Research ArticleSome Properties of Double Roman Domination

Hong Yang and Xiaoqing Zhou

School of Information Science and Engineering Chengdu University Chengdu 610106 China

Correspondence should be addressed to Hong Yang yanghong01cdueducn

Received 22 May 2020 Accepted 18 July 2020 Published 14 August 2020

Academic Editor Juan L G Guirao

Copyright copy 2020 Hong Yang and Xiaoqing Zhou is is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in anymedium provided the original work isproperly cited

A double Roman dominating function on a graph G is a function f V(G)⟶ 0 1 2 3 satisfying the conditions that every vertexu for which f(u) 0 is adjacent to at least one vertex v for which f(v) 3 or two vertices v1 and v2 for which f(v1) f(v2) 2and every vertex u for which f(u) 1 is adjacent to at least one vertex v for which f(v)ge 2 e weight of a double Romandominating function f is the value f(V) 1113936uisinVf(u) e minimum weight of a double Roman dominating function on a graphG is called the double Roman domination number cdR(G) of G A graph with cdR(G) 3c(G) is called a double Roman graph Inthis paper we study properties of double Roman domination in graphs Moreover we find a class of double Roman graphs andgive characterizations of trees with cdR(T) cR(T) + k for k 1 2

1 Introduction

In this paper we shall only consider graphs without multipleedges or loops Let G be a graph v isin V(G) and the neigh-borhood of v in G is denoted by N(v) at is to sayN(v) u | uv isin E(G) u isin V(G) e closed neighborhoodN[v] of v in G is defined as N[v] v cupN(v) e com-plementary graph ofG is denoted byG A vertex of degree one iscalled a leaf A graph is trivial if it has a single vertexe degreeof a vertex v is denoted by d(v) ie d(v) |N(v)| Denote byKn Pn and Cn the complete graph path and cycle on n

vertices respectively e maximum degree and the minimumdegree of a graph are denoted by Δ(G) and δ(G) respectivelyFor a set SsubeV(G) the graph induced by S is denoted by G[S]Let e isin E(G) and we denote byGe the graph obtained fromG

by contracting the edge e For an edge e isin E(G) we denote byG minus e the graph obtained from G by deleting e

A subset D of the vertex set of a graph G is a dominatingset if every vertex not in D has at least one neighbour in De domination number c(G) is the minimum cardinality ofa dominating set of G

e domination and its variations of graphs haveattracted considerable attention [1 2] Many varieties ofdominating sets are listed in the book Fundamentals ofDomination in Graphs [3] However Roman domination

and double Roman domination are not listed in this bookRoman domination and double Roman domination appearto be a new variety of interest [4ndash11]

A Roman dominating function (RDF) of a graph G is afