] and [81] which resulted in this version

References

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[2] S THedetniemi andR C Laskar ldquoBibliography on dominationin graphs and some basic definitions of domination parame-tersrdquo Discrete Mathematics vol 86 no 1ndash3 pp 257ndash277 1990

[3] TW Haynes S T Hedetniemi and P J Slater Fundamentals ofDomination in Graphs vol 208 ofMonographs and Textbooks inPure and Applied Mathematics Marcel Dekker New York NYUSA 1998

[4] TWHaynes S THedetniemi and P J Slater EdsDominationin Graphs vol 209 of Monographs and Textbooks in Pure andAppliedMathematicsMarcelDekker NewYorkNYUSA 1998

[5] M R Garey andD S JohnsonComputers and Intractability WH Freeman and Company San Francisco Calif USA 1979

[6] H B Walikar and B D Acharya ldquoDomination critical graphsrdquoNational Academy Science Letters vol 2 pp 70ndash72 1979

[7] D Bauer F Harary J Nieminen and C L Suffel ldquoDominationalteration sets in graphsrdquo Discrete Mathematics vol 47 no 2-3pp 153ndash161 1983

[8] J F Fink M S Jacobson L F Kinch and J Roberts ldquoThebondage number of a graphrdquo Discrete Mathematics vol 86 no1ndash3 pp 47ndash57 1990

[9] F-T Hu and J-M Xu ldquoThe bondage number of (119899 minus 3)-regulargraph G of order nrdquo Ars Combinatoria to appear

[10] U Teschner ldquoNew results about the bondage number of agraphrdquoDiscreteMathematics vol 171 no 1ndash3 pp 249ndash259 1997

[11] B L Hartnell and D F Rall ldquoA bound on the size of a graphwith given order and bondage numberrdquo Discrete Mathematicsvol 197-198 pp 409ndash413 1999 16th British CombinatorialConference (London 1997)

[12] B L Hartnell and D F Rall ldquoA characterization of trees inwhich no edge is essential to the domination numberrdquo ArsCombinatoria vol 33 pp 65ndash76 1992

[13] J Topp and P D Vestergaard ldquo120572119896- and 120574

119896-stable graphsrdquo

Discrete Mathematics vol 212 no 1-2 pp 149ndash160 2000[14] A Meir and J W Moon ldquoRelations between packing and

covering numbers of a treerdquo Pacific Journal of Mathematics vol61 no 1 pp 225ndash233 1975

[15] B L Hartnell L K Joslashrgensen P D Vestergaard and CA Whitehead ldquoEdge stability of the k-domination numberof treesrdquo Bulletin of the Institute of Combinatorics and ItsApplications vol 22 pp 31ndash40 1998

[16] F-T Hu and J-M Xu ldquoOn the complexity of the bondage andreinforcement problemsrdquo Journal of Complexity vol 28 no 2pp 192ndash201 2012

[17] B L Hartnell and D F Rall ldquoBounds on the bondage numberof a graphrdquo Discrete Mathematics vol 128 no 1ndash3 pp 173ndash1771994

[18] Y-L Wang ldquoOn the bondage number of a graphrdquo DiscreteMathematics vol 159 no 1ndash3 pp 291ndash294 1996

[19] G H Fricke TW Haynes S M Hedetniemi S T Hedetniemiand R C Laskar ldquoExcellent treesrdquo Bulletin of the Institute ofCombinatorics and Its Applications vol 34 pp 27ndash38 2002

[20] V Samodivkin ldquoThe bondage number of graphs good and badverticesrdquoDiscussiones Mathematicae GraphTheory vol 28 no3 pp 453ndash462 2008

[21] J E Dunbar T W Haynes U Teschner and L VolkmannldquoBondage insensitivity and reinforcementrdquo in Domination inGraphs vol 209 of Monographs and Textbooks in Pure andAppliedMathematics pp 471ndash489 Dekker NewYork NY USA1998

[22] H Liu and L Sun ldquoThe bondage and connectivity of a graphrdquoDiscrete Mathematics vol 263 no 1ndash3 pp 289ndash293 2003

[23] A-H Esfahanian and S L Hakimi ldquoOn computing a con-ditional edge-connectivity of a graphrdquo Information ProcessingLetters vol 27 no 4 pp 195ndash199 1988

[24] R C Brigham P Z Chinn and R D Dutton ldquoVertexdomination-critical graphsrdquoNetworks vol 18 no 3 pp 173ndash1791988

[25] V Samodivkin ldquoDomination with respect to nondegenerateproperties bondage numberrdquoThe Australasian Journal of Com-binatorics vol 45 pp 217ndash226 2009

[26] U Teschner ldquoA new upper bound for the bondage numberof graphs with small domination numberrdquo The AustralasianJournal of Combinatorics vol 12 pp 27ndash35 1995

32 International Journal of Combinatorics

[27] J Fulman D Hanson and G MacGillivray ldquoVertex dom-ination-critical graphsrdquoNetworks vol 25 no 2 pp 41ndash43 1995

[28] U Teschner ldquoA counterexample to a conjecture on the bondagenumber of a graphrdquo Discrete Mathematics vol 122 no 1ndash3 pp393ndash395 1993

[29] U Teschner ldquoThe bondage number of a graph G can be muchgreater than Δ(G)rdquo Ars Combinatoria vol 43 pp 81ndash87 1996

[30] L Volkmann ldquoOn graphs with equal domination and coveringnumbersrdquoDiscrete AppliedMathematics vol 51 no 1-2 pp 211ndash217 1994

[31] L A Sanchis ldquoMaximum number of edges in connected graphswith a given domination numberrdquoDiscreteMathematics vol 87no 1 pp 65ndash72 1991

[32] S Klavzar and N Seifter ldquoDominating Cartesian products ofcyclesrdquoDiscrete AppliedMathematics vol 59 no 2 pp 129ndash1361995

[33] H K Kim ldquoA note on Vizingrsquos conjecture about the bondagenumberrdquo Korean Journal of Mathematics vol 10 no 2 pp 39ndash42 2003

[34] M Y Sohn X Yuan and H S Jeong ldquoThe bondage number of1198623times119862

119899rdquo Journal of the KoreanMathematical Society vol 44 no

6 pp 1213ndash1231 2007[35] L Kang M Y Sohn and H K Kim ldquoBondage number of the

discrete torus 119862119899times 119862

4rdquo Discrete Mathematics vol 303 no 1ndash3

pp 80ndash86 2005[36] J Huang and J-M Xu ldquoThe bondage numbers and efficient

dominations of vertex-transitive graphsrdquoDiscrete Mathematicsvol 308 no 4 pp 571ndash582 2008

[37] C J Xiang X Yuan andM Y Sohn ldquoDomination and bondagenumber of119862

3times119862

119899rdquoArs Combinatoria vol 97 pp 299ndash310 2010

[38] M S Jacobson and L F Kinch ldquoOn the domination numberof products of graphs Irdquo Ars Combinatoria vol 18 pp 33ndash441984

[39] Y-Q Li ldquoA note on the bondage number of a graphrdquo ChineseQuarterly Journal of Mathematics vol 9 no 4 pp 1ndash3 1994

[40] F-T Hu and J-M Xu ldquoBondage number of mesh networksrdquoFrontiers ofMathematics inChina vol 7 no 5 pp 813ndash826 2012

[41] R Frucht and F Harary ldquoOn the corona of two graphsrdquoAequationes Mathematicae vol 4 pp 322ndash325 1970

[42] K Carlson and M Develin ldquoOn the bondage number of planarand directed graphsrdquoDiscreteMathematics vol 306 no 8-9 pp820ndash826 2006

[43] U Teschner ldquoOn the bondage number of block graphsrdquo ArsCombinatoria vol 46 pp 25ndash32 1997

[44] J Huang and J-M Xu ldquoNote on conjectures of bondagenumbers of planar graphsrdquo Applied Mathematical Sciences vol6 no 65ndash68 pp 3277ndash3287 2012

[45] L Kang and J Yuan ldquoBondage number of planar graphsrdquoDiscrete Mathematics vol 222 no 1ndash3 pp 191ndash198 2000

[46] M Fischermann D Rautenbach and L Volkmann ldquoRemarkson the bondage number of planar graphsrdquo Discrete Mathemat-ics vol 260 no 1ndash3 pp 57ndash67 2003

[47] J Hou G Liu and J Wu ldquoBondage number of planar graphswithout small cyclesrdquoUtilitasMathematica vol 84 pp 189ndash1992011

[48] J Huang and J-M Xu ldquoThe bondage numbers of graphs withsmall crossing numbersrdquo Discrete Mathematics vol 307 no 15pp 1881ndash1897 2007

[49] Q Ma S Zhang and J Wang ldquoBondage number of 1-planargraphrdquo Applied Mathematics vol 1 no 2 pp 101ndash103 2010

[50] J Hou and G Liu ldquoA bound on the bondage number of toroidalgraphsrdquoDiscreteMathematics Algorithms and Applications vol4 no 3 Article ID 1250046 5 pages 2012

[51] Y-C Cao J-M Xu and X-R Xu ldquoOn bondage number oftoroidal graphsrdquo Journal of University of Science and Technologyof China vol 39 no 3 pp 225ndash228 2009

[52] Y-C Cao J Huang and J-M Xu ldquoThe bondage number ofgraphs with crossing number less than fourrdquo Ars Combinatoriato appear

[53] H K Kim ldquoOn the bondage numbers of some graphsrdquo KoreanJournal of Mathematics vol 11 no 2 pp 11ndash15 2004

[54] A Gagarin and V Zverovich ldquoUpper bounds for the bondagenumber of graphs on topological surfacesrdquo Discrete Mathemat-ics 2012

[55] J Huang ldquoAn improved upper bound for the bondage numberof graphs on surfacesrdquoDiscrete Mathematics vol 312 no 18 pp2776ndash2781 2012

[56] A Gagarin and V Zverovich ldquoThe bondage number of graphson topological surfaces and Teschnerrsquos conjecturerdquo DiscreteMathematics vol 313 no 6 pp 796ndash808 2013

[57] V Samodivkin ldquoNote on the bondage number of graphs ontopological surfacesrdquo httparxivorgabs12086203

[58] E J Cockayne R M Dawes and S T Hedetniemi ldquoTotaldomination in graphsrdquoNetworks vol 10 no 3 pp 211ndash219 1980

[59] R Laskar J Pfaff S M Hedetniemi and S T HedetniemildquoOn the algorithmic complexity of total dominationrdquo Society forIndustrial and Applied Mathematics vol 5 no 3 pp 420ndash4251984

[60] J Pfaff R C Laskar and S T Hedetniemi ldquoNP-completeness oftotal and connected domination and irredundance for bipartitegraphsrdquo Tech Rep 428 Department of Mathematical SciencesClemson University 1983

[61] M A Henning ldquoA survey of selected recent results on totaldomination in graphsrdquoDiscrete Mathematics vol 309 no 1 pp32ndash63 2009

[62] V R Kulli and D K Patwari ldquoThe total bondage number ofa graphrdquo in Advances in Graph Theory pp 227ndash235 VishwaGulbarga India 1991

[63] F-T Hu Y Lu and J-M Xu ldquoThe total bondage number ofgrid graphsrdquo Discrete Applied Mathematics vol 160 no 16-17pp 2408ndash2418 2012

[64] J Cao M Shi and B Wu ldquoResearch on the total bondagenumber of a spectial networkrdquo in Proceedings of the 3rdInternational Joint Conference on Computational Sciences andOptimization (CSO rsquo10) pp 456ndash458 May 2010

[65] N Sridharan MD Elias and V S A Subramanian ldquoTotalbondage number of a graphrdquo AKCE International Journal ofGraphs and Combinatorics vol 4 no 2 pp 203ndash209 2007

[66] N Jafari Rad and J Raczek ldquoSome progress on total bondage ingraphsrdquo Graphs and Combinatorics 2011

[67] T W Haynes and P J Slater ldquoPaired-domination and thepaired-domatic numberrdquoCongressusNumerantium vol 109 pp65ndash72 1995

[68] T W Haynes and P J Slater ldquoPaired-domination in graphsrdquoNetworks vol 32 no 3 pp 199ndash206 1998

[69] X Hou ldquoA characterization of 2120574 120574119875-treesrdquoDiscrete Mathemat-

ics vol 308 no 15 pp 3420ndash3426 2008[70] K E Proffitt T W Haynes and P J Slater ldquoPaired-domination

in grid graphsrdquo Congressus Numerantium vol 150 pp 161ndash1722001

International Journal of Combinatorics 33

[71] H Qiao L KangM Cardei andD-Z Du ldquoPaired-dominationof treesrdquo Journal of Global Optimization vol 25 no 1 pp 43ndash542003

[72] E Shan L Kang and M A Henning ldquoA characterizationof trees with equal total domination and paired-dominationnumbersrdquo The Australasian Journal of Combinatorics vol 30pp 31ndash39 2004

[73] J Raczek ldquoPaired bondage in treesrdquo Discrete Mathematics vol308 no 23 pp 5570ndash5575 2008

[74] J A Telle and A Proskurowski ldquoAlgorithms for vertex parti-tioning problems on partial k-treesrdquo SIAM Journal on DiscreteMathematics vol 10 no 4 pp 529ndash550 1997

[75] G S Domke J H Hattingh M A Henning and L R MarkusldquoRestrained domination in treesrdquoDiscreteMathematics vol 211no 1ndash3 pp 1ndash9 2000

[76] G S Domke J H Hattingh M A Henning and L R MarkusldquoRestrained domination in graphs with minimum degree twordquoJournal of Combinatorial Mathematics and Combinatorial Com-puting vol 35 pp 239ndash254 2000

[77] G S Domke J H Hattingh S T Hedetniemi R C Laskarand L R Markus ldquoRestrained domination in graphsrdquo DiscreteMathematics vol 203 no 1ndash3 pp 61ndash69 1999

[78] J H Hattingh and E J Joubert ldquoAn upper bound for therestrained domination number of a graph with minimumdegree at least two in terms of order and minimum degreerdquoDiscrete Applied Mathematics vol 157 no 13 pp 2846ndash28582009

[79] M A Henning ldquoGraphs with large restrained dominationnumberrdquo Discrete Mathematics vol 197-198 pp 415ndash429 199916th British Combinatorial Conference (London 1997)

[80] J H Hattingh and A R Plummer ldquoRestrained bondage ingraphsrdquo Discrete Mathematics vol 308 no 23 pp 5446ndash54532008

[81] N Jafari Rad R Hasni J Raczek and L Volkmann ldquoTotalrestrained bondage in graphsrdquoActaMathematica Sinica vol 29no 6 pp 1033ndash1042 2013

[82] J Cyman and J Raczek ldquoOn the total restrained dominationnumber of a graphrdquoThe Australasian Journal of Combinatoricsvol 36 pp 91ndash100 2006

[83] M A Henning ldquoGraphs with large total domination numberrdquoJournal of Graph Theory vol 35 no 1 pp 21ndash45 2000

[84] D-X Ma X-G Chen and L Sun ldquoOn total restraineddomination in graphsrdquoCzechoslovakMathematical Journal vol55 no 1 pp 165ndash173 2005

[85] B Zelinka ldquoRemarks on restrained domination and totalrestrained domination in graphsrdquo Czechoslovak MathematicalJournal vol 55 no 2 pp 393ndash396 2005

[86] W Goddard and M A Henning ldquoIndependent domination ingraphs a survey and recent resultsrdquo Discrete Mathematics vol313 no 7 pp 839ndash854 2013

[87] J H Zhang H L Liu and L Sun ldquoIndependence bondagenumber and reinforcement number of some graphsrdquo Transac-tions of Beijing Institute of Technology vol 23 no 2 pp 140ndash1422003

[88] K D Dharmalingam Studies in Graph Theorymdashequitabledomination and bottleneck domination [PhD thesis] MaduraiKamaraj University Madurai India 2006

[89] GDeepakND Soner andAAlwardi ldquoThe equitable bondagenumber of a graphrdquo Research Journal of Pure Algebra vol 1 no9 pp 209ndash212 2011

[90] E Sampathkumar and H B Walikar ldquoThe connected domina-tion number of a graphrdquo Journal of Mathematical and PhysicalSciences vol 13 no 6 pp 607ndash613 1979

[91] K White M Farber and W Pulleyblank ldquoSteiner trees con-nected domination and strongly chordal graphsrdquoNetworks vol15 no 1 pp 109ndash124 1985

[92] M Cadei M X Cheng X Cheng and D-Z Du ldquoConnecteddomination in Ad Hoc wireless networksrdquo in Proceedings ofthe 6th International Conference on Computer Science andInformatics (CSampI rsquo02) 2002

[93] J F Fink and M S Jacobson ldquon-domination in graphsrdquo inGraph Theory with Applications to Algorithms and ComputerScience Wiley-Interscience Publications pp 283ndash300 WileyNew York NY USA 1985

[94] M Chellali O Favaron A Hansberg and L Volkmann ldquok-domination and k-independence in graphs a surveyrdquo Graphsand Combinatorics vol 28 no 1 pp 1ndash55 2012

[95] Y Lu X Hou J-M Xu andN Li ldquoTrees with uniqueminimump-dominating setsrdquo Utilitas Mathematica vol 86 pp 193ndash2052011

[96] Y Lu and J-M Xu ldquoThe p-bondage number of treesrdquo Graphsand Combinatorics vol 27 no 1 pp 129ndash141 2011

[97] Y Lu and J-M Xu ldquoThe 2-domination and 2-bondage numbersof grid graphs A manuscriptrdquo httparxivorgabs12044514

[98] M Krzywkowski ldquoNon-isolating 2-bondage in graphsrdquo TheJournal of the Mathematical Society of Japan vol 64 no 4 2012

[99] M A Henning O R Oellermann and H C Swart ldquoBoundson distance domination parametersrdquo Journal of CombinatoricsInformation amp System Sciences vol 16 no 1 pp 11ndash18 1991

[100] F Tian and J-M Xu ldquoBounds for distance domination numbersof graphsrdquo Journal of University of Science and Technology ofChina vol 34 no 5 pp 529ndash534 2004

[101] F Tian and J-M Xu ldquoDistance domination-critical graphsrdquoApplied Mathematics Letters vol 21 no 4 pp 416ndash420 2008

[102] F Tian and J-M Xu ldquoA note on distance domination numbersof graphsrdquo The Australasian Journal of Combinatorics vol 43pp 181ndash190 2009

[103] F Tian and J-M Xu ldquoAverage distances and distance domina-tion numbersrdquo Discrete Applied Mathematics vol 157 no 5 pp1113ndash1127 2009

[104] Z-L Liu F Tian and J-M Xu ldquoProbabilistic analysis of upperbounds for 2-connected distance k-dominating sets in graphsrdquoTheoretical Computer Science vol 410 no 38ndash40 pp 3804ndash3813 2009

[105] M A Henning O R Oellermann and H C Swart ldquoDistancedomination critical graphsrdquo Journal of Combinatorial Mathe-matics and Combinatorial Computing vol 44 pp 33ndash45 2003

[106] T J Bean M A Henning and H C Swart ldquoOn the integrityof distance domination in graphsrdquo The Australasian Journal ofCombinatorics vol 10 pp 29ndash43 1994

[107] M Fischermann and L Volkmann ldquoA remark on a conjecturefor the (kp)-domination numberrdquoUtilitasMathematica vol 67pp 223ndash227 2005

[108] T Korneffel D Meierling and L Volkmann ldquoA remark on the(22)-domination numberrdquo Discussiones Mathematicae GraphTheory vol 28 no 2 pp 361ndash366 2008

[109] Y Lu X Hou and J-M Xu ldquoOn the (22)-domination numberof treesrdquo Discussiones Mathematicae Graph Theory vol 30 no2 pp 185ndash199 2010

34 International Journal of Combinatorics

[110] H Li and J-M Xu ldquo(dm)-domination numbers of m-connected graphsrdquo Rapports de Recherche 1130 LRI UniversiteParis-Sud 1997

[111] S M Hedetniemi S T Hedetniemi and T V Wimer ldquoLineartime resource allocation algorithms for treesrdquo Tech Rep URI-014 Department of Mathematical Sciences Clemson Univer-sity 1987

[112] M Farber ldquoDomination independent domination and dualityin strongly chordal graphsrdquo Discrete Applied Mathematics vol7 no 2 pp 115ndash130 1984

[113] G S Domke and R C Laskar ldquoThe bondage and reinforcementnumbers of 120574

119891for some graphsrdquoDiscrete Mathematics vol 167-

168 pp 249ndash259 1997[114] G S Domke S T Hedetniemi and R C Laskar ldquoFractional

packings coverings and irredundance in graphsrdquo vol 66 pp227ndash238 1988

[115] E J Cockayne P A Dreyer Jr S M Hedetniemi andS T Hedetniemi ldquoRoman domination in graphsrdquo DiscreteMathematics vol 278 no 1ndash3 pp 11ndash22 2004

[116] I Stewart ldquoDefend the roman empirerdquo Scientific American vol281 pp 136ndash139 1999

[117] C S ReVelle and K E Rosing ldquoDefendens imperiumromanum a classical problem in military strategyrdquoThe Ameri-can Mathematical Monthly vol 107 no 7 pp 585ndash594 2000

[118] A Bahremandpour F-T Hu S M Sheikholeslami and J-MXu ldquoOn the Roman bondage number of a graphrdquo DiscreteMathematics Algorithms and Applications vol 5 no 1 ArticleID 1350001 15 pages 2013

[119] N Jafari Rad and L Volkmann ldquoRoman bondage in graphsrdquoDiscussiones Mathematicae Graph Theory vol 31 no 4 pp763ndash773 2011

[120] K Ebadi and L PushpaLatha ldquoRoman bondage and Romanreinforcement numbers of a graphrdquo International Journal ofContemporary Mathematical Sciences vol 5 pp 1487ndash14972010

[121] N Dehgardi S M Sheikholeslami and L Volkman ldquoOn theRoman k-bondage number of a graphrdquo AKCE InternationalJournal of Graphs and Combinatorics vol 8 pp 169ndash180 2011

[122] S Akbari and S Qajar ldquoA note on Roman bondage number ofgraphsrdquo Ars Combinatoria to Appear

[123] F-T Hu and J-M Xu ldquoRoman bondage numbers of somegraphsrdquo httparxivorgabs11093933

[124] N Jafari Rad and L Volkmann ldquoOn the Roman bondagenumber of planar graphsrdquo Graphs and Combinatorics vol 27no 4 pp 531ndash538 2011

[125] S Akbari M Khatirinejad and S Qajar ldquoA note on the romanbondage number of planar graphsrdquo Graphs and Combinatorics2012

[126] B Bresar M A Henning and D F Rall ldquoRainbow dominationin graphsrdquo Taiwanese Journal of Mathematics vol 12 no 1 pp213ndash225 2008

[127] B L Hartnell and D F Rall ldquoOn dominating the Cartesianproduct of a graph and 120572krdquo Discussiones Mathematicae GraphTheory vol 24 no 3 pp 389ndash402 2004

[128] N Dehgardi S M Sheikholeslami and L Volkman ldquoThe k-rainbow bondage number of a graphrdquo to appear in DiscreteApplied Mathematics

[129] J von Neumann and O Morgenstern Theory of Games andEconomic Behavior Princeton University Press Princeton NJUSA 1944

[130] C BergeTheorıe des Graphes et ses Applications 1958[131] O Ore Theory of Graphs vol 38 American Mathematical

Society Colloquium Publications 1962[132] E Shan and L Kang ldquoBondage number in oriented graphsrdquoArs

Combinatoria vol 84 pp 319ndash331 2007[133] J Huang and J-M Xu ldquoThe bondage numbers of extended

de Bruijn and Kautz digraphsrdquo Computers amp Mathematics withApplications vol 51 no 6-7 pp 1137ndash1147 2006

[134] J Huang and J-M Xu ldquoThe total domination and total bondagenumbers of extended de Bruijn and Kautz digraphsrdquoComputersamp Mathematics with Applications vol 53 no 8 pp 1206ndash12132007

[135] Y Shibata and Y Gonda ldquoExtension of de Bruijn graph andKautz graphrdquo Computers amp Mathematics with Applications vol30 no 9 pp 51ndash61 1995

[136] X Zhang J Liu and JMeng ldquoThebondage number in completet-partite digraphsrdquo Information Processing Letters vol 109 no17 pp 997ndash1000 2009

[137] D W Bange A E Barkauskas and P J Slater ldquoEfficient dom-inating sets in graphsrdquo in Applications of Discrete Mathematicspp 189ndash199 SIAM Philadelphia Pa USA 1988

[138] M Livingston and Q F Stout ldquoPerfect dominating setsrdquoCongressus Numerantium vol 79 pp 187ndash203 1990

[139] L Clark ldquoPerfect domination in random graphsrdquo Journal ofCombinatorial Mathematics and Combinatorial Computing vol14 pp 173ndash182 1993

[140] D W Bange A E Barkauskas L H Host and L H ClarkldquoEfficient domination of the orientations of a graphrdquo DiscreteMathematics vol 178 no 1ndash3 pp 1ndash14 1998

[141] A E Barkauskas and L H Host ldquoFinding efficient dominatingsets in oriented graphsrdquo Congressus Numerantium vol 98 pp27ndash32 1993

[142] I J Dejter and O Serra ldquoEfficient dominating sets in CayleygraphsrdquoDiscrete AppliedMathematics vol 129 no 2-3 pp 319ndash328 2003

[143] J Lee ldquoIndependent perfect domination sets in Cayley graphsrdquoJournal of Graph Theory vol 37 no 4 pp 213ndash219 2001

[144] WGu X Jia and J Shen ldquoIndependent perfect domination setsin meshes tori and treesrdquo Unpublished

[145] N Obradovic J Peters and G Ruzic ldquoEfficient domination incirculant graphswith two chord lengthsrdquo Information ProcessingLetters vol 102 no 6 pp 253ndash258 2007

[146] K Reji Kumar and G MacGillivray ldquoEfficient domination incirculant graphsrdquo Discrete Mathematics vol 313 no 6 pp 767ndash771 2013

[147] D Van Wieren M Livingston and Q F Stout ldquoPerfectdominating sets on cube-connected cyclesrdquo Congressus Numer-antium vol 97 pp 51ndash70 1993

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

International Journal of Combinatorics 25

Theorem 153 (Jafari Rad and Volkmann [119] 2011) For anypositive integers 119899 and119898 with119898 ⩽ 119899

119887RM (119875119899) = 2 119894119891 119899 equiv 2 (mod 3) 1 119900119905ℎ119890119903119908119894119904119890

119887RM (119862119899) =

3 119894119891 119899 = 2 (mod 3) 2 119900119905ℎ119890119903119908119894119904119890

119887RM (119870119899) = lceil

119899

2rceil 119891119900119903 119899 ⩾ 3

119887RM (119870119898119899) =

1 119894119891 119898 = 1 119886119899119889 119899 = 1

4 119894119891 119898 = 119899 = 3

119898 119900119905ℎ119890119903119908119894119904119890

(63)

Lemma 154 (Cockayne et al [115] 2004) If 119866 is a graph oforder 119899 and contains vertices of degree 119899 minus 1 then 120574RM(119866) = 2

Using Lemma 154 the third conclusion in Theorem 153can be generalized to more general case which is similar toLemma 49

Theorem 155 (Jafari Rad and Volkmann [119] 2011) Let119866 bea graph with order 119899 ge 3 and 119905 the number of vertices of degree119899 minus 1 in 119866 If 119905 ⩾ 1 then 119887RM(119866) = lceil1199052rceil

Ebadi and PushpaLatha [120] conjectured that 119887RM(119866) ⩽119899 minus 1 for any graph 119866 of order 119899 ⩾ 3 Dehgardi et al[121] Akbari andQajar [122] independently showed that thisconjecture is true

Theorem 156 119887RM(119866) ⩽ 119899 minus 1 for any connected graph 119866 oforder 119899 ⩾ 3

For a complete 119905-partite graph we have the followingresult

Theorem 157 (Hu and Xu [123] 2011) Let 119866 = 11987011989811198982119898119905

bea complete 119905-partite graph with 119898

1= sdot sdot sdot = 119898

119894lt 119898

119894+1le sdot sdot sdot ⩽

119898119905 119905 ⩾ 2 and 119899 = sum119905

119895=1119898

119895 Then

119887RM (119866) =

lceil119894

2rceil 119894119891 119898

119894= 1 119886119899119889 119899 ⩾ 3

2 119894119891 119898119894= 2 119886119899119889 119894 = 1

119894 119894119891 119898119894= 2 119886119899119889 119894 ⩾ 2

4 119894119891 119898119894= 3 119886119899119889 119894 = 119905 = 2

119899 minus 1 119894119891 119898119894= 3 119886119899119889 119894 = 119905 ⩾ 3

119899 minus 119898119905

119894119891 119898119894⩾ 3 119886119899119889 119898

119905⩾ 4

(64)

Consider a complete 119905-partite graph 119870119905(3) for 119905 ⩾ 3

119887RM(119870119905(3)) = 119899 minus 1 by Theorem 157 where 119899 = 3119905

which shows that this upper bound of 119899 minus 1 on 119887RM(119866) inTheorem 156 is sharp This fact and 120574

119877(119870

119905(3)) = 4 give

a negative answer to the following two problems posed byDehgardi et al [121]Prove or Disprove for any connected graph 119866 of order 119899 ge 3(i) 119887RM(119866) = 119899 minus 1 if and only if 119866 cong 119870

3 (ii) 119887RM(119866) ⩽ 119899 minus

120574119877(119866) + 1Note that a complete 119905-partite graph 119870

119905(3) is (119899 minus 3)-

regular and 119887RM(119870119905(3)) = 119899 minus 1 for 119905 ⩾ 3 where 119899 = 3119905

Hu and Xu further determined that 119887119877(119866) = 119899 minus 2 for any

(119899 minus 3)-regular graph 119866 of order 119899 ⩽ 5 except 119870119905(3)

Theorem 158 (Hu and Xu [123] 2011) For any (119899minus3)-regulargraph 119866 of order 119899 other than119870

119905(3) 119887RM(119866) = 119899 minus 2 for 119899 ⩾ 5

For a tree 119879 with order 119899 ⩾ 3 Ebadi and PushpaLatha[120] and Jafari Rad and Volkmann [119] independentlyobtained an upper bound on 119887RM(119879)

Theorem 159 119887RM(119879) ⩽ 3 for any tree 119879 with order 119899 ⩾ 3

Theorem 160 (Bahremandpour et al [118] 2013) 119887RM(1198752 times119875119899) = 2 for 119899 ⩾ 2

Theorem 161 (Jafari Rad and Volkmann [119] 2011) Let119866 bea graph with order at least three

(a) If (119909 119910 119911) is a path of length 2 in 119866 then 119887RM(119866) ⩽119889119866(119909) + 119889

119866(119910) + 119889

119866(119911) minus 3 minus |119873

119866(119909) cap 119873

119866(119910)|

(b) If 119866 is connected then 119887RM(119866) ⩽ 120582(119866) + 2Δ(119866) minus 3

Theorem 161 (b) implies 119887RM(119866) ⩽ 120575(119866) + 2Δ(119866) minus 3 for aconnected graph 119866 Note that for a planar graph 119866 120575(119866) ⩽ 5moreover 120575(119866) ⩽ 3 if the girth at least 4 and 120575(119866) ⩽ 2 if thegirth at least 6 These two facts show that 119887RM(119866) ⩽ 2Δ(119866) +2 for a connected planar graph 119866 Jafari Rad and Volkmann[124] improved this bound

Theorem 162 (Jafari Rad and Volkmann [124] 2011) For anyconnected planar graph 119866 of order 119899 ⩾ 3 with girth 119892(119866)

119887RM (119866)

⩽

2Δ (119866)

Δ (119866) + 6

Δ (119866) + 5 119894119891 119866 119888119900119899119905119886119894119899119904 119899119900 V119890119903119905119894119888119890119904 119900119891 119889119890119892119903119890119890 119891119894V119890

Δ (119866) + 4 119894119891 119892 (119866) ⩾ 4

Δ (119866) + 3 119894119891 119892 (119866) ⩾ 5

Δ (119866) + 2 119894119891 119892 (119866) ⩾ 6

Δ (119866) + 1 119894119891 119892 (119866) ⩾ 8

(65)

According toTheorem 157 119887RM(119862119899) = 3 = Δ(119862

119899)+1 for a

cycle 119862119899of length 119899 ⩾ 8 with 119899 equiv 2 (mod 3) and therefore

the last upper bound inTheorem 162 is sharp at least for Δ =2

Combining the fact that every planar graph 119866 withminimum degree 5 contains an edge 119909119910 with 119889

119866(119909) = 5

and 119889119866(119910) isin 5 6 with Theorem 161 (a) Akbari et al [125]

obtained the following result

26 International Journal of Combinatorics

middot middot middot

middot middot middot

middot middot middot

middot middot middot

119866

Figure 9 The graph 119866 is constructed from 119866

Theorem 163 (Akbari et al [125] 2012) 119887RM(119866) ⩽ 15 forevery planar graph 119866

It remains open to show whether the bound inTheorem 163 is sharp or not Though finding a planargraph 119866 with 119887RM(119866) = 15 seems to be difficult Akbari etal [125] constructed an infinite family of planar graphs withRoman bondage number equal to 7 by proving the followingresult

Theorem 164 (Akbari et al [125] 2012) Let 119866 be a graph oforder 119899 and 119866 is the graph of order 5119899 obtained from 119866 byattaching the central vertex of a copy of a path 119875

5to each vertex

of119866 (see Figure 9) Then 120574RM(119866) = 4119899 and 119887RM(119866) = 120575(119866)+2

ByTheorem 164 infinitemany planar graphswith Romanbondage number 7 can be obtained by considering any planargraph 119866 with 120575(119866) = 5 (eg the icosahedron graph)

Conjecture 165 (Akbari et al [125] 2012) The Romanbondage number of every planar graph is at most 7

For general bounds the following observation is directlyobtained which is similar to Observation 1

Observation 4 Let 119866 be a graph of order 119899 with maximumdegree at least two Assume that 119867 is a spanning subgraphobtained from 119866 by removing 119896 edges If 120574RM(119867) = 120574RM(119866)then 119887

119877(119867) ⩽ 119887RM(119866) ⩽ 119887RM(119867) + 119896

Theorem 166 (Bahremandpour et al [118] 2013) For anyconnected graph 119866 of order 119899 if 119899 ⩾ 3 and 120574RM(119866) = 120574(119866) + 1then

119887RM (119866) ⩽ min 119887 (119866) 119899Δ (66)

where 119899Δis the number of vertices with maximum degree Δ in

119866

Observation 5 For any graph 119866 if 120574(119866) = 120574RM(119866) then119887RM(119866) ⩽ 119887(119866)

Theorem 167 (Bahremandpour et al [118] 2013) For everyRoman graph 119866

119887RM (119866) ⩾ 119887 (119866) (67)

The bound is sharp for cycles on 119899 vertices where 119899 equiv 0 (mod3)

The strict inequality in Theorem 167 can hold for exam-ple 119887(119862

3119896+2) = 2 lt 3 = 119887RM(1198623119896+2

) byTheorem 157A graph 119866 is called to be vertex Roman domination-

critical if 120574RM(119866 minus 119909) lt 120574RM(119866) for every vertex 119909 in 119866 If119866 has no isolated vertices then 120574RM(119866) ⩽ 2120574(119866) ⩽ 2120573(119866)If 120574RM(119866) = 2120573(119866) then 120574RM(119866) = 2120574(119866) and hence 119866 is aRoman graph In [30] Volkmann gave a lot of graphs with120574(119866) = 120573(119866) The following result is similar to Theorem 57

Theorem 168 (Bahremandpour et al [118] 2013) For anygraph 119866 if 120574RM(119866) = 2120573(119866) then

(a) 119887RM(119866) ⩾ 120575(119866)

(b) 119887RM(119866) ⩾ 120575(119866)+1 if119866 is a vertex Roman domination-critical graph

If 120574RM(119866) = 2 then obviously 119887RM(119866) ⩽ 120575(119866) So weassume 120574RM(119866) ⩾ 3 Dehgardi et al [121] proved 119887RM(119866) ⩽(120574RM(119866) minus 2)Δ(119866) + 1 and posed the following problem if 119866is a connected graph of order 119899 ⩾ 4 with 120574RM(119866) ⩾ 3 then

119887RM (119866) ⩽ (120574RM (119866) minus 2) Δ (119866) (68)

Theorem 161 (a) shows that the inequality (68) holds if120574RM(119866) ⩾ 5 Thus the bound in (68) is of interest only when120574RM(119866) is 3 or 4

Lemma 169 (Bahremandpour et al [118] 2013) For anynonempty graph 119866 of order 119899 ⩾ 3 120574RM(119866) = 3 if and onlyif Δ(119866) = 119899 minus 2

The following result shows that (68) holds for all graphs119866 of order 119899 ⩾ 4 with 120574RM(119866) = 3 or 4 which improvesTheorem 161 (b)

Theorem 170 (Bahremandpour et al [118] 2013) For anyconnected graph 119866 of order 119899 ⩾ 4

119887RM (119866) le Δ (119866) = 119899 minus 2 119894119891 120574RM (119866) = 3

Δ (119866) + 120575 (119866) minus 1 119894119891 120574RM (119866) = 4(69)

with the first equality if and only if 119866 cong 1198624

Recently Akbari and Qajar [122] have proved the follow-ing result

Theorem 171 (Akbari and Qajar [122]) For any connectedgraph 119866 of order 119899 ⩾ 3

119887RM (119866) ⩽ 119899 minus 120574RM (119866) + 5 (70)

International Journal of Combinatorics 27

We conclude this section with the following problems

Problem 8 Characterize all connected graphs 119866 of order 119899 ⩾3 for which 119887RM(119866) = 119899 minus 1

Problem 9 Prove or disprove if 119866 is a connected graph oforder 119899 ⩾ 3 then

119887RM (119866) ⩽ 119899 minus 120574RM (119866) + 3 (71)

Note that 120574119877(119870

119905(3)) = 4 and 119887RM(119870119905

(3)) = 119899minus1 where 119899 =3119905 for 119905 ⩾ 3 The upper bound on 119887RM(119866) given in Problem 9is sharp if Problem 9 has positive answer

95 Rainbow Bondage Numbers For a positive integer 119896 a119896-rainbow dominating function of a graph 119866 is a function 119891from the vertex-set 119881(119866) to the set of all subsets of the set1 2 119896 such that for any vertex 119909 in 119866 with 119891(119909) = 0 thecondition

⋃

119910isin119873119866(119909)

119891 (119910) = 1 2 119896 (72)

is fulfilledThe weight of a 119896-rainbow dominating function 119891on 119866 is the value

119891 (119866) = sum

119909isin119881(119866)

1003816100381610038161003816119891 (119909)1003816100381610038161003816 (73)

The 119896-rainbow domination number of a graph 119866 denoted by120574119903119896(119866) is the minimum weight of a 119896-rainbow dominating

function 119891 of 119866Note that 120574

1199031(119866) is the classical domination number 120574(119866)

The 119896-rainbow domination number is introduced by Bresaret al [126] Rainbow domination of a graph 119866 coincides withordinary domination of the Cartesian product of 119866 with thecomplete graph in particular 120574

119903119896(119866) = 120574(119866 times 119870

119896) for any

positive integer 119896 and any graph 119866 [126] Hartnell and Rall[127] proved that 120574(119879) lt 120574

1199032(119879) for any tree 119879 no graph has

120574 = 120574119903119896for 119896 ⩾ 3 for any 119896 ⩾ 2 and any graph 119866 of order 119899

min 119899 120574 (119866) + 119896 minus 2 ⩽ 120574119903119896(119866) ⩽ 119896120574 (119866) (74)

The introduction of rainbow domination is motivatedby the study of paired domination in Cartesian productsof graphs where certain upper bounds can be expressed interms of rainbow domination that is for any graph 119866 andany graph 119867 if 119881(119867) can be partitioned into 119896120574

119875-sets then

120574119875(119866 times 119867) ⩽ (1119896)120592(119867)120574

119903119896(119866) proved by Bresar et al [126]

Dehgardi et al [128] introduced the concept of 119896-rainbowbondage number of graphs Let 119866 be a graph with maximumdegree at least two The 119896-rainbow bondage number 119887

119903119896(119866)

of 119866 is the minimum cardinality of all sets 119861 sube 119864(119866) forwhich 120574

119903119896(119866 minus 119861) gt 120574

119903119896(119866) Since in the study of 119896-rainbow

bondage number the assumption Δ(119866) ⩾ 2 is necessarywe always assume that when we discuss 119887

119903119896(119866) all graphs

involved satisfy Δ(119866) ⩾ 2The 2-rainbow bondage number of some special families

of graphs has been determined

Theorem 172 (Dehgardi et al [128]) For any integer 119899 ⩾ 3

1198871199032(119875

119899) = 1 119891119900119903 119899 ⩾ 2

1198871199032(119862

119899) =

1 119894119891 119899 equiv 0 (mod 4) 2 119900119905ℎ119890119903119908119894119904119890

1198871199032(119870

119899119899) =

1 119894119891 119899 = 2

3 119894119891 119899 = 3

6 119894119891 119899 = 4

119898 119894119891 119899 ⩾ 5

(75)

Theorem 173 (Dehgardi et al [128]) For any connected graph119866 of order 119899 ⩾ 3 119887

1199032(119866) ⩽ 120582(119866) + 2Δ(119866) minus 3

Theorem 174 (Dehgardi et al [128]) For any tree 119879 of order119899 ⩾ 3 119887

1199032(119879) ⩽ 2

Theorem 175 (Dehgardi et al [128]) If 119866 is a planar graphwithmaximumdegree at least two then 119887

1199032(119866) ⩽ 15Moreover

if the girth 119892(119866) ⩾ 4 then 1198871199032(119866) ⩽ 11 and if 119892(119866) ⩾ 6 then

1198871199032(119866) ⩽ 9

Theorem 176 (Dehgardi et al [128]) For a complete bipartitegraph 119870

119901119902 if 2119896 + 1 ⩽ 119901 ⩽ 119902 then 119887

119903119896(119870

119901119902) = 119901

Theorem177 (Dehgardi et al [128]) For a complete graph119870119899

if 119899 ⩾ 119896 + 1 then 119887119903119896(119870

119899) = lceil119896119899(119896 + 1)rceil

The special case 119896 = 1 of Theorem 177 can be found inTheorem 1 The following is a simple observation similar toObservation 1

Observation 6 (Dehgardi et al [128]) Let 119866 be a graphwith maximum degree at least two 119867 a spanning subgraphobtained from 119866 by removing 119896 edges If 120574

119903119896(119867) = 120574

119903119896(119866)

then 119887119903119896(119867) ⩽ 119887

119903119896(119866) ⩽ 119887

119903119896(119867) + 119896

Theorem 178 (Dehgardi et al [128]) For every graph 119866 with120574119903119896(119866) = 119896120574(119866) 119887

119903119896(119866) ⩾ 119887(119866)

A graph 119866 is said to be vertex 119896-rainbow domination-critical if 120574

119903119896(119866 minus 119909) lt 120574

119903119896(119866) for every vertex 119909 in 119866 It is

clear that if 119866 has no isolated vertices then 120574119903119896(119866) ⩽ 119896120574(119866) ⩽

119896120573(119866) where 120573(119866) is the vertex-covering number of119866 Thusif 120574

119903119896(119866) = 119896120573(119866) then 120574

119903119896(119866) = 119896120574(119866)

Theorem 179 (Dehgardi et al [128]) For any graph 119866 if120574119903119896(119866) = 119896120573(119866) then

(a) 119887119903119896(119866) ⩾ 120575(119866)

(b) 119887119903119896(119866) ⩾ 120575(119866)+1 if119866 is vertex 119896-rainbow domination-

critical

As far as we know there are no more results on the 119896-rainbow bondage number of graphs in the literature

28 International Journal of Combinatorics

10 Results on Digraphs

Although domination has been extensively studied in undi-rected graphs it is natural to think of a dominating setas a one-way relationship between vertices of the graphIndeed among the earliest literatures on this subject vonNeumann and Morgenstern [129] used what is now calleddomination in digraphs to find solution (or kernels whichare independent dominating sets) for cooperative 119899-persongames Most likely the first formulation of domination byBerge [130] in 1958 was given in the context of digraphs andonly some years latter by Ore [131] in 1962 for undirectedgraphs Despite this history examination of domination andits variants in digraphs has been essentially overlooked (see[4] for an overview of the domination literature) Thus thereare few if any such results on domination for digraphs in theliterature

The bondage number and its related topics for undirectedgraph have become one of major areas both in theoreticaland applied researches However until recently Carlsonand Develin [42] Shan and Kang [132] and Huang andXu [133 134] studied the bondage number for digraphsindependently In this section we introduce their resultsfor general digraphs Results for some special digraphs suchas vertex-transitive digraphs are introduced in the nextsection

101 Upper Bounds for General Digraphs Let 119866 = (119881 119864)

be a digraph without loops and parallel edges A digraph 119866is called to be asymmetric if whenever (119909 119910) isin 119864(119866) then(119910 119909) notin 119864(119866) and to be symmetric if (119909 119910) isin 119864(119866) implies(119910 119909) isin 119864(119866) For a vertex 119909 of 119881(119866) the sets of out-neighbors and in-neighbors of 119909 are respectively defined as119873

+

119866(119909) = 119910 isin 119881(119866) (119909 119910) isin 119864(119866) and 119873minus

119866(119909) = 119909 isin

119881(119866) (119909 119910) isin 119864(119866) the out-degree and the in-degree of119909 are respectively defined as 119889+

119866(119909) = |119873

+

119866(119909)| and 119889minus

119866(119909) =

|119873+

119866(119909)| Denote themaximum and theminimum out-degree

(resp in-degree) of 119866 by Δ+(119866) and 120575+(119866) (resp Δminus(119866) and120575minus

(119866))The degree 119889119866(119909) of 119909 is defined as 119889+

119866(119909)+119889

minus

119866(119909) and

the maximum and the minimum degrees of119866 are denoted byΔ(119866) and 120575(119866) respectively that is Δ(119866) = max119889

119866(119909) 119909 isin

119881(119866) and 120575(119866) = min119889119866(119909) 119909 isin 119881(119866) Note that the

definitions here are different from the ones in the textbookon digraphs see for example Xu [1]

A subset 119878 of119881(119866) is called a dominating set if119881(119866) = 119878cup119873

+

119866(119878) where119873+

119866(119878) is the set of out-neighbors of 119878Then just

as for undirected graphs 120574(119866) is the minimum cardinalityof a dominating set and the bondage number 119887(119866) is thesmallest cardinality of a set 119861 of edges such that 120574(119866 minus 119861) gt120574(119866) if such a subset 119861 sube 119864(119866) exists Otherwise we put119887(119866) = infin

Some basic results for undirected graphs stated inSection 3 can be generalized to digraphs For exampleTheorem 18 is generalized by Carlson and Develin [42]Huang andXu [133] and Shan andKang [132] independentlyas follows

Theorem 180 Let 119866 be a digraph and (119909 119910) isin 119864(119866) Then119887(119866) ⩽ 119889

119866(119910) + 119889

minus

119866(119909) minus |119873

minus

119866(119909) cap 119873

minus

119866(119910)|

Corollary 181 For a digraph 119866 119887(119866) ⩽ 120575minus(119866) + Δ(119866)

Since 120575minus

(119866) ⩽ (12)Δ(119866) from Corollary 181Conjecture 53 is valid for digraphs

Corollary 182 For a digraph 119866 119887(119866) ⩽ (32)Δ(119866)

In the case of undirected graphs the bondage number of119866119899= 119870

119899times 119870

119899achieves this bound However it was shown

by Carlson and Develin in [42] that if we take the symmetricdigraph119866

119899 we haveΔ(119866

119899) = 4(119899minus1) 120574(119866

119899) = 119899 and 119887(119866

119899) ⩽

4(119899 minus 1) + 1 = Δ(119866119899) + 1 So this family of digraphs cannot

show that the bound in Corollary 182 is tightCorresponding to Conjecture 51 which is discredited

for undirected graphs and is valid for digraphs the sameconjecture for digraphs can be proposed as follows

Conjecture 183 (Carlson and Develin [42] 2006) 119887(119866) ⩽Δ(119866) + 1 for any digraph 119866

If Conjecture 183 is true then the following results showsthat this upper bound is tight since 119887(119870

119899times119870

119899) = Δ(119870

119899times119870

119899)+1

for a complete digraph 119870119899

In 2007 Shan and Kang [132] gave some tight upperbounds on the bondage numbers for some asymmetricdigraphs For example 119887(119879) ⩽ Δ(119879) for any asymmetricdirected tree 119879 119887(119866) ⩽ Δ(119866) for any asymmetric digraph 119866with order at least 4 and 120574(119866) ⩽ 2 For planar digraphs theyobtained the following results

Theorem 184 (Shan and Kang [132] 2007) Let 119866 be aasymmetric planar digraphThen 119887(119866) ⩽ Δ(119866)+2 and 119887(119866) ⩽Δ(119866) + 1 if Δ(119866) ⩾ 5 and 119889minus

119866(119909) ⩾ 3 for every vertex 119909 with

119889119866(119909) ⩾ 4

102 Results for Some Special Digraphs The exact values andbounds on 119887(119866) for some standard digraphs are determined

Theorem185 (Huang andXu [133] 2006) For a directed cycle119862119899and a directed path 119875

119899

119887 (119862119899) =

3 119894119891 119899 119894119904 119900119889119889

2 119894119891 119899 119894119904 119890V119890119899

119887 (119875119899) =

2 119894119891 119899 119894119904 119900119889119889

1 119894119891 119899 119894119904 119890V119890119899

(76)

For the de Bruijn digraph 119861(119889 119899) and the Kautz digraph119870(119889 119899)

119887 (119861 (119889 119899)) = 119889 119894119891 119899 119894119904 119900119889119889

119889 ⩽ 119887 (119861 (119889 119899)) ⩽ 2119889 119894119891 119899 119894119904 119890V119890119899

119887 (119870 (119889 119899)) = 119889 + 1

(77)

Like undirected graphs we can define the total domi-nation number and the total bondage number On the totalbondage numbers for some special digraphs the knownresults are as follows

International Journal of Combinatorics 29

Theorem 186 (Huang and Xu [134] 2007) For a directedcycle 119862

119899and a directed path 119875

119899 119887

119879(119875

119899) and 119887

119879(119862

119899) all do not

exist For a complete digraph 119870119899

119887119879(119870

119899) = infin 119894119891 119899 = 1 2

119887119879(119870

119899) = 3 119894119891 119899 = 3

119899 ⩽ 119887119879(119870

119899) ⩽ 2119899 minus 3 119894119891 119899 ⩾ 4

(78)

The extended de Bruijn digraph EB(119889 119899 1199021 119902

119901) and

the extended Kautz digraph EK(119889 119899 1199021 119902

119901) are intro-

duced by Shibata and Gonda [135] If 119901 = 1 then theyare the de Bruijn digraph B(119889 119899) and the Kautz digraphK(119889 119899) respectively Huang and Xu [134] determined theirtotal domination numbers In particular their total bondagenumbers for general cases are determined as follows

Theorem 187 (Huang andXu [134] 2007) If 119889 ⩾ 2 and 119902119894⩾ 2

for each 119894 = 1 2 119901 then

119887119879(EB(119889 119899 119902

1 119902

119901)) = 119889

119901

minus 1

119887119879(EK(119889 119899 119902

1 119902

119901)) = 119889

119901

(79)

In particular for the de Bruijn digraph B(119889 119899) and the Kautzdigraph K(119889 119899)

119887119879(B (119889 119899)) = 119889 minus 1 119887

119879(K (119889 119899)) = 119889 (80)

Zhang et al [136] determined the bondage number incomplete 119905-partite digraphs

Theorem 188 (Zhang et al [136] 2009) For a complete 119905-partite digraph 119870

11989911198992119899119905

where 1198991⩽ 119899

2⩽ sdot sdot sdot ⩽ 119899

119905

119887 (11987011989911198992119899119905

)

=

119898 119894119891 119899119898= 1 119899

119898+1⩾ 2 119891119900119903 119904119900119898119890

119898 (1 ⩽ 119898 lt 119905)

4119905 minus 3 119894119891 1198991= 119899

2= sdot sdot sdot = 119899

119905= 2

119905minus1

sum

119894=1

119899119894+ 2 (119905 minus 1) 119900119905ℎ119890119903119908119894119904119890

(81)

Since an undirected graph can be thought of as a sym-metric digraph any result for digraphs has an analogy forundirected graphs in general In view of this point studyingthe bondage number for digraphs is more significant thanfor undirected graphs Thus we should further study thebondage number of digraphs and try to generalize knownresults on the bondage number and related variants for undi-rected graphs to digraphs prove or disprove Conjecture 183In particular determine the exact values of 119887(B(119889 119899)) for aneven 119899 and 119887

119879(119870

119899) for 119899 ⩾ 4

11 Efficient Dominating Sets

A dominating set 119878 of a graph 119866 is called to be efficient if forevery vertex 119909 in119866 |119873

119866[119909]cap119878| = 1 if119866 is a undirected graph

or |119873minus

119866[119909] cap 119878| = 1 if 119866 is a directed graph The concept of an

efficient set was proposed by Bange et al [137] in 1988 In theliterature an efficient set is also called a perfect dominatingset (see Livingston and Stout [138] for an explanation)

From definition if 119878 is an efficient dominating set of agraph 119866 then 119878 is certainly an independent set and everyvertex not in 119878 is adjacent to exactly one vertex in 119878

It is also clear from definition that a dominating set 119878 isefficient if and only if N

119866[119878] = 119873

119866[119909] 119909 isin 119878 for the

undirected graph 119866 or N+

119866[119878] = 119873

+

119866[119909] 119909 isin 119878 for the

digraph119866 is a partition of119881(119866) where the induced subgraphby119873

119866[119909] or119873+

119866[119909] is a star or an out-star with the root 119909

The efficient domination has important applications inmany areas such as error-correcting codes and receivesmuch attention in the late years

The concept of efficient dominating sets is a measureof the efficiency of domination in graphs Unfortunately asshown in [137] not every graph has an efficient dominatingset andmoreover it is anNP-complete problem to determinewhether a given graph has an efficient dominating set Inaddition it was shown by Clark [139] in 1993 that for a widerange of 119901 almost every random undirected graph 119866 isin

G(120592 119901) has no efficient dominating sets This means thatundirected graphs possessing an efficient dominating set arerare However it is easy to show that every undirected graphhas an orientationwith an efficient dominating set (see Bangeet al [140])

In 1993 Barkauskas and Host [141] showed that deter-mining whether an arbitrary oriented graph has an efficientdominating set is NP-complete Even so the existence ofefficient dominating sets for some special classes of graphs hasbeen examined see for example Dejter and Serra [142] andLee [143] for Cayley graph Gu et al [144] formeshes and toriObradovic et al [145] Huang and Xu [36] Reji Kumar andMacGillivray [146] for circulant graphs Harary graphs andtori and Van Wieren et al [147] for cube-connected cycles

In this section we introduce results of the bondagenumber for some graphs with efficient dominating sets dueto Huang and Xu [36]

111 Results for General Graphs In this subsection we intro-duce some results on bondage numbers obtained by applyingefficient dominating sets We first state the two followinglemmas

Lemma 189 For any 119896-regular graph or digraph 119866 of order 119899120574(119866) ⩾ 119899(119896 + 1) with equality if and only if 119866 has an efficientdominating set In addition if 119866 has an efficient dominatingset then every efficient dominating set is certainly a 120574-set andvice versa

Let 119890 be an edge and 119878 a dominating set in 119866 We say that119890 supports 119878 if 119890 isin (119878 119878) where (119878 119878) = (119909 119910) isin 119864(119866) 119909 isin 119878 119910 isin 119878 Denote by 119904(119866) the minimum number of edgeswhich support all 120574-sets in 119866

Lemma 190 For any graph or digraph 119866 119887(119866) ⩾ 119904(119866) withequality if 119866 is regular and has an efficient dominating set

30 International Journal of Combinatorics

A graph 119866 is called to be vertex-transitive if its automor-phism group Aut(119866) acts transitively on its vertex-set 119881(119866)A vertex-transitive graph is regular Applying Lemmas 189and 190 Huang and Xu obtained some results on bondagenumbers for vertex-transitive graphs or digraphs

Theorem 191 (Huang and Xu [36] 2008) For any vertex-transitive graph or digraph 119866 of order 119899

119887 (119866) ⩾

lceil119899

2120574 (119866)rceil 119894119891 119866 119894119904 119906119899119889119894119903119890119888119905119890119889

lceil119899

120574 (119866)rceil 119894119891 119866 119894119904 119889119894119903119890119888119905119890119889

(82)

Theorem192 (Huang andXu [36] 2008) Let119866 be a 119896-regulargraph of order 119899 Then

119887(119866) ⩽ 119896 if119866 is undirected and 119899 ⩾ 120574(119866)(119896+1)minus119896+1119887(119866) ⩽ 119896+1+ℓ if119866 is directed and 119899 ⩾ 120574(119866)(119896+1)minusℓwith 0 ⩽ ℓ ⩽ 119896 minus 1

Next we introduce a better upper bound on 119887(119866) To thisaim we need the following concept which is a generalizationof the concept of the edge-covering of a graph 119866 For 1198811015840

sube

119881(119866) and 1198641015840 sube 119864(119866) we say that 1198641015840covers 1198811015840 and call 1198641015840an edge-covering for 1198811015840 if there exists an edge (119909 119910) isin 1198641015840 forany vertex 119909 isin 1198811015840 For 119910 isin 119881(119866) let 1205731015840[119910] be the minimumcardinality over all edge-coverings for119873minus

119866[119910]

Theorem 193 (Huang and Xu [36] 2008) If 119866 is a 119896-regulargraph with order 120574(119866)(119896 + 1) then 119887(119866) ⩽ 1205731015840[119910] for any 119910 isin119881(119866)

Theupper bound on 119887(119866) given inTheorem 193 is tight inview of 119887(119862

119899) for a cycle or a directed cycle 119862

119899(seeTheorems

1 and 185 resp)It is easy to see that for a 119896-regular graph 119866 lceil(119896 + 1)2rceil ⩽

1205731015840

[119910] ⩽ 119896 when 119866 is undirected and 1205731015840[119910] = 119896 + 1 when 119866 isdirected By this fact and Lemma 189 the following theoremis merely a simple combination of Theorems 191 and 193

Theorem 194 (Huang and Xu [36] 2008) Let 119866 be a vertex-transitive graph of degree 119896 If 119866 has an efficient dominatingset then

lceil119896 + 1

2rceil ⩽ 119887 (119866) ⩽ 119896 119894119891 119866 119894119904 119906119899119889119894119903119890119888119905119890119889

119887 (119866) = 119896 + 1 119894119891 119866 119894119904 119889119894119903119890119888119905119890119889

(83)

Theorem 195 (Huang and Xu [36] 2008) If 119866 is an undi-rected vertex-transitive cubic graph with order 4120574(119866) and girth119892(119866) ⩽ 5 then 119887(119866) = 2

The proof of Theorem 195 leads to a byproduct If 119892 =5 let (119906

1 119906

2 119906

3 119906

4 119906

5) be a cycle and let D

119894be the family

of efficient dominating sets containing 119906119894for each 119894 isin

1 2 3 4 5 ThenD119894capD

119895= 0 for 119894 = 1 2 5 Then 119866 has

at least 5119904 efficient dominating sets where 119904 = 119904(119866) But thereare only 4s (=ns120574) distinct efficient dominating sets in119866This

contradiction implies that an undirected vertex-transitivecubic graph with girth five has no efficient dominating setsBut a similar argument for 119892(119866) = 3 4 or 119892(119866) ⩾ 6 couldnot give any contradiction This is consistent with the resultthat CCC(119899) a vertex-transitive cubic graph with girth 119899 if3 ⩽ 119899 ⩽ 8 or girth 8 if 119899 ⩾ 9 has efficient dominating setsfor all 119899 ⩾ 3 except 119899 = 5 (see Theorem 199 in the followingsubsection)

112 Results for Cayley Graphs In this subsection we useTheorem 194 to determine the exact values or approximativevalues of bondage numbers for some special vertex-transitivegraphs by characterizing the existence of efficient dominatingsets in these graphs

Let Γ be a nontrivial finite group 119878 a nonempty subset ofΓ without the identity element of Γ A digraph 119866 is defined asfollows

119881 (119866) = Γ (119909 119910) isin 119864 (119866) lArrrArr 119909minus1

119910 isin 119878

for any 119909 119910 isin Γ(84)

is called a Cayley digraph of the group Γ with respect to119878 denoted by 119862

Γ(119878) If 119878minus1 = 119904minus1 119904 isin 119878 = 119878 then

119862Γ(119878) is symmetric and is called a Cayley undirected graph

a Cayley graph for short Cayley graphs or digraphs arecertainly vertex-transitive (see Xu [1])

A circulant graph 119866(119899 119878) of order 119899 is a Cayley graph119862119885119899

(119878) where 119885119899= 0 119899 minus 1 is the addition group of

order 119899 and 119878 is a nonempty subset of119885119899without the identity

element and hence is a vertex-transitive digraph of degree|119878| If 119878minus1 = 119878 then119866(119899 119878) is an undirected graph If 119878 = 1 119896where 2 ⩽ 119896 ⩽ 119899 minus 2 we write 119866(119899 1 119896) for 119866(119899 1 119896) or119866(119899 plusmn1 plusmn119896) and call it a double-loop circulant graph

Huang and Xu [36] showed that for directed 119866 =

119866(119899 1 119896) lceil1198993rceil ⩽ 120574(119866) ⩽ lceil1198992rceil and 119866 has an efficientdominating set if and only if 3 | 119899 and 119896 equiv 2 (mod 3) fordirected 119866 = 119866(119899 1 119896) and 119896 = 1198992 lceil1198995rceil ⩽ 120574(119866) ⩽ lceil1198993rceiland 119866 has an efficient dominating set if and only if 5 | 119899 and119896 equiv plusmn2 (mod 5) By Theorem 194 we obtain the bondagenumber of a double loop circulant graph if it has an efficientdominating set

Theorem 196 (Huang and Xu [36] 2008) Let 119866 be a double-loop circulant graph 119866(119899 1 119896) If 119866 is directed with 3 | 119899and 119896 equiv 2 (mod 3) or 119866 is undirected with 5 | 119899 and119896 equiv plusmn2 (mod 5) then 119887(119866) = 3

The 119898 times 119899 torus is the Cartesian product 119862119898times 119862

119899of

two cycles and is a Cayley graph 119862119885119898times119885119899

(119878) where 119878 =

(0 1) (1 0) for directed cycles and 119878 = (0 plusmn1) (plusmn1 0) forundirected cycles and hence is vertex-transitive Gu et al[144] showed that the undirected torus119862

119898times119862

119899has an efficient

dominating set if and only if both 119898 and 119899 are multiplesof 5 Huang and Xu [36] showed that the directed torus119862119898times 119862

119899has an efficient dominating set if and only if both

119898 and 119899 are multiples of 3 Moreover they found a necessarycondition for a dominating set containing the vertex (0 0)in 119862

119898times 119862

119899to be efficient and obtained the following

result

International Journal of Combinatorics 31

Theorem 197 (Huang and Xu [36] 2008) Let 119866 = 119862119898times 119862

119899

If 119866 is undirected and both119898 and 119899 are multiples of 5 or if 119866 isdirected and both119898 and 119899 are multiples of 3 then 119887(119866) = 3

The hypercube 119876119899

is the Cayley graph 119862Γ(119878)

where Γ = 1198852times sdot sdot sdot times 119885

2= (119885

2)119899 and 119878 =

100 sdot sdot sdot 0 010 sdot sdot sdot 0 00 sdot sdot sdot 01 Lee [143] showed that119876119899has an efficient dominating set if and only if 119899 = 2119898 minus 1

for a positive integer119898 ByTheorem 194 the following resultis obtained immediately

Theorem 198 (Huang and Xu [36] 2008) If 119899 = 2119898 minus 1 for apositive integer119898 then 2119898minus1

⩽ 119887(119876119899) ⩽ 2

119898

minus 1

Problem 10 Determine the exact value of 119887(119876119899) for 119899 = 2119898minus1

The 119899-dimensional cube-connected cycle denoted byCCC(119899) is constructed from the 119899-dimensional hypercube119876119899by replacing each vertex 119909 in 119876

119899with an undirected cycle

119862119899of length 119899 and linking the 119894th vertex of the 119862

119899to the 119894th

neighbor of 119909 It has been proved that CCC(119899) is a Cayleygraph and hence is a vertex-transitive graph with degree 3Van Wieren et al [147] proved that CCC(119899) has an efficientdominating set if and only if 119899 = 5 From Theorems 194 and195 the following result can be derived

Theorem 199 (Huang and Xu [36] 2008) Let 119866 = 119862119862119862(119899)be the 119899-dimensional cube-connected cycles with 119899 ⩾ 3 and119899 = 5 Then 120574(119866) = 1198992119899minus2 and 2 ⩽ 119887(119866) ⩽ 3 In addition119887(119862119862119862(3)) = 119887(119862119862119862(4)) = 2

Problem 11 Determine the exact value of 119887(CCC(119899)) for 119899 ⩾5

Acknowledgments

This work was supported by NNSF of China (Grants nos11071233 61272008) The author would like to express hisgratitude to the anonymous referees for their kind sugges-tions and comments on the original paper to ProfessorSheikholeslami and Professor Jafari Rad for sending thereferences [128] and [81] which resulted in this version

References

[1] J-M Xu Theory and Application of Graphs vol 10 of NetworkTheory and Applications Kluwer Academic Dordrecht TheNetherlands 2003

[2] S THedetniemi andR C Laskar ldquoBibliography on dominationin graphs and some basic definitions of domination parame-tersrdquo Discrete Mathematics vol 86 no 1ndash3 pp 257ndash277 1990

[3] TW Haynes S T Hedetniemi and P J Slater Fundamentals ofDomination in Graphs vol 208 ofMonographs and Textbooks inPure and Applied Mathematics Marcel Dekker New York NYUSA 1998

[4] TWHaynes S THedetniemi and P J Slater EdsDominationin Graphs vol 209 of Monographs and Textbooks in Pure andAppliedMathematicsMarcelDekker NewYorkNYUSA 1998

[5] M R Garey andD S JohnsonComputers and Intractability WH Freeman and Company San Francisco Calif USA 1979

[6] H B Walikar and B D Acharya ldquoDomination critical graphsrdquoNational Academy Science Letters vol 2 pp 70ndash72 1979

[7] D Bauer F Harary J Nieminen and C L Suffel ldquoDominationalteration sets in graphsrdquo Discrete Mathematics vol 47 no 2-3pp 153ndash161 1983

[8] J F Fink M S Jacobson L F Kinch and J Roberts ldquoThebondage number of a graphrdquo Discrete Mathematics vol 86 no1ndash3 pp 47ndash57 1990

[9] F-T Hu and J-M Xu ldquoThe bondage number of (119899 minus 3)-regulargraph G of order nrdquo Ars Combinatoria to appear

[10] U Teschner ldquoNew results about the bondage number of agraphrdquoDiscreteMathematics vol 171 no 1ndash3 pp 249ndash259 1997

[11] B L Hartnell and D F Rall ldquoA bound on the size of a graphwith given order and bondage numberrdquo Discrete Mathematicsvol 197-198 pp 409ndash413 1999 16th British CombinatorialConference (London 1997)

[12] B L Hartnell and D F Rall ldquoA characterization of trees inwhich no edge is essential to the domination numberrdquo ArsCombinatoria vol 33 pp 65ndash76 1992

[13] J Topp and P D Vestergaard ldquo120572119896- and 120574

119896-stable graphsrdquo

Discrete Mathematics vol 212 no 1-2 pp 149ndash160 2000[14] A Meir and J W Moon ldquoRelations between packing and

covering numbers of a treerdquo Pacific Journal of Mathematics vol61 no 1 pp 225ndash233 1975

[15] B L Hartnell L K Joslashrgensen P D Vestergaard and CA Whitehead ldquoEdge stability of the k-domination numberof treesrdquo Bulletin of the Institute of Combinatorics and ItsApplications vol 22 pp 31ndash40 1998

[16] F-T Hu and J-M Xu ldquoOn the complexity of the bondage andreinforcement problemsrdquo Journal of Complexity vol 28 no 2pp 192ndash201 2012

[17] B L Hartnell and D F Rall ldquoBounds on the bondage numberof a graphrdquo Discrete Mathematics vol 128 no 1ndash3 pp 173ndash1771994

[18] Y-L Wang ldquoOn the bondage number of a graphrdquo DiscreteMathematics vol 159 no 1ndash3 pp 291ndash294 1996

[19] G H Fricke TW Haynes S M Hedetniemi S T Hedetniemiand R C Laskar ldquoExcellent treesrdquo Bulletin of the Institute ofCombinatorics and Its Applications vol 34 pp 27ndash38 2002

[20] V Samodivkin ldquoThe bondage number of graphs good and badverticesrdquoDiscussiones Mathematicae GraphTheory vol 28 no3 pp 453ndash462 2008

[21] J E Dunbar T W Haynes U Teschner and L VolkmannldquoBondage insensitivity and reinforcementrdquo in Domination inGraphs vol 209 of Monographs and Textbooks in Pure andAppliedMathematics pp 471ndash489 Dekker NewYork NY USA1998

[22] H Liu and L Sun ldquoThe bondage and connectivity of a graphrdquoDiscrete Mathematics vol 263 no 1ndash3 pp 289ndash293 2003

[23] A-H Esfahanian and S L Hakimi ldquoOn computing a con-ditional edge-connectivity of a graphrdquo Information ProcessingLetters vol 27 no 4 pp 195ndash199 1988

[24] R C Brigham P Z Chinn and R D Dutton ldquoVertexdomination-critical graphsrdquoNetworks vol 18 no 3 pp 173ndash1791988

[25] V Samodivkin ldquoDomination with respect to nondegenerateproperties bondage numberrdquoThe Australasian Journal of Com-binatorics vol 45 pp 217ndash226 2009

[26] U Teschner ldquoA new upper bound for the bondage numberof graphs with small domination numberrdquo The AustralasianJournal of Combinatorics vol 12 pp 27ndash35 1995

32 International Journal of Combinatorics

[27] J Fulman D Hanson and G MacGillivray ldquoVertex dom-ination-critical graphsrdquoNetworks vol 25 no 2 pp 41ndash43 1995

[28] U Teschner ldquoA counterexample to a conjecture on the bondagenumber of a graphrdquo Discrete Mathematics vol 122 no 1ndash3 pp393ndash395 1993

[29] U Teschner ldquoThe bondage number of a graph G can be muchgreater than Δ(G)rdquo Ars Combinatoria vol 43 pp 81ndash87 1996

[30] L Volkmann ldquoOn graphs with equal domination and coveringnumbersrdquoDiscrete AppliedMathematics vol 51 no 1-2 pp 211ndash217 1994

[31] L A Sanchis ldquoMaximum number of edges in connected graphswith a given domination numberrdquoDiscreteMathematics vol 87no 1 pp 65ndash72 1991

[32] S Klavzar and N Seifter ldquoDominating Cartesian products ofcyclesrdquoDiscrete AppliedMathematics vol 59 no 2 pp 129ndash1361995

[33] H K Kim ldquoA note on Vizingrsquos conjecture about the bondagenumberrdquo Korean Journal of Mathematics vol 10 no 2 pp 39ndash42 2003

[34] M Y Sohn X Yuan and H S Jeong ldquoThe bondage number of1198623times119862

119899rdquo Journal of the KoreanMathematical Society vol 44 no

6 pp 1213ndash1231 2007[35] L Kang M Y Sohn and H K Kim ldquoBondage number of the

discrete torus 119862119899times 119862

4rdquo Discrete Mathematics vol 303 no 1ndash3

pp 80ndash86 2005[36] J Huang and J-M Xu ldquoThe bondage numbers and efficient

dominations of vertex-transitive graphsrdquoDiscrete Mathematicsvol 308 no 4 pp 571ndash582 2008

[37] C J Xiang X Yuan andM Y Sohn ldquoDomination and bondagenumber of119862

3times119862

119899rdquoArs Combinatoria vol 97 pp 299ndash310 2010

[38] M S Jacobson and L F Kinch ldquoOn the domination numberof products of graphs Irdquo Ars Combinatoria vol 18 pp 33ndash441984

[39] Y-Q Li ldquoA note on the bondage number of a graphrdquo ChineseQuarterly Journal of Mathematics vol 9 no 4 pp 1ndash3 1994

[40] F-T Hu and J-M Xu ldquoBondage number of mesh networksrdquoFrontiers ofMathematics inChina vol 7 no 5 pp 813ndash826 2012

[41] R Frucht and F Harary ldquoOn the corona of two graphsrdquoAequationes Mathematicae vol 4 pp 322ndash325 1970

[42] K Carlson and M Develin ldquoOn the bondage number of planarand directed graphsrdquoDiscreteMathematics vol 306 no 8-9 pp820ndash826 2006

[43] U Teschner ldquoOn the bondage number of block graphsrdquo ArsCombinatoria vol 46 pp 25ndash32 1997

[44] J Huang and J-M Xu ldquoNote on conjectures of bondagenumbers of planar graphsrdquo Applied Mathematical Sciences vol6 no 65ndash68 pp 3277ndash3287 2012

[45] L Kang and J Yuan ldquoBondage number of planar graphsrdquoDiscrete Mathematics vol 222 no 1ndash3 pp 191ndash198 2000

[46] M Fischermann D Rautenbach and L Volkmann ldquoRemarkson the bondage number of planar graphsrdquo Discrete Mathemat-ics vol 260 no 1ndash3 pp 57ndash67 2003

[47] J Hou G Liu and J Wu ldquoBondage number of planar graphswithout small cyclesrdquoUtilitasMathematica vol 84 pp 189ndash1992011

[48] J Huang and J-M Xu ldquoThe bondage numbers of graphs withsmall crossing numbersrdquo Discrete Mathematics vol 307 no 15pp 1881ndash1897 2007

[49] Q Ma S Zhang and J Wang ldquoBondage number of 1-planargraphrdquo Applied Mathematics vol 1 no 2 pp 101ndash103 2010

[50] J Hou and G Liu ldquoA bound on the bondage number of toroidalgraphsrdquoDiscreteMathematics Algorithms and Applications vol4 no 3 Article ID 1250046 5 pages 2012

[51] Y-C Cao J-M Xu and X-R Xu ldquoOn bondage number oftoroidal graphsrdquo Journal of University of Science and Technologyof China vol 39 no 3 pp 225ndash228 2009

[52] Y-C Cao J Huang and J-M Xu ldquoThe bondage number ofgraphs with crossing number less than fourrdquo Ars Combinatoriato appear

[53] H K Kim ldquoOn the bondage numbers of some graphsrdquo KoreanJournal of Mathematics vol 11 no 2 pp 11ndash15 2004

[54] A Gagarin and V Zverovich ldquoUpper bounds for the bondagenumber of graphs on topological surfacesrdquo Discrete Mathemat-ics 2012

[55] J Huang ldquoAn improved upper bound for the bondage numberof graphs on surfacesrdquoDiscrete Mathematics vol 312 no 18 pp2776ndash2781 2012

[56] A Gagarin and V Zverovich ldquoThe bondage number of graphson topological surfaces and Teschnerrsquos conjecturerdquo DiscreteMathematics vol 313 no 6 pp 796ndash808 2013

[57] V Samodivkin ldquoNote on the bondage number of graphs ontopological surfacesrdquo httparxivorgabs12086203

[58] E J Cockayne R M Dawes and S T Hedetniemi ldquoTotaldomination in graphsrdquoNetworks vol 10 no 3 pp 211ndash219 1980

[59] R Laskar J Pfaff S M Hedetniemi and S T HedetniemildquoOn the algorithmic complexity of total dominationrdquo Society forIndustrial and Applied Mathematics vol 5 no 3 pp 420ndash4251984

[60] J Pfaff R C Laskar and S T Hedetniemi ldquoNP-completeness oftotal and connected domination and irredundance for bipartitegraphsrdquo Tech Rep 428 Department of Mathematical SciencesClemson University 1983

[61] M A Henning ldquoA survey of selected recent results on totaldomination in graphsrdquoDiscrete Mathematics vol 309 no 1 pp32ndash63 2009

[62] V R Kulli and D K Patwari ldquoThe total bondage number ofa graphrdquo in Advances in Graph Theory pp 227ndash235 VishwaGulbarga India 1991

[63] F-T Hu Y Lu and J-M Xu ldquoThe total bondage number ofgrid graphsrdquo Discrete Applied Mathematics vol 160 no 16-17pp 2408ndash2418 2012

[64] J Cao M Shi and B Wu ldquoResearch on the total bondagenumber of a spectial networkrdquo in Proceedings of the 3rdInternational Joint Conference on Computational Sciences andOptimization (CSO rsquo10) pp 456ndash458 May 2010

[65] N Sridharan MD Elias and V S A Subramanian ldquoTotalbondage number of a graphrdquo AKCE International Journal ofGraphs and Combinatorics vol 4 no 2 pp 203ndash209 2007

[66] N Jafari Rad and J Raczek ldquoSome progress on total bondage ingraphsrdquo Graphs and Combinatorics 2011

[67] T W Haynes and P J Slater ldquoPaired-domination and thepaired-domatic numberrdquoCongressusNumerantium vol 109 pp65ndash72 1995

[68] T W Haynes and P J Slater ldquoPaired-domination in graphsrdquoNetworks vol 32 no 3 pp 199ndash206 1998

[69] X Hou ldquoA characterization of 2120574 120574119875-treesrdquoDiscrete Mathemat-

ics vol 308 no 15 pp 3420ndash3426 2008[70] K E Proffitt T W Haynes and P J Slater ldquoPaired-domination

in grid graphsrdquo Congressus Numerantium vol 150 pp 161ndash1722001

International Journal of Combinatorics 33

[71] H Qiao L KangM Cardei andD-Z Du ldquoPaired-dominationof treesrdquo Journal of Global Optimization vol 25 no 1 pp 43ndash542003

[72] E Shan L Kang and M A Henning ldquoA characterizationof trees with equal total domination and paired-dominationnumbersrdquo The Australasian Journal of Combinatorics vol 30pp 31ndash39 2004

[73] J Raczek ldquoPaired bondage in treesrdquo Discrete Mathematics vol308 no 23 pp 5570ndash5575 2008

[74] J A Telle and A Proskurowski ldquoAlgorithms for vertex parti-tioning problems on partial k-treesrdquo SIAM Journal on DiscreteMathematics vol 10 no 4 pp 529ndash550 1997

[75] G S Domke J H Hattingh M A Henning and L R MarkusldquoRestrained domination in treesrdquoDiscreteMathematics vol 211no 1ndash3 pp 1ndash9 2000

[76] G S Domke J H Hattingh M A Henning and L R MarkusldquoRestrained domination in graphs with minimum degree twordquoJournal of Combinatorial Mathematics and Combinatorial Com-puting vol 35 pp 239ndash254 2000

[77] G S Domke J H Hattingh S T Hedetniemi R C Laskarand L R Markus ldquoRestrained domination in graphsrdquo DiscreteMathematics vol 203 no 1ndash3 pp 61ndash69 1999

[78] J H Hattingh and E J Joubert ldquoAn upper bound for therestrained domination number of a graph with minimumdegree at least two in terms of order and minimum degreerdquoDiscrete Applied Mathematics vol 157 no 13 pp 2846ndash28582009

[79] M A Henning ldquoGraphs with large restrained dominationnumberrdquo Discrete Mathematics vol 197-198 pp 415ndash429 199916th British Combinatorial Conference (London 1997)

[80] J H Hattingh and A R Plummer ldquoRestrained bondage ingraphsrdquo Discrete Mathematics vol 308 no 23 pp 5446ndash54532008

[81] N Jafari Rad R Hasni J Raczek and L Volkmann ldquoTotalrestrained bondage in graphsrdquoActaMathematica Sinica vol 29no 6 pp 1033ndash1042 2013

[82] J Cyman and J Raczek ldquoOn the total restrained dominationnumber of a graphrdquoThe Australasian Journal of Combinatoricsvol 36 pp 91ndash100 2006

[83] M A Henning ldquoGraphs with large total domination numberrdquoJournal of Graph Theory vol 35 no 1 pp 21ndash45 2000

[84] D-X Ma X-G Chen and L Sun ldquoOn total restraineddomination in graphsrdquoCzechoslovakMathematical Journal vol55 no 1 pp 165ndash173 2005

[85] B Zelinka ldquoRemarks on restrained domination and totalrestrained domination in graphsrdquo Czechoslovak MathematicalJournal vol 55 no 2 pp 393ndash396 2005

[86] W Goddard and M A Henning ldquoIndependent domination ingraphs a survey and recent resultsrdquo Discrete Mathematics vol313 no 7 pp 839ndash854 2013

[87] J H Zhang H L Liu and L Sun ldquoIndependence bondagenumber and reinforcement number of some graphsrdquo Transac-tions of Beijing Institute of Technology vol 23 no 2 pp 140ndash1422003

[88] K D Dharmalingam Studies in Graph Theorymdashequitabledomination and bottleneck domination [PhD thesis] MaduraiKamaraj University Madurai India 2006

[89] GDeepakND Soner andAAlwardi ldquoThe equitable bondagenumber of a graphrdquo Research Journal of Pure Algebra vol 1 no9 pp 209ndash212 2011

[90] E Sampathkumar and H B Walikar ldquoThe connected domina-tion number of a graphrdquo Journal of Mathematical and PhysicalSciences vol 13 no 6 pp 607ndash613 1979

[91] K White M Farber and W Pulleyblank ldquoSteiner trees con-nected domination and strongly chordal graphsrdquoNetworks vol15 no 1 pp 109ndash124 1985

[92] M Cadei M X Cheng X Cheng and D-Z Du ldquoConnecteddomination in Ad Hoc wireless networksrdquo in Proceedings ofthe 6th International Conference on Computer Science andInformatics (CSampI rsquo02) 2002

[93] J F Fink and M S Jacobson ldquon-domination in graphsrdquo inGraph Theory with Applications to Algorithms and ComputerScience Wiley-Interscience Publications pp 283ndash300 WileyNew York NY USA 1985

[94] M Chellali O Favaron A Hansberg and L Volkmann ldquok-domination and k-independence in graphs a surveyrdquo Graphsand Combinatorics vol 28 no 1 pp 1ndash55 2012

[95] Y Lu X Hou J-M Xu andN Li ldquoTrees with uniqueminimump-dominating setsrdquo Utilitas Mathematica vol 86 pp 193ndash2052011

[96] Y Lu and J-M Xu ldquoThe p-bondage number of treesrdquo Graphsand Combinatorics vol 27 no 1 pp 129ndash141 2011

[97] Y Lu and J-M Xu ldquoThe 2-domination and 2-bondage numbersof grid graphs A manuscriptrdquo httparxivorgabs12044514

[98] M Krzywkowski ldquoNon-isolating 2-bondage in graphsrdquo TheJournal of the Mathematical Society of Japan vol 64 no 4 2012

[99] M A Henning O R Oellermann and H C Swart ldquoBoundson distance domination parametersrdquo Journal of CombinatoricsInformation amp System Sciences vol 16 no 1 pp 11ndash18 1991

[100] F Tian and J-M Xu ldquoBounds for distance domination numbersof graphsrdquo Journal of University of Science and Technology ofChina vol 34 no 5 pp 529ndash534 2004

[101] F Tian and J-M Xu ldquoDistance domination-critical graphsrdquoApplied Mathematics Letters vol 21 no 4 pp 416ndash420 2008

[102] F Tian and J-M Xu ldquoA note on distance domination numbersof graphsrdquo The Australasian Journal of Combinatorics vol 43pp 181ndash190 2009

[103] F Tian and J-M Xu ldquoAverage distances and distance domina-tion numbersrdquo Discrete Applied Mathematics vol 157 no 5 pp1113ndash1127 2009

[104] Z-L Liu F Tian and J-M Xu ldquoProbabilistic analysis of upperbounds for 2-connected distance k-dominating sets in graphsrdquoTheoretical Computer Science vol 410 no 38ndash40 pp 3804ndash3813 2009

[105] M A Henning O R Oellermann and H C Swart ldquoDistancedomination critical graphsrdquo Journal of Combinatorial Mathe-matics and Combinatorial Computing vol 44 pp 33ndash45 2003

[106] T J Bean M A Henning and H C Swart ldquoOn the integrityof distance domination in graphsrdquo The Australasian Journal ofCombinatorics vol 10 pp 29ndash43 1994

[107] M Fischermann and L Volkmann ldquoA remark on a conjecturefor the (kp)-domination numberrdquoUtilitasMathematica vol 67pp 223ndash227 2005

[108] T Korneffel D Meierling and L Volkmann ldquoA remark on the(22)-domination numberrdquo Discussiones Mathematicae GraphTheory vol 28 no 2 pp 361ndash366 2008

[109] Y Lu X Hou and J-M Xu ldquoOn the (22)-domination numberof treesrdquo Discussiones Mathematicae Graph Theory vol 30 no2 pp 185ndash199 2010

34 International Journal of Combinatorics

[110] H Li and J-M Xu ldquo(dm)-domination numbers of m-connected graphsrdquo Rapports de Recherche 1130 LRI UniversiteParis-Sud 1997

[111] S M Hedetniemi S T Hedetniemi and T V Wimer ldquoLineartime resource allocation algorithms for treesrdquo Tech Rep URI-014 Department of Mathematical Sciences Clemson Univer-sity 1987

[112] M Farber ldquoDomination independent domination and dualityin strongly chordal graphsrdquo Discrete Applied Mathematics vol7 no 2 pp 115ndash130 1984

[113] G S Domke and R C Laskar ldquoThe bondage and reinforcementnumbers of 120574

119891for some graphsrdquoDiscrete Mathematics vol 167-

168 pp 249ndash259 1997[114] G S Domke S T Hedetniemi and R C Laskar ldquoFractional

packings coverings and irredundance in graphsrdquo vol 66 pp227ndash238 1988

[115] E J Cockayne P A Dreyer Jr S M Hedetniemi andS T Hedetniemi ldquoRoman domination in graphsrdquo DiscreteMathematics vol 278 no 1ndash3 pp 11ndash22 2004

[116] I Stewart ldquoDefend the roman empirerdquo Scientific American vol281 pp 136ndash139 1999

[117] C S ReVelle and K E Rosing ldquoDefendens imperiumromanum a classical problem in military strategyrdquoThe Ameri-can Mathematical Monthly vol 107 no 7 pp 585ndash594 2000

[118] A Bahremandpour F-T Hu S M Sheikholeslami and J-MXu ldquoOn the Roman bondage number of a graphrdquo DiscreteMathematics Algorithms and Applications vol 5 no 1 ArticleID 1350001 15 pages 2013

[119] N Jafari Rad and L Volkmann ldquoRoman bondage in graphsrdquoDiscussiones Mathematicae Graph Theory vol 31 no 4 pp763ndash773 2011

[120] K Ebadi and L PushpaLatha ldquoRoman bondage and Romanreinforcement numbers of a graphrdquo International Journal ofContemporary Mathematical Sciences vol 5 pp 1487ndash14972010

[121] N Dehgardi S M Sheikholeslami and L Volkman ldquoOn theRoman k-bondage number of a graphrdquo AKCE InternationalJournal of Graphs and Combinatorics vol 8 pp 169ndash180 2011

[122] S Akbari and S Qajar ldquoA note on Roman bondage number ofgraphsrdquo Ars Combinatoria to Appear

[123] F-T Hu and J-M Xu ldquoRoman bondage numbers of somegraphsrdquo httparxivorgabs11093933

[124] N Jafari Rad and L Volkmann ldquoOn the Roman bondagenumber of planar graphsrdquo Graphs and Combinatorics vol 27no 4 pp 531ndash538 2011

[125] S Akbari M Khatirinejad and S Qajar ldquoA note on the romanbondage number of planar graphsrdquo Graphs and Combinatorics2012

[126] B Bresar M A Henning and D F Rall ldquoRainbow dominationin graphsrdquo Taiwanese Journal of Mathematics vol 12 no 1 pp213ndash225 2008

[127] B L Hartnell and D F Rall ldquoOn dominating the Cartesianproduct of a graph and 120572krdquo Discussiones Mathematicae GraphTheory vol 24 no 3 pp 389ndash402 2004

[128] N Dehgardi S M Sheikholeslami and L Volkman ldquoThe k-rainbow bondage number of a graphrdquo to appear in DiscreteApplied Mathematics

[129] J von Neumann and O Morgenstern Theory of Games andEconomic Behavior Princeton University Press Princeton NJUSA 1944

[130] C BergeTheorıe des Graphes et ses Applications 1958[131] O Ore Theory of Graphs vol 38 American Mathematical

Society Colloquium Publications 1962[132] E Shan and L Kang ldquoBondage number in oriented graphsrdquoArs

Combinatoria vol 84 pp 319ndash331 2007[133] J Huang and J-M Xu ldquoThe bondage numbers of extended

de Bruijn and Kautz digraphsrdquo Computers amp Mathematics withApplications vol 51 no 6-7 pp 1137ndash1147 2006

[134] J Huang and J-M Xu ldquoThe total domination and total bondagenumbers of extended de Bruijn and Kautz digraphsrdquoComputersamp Mathematics with Applications vol 53 no 8 pp 1206ndash12132007

[135] Y Shibata and Y Gonda ldquoExtension of de Bruijn graph andKautz graphrdquo Computers amp Mathematics with Applications vol30 no 9 pp 51ndash61 1995

[136] X Zhang J Liu and JMeng ldquoThebondage number in completet-partite digraphsrdquo Information Processing Letters vol 109 no17 pp 997ndash1000 2009

[137] D W Bange A E Barkauskas and P J Slater ldquoEfficient dom-inating sets in graphsrdquo in Applications of Discrete Mathematicspp 189ndash199 SIAM Philadelphia Pa USA 1988

[138] M Livingston and Q F Stout ldquoPerfect dominating setsrdquoCongressus Numerantium vol 79 pp 187ndash203 1990

[139] L Clark ldquoPerfect domination in random graphsrdquo Journal ofCombinatorial Mathematics and Combinatorial Computing vol14 pp 173ndash182 1993

[140] D W Bange A E Barkauskas L H Host and L H ClarkldquoEfficient domination of the orientations of a graphrdquo DiscreteMathematics vol 178 no 1ndash3 pp 1ndash14 1998

[141] A E Barkauskas and L H Host ldquoFinding efficient dominatingsets in oriented graphsrdquo Congressus Numerantium vol 98 pp27ndash32 1993

[142] I J Dejter and O Serra ldquoEfficient dominating sets in CayleygraphsrdquoDiscrete AppliedMathematics vol 129 no 2-3 pp 319ndash328 2003

[143] J Lee ldquoIndependent perfect domination sets in Cayley graphsrdquoJournal of Graph Theory vol 37 no 4 pp 213ndash219 2001

[144] WGu X Jia and J Shen ldquoIndependent perfect domination setsin meshes tori and treesrdquo Unpublished

[145] N Obradovic J Peters and G Ruzic ldquoEfficient domination incirculant graphswith two chord lengthsrdquo Information ProcessingLetters vol 102 no 6 pp 253ndash258 2007

[146] K Reji Kumar and G MacGillivray ldquoEfficient domination incirculant graphsrdquo Discrete Mathematics vol 313 no 6 pp 767ndash771 2013

[147] D Van Wieren M Livingston and Q F Stout ldquoPerfectdominating sets on cube-connected cyclesrdquo Congressus Numer-antium vol 97 pp 51ndash70 1993

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

26 International Journal of Combinatorics

middot middot middot

middot middot middot

middot middot middot

middot middot middot

119866

Figure 9 The graph 119866 is constructed from 119866

Theorem 163 (Akbari et al [125] 2012) 119887RM(119866) ⩽ 15 forevery planar graph 119866

It remains open to show whether the bound inTheorem 163 is sharp or not Though finding a planargraph 119866 with 119887RM(119866) = 15 seems to be difficult Akbari etal [125] constructed an infinite family of planar graphs withRoman bondage number equal to 7 by proving the followingresult

Theorem 164 (Akbari et al [125] 2012) Let 119866 be a graph oforder 119899 and 119866 is the graph of order 5119899 obtained from 119866 byattaching the central vertex of a copy of a path 119875

5to each vertex

of119866 (see Figure 9) Then 120574RM(119866) = 4119899 and 119887RM(119866) = 120575(119866)+2

ByTheorem 164 infinitemany planar graphswith Romanbondage number 7 can be obtained by considering any planargraph 119866 with 120575(119866) = 5 (eg the icosahedron graph)

Conjecture 165 (Akbari et al [125] 2012) The Romanbondage number of every planar graph is at most 7

For general bounds the following observation is directlyobtained which is similar to Observation 1

Observation 4 Let 119866 be a graph of order 119899 with maximumdegree at least two Assume that 119867 is a spanning subgraphobtained from 119866 by removing 119896 edges If 120574RM(119867) = 120574RM(119866)then 119887

119877(119867) ⩽ 119887RM(119866) ⩽ 119887RM(119867) + 119896

Theorem 166 (Bahremandpour et al [118] 2013) For anyconnected graph 119866 of order 119899 if 119899 ⩾ 3 and 120574RM(119866) = 120574(119866) + 1then

119887RM (119866) ⩽ min 119887 (119866) 119899Δ (66)

where 119899Δis the number of vertices with maximum degree Δ in

119866

Observation 5 For any graph 119866 if 120574(119866) = 120574RM(119866) then119887RM(119866) ⩽ 119887(119866)

Theorem 167 (Bahremandpour et al [118] 2013) For everyRoman graph 119866

119887RM (119866) ⩾ 119887 (119866) (67)

The bound is sharp for cycles on 119899 vertices where 119899 equiv 0 (mod3)

The strict inequality in Theorem 167 can hold for exam-ple 119887(119862

3119896+2) = 2 lt 3 = 119887RM(1198623119896+2

) byTheorem 157A graph 119866 is called to be vertex Roman domination-

critical if 120574RM(119866 minus 119909) lt 120574RM(119866) for every vertex 119909 in 119866 If119866 has no isolated vertices then 120574RM(119866) ⩽ 2120574(119866) ⩽ 2120573(119866)If 120574RM(119866) = 2120573(119866) then 120574RM(119866) = 2120574(119866) and hence 119866 is aRoman graph In [30] Volkmann gave a lot of graphs with120574(119866) = 120573(119866) The following result is similar to Theorem 57

Theorem 168 (Bahremandpour et al [118] 2013) For anygraph 119866 if 120574RM(119866) = 2120573(119866) then

(a) 119887RM(119866) ⩾ 120575(119866)

(b) 119887RM(119866) ⩾ 120575(119866)+1 if119866 is a vertex Roman domination-critical graph

If 120574RM(119866) = 2 then obviously 119887RM(119866) ⩽ 120575(119866) So weassume 120574RM(119866) ⩾ 3 Dehgardi et al [121] proved 119887RM(119866) ⩽(120574RM(119866) minus 2)Δ(119866) + 1 and posed the following problem if 119866is a connected graph of order 119899 ⩾ 4 with 120574RM(119866) ⩾ 3 then

119887RM (119866) ⩽ (120574RM (119866) minus 2) Δ (119866) (68)

Theorem 161 (a) shows that the inequality (68) holds if120574RM(119866) ⩾ 5 Thus the bound in (68) is of interest only when120574RM(119866) is 3 or 4

Lemma 169 (Bahremandpour et al [118] 2013) For anynonempty graph 119866 of order 119899 ⩾ 3 120574RM(119866) = 3 if and onlyif Δ(119866) = 119899 minus 2

The following result shows that (68) holds for all graphs119866 of order 119899 ⩾ 4 with 120574RM(119866) = 3 or 4 which improvesTheorem 161 (b)

Theorem 170 (Bahremandpour et al [118] 2013) For anyconnected graph 119866 of order 119899 ⩾ 4

119887RM (119866) le Δ (119866) = 119899 minus 2 119894119891 120574RM (119866) = 3

Δ (119866) + 120575 (119866) minus 1 119894119891 120574RM (119866) = 4(69)

with the first equality if and only if 119866 cong 1198624

Recently Akbari and Qajar [122] have proved the follow-ing result

Theorem 171 (Akbari and Qajar [122]) For any connectedgraph 119866 of order 119899 ⩾ 3

119887RM (119866) ⩽ 119899 minus 120574RM (119866) + 5 (70)

International Journal of Combinatorics 27

We conclude this section with the following problems

Problem 8 Characterize all connected graphs 119866 of order 119899 ⩾3 for which 119887RM(119866) = 119899 minus 1

Problem 9 Prove or disprove if 119866 is a connected graph oforder 119899 ⩾ 3 then

119887RM (119866) ⩽ 119899 minus 120574RM (119866) + 3 (71)

Note that 120574119877(119870

119905(3)) = 4 and 119887RM(119870119905

(3)) = 119899minus1 where 119899 =3119905 for 119905 ⩾ 3 The upper bound on 119887RM(119866) given in Problem 9is sharp if Problem 9 has positive answer

95 Rainbow Bondage Numbers For a positive integer 119896 a119896-rainbow dominating function of a graph 119866 is a function 119891from the vertex-set 119881(119866) to the set of all subsets of the set1 2 119896 such that for any vertex 119909 in 119866 with 119891(119909) = 0 thecondition

⋃

119910isin119873119866(119909)

119891 (119910) = 1 2 119896 (72)

is fulfilledThe weight of a 119896-rainbow dominating function 119891on 119866 is the value

119891 (119866) = sum

119909isin119881(119866)

1003816100381610038161003816119891 (119909)1003816100381610038161003816 (73)

The 119896-rainbow domination number of a graph 119866 denoted by120574119903119896(119866) is the minimum weight of a 119896-rainbow dominating

function 119891 of 119866Note that 120574

1199031(119866) is the classical domination number 120574(119866)

The 119896-rainbow domination number is introduced by Bresaret al [126] Rainbow domination of a graph 119866 coincides withordinary domination of the Cartesian product of 119866 with thecomplete graph in particular 120574

119903119896(119866) = 120574(119866 times 119870

119896) for any

positive integer 119896 and any graph 119866 [126] Hartnell and Rall[127] proved that 120574(119879) lt 120574

1199032(119879) for any tree 119879 no graph has

120574 = 120574119903119896for 119896 ⩾ 3 for any 119896 ⩾ 2 and any graph 119866 of order 119899

min 119899 120574 (119866) + 119896 minus 2 ⩽ 120574119903119896(119866) ⩽ 119896120574 (119866) (74)

The introduction of rainbow domination is motivatedby the study of paired domination in Cartesian productsof graphs where certain upper bounds can be expressed interms of rainbow domination that is for any graph 119866 andany graph 119867 if 119881(119867) can be partitioned into 119896120574

119875-sets then

120574119875(119866 times 119867) ⩽ (1119896)120592(119867)120574

119903119896(119866) proved by Bresar et al [126]

Dehgardi et al [128] introduced the concept of 119896-rainbowbondage number of graphs Let 119866 be a graph with maximumdegree at least two The 119896-rainbow bondage number 119887

119903119896(119866)

of 119866 is the minimum cardinality of all sets 119861 sube 119864(119866) forwhich 120574

119903119896(119866 minus 119861) gt 120574

119903119896(119866) Since in the study of 119896-rainbow

bondage number the assumption Δ(119866) ⩾ 2 is necessarywe always assume that when we discuss 119887

119903119896(119866) all graphs

involved satisfy Δ(119866) ⩾ 2The 2-rainbow bondage number of some special families

of graphs has been determined

Theorem 172 (Dehgardi et al [128]) For any integer 119899 ⩾ 3

1198871199032(119875

119899) = 1 119891119900119903 119899 ⩾ 2

1198871199032(119862

119899) =

1 119894119891 119899 equiv 0 (mod 4) 2 119900119905ℎ119890119903119908119894119904119890

1198871199032(119870

119899119899) =

1 119894119891 119899 = 2

3 119894119891 119899 = 3

6 119894119891 119899 = 4

119898 119894119891 119899 ⩾ 5

(75)

Theorem 173 (Dehgardi et al [128]) For any connected graph119866 of order 119899 ⩾ 3 119887

1199032(119866) ⩽ 120582(119866) + 2Δ(119866) minus 3

Theorem 174 (Dehgardi et al [128]) For any tree 119879 of order119899 ⩾ 3 119887

1199032(119879) ⩽ 2

Theorem 175 (Dehgardi et al [128]) If 119866 is a planar graphwithmaximumdegree at least two then 119887

1199032(119866) ⩽ 15Moreover

if the girth 119892(119866) ⩾ 4 then 1198871199032(119866) ⩽ 11 and if 119892(119866) ⩾ 6 then

1198871199032(119866) ⩽ 9

Theorem 176 (Dehgardi et al [128]) For a complete bipartitegraph 119870

119901119902 if 2119896 + 1 ⩽ 119901 ⩽ 119902 then 119887

119903119896(119870

119901119902) = 119901

Theorem177 (Dehgardi et al [128]) For a complete graph119870119899

if 119899 ⩾ 119896 + 1 then 119887119903119896(119870

119899) = lceil119896119899(119896 + 1)rceil

The special case 119896 = 1 of Theorem 177 can be found inTheorem 1 The following is a simple observation similar toObservation 1

Observation 6 (Dehgardi et al [128]) Let 119866 be a graphwith maximum degree at least two 119867 a spanning subgraphobtained from 119866 by removing 119896 edges If 120574

119903119896(119867) = 120574

119903119896(119866)

then 119887119903119896(119867) ⩽ 119887

119903119896(119866) ⩽ 119887

119903119896(119867) + 119896

Theorem 178 (Dehgardi et al [128]) For every graph 119866 with120574119903119896(119866) = 119896120574(119866) 119887

119903119896(119866) ⩾ 119887(119866)

A graph 119866 is said to be vertex 119896-rainbow domination-critical if 120574

119903119896(119866 minus 119909) lt 120574

119903119896(119866) for every vertex 119909 in 119866 It is

clear that if 119866 has no isolated vertices then 120574119903119896(119866) ⩽ 119896120574(119866) ⩽

119896120573(119866) where 120573(119866) is the vertex-covering number of119866 Thusif 120574

119903119896(119866) = 119896120573(119866) then 120574

119903119896(119866) = 119896120574(119866)

Theorem 179 (Dehgardi et al [128]) For any graph 119866 if120574119903119896(119866) = 119896120573(119866) then

(a) 119887119903119896(119866) ⩾ 120575(119866)

(b) 119887119903119896(119866) ⩾ 120575(119866)+1 if119866 is vertex 119896-rainbow domination-

critical

As far as we know there are no more results on the 119896-rainbow bondage number of graphs in the literature

28 International Journal of Combinatorics

10 Results on Digraphs

Although domination has been extensively studied in undi-rected graphs it is natural to think of a dominating setas a one-way relationship between vertices of the graphIndeed among the earliest literatures on this subject vonNeumann and Morgenstern [129] used what is now calleddomination in digraphs to find solution (or kernels whichare independent dominating sets) for cooperative 119899-persongames Most likely the first formulation of domination byBerge [130] in 1958 was given in the context of digraphs andonly some years latter by Ore [131] in 1962 for undirectedgraphs Despite this history examination of domination andits variants in digraphs has been essentially overlooked (see[4] for an overview of the domination literature) Thus thereare few if any such results on domination for digraphs in theliterature

The bondage number and its related topics for undirectedgraph have become one of major areas both in theoreticaland applied researches However until recently Carlsonand Develin [42] Shan and Kang [132] and Huang andXu [133 134] studied the bondage number for digraphsindependently In this section we introduce their resultsfor general digraphs Results for some special digraphs suchas vertex-transitive digraphs are introduced in the nextsection

101 Upper Bounds for General Digraphs Let 119866 = (119881 119864)

be a digraph without loops and parallel edges A digraph 119866is called to be asymmetric if whenever (119909 119910) isin 119864(119866) then(119910 119909) notin 119864(119866) and to be symmetric if (119909 119910) isin 119864(119866) implies(119910 119909) isin 119864(119866) For a vertex 119909 of 119881(119866) the sets of out-neighbors and in-neighbors of 119909 are respectively defined as119873

+

119866(119909) = 119910 isin 119881(119866) (119909 119910) isin 119864(119866) and 119873minus

119866(119909) = 119909 isin

119881(119866) (119909 119910) isin 119864(119866) the out-degree and the in-degree of119909 are respectively defined as 119889+

119866(119909) = |119873

+

119866(119909)| and 119889minus

119866(119909) =

|119873+

119866(119909)| Denote themaximum and theminimum out-degree

(resp in-degree) of 119866 by Δ+(119866) and 120575+(119866) (resp Δminus(119866) and120575minus

(119866))The degree 119889119866(119909) of 119909 is defined as 119889+

119866(119909)+119889

minus

119866(119909) and

the maximum and the minimum degrees of119866 are denoted byΔ(119866) and 120575(119866) respectively that is Δ(119866) = max119889

119866(119909) 119909 isin

119881(119866) and 120575(119866) = min119889119866(119909) 119909 isin 119881(119866) Note that the

definitions here are different from the ones in the textbookon digraphs see for example Xu [1]

A subset 119878 of119881(119866) is called a dominating set if119881(119866) = 119878cup119873

+

119866(119878) where119873+

119866(119878) is the set of out-neighbors of 119878Then just

as for undirected graphs 120574(119866) is the minimum cardinalityof a dominating set and the bondage number 119887(119866) is thesmallest cardinality of a set 119861 of edges such that 120574(119866 minus 119861) gt120574(119866) if such a subset 119861 sube 119864(119866) exists Otherwise we put119887(119866) = infin

Some basic results for undirected graphs stated inSection 3 can be generalized to digraphs For exampleTheorem 18 is generalized by Carlson and Develin [42]Huang andXu [133] and Shan andKang [132] independentlyas follows

Theorem 180 Let 119866 be a digraph and (119909 119910) isin 119864(119866) Then119887(119866) ⩽ 119889

119866(119910) + 119889

minus

119866(119909) minus |119873

minus

119866(119909) cap 119873

minus

119866(119910)|

Corollary 181 For a digraph 119866 119887(119866) ⩽ 120575minus(119866) + Δ(119866)

Since 120575minus

(119866) ⩽ (12)Δ(119866) from Corollary 181Conjecture 53 is valid for digraphs

Corollary 182 For a digraph 119866 119887(119866) ⩽ (32)Δ(119866)

In the case of undirected graphs the bondage number of119866119899= 119870

119899times 119870

119899achieves this bound However it was shown

by Carlson and Develin in [42] that if we take the symmetricdigraph119866

119899 we haveΔ(119866

119899) = 4(119899minus1) 120574(119866

119899) = 119899 and 119887(119866

119899) ⩽

4(119899 minus 1) + 1 = Δ(119866119899) + 1 So this family of digraphs cannot

show that the bound in Corollary 182 is tightCorresponding to Conjecture 51 which is discredited

for undirected graphs and is valid for digraphs the sameconjecture for digraphs can be proposed as follows

Conjecture 183 (Carlson and Develin [42] 2006) 119887(119866) ⩽Δ(119866) + 1 for any digraph 119866

If Conjecture 183 is true then the following results showsthat this upper bound is tight since 119887(119870

119899times119870

119899) = Δ(119870

119899times119870

119899)+1

for a complete digraph 119870119899

In 2007 Shan and Kang [132] gave some tight upperbounds on the bondage numbers for some asymmetricdigraphs For example 119887(119879) ⩽ Δ(119879) for any asymmetricdirected tree 119879 119887(119866) ⩽ Δ(119866) for any asymmetric digraph 119866with order at least 4 and 120574(119866) ⩽ 2 For planar digraphs theyobtained the following results

Theorem 184 (Shan and Kang [132] 2007) Let 119866 be aasymmetric planar digraphThen 119887(119866) ⩽ Δ(119866)+2 and 119887(119866) ⩽Δ(119866) + 1 if Δ(119866) ⩾ 5 and 119889minus

119866(119909) ⩾ 3 for every vertex 119909 with

119889119866(119909) ⩾ 4

102 Results for Some Special Digraphs The exact values andbounds on 119887(119866) for some standard digraphs are determined

Theorem185 (Huang andXu [133] 2006) For a directed cycle119862119899and a directed path 119875

119899

119887 (119862119899) =

3 119894119891 119899 119894119904 119900119889119889

2 119894119891 119899 119894119904 119890V119890119899

119887 (119875119899) =

2 119894119891 119899 119894119904 119900119889119889

1 119894119891 119899 119894119904 119890V119890119899

(76)

For the de Bruijn digraph 119861(119889 119899) and the Kautz digraph119870(119889 119899)

119887 (119861 (119889 119899)) = 119889 119894119891 119899 119894119904 119900119889119889

119889 ⩽ 119887 (119861 (119889 119899)) ⩽ 2119889 119894119891 119899 119894119904 119890V119890119899

119887 (119870 (119889 119899)) = 119889 + 1

(77)

Like undirected graphs we can define the total domi-nation number and the total bondage number On the totalbondage numbers for some special digraphs the knownresults are as follows

International Journal of Combinatorics 29

Theorem 186 (Huang and Xu [134] 2007) For a directedcycle 119862

119899and a directed path 119875

119899 119887

119879(119875

119899) and 119887

119879(119862

119899) all do not

exist For a complete digraph 119870119899

119887119879(119870

119899) = infin 119894119891 119899 = 1 2

119887119879(119870

119899) = 3 119894119891 119899 = 3

119899 ⩽ 119887119879(119870

119899) ⩽ 2119899 minus 3 119894119891 119899 ⩾ 4

(78)

The extended de Bruijn digraph EB(119889 119899 1199021 119902

119901) and

the extended Kautz digraph EK(119889 119899 1199021 119902

119901) are intro-

duced by Shibata and Gonda [135] If 119901 = 1 then theyare the de Bruijn digraph B(119889 119899) and the Kautz digraphK(119889 119899) respectively Huang and Xu [134] determined theirtotal domination numbers In particular their total bondagenumbers for general cases are determined as follows

Theorem 187 (Huang andXu [134] 2007) If 119889 ⩾ 2 and 119902119894⩾ 2

for each 119894 = 1 2 119901 then

119887119879(EB(119889 119899 119902

1 119902

119901)) = 119889

119901

minus 1

119887119879(EK(119889 119899 119902

1 119902

119901)) = 119889

119901

(79)

In particular for the de Bruijn digraph B(119889 119899) and the Kautzdigraph K(119889 119899)

119887119879(B (119889 119899)) = 119889 minus 1 119887

119879(K (119889 119899)) = 119889 (80)

Zhang et al [136] determined the bondage number incomplete 119905-partite digraphs

Theorem 188 (Zhang et al [136] 2009) For a complete 119905-partite digraph 119870

11989911198992119899119905

where 1198991⩽ 119899

2⩽ sdot sdot sdot ⩽ 119899

119905

119887 (11987011989911198992119899119905

)

=

119898 119894119891 119899119898= 1 119899

119898+1⩾ 2 119891119900119903 119904119900119898119890

119898 (1 ⩽ 119898 lt 119905)

4119905 minus 3 119894119891 1198991= 119899

2= sdot sdot sdot = 119899

119905= 2

119905minus1

sum

119894=1

119899119894+ 2 (119905 minus 1) 119900119905ℎ119890119903119908119894119904119890

(81)

Since an undirected graph can be thought of as a sym-metric digraph any result for digraphs has an analogy forundirected graphs in general In view of this point studyingthe bondage number for digraphs is more significant thanfor undirected graphs Thus we should further study thebondage number of digraphs and try to generalize knownresults on the bondage number and related variants for undi-rected graphs to digraphs prove or disprove Conjecture 183In particular determine the exact values of 119887(B(119889 119899)) for aneven 119899 and 119887

119879(119870

119899) for 119899 ⩾ 4

11 Efficient Dominating Sets

A dominating set 119878 of a graph 119866 is called to be efficient if forevery vertex 119909 in119866 |119873

119866[119909]cap119878| = 1 if119866 is a undirected graph

or |119873minus

119866[119909] cap 119878| = 1 if 119866 is a directed graph The concept of an

efficient set was proposed by Bange et al [137] in 1988 In theliterature an efficient set is also called a perfect dominatingset (see Livingston and Stout [138] for an explanation)

From definition if 119878 is an efficient dominating set of agraph 119866 then 119878 is certainly an independent set and everyvertex not in 119878 is adjacent to exactly one vertex in 119878

It is also clear from definition that a dominating set 119878 isefficient if and only if N

119866[119878] = 119873

119866[119909] 119909 isin 119878 for the

undirected graph 119866 or N+

119866[119878] = 119873

+

119866[119909] 119909 isin 119878 for the

digraph119866 is a partition of119881(119866) where the induced subgraphby119873

119866[119909] or119873+

119866[119909] is a star or an out-star with the root 119909

The efficient domination has important applications inmany areas such as error-correcting codes and receivesmuch attention in the late years

The concept of efficient dominating sets is a measureof the efficiency of domination in graphs Unfortunately asshown in [137] not every graph has an efficient dominatingset andmoreover it is anNP-complete problem to determinewhether a given graph has an efficient dominating set Inaddition it was shown by Clark [139] in 1993 that for a widerange of 119901 almost every random undirected graph 119866 isin

G(120592 119901) has no efficient dominating sets This means thatundirected graphs possessing an efficient dominating set arerare However it is easy to show that every undirected graphhas an orientationwith an efficient dominating set (see Bangeet al [140])

In 1993 Barkauskas and Host [141] showed that deter-mining whether an arbitrary oriented graph has an efficientdominating set is NP-complete Even so the existence ofefficient dominating sets for some special classes of graphs hasbeen examined see for example Dejter and Serra [142] andLee [143] for Cayley graph Gu et al [144] formeshes and toriObradovic et al [145] Huang and Xu [36] Reji Kumar andMacGillivray [146] for circulant graphs Harary graphs andtori and Van Wieren et al [147] for cube-connected cycles

In this section we introduce results of the bondagenumber for some graphs with efficient dominating sets dueto Huang and Xu [36]

111 Results for General Graphs In this subsection we intro-duce some results on bondage numbers obtained by applyingefficient dominating sets We first state the two followinglemmas

Lemma 189 For any 119896-regular graph or digraph 119866 of order 119899120574(119866) ⩾ 119899(119896 + 1) with equality if and only if 119866 has an efficientdominating set In addition if 119866 has an efficient dominatingset then every efficient dominating set is certainly a 120574-set andvice versa

Let 119890 be an edge and 119878 a dominating set in 119866 We say that119890 supports 119878 if 119890 isin (119878 119878) where (119878 119878) = (119909 119910) isin 119864(119866) 119909 isin 119878 119910 isin 119878 Denote by 119904(119866) the minimum number of edgeswhich support all 120574-sets in 119866

Lemma 190 For any graph or digraph 119866 119887(119866) ⩾ 119904(119866) withequality if 119866 is regular and has an efficient dominating set

30 International Journal of Combinatorics

A graph 119866 is called to be vertex-transitive if its automor-phism group Aut(119866) acts transitively on its vertex-set 119881(119866)A vertex-transitive graph is regular Applying Lemmas 189and 190 Huang and Xu obtained some results on bondagenumbers for vertex-transitive graphs or digraphs

Theorem 191 (Huang and Xu [36] 2008) For any vertex-transitive graph or digraph 119866 of order 119899

119887 (119866) ⩾

lceil119899

2120574 (119866)rceil 119894119891 119866 119894119904 119906119899119889119894119903119890119888119905119890119889

lceil119899

120574 (119866)rceil 119894119891 119866 119894119904 119889119894119903119890119888119905119890119889

(82)

Theorem192 (Huang andXu [36] 2008) Let119866 be a 119896-regulargraph of order 119899 Then

119887(119866) ⩽ 119896 if119866 is undirected and 119899 ⩾ 120574(119866)(119896+1)minus119896+1119887(119866) ⩽ 119896+1+ℓ if119866 is directed and 119899 ⩾ 120574(119866)(119896+1)minusℓwith 0 ⩽ ℓ ⩽ 119896 minus 1

Next we introduce a better upper bound on 119887(119866) To thisaim we need the following concept which is a generalizationof the concept of the edge-covering of a graph 119866 For 1198811015840

sube

119881(119866) and 1198641015840 sube 119864(119866) we say that 1198641015840covers 1198811015840 and call 1198641015840an edge-covering for 1198811015840 if there exists an edge (119909 119910) isin 1198641015840 forany vertex 119909 isin 1198811015840 For 119910 isin 119881(119866) let 1205731015840[119910] be the minimumcardinality over all edge-coverings for119873minus

119866[119910]

Theorem 193 (Huang and Xu [36] 2008) If 119866 is a 119896-regulargraph with order 120574(119866)(119896 + 1) then 119887(119866) ⩽ 1205731015840[119910] for any 119910 isin119881(119866)

Theupper bound on 119887(119866) given inTheorem 193 is tight inview of 119887(119862

119899) for a cycle or a directed cycle 119862

119899(seeTheorems

1 and 185 resp)It is easy to see that for a 119896-regular graph 119866 lceil(119896 + 1)2rceil ⩽

1205731015840

[119910] ⩽ 119896 when 119866 is undirected and 1205731015840[119910] = 119896 + 1 when 119866 isdirected By this fact and Lemma 189 the following theoremis merely a simple combination of Theorems 191 and 193

Theorem 194 (Huang and Xu [36] 2008) Let 119866 be a vertex-transitive graph of degree 119896 If 119866 has an efficient dominatingset then

lceil119896 + 1

2rceil ⩽ 119887 (119866) ⩽ 119896 119894119891 119866 119894119904 119906119899119889119894119903119890119888119905119890119889

119887 (119866) = 119896 + 1 119894119891 119866 119894119904 119889119894119903119890119888119905119890119889

(83)

Theorem 195 (Huang and Xu [36] 2008) If 119866 is an undi-rected vertex-transitive cubic graph with order 4120574(119866) and girth119892(119866) ⩽ 5 then 119887(119866) = 2

The proof of Theorem 195 leads to a byproduct If 119892 =5 let (119906

1 119906

2 119906

3 119906

4 119906

5) be a cycle and let D

119894be the family

of efficient dominating sets containing 119906119894for each 119894 isin

1 2 3 4 5 ThenD119894capD

119895= 0 for 119894 = 1 2 5 Then 119866 has

at least 5119904 efficient dominating sets where 119904 = 119904(119866) But thereare only 4s (=ns120574) distinct efficient dominating sets in119866This

contradiction implies that an undirected vertex-transitivecubic graph with girth five has no efficient dominating setsBut a similar argument for 119892(119866) = 3 4 or 119892(119866) ⩾ 6 couldnot give any contradiction This is consistent with the resultthat CCC(119899) a vertex-transitive cubic graph with girth 119899 if3 ⩽ 119899 ⩽ 8 or girth 8 if 119899 ⩾ 9 has efficient dominating setsfor all 119899 ⩾ 3 except 119899 = 5 (see Theorem 199 in the followingsubsection)

112 Results for Cayley Graphs In this subsection we useTheorem 194 to determine the exact values or approximativevalues of bondage numbers for some special vertex-transitivegraphs by characterizing the existence of efficient dominatingsets in these graphs

Let Γ be a nontrivial finite group 119878 a nonempty subset ofΓ without the identity element of Γ A digraph 119866 is defined asfollows

119881 (119866) = Γ (119909 119910) isin 119864 (119866) lArrrArr 119909minus1

119910 isin 119878

for any 119909 119910 isin Γ(84)

is called a Cayley digraph of the group Γ with respect to119878 denoted by 119862

Γ(119878) If 119878minus1 = 119904minus1 119904 isin 119878 = 119878 then

119862Γ(119878) is symmetric and is called a Cayley undirected graph

a Cayley graph for short Cayley graphs or digraphs arecertainly vertex-transitive (see Xu [1])

A circulant graph 119866(119899 119878) of order 119899 is a Cayley graph119862119885119899

(119878) where 119885119899= 0 119899 minus 1 is the addition group of

order 119899 and 119878 is a nonempty subset of119885119899without the identity

element and hence is a vertex-transitive digraph of degree|119878| If 119878minus1 = 119878 then119866(119899 119878) is an undirected graph If 119878 = 1 119896where 2 ⩽ 119896 ⩽ 119899 minus 2 we write 119866(119899 1 119896) for 119866(119899 1 119896) or119866(119899 plusmn1 plusmn119896) and call it a double-loop circulant graph

Huang and Xu [36] showed that for directed 119866 =

119866(119899 1 119896) lceil1198993rceil ⩽ 120574(119866) ⩽ lceil1198992rceil and 119866 has an efficientdominating set if and only if 3 | 119899 and 119896 equiv 2 (mod 3) fordirected 119866 = 119866(119899 1 119896) and 119896 = 1198992 lceil1198995rceil ⩽ 120574(119866) ⩽ lceil1198993rceiland 119866 has an efficient dominating set if and only if 5 | 119899 and119896 equiv plusmn2 (mod 5) By Theorem 194 we obtain the bondagenumber of a double loop circulant graph if it has an efficientdominating set

Theorem 196 (Huang and Xu [36] 2008) Let 119866 be a double-loop circulant graph 119866(119899 1 119896) If 119866 is directed with 3 | 119899and 119896 equiv 2 (mod 3) or 119866 is undirected with 5 | 119899 and119896 equiv plusmn2 (mod 5) then 119887(119866) = 3

The 119898 times 119899 torus is the Cartesian product 119862119898times 119862

119899of

two cycles and is a Cayley graph 119862119885119898times119885119899

(119878) where 119878 =

(0 1) (1 0) for directed cycles and 119878 = (0 plusmn1) (plusmn1 0) forundirected cycles and hence is vertex-transitive Gu et al[144] showed that the undirected torus119862

119898times119862

119899has an efficient

dominating set if and only if both 119898 and 119899 are multiplesof 5 Huang and Xu [36] showed that the directed torus119862119898times 119862

119899has an efficient dominating set if and only if both

119898 and 119899 are multiples of 3 Moreover they found a necessarycondition for a dominating set containing the vertex (0 0)in 119862

119898times 119862

119899to be efficient and obtained the following

result

International Journal of Combinatorics 31

Theorem 197 (Huang and Xu [36] 2008) Let 119866 = 119862119898times 119862

119899

If 119866 is undirected and both119898 and 119899 are multiples of 5 or if 119866 isdirected and both119898 and 119899 are multiples of 3 then 119887(119866) = 3

The hypercube 119876119899

is the Cayley graph 119862Γ(119878)

where Γ = 1198852times sdot sdot sdot times 119885

2= (119885

2)119899 and 119878 =

100 sdot sdot sdot 0 010 sdot sdot sdot 0 00 sdot sdot sdot 01 Lee [143] showed that119876119899has an efficient dominating set if and only if 119899 = 2119898 minus 1

for a positive integer119898 ByTheorem 194 the following resultis obtained immediately

Theorem 198 (Huang and Xu [36] 2008) If 119899 = 2119898 minus 1 for apositive integer119898 then 2119898minus1

⩽ 119887(119876119899) ⩽ 2

119898

minus 1

Problem 10 Determine the exact value of 119887(119876119899) for 119899 = 2119898minus1

The 119899-dimensional cube-connected cycle denoted byCCC(119899) is constructed from the 119899-dimensional hypercube119876119899by replacing each vertex 119909 in 119876

119899with an undirected cycle

119862119899of length 119899 and linking the 119894th vertex of the 119862

119899to the 119894th

neighbor of 119909 It has been proved that CCC(119899) is a Cayleygraph and hence is a vertex-transitive graph with degree 3Van Wieren et al [147] proved that CCC(119899) has an efficientdominating set if and only if 119899 = 5 From Theorems 194 and195 the following result can be derived

Theorem 199 (Huang and Xu [36] 2008) Let 119866 = 119862119862119862(119899)be the 119899-dimensional cube-connected cycles with 119899 ⩾ 3 and119899 = 5 Then 120574(119866) = 1198992119899minus2 and 2 ⩽ 119887(119866) ⩽ 3 In addition119887(119862119862119862(3)) = 119887(119862119862119862(4)) = 2

Problem 11 Determine the exact value of 119887(CCC(119899)) for 119899 ⩾5

Acknowledgments

This work was supported by NNSF of China (Grants nos11071233 61272008) The author would like to express hisgratitude to the anonymous referees for their kind sugges-tions and comments on the original paper to ProfessorSheikholeslami and Professor Jafari Rad for sending thereferences [128] and [81] which resulted in this version

References

[1] J-M Xu Theory and Application of Graphs vol 10 of NetworkTheory and Applications Kluwer Academic Dordrecht TheNetherlands 2003

[2] S THedetniemi andR C Laskar ldquoBibliography on dominationin graphs and some basic definitions of domination parame-tersrdquo Discrete Mathematics vol 86 no 1ndash3 pp 257ndash277 1990

[3] TW Haynes S T Hedetniemi and P J Slater Fundamentals ofDomination in Graphs vol 208 ofMonographs and Textbooks inPure and Applied Mathematics Marcel Dekker New York NYUSA 1998

[4] TWHaynes S THedetniemi and P J Slater EdsDominationin Graphs vol 209 of Monographs and Textbooks in Pure andAppliedMathematicsMarcelDekker NewYorkNYUSA 1998

[5] M R Garey andD S JohnsonComputers and Intractability WH Freeman and Company San Francisco Calif USA 1979

[6] H B Walikar and B D Acharya ldquoDomination critical graphsrdquoNational Academy Science Letters vol 2 pp 70ndash72 1979

[7] D Bauer F Harary J Nieminen and C L Suffel ldquoDominationalteration sets in graphsrdquo Discrete Mathematics vol 47 no 2-3pp 153ndash161 1983

[8] J F Fink M S Jacobson L F Kinch and J Roberts ldquoThebondage number of a graphrdquo Discrete Mathematics vol 86 no1ndash3 pp 47ndash57 1990

[9] F-T Hu and J-M Xu ldquoThe bondage number of (119899 minus 3)-regulargraph G of order nrdquo Ars Combinatoria to appear

[10] U Teschner ldquoNew results about the bondage number of agraphrdquoDiscreteMathematics vol 171 no 1ndash3 pp 249ndash259 1997

[11] B L Hartnell and D F Rall ldquoA bound on the size of a graphwith given order and bondage numberrdquo Discrete Mathematicsvol 197-198 pp 409ndash413 1999 16th British CombinatorialConference (London 1997)

[12] B L Hartnell and D F Rall ldquoA characterization of trees inwhich no edge is essential to the domination numberrdquo ArsCombinatoria vol 33 pp 65ndash76 1992

[13] J Topp and P D Vestergaard ldquo120572119896- and 120574

119896-stable graphsrdquo

Discrete Mathematics vol 212 no 1-2 pp 149ndash160 2000[14] A Meir and J W Moon ldquoRelations between packing and

covering numbers of a treerdquo Pacific Journal of Mathematics vol61 no 1 pp 225ndash233 1975

[15] B L Hartnell L K Joslashrgensen P D Vestergaard and CA Whitehead ldquoEdge stability of the k-domination numberof treesrdquo Bulletin of the Institute of Combinatorics and ItsApplications vol 22 pp 31ndash40 1998

[16] F-T Hu and J-M Xu ldquoOn the complexity of the bondage andreinforcement problemsrdquo Journal of Complexity vol 28 no 2pp 192ndash201 2012

[17] B L Hartnell and D F Rall ldquoBounds on the bondage numberof a graphrdquo Discrete Mathematics vol 128 no 1ndash3 pp 173ndash1771994

[18] Y-L Wang ldquoOn the bondage number of a graphrdquo DiscreteMathematics vol 159 no 1ndash3 pp 291ndash294 1996

[19] G H Fricke TW Haynes S M Hedetniemi S T Hedetniemiand R C Laskar ldquoExcellent treesrdquo Bulletin of the Institute ofCombinatorics and Its Applications vol 34 pp 27ndash38 2002

[20] V Samodivkin ldquoThe bondage number of graphs good and badverticesrdquoDiscussiones Mathematicae GraphTheory vol 28 no3 pp 453ndash462 2008

[21] J E Dunbar T W Haynes U Teschner and L VolkmannldquoBondage insensitivity and reinforcementrdquo in Domination inGraphs vol 209 of Monographs and Textbooks in Pure andAppliedMathematics pp 471ndash489 Dekker NewYork NY USA1998

[22] H Liu and L Sun ldquoThe bondage and connectivity of a graphrdquoDiscrete Mathematics vol 263 no 1ndash3 pp 289ndash293 2003

[23] A-H Esfahanian and S L Hakimi ldquoOn computing a con-ditional edge-connectivity of a graphrdquo Information ProcessingLetters vol 27 no 4 pp 195ndash199 1988

[24] R C Brigham P Z Chinn and R D Dutton ldquoVertexdomination-critical graphsrdquoNetworks vol 18 no 3 pp 173ndash1791988

[25] V Samodivkin ldquoDomination with respect to nondegenerateproperties bondage numberrdquoThe Australasian Journal of Com-binatorics vol 45 pp 217ndash226 2009

[26] U Teschner ldquoA new upper bound for the bondage numberof graphs with small domination numberrdquo The AustralasianJournal of Combinatorics vol 12 pp 27ndash35 1995

32 International Journal of Combinatorics

[27] J Fulman D Hanson and G MacGillivray ldquoVertex dom-ination-critical graphsrdquoNetworks vol 25 no 2 pp 41ndash43 1995

[28] U Teschner ldquoA counterexample to a conjecture on the bondagenumber of a graphrdquo Discrete Mathematics vol 122 no 1ndash3 pp393ndash395 1993

[29] U Teschner ldquoThe bondage number of a graph G can be muchgreater than Δ(G)rdquo Ars Combinatoria vol 43 pp 81ndash87 1996

[30] L Volkmann ldquoOn graphs with equal domination and coveringnumbersrdquoDiscrete AppliedMathematics vol 51 no 1-2 pp 211ndash217 1994

[31] L A Sanchis ldquoMaximum number of edges in connected graphswith a given domination numberrdquoDiscreteMathematics vol 87no 1 pp 65ndash72 1991

[32] S Klavzar and N Seifter ldquoDominating Cartesian products ofcyclesrdquoDiscrete AppliedMathematics vol 59 no 2 pp 129ndash1361995

[33] H K Kim ldquoA note on Vizingrsquos conjecture about the bondagenumberrdquo Korean Journal of Mathematics vol 10 no 2 pp 39ndash42 2003

[34] M Y Sohn X Yuan and H S Jeong ldquoThe bondage number of1198623times119862

119899rdquo Journal of the KoreanMathematical Society vol 44 no

6 pp 1213ndash1231 2007[35] L Kang M Y Sohn and H K Kim ldquoBondage number of the

discrete torus 119862119899times 119862

4rdquo Discrete Mathematics vol 303 no 1ndash3

pp 80ndash86 2005[36] J Huang and J-M Xu ldquoThe bondage numbers and efficient

dominations of vertex-transitive graphsrdquoDiscrete Mathematicsvol 308 no 4 pp 571ndash582 2008

[37] C J Xiang X Yuan andM Y Sohn ldquoDomination and bondagenumber of119862

3times119862

119899rdquoArs Combinatoria vol 97 pp 299ndash310 2010

[38] M S Jacobson and L F Kinch ldquoOn the domination numberof products of graphs Irdquo Ars Combinatoria vol 18 pp 33ndash441984

[39] Y-Q Li ldquoA note on the bondage number of a graphrdquo ChineseQuarterly Journal of Mathematics vol 9 no 4 pp 1ndash3 1994

[40] F-T Hu and J-M Xu ldquoBondage number of mesh networksrdquoFrontiers ofMathematics inChina vol 7 no 5 pp 813ndash826 2012

[41] R Frucht and F Harary ldquoOn the corona of two graphsrdquoAequationes Mathematicae vol 4 pp 322ndash325 1970

[42] K Carlson and M Develin ldquoOn the bondage number of planarand directed graphsrdquoDiscreteMathematics vol 306 no 8-9 pp820ndash826 2006

[43] U Teschner ldquoOn the bondage number of block graphsrdquo ArsCombinatoria vol 46 pp 25ndash32 1997

[44] J Huang and J-M Xu ldquoNote on conjectures of bondagenumbers of planar graphsrdquo Applied Mathematical Sciences vol6 no 65ndash68 pp 3277ndash3287 2012

[45] L Kang and J Yuan ldquoBondage number of planar graphsrdquoDiscrete Mathematics vol 222 no 1ndash3 pp 191ndash198 2000

[46] M Fischermann D Rautenbach and L Volkmann ldquoRemarkson the bondage number of planar graphsrdquo Discrete Mathemat-ics vol 260 no 1ndash3 pp 57ndash67 2003

[47] J Hou G Liu and J Wu ldquoBondage number of planar graphswithout small cyclesrdquoUtilitasMathematica vol 84 pp 189ndash1992011

[48] J Huang and J-M Xu ldquoThe bondage numbers of graphs withsmall crossing numbersrdquo Discrete Mathematics vol 307 no 15pp 1881ndash1897 2007

[49] Q Ma S Zhang and J Wang ldquoBondage number of 1-planargraphrdquo Applied Mathematics vol 1 no 2 pp 101ndash103 2010

[50] J Hou and G Liu ldquoA bound on the bondage number of toroidalgraphsrdquoDiscreteMathematics Algorithms and Applications vol4 no 3 Article ID 1250046 5 pages 2012

[51] Y-C Cao J-M Xu and X-R Xu ldquoOn bondage number oftoroidal graphsrdquo Journal of University of Science and Technologyof China vol 39 no 3 pp 225ndash228 2009

[52] Y-C Cao J Huang and J-M Xu ldquoThe bondage number ofgraphs with crossing number less than fourrdquo Ars Combinatoriato appear

[53] H K Kim ldquoOn the bondage numbers of some graphsrdquo KoreanJournal of Mathematics vol 11 no 2 pp 11ndash15 2004

[54] A Gagarin and V Zverovich ldquoUpper bounds for the bondagenumber of graphs on topological surfacesrdquo Discrete Mathemat-ics 2012

[55] J Huang ldquoAn improved upper bound for the bondage numberof graphs on surfacesrdquoDiscrete Mathematics vol 312 no 18 pp2776ndash2781 2012

[56] A Gagarin and V Zverovich ldquoThe bondage number of graphson topological surfaces and Teschnerrsquos conjecturerdquo DiscreteMathematics vol 313 no 6 pp 796ndash808 2013

[57] V Samodivkin ldquoNote on the bondage number of graphs ontopological surfacesrdquo httparxivorgabs12086203

[58] E J Cockayne R M Dawes and S T Hedetniemi ldquoTotaldomination in graphsrdquoNetworks vol 10 no 3 pp 211ndash219 1980

[59] R Laskar J Pfaff S M Hedetniemi and S T HedetniemildquoOn the algorithmic complexity of total dominationrdquo Society forIndustrial and Applied Mathematics vol 5 no 3 pp 420ndash4251984

[60] J Pfaff R C Laskar and S T Hedetniemi ldquoNP-completeness oftotal and connected domination and irredundance for bipartitegraphsrdquo Tech Rep 428 Department of Mathematical SciencesClemson University 1983

[61] M A Henning ldquoA survey of selected recent results on totaldomination in graphsrdquoDiscrete Mathematics vol 309 no 1 pp32ndash63 2009

[62] V R Kulli and D K Patwari ldquoThe total bondage number ofa graphrdquo in Advances in Graph Theory pp 227ndash235 VishwaGulbarga India 1991

[63] F-T Hu Y Lu and J-M Xu ldquoThe total bondage number ofgrid graphsrdquo Discrete Applied Mathematics vol 160 no 16-17pp 2408ndash2418 2012

[64] J Cao M Shi and B Wu ldquoResearch on the total bondagenumber of a spectial networkrdquo in Proceedings of the 3rdInternational Joint Conference on Computational Sciences andOptimization (CSO rsquo10) pp 456ndash458 May 2010

[65] N Sridharan MD Elias and V S A Subramanian ldquoTotalbondage number of a graphrdquo AKCE International Journal ofGraphs and Combinatorics vol 4 no 2 pp 203ndash209 2007

[66] N Jafari Rad and J Raczek ldquoSome progress on total bondage ingraphsrdquo Graphs and Combinatorics 2011

[67] T W Haynes and P J Slater ldquoPaired-domination and thepaired-domatic numberrdquoCongressusNumerantium vol 109 pp65ndash72 1995

[68] T W Haynes and P J Slater ldquoPaired-domination in graphsrdquoNetworks vol 32 no 3 pp 199ndash206 1998

[69] X Hou ldquoA characterization of 2120574 120574119875-treesrdquoDiscrete Mathemat-

ics vol 308 no 15 pp 3420ndash3426 2008[70] K E Proffitt T W Haynes and P J Slater ldquoPaired-domination

in grid graphsrdquo Congressus Numerantium vol 150 pp 161ndash1722001

International Journal of Combinatorics 33

[71] H Qiao L KangM Cardei andD-Z Du ldquoPaired-dominationof treesrdquo Journal of Global Optimization vol 25 no 1 pp 43ndash542003

[72] E Shan L Kang and M A Henning ldquoA characterizationof trees with equal total domination and paired-dominationnumbersrdquo The Australasian Journal of Combinatorics vol 30pp 31ndash39 2004

[73] J Raczek ldquoPaired bondage in treesrdquo Discrete Mathematics vol308 no 23 pp 5570ndash5575 2008

[74] J A Telle and A Proskurowski ldquoAlgorithms for vertex parti-tioning problems on partial k-treesrdquo SIAM Journal on DiscreteMathematics vol 10 no 4 pp 529ndash550 1997

[75] G S Domke J H Hattingh M A Henning and L R MarkusldquoRestrained domination in treesrdquoDiscreteMathematics vol 211no 1ndash3 pp 1ndash9 2000

[76] G S Domke J H Hattingh M A Henning and L R MarkusldquoRestrained domination in graphs with minimum degree twordquoJournal of Combinatorial Mathematics and Combinatorial Com-puting vol 35 pp 239ndash254 2000

[77] G S Domke J H Hattingh S T Hedetniemi R C Laskarand L R Markus ldquoRestrained domination in graphsrdquo DiscreteMathematics vol 203 no 1ndash3 pp 61ndash69 1999

[78] J H Hattingh and E J Joubert ldquoAn upper bound for therestrained domination number of a graph with minimumdegree at least two in terms of order and minimum degreerdquoDiscrete Applied Mathematics vol 157 no 13 pp 2846ndash28582009

[79] M A Henning ldquoGraphs with large restrained dominationnumberrdquo Discrete Mathematics vol 197-198 pp 415ndash429 199916th British Combinatorial Conference (London 1997)

[80] J H Hattingh and A R Plummer ldquoRestrained bondage ingraphsrdquo Discrete Mathematics vol 308 no 23 pp 5446ndash54532008

[81] N Jafari Rad R Hasni J Raczek and L Volkmann ldquoTotalrestrained bondage in graphsrdquoActaMathematica Sinica vol 29no 6 pp 1033ndash1042 2013

[82] J Cyman and J Raczek ldquoOn the total restrained dominationnumber of a graphrdquoThe Australasian Journal of Combinatoricsvol 36 pp 91ndash100 2006

[83] M A Henning ldquoGraphs with large total domination numberrdquoJournal of Graph Theory vol 35 no 1 pp 21ndash45 2000

[84] D-X Ma X-G Chen and L Sun ldquoOn total restraineddomination in graphsrdquoCzechoslovakMathematical Journal vol55 no 1 pp 165ndash173 2005

[85] B Zelinka ldquoRemarks on restrained domination and totalrestrained domination in graphsrdquo Czechoslovak MathematicalJournal vol 55 no 2 pp 393ndash396 2005

[86] W Goddard and M A Henning ldquoIndependent domination ingraphs a survey and recent resultsrdquo Discrete Mathematics vol313 no 7 pp 839ndash854 2013

[87] J H Zhang H L Liu and L Sun ldquoIndependence bondagenumber and reinforcement number of some graphsrdquo Transac-tions of Beijing Institute of Technology vol 23 no 2 pp 140ndash1422003

[88] K D Dharmalingam Studies in Graph Theorymdashequitabledomination and bottleneck domination [PhD thesis] MaduraiKamaraj University Madurai India 2006

[89] GDeepakND Soner andAAlwardi ldquoThe equitable bondagenumber of a graphrdquo Research Journal of Pure Algebra vol 1 no9 pp 209ndash212 2011

[90] E Sampathkumar and H B Walikar ldquoThe connected domina-tion number of a graphrdquo Journal of Mathematical and PhysicalSciences vol 13 no 6 pp 607ndash613 1979

[91] K White M Farber and W Pulleyblank ldquoSteiner trees con-nected domination and strongly chordal graphsrdquoNetworks vol15 no 1 pp 109ndash124 1985

[92] M Cadei M X Cheng X Cheng and D-Z Du ldquoConnecteddomination in Ad Hoc wireless networksrdquo in Proceedings ofthe 6th International Conference on Computer Science andInformatics (CSampI rsquo02) 2002

[93] J F Fink and M S Jacobson ldquon-domination in graphsrdquo inGraph Theory with Applications to Algorithms and ComputerScience Wiley-Interscience Publications pp 283ndash300 WileyNew York NY USA 1985

[94] M Chellali O Favaron A Hansberg and L Volkmann ldquok-domination and k-independence in graphs a surveyrdquo Graphsand Combinatorics vol 28 no 1 pp 1ndash55 2012

[95] Y Lu X Hou J-M Xu andN Li ldquoTrees with uniqueminimump-dominating setsrdquo Utilitas Mathematica vol 86 pp 193ndash2052011

[96] Y Lu and J-M Xu ldquoThe p-bondage number of treesrdquo Graphsand Combinatorics vol 27 no 1 pp 129ndash141 2011

[97] Y Lu and J-M Xu ldquoThe 2-domination and 2-bondage numbersof grid graphs A manuscriptrdquo httparxivorgabs12044514

[98] M Krzywkowski ldquoNon-isolating 2-bondage in graphsrdquo TheJournal of the Mathematical Society of Japan vol 64 no 4 2012

[99] M A Henning O R Oellermann and H C Swart ldquoBoundson distance domination parametersrdquo Journal of CombinatoricsInformation amp System Sciences vol 16 no 1 pp 11ndash18 1991

[100] F Tian and J-M Xu ldquoBounds for distance domination numbersof graphsrdquo Journal of University of Science and Technology ofChina vol 34 no 5 pp 529ndash534 2004

[101] F Tian and J-M Xu ldquoDistance domination-critical graphsrdquoApplied Mathematics Letters vol 21 no 4 pp 416ndash420 2008

[102] F Tian and J-M Xu ldquoA note on distance domination numbersof graphsrdquo The Australasian Journal of Combinatorics vol 43pp 181ndash190 2009

[103] F Tian and J-M Xu ldquoAverage distances and distance domina-tion numbersrdquo Discrete Applied Mathematics vol 157 no 5 pp1113ndash1127 2009

[104] Z-L Liu F Tian and J-M Xu ldquoProbabilistic analysis of upperbounds for 2-connected distance k-dominating sets in graphsrdquoTheoretical Computer Science vol 410 no 38ndash40 pp 3804ndash3813 2009

[105] M A Henning O R Oellermann and H C Swart ldquoDistancedomination critical graphsrdquo Journal of Combinatorial Mathe-matics and Combinatorial Computing vol 44 pp 33ndash45 2003

[106] T J Bean M A Henning and H C Swart ldquoOn the integrityof distance domination in graphsrdquo The Australasian Journal ofCombinatorics vol 10 pp 29ndash43 1994

[107] M Fischermann and L Volkmann ldquoA remark on a conjecturefor the (kp)-domination numberrdquoUtilitasMathematica vol 67pp 223ndash227 2005

[108] T Korneffel D Meierling and L Volkmann ldquoA remark on the(22)-domination numberrdquo Discussiones Mathematicae GraphTheory vol 28 no 2 pp 361ndash366 2008

[109] Y Lu X Hou and J-M Xu ldquoOn the (22)-domination numberof treesrdquo Discussiones Mathematicae Graph Theory vol 30 no2 pp 185ndash199 2010

34 International Journal of Combinatorics

[110] H Li and J-M Xu ldquo(dm)-domination numbers of m-connected graphsrdquo Rapports de Recherche 1130 LRI UniversiteParis-Sud 1997

[111] S M Hedetniemi S T Hedetniemi and T V Wimer ldquoLineartime resource allocation algorithms for treesrdquo Tech Rep URI-014 Department of Mathematical Sciences Clemson Univer-sity 1987

[112] M Farber ldquoDomination independent domination and dualityin strongly chordal graphsrdquo Discrete Applied Mathematics vol7 no 2 pp 115ndash130 1984

[113] G S Domke and R C Laskar ldquoThe bondage and reinforcementnumbers of 120574

119891for some graphsrdquoDiscrete Mathematics vol 167-

168 pp 249ndash259 1997[114] G S Domke S T Hedetniemi and R C Laskar ldquoFractional

packings coverings and irredundance in graphsrdquo vol 66 pp227ndash238 1988

[115] E J Cockayne P A Dreyer Jr S M Hedetniemi andS T Hedetniemi ldquoRoman domination in graphsrdquo DiscreteMathematics vol 278 no 1ndash3 pp 11ndash22 2004

[116] I Stewart ldquoDefend the roman empirerdquo Scientific American vol281 pp 136ndash139 1999

[117] C S ReVelle and K E Rosing ldquoDefendens imperiumromanum a classical problem in military strategyrdquoThe Ameri-can Mathematical Monthly vol 107 no 7 pp 585ndash594 2000

[118] A Bahremandpour F-T Hu S M Sheikholeslami and J-MXu ldquoOn the Roman bondage number of a graphrdquo DiscreteMathematics Algorithms and Applications vol 5 no 1 ArticleID 1350001 15 pages 2013

[119] N Jafari Rad and L Volkmann ldquoRoman bondage in graphsrdquoDiscussiones Mathematicae Graph Theory vol 31 no 4 pp763ndash773 2011

[120] K Ebadi and L PushpaLatha ldquoRoman bondage and Romanreinforcement numbers of a graphrdquo International Journal ofContemporary Mathematical Sciences vol 5 pp 1487ndash14972010

[121] N Dehgardi S M Sheikholeslami and L Volkman ldquoOn theRoman k-bondage number of a graphrdquo AKCE InternationalJournal of Graphs and Combinatorics vol 8 pp 169ndash180 2011

[122] S Akbari and S Qajar ldquoA note on Roman bondage number ofgraphsrdquo Ars Combinatoria to Appear

[123] F-T Hu and J-M Xu ldquoRoman bondage numbers of somegraphsrdquo httparxivorgabs11093933

[124] N Jafari Rad and L Volkmann ldquoOn the Roman bondagenumber of planar graphsrdquo Graphs and Combinatorics vol 27no 4 pp 531ndash538 2011

[125] S Akbari M Khatirinejad and S Qajar ldquoA note on the romanbondage number of planar graphsrdquo Graphs and Combinatorics2012

[126] B Bresar M A Henning and D F Rall ldquoRainbow dominationin graphsrdquo Taiwanese Journal of Mathematics vol 12 no 1 pp213ndash225 2008

[127] B L Hartnell and D F Rall ldquoOn dominating the Cartesianproduct of a graph and 120572krdquo Discussiones Mathematicae GraphTheory vol 24 no 3 pp 389ndash402 2004

[128] N Dehgardi S M Sheikholeslami and L Volkman ldquoThe k-rainbow bondage number of a graphrdquo to appear in DiscreteApplied Mathematics

[129] J von Neumann and O Morgenstern Theory of Games andEconomic Behavior Princeton University Press Princeton NJUSA 1944

[130] C BergeTheorıe des Graphes et ses Applications 1958[131] O Ore Theory of Graphs vol 38 American Mathematical

Society Colloquium Publications 1962[132] E Shan and L Kang ldquoBondage number in oriented graphsrdquoArs

Combinatoria vol 84 pp 319ndash331 2007[133] J Huang and J-M Xu ldquoThe bondage numbers of extended

de Bruijn and Kautz digraphsrdquo Computers amp Mathematics withApplications vol 51 no 6-7 pp 1137ndash1147 2006

[134] J Huang and J-M Xu ldquoThe total domination and total bondagenumbers of extended de Bruijn and Kautz digraphsrdquoComputersamp Mathematics with Applications vol 53 no 8 pp 1206ndash12132007

[135] Y Shibata and Y Gonda ldquoExtension of de Bruijn graph andKautz graphrdquo Computers amp Mathematics with Applications vol30 no 9 pp 51ndash61 1995

[136] X Zhang J Liu and JMeng ldquoThebondage number in completet-partite digraphsrdquo Information Processing Letters vol 109 no17 pp 997ndash1000 2009

[137] D W Bange A E Barkauskas and P J Slater ldquoEfficient dom-inating sets in graphsrdquo in Applications of Discrete Mathematicspp 189ndash199 SIAM Philadelphia Pa USA 1988

[138] M Livingston and Q F Stout ldquoPerfect dominating setsrdquoCongressus Numerantium vol 79 pp 187ndash203 1990

[139] L Clark ldquoPerfect domination in random graphsrdquo Journal ofCombinatorial Mathematics and Combinatorial Computing vol14 pp 173ndash182 1993

[140] D W Bange A E Barkauskas L H Host and L H ClarkldquoEfficient domination of the orientations of a graphrdquo DiscreteMathematics vol 178 no 1ndash3 pp 1ndash14 1998

[141] A E Barkauskas and L H Host ldquoFinding efficient dominatingsets in oriented graphsrdquo Congressus Numerantium vol 98 pp27ndash32 1993

[142] I J Dejter and O Serra ldquoEfficient dominating sets in CayleygraphsrdquoDiscrete AppliedMathematics vol 129 no 2-3 pp 319ndash328 2003

[143] J Lee ldquoIndependent perfect domination sets in Cayley graphsrdquoJournal of Graph Theory vol 37 no 4 pp 213ndash219 2001

[144] WGu X Jia and J Shen ldquoIndependent perfect domination setsin meshes tori and treesrdquo Unpublished

[145] N Obradovic J Peters and G Ruzic ldquoEfficient domination incirculant graphswith two chord lengthsrdquo Information ProcessingLetters vol 102 no 6 pp 253ndash258 2007

[146] K Reji Kumar and G MacGillivray ldquoEfficient domination incirculant graphsrdquo Discrete Mathematics vol 313 no 6 pp 767ndash771 2013

[147] D Van Wieren M Livingston and Q F Stout ldquoPerfectdominating sets on cube-connected cyclesrdquo Congressus Numer-antium vol 97 pp 51ndash70 1993

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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International Journal of Combinatorics 27

We conclude this section with the following problems

Problem 8 Characterize all connected graphs 119866 of order 119899 ⩾3 for which 119887RM(119866) = 119899 minus 1

Problem 9 Prove or disprove if 119866 is a connected graph oforder 119899 ⩾ 3 then

119887RM (119866) ⩽ 119899 minus 120574RM (119866) + 3 (71)

Note that 120574119877(119870

119905(3)) = 4 and 119887RM(119870119905

(3)) = 119899minus1 where 119899 =3119905 for 119905 ⩾ 3 The upper bound on 119887RM(119866) given in Problem 9is sharp if Problem 9 has positive answer

95 Rainbow Bondage Numbers For a positive integer 119896 a119896-rainbow dominating function of a graph 119866 is a function 119891from the vertex-set 119881(119866) to the set of all subsets of the set1 2 119896 such that for any vertex 119909 in 119866 with 119891(119909) = 0 thecondition

⋃

119910isin119873119866(119909)

119891 (119910) = 1 2 119896 (72)

is fulfilledThe weight of a 119896-rainbow dominating function 119891on 119866 is the value

119891 (119866) = sum

119909isin119881(119866)

1003816100381610038161003816119891 (119909)1003816100381610038161003816 (73)

The 119896-rainbow domination number of a graph 119866 denoted by120574119903119896(119866) is the minimum weight of a 119896-rainbow dominating

function 119891 of 119866Note that 120574

1199031(119866) is the classical domination number 120574(119866)

The 119896-rainbow domination number is introduced by Bresaret al [126] Rainbow domination of a graph 119866 coincides withordinary domination of the Cartesian product of 119866 with thecomplete graph in particular 120574

119903119896(119866) = 120574(119866 times 119870

119896) for any

positive integer 119896 and any graph 119866 [126] Hartnell and Rall[127] proved that 120574(119879) lt 120574

1199032(119879) for any tree 119879 no graph has

120574 = 120574119903119896for 119896 ⩾ 3 for any 119896 ⩾ 2 and any graph 119866 of order 119899

min 119899 120574 (119866) + 119896 minus 2 ⩽ 120574119903119896(119866) ⩽ 119896120574 (119866) (74)

The introduction of rainbow domination is motivatedby the study of paired domination in Cartesian productsof graphs where certain upper bounds can be expressed interms of rainbow domination that is for any graph 119866 andany graph 119867 if 119881(119867) can be partitioned into 119896120574

119875-sets then

120574119875(119866 times 119867) ⩽ (1119896)120592(119867)120574

119903119896(119866) proved by Bresar et al [126]

Dehgardi et al [128] introduced the concept of 119896-rainbowbondage number of graphs Let 119866 be a graph with maximumdegree at least two The 119896-rainbow bondage number 119887

119903119896(119866)

of 119866 is the minimum cardinality of all sets 119861 sube 119864(119866) forwhich 120574

119903119896(119866 minus 119861) gt 120574

119903119896(119866) Since in the study of 119896-rainbow

bondage number the assumption Δ(119866) ⩾ 2 is necessarywe always assume that when we discuss 119887

119903119896(119866) all graphs

involved satisfy Δ(119866) ⩾ 2The 2-rainbow bondage number of some special families

of graphs has been determined

Theorem 172 (Dehgardi et al [128]) For any integer 119899 ⩾ 3

1198871199032(119875

119899) = 1 119891119900119903 119899 ⩾ 2

1198871199032(119862

119899) =

1 119894119891 119899 equiv 0 (mod 4) 2 119900119905ℎ119890119903119908119894119904119890

1198871199032(119870

119899119899) =

1 119894119891 119899 = 2

3 119894119891 119899 = 3

6 119894119891 119899 = 4

119898 119894119891 119899 ⩾ 5

(75)

Theorem 173 (Dehgardi et al [128]) For any connected graph119866 of order 119899 ⩾ 3 119887

1199032(119866) ⩽ 120582(119866) + 2Δ(119866) minus 3

Theorem 174 (Dehgardi et al [128]) For any tree 119879 of order119899 ⩾ 3 119887

1199032(119879) ⩽ 2

Theorem 175 (Dehgardi et al [128]) If 119866 is a planar graphwithmaximumdegree at least two then 119887

1199032(119866) ⩽ 15Moreover

if the girth 119892(119866) ⩾ 4 then 1198871199032(119866) ⩽ 11 and if 119892(119866) ⩾ 6 then

1198871199032(119866) ⩽ 9

Theorem 176 (Dehgardi et al [128]) For a complete bipartitegraph 119870

119901119902 if 2119896 + 1 ⩽ 119901 ⩽ 119902 then 119887

119903119896(119870

119901119902) = 119901

Theorem177 (Dehgardi et al [128]) For a complete graph119870119899

if 119899 ⩾ 119896 + 1 then 119887119903119896(119870

119899) = lceil119896119899(119896 + 1)rceil

The special case 119896 = 1 of Theorem 177 can be found inTheorem 1 The following is a simple observation similar toObservation 1

Observation 6 (Dehgardi et al [128]) Let 119866 be a graphwith maximum degree at least two 119867 a spanning subgraphobtained from 119866 by removing 119896 edges If 120574

119903119896(119867) = 120574

119903119896(119866)

then 119887119903119896(119867) ⩽ 119887

119903119896(119866) ⩽ 119887

119903119896(119867) + 119896

Theorem 178 (Dehgardi et al [128]) For every graph 119866 with120574119903119896(119866) = 119896120574(119866) 119887

119903119896(119866) ⩾ 119887(119866)

A graph 119866 is said to be vertex 119896-rainbow domination-critical if 120574

119903119896(119866 minus 119909) lt 120574

119903119896(119866) for every vertex 119909 in 119866 It is

clear that if 119866 has no isolated vertices then 120574119903119896(119866) ⩽ 119896120574(119866) ⩽

119896120573(119866) where 120573(119866) is the vertex-covering number of119866 Thusif 120574

119903119896(119866) = 119896120573(119866) then 120574

119903119896(119866) = 119896120574(119866)

Theorem 179 (Dehgardi et al [128]) For any graph 119866 if120574119903119896(119866) = 119896120573(119866) then

(a) 119887119903119896(119866) ⩾ 120575(119866)

(b) 119887119903119896(119866) ⩾ 120575(119866)+1 if119866 is vertex 119896-rainbow domination-

critical

As far as we know there are no more results on the 119896-rainbow bondage number of graphs in the literature

28 International Journal of Combinatorics

10 Results on Digraphs

Although domination has been extensively studied in undi-rected graphs it is natural to think of a dominating setas a one-way relationship between vertices of the graphIndeed among the earliest literatures on this subject vonNeumann and Morgenstern [129] used what is now calleddomination in digraphs to find solution (or kernels whichare independent dominating sets) for cooperative 119899-persongames Most likely the first formulation of domination byBerge [130] in 1958 was given in the context of digraphs andonly some years latter by Ore [131] in 1962 for undirectedgraphs Despite this history examination of domination andits variants in digraphs has been essentially overlooked (see[4] for an overview of the domination literature) Thus thereare few if any such results on domination for digraphs in theliterature

The bondage number and its related topics for undirectedgraph have become one of major areas both in theoreticaland applied researches However until recently Carlsonand Develin [42] Shan and Kang [132] and Huang andXu [133 134] studied the bondage number for digraphsindependently In this section we introduce their resultsfor general digraphs Results for some special digraphs suchas vertex-transitive digraphs are introduced in the nextsection

101 Upper Bounds for General Digraphs Let 119866 = (119881 119864)

be a digraph without loops and parallel edges A digraph 119866is called to be asymmetric if whenever (119909 119910) isin 119864(119866) then(119910 119909) notin 119864(119866) and to be symmetric if (119909 119910) isin 119864(119866) implies(119910 119909) isin 119864(119866) For a vertex 119909 of 119881(119866) the sets of out-neighbors and in-neighbors of 119909 are respectively defined as119873

+

119866(119909) = 119910 isin 119881(119866) (119909 119910) isin 119864(119866) and 119873minus

119866(119909) = 119909 isin

119881(119866) (119909 119910) isin 119864(119866) the out-degree and the in-degree of119909 are respectively defined as 119889+

119866(119909) = |119873

+

119866(119909)| and 119889minus

119866(119909) =

|119873+

119866(119909)| Denote themaximum and theminimum out-degree

(resp in-degree) of 119866 by Δ+(119866) and 120575+(119866) (resp Δminus(119866) and120575minus

(119866))The degree 119889119866(119909) of 119909 is defined as 119889+

119866(119909)+119889

minus

119866(119909) and

the maximum and the minimum degrees of119866 are denoted byΔ(119866) and 120575(119866) respectively that is Δ(119866) = max119889

119866(119909) 119909 isin

119881(119866) and 120575(119866) = min119889119866(119909) 119909 isin 119881(119866) Note that the

definitions here are different from the ones in the textbookon digraphs see for example Xu [1]

A subset 119878 of119881(119866) is called a dominating set if119881(119866) = 119878cup119873

+

119866(119878) where119873+

119866(119878) is the set of out-neighbors of 119878Then just

as for undirected graphs 120574(119866) is the minimum cardinalityof a dominating set and the bondage number 119887(119866) is thesmallest cardinality of a set 119861 of edges such that 120574(119866 minus 119861) gt120574(119866) if such a subset 119861 sube 119864(119866) exists Otherwise we put119887(119866) = infin

Some basic results for undirected graphs stated inSection 3 can be generalized to digraphs For exampleTheorem 18 is generalized by Carlson and Develin [42]Huang andXu [133] and Shan andKang [132] independentlyas follows

Theorem 180 Let 119866 be a digraph and (119909 119910) isin 119864(119866) Then119887(119866) ⩽ 119889

119866(119910) + 119889

minus

119866(119909) minus |119873

minus

119866(119909) cap 119873

minus

119866(119910)|

Corollary 181 For a digraph 119866 119887(119866) ⩽ 120575minus(119866) + Δ(119866)

Since 120575minus

(119866) ⩽ (12)Δ(119866) from Corollary 181Conjecture 53 is valid for digraphs

Corollary 182 For a digraph 119866 119887(119866) ⩽ (32)Δ(119866)

In the case of undirected graphs the bondage number of119866119899= 119870

119899times 119870

119899achieves this bound However it was shown

by Carlson and Develin in [42] that if we take the symmetricdigraph119866

119899 we haveΔ(119866

119899) = 4(119899minus1) 120574(119866

119899) = 119899 and 119887(119866

119899) ⩽

4(119899 minus 1) + 1 = Δ(119866119899) + 1 So this family of digraphs cannot

show that the bound in Corollary 182 is tightCorresponding to Conjecture 51 which is discredited

for undirected graphs and is valid for digraphs the sameconjecture for digraphs can be proposed as follows

Conjecture 183 (Carlson and Develin [42] 2006) 119887(119866) ⩽Δ(119866) + 1 for any digraph 119866

If Conjecture 183 is true then the following results showsthat this upper bound is tight since 119887(119870

119899times119870

119899) = Δ(119870

119899times119870

119899)+1

for a complete digraph 119870119899

In 2007 Shan and Kang [132] gave some tight upperbounds on the bondage numbers for some asymmetricdigraphs For example 119887(119879) ⩽ Δ(119879) for any asymmetricdirected tree 119879 119887(119866) ⩽ Δ(119866) for any asymmetric digraph 119866with order at least 4 and 120574(119866) ⩽ 2 For planar digraphs theyobtained the following results

Theorem 184 (Shan and Kang [132] 2007) Let 119866 be aasymmetric planar digraphThen 119887(119866) ⩽ Δ(119866)+2 and 119887(119866) ⩽Δ(119866) + 1 if Δ(119866) ⩾ 5 and 119889minus

119866(119909) ⩾ 3 for every vertex 119909 with

119889119866(119909) ⩾ 4

102 Results for Some Special Digraphs The exact values andbounds on 119887(119866) for some standard digraphs are determined

Theorem185 (Huang andXu [133] 2006) For a directed cycle119862119899and a directed path 119875

119899

119887 (119862119899) =

3 119894119891 119899 119894119904 119900119889119889

2 119894119891 119899 119894119904 119890V119890119899

119887 (119875119899) =

2 119894119891 119899 119894119904 119900119889119889

1 119894119891 119899 119894119904 119890V119890119899

(76)

For the de Bruijn digraph 119861(119889 119899) and the Kautz digraph119870(119889 119899)

119887 (119861 (119889 119899)) = 119889 119894119891 119899 119894119904 119900119889119889

119889 ⩽ 119887 (119861 (119889 119899)) ⩽ 2119889 119894119891 119899 119894119904 119890V119890119899

119887 (119870 (119889 119899)) = 119889 + 1

(77)

Like undirected graphs we can define the total domi-nation number and the total bondage number On the totalbondage numbers for some special digraphs the knownresults are as follows

International Journal of Combinatorics 29

Theorem 186 (Huang and Xu [134] 2007) For a directedcycle 119862

119899and a directed path 119875

119899 119887

119879(119875

119899) and 119887

119879(119862

119899) all do not

exist For a complete digraph 119870119899

119887119879(119870

119899) = infin 119894119891 119899 = 1 2

119887119879(119870

119899) = 3 119894119891 119899 = 3

119899 ⩽ 119887119879(119870

119899) ⩽ 2119899 minus 3 119894119891 119899 ⩾ 4

(78)

The extended de Bruijn digraph EB(119889 119899 1199021 119902

119901) and

the extended Kautz digraph EK(119889 119899 1199021 119902

119901) are intro-

duced by Shibata and Gonda [135] If 119901 = 1 then theyare the de Bruijn digraph B(119889 119899) and the Kautz digraphK(119889 119899) respectively Huang and Xu [134] determined theirtotal domination numbers In particular their total bondagenumbers for general cases are determined as follows

Theorem 187 (Huang andXu [134] 2007) If 119889 ⩾ 2 and 119902119894⩾ 2

for each 119894 = 1 2 119901 then

119887119879(EB(119889 119899 119902

1 119902

119901)) = 119889

119901

minus 1

119887119879(EK(119889 119899 119902

1 119902

119901)) = 119889

119901

(79)

In particular for the de Bruijn digraph B(119889 119899) and the Kautzdigraph K(119889 119899)

119887119879(B (119889 119899)) = 119889 minus 1 119887

119879(K (119889 119899)) = 119889 (80)

Zhang et al [136] determined the bondage number incomplete 119905-partite digraphs

Theorem 188 (Zhang et al [136] 2009) For a complete 119905-partite digraph 119870

11989911198992119899119905

where 1198991⩽ 119899

2⩽ sdot sdot sdot ⩽ 119899

119905

119887 (11987011989911198992119899119905

)

=

119898 119894119891 119899119898= 1 119899

119898+1⩾ 2 119891119900119903 119904119900119898119890

119898 (1 ⩽ 119898 lt 119905)

4119905 minus 3 119894119891 1198991= 119899

2= sdot sdot sdot = 119899

119905= 2

119905minus1

sum

119894=1

119899119894+ 2 (119905 minus 1) 119900119905ℎ119890119903119908119894119904119890

(81)

Since an undirected graph can be thought of as a sym-metric digraph any result for digraphs has an analogy forundirected graphs in general In view of this point studyingthe bondage number for digraphs is more significant thanfor undirected graphs Thus we should further study thebondage number of digraphs and try to generalize knownresults on the bondage number and related variants for undi-rected graphs to digraphs prove or disprove Conjecture 183In particular determine the exact values of 119887(B(119889 119899)) for aneven 119899 and 119887

119879(119870

119899) for 119899 ⩾ 4

11 Efficient Dominating Sets

A dominating set 119878 of a graph 119866 is called to be efficient if forevery vertex 119909 in119866 |119873

119866[119909]cap119878| = 1 if119866 is a undirected graph

or |119873minus

119866[119909] cap 119878| = 1 if 119866 is a directed graph The concept of an

efficient set was proposed by Bange et al [137] in 1988 In theliterature an efficient set is also called a perfect dominatingset (see Livingston and Stout [138] for an explanation)

From definition if 119878 is an efficient dominating set of agraph 119866 then 119878 is certainly an independent set and everyvertex not in 119878 is adjacent to exactly one vertex in 119878

It is also clear from definition that a dominating set 119878 isefficient if and only if N

119866[119878] = 119873

119866[119909] 119909 isin 119878 for the

undirected graph 119866 or N+

119866[119878] = 119873

+

119866[119909] 119909 isin 119878 for the

digraph119866 is a partition of119881(119866) where the induced subgraphby119873

119866[119909] or119873+

119866[119909] is a star or an out-star with the root 119909

The efficient domination has important applications inmany areas such as error-correcting codes and receivesmuch attention in the late years

The concept of efficient dominating sets is a measureof the efficiency of domination in graphs Unfortunately asshown in [137] not every graph has an efficient dominatingset andmoreover it is anNP-complete problem to determinewhether a given graph has an efficient dominating set Inaddition it was shown by Clark [139] in 1993 that for a widerange of 119901 almost every random undirected graph 119866 isin

G(120592 119901) has no efficient dominating sets This means thatundirected graphs possessing an efficient dominating set arerare However it is easy to show that every undirected graphhas an orientationwith an efficient dominating set (see Bangeet al [140])

In 1993 Barkauskas and Host [141] showed that deter-mining whether an arbitrary oriented graph has an efficientdominating set is NP-complete Even so the existence ofefficient dominating sets for some special classes of graphs hasbeen examined see for example Dejter and Serra [142] andLee [143] for Cayley graph Gu et al [144] formeshes and toriObradovic et al [145] Huang and Xu [36] Reji Kumar andMacGillivray [146] for circulant graphs Harary graphs andtori and Van Wieren et al [147] for cube-connected cycles

In this section we introduce results of the bondagenumber for some graphs with efficient dominating sets dueto Huang and Xu [36]

111 Results for General Graphs In this subsection we intro-duce some results on bondage numbers obtained by applyingefficient dominating sets We first state the two followinglemmas

Lemma 189 For any 119896-regular graph or digraph 119866 of order 119899120574(119866) ⩾ 119899(119896 + 1) with equality if and only if 119866 has an efficientdominating set In addition if 119866 has an efficient dominatingset then every efficient dominating set is certainly a 120574-set andvice versa

Let 119890 be an edge and 119878 a dominating set in 119866 We say that119890 supports 119878 if 119890 isin (119878 119878) where (119878 119878) = (119909 119910) isin 119864(119866) 119909 isin 119878 119910 isin 119878 Denote by 119904(119866) the minimum number of edgeswhich support all 120574-sets in 119866

Lemma 190 For any graph or digraph 119866 119887(119866) ⩾ 119904(119866) withequality if 119866 is regular and has an efficient dominating set

30 International Journal of Combinatorics

A graph 119866 is called to be vertex-transitive if its automor-phism group Aut(119866) acts transitively on its vertex-set 119881(119866)A vertex-transitive graph is regular Applying Lemmas 189and 190 Huang and Xu obtained some results on bondagenumbers for vertex-transitive graphs or digraphs

Theorem 191 (Huang and Xu [36] 2008) For any vertex-transitive graph or digraph 119866 of order 119899

119887 (119866) ⩾

lceil119899

2120574 (119866)rceil 119894119891 119866 119894119904 119906119899119889119894119903119890119888119905119890119889

lceil119899

120574 (119866)rceil 119894119891 119866 119894119904 119889119894119903119890119888119905119890119889

(82)

Theorem192 (Huang andXu [36] 2008) Let119866 be a 119896-regulargraph of order 119899 Then

119887(119866) ⩽ 119896 if119866 is undirected and 119899 ⩾ 120574(119866)(119896+1)minus119896+1119887(119866) ⩽ 119896+1+ℓ if119866 is directed and 119899 ⩾ 120574(119866)(119896+1)minusℓwith 0 ⩽ ℓ ⩽ 119896 minus 1

Next we introduce a better upper bound on 119887(119866) To thisaim we need the following concept which is a generalizationof the concept of the edge-covering of a graph 119866 For 1198811015840

sube

119881(119866) and 1198641015840 sube 119864(119866) we say that 1198641015840covers 1198811015840 and call 1198641015840an edge-covering for 1198811015840 if there exists an edge (119909 119910) isin 1198641015840 forany vertex 119909 isin 1198811015840 For 119910 isin 119881(119866) let 1205731015840[119910] be the minimumcardinality over all edge-coverings for119873minus

119866[119910]

Theorem 193 (Huang and Xu [36] 2008) If 119866 is a 119896-regulargraph with order 120574(119866)(119896 + 1) then 119887(119866) ⩽ 1205731015840[119910] for any 119910 isin119881(119866)

Theupper bound on 119887(119866) given inTheorem 193 is tight inview of 119887(119862

119899) for a cycle or a directed cycle 119862

119899(seeTheorems

1 and 185 resp)It is easy to see that for a 119896-regular graph 119866 lceil(119896 + 1)2rceil ⩽

1205731015840

[119910] ⩽ 119896 when 119866 is undirected and 1205731015840[119910] = 119896 + 1 when 119866 isdirected By this fact and Lemma 189 the following theoremis merely a simple combination of Theorems 191 and 193

Theorem 194 (Huang and Xu [36] 2008) Let 119866 be a vertex-transitive graph of degree 119896 If 119866 has an efficient dominatingset then

lceil119896 + 1

2rceil ⩽ 119887 (119866) ⩽ 119896 119894119891 119866 119894119904 119906119899119889119894119903119890119888119905119890119889

119887 (119866) = 119896 + 1 119894119891 119866 119894119904 119889119894119903119890119888119905119890119889

(83)

Theorem 195 (Huang and Xu [36] 2008) If 119866 is an undi-rected vertex-transitive cubic graph with order 4120574(119866) and girth119892(119866) ⩽ 5 then 119887(119866) = 2

The proof of Theorem 195 leads to a byproduct If 119892 =5 let (119906

1 119906

2 119906

3 119906

4 119906

5) be a cycle and let D

119894be the family

of efficient dominating sets containing 119906119894for each 119894 isin

1 2 3 4 5 ThenD119894capD

119895= 0 for 119894 = 1 2 5 Then 119866 has

at least 5119904 efficient dominating sets where 119904 = 119904(119866) But thereare only 4s (=ns120574) distinct efficient dominating sets in119866This

contradiction implies that an undirected vertex-transitivecubic graph with girth five has no efficient dominating setsBut a similar argument for 119892(119866) = 3 4 or 119892(119866) ⩾ 6 couldnot give any contradiction This is consistent with the resultthat CCC(119899) a vertex-transitive cubic graph with girth 119899 if3 ⩽ 119899 ⩽ 8 or girth 8 if 119899 ⩾ 9 has efficient dominating setsfor all 119899 ⩾ 3 except 119899 = 5 (see Theorem 199 in the followingsubsection)

112 Results for Cayley Graphs In this subsection we useTheorem 194 to determine the exact values or approximativevalues of bondage numbers for some special vertex-transitivegraphs by characterizing the existence of efficient dominatingsets in these graphs

Let Γ be a nontrivial finite group 119878 a nonempty subset ofΓ without the identity element of Γ A digraph 119866 is defined asfollows

119881 (119866) = Γ (119909 119910) isin 119864 (119866) lArrrArr 119909minus1

119910 isin 119878

for any 119909 119910 isin Γ(84)

is called a Cayley digraph of the group Γ with respect to119878 denoted by 119862

Γ(119878) If 119878minus1 = 119904minus1 119904 isin 119878 = 119878 then

119862Γ(119878) is symmetric and is called a Cayley undirected graph

a Cayley graph for short Cayley graphs or digraphs arecertainly vertex-transitive (see Xu [1])

A circulant graph 119866(119899 119878) of order 119899 is a Cayley graph119862119885119899

(119878) where 119885119899= 0 119899 minus 1 is the addition group of

order 119899 and 119878 is a nonempty subset of119885119899without the identity

element and hence is a vertex-transitive digraph of degree|119878| If 119878minus1 = 119878 then119866(119899 119878) is an undirected graph If 119878 = 1 119896where 2 ⩽ 119896 ⩽ 119899 minus 2 we write 119866(119899 1 119896) for 119866(119899 1 119896) or119866(119899 plusmn1 plusmn119896) and call it a double-loop circulant graph

Huang and Xu [36] showed that for directed 119866 =

119866(119899 1 119896) lceil1198993rceil ⩽ 120574(119866) ⩽ lceil1198992rceil and 119866 has an efficientdominating set if and only if 3 | 119899 and 119896 equiv 2 (mod 3) fordirected 119866 = 119866(119899 1 119896) and 119896 = 1198992 lceil1198995rceil ⩽ 120574(119866) ⩽ lceil1198993rceiland 119866 has an efficient dominating set if and only if 5 | 119899 and119896 equiv plusmn2 (mod 5) By Theorem 194 we obtain the bondagenumber of a double loop circulant graph if it has an efficientdominating set

Theorem 196 (Huang and Xu [36] 2008) Let 119866 be a double-loop circulant graph 119866(119899 1 119896) If 119866 is directed with 3 | 119899and 119896 equiv 2 (mod 3) or 119866 is undirected with 5 | 119899 and119896 equiv plusmn2 (mod 5) then 119887(119866) = 3

The 119898 times 119899 torus is the Cartesian product 119862119898times 119862

119899of

two cycles and is a Cayley graph 119862119885119898times119885119899

(119878) where 119878 =

(0 1) (1 0) for directed cycles and 119878 = (0 plusmn1) (plusmn1 0) forundirected cycles and hence is vertex-transitive Gu et al[144] showed that the undirected torus119862

119898times119862

119899has an efficient

dominating set if and only if both 119898 and 119899 are multiplesof 5 Huang and Xu [36] showed that the directed torus119862119898times 119862

119899has an efficient dominating set if and only if both

119898 and 119899 are multiples of 3 Moreover they found a necessarycondition for a dominating set containing the vertex (0 0)in 119862

119898times 119862

119899to be efficient and obtained the following

result

International Journal of Combinatorics 31

Theorem 197 (Huang and Xu [36] 2008) Let 119866 = 119862119898times 119862

119899

If 119866 is undirected and both119898 and 119899 are multiples of 5 or if 119866 isdirected and both119898 and 119899 are multiples of 3 then 119887(119866) = 3

The hypercube 119876119899

is the Cayley graph 119862Γ(119878)

where Γ = 1198852times sdot sdot sdot times 119885

2= (119885

2)119899 and 119878 =

100 sdot sdot sdot 0 010 sdot sdot sdot 0 00 sdot sdot sdot 01 Lee [143] showed that119876119899has an efficient dominating set if and only if 119899 = 2119898 minus 1

for a positive integer119898 ByTheorem 194 the following resultis obtained immediately

Theorem 198 (Huang and Xu [36] 2008) If 119899 = 2119898 minus 1 for apositive integer119898 then 2119898minus1

⩽ 119887(119876119899) ⩽ 2

119898

minus 1

Problem 10 Determine the exact value of 119887(119876119899) for 119899 = 2119898minus1

The 119899-dimensional cube-connected cycle denoted byCCC(119899) is constructed from the 119899-dimensional hypercube119876119899by replacing each vertex 119909 in 119876

119899with an undirected cycle

119862119899of length 119899 and linking the 119894th vertex of the 119862

119899to the 119894th

neighbor of 119909 It has been proved that CCC(119899) is a Cayleygraph and hence is a vertex-transitive graph with degree 3Van Wieren et al [147] proved that CCC(119899) has an efficientdominating set if and only if 119899 = 5 From Theorems 194 and195 the following result can be derived

Theorem 199 (Huang and Xu [36] 2008) Let 119866 = 119862119862119862(119899)be the 119899-dimensional cube-connected cycles with 119899 ⩾ 3 and119899 = 5 Then 120574(119866) = 1198992119899minus2 and 2 ⩽ 119887(119866) ⩽ 3 In addition119887(119862119862119862(3)) = 119887(119862119862119862(4)) = 2

Problem 11 Determine the exact value of 119887(CCC(119899)) for 119899 ⩾5

Acknowledgments

This work was supported by NNSF of China (Grants nos11071233 61272008) The author would like to express hisgratitude to the anonymous referees for their kind sugges-tions and comments on the original paper to ProfessorSheikholeslami and Professor Jafari Rad for sending thereferences [128] and [81] which resulted in this version

References

[1] J-M Xu Theory and Application of Graphs vol 10 of NetworkTheory and Applications Kluwer Academic Dordrecht TheNetherlands 2003

[2] S THedetniemi andR C Laskar ldquoBibliography on dominationin graphs and some basic definitions of domination parame-tersrdquo Discrete Mathematics vol 86 no 1ndash3 pp 257ndash277 1990

[3] TW Haynes S T Hedetniemi and P J Slater Fundamentals ofDomination in Graphs vol 208 ofMonographs and Textbooks inPure and Applied Mathematics Marcel Dekker New York NYUSA 1998

[4] TWHaynes S THedetniemi and P J Slater EdsDominationin Graphs vol 209 of Monographs and Textbooks in Pure andAppliedMathematicsMarcelDekker NewYorkNYUSA 1998

[5] M R Garey andD S JohnsonComputers and Intractability WH Freeman and Company San Francisco Calif USA 1979

[6] H B Walikar and B D Acharya ldquoDomination critical graphsrdquoNational Academy Science Letters vol 2 pp 70ndash72 1979

[7] D Bauer F Harary J Nieminen and C L Suffel ldquoDominationalteration sets in graphsrdquo Discrete Mathematics vol 47 no 2-3pp 153ndash161 1983

[8] J F Fink M S Jacobson L F Kinch and J Roberts ldquoThebondage number of a graphrdquo Discrete Mathematics vol 86 no1ndash3 pp 47ndash57 1990

[9] F-T Hu and J-M Xu ldquoThe bondage number of (119899 minus 3)-regulargraph G of order nrdquo Ars Combinatoria to appear

[10] U Teschner ldquoNew results about the bondage number of agraphrdquoDiscreteMathematics vol 171 no 1ndash3 pp 249ndash259 1997

[11] B L Hartnell and D F Rall ldquoA bound on the size of a graphwith given order and bondage numberrdquo Discrete Mathematicsvol 197-198 pp 409ndash413 1999 16th British CombinatorialConference (London 1997)

[12] B L Hartnell and D F Rall ldquoA characterization of trees inwhich no edge is essential to the domination numberrdquo ArsCombinatoria vol 33 pp 65ndash76 1992

[13] J Topp and P D Vestergaard ldquo120572119896- and 120574

119896-stable graphsrdquo

Discrete Mathematics vol 212 no 1-2 pp 149ndash160 2000[14] A Meir and J W Moon ldquoRelations between packing and

covering numbers of a treerdquo Pacific Journal of Mathematics vol61 no 1 pp 225ndash233 1975

[15] B L Hartnell L K Joslashrgensen P D Vestergaard and CA Whitehead ldquoEdge stability of the k-domination numberof treesrdquo Bulletin of the Institute of Combinatorics and ItsApplications vol 22 pp 31ndash40 1998

[16] F-T Hu and J-M Xu ldquoOn the complexity of the bondage andreinforcement problemsrdquo Journal of Complexity vol 28 no 2pp 192ndash201 2012

[17] B L Hartnell and D F Rall ldquoBounds on the bondage numberof a graphrdquo Discrete Mathematics vol 128 no 1ndash3 pp 173ndash1771994

[18] Y-L Wang ldquoOn the bondage number of a graphrdquo DiscreteMathematics vol 159 no 1ndash3 pp 291ndash294 1996

[19] G H Fricke TW Haynes S M Hedetniemi S T Hedetniemiand R C Laskar ldquoExcellent treesrdquo Bulletin of the Institute ofCombinatorics and Its Applications vol 34 pp 27ndash38 2002

[20] V Samodivkin ldquoThe bondage number of graphs good and badverticesrdquoDiscussiones Mathematicae GraphTheory vol 28 no3 pp 453ndash462 2008

[21] J E Dunbar T W Haynes U Teschner and L VolkmannldquoBondage insensitivity and reinforcementrdquo in Domination inGraphs vol 209 of Monographs and Textbooks in Pure andAppliedMathematics pp 471ndash489 Dekker NewYork NY USA1998

[22] H Liu and L Sun ldquoThe bondage and connectivity of a graphrdquoDiscrete Mathematics vol 263 no 1ndash3 pp 289ndash293 2003

[23] A-H Esfahanian and S L Hakimi ldquoOn computing a con-ditional edge-connectivity of a graphrdquo Information ProcessingLetters vol 27 no 4 pp 195ndash199 1988

[24] R C Brigham P Z Chinn and R D Dutton ldquoVertexdomination-critical graphsrdquoNetworks vol 18 no 3 pp 173ndash1791988

[25] V Samodivkin ldquoDomination with respect to nondegenerateproperties bondage numberrdquoThe Australasian Journal of Com-binatorics vol 45 pp 217ndash226 2009

[26] U Teschner ldquoA new upper bound for the bondage numberof graphs with small domination numberrdquo The AustralasianJournal of Combinatorics vol 12 pp 27ndash35 1995

32 International Journal of Combinatorics

[27] J Fulman D Hanson and G MacGillivray ldquoVertex dom-ination-critical graphsrdquoNetworks vol 25 no 2 pp 41ndash43 1995

[28] U Teschner ldquoA counterexample to a conjecture on the bondagenumber of a graphrdquo Discrete Mathematics vol 122 no 1ndash3 pp393ndash395 1993

[29] U Teschner ldquoThe bondage number of a graph G can be muchgreater than Δ(G)rdquo Ars Combinatoria vol 43 pp 81ndash87 1996

[30] L Volkmann ldquoOn graphs with equal domination and coveringnumbersrdquoDiscrete AppliedMathematics vol 51 no 1-2 pp 211ndash217 1994

[31] L A Sanchis ldquoMaximum number of edges in connected graphswith a given domination numberrdquoDiscreteMathematics vol 87no 1 pp 65ndash72 1991

[32] S Klavzar and N Seifter ldquoDominating Cartesian products ofcyclesrdquoDiscrete AppliedMathematics vol 59 no 2 pp 129ndash1361995

[33] H K Kim ldquoA note on Vizingrsquos conjecture about the bondagenumberrdquo Korean Journal of Mathematics vol 10 no 2 pp 39ndash42 2003

[34] M Y Sohn X Yuan and H S Jeong ldquoThe bondage number of1198623times119862

119899rdquo Journal of the KoreanMathematical Society vol 44 no

6 pp 1213ndash1231 2007[35] L Kang M Y Sohn and H K Kim ldquoBondage number of the

discrete torus 119862119899times 119862

4rdquo Discrete Mathematics vol 303 no 1ndash3

pp 80ndash86 2005[36] J Huang and J-M Xu ldquoThe bondage numbers and efficient

dominations of vertex-transitive graphsrdquoDiscrete Mathematicsvol 308 no 4 pp 571ndash582 2008

[37] C J Xiang X Yuan andM Y Sohn ldquoDomination and bondagenumber of119862

3times119862

119899rdquoArs Combinatoria vol 97 pp 299ndash310 2010

[38] M S Jacobson and L F Kinch ldquoOn the domination numberof products of graphs Irdquo Ars Combinatoria vol 18 pp 33ndash441984

[39] Y-Q Li ldquoA note on the bondage number of a graphrdquo ChineseQuarterly Journal of Mathematics vol 9 no 4 pp 1ndash3 1994

[40] F-T Hu and J-M Xu ldquoBondage number of mesh networksrdquoFrontiers ofMathematics inChina vol 7 no 5 pp 813ndash826 2012

[41] R Frucht and F Harary ldquoOn the corona of two graphsrdquoAequationes Mathematicae vol 4 pp 322ndash325 1970

[42] K Carlson and M Develin ldquoOn the bondage number of planarand directed graphsrdquoDiscreteMathematics vol 306 no 8-9 pp820ndash826 2006

[43] U Teschner ldquoOn the bondage number of block graphsrdquo ArsCombinatoria vol 46 pp 25ndash32 1997

[44] J Huang and J-M Xu ldquoNote on conjectures of bondagenumbers of planar graphsrdquo Applied Mathematical Sciences vol6 no 65ndash68 pp 3277ndash3287 2012

[45] L Kang and J Yuan ldquoBondage number of planar graphsrdquoDiscrete Mathematics vol 222 no 1ndash3 pp 191ndash198 2000

[46] M Fischermann D Rautenbach and L Volkmann ldquoRemarkson the bondage number of planar graphsrdquo Discrete Mathemat-ics vol 260 no 1ndash3 pp 57ndash67 2003

[47] J Hou G Liu and J Wu ldquoBondage number of planar graphswithout small cyclesrdquoUtilitasMathematica vol 84 pp 189ndash1992011

[48] J Huang and J-M Xu ldquoThe bondage numbers of graphs withsmall crossing numbersrdquo Discrete Mathematics vol 307 no 15pp 1881ndash1897 2007

[49] Q Ma S Zhang and J Wang ldquoBondage number of 1-planargraphrdquo Applied Mathematics vol 1 no 2 pp 101ndash103 2010

[50] J Hou and G Liu ldquoA bound on the bondage number of toroidalgraphsrdquoDiscreteMathematics Algorithms and Applications vol4 no 3 Article ID 1250046 5 pages 2012

[51] Y-C Cao J-M Xu and X-R Xu ldquoOn bondage number oftoroidal graphsrdquo Journal of University of Science and Technologyof China vol 39 no 3 pp 225ndash228 2009

[52] Y-C Cao J Huang and J-M Xu ldquoThe bondage number ofgraphs with crossing number less than fourrdquo Ars Combinatoriato appear

[53] H K Kim ldquoOn the bondage numbers of some graphsrdquo KoreanJournal of Mathematics vol 11 no 2 pp 11ndash15 2004

[54] A Gagarin and V Zverovich ldquoUpper bounds for the bondagenumber of graphs on topological surfacesrdquo Discrete Mathemat-ics 2012

[55] J Huang ldquoAn improved upper bound for the bondage numberof graphs on surfacesrdquoDiscrete Mathematics vol 312 no 18 pp2776ndash2781 2012

[56] A Gagarin and V Zverovich ldquoThe bondage number of graphson topological surfaces and Teschnerrsquos conjecturerdquo DiscreteMathematics vol 313 no 6 pp 796ndash808 2013

[57] V Samodivkin ldquoNote on the bondage number of graphs ontopological surfacesrdquo httparxivorgabs12086203

[58] E J Cockayne R M Dawes and S T Hedetniemi ldquoTotaldomination in graphsrdquoNetworks vol 10 no 3 pp 211ndash219 1980

[59] R Laskar J Pfaff S M Hedetniemi and S T HedetniemildquoOn the algorithmic complexity of total dominationrdquo Society forIndustrial and Applied Mathematics vol 5 no 3 pp 420ndash4251984

[60] J Pfaff R C Laskar and S T Hedetniemi ldquoNP-completeness oftotal and connected domination and irredundance for bipartitegraphsrdquo Tech Rep 428 Department of Mathematical SciencesClemson University 1983

[61] M A Henning ldquoA survey of selected recent results on totaldomination in graphsrdquoDiscrete Mathematics vol 309 no 1 pp32ndash63 2009

[62] V R Kulli and D K Patwari ldquoThe total bondage number ofa graphrdquo in Advances in Graph Theory pp 227ndash235 VishwaGulbarga India 1991

[63] F-T Hu Y Lu and J-M Xu ldquoThe total bondage number ofgrid graphsrdquo Discrete Applied Mathematics vol 160 no 16-17pp 2408ndash2418 2012

[64] J Cao M Shi and B Wu ldquoResearch on the total bondagenumber of a spectial networkrdquo in Proceedings of the 3rdInternational Joint Conference on Computational Sciences andOptimization (CSO rsquo10) pp 456ndash458 May 2010

[65] N Sridharan MD Elias and V S A Subramanian ldquoTotalbondage number of a graphrdquo AKCE International Journal ofGraphs and Combinatorics vol 4 no 2 pp 203ndash209 2007

[66] N Jafari Rad and J Raczek ldquoSome progress on total bondage ingraphsrdquo Graphs and Combinatorics 2011

[67] T W Haynes and P J Slater ldquoPaired-domination and thepaired-domatic numberrdquoCongressusNumerantium vol 109 pp65ndash72 1995

[68] T W Haynes and P J Slater ldquoPaired-domination in graphsrdquoNetworks vol 32 no 3 pp 199ndash206 1998

[69] X Hou ldquoA characterization of 2120574 120574119875-treesrdquoDiscrete Mathemat-

ics vol 308 no 15 pp 3420ndash3426 2008[70] K E Proffitt T W Haynes and P J Slater ldquoPaired-domination

in grid graphsrdquo Congressus Numerantium vol 150 pp 161ndash1722001

International Journal of Combinatorics 33

[71] H Qiao L KangM Cardei andD-Z Du ldquoPaired-dominationof treesrdquo Journal of Global Optimization vol 25 no 1 pp 43ndash542003

[72] E Shan L Kang and M A Henning ldquoA characterizationof trees with equal total domination and paired-dominationnumbersrdquo The Australasian Journal of Combinatorics vol 30pp 31ndash39 2004

[73] J Raczek ldquoPaired bondage in treesrdquo Discrete Mathematics vol308 no 23 pp 5570ndash5575 2008

[74] J A Telle and A Proskurowski ldquoAlgorithms for vertex parti-tioning problems on partial k-treesrdquo SIAM Journal on DiscreteMathematics vol 10 no 4 pp 529ndash550 1997

[75] G S Domke J H Hattingh M A Henning and L R MarkusldquoRestrained domination in treesrdquoDiscreteMathematics vol 211no 1ndash3 pp 1ndash9 2000

[76] G S Domke J H Hattingh M A Henning and L R MarkusldquoRestrained domination in graphs with minimum degree twordquoJournal of Combinatorial Mathematics and Combinatorial Com-puting vol 35 pp 239ndash254 2000

[77] G S Domke J H Hattingh S T Hedetniemi R C Laskarand L R Markus ldquoRestrained domination in graphsrdquo DiscreteMathematics vol 203 no 1ndash3 pp 61ndash69 1999

[78] J H Hattingh and E J Joubert ldquoAn upper bound for therestrained domination number of a graph with minimumdegree at least two in terms of order and minimum degreerdquoDiscrete Applied Mathematics vol 157 no 13 pp 2846ndash28582009

[79] M A Henning ldquoGraphs with large restrained dominationnumberrdquo Discrete Mathematics vol 197-198 pp 415ndash429 199916th British Combinatorial Conference (London 1997)

[80] J H Hattingh and A R Plummer ldquoRestrained bondage ingraphsrdquo Discrete Mathematics vol 308 no 23 pp 5446ndash54532008

[81] N Jafari Rad R Hasni J Raczek and L Volkmann ldquoTotalrestrained bondage in graphsrdquoActaMathematica Sinica vol 29no 6 pp 1033ndash1042 2013

[82] J Cyman and J Raczek ldquoOn the total restrained dominationnumber of a graphrdquoThe Australasian Journal of Combinatoricsvol 36 pp 91ndash100 2006

[83] M A Henning ldquoGraphs with large total domination numberrdquoJournal of Graph Theory vol 35 no 1 pp 21ndash45 2000

[84] D-X Ma X-G Chen and L Sun ldquoOn total restraineddomination in graphsrdquoCzechoslovakMathematical Journal vol55 no 1 pp 165ndash173 2005

[85] B Zelinka ldquoRemarks on restrained domination and totalrestrained domination in graphsrdquo Czechoslovak MathematicalJournal vol 55 no 2 pp 393ndash396 2005

[86] W Goddard and M A Henning ldquoIndependent domination ingraphs a survey and recent resultsrdquo Discrete Mathematics vol313 no 7 pp 839ndash854 2013

[87] J H Zhang H L Liu and L Sun ldquoIndependence bondagenumber and reinforcement number of some graphsrdquo Transac-tions of Beijing Institute of Technology vol 23 no 2 pp 140ndash1422003

[88] K D Dharmalingam Studies in Graph Theorymdashequitabledomination and bottleneck domination [PhD thesis] MaduraiKamaraj University Madurai India 2006

[89] GDeepakND Soner andAAlwardi ldquoThe equitable bondagenumber of a graphrdquo Research Journal of Pure Algebra vol 1 no9 pp 209ndash212 2011

[90] E Sampathkumar and H B Walikar ldquoThe connected domina-tion number of a graphrdquo Journal of Mathematical and PhysicalSciences vol 13 no 6 pp 607ndash613 1979

[91] K White M Farber and W Pulleyblank ldquoSteiner trees con-nected domination and strongly chordal graphsrdquoNetworks vol15 no 1 pp 109ndash124 1985

[92] M Cadei M X Cheng X Cheng and D-Z Du ldquoConnecteddomination in Ad Hoc wireless networksrdquo in Proceedings ofthe 6th International Conference on Computer Science andInformatics (CSampI rsquo02) 2002

[93] J F Fink and M S Jacobson ldquon-domination in graphsrdquo inGraph Theory with Applications to Algorithms and ComputerScience Wiley-Interscience Publications pp 283ndash300 WileyNew York NY USA 1985

[94] M Chellali O Favaron A Hansberg and L Volkmann ldquok-domination and k-independence in graphs a surveyrdquo Graphsand Combinatorics vol 28 no 1 pp 1ndash55 2012

[95] Y Lu X Hou J-M Xu andN Li ldquoTrees with uniqueminimump-dominating setsrdquo Utilitas Mathematica vol 86 pp 193ndash2052011

[96] Y Lu and J-M Xu ldquoThe p-bondage number of treesrdquo Graphsand Combinatorics vol 27 no 1 pp 129ndash141 2011

[97] Y Lu and J-M Xu ldquoThe 2-domination and 2-bondage numbersof grid graphs A manuscriptrdquo httparxivorgabs12044514

[98] M Krzywkowski ldquoNon-isolating 2-bondage in graphsrdquo TheJournal of the Mathematical Society of Japan vol 64 no 4 2012

[99] M A Henning O R Oellermann and H C Swart ldquoBoundson distance domination parametersrdquo Journal of CombinatoricsInformation amp System Sciences vol 16 no 1 pp 11ndash18 1991

[100] F Tian and J-M Xu ldquoBounds for distance domination numbersof graphsrdquo Journal of University of Science and Technology ofChina vol 34 no 5 pp 529ndash534 2004

[101] F Tian and J-M Xu ldquoDistance domination-critical graphsrdquoApplied Mathematics Letters vol 21 no 4 pp 416ndash420 2008

[102] F Tian and J-M Xu ldquoA note on distance domination numbersof graphsrdquo The Australasian Journal of Combinatorics vol 43pp 181ndash190 2009

[103] F Tian and J-M Xu ldquoAverage distances and distance domina-tion numbersrdquo Discrete Applied Mathematics vol 157 no 5 pp1113ndash1127 2009

[104] Z-L Liu F Tian and J-M Xu ldquoProbabilistic analysis of upperbounds for 2-connected distance k-dominating sets in graphsrdquoTheoretical Computer Science vol 410 no 38ndash40 pp 3804ndash3813 2009

[105] M A Henning O R Oellermann and H C Swart ldquoDistancedomination critical graphsrdquo Journal of Combinatorial Mathe-matics and Combinatorial Computing vol 44 pp 33ndash45 2003

[106] T J Bean M A Henning and H C Swart ldquoOn the integrityof distance domination in graphsrdquo The Australasian Journal ofCombinatorics vol 10 pp 29ndash43 1994

[107] M Fischermann and L Volkmann ldquoA remark on a conjecturefor the (kp)-domination numberrdquoUtilitasMathematica vol 67pp 223ndash227 2005

[108] T Korneffel D Meierling and L Volkmann ldquoA remark on the(22)-domination numberrdquo Discussiones Mathematicae GraphTheory vol 28 no 2 pp 361ndash366 2008

[109] Y Lu X Hou and J-M Xu ldquoOn the (22)-domination numberof treesrdquo Discussiones Mathematicae Graph Theory vol 30 no2 pp 185ndash199 2010

34 International Journal of Combinatorics

[110] H Li and J-M Xu ldquo(dm)-domination numbers of m-connected graphsrdquo Rapports de Recherche 1130 LRI UniversiteParis-Sud 1997

[111] S M Hedetniemi S T Hedetniemi and T V Wimer ldquoLineartime resource allocation algorithms for treesrdquo Tech Rep URI-014 Department of Mathematical Sciences Clemson Univer-sity 1987

[112] M Farber ldquoDomination independent domination and dualityin strongly chordal graphsrdquo Discrete Applied Mathematics vol7 no 2 pp 115ndash130 1984

[113] G S Domke and R C Laskar ldquoThe bondage and reinforcementnumbers of 120574

119891for some graphsrdquoDiscrete Mathematics vol 167-

168 pp 249ndash259 1997[114] G S Domke S T Hedetniemi and R C Laskar ldquoFractional

packings coverings and irredundance in graphsrdquo vol 66 pp227ndash238 1988

[115] E J Cockayne P A Dreyer Jr S M Hedetniemi andS T Hedetniemi ldquoRoman domination in graphsrdquo DiscreteMathematics vol 278 no 1ndash3 pp 11ndash22 2004

[116] I Stewart ldquoDefend the roman empirerdquo Scientific American vol281 pp 136ndash139 1999

[117] C S ReVelle and K E Rosing ldquoDefendens imperiumromanum a classical problem in military strategyrdquoThe Ameri-can Mathematical Monthly vol 107 no 7 pp 585ndash594 2000

[118] A Bahremandpour F-T Hu S M Sheikholeslami and J-MXu ldquoOn the Roman bondage number of a graphrdquo DiscreteMathematics Algorithms and Applications vol 5 no 1 ArticleID 1350001 15 pages 2013

[119] N Jafari Rad and L Volkmann ldquoRoman bondage in graphsrdquoDiscussiones Mathematicae Graph Theory vol 31 no 4 pp763ndash773 2011

[120] K Ebadi and L PushpaLatha ldquoRoman bondage and Romanreinforcement numbers of a graphrdquo International Journal ofContemporary Mathematical Sciences vol 5 pp 1487ndash14972010

[121] N Dehgardi S M Sheikholeslami and L Volkman ldquoOn theRoman k-bondage number of a graphrdquo AKCE InternationalJournal of Graphs and Combinatorics vol 8 pp 169ndash180 2011

[122] S Akbari and S Qajar ldquoA note on Roman bondage number ofgraphsrdquo Ars Combinatoria to Appear

[123] F-T Hu and J-M Xu ldquoRoman bondage numbers of somegraphsrdquo httparxivorgabs11093933

[124] N Jafari Rad and L Volkmann ldquoOn the Roman bondagenumber of planar graphsrdquo Graphs and Combinatorics vol 27no 4 pp 531ndash538 2011

[125] S Akbari M Khatirinejad and S Qajar ldquoA note on the romanbondage number of planar graphsrdquo Graphs and Combinatorics2012

[126] B Bresar M A Henning and D F Rall ldquoRainbow dominationin graphsrdquo Taiwanese Journal of Mathematics vol 12 no 1 pp213ndash225 2008

[127] B L Hartnell and D F Rall ldquoOn dominating the Cartesianproduct of a graph and 120572krdquo Discussiones Mathematicae GraphTheory vol 24 no 3 pp 389ndash402 2004

[128] N Dehgardi S M Sheikholeslami and L Volkman ldquoThe k-rainbow bondage number of a graphrdquo to appear in DiscreteApplied Mathematics

[129] J von Neumann and O Morgenstern Theory of Games andEconomic Behavior Princeton University Press Princeton NJUSA 1944

[130] C BergeTheorıe des Graphes et ses Applications 1958[131] O Ore Theory of Graphs vol 38 American Mathematical

Society Colloquium Publications 1962[132] E Shan and L Kang ldquoBondage number in oriented graphsrdquoArs

Combinatoria vol 84 pp 319ndash331 2007[133] J Huang and J-M Xu ldquoThe bondage numbers of extended

de Bruijn and Kautz digraphsrdquo Computers amp Mathematics withApplications vol 51 no 6-7 pp 1137ndash1147 2006

[134] J Huang and J-M Xu ldquoThe total domination and total bondagenumbers of extended de Bruijn and Kautz digraphsrdquoComputersamp Mathematics with Applications vol 53 no 8 pp 1206ndash12132007

[135] Y Shibata and Y Gonda ldquoExtension of de Bruijn graph andKautz graphrdquo Computers amp Mathematics with Applications vol30 no 9 pp 51ndash61 1995

[136] X Zhang J Liu and JMeng ldquoThebondage number in completet-partite digraphsrdquo Information Processing Letters vol 109 no17 pp 997ndash1000 2009

[137] D W Bange A E Barkauskas and P J Slater ldquoEfficient dom-inating sets in graphsrdquo in Applications of Discrete Mathematicspp 189ndash199 SIAM Philadelphia Pa USA 1988

[138] M Livingston and Q F Stout ldquoPerfect dominating setsrdquoCongressus Numerantium vol 79 pp 187ndash203 1990

[139] L Clark ldquoPerfect domination in random graphsrdquo Journal ofCombinatorial Mathematics and Combinatorial Computing vol14 pp 173ndash182 1993

[140] D W Bange A E Barkauskas L H Host and L H ClarkldquoEfficient domination of the orientations of a graphrdquo DiscreteMathematics vol 178 no 1ndash3 pp 1ndash14 1998

[141] A E Barkauskas and L H Host ldquoFinding efficient dominatingsets in oriented graphsrdquo Congressus Numerantium vol 98 pp27ndash32 1993

[142] I J Dejter and O Serra ldquoEfficient dominating sets in CayleygraphsrdquoDiscrete AppliedMathematics vol 129 no 2-3 pp 319ndash328 2003

[143] J Lee ldquoIndependent perfect domination sets in Cayley graphsrdquoJournal of Graph Theory vol 37 no 4 pp 213ndash219 2001

[144] WGu X Jia and J Shen ldquoIndependent perfect domination setsin meshes tori and treesrdquo Unpublished

[145] N Obradovic J Peters and G Ruzic ldquoEfficient domination incirculant graphswith two chord lengthsrdquo Information ProcessingLetters vol 102 no 6 pp 253ndash258 2007

[146] K Reji Kumar and G MacGillivray ldquoEfficient domination incirculant graphsrdquo Discrete Mathematics vol 313 no 6 pp 767ndash771 2013

[147] D Van Wieren M Livingston and Q F Stout ldquoPerfectdominating sets on cube-connected cyclesrdquo Congressus Numer-antium vol 97 pp 51ndash70 1993

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

28 International Journal of Combinatorics

10 Results on Digraphs

Although domination has been extensively studied in undi-rected graphs it is natural to think of a dominating setas a one-way relationship between vertices of the graphIndeed among the earliest literatures on this subject vonNeumann and Morgenstern [129] used what is now calleddomination in digraphs to find solution (or kernels whichare independent dominating sets) for cooperative 119899-persongames Most likely the first formulation of domination byBerge [130] in 1958 was given in the context of digraphs andonly some years latter by Ore [131] in 1962 for undirectedgraphs Despite this history examination of domination andits variants in digraphs has been essentially overlooked (see[4] for an overview of the domination literature) Thus thereare few if any such results on domination for digraphs in theliterature

The bondage number and its related topics for undirectedgraph have become one of major areas both in theoreticaland applied researches However until recently Carlsonand Develin [42] Shan and Kang [132] and Huang andXu [133 134] studied the bondage number for digraphsindependently In this section we introduce their resultsfor general digraphs Results for some special digraphs suchas vertex-transitive digraphs are introduced in the nextsection

101 Upper Bounds for General Digraphs Let 119866 = (119881 119864)

be a digraph without loops and parallel edges A digraph 119866is called to be asymmetric if whenever (119909 119910) isin 119864(119866) then(119910 119909) notin 119864(119866) and to be symmetric if (119909 119910) isin 119864(119866) implies(119910 119909) isin 119864(119866) For a vertex 119909 of 119881(119866) the sets of out-neighbors and in-neighbors of 119909 are respectively defined as119873

+

119866(119909) = 119910 isin 119881(119866) (119909 119910) isin 119864(119866) and 119873minus

119866(119909) = 119909 isin

119881(119866) (119909 119910) isin 119864(119866) the out-degree and the in-degree of119909 are respectively defined as 119889+

119866(119909) = |119873

+

119866(119909)| and 119889minus

119866(119909) =

|119873+

119866(119909)| Denote themaximum and theminimum out-degree

(resp in-degree) of 119866 by Δ+(119866) and 120575+(119866) (resp Δminus(119866) and120575minus

(119866))The degree 119889119866(119909) of 119909 is defined as 119889+

119866(119909)+119889

minus

119866(119909) and

the maximum and the minimum degrees of119866 are denoted byΔ(119866) and 120575(119866) respectively that is Δ(119866) = max119889

119866(119909) 119909 isin

119881(119866) and 120575(119866) = min119889119866(119909) 119909 isin 119881(119866) Note that the

definitions here are different from the ones in the textbookon digraphs see for example Xu [1]

A subset 119878 of119881(119866) is called a dominating set if119881(119866) = 119878cup119873

+

119866(119878) where119873+

119866(119878) is the set of out-neighbors of 119878Then just

as for undirected graphs 120574(119866) is the minimum cardinalityof a dominating set and the bondage number 119887(119866) is thesmallest cardinality of a set 119861 of edges such that 120574(119866 minus 119861) gt120574(119866) if such a subset 119861 sube 119864(119866) exists Otherwise we put119887(119866) = infin

Some basic results for undirected graphs stated inSection 3 can be generalized to digraphs For exampleTheorem 18 is generalized by Carlson and Develin [42]Huang andXu [133] and Shan andKang [132] independentlyas follows

Theorem 180 Let 119866 be a digraph and (119909 119910) isin 119864(119866) Then119887(119866) ⩽ 119889

119866(119910) + 119889

minus

119866(119909) minus |119873

minus

119866(119909) cap 119873

minus

119866(119910)|

Corollary 181 For a digraph 119866 119887(119866) ⩽ 120575minus(119866) + Δ(119866)

Since 120575minus

(119866) ⩽ (12)Δ(119866) from Corollary 181Conjecture 53 is valid for digraphs

Corollary 182 For a digraph 119866 119887(119866) ⩽ (32)Δ(119866)

In the case of undirected graphs the bondage number of119866119899= 119870

119899times 119870

119899achieves this bound However it was shown

by Carlson and Develin in [42] that if we take the symmetricdigraph119866

119899 we haveΔ(119866

119899) = 4(119899minus1) 120574(119866

119899) = 119899 and 119887(119866

119899) ⩽

4(119899 minus 1) + 1 = Δ(119866119899) + 1 So this family of digraphs cannot

show that the bound in Corollary 182 is tightCorresponding to Conjecture 51 which is discredited

for undirected graphs and is valid for digraphs the sameconjecture for digraphs can be proposed as follows

Conjecture 183 (Carlson and Develin [42] 2006) 119887(119866) ⩽Δ(119866) + 1 for any digraph 119866

If Conjecture 183 is true then the following results showsthat this upper bound is tight since 119887(119870

119899times119870

119899) = Δ(119870

119899times119870

119899)+1

for a complete digraph 119870119899

In 2007 Shan and Kang [132] gave some tight upperbounds on the bondage numbers for some asymmetricdigraphs For example 119887(119879) ⩽ Δ(119879) for any asymmetricdirected tree 119879 119887(119866) ⩽ Δ(119866) for any asymmetric digraph 119866with order at least 4 and 120574(119866) ⩽ 2 For planar digraphs theyobtained the following results

Theorem 184 (Shan and Kang [132] 2007) Let 119866 be aasymmetric planar digraphThen 119887(119866) ⩽ Δ(119866)+2 and 119887(119866) ⩽Δ(119866) + 1 if Δ(119866) ⩾ 5 and 119889minus

119866(119909) ⩾ 3 for every vertex 119909 with

119889119866(119909) ⩾ 4

102 Results for Some Special Digraphs The exact values andbounds on 119887(119866) for some standard digraphs are determined

Theorem185 (Huang andXu [133] 2006) For a directed cycle119862119899and a directed path 119875

119899

119887 (119862119899) =

3 119894119891 119899 119894119904 119900119889119889

2 119894119891 119899 119894119904 119890V119890119899

119887 (119875119899) =

2 119894119891 119899 119894119904 119900119889119889

1 119894119891 119899 119894119904 119890V119890119899

(76)

For the de Bruijn digraph 119861(119889 119899) and the Kautz digraph119870(119889 119899)

119887 (119861 (119889 119899)) = 119889 119894119891 119899 119894119904 119900119889119889

119889 ⩽ 119887 (119861 (119889 119899)) ⩽ 2119889 119894119891 119899 119894119904 119890V119890119899

119887 (119870 (119889 119899)) = 119889 + 1

(77)

Like undirected graphs we can define the total domi-nation number and the total bondage number On the totalbondage numbers for some special digraphs the knownresults are as follows

International Journal of Combinatorics 29

Theorem 186 (Huang and Xu [134] 2007) For a directedcycle 119862

119899and a directed path 119875

119899 119887

119879(119875

119899) and 119887

119879(119862

119899) all do not

exist For a complete digraph 119870119899

119887119879(119870

119899) = infin 119894119891 119899 = 1 2

119887119879(119870

119899) = 3 119894119891 119899 = 3

119899 ⩽ 119887119879(119870

119899) ⩽ 2119899 minus 3 119894119891 119899 ⩾ 4

(78)

The extended de Bruijn digraph EB(119889 119899 1199021 119902

119901) and

the extended Kautz digraph EK(119889 119899 1199021 119902

119901) are intro-

duced by Shibata and Gonda [135] If 119901 = 1 then theyare the de Bruijn digraph B(119889 119899) and the Kautz digraphK(119889 119899) respectively Huang and Xu [134] determined theirtotal domination numbers In particular their total bondagenumbers for general cases are determined as follows

Theorem 187 (Huang andXu [134] 2007) If 119889 ⩾ 2 and 119902119894⩾ 2

for each 119894 = 1 2 119901 then

119887119879(EB(119889 119899 119902

1 119902

119901)) = 119889

119901

minus 1

119887119879(EK(119889 119899 119902

1 119902

119901)) = 119889

119901

(79)

In particular for the de Bruijn digraph B(119889 119899) and the Kautzdigraph K(119889 119899)

119887119879(B (119889 119899)) = 119889 minus 1 119887

119879(K (119889 119899)) = 119889 (80)

Zhang et al [136] determined the bondage number incomplete 119905-partite digraphs

Theorem 188 (Zhang et al [136] 2009) For a complete 119905-partite digraph 119870

11989911198992119899119905

where 1198991⩽ 119899

2⩽ sdot sdot sdot ⩽ 119899

119905

119887 (11987011989911198992119899119905

)

=

119898 119894119891 119899119898= 1 119899

119898+1⩾ 2 119891119900119903 119904119900119898119890

119898 (1 ⩽ 119898 lt 119905)

4119905 minus 3 119894119891 1198991= 119899

2= sdot sdot sdot = 119899

119905= 2

119905minus1

sum

119894=1

119899119894+ 2 (119905 minus 1) 119900119905ℎ119890119903119908119894119904119890

(81)

Since an undirected graph can be thought of as a sym-metric digraph any result for digraphs has an analogy forundirected graphs in general In view of this point studyingthe bondage number for digraphs is more significant thanfor undirected graphs Thus we should further study thebondage number of digraphs and try to generalize knownresults on the bondage number and related variants for undi-rected graphs to digraphs prove or disprove Conjecture 183In particular determine the exact values of 119887(B(119889 119899)) for aneven 119899 and 119887

119879(119870

119899) for 119899 ⩾ 4

11 Efficient Dominating Sets

A dominating set 119878 of a graph 119866 is called to be efficient if forevery vertex 119909 in119866 |119873

119866[119909]cap119878| = 1 if119866 is a undirected graph

or |119873minus

119866[119909] cap 119878| = 1 if 119866 is a directed graph The concept of an

efficient set was proposed by Bange et al [137] in 1988 In theliterature an efficient set is also called a perfect dominatingset (see Livingston and Stout [138] for an explanation)

From definition if 119878 is an efficient dominating set of agraph 119866 then 119878 is certainly an independent set and everyvertex not in 119878 is adjacent to exactly one vertex in 119878

It is also clear from definition that a dominating set 119878 isefficient if and only if N

119866[119878] = 119873

119866[119909] 119909 isin 119878 for the

undirected graph 119866 or N+

119866[119878] = 119873

+

119866[119909] 119909 isin 119878 for the

digraph119866 is a partition of119881(119866) where the induced subgraphby119873

119866[119909] or119873+

119866[119909] is a star or an out-star with the root 119909

The efficient domination has important applications inmany areas such as error-correcting codes and receivesmuch attention in the late years

The concept of efficient dominating sets is a measureof the efficiency of domination in graphs Unfortunately asshown in [137] not every graph has an efficient dominatingset andmoreover it is anNP-complete problem to determinewhether a given graph has an efficient dominating set Inaddition it was shown by Clark [139] in 1993 that for a widerange of 119901 almost every random undirected graph 119866 isin

G(120592 119901) has no efficient dominating sets This means thatundirected graphs possessing an efficient dominating set arerare However it is easy to show that every undirected graphhas an orientationwith an efficient dominating set (see Bangeet al [140])

In 1993 Barkauskas and Host [141] showed that deter-mining whether an arbitrary oriented graph has an efficientdominating set is NP-complete Even so the existence ofefficient dominating sets for some special classes of graphs hasbeen examined see for example Dejter and Serra [142] andLee [143] for Cayley graph Gu et al [144] formeshes and toriObradovic et al [145] Huang and Xu [36] Reji Kumar andMacGillivray [146] for circulant graphs Harary graphs andtori and Van Wieren et al [147] for cube-connected cycles

In this section we introduce results of the bondagenumber for some graphs with efficient dominating sets dueto Huang and Xu [36]

111 Results for General Graphs In this subsection we intro-duce some results on bondage numbers obtained by applyingefficient dominating sets We first state the two followinglemmas

Lemma 189 For any 119896-regular graph or digraph 119866 of order 119899120574(119866) ⩾ 119899(119896 + 1) with equality if and only if 119866 has an efficientdominating set In addition if 119866 has an efficient dominatingset then every efficient dominating set is certainly a 120574-set andvice versa

Let 119890 be an edge and 119878 a dominating set in 119866 We say that119890 supports 119878 if 119890 isin (119878 119878) where (119878 119878) = (119909 119910) isin 119864(119866) 119909 isin 119878 119910 isin 119878 Denote by 119904(119866) the minimum number of edgeswhich support all 120574-sets in 119866

Lemma 190 For any graph or digraph 119866 119887(119866) ⩾ 119904(119866) withequality if 119866 is regular and has an efficient dominating set

30 International Journal of Combinatorics

A graph 119866 is called to be vertex-transitive if its automor-phism group Aut(119866) acts transitively on its vertex-set 119881(119866)A vertex-transitive graph is regular Applying Lemmas 189and 190 Huang and Xu obtained some results on bondagenumbers for vertex-transitive graphs or digraphs

Theorem 191 (Huang and Xu [36] 2008) For any vertex-transitive graph or digraph 119866 of order 119899

119887 (119866) ⩾

lceil119899

2120574 (119866)rceil 119894119891 119866 119894119904 119906119899119889119894119903119890119888119905119890119889

lceil119899

120574 (119866)rceil 119894119891 119866 119894119904 119889119894119903119890119888119905119890119889

(82)

Theorem192 (Huang andXu [36] 2008) Let119866 be a 119896-regulargraph of order 119899 Then

119887(119866) ⩽ 119896 if119866 is undirected and 119899 ⩾ 120574(119866)(119896+1)minus119896+1119887(119866) ⩽ 119896+1+ℓ if119866 is directed and 119899 ⩾ 120574(119866)(119896+1)minusℓwith 0 ⩽ ℓ ⩽ 119896 minus 1

Next we introduce a better upper bound on 119887(119866) To thisaim we need the following concept which is a generalizationof the concept of the edge-covering of a graph 119866 For 1198811015840

sube

119881(119866) and 1198641015840 sube 119864(119866) we say that 1198641015840covers 1198811015840 and call 1198641015840an edge-covering for 1198811015840 if there exists an edge (119909 119910) isin 1198641015840 forany vertex 119909 isin 1198811015840 For 119910 isin 119881(119866) let 1205731015840[119910] be the minimumcardinality over all edge-coverings for119873minus

119866[119910]

Theorem 193 (Huang and Xu [36] 2008) If 119866 is a 119896-regulargraph with order 120574(119866)(119896 + 1) then 119887(119866) ⩽ 1205731015840[119910] for any 119910 isin119881(119866)

Theupper bound on 119887(119866) given inTheorem 193 is tight inview of 119887(119862

119899) for a cycle or a directed cycle 119862

119899(seeTheorems

1 and 185 resp)It is easy to see that for a 119896-regular graph 119866 lceil(119896 + 1)2rceil ⩽

1205731015840

[119910] ⩽ 119896 when 119866 is undirected and 1205731015840[119910] = 119896 + 1 when 119866 isdirected By this fact and Lemma 189 the following theoremis merely a simple combination of Theorems 191 and 193

Theorem 194 (Huang and Xu [36] 2008) Let 119866 be a vertex-transitive graph of degree 119896 If 119866 has an efficient dominatingset then

lceil119896 + 1

2rceil ⩽ 119887 (119866) ⩽ 119896 119894119891 119866 119894119904 119906119899119889119894119903119890119888119905119890119889

119887 (119866) = 119896 + 1 119894119891 119866 119894119904 119889119894119903119890119888119905119890119889

(83)

Theorem 195 (Huang and Xu [36] 2008) If 119866 is an undi-rected vertex-transitive cubic graph with order 4120574(119866) and girth119892(119866) ⩽ 5 then 119887(119866) = 2

The proof of Theorem 195 leads to a byproduct If 119892 =5 let (119906

1 119906

2 119906

3 119906

4 119906

5) be a cycle and let D

119894be the family

of efficient dominating sets containing 119906119894for each 119894 isin

1 2 3 4 5 ThenD119894capD

119895= 0 for 119894 = 1 2 5 Then 119866 has

at least 5119904 efficient dominating sets where 119904 = 119904(119866) But thereare only 4s (=ns120574) distinct efficient dominating sets in119866This

contradiction implies that an undirected vertex-transitivecubic graph with girth five has no efficient dominating setsBut a similar argument for 119892(119866) = 3 4 or 119892(119866) ⩾ 6 couldnot give any contradiction This is consistent with the resultthat CCC(119899) a vertex-transitive cubic graph with girth 119899 if3 ⩽ 119899 ⩽ 8 or girth 8 if 119899 ⩾ 9 has efficient dominating setsfor all 119899 ⩾ 3 except 119899 = 5 (see Theorem 199 in the followingsubsection)

112 Results for Cayley Graphs In this subsection we useTheorem 194 to determine the exact values or approximativevalues of bondage numbers for some special vertex-transitivegraphs by characterizing the existence of efficient dominatingsets in these graphs

Let Γ be a nontrivial finite group 119878 a nonempty subset ofΓ without the identity element of Γ A digraph 119866 is defined asfollows

119881 (119866) = Γ (119909 119910) isin 119864 (119866) lArrrArr 119909minus1

119910 isin 119878

for any 119909 119910 isin Γ(84)

is called a Cayley digraph of the group Γ with respect to119878 denoted by 119862

Γ(119878) If 119878minus1 = 119904minus1 119904 isin 119878 = 119878 then

119862Γ(119878) is symmetric and is called a Cayley undirected graph

a Cayley graph for short Cayley graphs or digraphs arecertainly vertex-transitive (see Xu [1])

A circulant graph 119866(119899 119878) of order 119899 is a Cayley graph119862119885119899

(119878) where 119885119899= 0 119899 minus 1 is the addition group of

order 119899 and 119878 is a nonempty subset of119885119899without the identity

element and hence is a vertex-transitive digraph of degree|119878| If 119878minus1 = 119878 then119866(119899 119878) is an undirected graph If 119878 = 1 119896where 2 ⩽ 119896 ⩽ 119899 minus 2 we write 119866(119899 1 119896) for 119866(119899 1 119896) or119866(119899 plusmn1 plusmn119896) and call it a double-loop circulant graph

Huang and Xu [36] showed that for directed 119866 =

119866(119899 1 119896) lceil1198993rceil ⩽ 120574(119866) ⩽ lceil1198992rceil and 119866 has an efficientdominating set if and only if 3 | 119899 and 119896 equiv 2 (mod 3) fordirected 119866 = 119866(119899 1 119896) and 119896 = 1198992 lceil1198995rceil ⩽ 120574(119866) ⩽ lceil1198993rceiland 119866 has an efficient dominating set if and only if 5 | 119899 and119896 equiv plusmn2 (mod 5) By Theorem 194 we obtain the bondagenumber of a double loop circulant graph if it has an efficientdominating set

Theorem 196 (Huang and Xu [36] 2008) Let 119866 be a double-loop circulant graph 119866(119899 1 119896) If 119866 is directed with 3 | 119899and 119896 equiv 2 (mod 3) or 119866 is undirected with 5 | 119899 and119896 equiv plusmn2 (mod 5) then 119887(119866) = 3

The 119898 times 119899 torus is the Cartesian product 119862119898times 119862

119899of

two cycles and is a Cayley graph 119862119885119898times119885119899

(119878) where 119878 =

(0 1) (1 0) for directed cycles and 119878 = (0 plusmn1) (plusmn1 0) forundirected cycles and hence is vertex-transitive Gu et al[144] showed that the undirected torus119862

119898times119862

119899has an efficient

dominating set if and only if both 119898 and 119899 are multiplesof 5 Huang and Xu [36] showed that the directed torus119862119898times 119862

119899has an efficient dominating set if and only if both

119898 and 119899 are multiples of 3 Moreover they found a necessarycondition for a dominating set containing the vertex (0 0)in 119862

119898times 119862

119899to be efficient and obtained the following

result

International Journal of Combinatorics 31

Theorem 197 (Huang and Xu [36] 2008) Let 119866 = 119862119898times 119862

119899

If 119866 is undirected and both119898 and 119899 are multiples of 5 or if 119866 isdirected and both119898 and 119899 are multiples of 3 then 119887(119866) = 3

The hypercube 119876119899

is the Cayley graph 119862Γ(119878)

where Γ = 1198852times sdot sdot sdot times 119885

2= (119885

2)119899 and 119878 =

100 sdot sdot sdot 0 010 sdot sdot sdot 0 00 sdot sdot sdot 01 Lee [143] showed that119876119899has an efficient dominating set if and only if 119899 = 2119898 minus 1

for a positive integer119898 ByTheorem 194 the following resultis obtained immediately

Theorem 198 (Huang and Xu [36] 2008) If 119899 = 2119898 minus 1 for apositive integer119898 then 2119898minus1

⩽ 119887(119876119899) ⩽ 2

119898

minus 1

Problem 10 Determine the exact value of 119887(119876119899) for 119899 = 2119898minus1

The 119899-dimensional cube-connected cycle denoted byCCC(119899) is constructed from the 119899-dimensional hypercube119876119899by replacing each vertex 119909 in 119876

119899with an undirected cycle

119862119899of length 119899 and linking the 119894th vertex of the 119862

119899to the 119894th

neighbor of 119909 It has been proved that CCC(119899) is a Cayleygraph and hence is a vertex-transitive graph with degree 3Van Wieren et al [147] proved that CCC(119899) has an efficientdominating set if and only if 119899 = 5 From Theorems 194 and195 the following result can be derived

Theorem 199 (Huang and Xu [36] 2008) Let 119866 = 119862119862119862(119899)be the 119899-dimensional cube-connected cycles with 119899 ⩾ 3 and119899 = 5 Then 120574(119866) = 1198992119899minus2 and 2 ⩽ 119887(119866) ⩽ 3 In addition119887(119862119862119862(3)) = 119887(119862119862119862(4)) = 2

Problem 11 Determine the exact value of 119887(CCC(119899)) for 119899 ⩾5

Acknowledgments

This work was supported by NNSF of China (Grants nos11071233 61272008) The author would like to express hisgratitude to the anonymous referees for their kind sugges-tions and comments on the original paper to ProfessorSheikholeslami and Professor Jafari Rad for sending thereferences [128] and [81] which resulted in this version

References

[1] J-M Xu Theory and Application of Graphs vol 10 of NetworkTheory and Applications Kluwer Academic Dordrecht TheNetherlands 2003

[2] S THedetniemi andR C Laskar ldquoBibliography on dominationin graphs and some basic definitions of domination parame-tersrdquo Discrete Mathematics vol 86 no 1ndash3 pp 257ndash277 1990

[3] TW Haynes S T Hedetniemi and P J Slater Fundamentals ofDomination in Graphs vol 208 ofMonographs and Textbooks inPure and Applied Mathematics Marcel Dekker New York NYUSA 1998

[4] TWHaynes S THedetniemi and P J Slater EdsDominationin Graphs vol 209 of Monographs and Textbooks in Pure andAppliedMathematicsMarcelDekker NewYorkNYUSA 1998

[5] M R Garey andD S JohnsonComputers and Intractability WH Freeman and Company San Francisco Calif USA 1979

[6] H B Walikar and B D Acharya ldquoDomination critical graphsrdquoNational Academy Science Letters vol 2 pp 70ndash72 1979

[7] D Bauer F Harary J Nieminen and C L Suffel ldquoDominationalteration sets in graphsrdquo Discrete Mathematics vol 47 no 2-3pp 153ndash161 1983

[8] J F Fink M S Jacobson L F Kinch and J Roberts ldquoThebondage number of a graphrdquo Discrete Mathematics vol 86 no1ndash3 pp 47ndash57 1990

[9] F-T Hu and J-M Xu ldquoThe bondage number of (119899 minus 3)-regulargraph G of order nrdquo Ars Combinatoria to appear

[10] U Teschner ldquoNew results about the bondage number of agraphrdquoDiscreteMathematics vol 171 no 1ndash3 pp 249ndash259 1997

[11] B L Hartnell and D F Rall ldquoA bound on the size of a graphwith given order and bondage numberrdquo Discrete Mathematicsvol 197-198 pp 409ndash413 1999 16th British CombinatorialConference (London 1997)

[12] B L Hartnell and D F Rall ldquoA characterization of trees inwhich no edge is essential to the domination numberrdquo ArsCombinatoria vol 33 pp 65ndash76 1992

[13] J Topp and P D Vestergaard ldquo120572119896- and 120574

119896-stable graphsrdquo

Discrete Mathematics vol 212 no 1-2 pp 149ndash160 2000[14] A Meir and J W Moon ldquoRelations between packing and

covering numbers of a treerdquo Pacific Journal of Mathematics vol61 no 1 pp 225ndash233 1975

[15] B L Hartnell L K Joslashrgensen P D Vestergaard and CA Whitehead ldquoEdge stability of the k-domination numberof treesrdquo Bulletin of the Institute of Combinatorics and ItsApplications vol 22 pp 31ndash40 1998

[16] F-T Hu and J-M Xu ldquoOn the complexity of the bondage andreinforcement problemsrdquo Journal of Complexity vol 28 no 2pp 192ndash201 2012

[17] B L Hartnell and D F Rall ldquoBounds on the bondage numberof a graphrdquo Discrete Mathematics vol 128 no 1ndash3 pp 173ndash1771994

[18] Y-L Wang ldquoOn the bondage number of a graphrdquo DiscreteMathematics vol 159 no 1ndash3 pp 291ndash294 1996

[19] G H Fricke TW Haynes S M Hedetniemi S T Hedetniemiand R C Laskar ldquoExcellent treesrdquo Bulletin of the Institute ofCombinatorics and Its Applications vol 34 pp 27ndash38 2002

[20] V Samodivkin ldquoThe bondage number of graphs good and badverticesrdquoDiscussiones Mathematicae GraphTheory vol 28 no3 pp 453ndash462 2008

[21] J E Dunbar T W Haynes U Teschner and L VolkmannldquoBondage insensitivity and reinforcementrdquo in Domination inGraphs vol 209 of Monographs and Textbooks in Pure andAppliedMathematics pp 471ndash489 Dekker NewYork NY USA1998

[22] H Liu and L Sun ldquoThe bondage and connectivity of a graphrdquoDiscrete Mathematics vol 263 no 1ndash3 pp 289ndash293 2003

[23] A-H Esfahanian and S L Hakimi ldquoOn computing a con-ditional edge-connectivity of a graphrdquo Information ProcessingLetters vol 27 no 4 pp 195ndash199 1988

[24] R C Brigham P Z Chinn and R D Dutton ldquoVertexdomination-critical graphsrdquoNetworks vol 18 no 3 pp 173ndash1791988

[25] V Samodivkin ldquoDomination with respect to nondegenerateproperties bondage numberrdquoThe Australasian Journal of Com-binatorics vol 45 pp 217ndash226 2009

[26] U Teschner ldquoA new upper bound for the bondage numberof graphs with small domination numberrdquo The AustralasianJournal of Combinatorics vol 12 pp 27ndash35 1995

32 International Journal of Combinatorics

[27] J Fulman D Hanson and G MacGillivray ldquoVertex dom-ination-critical graphsrdquoNetworks vol 25 no 2 pp 41ndash43 1995

[28] U Teschner ldquoA counterexample to a conjecture on the bondagenumber of a graphrdquo Discrete Mathematics vol 122 no 1ndash3 pp393ndash395 1993

[29] U Teschner ldquoThe bondage number of a graph G can be muchgreater than Δ(G)rdquo Ars Combinatoria vol 43 pp 81ndash87 1996

[30] L Volkmann ldquoOn graphs with equal domination and coveringnumbersrdquoDiscrete AppliedMathematics vol 51 no 1-2 pp 211ndash217 1994

[31] L A Sanchis ldquoMaximum number of edges in connected graphswith a given domination numberrdquoDiscreteMathematics vol 87no 1 pp 65ndash72 1991

[32] S Klavzar and N Seifter ldquoDominating Cartesian products ofcyclesrdquoDiscrete AppliedMathematics vol 59 no 2 pp 129ndash1361995

[33] H K Kim ldquoA note on Vizingrsquos conjecture about the bondagenumberrdquo Korean Journal of Mathematics vol 10 no 2 pp 39ndash42 2003

[34] M Y Sohn X Yuan and H S Jeong ldquoThe bondage number of1198623times119862

119899rdquo Journal of the KoreanMathematical Society vol 44 no

6 pp 1213ndash1231 2007[35] L Kang M Y Sohn and H K Kim ldquoBondage number of the

discrete torus 119862119899times 119862

4rdquo Discrete Mathematics vol 303 no 1ndash3

pp 80ndash86 2005[36] J Huang and J-M Xu ldquoThe bondage numbers and efficient

dominations of vertex-transitive graphsrdquoDiscrete Mathematicsvol 308 no 4 pp 571ndash582 2008

[37] C J Xiang X Yuan andM Y Sohn ldquoDomination and bondagenumber of119862

3times119862

119899rdquoArs Combinatoria vol 97 pp 299ndash310 2010

[38] M S Jacobson and L F Kinch ldquoOn the domination numberof products of graphs Irdquo Ars Combinatoria vol 18 pp 33ndash441984

[39] Y-Q Li ldquoA note on the bondage number of a graphrdquo ChineseQuarterly Journal of Mathematics vol 9 no 4 pp 1ndash3 1994

[40] F-T Hu and J-M Xu ldquoBondage number of mesh networksrdquoFrontiers ofMathematics inChina vol 7 no 5 pp 813ndash826 2012

[41] R Frucht and F Harary ldquoOn the corona of two graphsrdquoAequationes Mathematicae vol 4 pp 322ndash325 1970

[42] K Carlson and M Develin ldquoOn the bondage number of planarand directed graphsrdquoDiscreteMathematics vol 306 no 8-9 pp820ndash826 2006

[43] U Teschner ldquoOn the bondage number of block graphsrdquo ArsCombinatoria vol 46 pp 25ndash32 1997

[44] J Huang and J-M Xu ldquoNote on conjectures of bondagenumbers of planar graphsrdquo Applied Mathematical Sciences vol6 no 65ndash68 pp 3277ndash3287 2012

[45] L Kang and J Yuan ldquoBondage number of planar graphsrdquoDiscrete Mathematics vol 222 no 1ndash3 pp 191ndash198 2000

[46] M Fischermann D Rautenbach and L Volkmann ldquoRemarkson the bondage number of planar graphsrdquo Discrete Mathemat-ics vol 260 no 1ndash3 pp 57ndash67 2003

[47] J Hou G Liu and J Wu ldquoBondage number of planar graphswithout small cyclesrdquoUtilitasMathematica vol 84 pp 189ndash1992011

[48] J Huang and J-M Xu ldquoThe bondage numbers of graphs withsmall crossing numbersrdquo Discrete Mathematics vol 307 no 15pp 1881ndash1897 2007

[49] Q Ma S Zhang and J Wang ldquoBondage number of 1-planargraphrdquo Applied Mathematics vol 1 no 2 pp 101ndash103 2010

[50] J Hou and G Liu ldquoA bound on the bondage number of toroidalgraphsrdquoDiscreteMathematics Algorithms and Applications vol4 no 3 Article ID 1250046 5 pages 2012

[51] Y-C Cao J-M Xu and X-R Xu ldquoOn bondage number oftoroidal graphsrdquo Journal of University of Science and Technologyof China vol 39 no 3 pp 225ndash228 2009

[52] Y-C Cao J Huang and J-M Xu ldquoThe bondage number ofgraphs with crossing number less than fourrdquo Ars Combinatoriato appear

[53] H K Kim ldquoOn the bondage numbers of some graphsrdquo KoreanJournal of Mathematics vol 11 no 2 pp 11ndash15 2004

[54] A Gagarin and V Zverovich ldquoUpper bounds for the bondagenumber of graphs on topological surfacesrdquo Discrete Mathemat-ics 2012

[55] J Huang ldquoAn improved upper bound for the bondage numberof graphs on surfacesrdquoDiscrete Mathematics vol 312 no 18 pp2776ndash2781 2012

[56] A Gagarin and V Zverovich ldquoThe bondage number of graphson topological surfaces and Teschnerrsquos conjecturerdquo DiscreteMathematics vol 313 no 6 pp 796ndash808 2013

[57] V Samodivkin ldquoNote on the bondage number of graphs ontopological surfacesrdquo httparxivorgabs12086203

[58] E J Cockayne R M Dawes and S T Hedetniemi ldquoTotaldomination in graphsrdquoNetworks vol 10 no 3 pp 211ndash219 1980

[59] R Laskar J Pfaff S M Hedetniemi and S T HedetniemildquoOn the algorithmic complexity of total dominationrdquo Society forIndustrial and Applied Mathematics vol 5 no 3 pp 420ndash4251984

[60] J Pfaff R C Laskar and S T Hedetniemi ldquoNP-completeness oftotal and connected domination and irredundance for bipartitegraphsrdquo Tech Rep 428 Department of Mathematical SciencesClemson University 1983

[61] M A Henning ldquoA survey of selected recent results on totaldomination in graphsrdquoDiscrete Mathematics vol 309 no 1 pp32ndash63 2009

[62] V R Kulli and D K Patwari ldquoThe total bondage number ofa graphrdquo in Advances in Graph Theory pp 227ndash235 VishwaGulbarga India 1991

[63] F-T Hu Y Lu and J-M Xu ldquoThe total bondage number ofgrid graphsrdquo Discrete Applied Mathematics vol 160 no 16-17pp 2408ndash2418 2012

[64] J Cao M Shi and B Wu ldquoResearch on the total bondagenumber of a spectial networkrdquo in Proceedings of the 3rdInternational Joint Conference on Computational Sciences andOptimization (CSO rsquo10) pp 456ndash458 May 2010

[65] N Sridharan MD Elias and V S A Subramanian ldquoTotalbondage number of a graphrdquo AKCE International Journal ofGraphs and Combinatorics vol 4 no 2 pp 203ndash209 2007

[66] N Jafari Rad and J Raczek ldquoSome progress on total bondage ingraphsrdquo Graphs and Combinatorics 2011

[67] T W Haynes and P J Slater ldquoPaired-domination and thepaired-domatic numberrdquoCongressusNumerantium vol 109 pp65ndash72 1995

[68] T W Haynes and P J Slater ldquoPaired-domination in graphsrdquoNetworks vol 32 no 3 pp 199ndash206 1998

[69] X Hou ldquoA characterization of 2120574 120574119875-treesrdquoDiscrete Mathemat-

ics vol 308 no 15 pp 3420ndash3426 2008[70] K E Proffitt T W Haynes and P J Slater ldquoPaired-domination

in grid graphsrdquo Congressus Numerantium vol 150 pp 161ndash1722001

International Journal of Combinatorics 33

[71] H Qiao L KangM Cardei andD-Z Du ldquoPaired-dominationof treesrdquo Journal of Global Optimization vol 25 no 1 pp 43ndash542003

[72] E Shan L Kang and M A Henning ldquoA characterizationof trees with equal total domination and paired-dominationnumbersrdquo The Australasian Journal of Combinatorics vol 30pp 31ndash39 2004

[73] J Raczek ldquoPaired bondage in treesrdquo Discrete Mathematics vol308 no 23 pp 5570ndash5575 2008

[74] J A Telle and A Proskurowski ldquoAlgorithms for vertex parti-tioning problems on partial k-treesrdquo SIAM Journal on DiscreteMathematics vol 10 no 4 pp 529ndash550 1997

[75] G S Domke J H Hattingh M A Henning and L R MarkusldquoRestrained domination in treesrdquoDiscreteMathematics vol 211no 1ndash3 pp 1ndash9 2000

[76] G S Domke J H Hattingh M A Henning and L R MarkusldquoRestrained domination in graphs with minimum degree twordquoJournal of Combinatorial Mathematics and Combinatorial Com-puting vol 35 pp 239ndash254 2000

[77] G S Domke J H Hattingh S T Hedetniemi R C Laskarand L R Markus ldquoRestrained domination in graphsrdquo DiscreteMathematics vol 203 no 1ndash3 pp 61ndash69 1999

[78] J H Hattingh and E J Joubert ldquoAn upper bound for therestrained domination number of a graph with minimumdegree at least two in terms of order and minimum degreerdquoDiscrete Applied Mathematics vol 157 no 13 pp 2846ndash28582009

[79] M A Henning ldquoGraphs with large restrained dominationnumberrdquo Discrete Mathematics vol 197-198 pp 415ndash429 199916th British Combinatorial Conference (London 1997)

[80] J H Hattingh and A R Plummer ldquoRestrained bondage ingraphsrdquo Discrete Mathematics vol 308 no 23 pp 5446ndash54532008

[81] N Jafari Rad R Hasni J Raczek and L Volkmann ldquoTotalrestrained bondage in graphsrdquoActaMathematica Sinica vol 29no 6 pp 1033ndash1042 2013

[82] J Cyman and J Raczek ldquoOn the total restrained dominationnumber of a graphrdquoThe Australasian Journal of Combinatoricsvol 36 pp 91ndash100 2006

[83] M A Henning ldquoGraphs with large total domination numberrdquoJournal of Graph Theory vol 35 no 1 pp 21ndash45 2000

[84] D-X Ma X-G Chen and L Sun ldquoOn total restraineddomination in graphsrdquoCzechoslovakMathematical Journal vol55 no 1 pp 165ndash173 2005

[85] B Zelinka ldquoRemarks on restrained domination and totalrestrained domination in graphsrdquo Czechoslovak MathematicalJournal vol 55 no 2 pp 393ndash396 2005

[86] W Goddard and M A Henning ldquoIndependent domination ingraphs a survey and recent resultsrdquo Discrete Mathematics vol313 no 7 pp 839ndash854 2013

[87] J H Zhang H L Liu and L Sun ldquoIndependence bondagenumber and reinforcement number of some graphsrdquo Transac-tions of Beijing Institute of Technology vol 23 no 2 pp 140ndash1422003

[88] K D Dharmalingam Studies in Graph Theorymdashequitabledomination and bottleneck domination [PhD thesis] MaduraiKamaraj University Madurai India 2006

[89] GDeepakND Soner andAAlwardi ldquoThe equitable bondagenumber of a graphrdquo Research Journal of Pure Algebra vol 1 no9 pp 209ndash212 2011

[90] E Sampathkumar and H B Walikar ldquoThe connected domina-tion number of a graphrdquo Journal of Mathematical and PhysicalSciences vol 13 no 6 pp 607ndash613 1979

[91] K White M Farber and W Pulleyblank ldquoSteiner trees con-nected domination and strongly chordal graphsrdquoNetworks vol15 no 1 pp 109ndash124 1985

[92] M Cadei M X Cheng X Cheng and D-Z Du ldquoConnecteddomination in Ad Hoc wireless networksrdquo in Proceedings ofthe 6th International Conference on Computer Science andInformatics (CSampI rsquo02) 2002

[93] J F Fink and M S Jacobson ldquon-domination in graphsrdquo inGraph Theory with Applications to Algorithms and ComputerScience Wiley-Interscience Publications pp 283ndash300 WileyNew York NY USA 1985

[94] M Chellali O Favaron A Hansberg and L Volkmann ldquok-domination and k-independence in graphs a surveyrdquo Graphsand Combinatorics vol 28 no 1 pp 1ndash55 2012

[95] Y Lu X Hou J-M Xu andN Li ldquoTrees with uniqueminimump-dominating setsrdquo Utilitas Mathematica vol 86 pp 193ndash2052011

[96] Y Lu and J-M Xu ldquoThe p-bondage number of treesrdquo Graphsand Combinatorics vol 27 no 1 pp 129ndash141 2011

[97] Y Lu and J-M Xu ldquoThe 2-domination and 2-bondage numbersof grid graphs A manuscriptrdquo httparxivorgabs12044514

[98] M Krzywkowski ldquoNon-isolating 2-bondage in graphsrdquo TheJournal of the Mathematical Society of Japan vol 64 no 4 2012

[99] M A Henning O R Oellermann and H C Swart ldquoBoundson distance domination parametersrdquo Journal of CombinatoricsInformation amp System Sciences vol 16 no 1 pp 11ndash18 1991

[100] F Tian and J-M Xu ldquoBounds for distance domination numbersof graphsrdquo Journal of University of Science and Technology ofChina vol 34 no 5 pp 529ndash534 2004

[101] F Tian and J-M Xu ldquoDistance domination-critical graphsrdquoApplied Mathematics Letters vol 21 no 4 pp 416ndash420 2008

[102] F Tian and J-M Xu ldquoA note on distance domination numbersof graphsrdquo The Australasian Journal of Combinatorics vol 43pp 181ndash190 2009

[103] F Tian and J-M Xu ldquoAverage distances and distance domina-tion numbersrdquo Discrete Applied Mathematics vol 157 no 5 pp1113ndash1127 2009

[104] Z-L Liu F Tian and J-M Xu ldquoProbabilistic analysis of upperbounds for 2-connected distance k-dominating sets in graphsrdquoTheoretical Computer Science vol 410 no 38ndash40 pp 3804ndash3813 2009

[105] M A Henning O R Oellermann and H C Swart ldquoDistancedomination critical graphsrdquo Journal of Combinatorial Mathe-matics and Combinatorial Computing vol 44 pp 33ndash45 2003

[106] T J Bean M A Henning and H C Swart ldquoOn the integrityof distance domination in graphsrdquo The Australasian Journal ofCombinatorics vol 10 pp 29ndash43 1994

[107] M Fischermann and L Volkmann ldquoA remark on a conjecturefor the (kp)-domination numberrdquoUtilitasMathematica vol 67pp 223ndash227 2005

[108] T Korneffel D Meierling and L Volkmann ldquoA remark on the(22)-domination numberrdquo Discussiones Mathematicae GraphTheory vol 28 no 2 pp 361ndash366 2008

[109] Y Lu X Hou and J-M Xu ldquoOn the (22)-domination numberof treesrdquo Discussiones Mathematicae Graph Theory vol 30 no2 pp 185ndash199 2010

34 International Journal of Combinatorics

[110] H Li and J-M Xu ldquo(dm)-domination numbers of m-connected graphsrdquo Rapports de Recherche 1130 LRI UniversiteParis-Sud 1997

[111] S M Hedetniemi S T Hedetniemi and T V Wimer ldquoLineartime resource allocation algorithms for treesrdquo Tech Rep URI-014 Department of Mathematical Sciences Clemson Univer-sity 1987

[112] M Farber ldquoDomination independent domination and dualityin strongly chordal graphsrdquo Discrete Applied Mathematics vol7 no 2 pp 115ndash130 1984

[113] G S Domke and R C Laskar ldquoThe bondage and reinforcementnumbers of 120574

119891for some graphsrdquoDiscrete Mathematics vol 167-

168 pp 249ndash259 1997[114] G S Domke S T Hedetniemi and R C Laskar ldquoFractional

packings coverings and irredundance in graphsrdquo vol 66 pp227ndash238 1988

[115] E J Cockayne P A Dreyer Jr S M Hedetniemi andS T Hedetniemi ldquoRoman domination in graphsrdquo DiscreteMathematics vol 278 no 1ndash3 pp 11ndash22 2004

[116] I Stewart ldquoDefend the roman empirerdquo Scientific American vol281 pp 136ndash139 1999

[117] C S ReVelle and K E Rosing ldquoDefendens imperiumromanum a classical problem in military strategyrdquoThe Ameri-can Mathematical Monthly vol 107 no 7 pp 585ndash594 2000

[118] A Bahremandpour F-T Hu S M Sheikholeslami and J-MXu ldquoOn the Roman bondage number of a graphrdquo DiscreteMathematics Algorithms and Applications vol 5 no 1 ArticleID 1350001 15 pages 2013

[119] N Jafari Rad and L Volkmann ldquoRoman bondage in graphsrdquoDiscussiones Mathematicae Graph Theory vol 31 no 4 pp763ndash773 2011

[120] K Ebadi and L PushpaLatha ldquoRoman bondage and Romanreinforcement numbers of a graphrdquo International Journal ofContemporary Mathematical Sciences vol 5 pp 1487ndash14972010

[121] N Dehgardi S M Sheikholeslami and L Volkman ldquoOn theRoman k-bondage number of a graphrdquo AKCE InternationalJournal of Graphs and Combinatorics vol 8 pp 169ndash180 2011

[122] S Akbari and S Qajar ldquoA note on Roman bondage number ofgraphsrdquo Ars Combinatoria to Appear

[123] F-T Hu and J-M Xu ldquoRoman bondage numbers of somegraphsrdquo httparxivorgabs11093933

[124] N Jafari Rad and L Volkmann ldquoOn the Roman bondagenumber of planar graphsrdquo Graphs and Combinatorics vol 27no 4 pp 531ndash538 2011

[125] S Akbari M Khatirinejad and S Qajar ldquoA note on the romanbondage number of planar graphsrdquo Graphs and Combinatorics2012

[126] B Bresar M A Henning and D F Rall ldquoRainbow dominationin graphsrdquo Taiwanese Journal of Mathematics vol 12 no 1 pp213ndash225 2008

[127] B L Hartnell and D F Rall ldquoOn dominating the Cartesianproduct of a graph and 120572krdquo Discussiones Mathematicae GraphTheory vol 24 no 3 pp 389ndash402 2004

[128] N Dehgardi S M Sheikholeslami and L Volkman ldquoThe k-rainbow bondage number of a graphrdquo to appear in DiscreteApplied Mathematics

[129] J von Neumann and O Morgenstern Theory of Games andEconomic Behavior Princeton University Press Princeton NJUSA 1944

[130] C BergeTheorıe des Graphes et ses Applications 1958[131] O Ore Theory of Graphs vol 38 American Mathematical

Society Colloquium Publications 1962[132] E Shan and L Kang ldquoBondage number in oriented graphsrdquoArs

Combinatoria vol 84 pp 319ndash331 2007[133] J Huang and J-M Xu ldquoThe bondage numbers of extended

de Bruijn and Kautz digraphsrdquo Computers amp Mathematics withApplications vol 51 no 6-7 pp 1137ndash1147 2006

[134] J Huang and J-M Xu ldquoThe total domination and total bondagenumbers of extended de Bruijn and Kautz digraphsrdquoComputersamp Mathematics with Applications vol 53 no 8 pp 1206ndash12132007

[135] Y Shibata and Y Gonda ldquoExtension of de Bruijn graph andKautz graphrdquo Computers amp Mathematics with Applications vol30 no 9 pp 51ndash61 1995

[136] X Zhang J Liu and JMeng ldquoThebondage number in completet-partite digraphsrdquo Information Processing Letters vol 109 no17 pp 997ndash1000 2009

[137] D W Bange A E Barkauskas and P J Slater ldquoEfficient dom-inating sets in graphsrdquo in Applications of Discrete Mathematicspp 189ndash199 SIAM Philadelphia Pa USA 1988

[138] M Livingston and Q F Stout ldquoPerfect dominating setsrdquoCongressus Numerantium vol 79 pp 187ndash203 1990

[139] L Clark ldquoPerfect domination in random graphsrdquo Journal ofCombinatorial Mathematics and Combinatorial Computing vol14 pp 173ndash182 1993

[140] D W Bange A E Barkauskas L H Host and L H ClarkldquoEfficient domination of the orientations of a graphrdquo DiscreteMathematics vol 178 no 1ndash3 pp 1ndash14 1998

[141] A E Barkauskas and L H Host ldquoFinding efficient dominatingsets in oriented graphsrdquo Congressus Numerantium vol 98 pp27ndash32 1993

[142] I J Dejter and O Serra ldquoEfficient dominating sets in CayleygraphsrdquoDiscrete AppliedMathematics vol 129 no 2-3 pp 319ndash328 2003

[143] J Lee ldquoIndependent perfect domination sets in Cayley graphsrdquoJournal of Graph Theory vol 37 no 4 pp 213ndash219 2001

[144] WGu X Jia and J Shen ldquoIndependent perfect domination setsin meshes tori and treesrdquo Unpublished

[145] N Obradovic J Peters and G Ruzic ldquoEfficient domination incirculant graphswith two chord lengthsrdquo Information ProcessingLetters vol 102 no 6 pp 253ndash258 2007

[146] K Reji Kumar and G MacGillivray ldquoEfficient domination incirculant graphsrdquo Discrete Mathematics vol 313 no 6 pp 767ndash771 2013

[147] D Van Wieren M Livingston and Q F Stout ldquoPerfectdominating sets on cube-connected cyclesrdquo Congressus Numer-antium vol 97 pp 51ndash70 1993

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

International Journal of Combinatorics 29

Theorem 186 (Huang and Xu [134] 2007) For a directedcycle 119862

119899and a directed path 119875

119899 119887

119879(119875

119899) and 119887

119879(119862

119899) all do not

exist For a complete digraph 119870119899

119887119879(119870

119899) = infin 119894119891 119899 = 1 2

119887119879(119870

119899) = 3 119894119891 119899 = 3

119899 ⩽ 119887119879(119870

119899) ⩽ 2119899 minus 3 119894119891 119899 ⩾ 4

(78)

The extended de Bruijn digraph EB(119889 119899 1199021 119902

119901) and

the extended Kautz digraph EK(119889 119899 1199021 119902

119901) are intro-

duced by Shibata and Gonda [135] If 119901 = 1 then theyare the de Bruijn digraph B(119889 119899) and the Kautz digraphK(119889 119899) respectively Huang and Xu [134] determined theirtotal domination numbers In particular their total bondagenumbers for general cases are determined as follows

Theorem 187 (Huang andXu [134] 2007) If 119889 ⩾ 2 and 119902119894⩾ 2

for each 119894 = 1 2 119901 then

119887119879(EB(119889 119899 119902

1 119902

119901)) = 119889

119901

minus 1

119887119879(EK(119889 119899 119902

1 119902

119901)) = 119889

119901

(79)

In particular for the de Bruijn digraph B(119889 119899) and the Kautzdigraph K(119889 119899)

119887119879(B (119889 119899)) = 119889 minus 1 119887

119879(K (119889 119899)) = 119889 (80)

Zhang et al [136] determined the bondage number incomplete 119905-partite digraphs

Theorem 188 (Zhang et al [136] 2009) For a complete 119905-partite digraph 119870

11989911198992119899119905

where 1198991⩽ 119899

2⩽ sdot sdot sdot ⩽ 119899

119905

119887 (11987011989911198992119899119905

)

=

119898 119894119891 119899119898= 1 119899

119898+1⩾ 2 119891119900119903 119904119900119898119890

119898 (1 ⩽ 119898 lt 119905)

4119905 minus 3 119894119891 1198991= 119899

2= sdot sdot sdot = 119899

119905= 2

119905minus1

sum

119894=1

119899119894+ 2 (119905 minus 1) 119900119905ℎ119890119903119908119894119904119890

(81)

Since an undirected graph can be thought of as a sym-metric digraph any result for digraphs has an analogy forundirected graphs in general In view of this point studyingthe bondage number for digraphs is more significant thanfor undirected graphs Thus we should further study thebondage number of digraphs and try to generalize knownresults on the bondage number and related variants for undi-rected graphs to digraphs prove or disprove Conjecture 183In particular determine the exact values of 119887(B(119889 119899)) for aneven 119899 and 119887

119879(119870

119899) for 119899 ⩾ 4

11 Efficient Dominating Sets

A dominating set 119878 of a graph 119866 is called to be efficient if forevery vertex 119909 in119866 |119873

119866[119909]cap119878| = 1 if119866 is a undirected graph

or |119873minus

119866[119909] cap 119878| = 1 if 119866 is a directed graph The concept of an

efficient set was proposed by Bange et al [137] in 1988 In theliterature an efficient set is also called a perfect dominatingset (see Livingston and Stout [138] for an explanation)

From definition if 119878 is an efficient dominating set of agraph 119866 then 119878 is certainly an independent set and everyvertex not in 119878 is adjacent to exactly one vertex in 119878

It is also clear from definition that a dominating set 119878 isefficient if and only if N

119866[119878] = 119873

119866[119909] 119909 isin 119878 for the

undirected graph 119866 or N+

119866[119878] = 119873

+

119866[119909] 119909 isin 119878 for the

digraph119866 is a partition of119881(119866) where the induced subgraphby119873

119866[119909] or119873+

119866[119909] is a star or an out-star with the root 119909

The efficient domination has important applications inmany areas such as error-correcting codes and receivesmuch attention in the late years

The concept of efficient dominating sets is a measureof the efficiency of domination in graphs Unfortunately asshown in [137] not every graph has an efficient dominatingset andmoreover it is anNP-complete problem to determinewhether a given graph has an efficient dominating set Inaddition it was shown by Clark [139] in 1993 that for a widerange of 119901 almost every random undirected graph 119866 isin

G(120592 119901) has no efficient dominating sets This means thatundirected graphs possessing an efficient dominating set arerare However it is easy to show that every undirected graphhas an orientationwith an efficient dominating set (see Bangeet al [140])

In 1993 Barkauskas and Host [141] showed that deter-mining whether an arbitrary oriented graph has an efficientdominating set is NP-complete Even so the existence ofefficient dominating sets for some special classes of graphs hasbeen examined see for example Dejter and Serra [142] andLee [143] for Cayley graph Gu et al [144] formeshes and toriObradovic et al [145] Huang and Xu [36] Reji Kumar andMacGillivray [146] for circulant graphs Harary graphs andtori and Van Wieren et al [147] for cube-connected cycles

In this section we introduce results of the bondagenumber for some graphs with efficient dominating sets dueto Huang and Xu [36]

111 Results for General Graphs In this subsection we intro-duce some results on bondage numbers obtained by applyingefficient dominating sets We first state the two followinglemmas

Lemma 189 For any 119896-regular graph or digraph 119866 of order 119899120574(119866) ⩾ 119899(119896 + 1) with equality if and only if 119866 has an efficientdominating set In addition if 119866 has an efficient dominatingset then every efficient dominating set is certainly a 120574-set andvice versa

Let 119890 be an edge and 119878 a dominating set in 119866 We say that119890 supports 119878 if 119890 isin (119878 119878) where (119878 119878) = (119909 119910) isin 119864(119866) 119909 isin 119878 119910 isin 119878 Denote by 119904(119866) the minimum number of edgeswhich support all 120574-sets in 119866

Lemma 190 For any graph or digraph 119866 119887(119866) ⩾ 119904(119866) withequality if 119866 is regular and has an efficient dominating set

30 International Journal of Combinatorics

A graph 119866 is called to be vertex-transitive if its automor-phism group Aut(119866) acts transitively on its vertex-set 119881(119866)A vertex-transitive graph is regular Applying Lemmas 189and 190 Huang and Xu obtained some results on bondagenumbers for vertex-transitive graphs or digraphs

Theorem 191 (Huang and Xu [36] 2008) For any vertex-transitive graph or digraph 119866 of order 119899

119887 (119866) ⩾

lceil119899

2120574 (119866)rceil 119894119891 119866 119894119904 119906119899119889119894119903119890119888119905119890119889

lceil119899

120574 (119866)rceil 119894119891 119866 119894119904 119889119894119903119890119888119905119890119889

(82)

Theorem192 (Huang andXu [36] 2008) Let119866 be a 119896-regulargraph of order 119899 Then

119887(119866) ⩽ 119896 if119866 is undirected and 119899 ⩾ 120574(119866)(119896+1)minus119896+1119887(119866) ⩽ 119896+1+ℓ if119866 is directed and 119899 ⩾ 120574(119866)(119896+1)minusℓwith 0 ⩽ ℓ ⩽ 119896 minus 1

Next we introduce a better upper bound on 119887(119866) To thisaim we need the following concept which is a generalizationof the concept of the edge-covering of a graph 119866 For 1198811015840

sube

119881(119866) and 1198641015840 sube 119864(119866) we say that 1198641015840covers 1198811015840 and call 1198641015840an edge-covering for 1198811015840 if there exists an edge (119909 119910) isin 1198641015840 forany vertex 119909 isin 1198811015840 For 119910 isin 119881(119866) let 1205731015840[119910] be the minimumcardinality over all edge-coverings for119873minus

119866[119910]

Theorem 193 (Huang and Xu [36] 2008) If 119866 is a 119896-regulargraph with order 120574(119866)(119896 + 1) then 119887(119866) ⩽ 1205731015840[119910] for any 119910 isin119881(119866)

Theupper bound on 119887(119866) given inTheorem 193 is tight inview of 119887(119862

119899) for a cycle or a directed cycle 119862

119899(seeTheorems

1 and 185 resp)It is easy to see that for a 119896-regular graph 119866 lceil(119896 + 1)2rceil ⩽

1205731015840

[119910] ⩽ 119896 when 119866 is undirected and 1205731015840[119910] = 119896 + 1 when 119866 isdirected By this fact and Lemma 189 the following theoremis merely a simple combination of Theorems 191 and 193

Theorem 194 (Huang and Xu [36] 2008) Let 119866 be a vertex-transitive graph of degree 119896 If 119866 has an efficient dominatingset then

lceil119896 + 1

2rceil ⩽ 119887 (119866) ⩽ 119896 119894119891 119866 119894119904 119906119899119889119894119903119890119888119905119890119889

119887 (119866) = 119896 + 1 119894119891 119866 119894119904 119889119894119903119890119888119905119890119889

(83)

Theorem 195 (Huang and Xu [36] 2008) If 119866 is an undi-rected vertex-transitive cubic graph with order 4120574(119866) and girth119892(119866) ⩽ 5 then 119887(119866) = 2

The proof of Theorem 195 leads to a byproduct If 119892 =5 let (119906

1 119906

2 119906

3 119906

4 119906

5) be a cycle and let D

119894be the family

of efficient dominating sets containing 119906119894for each 119894 isin

1 2 3 4 5 ThenD119894capD

119895= 0 for 119894 = 1 2 5 Then 119866 has

at least 5119904 efficient dominating sets where 119904 = 119904(119866) But thereare only 4s (=ns120574) distinct efficient dominating sets in119866This

contradiction implies that an undirected vertex-transitivecubic graph with girth five has no efficient dominating setsBut a similar argument for 119892(119866) = 3 4 or 119892(119866) ⩾ 6 couldnot give any contradiction This is consistent with the resultthat CCC(119899) a vertex-transitive cubic graph with girth 119899 if3 ⩽ 119899 ⩽ 8 or girth 8 if 119899 ⩾ 9 has efficient dominating setsfor all 119899 ⩾ 3 except 119899 = 5 (see Theorem 199 in the followingsubsection)

112 Results for Cayley Graphs In this subsection we useTheorem 194 to determine the exact values or approximativevalues of bondage numbers for some special vertex-transitivegraphs by characterizing the existence of efficient dominatingsets in these graphs

Let Γ be a nontrivial finite group 119878 a nonempty subset ofΓ without the identity element of Γ A digraph 119866 is defined asfollows

119881 (119866) = Γ (119909 119910) isin 119864 (119866) lArrrArr 119909minus1

119910 isin 119878

for any 119909 119910 isin Γ(84)

is called a Cayley digraph of the group Γ with respect to119878 denoted by 119862

Γ(119878) If 119878minus1 = 119904minus1 119904 isin 119878 = 119878 then

119862Γ(119878) is symmetric and is called a Cayley undirected graph

a Cayley graph for short Cayley graphs or digraphs arecertainly vertex-transitive (see Xu [1])

A circulant graph 119866(119899 119878) of order 119899 is a Cayley graph119862119885119899

(119878) where 119885119899= 0 119899 minus 1 is the addition group of

order 119899 and 119878 is a nonempty subset of119885119899without the identity

element and hence is a vertex-transitive digraph of degree|119878| If 119878minus1 = 119878 then119866(119899 119878) is an undirected graph If 119878 = 1 119896where 2 ⩽ 119896 ⩽ 119899 minus 2 we write 119866(119899 1 119896) for 119866(119899 1 119896) or119866(119899 plusmn1 plusmn119896) and call it a double-loop circulant graph

Huang and Xu [36] showed that for directed 119866 =

119866(119899 1 119896) lceil1198993rceil ⩽ 120574(119866) ⩽ lceil1198992rceil and 119866 has an efficientdominating set if and only if 3 | 119899 and 119896 equiv 2 (mod 3) fordirected 119866 = 119866(119899 1 119896) and 119896 = 1198992 lceil1198995rceil ⩽ 120574(119866) ⩽ lceil1198993rceiland 119866 has an efficient dominating set if and only if 5 | 119899 and119896 equiv plusmn2 (mod 5) By Theorem 194 we obtain the bondagenumber of a double loop circulant graph if it has an efficientdominating set

Theorem 196 (Huang and Xu [36] 2008) Let 119866 be a double-loop circulant graph 119866(119899 1 119896) If 119866 is directed with 3 | 119899and 119896 equiv 2 (mod 3) or 119866 is undirected with 5 | 119899 and119896 equiv plusmn2 (mod 5) then 119887(119866) = 3

The 119898 times 119899 torus is the Cartesian product 119862119898times 119862

119899of

two cycles and is a Cayley graph 119862119885119898times119885119899

(119878) where 119878 =

(0 1) (1 0) for directed cycles and 119878 = (0 plusmn1) (plusmn1 0) forundirected cycles and hence is vertex-transitive Gu et al[144] showed that the undirected torus119862

119898times119862

119899has an efficient

dominating set if and only if both 119898 and 119899 are multiplesof 5 Huang and Xu [36] showed that the directed torus119862119898times 119862

119899has an efficient dominating set if and only if both

119898 and 119899 are multiples of 3 Moreover they found a necessarycondition for a dominating set containing the vertex (0 0)in 119862

119898times 119862

119899to be efficient and obtained the following

result

International Journal of Combinatorics 31

Theorem 197 (Huang and Xu [36] 2008) Let 119866 = 119862119898times 119862

119899

If 119866 is undirected and both119898 and 119899 are multiples of 5 or if 119866 isdirected and both119898 and 119899 are multiples of 3 then 119887(119866) = 3

The hypercube 119876119899

is the Cayley graph 119862Γ(119878)

where Γ = 1198852times sdot sdot sdot times 119885

2= (119885

2)119899 and 119878 =

100 sdot sdot sdot 0 010 sdot sdot sdot 0 00 sdot sdot sdot 01 Lee [143] showed that119876119899has an efficient dominating set if and only if 119899 = 2119898 minus 1

for a positive integer119898 ByTheorem 194 the following resultis obtained immediately

Theorem 198 (Huang and Xu [36] 2008) If 119899 = 2119898 minus 1 for apositive integer119898 then 2119898minus1

⩽ 119887(119876119899) ⩽ 2

119898

minus 1

Problem 10 Determine the exact value of 119887(119876119899) for 119899 = 2119898minus1

The 119899-dimensional cube-connected cycle denoted byCCC(119899) is constructed from the 119899-dimensional hypercube119876119899by replacing each vertex 119909 in 119876

119899with an undirected cycle

119862119899of length 119899 and linking the 119894th vertex of the 119862

119899to the 119894th

neighbor of 119909 It has been proved that CCC(119899) is a Cayleygraph and hence is a vertex-transitive graph with degree 3Van Wieren et al [147] proved that CCC(119899) has an efficientdominating set if and only if 119899 = 5 From Theorems 194 and195 the following result can be derived

Theorem 199 (Huang and Xu [36] 2008) Let 119866 = 119862119862119862(119899)be the 119899-dimensional cube-connected cycles with 119899 ⩾ 3 and119899 = 5 Then 120574(119866) = 1198992119899minus2 and 2 ⩽ 119887(119866) ⩽ 3 In addition119887(119862119862119862(3)) = 119887(119862119862119862(4)) = 2

Problem 11 Determine the exact value of 119887(CCC(119899)) for 119899 ⩾5

Acknowledgments

This work was supported by NNSF of China (Grants nos11071233 61272008) The author would like to express hisgratitude to the anonymous referees for their kind sugges-tions and comments on the original paper to ProfessorSheikholeslami and Professor Jafari Rad for sending thereferences [128] and [81] which resulted in this version

References

[1] J-M Xu Theory and Application of Graphs vol 10 of NetworkTheory and Applications Kluwer Academic Dordrecht TheNetherlands 2003

[2] S THedetniemi andR C Laskar ldquoBibliography on dominationin graphs and some basic definitions of domination parame-tersrdquo Discrete Mathematics vol 86 no 1ndash3 pp 257ndash277 1990

[3] TW Haynes S T Hedetniemi and P J Slater Fundamentals ofDomination in Graphs vol 208 ofMonographs and Textbooks inPure and Applied Mathematics Marcel Dekker New York NYUSA 1998

[4] TWHaynes S THedetniemi and P J Slater EdsDominationin Graphs vol 209 of Monographs and Textbooks in Pure andAppliedMathematicsMarcelDekker NewYorkNYUSA 1998

[5] M R Garey andD S JohnsonComputers and Intractability WH Freeman and Company San Francisco Calif USA 1979

[6] H B Walikar and B D Acharya ldquoDomination critical graphsrdquoNational Academy Science Letters vol 2 pp 70ndash72 1979

[7] D Bauer F Harary J Nieminen and C L Suffel ldquoDominationalteration sets in graphsrdquo Discrete Mathematics vol 47 no 2-3pp 153ndash161 1983

[8] J F Fink M S Jacobson L F Kinch and J Roberts ldquoThebondage number of a graphrdquo Discrete Mathematics vol 86 no1ndash3 pp 47ndash57 1990

[9] F-T Hu and J-M Xu ldquoThe bondage number of (119899 minus 3)-regulargraph G of order nrdquo Ars Combinatoria to appear

[10] U Teschner ldquoNew results about the bondage number of agraphrdquoDiscreteMathematics vol 171 no 1ndash3 pp 249ndash259 1997

[11] B L Hartnell and D F Rall ldquoA bound on the size of a graphwith given order and bondage numberrdquo Discrete Mathematicsvol 197-198 pp 409ndash413 1999 16th British CombinatorialConference (London 1997)

[12] B L Hartnell and D F Rall ldquoA characterization of trees inwhich no edge is essential to the domination numberrdquo ArsCombinatoria vol 33 pp 65ndash76 1992

[13] J Topp and P D Vestergaard ldquo120572119896- and 120574

119896-stable graphsrdquo

Discrete Mathematics vol 212 no 1-2 pp 149ndash160 2000[14] A Meir and J W Moon ldquoRelations between packing and

covering numbers of a treerdquo Pacific Journal of Mathematics vol61 no 1 pp 225ndash233 1975

[15] B L Hartnell L K Joslashrgensen P D Vestergaard and CA Whitehead ldquoEdge stability of the k-domination numberof treesrdquo Bulletin of the Institute of Combinatorics and ItsApplications vol 22 pp 31ndash40 1998

[16] F-T Hu and J-M Xu ldquoOn the complexity of the bondage andreinforcement problemsrdquo Journal of Complexity vol 28 no 2pp 192ndash201 2012

[17] B L Hartnell and D F Rall ldquoBounds on the bondage numberof a graphrdquo Discrete Mathematics vol 128 no 1ndash3 pp 173ndash1771994

[18] Y-L Wang ldquoOn the bondage number of a graphrdquo DiscreteMathematics vol 159 no 1ndash3 pp 291ndash294 1996

[19] G H Fricke TW Haynes S M Hedetniemi S T Hedetniemiand R C Laskar ldquoExcellent treesrdquo Bulletin of the Institute ofCombinatorics and Its Applications vol 34 pp 27ndash38 2002

[20] V Samodivkin ldquoThe bondage number of graphs good and badverticesrdquoDiscussiones Mathematicae GraphTheory vol 28 no3 pp 453ndash462 2008

[21] J E Dunbar T W Haynes U Teschner and L VolkmannldquoBondage insensitivity and reinforcementrdquo in Domination inGraphs vol 209 of Monographs and Textbooks in Pure andAppliedMathematics pp 471ndash489 Dekker NewYork NY USA1998

[22] H Liu and L Sun ldquoThe bondage and connectivity of a graphrdquoDiscrete Mathematics vol 263 no 1ndash3 pp 289ndash293 2003

[23] A-H Esfahanian and S L Hakimi ldquoOn computing a con-ditional edge-connectivity of a graphrdquo Information ProcessingLetters vol 27 no 4 pp 195ndash199 1988

[24] R C Brigham P Z Chinn and R D Dutton ldquoVertexdomination-critical graphsrdquoNetworks vol 18 no 3 pp 173ndash1791988

[25] V Samodivkin ldquoDomination with respect to nondegenerateproperties bondage numberrdquoThe Australasian Journal of Com-binatorics vol 45 pp 217ndash226 2009

[26] U Teschner ldquoA new upper bound for the bondage numberof graphs with small domination numberrdquo The AustralasianJournal of Combinatorics vol 12 pp 27ndash35 1995

32 International Journal of Combinatorics

[27] J Fulman D Hanson and G MacGillivray ldquoVertex dom-ination-critical graphsrdquoNetworks vol 25 no 2 pp 41ndash43 1995

[28] U Teschner ldquoA counterexample to a conjecture on the bondagenumber of a graphrdquo Discrete Mathematics vol 122 no 1ndash3 pp393ndash395 1993

[29] U Teschner ldquoThe bondage number of a graph G can be muchgreater than Δ(G)rdquo Ars Combinatoria vol 43 pp 81ndash87 1996

[30] L Volkmann ldquoOn graphs with equal domination and coveringnumbersrdquoDiscrete AppliedMathematics vol 51 no 1-2 pp 211ndash217 1994

[31] L A Sanchis ldquoMaximum number of edges in connected graphswith a given domination numberrdquoDiscreteMathematics vol 87no 1 pp 65ndash72 1991

[32] S Klavzar and N Seifter ldquoDominating Cartesian products ofcyclesrdquoDiscrete AppliedMathematics vol 59 no 2 pp 129ndash1361995

[33] H K Kim ldquoA note on Vizingrsquos conjecture about the bondagenumberrdquo Korean Journal of Mathematics vol 10 no 2 pp 39ndash42 2003

[34] M Y Sohn X Yuan and H S Jeong ldquoThe bondage number of1198623times119862

119899rdquo Journal of the KoreanMathematical Society vol 44 no

6 pp 1213ndash1231 2007[35] L Kang M Y Sohn and H K Kim ldquoBondage number of the

discrete torus 119862119899times 119862

4rdquo Discrete Mathematics vol 303 no 1ndash3

pp 80ndash86 2005[36] J Huang and J-M Xu ldquoThe bondage numbers and efficient

dominations of vertex-transitive graphsrdquoDiscrete Mathematicsvol 308 no 4 pp 571ndash582 2008

[37] C J Xiang X Yuan andM Y Sohn ldquoDomination and bondagenumber of119862

3times119862

119899rdquoArs Combinatoria vol 97 pp 299ndash310 2010

[38] M S Jacobson and L F Kinch ldquoOn the domination numberof products of graphs Irdquo Ars Combinatoria vol 18 pp 33ndash441984

[39] Y-Q Li ldquoA note on the bondage number of a graphrdquo ChineseQuarterly Journal of Mathematics vol 9 no 4 pp 1ndash3 1994

[40] F-T Hu and J-M Xu ldquoBondage number of mesh networksrdquoFrontiers ofMathematics inChina vol 7 no 5 pp 813ndash826 2012

[41] R Frucht and F Harary ldquoOn the corona of two graphsrdquoAequationes Mathematicae vol 4 pp 322ndash325 1970

[42] K Carlson and M Develin ldquoOn the bondage number of planarand directed graphsrdquoDiscreteMathematics vol 306 no 8-9 pp820ndash826 2006

[43] U Teschner ldquoOn the bondage number of block graphsrdquo ArsCombinatoria vol 46 pp 25ndash32 1997

[44] J Huang and J-M Xu ldquoNote on conjectures of bondagenumbers of planar graphsrdquo Applied Mathematical Sciences vol6 no 65ndash68 pp 3277ndash3287 2012

[45] L Kang and J Yuan ldquoBondage number of planar graphsrdquoDiscrete Mathematics vol 222 no 1ndash3 pp 191ndash198 2000

[46] M Fischermann D Rautenbach and L Volkmann ldquoRemarkson the bondage number of planar graphsrdquo Discrete Mathemat-ics vol 260 no 1ndash3 pp 57ndash67 2003

[47] J Hou G Liu and J Wu ldquoBondage number of planar graphswithout small cyclesrdquoUtilitasMathematica vol 84 pp 189ndash1992011

[48] J Huang and J-M Xu ldquoThe bondage numbers of graphs withsmall crossing numbersrdquo Discrete Mathematics vol 307 no 15pp 1881ndash1897 2007

[49] Q Ma S Zhang and J Wang ldquoBondage number of 1-planargraphrdquo Applied Mathematics vol 1 no 2 pp 101ndash103 2010

[50] J Hou and G Liu ldquoA bound on the bondage number of toroidalgraphsrdquoDiscreteMathematics Algorithms and Applications vol4 no 3 Article ID 1250046 5 pages 2012

[51] Y-C Cao J-M Xu and X-R Xu ldquoOn bondage number oftoroidal graphsrdquo Journal of University of Science and Technologyof China vol 39 no 3 pp 225ndash228 2009

[52] Y-C Cao J Huang and J-M Xu ldquoThe bondage number ofgraphs with crossing number less than fourrdquo Ars Combinatoriato appear

[53] H K Kim ldquoOn the bondage numbers of some graphsrdquo KoreanJournal of Mathematics vol 11 no 2 pp 11ndash15 2004

[54] A Gagarin and V Zverovich ldquoUpper bounds for the bondagenumber of graphs on topological surfacesrdquo Discrete Mathemat-ics 2012

[55] J Huang ldquoAn improved upper bound for the bondage numberof graphs on surfacesrdquoDiscrete Mathematics vol 312 no 18 pp2776ndash2781 2012

[56] A Gagarin and V Zverovich ldquoThe bondage number of graphson topological surfaces and Teschnerrsquos conjecturerdquo DiscreteMathematics vol 313 no 6 pp 796ndash808 2013

[57] V Samodivkin ldquoNote on the bondage number of graphs ontopological surfacesrdquo httparxivorgabs12086203

[58] E J Cockayne R M Dawes and S T Hedetniemi ldquoTotaldomination in graphsrdquoNetworks vol 10 no 3 pp 211ndash219 1980

[59] R Laskar J Pfaff S M Hedetniemi and S T HedetniemildquoOn the algorithmic complexity of total dominationrdquo Society forIndustrial and Applied Mathematics vol 5 no 3 pp 420ndash4251984

[60] J Pfaff R C Laskar and S T Hedetniemi ldquoNP-completeness oftotal and connected domination and irredundance for bipartitegraphsrdquo Tech Rep 428 Department of Mathematical SciencesClemson University 1983

[61] M A Henning ldquoA survey of selected recent results on totaldomination in graphsrdquoDiscrete Mathematics vol 309 no 1 pp32ndash63 2009

[62] V R Kulli and D K Patwari ldquoThe total bondage number ofa graphrdquo in Advances in Graph Theory pp 227ndash235 VishwaGulbarga India 1991

[63] F-T Hu Y Lu and J-M Xu ldquoThe total bondage number ofgrid graphsrdquo Discrete Applied Mathematics vol 160 no 16-17pp 2408ndash2418 2012

[64] J Cao M Shi and B Wu ldquoResearch on the total bondagenumber of a spectial networkrdquo in Proceedings of the 3rdInternational Joint Conference on Computational Sciences andOptimization (CSO rsquo10) pp 456ndash458 May 2010

[65] N Sridharan MD Elias and V S A Subramanian ldquoTotalbondage number of a graphrdquo AKCE International Journal ofGraphs and Combinatorics vol 4 no 2 pp 203ndash209 2007

[66] N Jafari Rad and J Raczek ldquoSome progress on total bondage ingraphsrdquo Graphs and Combinatorics 2011

[67] T W Haynes and P J Slater ldquoPaired-domination and thepaired-domatic numberrdquoCongressusNumerantium vol 109 pp65ndash72 1995

[68] T W Haynes and P J Slater ldquoPaired-domination in graphsrdquoNetworks vol 32 no 3 pp 199ndash206 1998

[69] X Hou ldquoA characterization of 2120574 120574119875-treesrdquoDiscrete Mathemat-

ics vol 308 no 15 pp 3420ndash3426 2008[70] K E Proffitt T W Haynes and P J Slater ldquoPaired-domination

in grid graphsrdquo Congressus Numerantium vol 150 pp 161ndash1722001

International Journal of Combinatorics 33

[71] H Qiao L KangM Cardei andD-Z Du ldquoPaired-dominationof treesrdquo Journal of Global Optimization vol 25 no 1 pp 43ndash542003

[72] E Shan L Kang and M A Henning ldquoA characterizationof trees with equal total domination and paired-dominationnumbersrdquo The Australasian Journal of Combinatorics vol 30pp 31ndash39 2004

[73] J Raczek ldquoPaired bondage in treesrdquo Discrete Mathematics vol308 no 23 pp 5570ndash5575 2008

[74] J A Telle and A Proskurowski ldquoAlgorithms for vertex parti-tioning problems on partial k-treesrdquo SIAM Journal on DiscreteMathematics vol 10 no 4 pp 529ndash550 1997

[75] G S Domke J H Hattingh M A Henning and L R MarkusldquoRestrained domination in treesrdquoDiscreteMathematics vol 211no 1ndash3 pp 1ndash9 2000

[76] G S Domke J H Hattingh M A Henning and L R MarkusldquoRestrained domination in graphs with minimum degree twordquoJournal of Combinatorial Mathematics and Combinatorial Com-puting vol 35 pp 239ndash254 2000

[77] G S Domke J H Hattingh S T Hedetniemi R C Laskarand L R Markus ldquoRestrained domination in graphsrdquo DiscreteMathematics vol 203 no 1ndash3 pp 61ndash69 1999

[78] J H Hattingh and E J Joubert ldquoAn upper bound for therestrained domination number of a graph with minimumdegree at least two in terms of order and minimum degreerdquoDiscrete Applied Mathematics vol 157 no 13 pp 2846ndash28582009

[79] M A Henning ldquoGraphs with large restrained dominationnumberrdquo Discrete Mathematics vol 197-198 pp 415ndash429 199916th British Combinatorial Conference (London 1997)

[80] J H Hattingh and A R Plummer ldquoRestrained bondage ingraphsrdquo Discrete Mathematics vol 308 no 23 pp 5446ndash54532008

[81] N Jafari Rad R Hasni J Raczek and L Volkmann ldquoTotalrestrained bondage in graphsrdquoActaMathematica Sinica vol 29no 6 pp 1033ndash1042 2013

[82] J Cyman and J Raczek ldquoOn the total restrained dominationnumber of a graphrdquoThe Australasian Journal of Combinatoricsvol 36 pp 91ndash100 2006

[83] M A Henning ldquoGraphs with large total domination numberrdquoJournal of Graph Theory vol 35 no 1 pp 21ndash45 2000

[84] D-X Ma X-G Chen and L Sun ldquoOn total restraineddomination in graphsrdquoCzechoslovakMathematical Journal vol55 no 1 pp 165ndash173 2005

[85] B Zelinka ldquoRemarks on restrained domination and totalrestrained domination in graphsrdquo Czechoslovak MathematicalJournal vol 55 no 2 pp 393ndash396 2005

[86] W Goddard and M A Henning ldquoIndependent domination ingraphs a survey and recent resultsrdquo Discrete Mathematics vol313 no 7 pp 839ndash854 2013

[87] J H Zhang H L Liu and L Sun ldquoIndependence bondagenumber and reinforcement number of some graphsrdquo Transac-tions of Beijing Institute of Technology vol 23 no 2 pp 140ndash1422003

[88] K D Dharmalingam Studies in Graph Theorymdashequitabledomination and bottleneck domination [PhD thesis] MaduraiKamaraj University Madurai India 2006

[89] GDeepakND Soner andAAlwardi ldquoThe equitable bondagenumber of a graphrdquo Research Journal of Pure Algebra vol 1 no9 pp 209ndash212 2011

[90] E Sampathkumar and H B Walikar ldquoThe connected domina-tion number of a graphrdquo Journal of Mathematical and PhysicalSciences vol 13 no 6 pp 607ndash613 1979

[91] K White M Farber and W Pulleyblank ldquoSteiner trees con-nected domination and strongly chordal graphsrdquoNetworks vol15 no 1 pp 109ndash124 1985

[92] M Cadei M X Cheng X Cheng and D-Z Du ldquoConnecteddomination in Ad Hoc wireless networksrdquo in Proceedings ofthe 6th International Conference on Computer Science andInformatics (CSampI rsquo02) 2002

[93] J F Fink and M S Jacobson ldquon-domination in graphsrdquo inGraph Theory with Applications to Algorithms and ComputerScience Wiley-Interscience Publications pp 283ndash300 WileyNew York NY USA 1985

[94] M Chellali O Favaron A Hansberg and L Volkmann ldquok-domination and k-independence in graphs a surveyrdquo Graphsand Combinatorics vol 28 no 1 pp 1ndash55 2012

[95] Y Lu X Hou J-M Xu andN Li ldquoTrees with uniqueminimump-dominating setsrdquo Utilitas Mathematica vol 86 pp 193ndash2052011

[96] Y Lu and J-M Xu ldquoThe p-bondage number of treesrdquo Graphsand Combinatorics vol 27 no 1 pp 129ndash141 2011

[97] Y Lu and J-M Xu ldquoThe 2-domination and 2-bondage numbersof grid graphs A manuscriptrdquo httparxivorgabs12044514

[98] M Krzywkowski ldquoNon-isolating 2-bondage in graphsrdquo TheJournal of the Mathematical Society of Japan vol 64 no 4 2012

[99] M A Henning O R Oellermann and H C Swart ldquoBoundson distance domination parametersrdquo Journal of CombinatoricsInformation amp System Sciences vol 16 no 1 pp 11ndash18 1991

[100] F Tian and J-M Xu ldquoBounds for distance domination numbersof graphsrdquo Journal of University of Science and Technology ofChina vol 34 no 5 pp 529ndash534 2004

[101] F Tian and J-M Xu ldquoDistance domination-critical graphsrdquoApplied Mathematics Letters vol 21 no 4 pp 416ndash420 2008

[102] F Tian and J-M Xu ldquoA note on distance domination numbersof graphsrdquo The Australasian Journal of Combinatorics vol 43pp 181ndash190 2009

[103] F Tian and J-M Xu ldquoAverage distances and distance domina-tion numbersrdquo Discrete Applied Mathematics vol 157 no 5 pp1113ndash1127 2009

[104] Z-L Liu F Tian and J-M Xu ldquoProbabilistic analysis of upperbounds for 2-connected distance k-dominating sets in graphsrdquoTheoretical Computer Science vol 410 no 38ndash40 pp 3804ndash3813 2009

[105] M A Henning O R Oellermann and H C Swart ldquoDistancedomination critical graphsrdquo Journal of Combinatorial Mathe-matics and Combinatorial Computing vol 44 pp 33ndash45 2003

[106] T J Bean M A Henning and H C Swart ldquoOn the integrityof distance domination in graphsrdquo The Australasian Journal ofCombinatorics vol 10 pp 29ndash43 1994

[107] M Fischermann and L Volkmann ldquoA remark on a conjecturefor the (kp)-domination numberrdquoUtilitasMathematica vol 67pp 223ndash227 2005

[108] T Korneffel D Meierling and L Volkmann ldquoA remark on the(22)-domination numberrdquo Discussiones Mathematicae GraphTheory vol 28 no 2 pp 361ndash366 2008

[109] Y Lu X Hou and J-M Xu ldquoOn the (22)-domination numberof treesrdquo Discussiones Mathematicae Graph Theory vol 30 no2 pp 185ndash199 2010

34 International Journal of Combinatorics

[110] H Li and J-M Xu ldquo(dm)-domination numbers of m-connected graphsrdquo Rapports de Recherche 1130 LRI UniversiteParis-Sud 1997

[111] S M Hedetniemi S T Hedetniemi and T V Wimer ldquoLineartime resource allocation algorithms for treesrdquo Tech Rep URI-014 Department of Mathematical Sciences Clemson Univer-sity 1987

[112] M Farber ldquoDomination independent domination and dualityin strongly chordal graphsrdquo Discrete Applied Mathematics vol7 no 2 pp 115ndash130 1984

[113] G S Domke and R C Laskar ldquoThe bondage and reinforcementnumbers of 120574

119891for some graphsrdquoDiscrete Mathematics vol 167-

168 pp 249ndash259 1997[114] G S Domke S T Hedetniemi and R C Laskar ldquoFractional

packings coverings and irredundance in graphsrdquo vol 66 pp227ndash238 1988

[115] E J Cockayne P A Dreyer Jr S M Hedetniemi andS T Hedetniemi ldquoRoman domination in graphsrdquo DiscreteMathematics vol 278 no 1ndash3 pp 11ndash22 2004

[116] I Stewart ldquoDefend the roman empirerdquo Scientific American vol281 pp 136ndash139 1999

[117] C S ReVelle and K E Rosing ldquoDefendens imperiumromanum a classical problem in military strategyrdquoThe Ameri-can Mathematical Monthly vol 107 no 7 pp 585ndash594 2000

[118] A Bahremandpour F-T Hu S M Sheikholeslami and J-MXu ldquoOn the Roman bondage number of a graphrdquo DiscreteMathematics Algorithms and Applications vol 5 no 1 ArticleID 1350001 15 pages 2013

[119] N Jafari Rad and L Volkmann ldquoRoman bondage in graphsrdquoDiscussiones Mathematicae Graph Theory vol 31 no 4 pp763ndash773 2011

[120] K Ebadi and L PushpaLatha ldquoRoman bondage and Romanreinforcement numbers of a graphrdquo International Journal ofContemporary Mathematical Sciences vol 5 pp 1487ndash14972010

[121] N Dehgardi S M Sheikholeslami and L Volkman ldquoOn theRoman k-bondage number of a graphrdquo AKCE InternationalJournal of Graphs and Combinatorics vol 8 pp 169ndash180 2011

[122] S Akbari and S Qajar ldquoA note on Roman bondage number ofgraphsrdquo Ars Combinatoria to Appear

[123] F-T Hu and J-M Xu ldquoRoman bondage numbers of somegraphsrdquo httparxivorgabs11093933

[124] N Jafari Rad and L Volkmann ldquoOn the Roman bondagenumber of planar graphsrdquo Graphs and Combinatorics vol 27no 4 pp 531ndash538 2011

[125] S Akbari M Khatirinejad and S Qajar ldquoA note on the romanbondage number of planar graphsrdquo Graphs and Combinatorics2012

[126] B Bresar M A Henning and D F Rall ldquoRainbow dominationin graphsrdquo Taiwanese Journal of Mathematics vol 12 no 1 pp213ndash225 2008

[127] B L Hartnell and D F Rall ldquoOn dominating the Cartesianproduct of a graph and 120572krdquo Discussiones Mathematicae GraphTheory vol 24 no 3 pp 389ndash402 2004

[128] N Dehgardi S M Sheikholeslami and L Volkman ldquoThe k-rainbow bondage number of a graphrdquo to appear in DiscreteApplied Mathematics

[129] J von Neumann and O Morgenstern Theory of Games andEconomic Behavior Princeton University Press Princeton NJUSA 1944

[130] C BergeTheorıe des Graphes et ses Applications 1958[131] O Ore Theory of Graphs vol 38 American Mathematical

Society Colloquium Publications 1962[132] E Shan and L Kang ldquoBondage number in oriented graphsrdquoArs

Combinatoria vol 84 pp 319ndash331 2007[133] J Huang and J-M Xu ldquoThe bondage numbers of extended

de Bruijn and Kautz digraphsrdquo Computers amp Mathematics withApplications vol 51 no 6-7 pp 1137ndash1147 2006

[134] J Huang and J-M Xu ldquoThe total domination and total bondagenumbers of extended de Bruijn and Kautz digraphsrdquoComputersamp Mathematics with Applications vol 53 no 8 pp 1206ndash12132007

[135] Y Shibata and Y Gonda ldquoExtension of de Bruijn graph andKautz graphrdquo Computers amp Mathematics with Applications vol30 no 9 pp 51ndash61 1995

[136] X Zhang J Liu and JMeng ldquoThebondage number in completet-partite digraphsrdquo Information Processing Letters vol 109 no17 pp 997ndash1000 2009

[137] D W Bange A E Barkauskas and P J Slater ldquoEfficient dom-inating sets in graphsrdquo in Applications of Discrete Mathematicspp 189ndash199 SIAM Philadelphia Pa USA 1988

[138] M Livingston and Q F Stout ldquoPerfect dominating setsrdquoCongressus Numerantium vol 79 pp 187ndash203 1990

[139] L Clark ldquoPerfect domination in random graphsrdquo Journal ofCombinatorial Mathematics and Combinatorial Computing vol14 pp 173ndash182 1993

[140] D W Bange A E Barkauskas L H Host and L H ClarkldquoEfficient domination of the orientations of a graphrdquo DiscreteMathematics vol 178 no 1ndash3 pp 1ndash14 1998

[141] A E Barkauskas and L H Host ldquoFinding efficient dominatingsets in oriented graphsrdquo Congressus Numerantium vol 98 pp27ndash32 1993

[142] I J Dejter and O Serra ldquoEfficient dominating sets in CayleygraphsrdquoDiscrete AppliedMathematics vol 129 no 2-3 pp 319ndash328 2003

[143] J Lee ldquoIndependent perfect domination sets in Cayley graphsrdquoJournal of Graph Theory vol 37 no 4 pp 213ndash219 2001

[144] WGu X Jia and J Shen ldquoIndependent perfect domination setsin meshes tori and treesrdquo Unpublished

[145] N Obradovic J Peters and G Ruzic ldquoEfficient domination incirculant graphswith two chord lengthsrdquo Information ProcessingLetters vol 102 no 6 pp 253ndash258 2007

[146] K Reji Kumar and G MacGillivray ldquoEfficient domination incirculant graphsrdquo Discrete Mathematics vol 313 no 6 pp 767ndash771 2013

[147] D Van Wieren M Livingston and Q F Stout ldquoPerfectdominating sets on cube-connected cyclesrdquo Congressus Numer-antium vol 97 pp 51ndash70 1993

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

30 International Journal of Combinatorics

A graph 119866 is called to be vertex-transitive if its automor-phism group Aut(119866) acts transitively on its vertex-set 119881(119866)A vertex-transitive graph is regular Applying Lemmas 189and 190 Huang and Xu obtained some results on bondagenumbers for vertex-transitive graphs or digraphs

Theorem 191 (Huang and Xu [36] 2008) For any vertex-transitive graph or digraph 119866 of order 119899

119887 (119866) ⩾

lceil119899

2120574 (119866)rceil 119894119891 119866 119894119904 119906119899119889119894119903119890119888119905119890119889

lceil119899

120574 (119866)rceil 119894119891 119866 119894119904 119889119894119903119890119888119905119890119889

(82)

Theorem192 (Huang andXu [36] 2008) Let119866 be a 119896-regulargraph of order 119899 Then

119887(119866) ⩽ 119896 if119866 is undirected and 119899 ⩾ 120574(119866)(119896+1)minus119896+1119887(119866) ⩽ 119896+1+ℓ if119866 is directed and 119899 ⩾ 120574(119866)(119896+1)minusℓwith 0 ⩽ ℓ ⩽ 119896 minus 1

Next we introduce a better upper bound on 119887(119866) To thisaim we need the following concept which is a generalizationof the concept of the edge-covering of a graph 119866 For 1198811015840

sube

119881(119866) and 1198641015840 sube 119864(119866) we say that 1198641015840covers 1198811015840 and call 1198641015840an edge-covering for 1198811015840 if there exists an edge (119909 119910) isin 1198641015840 forany vertex 119909 isin 1198811015840 For 119910 isin 119881(119866) let 1205731015840[119910] be the minimumcardinality over all edge-coverings for119873minus

119866[119910]

Theorem 193 (Huang and Xu [36] 2008) If 119866 is a 119896-regulargraph with order 120574(119866)(119896 + 1) then 119887(119866) ⩽ 1205731015840[119910] for any 119910 isin119881(119866)

Theupper bound on 119887(119866) given inTheorem 193 is tight inview of 119887(119862

119899) for a cycle or a directed cycle 119862

119899(seeTheorems

1 and 185 resp)It is easy to see that for a 119896-regular graph 119866 lceil(119896 + 1)2rceil ⩽

1205731015840

[119910] ⩽ 119896 when 119866 is undirected and 1205731015840[119910] = 119896 + 1 when 119866 isdirected By this fact and Lemma 189 the following theoremis merely a simple combination of Theorems 191 and 193

Theorem 194 (Huang and Xu [36] 2008) Let 119866 be a vertex-transitive graph of degree 119896 If 119866 has an efficient dominatingset then

lceil119896 + 1

2rceil ⩽ 119887 (119866) ⩽ 119896 119894119891 119866 119894119904 119906119899119889119894119903119890119888119905119890119889

119887 (119866) = 119896 + 1 119894119891 119866 119894119904 119889119894119903119890119888119905119890119889

(83)

Theorem 195 (Huang and Xu [36] 2008) If 119866 is an undi-rected vertex-transitive cubic graph with order 4120574(119866) and girth119892(119866) ⩽ 5 then 119887(119866) = 2

The proof of Theorem 195 leads to a byproduct If 119892 =5 let (119906

1 119906

2 119906

3 119906

4 119906

5) be a cycle and let D

119894be the family

of efficient dominating sets containing 119906119894for each 119894 isin

1 2 3 4 5 ThenD119894capD

119895= 0 for 119894 = 1 2 5 Then 119866 has

at least 5119904 efficient dominating sets where 119904 = 119904(119866) But thereare only 4s (=ns120574) distinct efficient dominating sets in119866This

contradiction implies that an undirected vertex-transitivecubic graph with girth five has no efficient dominating setsBut a similar argument for 119892(119866) = 3 4 or 119892(119866) ⩾ 6 couldnot give any contradiction This is consistent with the resultthat CCC(119899) a vertex-transitive cubic graph with girth 119899 if3 ⩽ 119899 ⩽ 8 or girth 8 if 119899 ⩾ 9 has efficient dominating setsfor all 119899 ⩾ 3 except 119899 = 5 (see Theorem 199 in the followingsubsection)

112 Results for Cayley Graphs In this subsection we useTheorem 194 to determine the exact values or approximativevalues of bondage numbers for some special vertex-transitivegraphs by characterizing the existence of efficient dominatingsets in these graphs

Let Γ be a nontrivial finite group 119878 a nonempty subset ofΓ without the identity element of Γ A digraph 119866 is defined asfollows

119881 (119866) = Γ (119909 119910) isin 119864 (119866) lArrrArr 119909minus1

119910 isin 119878

for any 119909 119910 isin Γ(84)

is called a Cayley digraph of the group Γ with respect to119878 denoted by 119862

Γ(119878) If 119878minus1 = 119904minus1 119904 isin 119878 = 119878 then

119862Γ(119878) is symmetric and is called a Cayley undirected graph

a Cayley graph for short Cayley graphs or digraphs arecertainly vertex-transitive (see Xu [1])

A circulant graph 119866(119899 119878) of order 119899 is a Cayley graph119862119885119899

(119878) where 119885119899= 0 119899 minus 1 is the addition group of

order 119899 and 119878 is a nonempty subset of119885119899without the identity

element and hence is a vertex-transitive digraph of degree|119878| If 119878minus1 = 119878 then119866(119899 119878) is an undirected graph If 119878 = 1 119896where 2 ⩽ 119896 ⩽ 119899 minus 2 we write 119866(119899 1 119896) for 119866(119899 1 119896) or119866(119899 plusmn1 plusmn119896) and call it a double-loop circulant graph

Huang and Xu [36] showed that for directed 119866 =

119866(119899 1 119896) lceil1198993rceil ⩽ 120574(119866) ⩽ lceil1198992rceil and 119866 has an efficientdominating set if and only if 3 | 119899 and 119896 equiv 2 (mod 3) fordirected 119866 = 119866(119899 1 119896) and 119896 = 1198992 lceil1198995rceil ⩽ 120574(119866) ⩽ lceil1198993rceiland 119866 has an efficient dominating set if and only if 5 | 119899 and119896 equiv plusmn2 (mod 5) By Theorem 194 we obtain the bondagenumber of a double loop circulant graph if it has an efficientdominating set

Theorem 196 (Huang and Xu [36] 2008) Let 119866 be a double-loop circulant graph 119866(119899 1 119896) If 119866 is directed with 3 | 119899and 119896 equiv 2 (mod 3) or 119866 is undirected with 5 | 119899 and119896 equiv plusmn2 (mod 5) then 119887(119866) = 3

The 119898 times 119899 torus is the Cartesian product 119862119898times 119862

119899of

two cycles and is a Cayley graph 119862119885119898times119885119899

(119878) where 119878 =

(0 1) (1 0) for directed cycles and 119878 = (0 plusmn1) (plusmn1 0) forundirected cycles and hence is vertex-transitive Gu et al[144] showed that the undirected torus119862

119898times119862

119899has an efficient

dominating set if and only if both 119898 and 119899 are multiplesof 5 Huang and Xu [36] showed that the directed torus119862119898times 119862

119899has an efficient dominating set if and only if both

119898 and 119899 are multiples of 3 Moreover they found a necessarycondition for a dominating set containing the vertex (0 0)in 119862

119898times 119862

119899to be efficient and obtained the following

result

International Journal of Combinatorics 31

Theorem 197 (Huang and Xu [36] 2008) Let 119866 = 119862119898times 119862

119899

If 119866 is undirected and both119898 and 119899 are multiples of 5 or if 119866 isdirected and both119898 and 119899 are multiples of 3 then 119887(119866) = 3

The hypercube 119876119899

is the Cayley graph 119862Γ(119878)

where Γ = 1198852times sdot sdot sdot times 119885

2= (119885

2)119899 and 119878 =

100 sdot sdot sdot 0 010 sdot sdot sdot 0 00 sdot sdot sdot 01 Lee [143] showed that119876119899has an efficient dominating set if and only if 119899 = 2119898 minus 1

for a positive integer119898 ByTheorem 194 the following resultis obtained immediately

Theorem 198 (Huang and Xu [36] 2008) If 119899 = 2119898 minus 1 for apositive integer119898 then 2119898minus1

⩽ 119887(119876119899) ⩽ 2

119898

minus 1

Problem 10 Determine the exact value of 119887(119876119899) for 119899 = 2119898minus1

The 119899-dimensional cube-connected cycle denoted byCCC(119899) is constructed from the 119899-dimensional hypercube119876119899by replacing each vertex 119909 in 119876

119899with an undirected cycle

119862119899of length 119899 and linking the 119894th vertex of the 119862

119899to the 119894th

neighbor of 119909 It has been proved that CCC(119899) is a Cayleygraph and hence is a vertex-transitive graph with degree 3Van Wieren et al [147] proved that CCC(119899) has an efficientdominating set if and only if 119899 = 5 From Theorems 194 and195 the following result can be derived

Theorem 199 (Huang and Xu [36] 2008) Let 119866 = 119862119862119862(119899)be the 119899-dimensional cube-connected cycles with 119899 ⩾ 3 and119899 = 5 Then 120574(119866) = 1198992119899minus2 and 2 ⩽ 119887(119866) ⩽ 3 In addition119887(119862119862119862(3)) = 119887(119862119862119862(4)) = 2

Problem 11 Determine the exact value of 119887(CCC(119899)) for 119899 ⩾5

Acknowledgments

This work was supported by NNSF of China (Grants nos11071233 61272008) The author would like to express hisgratitude to the anonymous referees for their kind sugges-tions and comments on the original paper to ProfessorSheikholeslami and Professor Jafari Rad for sending thereferences [128] and [81] which resulted in this version

References

[1] J-M Xu Theory and Application of Graphs vol 10 of NetworkTheory and Applications Kluwer Academic Dordrecht TheNetherlands 2003

[2] S THedetniemi andR C Laskar ldquoBibliography on dominationin graphs and some basic definitions of domination parame-tersrdquo Discrete Mathematics vol 86 no 1ndash3 pp 257ndash277 1990

[3] TW Haynes S T Hedetniemi and P J Slater Fundamentals ofDomination in Graphs vol 208 ofMonographs and Textbooks inPure and Applied Mathematics Marcel Dekker New York NYUSA 1998

[4] TWHaynes S THedetniemi and P J Slater EdsDominationin Graphs vol 209 of Monographs and Textbooks in Pure andAppliedMathematicsMarcelDekker NewYorkNYUSA 1998

[5] M R Garey andD S JohnsonComputers and Intractability WH Freeman and Company San Francisco Calif USA 1979

[6] H B Walikar and B D Acharya ldquoDomination critical graphsrdquoNational Academy Science Letters vol 2 pp 70ndash72 1979

[7] D Bauer F Harary J Nieminen and C L Suffel ldquoDominationalteration sets in graphsrdquo Discrete Mathematics vol 47 no 2-3pp 153ndash161 1983

[8] J F Fink M S Jacobson L F Kinch and J Roberts ldquoThebondage number of a graphrdquo Discrete Mathematics vol 86 no1ndash3 pp 47ndash57 1990

[9] F-T Hu and J-M Xu ldquoThe bondage number of (119899 minus 3)-regulargraph G of order nrdquo Ars Combinatoria to appear

[10] U Teschner ldquoNew results about the bondage number of agraphrdquoDiscreteMathematics vol 171 no 1ndash3 pp 249ndash259 1997

[11] B L Hartnell and D F Rall ldquoA bound on the size of a graphwith given order and bondage numberrdquo Discrete Mathematicsvol 197-198 pp 409ndash413 1999 16th British CombinatorialConference (London 1997)

[12] B L Hartnell and D F Rall ldquoA characterization of trees inwhich no edge is essential to the domination numberrdquo ArsCombinatoria vol 33 pp 65ndash76 1992

[13] J Topp and P D Vestergaard ldquo120572119896- and 120574

119896-stable graphsrdquo

Discrete Mathematics vol 212 no 1-2 pp 149ndash160 2000[14] A Meir and J W Moon ldquoRelations between packing and

covering numbers of a treerdquo Pacific Journal of Mathematics vol61 no 1 pp 225ndash233 1975

[15] B L Hartnell L K Joslashrgensen P D Vestergaard and CA Whitehead ldquoEdge stability of the k-domination numberof treesrdquo Bulletin of the Institute of Combinatorics and ItsApplications vol 22 pp 31ndash40 1998

[16] F-T Hu and J-M Xu ldquoOn the complexity of the bondage andreinforcement problemsrdquo Journal of Complexity vol 28 no 2pp 192ndash201 2012

[17] B L Hartnell and D F Rall ldquoBounds on the bondage numberof a graphrdquo Discrete Mathematics vol 128 no 1ndash3 pp 173ndash1771994

[18] Y-L Wang ldquoOn the bondage number of a graphrdquo DiscreteMathematics vol 159 no 1ndash3 pp 291ndash294 1996

[19] G H Fricke TW Haynes S M Hedetniemi S T Hedetniemiand R C Laskar ldquoExcellent treesrdquo Bulletin of the Institute ofCombinatorics and Its Applications vol 34 pp 27ndash38 2002

[20] V Samodivkin ldquoThe bondage number of graphs good and badverticesrdquoDiscussiones Mathematicae GraphTheory vol 28 no3 pp 453ndash462 2008

[21] J E Dunbar T W Haynes U Teschner and L VolkmannldquoBondage insensitivity and reinforcementrdquo in Domination inGraphs vol 209 of Monographs and Textbooks in Pure andAppliedMathematics pp 471ndash489 Dekker NewYork NY USA1998

[22] H Liu and L Sun ldquoThe bondage and connectivity of a graphrdquoDiscrete Mathematics vol 263 no 1ndash3 pp 289ndash293 2003

[23] A-H Esfahanian and S L Hakimi ldquoOn computing a con-ditional edge-connectivity of a graphrdquo Information ProcessingLetters vol 27 no 4 pp 195ndash199 1988

[24] R C Brigham P Z Chinn and R D Dutton ldquoVertexdomination-critical graphsrdquoNetworks vol 18 no 3 pp 173ndash1791988

[25] V Samodivkin ldquoDomination with respect to nondegenerateproperties bondage numberrdquoThe Australasian Journal of Com-binatorics vol 45 pp 217ndash226 2009

[26] U Teschner ldquoA new upper bound for the bondage numberof graphs with small domination numberrdquo The AustralasianJournal of Combinatorics vol 12 pp 27ndash35 1995

32 International Journal of Combinatorics

[27] J Fulman D Hanson and G MacGillivray ldquoVertex dom-ination-critical graphsrdquoNetworks vol 25 no 2 pp 41ndash43 1995

[28] U Teschner ldquoA counterexample to a conjecture on the bondagenumber of a graphrdquo Discrete Mathematics vol 122 no 1ndash3 pp393ndash395 1993

[29] U Teschner ldquoThe bondage number of a graph G can be muchgreater than Δ(G)rdquo Ars Combinatoria vol 43 pp 81ndash87 1996

[30] L Volkmann ldquoOn graphs with equal domination and coveringnumbersrdquoDiscrete AppliedMathematics vol 51 no 1-2 pp 211ndash217 1994

[31] L A Sanchis ldquoMaximum number of edges in connected graphswith a given domination numberrdquoDiscreteMathematics vol 87no 1 pp 65ndash72 1991

[32] S Klavzar and N Seifter ldquoDominating Cartesian products ofcyclesrdquoDiscrete AppliedMathematics vol 59 no 2 pp 129ndash1361995

[33] H K Kim ldquoA note on Vizingrsquos conjecture about the bondagenumberrdquo Korean Journal of Mathematics vol 10 no 2 pp 39ndash42 2003

[34] M Y Sohn X Yuan and H S Jeong ldquoThe bondage number of1198623times119862

119899rdquo Journal of the KoreanMathematical Society vol 44 no

6 pp 1213ndash1231 2007[35] L Kang M Y Sohn and H K Kim ldquoBondage number of the

discrete torus 119862119899times 119862

4rdquo Discrete Mathematics vol 303 no 1ndash3

pp 80ndash86 2005[36] J Huang and J-M Xu ldquoThe bondage numbers and efficient

dominations of vertex-transitive graphsrdquoDiscrete Mathematicsvol 308 no 4 pp 571ndash582 2008

[37] C J Xiang X Yuan andM Y Sohn ldquoDomination and bondagenumber of119862

3times119862

119899rdquoArs Combinatoria vol 97 pp 299ndash310 2010

[38] M S Jacobson and L F Kinch ldquoOn the domination numberof products of graphs Irdquo Ars Combinatoria vol 18 pp 33ndash441984

[39] Y-Q Li ldquoA note on the bondage number of a graphrdquo ChineseQuarterly Journal of Mathematics vol 9 no 4 pp 1ndash3 1994

[40] F-T Hu and J-M Xu ldquoBondage number of mesh networksrdquoFrontiers ofMathematics inChina vol 7 no 5 pp 813ndash826 2012

[41] R Frucht and F Harary ldquoOn the corona of two graphsrdquoAequationes Mathematicae vol 4 pp 322ndash325 1970

[42] K Carlson and M Develin ldquoOn the bondage number of planarand directed graphsrdquoDiscreteMathematics vol 306 no 8-9 pp820ndash826 2006

[43] U Teschner ldquoOn the bondage number of block graphsrdquo ArsCombinatoria vol 46 pp 25ndash32 1997

[44] J Huang and J-M Xu ldquoNote on conjectures of bondagenumbers of planar graphsrdquo Applied Mathematical Sciences vol6 no 65ndash68 pp 3277ndash3287 2012

[45] L Kang and J Yuan ldquoBondage number of planar graphsrdquoDiscrete Mathematics vol 222 no 1ndash3 pp 191ndash198 2000

[46] M Fischermann D Rautenbach and L Volkmann ldquoRemarkson the bondage number of planar graphsrdquo Discrete Mathemat-ics vol 260 no 1ndash3 pp 57ndash67 2003

[47] J Hou G Liu and J Wu ldquoBondage number of planar graphswithout small cyclesrdquoUtilitasMathematica vol 84 pp 189ndash1992011

[48] J Huang and J-M Xu ldquoThe bondage numbers of graphs withsmall crossing numbersrdquo Discrete Mathematics vol 307 no 15pp 1881ndash1897 2007

[49] Q Ma S Zhang and J Wang ldquoBondage number of 1-planargraphrdquo Applied Mathematics vol 1 no 2 pp 101ndash103 2010

[50] J Hou and G Liu ldquoA bound on the bondage number of toroidalgraphsrdquoDiscreteMathematics Algorithms and Applications vol4 no 3 Article ID 1250046 5 pages 2012

[51] Y-C Cao J-M Xu and X-R Xu ldquoOn bondage number oftoroidal graphsrdquo Journal of University of Science and Technologyof China vol 39 no 3 pp 225ndash228 2009

[52] Y-C Cao J Huang and J-M Xu ldquoThe bondage number ofgraphs with crossing number less than fourrdquo Ars Combinatoriato appear

[53] H K Kim ldquoOn the bondage numbers of some graphsrdquo KoreanJournal of Mathematics vol 11 no 2 pp 11ndash15 2004

[54] A Gagarin and V Zverovich ldquoUpper bounds for the bondagenumber of graphs on topological surfacesrdquo Discrete Mathemat-ics 2012

[55] J Huang ldquoAn improved upper bound for the bondage numberof graphs on surfacesrdquoDiscrete Mathematics vol 312 no 18 pp2776ndash2781 2012

[56] A Gagarin and V Zverovich ldquoThe bondage number of graphson topological surfaces and Teschnerrsquos conjecturerdquo DiscreteMathematics vol 313 no 6 pp 796ndash808 2013

[57] V Samodivkin ldquoNote on the bondage number of graphs ontopological surfacesrdquo httparxivorgabs12086203

[58] E J Cockayne R M Dawes and S T Hedetniemi ldquoTotaldomination in graphsrdquoNetworks vol 10 no 3 pp 211ndash219 1980

[59] R Laskar J Pfaff S M Hedetniemi and S T HedetniemildquoOn the algorithmic complexity of total dominationrdquo Society forIndustrial and Applied Mathematics vol 5 no 3 pp 420ndash4251984

[60] J Pfaff R C Laskar and S T Hedetniemi ldquoNP-completeness oftotal and connected domination and irredundance for bipartitegraphsrdquo Tech Rep 428 Department of Mathematical SciencesClemson University 1983

[61] M A Henning ldquoA survey of selected recent results on totaldomination in graphsrdquoDiscrete Mathematics vol 309 no 1 pp32ndash63 2009

[62] V R Kulli and D K Patwari ldquoThe total bondage number ofa graphrdquo in Advances in Graph Theory pp 227ndash235 VishwaGulbarga India 1991

[63] F-T Hu Y Lu and J-M Xu ldquoThe total bondage number ofgrid graphsrdquo Discrete Applied Mathematics vol 160 no 16-17pp 2408ndash2418 2012

[64] J Cao M Shi and B Wu ldquoResearch on the total bondagenumber of a spectial networkrdquo in Proceedings of the 3rdInternational Joint Conference on Computational Sciences andOptimization (CSO rsquo10) pp 456ndash458 May 2010

[65] N Sridharan MD Elias and V S A Subramanian ldquoTotalbondage number of a graphrdquo AKCE International Journal ofGraphs and Combinatorics vol 4 no 2 pp 203ndash209 2007

[66] N Jafari Rad and J Raczek ldquoSome progress on total bondage ingraphsrdquo Graphs and Combinatorics 2011

[67] T W Haynes and P J Slater ldquoPaired-domination and thepaired-domatic numberrdquoCongressusNumerantium vol 109 pp65ndash72 1995

[68] T W Haynes and P J Slater ldquoPaired-domination in graphsrdquoNetworks vol 32 no 3 pp 199ndash206 1998

[69] X Hou ldquoA characterization of 2120574 120574119875-treesrdquoDiscrete Mathemat-

ics vol 308 no 15 pp 3420ndash3426 2008[70] K E Proffitt T W Haynes and P J Slater ldquoPaired-domination

in grid graphsrdquo Congressus Numerantium vol 150 pp 161ndash1722001

International Journal of Combinatorics 33

[71] H Qiao L KangM Cardei andD-Z Du ldquoPaired-dominationof treesrdquo Journal of Global Optimization vol 25 no 1 pp 43ndash542003

[72] E Shan L Kang and M A Henning ldquoA characterizationof trees with equal total domination and paired-dominationnumbersrdquo The Australasian Journal of Combinatorics vol 30pp 31ndash39 2004

[73] J Raczek ldquoPaired bondage in treesrdquo Discrete Mathematics vol308 no 23 pp 5570ndash5575 2008

[74] J A Telle and A Proskurowski ldquoAlgorithms for vertex parti-tioning problems on partial k-treesrdquo SIAM Journal on DiscreteMathematics vol 10 no 4 pp 529ndash550 1997

[75] G S Domke J H Hattingh M A Henning and L R MarkusldquoRestrained domination in treesrdquoDiscreteMathematics vol 211no 1ndash3 pp 1ndash9 2000

[76] G S Domke J H Hattingh M A Henning and L R MarkusldquoRestrained domination in graphs with minimum degree twordquoJournal of Combinatorial Mathematics and Combinatorial Com-puting vol 35 pp 239ndash254 2000

[77] G S Domke J H Hattingh S T Hedetniemi R C Laskarand L R Markus ldquoRestrained domination in graphsrdquo DiscreteMathematics vol 203 no 1ndash3 pp 61ndash69 1999

[78] J H Hattingh and E J Joubert ldquoAn upper bound for therestrained domination number of a graph with minimumdegree at least two in terms of order and minimum degreerdquoDiscrete Applied Mathematics vol 157 no 13 pp 2846ndash28582009

[79] M A Henning ldquoGraphs with large restrained dominationnumberrdquo Discrete Mathematics vol 197-198 pp 415ndash429 199916th British Combinatorial Conference (London 1997)

[80] J H Hattingh and A R Plummer ldquoRestrained bondage ingraphsrdquo Discrete Mathematics vol 308 no 23 pp 5446ndash54532008

[81] N Jafari Rad R Hasni J Raczek and L Volkmann ldquoTotalrestrained bondage in graphsrdquoActaMathematica Sinica vol 29no 6 pp 1033ndash1042 2013

[82] J Cyman and J Raczek ldquoOn the total restrained dominationnumber of a graphrdquoThe Australasian Journal of Combinatoricsvol 36 pp 91ndash100 2006

[83] M A Henning ldquoGraphs with large total domination numberrdquoJournal of Graph Theory vol 35 no 1 pp 21ndash45 2000

[84] D-X Ma X-G Chen and L Sun ldquoOn total restraineddomination in graphsrdquoCzechoslovakMathematical Journal vol55 no 1 pp 165ndash173 2005

[85] B Zelinka ldquoRemarks on restrained domination and totalrestrained domination in graphsrdquo Czechoslovak MathematicalJournal vol 55 no 2 pp 393ndash396 2005

[86] W Goddard and M A Henning ldquoIndependent domination ingraphs a survey and recent resultsrdquo Discrete Mathematics vol313 no 7 pp 839ndash854 2013

[87] J H Zhang H L Liu and L Sun ldquoIndependence bondagenumber and reinforcement number of some graphsrdquo Transac-tions of Beijing Institute of Technology vol 23 no 2 pp 140ndash1422003

[88] K D Dharmalingam Studies in Graph Theorymdashequitabledomination and bottleneck domination [PhD thesis] MaduraiKamaraj University Madurai India 2006

[89] GDeepakND Soner andAAlwardi ldquoThe equitable bondagenumber of a graphrdquo Research Journal of Pure Algebra vol 1 no9 pp 209ndash212 2011

[90] E Sampathkumar and H B Walikar ldquoThe connected domina-tion number of a graphrdquo Journal of Mathematical and PhysicalSciences vol 13 no 6 pp 607ndash613 1979

[91] K White M Farber and W Pulleyblank ldquoSteiner trees con-nected domination and strongly chordal graphsrdquoNetworks vol15 no 1 pp 109ndash124 1985

[92] M Cadei M X Cheng X Cheng and D-Z Du ldquoConnecteddomination in Ad Hoc wireless networksrdquo in Proceedings ofthe 6th International Conference on Computer Science andInformatics (CSampI rsquo02) 2002

[93] J F Fink and M S Jacobson ldquon-domination in graphsrdquo inGraph Theory with Applications to Algorithms and ComputerScience Wiley-Interscience Publications pp 283ndash300 WileyNew York NY USA 1985

[94] M Chellali O Favaron A Hansberg and L Volkmann ldquok-domination and k-independence in graphs a surveyrdquo Graphsand Combinatorics vol 28 no 1 pp 1ndash55 2012

[95] Y Lu X Hou J-M Xu andN Li ldquoTrees with uniqueminimump-dominating setsrdquo Utilitas Mathematica vol 86 pp 193ndash2052011

[96] Y Lu and J-M Xu ldquoThe p-bondage number of treesrdquo Graphsand Combinatorics vol 27 no 1 pp 129ndash141 2011

[97] Y Lu and J-M Xu ldquoThe 2-domination and 2-bondage numbersof grid graphs A manuscriptrdquo httparxivorgabs12044514

[98] M Krzywkowski ldquoNon-isolating 2-bondage in graphsrdquo TheJournal of the Mathematical Society of Japan vol 64 no 4 2012

[99] M A Henning O R Oellermann and H C Swart ldquoBoundson distance domination parametersrdquo Journal of CombinatoricsInformation amp System Sciences vol 16 no 1 pp 11ndash18 1991

[100] F Tian and J-M Xu ldquoBounds for distance domination numbersof graphsrdquo Journal of University of Science and Technology ofChina vol 34 no 5 pp 529ndash534 2004

[101] F Tian and J-M Xu ldquoDistance domination-critical graphsrdquoApplied Mathematics Letters vol 21 no 4 pp 416ndash420 2008

[102] F Tian and J-M Xu ldquoA note on distance domination numbersof graphsrdquo The Australasian Journal of Combinatorics vol 43pp 181ndash190 2009

[103] F Tian and J-M Xu ldquoAverage distances and distance domina-tion numbersrdquo Discrete Applied Mathematics vol 157 no 5 pp1113ndash1127 2009

[104] Z-L Liu F Tian and J-M Xu ldquoProbabilistic analysis of upperbounds for 2-connected distance k-dominating sets in graphsrdquoTheoretical Computer Science vol 410 no 38ndash40 pp 3804ndash3813 2009

[105] M A Henning O R Oellermann and H C Swart ldquoDistancedomination critical graphsrdquo Journal of Combinatorial Mathe-matics and Combinatorial Computing vol 44 pp 33ndash45 2003

[106] T J Bean M A Henning and H C Swart ldquoOn the integrityof distance domination in graphsrdquo The Australasian Journal ofCombinatorics vol 10 pp 29ndash43 1994

[107] M Fischermann and L Volkmann ldquoA remark on a conjecturefor the (kp)-domination numberrdquoUtilitasMathematica vol 67pp 223ndash227 2005

[108] T Korneffel D Meierling and L Volkmann ldquoA remark on the(22)-domination numberrdquo Discussiones Mathematicae GraphTheory vol 28 no 2 pp 361ndash366 2008

[109] Y Lu X Hou and J-M Xu ldquoOn the (22)-domination numberof treesrdquo Discussiones Mathematicae Graph Theory vol 30 no2 pp 185ndash199 2010

34 International Journal of Combinatorics

[110] H Li and J-M Xu ldquo(dm)-domination numbers of m-connected graphsrdquo Rapports de Recherche 1130 LRI UniversiteParis-Sud 1997

[111] S M Hedetniemi S T Hedetniemi and T V Wimer ldquoLineartime resource allocation algorithms for treesrdquo Tech Rep URI-014 Department of Mathematical Sciences Clemson Univer-sity 1987

[112] M Farber ldquoDomination independent domination and dualityin strongly chordal graphsrdquo Discrete Applied Mathematics vol7 no 2 pp 115ndash130 1984

[113] G S Domke and R C Laskar ldquoThe bondage and reinforcementnumbers of 120574

119891for some graphsrdquoDiscrete Mathematics vol 167-

168 pp 249ndash259 1997[114] G S Domke S T Hedetniemi and R C Laskar ldquoFractional

packings coverings and irredundance in graphsrdquo vol 66 pp227ndash238 1988

[115] E J Cockayne P A Dreyer Jr S M Hedetniemi andS T Hedetniemi ldquoRoman domination in graphsrdquo DiscreteMathematics vol 278 no 1ndash3 pp 11ndash22 2004

[116] I Stewart ldquoDefend the roman empirerdquo Scientific American vol281 pp 136ndash139 1999

[117] C S ReVelle and K E Rosing ldquoDefendens imperiumromanum a classical problem in military strategyrdquoThe Ameri-can Mathematical Monthly vol 107 no 7 pp 585ndash594 2000

[118] A Bahremandpour F-T Hu S M Sheikholeslami and J-MXu ldquoOn the Roman bondage number of a graphrdquo DiscreteMathematics Algorithms and Applications vol 5 no 1 ArticleID 1350001 15 pages 2013

[119] N Jafari Rad and L Volkmann ldquoRoman bondage in graphsrdquoDiscussiones Mathematicae Graph Theory vol 31 no 4 pp763ndash773 2011

[120] K Ebadi and L PushpaLatha ldquoRoman bondage and Romanreinforcement numbers of a graphrdquo International Journal ofContemporary Mathematical Sciences vol 5 pp 1487ndash14972010

[121] N Dehgardi S M Sheikholeslami and L Volkman ldquoOn theRoman k-bondage number of a graphrdquo AKCE InternationalJournal of Graphs and Combinatorics vol 8 pp 169ndash180 2011

[122] S Akbari and S Qajar ldquoA note on Roman bondage number ofgraphsrdquo Ars Combinatoria to Appear

[123] F-T Hu and J-M Xu ldquoRoman bondage numbers of somegraphsrdquo httparxivorgabs11093933

[124] N Jafari Rad and L Volkmann ldquoOn the Roman bondagenumber of planar graphsrdquo Graphs and Combinatorics vol 27no 4 pp 531ndash538 2011

[125] S Akbari M Khatirinejad and S Qajar ldquoA note on the romanbondage number of planar graphsrdquo Graphs and Combinatorics2012

[126] B Bresar M A Henning and D F Rall ldquoRainbow dominationin graphsrdquo Taiwanese Journal of Mathematics vol 12 no 1 pp213ndash225 2008

[127] B L Hartnell and D F Rall ldquoOn dominating the Cartesianproduct of a graph and 120572krdquo Discussiones Mathematicae GraphTheory vol 24 no 3 pp 389ndash402 2004

[128] N Dehgardi S M Sheikholeslami and L Volkman ldquoThe k-rainbow bondage number of a graphrdquo to appear in DiscreteApplied Mathematics

[129] J von Neumann and O Morgenstern Theory of Games andEconomic Behavior Princeton University Press Princeton NJUSA 1944

[130] C BergeTheorıe des Graphes et ses Applications 1958[131] O Ore Theory of Graphs vol 38 American Mathematical

Society Colloquium Publications 1962[132] E Shan and L Kang ldquoBondage number in oriented graphsrdquoArs

Combinatoria vol 84 pp 319ndash331 2007[133] J Huang and J-M Xu ldquoThe bondage numbers of extended

de Bruijn and Kautz digraphsrdquo Computers amp Mathematics withApplications vol 51 no 6-7 pp 1137ndash1147 2006

[134] J Huang and J-M Xu ldquoThe total domination and total bondagenumbers of extended de Bruijn and Kautz digraphsrdquoComputersamp Mathematics with Applications vol 53 no 8 pp 1206ndash12132007

[135] Y Shibata and Y Gonda ldquoExtension of de Bruijn graph andKautz graphrdquo Computers amp Mathematics with Applications vol30 no 9 pp 51ndash61 1995

[136] X Zhang J Liu and JMeng ldquoThebondage number in completet-partite digraphsrdquo Information Processing Letters vol 109 no17 pp 997ndash1000 2009

[137] D W Bange A E Barkauskas and P J Slater ldquoEfficient dom-inating sets in graphsrdquo in Applications of Discrete Mathematicspp 189ndash199 SIAM Philadelphia Pa USA 1988

[138] M Livingston and Q F Stout ldquoPerfect dominating setsrdquoCongressus Numerantium vol 79 pp 187ndash203 1990

[139] L Clark ldquoPerfect domination in random graphsrdquo Journal ofCombinatorial Mathematics and Combinatorial Computing vol14 pp 173ndash182 1993

[140] D W Bange A E Barkauskas L H Host and L H ClarkldquoEfficient domination of the orientations of a graphrdquo DiscreteMathematics vol 178 no 1ndash3 pp 1ndash14 1998

[141] A E Barkauskas and L H Host ldquoFinding efficient dominatingsets in oriented graphsrdquo Congressus Numerantium vol 98 pp27ndash32 1993

[142] I J Dejter and O Serra ldquoEfficient dominating sets in CayleygraphsrdquoDiscrete AppliedMathematics vol 129 no 2-3 pp 319ndash328 2003

[143] J Lee ldquoIndependent perfect domination sets in Cayley graphsrdquoJournal of Graph Theory vol 37 no 4 pp 213ndash219 2001

[144] WGu X Jia and J Shen ldquoIndependent perfect domination setsin meshes tori and treesrdquo Unpublished

[145] N Obradovic J Peters and G Ruzic ldquoEfficient domination incirculant graphswith two chord lengthsrdquo Information ProcessingLetters vol 102 no 6 pp 253ndash258 2007

[146] K Reji Kumar and G MacGillivray ldquoEfficient domination incirculant graphsrdquo Discrete Mathematics vol 313 no 6 pp 767ndash771 2013

[147] D Van Wieren M Livingston and Q F Stout ldquoPerfectdominating sets on cube-connected cyclesrdquo Congressus Numer-antium vol 97 pp 51ndash70 1993

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

International Journal of Combinatorics 31

Theorem 197 (Huang and Xu [36] 2008) Let 119866 = 119862119898times 119862

119899

If 119866 is undirected and both119898 and 119899 are multiples of 5 or if 119866 isdirected and both119898 and 119899 are multiples of 3 then 119887(119866) = 3

The hypercube 119876119899

is the Cayley graph 119862Γ(119878)

where Γ = 1198852times sdot sdot sdot times 119885

2= (119885

2)119899 and 119878 =

100 sdot sdot sdot 0 010 sdot sdot sdot 0 00 sdot sdot sdot 01 Lee [143] showed that119876119899has an efficient dominating set if and only if 119899 = 2119898 minus 1

for a positive integer119898 ByTheorem 194 the following resultis obtained immediately

Theorem 198 (Huang and Xu [36] 2008) If 119899 = 2119898 minus 1 for apositive integer119898 then 2119898minus1

⩽ 119887(119876119899) ⩽ 2

119898

minus 1

Problem 10 Determine the exact value of 119887(119876119899) for 119899 = 2119898minus1

The 119899-dimensional cube-connected cycle denoted byCCC(119899) is constructed from the 119899-dimensional hypercube119876119899by replacing each vertex 119909 in 119876

119899with an undirected cycle

119862119899of length 119899 and linking the 119894th vertex of the 119862

119899to the 119894th

neighbor of 119909 It has been proved that CCC(119899) is a Cayleygraph and hence is a vertex-transitive graph with degree 3Van Wieren et al [147] proved that CCC(119899) has an efficientdominating set if and only if 119899 = 5 From Theorems 194 and195 the following result can be derived

Theorem 199 (Huang and Xu [36] 2008) Let 119866 = 119862119862119862(119899)be the 119899-dimensional cube-connected cycles with 119899 ⩾ 3 and119899 = 5 Then 120574(119866) = 1198992119899minus2 and 2 ⩽ 119887(119866) ⩽ 3 In addition119887(119862119862119862(3)) = 119887(119862119862119862(4)) = 2

Problem 11 Determine the exact value of 119887(CCC(119899)) for 119899 ⩾5

Acknowledgments

This work was supported by NNSF of China (Grants nos11071233 61272008) The author would like to express hisgratitude to the anonymous referees for their kind sugges-tions and comments on the original paper to ProfessorSheikholeslami and Professor Jafari Rad for sending thereferences [128] and [81] which resulted in this version

References

[1] J-M Xu Theory and Application of Graphs vol 10 of NetworkTheory and Applications Kluwer Academic Dordrecht TheNetherlands 2003

[2] S THedetniemi andR C Laskar ldquoBibliography on dominationin graphs and some basic definitions of domination parame-tersrdquo Discrete Mathematics vol 86 no 1ndash3 pp 257ndash277 1990

[3] TW Haynes S T Hedetniemi and P J Slater Fundamentals ofDomination in Graphs vol 208 ofMonographs and Textbooks inPure and Applied Mathematics Marcel Dekker New York NYUSA 1998

[4] TWHaynes S THedetniemi and P J Slater EdsDominationin Graphs vol 209 of Monographs and Textbooks in Pure andAppliedMathematicsMarcelDekker NewYorkNYUSA 1998

[5] M R Garey andD S JohnsonComputers and Intractability WH Freeman and Company San Francisco Calif USA 1979

[6] H B Walikar and B D Acharya ldquoDomination critical graphsrdquoNational Academy Science Letters vol 2 pp 70ndash72 1979

[7] D Bauer F Harary J Nieminen and C L Suffel ldquoDominationalteration sets in graphsrdquo Discrete Mathematics vol 47 no 2-3pp 153ndash161 1983

[8] J F Fink M S Jacobson L F Kinch and J Roberts ldquoThebondage number of a graphrdquo Discrete Mathematics vol 86 no1ndash3 pp 47ndash57 1990

[9] F-T Hu and J-M Xu ldquoThe bondage number of (119899 minus 3)-regulargraph G of order nrdquo Ars Combinatoria to appear

[10] U Teschner ldquoNew results about the bondage number of agraphrdquoDiscreteMathematics vol 171 no 1ndash3 pp 249ndash259 1997

[11] B L Hartnell and D F Rall ldquoA bound on the size of a graphwith given order and bondage numberrdquo Discrete Mathematicsvol 197-198 pp 409ndash413 1999 16th British CombinatorialConference (London 1997)

[12] B L Hartnell and D F Rall ldquoA characterization of trees inwhich no edge is essential to the domination numberrdquo ArsCombinatoria vol 33 pp 65ndash76 1992

[13] J Topp and P D Vestergaard ldquo120572119896- and 120574

119896-stable graphsrdquo

Discrete Mathematics vol 212 no 1-2 pp 149ndash160 2000[14] A Meir and J W Moon ldquoRelations between packing and

covering numbers of a treerdquo Pacific Journal of Mathematics vol61 no 1 pp 225ndash233 1975

[15] B L Hartnell L K Joslashrgensen P D Vestergaard and CA Whitehead ldquoEdge stability of the k-domination numberof treesrdquo Bulletin of the Institute of Combinatorics and ItsApplications vol 22 pp 31ndash40 1998

[16] F-T Hu and J-M Xu ldquoOn the complexity of the bondage andreinforcement problemsrdquo Journal of Complexity vol 28 no 2pp 192ndash201 2012

[17] B L Hartnell and D F Rall ldquoBounds on the bondage numberof a graphrdquo Discrete Mathematics vol 128 no 1ndash3 pp 173ndash1771994

[18] Y-L Wang ldquoOn the bondage number of a graphrdquo DiscreteMathematics vol 159 no 1ndash3 pp 291ndash294 1996

[19] G H Fricke TW Haynes S M Hedetniemi S T Hedetniemiand R C Laskar ldquoExcellent treesrdquo Bulletin of the Institute ofCombinatorics and Its Applications vol 34 pp 27ndash38 2002

[20] V Samodivkin ldquoThe bondage number of graphs good and badverticesrdquoDiscussiones Mathematicae GraphTheory vol 28 no3 pp 453ndash462 2008

[21] J E Dunbar T W Haynes U Teschner and L VolkmannldquoBondage insensitivity and reinforcementrdquo in Domination inGraphs vol 209 of Monographs and Textbooks in Pure andAppliedMathematics pp 471ndash489 Dekker NewYork NY USA1998

[22] H Liu and L Sun ldquoThe bondage and connectivity of a graphrdquoDiscrete Mathematics vol 263 no 1ndash3 pp 289ndash293 2003

[23] A-H Esfahanian and S L Hakimi ldquoOn computing a con-ditional edge-connectivity of a graphrdquo Information ProcessingLetters vol 27 no 4 pp 195ndash199 1988

[24] R C Brigham P Z Chinn and R D Dutton ldquoVertexdomination-critical graphsrdquoNetworks vol 18 no 3 pp 173ndash1791988

[25] V Samodivkin ldquoDomination with respect to nondegenerateproperties bondage numberrdquoThe Australasian Journal of Com-binatorics vol 45 pp 217ndash226 2009

[26] U Teschner ldquoA new upper bound for the bondage numberof graphs with small domination numberrdquo The AustralasianJournal of Combinatorics vol 12 pp 27ndash35 1995

32 International Journal of Combinatorics

[27] J Fulman D Hanson and G MacGillivray ldquoVertex dom-ination-critical graphsrdquoNetworks vol 25 no 2 pp 41ndash43 1995

[28] U Teschner ldquoA counterexample to a conjecture on the bondagenumber of a graphrdquo Discrete Mathematics vol 122 no 1ndash3 pp393ndash395 1993

[29] U Teschner ldquoThe bondage number of a graph G can be muchgreater than Δ(G)rdquo Ars Combinatoria vol 43 pp 81ndash87 1996

[30] L Volkmann ldquoOn graphs with equal domination and coveringnumbersrdquoDiscrete AppliedMathematics vol 51 no 1-2 pp 211ndash217 1994

[31] L A Sanchis ldquoMaximum number of edges in connected graphswith a given domination numberrdquoDiscreteMathematics vol 87no 1 pp 65ndash72 1991

[32] S Klavzar and N Seifter ldquoDominating Cartesian products ofcyclesrdquoDiscrete AppliedMathematics vol 59 no 2 pp 129ndash1361995

[33] H K Kim ldquoA note on Vizingrsquos conjecture about the bondagenumberrdquo Korean Journal of Mathematics vol 10 no 2 pp 39ndash42 2003

[34] M Y Sohn X Yuan and H S Jeong ldquoThe bondage number of1198623times119862

119899rdquo Journal of the KoreanMathematical Society vol 44 no

6 pp 1213ndash1231 2007[35] L Kang M Y Sohn and H K Kim ldquoBondage number of the

discrete torus 119862119899times 119862

4rdquo Discrete Mathematics vol 303 no 1ndash3

pp 80ndash86 2005[36] J Huang and J-M Xu ldquoThe bondage numbers and efficient

dominations of vertex-transitive graphsrdquoDiscrete Mathematicsvol 308 no 4 pp 571ndash582 2008

[37] C J Xiang X Yuan andM Y Sohn ldquoDomination and bondagenumber of119862

3times119862

119899rdquoArs Combinatoria vol 97 pp 299ndash310 2010

[38] M S Jacobson and L F Kinch ldquoOn the domination numberof products of graphs Irdquo Ars Combinatoria vol 18 pp 33ndash441984

[39] Y-Q Li ldquoA note on the bondage number of a graphrdquo ChineseQuarterly Journal of Mathematics vol 9 no 4 pp 1ndash3 1994

[40] F-T Hu and J-M Xu ldquoBondage number of mesh networksrdquoFrontiers ofMathematics inChina vol 7 no 5 pp 813ndash826 2012

[41] R Frucht and F Harary ldquoOn the corona of two graphsrdquoAequationes Mathematicae vol 4 pp 322ndash325 1970

[42] K Carlson and M Develin ldquoOn the bondage number of planarand directed graphsrdquoDiscreteMathematics vol 306 no 8-9 pp820ndash826 2006

[43] U Teschner ldquoOn the bondage number of block graphsrdquo ArsCombinatoria vol 46 pp 25ndash32 1997

[44] J Huang and J-M Xu ldquoNote on conjectures of bondagenumbers of planar graphsrdquo Applied Mathematical Sciences vol6 no 65ndash68 pp 3277ndash3287 2012

[45] L Kang and J Yuan ldquoBondage number of planar graphsrdquoDiscrete Mathematics vol 222 no 1ndash3 pp 191ndash198 2000

[46] M Fischermann D Rautenbach and L Volkmann ldquoRemarkson the bondage number of planar graphsrdquo Discrete Mathemat-ics vol 260 no 1ndash3 pp 57ndash67 2003

[47] J Hou G Liu and J Wu ldquoBondage number of planar graphswithout small cyclesrdquoUtilitasMathematica vol 84 pp 189ndash1992011

[48] J Huang and J-M Xu ldquoThe bondage numbers of graphs withsmall crossing numbersrdquo Discrete Mathematics vol 307 no 15pp 1881ndash1897 2007

[49] Q Ma S Zhang and J Wang ldquoBondage number of 1-planargraphrdquo Applied Mathematics vol 1 no 2 pp 101ndash103 2010

[50] J Hou and G Liu ldquoA bound on the bondage number of toroidalgraphsrdquoDiscreteMathematics Algorithms and Applications vol4 no 3 Article ID 1250046 5 pages 2012

[51] Y-C Cao J-M Xu and X-R Xu ldquoOn bondage number oftoroidal graphsrdquo Journal of University of Science and Technologyof China vol 39 no 3 pp 225ndash228 2009

[52] Y-C Cao J Huang and J-M Xu ldquoThe bondage number ofgraphs with crossing number less than fourrdquo Ars Combinatoriato appear

[53] H K Kim ldquoOn the bondage numbers of some graphsrdquo KoreanJournal of Mathematics vol 11 no 2 pp 11ndash15 2004

[54] A Gagarin and V Zverovich ldquoUpper bounds for the bondagenumber of graphs on topological surfacesrdquo Discrete Mathemat-ics 2012

[55] J Huang ldquoAn improved upper bound for the bondage numberof graphs on surfacesrdquoDiscrete Mathematics vol 312 no 18 pp2776ndash2781 2012

[56] A Gagarin and V Zverovich ldquoThe bondage number of graphson topological surfaces and Teschnerrsquos conjecturerdquo DiscreteMathematics vol 313 no 6 pp 796ndash808 2013

[57] V Samodivkin ldquoNote on the bondage number of graphs ontopological surfacesrdquo httparxivorgabs12086203

[58] E J Cockayne R M Dawes and S T Hedetniemi ldquoTotaldomination in graphsrdquoNetworks vol 10 no 3 pp 211ndash219 1980

[59] R Laskar J Pfaff S M Hedetniemi and S T HedetniemildquoOn the algorithmic complexity of total dominationrdquo Society forIndustrial and Applied Mathematics vol 5 no 3 pp 420ndash4251984

[60] J Pfaff R C Laskar and S T Hedetniemi ldquoNP-completeness oftotal and connected domination and irredundance for bipartitegraphsrdquo Tech Rep 428 Department of Mathematical SciencesClemson University 1983

[61] M A Henning ldquoA survey of selected recent results on totaldomination in graphsrdquoDiscrete Mathematics vol 309 no 1 pp32ndash63 2009

[62] V R Kulli and D K Patwari ldquoThe total bondage number ofa graphrdquo in Advances in Graph Theory pp 227ndash235 VishwaGulbarga India 1991

[63] F-T Hu Y Lu and J-M Xu ldquoThe total bondage number ofgrid graphsrdquo Discrete Applied Mathematics vol 160 no 16-17pp 2408ndash2418 2012

[64] J Cao M Shi and B Wu ldquoResearch on the total bondagenumber of a spectial networkrdquo in Proceedings of the 3rdInternational Joint Conference on Computational Sciences andOptimization (CSO rsquo10) pp 456ndash458 May 2010

[65] N Sridharan MD Elias and V S A Subramanian ldquoTotalbondage number of a graphrdquo AKCE International Journal ofGraphs and Combinatorics vol 4 no 2 pp 203ndash209 2007

[66] N Jafari Rad and J Raczek ldquoSome progress on total bondage ingraphsrdquo Graphs and Combinatorics 2011

[67] T W Haynes and P J Slater ldquoPaired-domination and thepaired-domatic numberrdquoCongressusNumerantium vol 109 pp65ndash72 1995

[68] T W Haynes and P J Slater ldquoPaired-domination in graphsrdquoNetworks vol 32 no 3 pp 199ndash206 1998

[69] X Hou ldquoA characterization of 2120574 120574119875-treesrdquoDiscrete Mathemat-

ics vol 308 no 15 pp 3420ndash3426 2008[70] K E Proffitt T W Haynes and P J Slater ldquoPaired-domination

in grid graphsrdquo Congressus Numerantium vol 150 pp 161ndash1722001

International Journal of Combinatorics 33

[71] H Qiao L KangM Cardei andD-Z Du ldquoPaired-dominationof treesrdquo Journal of Global Optimization vol 25 no 1 pp 43ndash542003

[72] E Shan L Kang and M A Henning ldquoA characterizationof trees with equal total domination and paired-dominationnumbersrdquo The Australasian Journal of Combinatorics vol 30pp 31ndash39 2004

[73] J Raczek ldquoPaired bondage in treesrdquo Discrete Mathematics vol308 no 23 pp 5570ndash5575 2008

[74] J A Telle and A Proskurowski ldquoAlgorithms for vertex parti-tioning problems on partial k-treesrdquo SIAM Journal on DiscreteMathematics vol 10 no 4 pp 529ndash550 1997

[75] G S Domke J H Hattingh M A Henning and L R MarkusldquoRestrained domination in treesrdquoDiscreteMathematics vol 211no 1ndash3 pp 1ndash9 2000

[76] G S Domke J H Hattingh M A Henning and L R MarkusldquoRestrained domination in graphs with minimum degree twordquoJournal of Combinatorial Mathematics and Combinatorial Com-puting vol 35 pp 239ndash254 2000

[77] G S Domke J H Hattingh S T Hedetniemi R C Laskarand L R Markus ldquoRestrained domination in graphsrdquo DiscreteMathematics vol 203 no 1ndash3 pp 61ndash69 1999

[78] J H Hattingh and E J Joubert ldquoAn upper bound for therestrained domination number of a graph with minimumdegree at least two in terms of order and minimum degreerdquoDiscrete Applied Mathematics vol 157 no 13 pp 2846ndash28582009

[79] M A Henning ldquoGraphs with large restrained dominationnumberrdquo Discrete Mathematics vol 197-198 pp 415ndash429 199916th British Combinatorial Conference (London 1997)

[80] J H Hattingh and A R Plummer ldquoRestrained bondage ingraphsrdquo Discrete Mathematics vol 308 no 23 pp 5446ndash54532008

[81] N Jafari Rad R Hasni J Raczek and L Volkmann ldquoTotalrestrained bondage in graphsrdquoActaMathematica Sinica vol 29no 6 pp 1033ndash1042 2013

[82] J Cyman and J Raczek ldquoOn the total restrained dominationnumber of a graphrdquoThe Australasian Journal of Combinatoricsvol 36 pp 91ndash100 2006

[83] M A Henning ldquoGraphs with large total domination numberrdquoJournal of Graph Theory vol 35 no 1 pp 21ndash45 2000

[84] D-X Ma X-G Chen and L Sun ldquoOn total restraineddomination in graphsrdquoCzechoslovakMathematical Journal vol55 no 1 pp 165ndash173 2005

[85] B Zelinka ldquoRemarks on restrained domination and totalrestrained domination in graphsrdquo Czechoslovak MathematicalJournal vol 55 no 2 pp 393ndash396 2005

[86] W Goddard and M A Henning ldquoIndependent domination ingraphs a survey and recent resultsrdquo Discrete Mathematics vol313 no 7 pp 839ndash854 2013

[87] J H Zhang H L Liu and L Sun ldquoIndependence bondagenumber and reinforcement number of some graphsrdquo Transac-tions of Beijing Institute of Technology vol 23 no 2 pp 140ndash1422003

[88] K D Dharmalingam Studies in Graph Theorymdashequitabledomination and bottleneck domination [PhD thesis] MaduraiKamaraj University Madurai India 2006

[89] GDeepakND Soner andAAlwardi ldquoThe equitable bondagenumber of a graphrdquo Research Journal of Pure Algebra vol 1 no9 pp 209ndash212 2011

[90] E Sampathkumar and H B Walikar ldquoThe connected domina-tion number of a graphrdquo Journal of Mathematical and PhysicalSciences vol 13 no 6 pp 607ndash613 1979

[91] K White M Farber and W Pulleyblank ldquoSteiner trees con-nected domination and strongly chordal graphsrdquoNetworks vol15 no 1 pp 109ndash124 1985

[92] M Cadei M X Cheng X Cheng and D-Z Du ldquoConnecteddomination in Ad Hoc wireless networksrdquo in Proceedings ofthe 6th International Conference on Computer Science andInformatics (CSampI rsquo02) 2002

[93] J F Fink and M S Jacobson ldquon-domination in graphsrdquo inGraph Theory with Applications to Algorithms and ComputerScience Wiley-Interscience Publications pp 283ndash300 WileyNew York NY USA 1985

[94] M Chellali O Favaron A Hansberg and L Volkmann ldquok-domination and k-independence in graphs a surveyrdquo Graphsand Combinatorics vol 28 no 1 pp 1ndash55 2012

[95] Y Lu X Hou J-M Xu andN Li ldquoTrees with uniqueminimump-dominating setsrdquo Utilitas Mathematica vol 86 pp 193ndash2052011

[96] Y Lu and J-M Xu ldquoThe p-bondage number of treesrdquo Graphsand Combinatorics vol 27 no 1 pp 129ndash141 2011

[97] Y Lu and J-M Xu ldquoThe 2-domination and 2-bondage numbersof grid graphs A manuscriptrdquo httparxivorgabs12044514

[98] M Krzywkowski ldquoNon-isolating 2-bondage in graphsrdquo TheJournal of the Mathematical Society of Japan vol 64 no 4 2012

[99] M A Henning O R Oellermann and H C Swart ldquoBoundson distance domination parametersrdquo Journal of CombinatoricsInformation amp System Sciences vol 16 no 1 pp 11ndash18 1991

[100] F Tian and J-M Xu ldquoBounds for distance domination numbersof graphsrdquo Journal of University of Science and Technology ofChina vol 34 no 5 pp 529ndash534 2004

[101] F Tian and J-M Xu ldquoDistance domination-critical graphsrdquoApplied Mathematics Letters vol 21 no 4 pp 416ndash420 2008

[102] F Tian and J-M Xu ldquoA note on distance domination numbersof graphsrdquo The Australasian Journal of Combinatorics vol 43pp 181ndash190 2009

[103] F Tian and J-M Xu ldquoAverage distances and distance domina-tion numbersrdquo Discrete Applied Mathematics vol 157 no 5 pp1113ndash1127 2009

[104] Z-L Liu F Tian and J-M Xu ldquoProbabilistic analysis of upperbounds for 2-connected distance k-dominating sets in graphsrdquoTheoretical Computer Science vol 410 no 38ndash40 pp 3804ndash3813 2009

[105] M A Henning O R Oellermann and H C Swart ldquoDistancedomination critical graphsrdquo Journal of Combinatorial Mathe-matics and Combinatorial Computing vol 44 pp 33ndash45 2003

[106] T J Bean M A Henning and H C Swart ldquoOn the integrityof distance domination in graphsrdquo The Australasian Journal ofCombinatorics vol 10 pp 29ndash43 1994

[107] M Fischermann and L Volkmann ldquoA remark on a conjecturefor the (kp)-domination numberrdquoUtilitasMathematica vol 67pp 223ndash227 2005

[108] T Korneffel D Meierling and L Volkmann ldquoA remark on the(22)-domination numberrdquo Discussiones Mathematicae GraphTheory vol 28 no 2 pp 361ndash366 2008

[109] Y Lu X Hou and J-M Xu ldquoOn the (22)-domination numberof treesrdquo Discussiones Mathematicae Graph Theory vol 30 no2 pp 185ndash199 2010

34 International Journal of Combinatorics

[110] H Li and J-M Xu ldquo(dm)-domination numbers of m-connected graphsrdquo Rapports de Recherche 1130 LRI UniversiteParis-Sud 1997

[111] S M Hedetniemi S T Hedetniemi and T V Wimer ldquoLineartime resource allocation algorithms for treesrdquo Tech Rep URI-014 Department of Mathematical Sciences Clemson Univer-sity 1987

[112] M Farber ldquoDomination independent domination and dualityin strongly chordal graphsrdquo Discrete Applied Mathematics vol7 no 2 pp 115ndash130 1984

[113] G S Domke and R C Laskar ldquoThe bondage and reinforcementnumbers of 120574

119891for some graphsrdquoDiscrete Mathematics vol 167-

168 pp 249ndash259 1997[114] G S Domke S T Hedetniemi and R C Laskar ldquoFractional

packings coverings and irredundance in graphsrdquo vol 66 pp227ndash238 1988

[115] E J Cockayne P A Dreyer Jr S M Hedetniemi andS T Hedetniemi ldquoRoman domination in graphsrdquo DiscreteMathematics vol 278 no 1ndash3 pp 11ndash22 2004

[116] I Stewart ldquoDefend the roman empirerdquo Scientific American vol281 pp 136ndash139 1999

[117] C S ReVelle and K E Rosing ldquoDefendens imperiumromanum a classical problem in military strategyrdquoThe Ameri-can Mathematical Monthly vol 107 no 7 pp 585ndash594 2000

[118] A Bahremandpour F-T Hu S M Sheikholeslami and J-MXu ldquoOn the Roman bondage number of a graphrdquo DiscreteMathematics Algorithms and Applications vol 5 no 1 ArticleID 1350001 15 pages 2013

[119] N Jafari Rad and L Volkmann ldquoRoman bondage in graphsrdquoDiscussiones Mathematicae Graph Theory vol 31 no 4 pp763ndash773 2011

[120] K Ebadi and L PushpaLatha ldquoRoman bondage and Romanreinforcement numbers of a graphrdquo International Journal ofContemporary Mathematical Sciences vol 5 pp 1487ndash14972010

[121] N Dehgardi S M Sheikholeslami and L Volkman ldquoOn theRoman k-bondage number of a graphrdquo AKCE InternationalJournal of Graphs and Combinatorics vol 8 pp 169ndash180 2011

[122] S Akbari and S Qajar ldquoA note on Roman bondage number ofgraphsrdquo Ars Combinatoria to Appear

[123] F-T Hu and J-M Xu ldquoRoman bondage numbers of somegraphsrdquo httparxivorgabs11093933

[124] N Jafari Rad and L Volkmann ldquoOn the Roman bondagenumber of planar graphsrdquo Graphs and Combinatorics vol 27no 4 pp 531ndash538 2011

[125] S Akbari M Khatirinejad and S Qajar ldquoA note on the romanbondage number of planar graphsrdquo Graphs and Combinatorics2012

[126] B Bresar M A Henning and D F Rall ldquoRainbow dominationin graphsrdquo Taiwanese Journal of Mathematics vol 12 no 1 pp213ndash225 2008

[127] B L Hartnell and D F Rall ldquoOn dominating the Cartesianproduct of a graph and 120572krdquo Discussiones Mathematicae GraphTheory vol 24 no 3 pp 389ndash402 2004

[128] N Dehgardi S M Sheikholeslami and L Volkman ldquoThe k-rainbow bondage number of a graphrdquo to appear in DiscreteApplied Mathematics

[129] J von Neumann and O Morgenstern Theory of Games andEconomic Behavior Princeton University Press Princeton NJUSA 1944

[130] C BergeTheorıe des Graphes et ses Applications 1958[131] O Ore Theory of Graphs vol 38 American Mathematical

Society Colloquium Publications 1962[132] E Shan and L Kang ldquoBondage number in oriented graphsrdquoArs

Combinatoria vol 84 pp 319ndash331 2007[133] J Huang and J-M Xu ldquoThe bondage numbers of extended

de Bruijn and Kautz digraphsrdquo Computers amp Mathematics withApplications vol 51 no 6-7 pp 1137ndash1147 2006

[134] J Huang and J-M Xu ldquoThe total domination and total bondagenumbers of extended de Bruijn and Kautz digraphsrdquoComputersamp Mathematics with Applications vol 53 no 8 pp 1206ndash12132007

[135] Y Shibata and Y Gonda ldquoExtension of de Bruijn graph andKautz graphrdquo Computers amp Mathematics with Applications vol30 no 9 pp 51ndash61 1995

[136] X Zhang J Liu and JMeng ldquoThebondage number in completet-partite digraphsrdquo Information Processing Letters vol 109 no17 pp 997ndash1000 2009

[137] D W Bange A E Barkauskas and P J Slater ldquoEfficient dom-inating sets in graphsrdquo in Applications of Discrete Mathematicspp 189ndash199 SIAM Philadelphia Pa USA 1988

[138] M Livingston and Q F Stout ldquoPerfect dominating setsrdquoCongressus Numerantium vol 79 pp 187ndash203 1990

[139] L Clark ldquoPerfect domination in random graphsrdquo Journal ofCombinatorial Mathematics and Combinatorial Computing vol14 pp 173ndash182 1993

[140] D W Bange A E Barkauskas L H Host and L H ClarkldquoEfficient domination of the orientations of a graphrdquo DiscreteMathematics vol 178 no 1ndash3 pp 1ndash14 1998

[141] A E Barkauskas and L H Host ldquoFinding efficient dominatingsets in oriented graphsrdquo Congressus Numerantium vol 98 pp27ndash32 1993

[142] I J Dejter and O Serra ldquoEfficient dominating sets in CayleygraphsrdquoDiscrete AppliedMathematics vol 129 no 2-3 pp 319ndash328 2003

[143] J Lee ldquoIndependent perfect domination sets in Cayley graphsrdquoJournal of Graph Theory vol 37 no 4 pp 213ndash219 2001

[144] WGu X Jia and J Shen ldquoIndependent perfect domination setsin meshes tori and treesrdquo Unpublished

[145] N Obradovic J Peters and G Ruzic ldquoEfficient domination incirculant graphswith two chord lengthsrdquo Information ProcessingLetters vol 102 no 6 pp 253ndash258 2007

[146] K Reji Kumar and G MacGillivray ldquoEfficient domination incirculant graphsrdquo Discrete Mathematics vol 313 no 6 pp 767ndash771 2013

[147] D Van Wieren M Livingston and Q F Stout ldquoPerfectdominating sets on cube-connected cyclesrdquo Congressus Numer-antium vol 97 pp 51ndash70 1993

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

32 International Journal of Combinatorics

[27] J Fulman D Hanson and G MacGillivray ldquoVertex dom-ination-critical graphsrdquoNetworks vol 25 no 2 pp 41ndash43 1995

[28] U Teschner ldquoA counterexample to a conjecture on the bondagenumber of a graphrdquo Discrete Mathematics vol 122 no 1ndash3 pp393ndash395 1993

[29] U Teschner ldquoThe bondage number of a graph G can be muchgreater than Δ(G)rdquo Ars Combinatoria vol 43 pp 81ndash87 1996

[30] L Volkmann ldquoOn graphs with equal domination and coveringnumbersrdquoDiscrete AppliedMathematics vol 51 no 1-2 pp 211ndash217 1994

[31] L A Sanchis ldquoMaximum number of edges in connected graphswith a given domination numberrdquoDiscreteMathematics vol 87no 1 pp 65ndash72 1991

[32] S Klavzar and N Seifter ldquoDominating Cartesian products ofcyclesrdquoDiscrete AppliedMathematics vol 59 no 2 pp 129ndash1361995

[33] H K Kim ldquoA note on Vizingrsquos conjecture about the bondagenumberrdquo Korean Journal of Mathematics vol 10 no 2 pp 39ndash42 2003

[34] M Y Sohn X Yuan and H S Jeong ldquoThe bondage number of1198623times119862

119899rdquo Journal of the KoreanMathematical Society vol 44 no

6 pp 1213ndash1231 2007[35] L Kang M Y Sohn and H K Kim ldquoBondage number of the

discrete torus 119862119899times 119862

4rdquo Discrete Mathematics vol 303 no 1ndash3

pp 80ndash86 2005[36] J Huang and J-M Xu ldquoThe bondage numbers and efficient

dominations of vertex-transitive graphsrdquoDiscrete Mathematicsvol 308 no 4 pp 571ndash582 2008

[37] C J Xiang X Yuan andM Y Sohn ldquoDomination and bondagenumber of119862

3times119862

119899rdquoArs Combinatoria vol 97 pp 299ndash310 2010

[38] M S Jacobson and L F Kinch ldquoOn the domination numberof products of graphs Irdquo Ars Combinatoria vol 18 pp 33ndash441984

[39] Y-Q Li ldquoA note on the bondage number of a graphrdquo ChineseQuarterly Journal of Mathematics vol 9 no 4 pp 1ndash3 1994

[40] F-T Hu and J-M Xu ldquoBondage number of mesh networksrdquoFrontiers ofMathematics inChina vol 7 no 5 pp 813ndash826 2012

[41] R Frucht and F Harary ldquoOn the corona of two graphsrdquoAequationes Mathematicae vol 4 pp 322ndash325 1970

[42] K Carlson and M Develin ldquoOn the bondage number of planarand directed graphsrdquoDiscreteMathematics vol 306 no 8-9 pp820ndash826 2006

[43] U Teschner ldquoOn the bondage number of block graphsrdquo ArsCombinatoria vol 46 pp 25ndash32 1997

[44] J Huang and J-M Xu ldquoNote on conjectures of bondagenumbers of planar graphsrdquo Applied Mathematical Sciences vol6 no 65ndash68 pp 3277ndash3287 2012

[45] L Kang and J Yuan ldquoBondage number of planar graphsrdquoDiscrete Mathematics vol 222 no 1ndash3 pp 191ndash198 2000

[46] M Fischermann D Rautenbach and L Volkmann ldquoRemarkson the bondage number of planar graphsrdquo Discrete Mathemat-ics vol 260 no 1ndash3 pp 57ndash67 2003

[47] J Hou G Liu and J Wu ldquoBondage number of planar graphswithout small cyclesrdquoUtilitasMathematica vol 84 pp 189ndash1992011

[48] J Huang and J-M Xu ldquoThe bondage numbers of graphs withsmall crossing numbersrdquo Discrete Mathematics vol 307 no 15pp 1881ndash1897 2007

[49] Q Ma S Zhang and J Wang ldquoBondage number of 1-planargraphrdquo Applied Mathematics vol 1 no 2 pp 101ndash103 2010

[50] J Hou and G Liu ldquoA bound on the bondage number of toroidalgraphsrdquoDiscreteMathematics Algorithms and Applications vol4 no 3 Article ID 1250046 5 pages 2012

[51] Y-C Cao J-M Xu and X-R Xu ldquoOn bondage number oftoroidal graphsrdquo Journal of University of Science and Technologyof China vol 39 no 3 pp 225ndash228 2009

[52] Y-C Cao J Huang and J-M Xu ldquoThe bondage number ofgraphs with crossing number less than fourrdquo Ars Combinatoriato appear

[53] H K Kim ldquoOn the bondage numbers of some graphsrdquo KoreanJournal of Mathematics vol 11 no 2 pp 11ndash15 2004

[54] A Gagarin and V Zverovich ldquoUpper bounds for the bondagenumber of graphs on topological surfacesrdquo Discrete Mathemat-ics 2012

[55] J Huang ldquoAn improved upper bound for the bondage numberof graphs on surfacesrdquoDiscrete Mathematics vol 312 no 18 pp2776ndash2781 2012

[56] A Gagarin and V Zverovich ldquoThe bondage number of graphson topological surfaces and Teschnerrsquos conjecturerdquo DiscreteMathematics vol 313 no 6 pp 796ndash808 2013

[57] V Samodivkin ldquoNote on the bondage number of graphs ontopological surfacesrdquo httparxivorgabs12086203

[58] E J Cockayne R M Dawes and S T Hedetniemi ldquoTotaldomination in graphsrdquoNetworks vol 10 no 3 pp 211ndash219 1980

[59] R Laskar J Pfaff S M Hedetniemi and S T HedetniemildquoOn the algorithmic complexity of total dominationrdquo Society forIndustrial and Applied Mathematics vol 5 no 3 pp 420ndash4251984

[60] J Pfaff R C Laskar and S T Hedetniemi ldquoNP-completeness oftotal and connected domination and irredundance for bipartitegraphsrdquo Tech Rep 428 Department of Mathematical SciencesClemson University 1983

[61] M A Henning ldquoA survey of selected recent results on totaldomination in graphsrdquoDiscrete Mathematics vol 309 no 1 pp32ndash63 2009

[62] V R Kulli and D K Patwari ldquoThe total bondage number ofa graphrdquo in Advances in Graph Theory pp 227ndash235 VishwaGulbarga India 1991

[63] F-T Hu Y Lu and J-M Xu ldquoThe total bondage number ofgrid graphsrdquo Discrete Applied Mathematics vol 160 no 16-17pp 2408ndash2418 2012

[64] J Cao M Shi and B Wu ldquoResearch on the total bondagenumber of a spectial networkrdquo in Proceedings of the 3rdInternational Joint Conference on Computational Sciences andOptimization (CSO rsquo10) pp 456ndash458 May 2010

[65] N Sridharan MD Elias and V S A Subramanian ldquoTotalbondage number of a graphrdquo AKCE International Journal ofGraphs and Combinatorics vol 4 no 2 pp 203ndash209 2007

[66] N Jafari Rad and J Raczek ldquoSome progress on total bondage ingraphsrdquo Graphs and Combinatorics 2011

[67] T W Haynes and P J Slater ldquoPaired-domination and thepaired-domatic numberrdquoCongressusNumerantium vol 109 pp65ndash72 1995

[68] T W Haynes and P J Slater ldquoPaired-domination in graphsrdquoNetworks vol 32 no 3 pp 199ndash206 1998

[69] X Hou ldquoA characterization of 2120574 120574119875-treesrdquoDiscrete Mathemat-

ics vol 308 no 15 pp 3420ndash3426 2008[70] K E Proffitt T W Haynes and P J Slater ldquoPaired-domination

in grid graphsrdquo Congressus Numerantium vol 150 pp 161ndash1722001

International Journal of Combinatorics 33

[71] H Qiao L KangM Cardei andD-Z Du ldquoPaired-dominationof treesrdquo Journal of Global Optimization vol 25 no 1 pp 43ndash542003

[72] E Shan L Kang and M A Henning ldquoA characterizationof trees with equal total domination and paired-dominationnumbersrdquo The Australasian Journal of Combinatorics vol 30pp 31ndash39 2004

[73] J Raczek ldquoPaired bondage in treesrdquo Discrete Mathematics vol308 no 23 pp 5570ndash5575 2008

[74] J A Telle and A Proskurowski ldquoAlgorithms for vertex parti-tioning problems on partial k-treesrdquo SIAM Journal on DiscreteMathematics vol 10 no 4 pp 529ndash550 1997

[75] G S Domke J H Hattingh M A Henning and L R MarkusldquoRestrained domination in treesrdquoDiscreteMathematics vol 211no 1ndash3 pp 1ndash9 2000

[76] G S Domke J H Hattingh M A Henning and L R MarkusldquoRestrained domination in graphs with minimum degree twordquoJournal of Combinatorial Mathematics and Combinatorial Com-puting vol 35 pp 239ndash254 2000

[77] G S Domke J H Hattingh S T Hedetniemi R C Laskarand L R Markus ldquoRestrained domination in graphsrdquo DiscreteMathematics vol 203 no 1ndash3 pp 61ndash69 1999

[78] J H Hattingh and E J Joubert ldquoAn upper bound for therestrained domination number of a graph with minimumdegree at least two in terms of order and minimum degreerdquoDiscrete Applied Mathematics vol 157 no 13 pp 2846ndash28582009

[79] M A Henning ldquoGraphs with large restrained dominationnumberrdquo Discrete Mathematics vol 197-198 pp 415ndash429 199916th British Combinatorial Conference (London 1997)

[80] J H Hattingh and A R Plummer ldquoRestrained bondage ingraphsrdquo Discrete Mathematics vol 308 no 23 pp 5446ndash54532008

[81] N Jafari Rad R Hasni J Raczek and L Volkmann ldquoTotalrestrained bondage in graphsrdquoActaMathematica Sinica vol 29no 6 pp 1033ndash1042 2013

[82] J Cyman and J Raczek ldquoOn the total restrained dominationnumber of a graphrdquoThe Australasian Journal of Combinatoricsvol 36 pp 91ndash100 2006

[83] M A Henning ldquoGraphs with large total domination numberrdquoJournal of Graph Theory vol 35 no 1 pp 21ndash45 2000

[84] D-X Ma X-G Chen and L Sun ldquoOn total restraineddomination in graphsrdquoCzechoslovakMathematical Journal vol55 no 1 pp 165ndash173 2005

[85] B Zelinka ldquoRemarks on restrained domination and totalrestrained domination in graphsrdquo Czechoslovak MathematicalJournal vol 55 no 2 pp 393ndash396 2005

[86] W Goddard and M A Henning ldquoIndependent domination ingraphs a survey and recent resultsrdquo Discrete Mathematics vol313 no 7 pp 839ndash854 2013

[87] J H Zhang H L Liu and L Sun ldquoIndependence bondagenumber and reinforcement number of some graphsrdquo Transac-tions of Beijing Institute of Technology vol 23 no 2 pp 140ndash1422003

[88] K D Dharmalingam Studies in Graph Theorymdashequitabledomination and bottleneck domination [PhD thesis] MaduraiKamaraj University Madurai India 2006

[89] GDeepakND Soner andAAlwardi ldquoThe equitable bondagenumber of a graphrdquo Research Journal of Pure Algebra vol 1 no9 pp 209ndash212 2011

[90] E Sampathkumar and H B Walikar ldquoThe connected domina-tion number of a graphrdquo Journal of Mathematical and PhysicalSciences vol 13 no 6 pp 607ndash613 1979

[91] K White M Farber and W Pulleyblank ldquoSteiner trees con-nected domination and strongly chordal graphsrdquoNetworks vol15 no 1 pp 109ndash124 1985

[92] M Cadei M X Cheng X Cheng and D-Z Du ldquoConnecteddomination in Ad Hoc wireless networksrdquo in Proceedings ofthe 6th International Conference on Computer Science andInformatics (CSampI rsquo02) 2002

[93] J F Fink and M S Jacobson ldquon-domination in graphsrdquo inGraph Theory with Applications to Algorithms and ComputerScience Wiley-Interscience Publications pp 283ndash300 WileyNew York NY USA 1985

[94] M Chellali O Favaron A Hansberg and L Volkmann ldquok-domination and k-independence in graphs a surveyrdquo Graphsand Combinatorics vol 28 no 1 pp 1ndash55 2012

[95] Y Lu X Hou J-M Xu andN Li ldquoTrees with uniqueminimump-dominating setsrdquo Utilitas Mathematica vol 86 pp 193ndash2052011

[96] Y Lu and J-M Xu ldquoThe p-bondage number of treesrdquo Graphsand Combinatorics vol 27 no 1 pp 129ndash141 2011

[97] Y Lu and J-M Xu ldquoThe 2-domination and 2-bondage numbersof grid graphs A manuscriptrdquo httparxivorgabs12044514

[98] M Krzywkowski ldquoNon-isolating 2-bondage in graphsrdquo TheJournal of the Mathematical Society of Japan vol 64 no 4 2012

[99] M A Henning O R Oellermann and H C Swart ldquoBoundson distance domination parametersrdquo Journal of CombinatoricsInformation amp System Sciences vol 16 no 1 pp 11ndash18 1991

[100] F Tian and J-M Xu ldquoBounds for distance domination numbersof graphsrdquo Journal of University of Science and Technology ofChina vol 34 no 5 pp 529ndash534 2004

[101] F Tian and J-M Xu ldquoDistance domination-critical graphsrdquoApplied Mathematics Letters vol 21 no 4 pp 416ndash420 2008

[102] F Tian and J-M Xu ldquoA note on distance domination numbersof graphsrdquo The Australasian Journal of Combinatorics vol 43pp 181ndash190 2009

[103] F Tian and J-M Xu ldquoAverage distances and distance domina-tion numbersrdquo Discrete Applied Mathematics vol 157 no 5 pp1113ndash1127 2009

[104] Z-L Liu F Tian and J-M Xu ldquoProbabilistic analysis of upperbounds for 2-connected distance k-dominating sets in graphsrdquoTheoretical Computer Science vol 410 no 38ndash40 pp 3804ndash3813 2009

[105] M A Henning O R Oellermann and H C Swart ldquoDistancedomination critical graphsrdquo Journal of Combinatorial Mathe-matics and Combinatorial Computing vol 44 pp 33ndash45 2003

[106] T J Bean M A Henning and H C Swart ldquoOn the integrityof distance domination in graphsrdquo The Australasian Journal ofCombinatorics vol 10 pp 29ndash43 1994

[107] M Fischermann and L Volkmann ldquoA remark on a conjecturefor the (kp)-domination numberrdquoUtilitasMathematica vol 67pp 223ndash227 2005

[108] T Korneffel D Meierling and L Volkmann ldquoA remark on the(22)-domination numberrdquo Discussiones Mathematicae GraphTheory vol 28 no 2 pp 361ndash366 2008

[109] Y Lu X Hou and J-M Xu ldquoOn the (22)-domination numberof treesrdquo Discussiones Mathematicae Graph Theory vol 30 no2 pp 185ndash199 2010

34 International Journal of Combinatorics

[110] H Li and J-M Xu ldquo(dm)-domination numbers of m-connected graphsrdquo Rapports de Recherche 1130 LRI UniversiteParis-Sud 1997

[111] S M Hedetniemi S T Hedetniemi and T V Wimer ldquoLineartime resource allocation algorithms for treesrdquo Tech Rep URI-014 Department of Mathematical Sciences Clemson Univer-sity 1987

[112] M Farber ldquoDomination independent domination and dualityin strongly chordal graphsrdquo Discrete Applied Mathematics vol7 no 2 pp 115ndash130 1984

[113] G S Domke and R C Laskar ldquoThe bondage and reinforcementnumbers of 120574

119891for some graphsrdquoDiscrete Mathematics vol 167-

168 pp 249ndash259 1997[114] G S Domke S T Hedetniemi and R C Laskar ldquoFractional

packings coverings and irredundance in graphsrdquo vol 66 pp227ndash238 1988

[115] E J Cockayne P A Dreyer Jr S M Hedetniemi andS T Hedetniemi ldquoRoman domination in graphsrdquo DiscreteMathematics vol 278 no 1ndash3 pp 11ndash22 2004

[116] I Stewart ldquoDefend the roman empirerdquo Scientific American vol281 pp 136ndash139 1999

[117] C S ReVelle and K E Rosing ldquoDefendens imperiumromanum a classical problem in military strategyrdquoThe Ameri-can Mathematical Monthly vol 107 no 7 pp 585ndash594 2000

[118] A Bahremandpour F-T Hu S M Sheikholeslami and J-MXu ldquoOn the Roman bondage number of a graphrdquo DiscreteMathematics Algorithms and Applications vol 5 no 1 ArticleID 1350001 15 pages 2013

[119] N Jafari Rad and L Volkmann ldquoRoman bondage in graphsrdquoDiscussiones Mathematicae Graph Theory vol 31 no 4 pp763ndash773 2011

[120] K Ebadi and L PushpaLatha ldquoRoman bondage and Romanreinforcement numbers of a graphrdquo International Journal ofContemporary Mathematical Sciences vol 5 pp 1487ndash14972010

[121] N Dehgardi S M Sheikholeslami and L Volkman ldquoOn theRoman k-bondage number of a graphrdquo AKCE InternationalJournal of Graphs and Combinatorics vol 8 pp 169ndash180 2011

[122] S Akbari and S Qajar ldquoA note on Roman bondage number ofgraphsrdquo Ars Combinatoria to Appear

[123] F-T Hu and J-M Xu ldquoRoman bondage numbers of somegraphsrdquo httparxivorgabs11093933

[124] N Jafari Rad and L Volkmann ldquoOn the Roman bondagenumber of planar graphsrdquo Graphs and Combinatorics vol 27no 4 pp 531ndash538 2011

[125] S Akbari M Khatirinejad and S Qajar ldquoA note on the romanbondage number of planar graphsrdquo Graphs and Combinatorics2012

[126] B Bresar M A Henning and D F Rall ldquoRainbow dominationin graphsrdquo Taiwanese Journal of Mathematics vol 12 no 1 pp213ndash225 2008

[127] B L Hartnell and D F Rall ldquoOn dominating the Cartesianproduct of a graph and 120572krdquo Discussiones Mathematicae GraphTheory vol 24 no 3 pp 389ndash402 2004

[128] N Dehgardi S M Sheikholeslami and L Volkman ldquoThe k-rainbow bondage number of a graphrdquo to appear in DiscreteApplied Mathematics

[129] J von Neumann and O Morgenstern Theory of Games andEconomic Behavior Princeton University Press Princeton NJUSA 1944

[130] C BergeTheorıe des Graphes et ses Applications 1958[131] O Ore Theory of Graphs vol 38 American Mathematical

Society Colloquium Publications 1962[132] E Shan and L Kang ldquoBondage number in oriented graphsrdquoArs

Combinatoria vol 84 pp 319ndash331 2007[133] J Huang and J-M Xu ldquoThe bondage numbers of extended

de Bruijn and Kautz digraphsrdquo Computers amp Mathematics withApplications vol 51 no 6-7 pp 1137ndash1147 2006

[134] J Huang and J-M Xu ldquoThe total domination and total bondagenumbers of extended de Bruijn and Kautz digraphsrdquoComputersamp Mathematics with Applications vol 53 no 8 pp 1206ndash12132007

[135] Y Shibata and Y Gonda ldquoExtension of de Bruijn graph andKautz graphrdquo Computers amp Mathematics with Applications vol30 no 9 pp 51ndash61 1995

[136] X Zhang J Liu and JMeng ldquoThebondage number in completet-partite digraphsrdquo Information Processing Letters vol 109 no17 pp 997ndash1000 2009

[137] D W Bange A E Barkauskas and P J Slater ldquoEfficient dom-inating sets in graphsrdquo in Applications of Discrete Mathematicspp 189ndash199 SIAM Philadelphia Pa USA 1988

[138] M Livingston and Q F Stout ldquoPerfect dominating setsrdquoCongressus Numerantium vol 79 pp 187ndash203 1990

[139] L Clark ldquoPerfect domination in random graphsrdquo Journal ofCombinatorial Mathematics and Combinatorial Computing vol14 pp 173ndash182 1993

[140] D W Bange A E Barkauskas L H Host and L H ClarkldquoEfficient domination of the orientations of a graphrdquo DiscreteMathematics vol 178 no 1ndash3 pp 1ndash14 1998

[141] A E Barkauskas and L H Host ldquoFinding efficient dominatingsets in oriented graphsrdquo Congressus Numerantium vol 98 pp27ndash32 1993

[142] I J Dejter and O Serra ldquoEfficient dominating sets in CayleygraphsrdquoDiscrete AppliedMathematics vol 129 no 2-3 pp 319ndash328 2003

[143] J Lee ldquoIndependent perfect domination sets in Cayley graphsrdquoJournal of Graph Theory vol 37 no 4 pp 213ndash219 2001

[144] WGu X Jia and J Shen ldquoIndependent perfect domination setsin meshes tori and treesrdquo Unpublished

[145] N Obradovic J Peters and G Ruzic ldquoEfficient domination incirculant graphswith two chord lengthsrdquo Information ProcessingLetters vol 102 no 6 pp 253ndash258 2007

[146] K Reji Kumar and G MacGillivray ldquoEfficient domination incirculant graphsrdquo Discrete Mathematics vol 313 no 6 pp 767ndash771 2013

[147] D Van Wieren M Livingston and Q F Stout ldquoPerfectdominating sets on cube-connected cyclesrdquo Congressus Numer-antium vol 97 pp 51ndash70 1993

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

International Journal of Combinatorics 33

[71] H Qiao L KangM Cardei andD-Z Du ldquoPaired-dominationof treesrdquo Journal of Global Optimization vol 25 no 1 pp 43ndash542003

[72] E Shan L Kang and M A Henning ldquoA characterizationof trees with equal total domination and paired-dominationnumbersrdquo The Australasian Journal of Combinatorics vol 30pp 31ndash39 2004

[73] J Raczek ldquoPaired bondage in treesrdquo Discrete Mathematics vol308 no 23 pp 5570ndash5575 2008

[74] J A Telle and A Proskurowski ldquoAlgorithms for vertex parti-tioning problems on partial k-treesrdquo SIAM Journal on DiscreteMathematics vol 10 no 4 pp 529ndash550 1997

[75] G S Domke J H Hattingh M A Henning and L R MarkusldquoRestrained domination in treesrdquoDiscreteMathematics vol 211no 1ndash3 pp 1ndash9 2000

[76] G S Domke J H Hattingh M A Henning and L R MarkusldquoRestrained domination in graphs with minimum degree twordquoJournal of Combinatorial Mathematics and Combinatorial Com-puting vol 35 pp 239ndash254 2000

[77] G S Domke J H Hattingh S T Hedetniemi R C Laskarand L R Markus ldquoRestrained domination in graphsrdquo DiscreteMathematics vol 203 no 1ndash3 pp 61ndash69 1999

[78] J H Hattingh and E J Joubert ldquoAn upper bound for therestrained domination number of a graph with minimumdegree at least two in terms of order and minimum degreerdquoDiscrete Applied Mathematics vol 157 no 13 pp 2846ndash28582009

[79] M A Henning ldquoGraphs with large restrained dominationnumberrdquo Discrete Mathematics vol 197-198 pp 415ndash429 199916th British Combinatorial Conference (London 1997)

[80] J H Hattingh and A R Plummer ldquoRestrained bondage ingraphsrdquo Discrete Mathematics vol 308 no 23 pp 5446ndash54532008

[81] N Jafari Rad R Hasni J Raczek and L Volkmann ldquoTotalrestrained bondage in graphsrdquoActaMathematica Sinica vol 29no 6 pp 1033ndash1042 2013

[82] J Cyman and J Raczek ldquoOn the total restrained dominationnumber of a graphrdquoThe Australasian Journal of Combinatoricsvol 36 pp 91ndash100 2006

[83] M A Henning ldquoGraphs with large total domination numberrdquoJournal of Graph Theory vol 35 no 1 pp 21ndash45 2000

[84] D-X Ma X-G Chen and L Sun ldquoOn total restraineddomination in graphsrdquoCzechoslovakMathematical Journal vol55 no 1 pp 165ndash173 2005

[85] B Zelinka ldquoRemarks on restrained domination and totalrestrained domination in graphsrdquo Czechoslovak MathematicalJournal vol 55 no 2 pp 393ndash396 2005

[86] W Goddard and M A Henning ldquoIndependent domination ingraphs a survey and recent resultsrdquo Discrete Mathematics vol313 no 7 pp 839ndash854 2013

[87] J H Zhang H L Liu and L Sun ldquoIndependence bondagenumber and reinforcement number of some graphsrdquo Transac-tions of Beijing Institute of Technology vol 23 no 2 pp 140ndash1422003

[88] K D Dharmalingam Studies in Graph Theorymdashequitabledomination and bottleneck domination [PhD thesis] MaduraiKamaraj University Madurai India 2006

[89] GDeepakND Soner andAAlwardi ldquoThe equitable bondagenumber of a graphrdquo Research Journal of Pure Algebra vol 1 no9 pp 209ndash212 2011

[90] E Sampathkumar and H B Walikar ldquoThe connected domina-tion number of a graphrdquo Journal of Mathematical and PhysicalSciences vol 13 no 6 pp 607ndash613 1979

[91] K White M Farber and W Pulleyblank ldquoSteiner trees con-nected domination and strongly chordal graphsrdquoNetworks vol15 no 1 pp 109ndash124 1985

[92] M Cadei M X Cheng X Cheng and D-Z Du ldquoConnecteddomination in Ad Hoc wireless networksrdquo in Proceedings ofthe 6th International Conference on Computer Science andInformatics (CSampI rsquo02) 2002

[93] J F Fink and M S Jacobson ldquon-domination in graphsrdquo inGraph Theory with Applications to Algorithms and ComputerScience Wiley-Interscience Publications pp 283ndash300 WileyNew York NY USA 1985

[94] M Chellali O Favaron A Hansberg and L Volkmann ldquok-domination and k-independence in graphs a surveyrdquo Graphsand Combinatorics vol 28 no 1 pp 1ndash55 2012

[95] Y Lu X Hou J-M Xu andN Li ldquoTrees with uniqueminimump-dominating setsrdquo Utilitas Mathematica vol 86 pp 193ndash2052011

[96] Y Lu and J-M Xu ldquoThe p-bondage number of treesrdquo Graphsand Combinatorics vol 27 no 1 pp 129ndash141 2011

[97] Y Lu and J-M Xu ldquoThe 2-domination and 2-bondage numbersof grid graphs A manuscriptrdquo httparxivorgabs12044514

[98] M Krzywkowski ldquoNon-isolating 2-bondage in graphsrdquo TheJournal of the Mathematical Society of Japan vol 64 no 4 2012

[99] M A Henning O R Oellermann and H C Swart ldquoBoundson distance domination parametersrdquo Journal of CombinatoricsInformation amp System Sciences vol 16 no 1 pp 11ndash18 1991

[100] F Tian and J-M Xu ldquoBounds for distance domination numbersof graphsrdquo Journal of University of Science and Technology ofChina vol 34 no 5 pp 529ndash534 2004

[101] F Tian and J-M Xu ldquoDistance domination-critical graphsrdquoApplied Mathematics Letters vol 21 no 4 pp 416ndash420 2008

[102] F Tian and J-M Xu ldquoA note on distance domination numbersof graphsrdquo The Australasian Journal of Combinatorics vol 43pp 181ndash190 2009

[103] F Tian and J-M Xu ldquoAverage distances and distance domina-tion numbersrdquo Discrete Applied Mathematics vol 157 no 5 pp1113ndash1127 2009

[104] Z-L Liu F Tian and J-M Xu ldquoProbabilistic analysis of upperbounds for 2-connected distance k-dominating sets in graphsrdquoTheoretical Computer Science vol 410 no 38ndash40 pp 3804ndash3813 2009

[105] M A Henning O R Oellermann and H C Swart ldquoDistancedomination critical graphsrdquo Journal of Combinatorial Mathe-matics and Combinatorial Computing vol 44 pp 33ndash45 2003

[106] T J Bean M A Henning and H C Swart ldquoOn the integrityof distance domination in graphsrdquo The Australasian Journal ofCombinatorics vol 10 pp 29ndash43 1994

[107] M Fischermann and L Volkmann ldquoA remark on a conjecturefor the (kp)-domination numberrdquoUtilitasMathematica vol 67pp 223ndash227 2005

[108] T Korneffel D Meierling and L Volkmann ldquoA remark on the(22)-domination numberrdquo Discussiones Mathematicae GraphTheory vol 28 no 2 pp 361ndash366 2008

[109] Y Lu X Hou and J-M Xu ldquoOn the (22)-domination numberof treesrdquo Discussiones Mathematicae Graph Theory vol 30 no2 pp 185ndash199 2010

34 International Journal of Combinatorics

[110] H Li and J-M Xu ldquo(dm)-domination numbers of m-connected graphsrdquo Rapports de Recherche 1130 LRI UniversiteParis-Sud 1997

[111] S M Hedetniemi S T Hedetniemi and T V Wimer ldquoLineartime resource allocation algorithms for treesrdquo Tech Rep URI-014 Department of Mathematical Sciences Clemson Univer-sity 1987

[112] M Farber ldquoDomination independent domination and dualityin strongly chordal graphsrdquo Discrete Applied Mathematics vol7 no 2 pp 115ndash130 1984

[113] G S Domke and R C Laskar ldquoThe bondage and reinforcementnumbers of 120574

119891for some graphsrdquoDiscrete Mathematics vol 167-

168 pp 249ndash259 1997[114] G S Domke S T Hedetniemi and R C Laskar ldquoFractional

packings coverings and irredundance in graphsrdquo vol 66 pp227ndash238 1988

[115] E J Cockayne P A Dreyer Jr S M Hedetniemi andS T Hedetniemi ldquoRoman domination in graphsrdquo DiscreteMathematics vol 278 no 1ndash3 pp 11ndash22 2004

[116] I Stewart ldquoDefend the roman empirerdquo Scientific American vol281 pp 136ndash139 1999

[117] C S ReVelle and K E Rosing ldquoDefendens imperiumromanum a classical problem in military strategyrdquoThe Ameri-can Mathematical Monthly vol 107 no 7 pp 585ndash594 2000

[118] A Bahremandpour F-T Hu S M Sheikholeslami and J-MXu ldquoOn the Roman bondage number of a graphrdquo DiscreteMathematics Algorithms and Applications vol 5 no 1 ArticleID 1350001 15 pages 2013

[119] N Jafari Rad and L Volkmann ldquoRoman bondage in graphsrdquoDiscussiones Mathematicae Graph Theory vol 31 no 4 pp763ndash773 2011

[120] K Ebadi and L PushpaLatha ldquoRoman bondage and Romanreinforcement numbers of a graphrdquo International Journal ofContemporary Mathematical Sciences vol 5 pp 1487ndash14972010

[121] N Dehgardi S M Sheikholeslami and L Volkman ldquoOn theRoman k-bondage number of a graphrdquo AKCE InternationalJournal of Graphs and Combinatorics vol 8 pp 169ndash180 2011

[122] S Akbari and S Qajar ldquoA note on Roman bondage number ofgraphsrdquo Ars Combinatoria to Appear

[123] F-T Hu and J-M Xu ldquoRoman bondage numbers of somegraphsrdquo httparxivorgabs11093933

[124] N Jafari Rad and L Volkmann ldquoOn the Roman bondagenumber of planar graphsrdquo Graphs and Combinatorics vol 27no 4 pp 531ndash538 2011

[125] S Akbari M Khatirinejad and S Qajar ldquoA note on the romanbondage number of planar graphsrdquo Graphs and Combinatorics2012

[126] B Bresar M A Henning and D F Rall ldquoRainbow dominationin graphsrdquo Taiwanese Journal of Mathematics vol 12 no 1 pp213ndash225 2008

[127] B L Hartnell and D F Rall ldquoOn dominating the Cartesianproduct of a graph and 120572krdquo Discussiones Mathematicae GraphTheory vol 24 no 3 pp 389ndash402 2004

[128] N Dehgardi S M Sheikholeslami and L Volkman ldquoThe k-rainbow bondage number of a graphrdquo to appear in DiscreteApplied Mathematics

[129] J von Neumann and O Morgenstern Theory of Games andEconomic Behavior Princeton University Press Princeton NJUSA 1944

[130] C BergeTheorıe des Graphes et ses Applications 1958[131] O Ore Theory of Graphs vol 38 American Mathematical

Society Colloquium Publications 1962[132] E Shan and L Kang ldquoBondage number in oriented graphsrdquoArs

Combinatoria vol 84 pp 319ndash331 2007[133] J Huang and J-M Xu ldquoThe bondage numbers of extended

de Bruijn and Kautz digraphsrdquo Computers amp Mathematics withApplications vol 51 no 6-7 pp 1137ndash1147 2006

[134] J Huang and J-M Xu ldquoThe total domination and total bondagenumbers of extended de Bruijn and Kautz digraphsrdquoComputersamp Mathematics with Applications vol 53 no 8 pp 1206ndash12132007

[135] Y Shibata and Y Gonda ldquoExtension of de Bruijn graph andKautz graphrdquo Computers amp Mathematics with Applications vol30 no 9 pp 51ndash61 1995

[136] X Zhang J Liu and JMeng ldquoThebondage number in completet-partite digraphsrdquo Information Processing Letters vol 109 no17 pp 997ndash1000 2009

[137] D W Bange A E Barkauskas and P J Slater ldquoEfficient dom-inating sets in graphsrdquo in Applications of Discrete Mathematicspp 189ndash199 SIAM Philadelphia Pa USA 1988

[138] M Livingston and Q F Stout ldquoPerfect dominating setsrdquoCongressus Numerantium vol 79 pp 187ndash203 1990

[139] L Clark ldquoPerfect domination in random graphsrdquo Journal ofCombinatorial Mathematics and Combinatorial Computing vol14 pp 173ndash182 1993

[140] D W Bange A E Barkauskas L H Host and L H ClarkldquoEfficient domination of the orientations of a graphrdquo DiscreteMathematics vol 178 no 1ndash3 pp 1ndash14 1998

[141] A E Barkauskas and L H Host ldquoFinding efficient dominatingsets in oriented graphsrdquo Congressus Numerantium vol 98 pp27ndash32 1993

[142] I J Dejter and O Serra ldquoEfficient dominating sets in CayleygraphsrdquoDiscrete AppliedMathematics vol 129 no 2-3 pp 319ndash328 2003

[143] J Lee ldquoIndependent perfect domination sets in Cayley graphsrdquoJournal of Graph Theory vol 37 no 4 pp 213ndash219 2001

[144] WGu X Jia and J Shen ldquoIndependent perfect domination setsin meshes tori and treesrdquo Unpublished

[145] N Obradovic J Peters and G Ruzic ldquoEfficient domination incirculant graphswith two chord lengthsrdquo Information ProcessingLetters vol 102 no 6 pp 253ndash258 2007

[146] K Reji Kumar and G MacGillivray ldquoEfficient domination incirculant graphsrdquo Discrete Mathematics vol 313 no 6 pp 767ndash771 2013

[147] D Van Wieren M Livingston and Q F Stout ldquoPerfectdominating sets on cube-connected cyclesrdquo Congressus Numer-antium vol 97 pp 51ndash70 1993

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

34 International Journal of Combinatorics

[110] H Li and J-M Xu ldquo(dm)-domination numbers of m-connected graphsrdquo Rapports de Recherche 1130 LRI UniversiteParis-Sud 1997

[111] S M Hedetniemi S T Hedetniemi and T V Wimer ldquoLineartime resource allocation algorithms for treesrdquo Tech Rep URI-014 Department of Mathematical Sciences Clemson Univer-sity 1987

[112] M Farber ldquoDomination independent domination and dualityin strongly chordal graphsrdquo Discrete Applied Mathematics vol7 no 2 pp 115ndash130 1984

[113] G S Domke and R C Laskar ldquoThe bondage and reinforcementnumbers of 120574

119891for some graphsrdquoDiscrete Mathematics vol 167-

168 pp 249ndash259 1997[114] G S Domke S T Hedetniemi and R C Laskar ldquoFractional

packings coverings and irredundance in graphsrdquo vol 66 pp227ndash238 1988

[115] E J Cockayne P A Dreyer Jr S M Hedetniemi andS T Hedetniemi ldquoRoman domination in graphsrdquo DiscreteMathematics vol 278 no 1ndash3 pp 11ndash22 2004

[116] I Stewart ldquoDefend the roman empirerdquo Scientific American vol281 pp 136ndash139 1999

[117] C S ReVelle and K E Rosing ldquoDefendens imperiumromanum a classical problem in military strategyrdquoThe Ameri-can Mathematical Monthly vol 107 no 7 pp 585ndash594 2000

[118] A Bahremandpour F-T Hu S M Sheikholeslami and J-MXu ldquoOn the Roman bondage number of a graphrdquo DiscreteMathematics Algorithms and Applications vol 5 no 1 ArticleID 1350001 15 pages 2013

[119] N Jafari Rad and L Volkmann ldquoRoman bondage in graphsrdquoDiscussiones Mathematicae Graph Theory vol 31 no 4 pp763ndash773 2011

[120] K Ebadi and L PushpaLatha ldquoRoman bondage and Romanreinforcement numbers of a graphrdquo International Journal ofContemporary Mathematical Sciences vol 5 pp 1487ndash14972010

[121] N Dehgardi S M Sheikholeslami and L Volkman ldquoOn theRoman k-bondage number of a graphrdquo AKCE InternationalJournal of Graphs and Combinatorics vol 8 pp 169ndash180 2011

[122] S Akbari and S Qajar ldquoA note on Roman bondage number ofgraphsrdquo Ars Combinatoria to Appear

[123] F-T Hu and J-M Xu ldquoRoman bondage numbers of somegraphsrdquo httparxivorgabs11093933

[124] N Jafari Rad and L Volkmann ldquoOn the Roman bondagenumber of planar graphsrdquo Graphs and Combinatorics vol 27no 4 pp 531ndash538 2011

[125] S Akbari M Khatirinejad and S Qajar ldquoA note on the romanbondage number of planar graphsrdquo Graphs and Combinatorics2012

[126] B Bresar M A Henning and D F Rall ldquoRainbow dominationin graphsrdquo Taiwanese Journal of Mathematics vol 12 no 1 pp213ndash225 2008

[127] B L Hartnell and D F Rall ldquoOn dominating the Cartesianproduct of a graph and 120572krdquo Discussiones Mathematicae GraphTheory vol 24 no 3 pp 389ndash402 2004

[128] N Dehgardi S M Sheikholeslami and L Volkman ldquoThe k-rainbow bondage number of a graphrdquo to appear in DiscreteApplied Mathematics

[129] J von Neumann and O Morgenstern Theory of Games andEconomic Behavior Princeton University Press Princeton NJUSA 1944

[130] C BergeTheorıe des Graphes et ses Applications 1958[131] O Ore Theory of Graphs vol 38 American Mathematical

Society Colloquium Publications 1962[132] E Shan and L Kang ldquoBondage number in oriented graphsrdquoArs

Combinatoria vol 84 pp 319ndash331 2007[133] J Huang and J-M Xu ldquoThe bondage numbers of extended

de Bruijn and Kautz digraphsrdquo Computers amp Mathematics withApplications vol 51 no 6-7 pp 1137ndash1147 2006

[134] J Huang and J-M Xu ldquoThe total domination and total bondagenumbers of extended de Bruijn and Kautz digraphsrdquoComputersamp Mathematics with Applications vol 53 no 8 pp 1206ndash12132007

[135] Y Shibata and Y Gonda ldquoExtension of de Bruijn graph andKautz graphrdquo Computers amp Mathematics with Applications vol30 no 9 pp 51ndash61 1995

[136] X Zhang J Liu and JMeng ldquoThebondage number in completet-partite digraphsrdquo Information Processing Letters vol 109 no17 pp 997ndash1000 2009

[137] D W Bange A E Barkauskas and P J Slater ldquoEfficient dom-inating sets in graphsrdquo in Applications of Discrete Mathematicspp 189ndash199 SIAM Philadelphia Pa USA 1988

[138] M Livingston and Q F Stout ldquoPerfect dominating setsrdquoCongressus Numerantium vol 79 pp 187ndash203 1990

[139] L Clark ldquoPerfect domination in random graphsrdquo Journal ofCombinatorial Mathematics and Combinatorial Computing vol14 pp 173ndash182 1993

[140] D W Bange A E Barkauskas L H Host and L H ClarkldquoEfficient domination of the orientations of a graphrdquo DiscreteMathematics vol 178 no 1ndash3 pp 1ndash14 1998

[141] A E Barkauskas and L H Host ldquoFinding efficient dominatingsets in oriented graphsrdquo Congressus Numerantium vol 98 pp27ndash32 1993

[142] I J Dejter and O Serra ldquoEfficient dominating sets in CayleygraphsrdquoDiscrete AppliedMathematics vol 129 no 2-3 pp 319ndash328 2003

[143] J Lee ldquoIndependent perfect domination sets in Cayley graphsrdquoJournal of Graph Theory vol 37 no 4 pp 213ndash219 2001

[144] WGu X Jia and J Shen ldquoIndependent perfect domination setsin meshes tori and treesrdquo Unpublished

[145] N Obradovic J Peters and G Ruzic ldquoEfficient domination incirculant graphswith two chord lengthsrdquo Information ProcessingLetters vol 102 no 6 pp 253ndash258 2007

[146] K Reji Kumar and G MacGillivray ldquoEfficient domination incirculant graphsrdquo Discrete Mathematics vol 313 no 6 pp 767ndash771 2013

[147] D Van Wieren M Livingston and Q F Stout ldquoPerfectdominating sets on cube-connected cyclesrdquo Congressus Numer-antium vol 97 pp 51ndash70 1993

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of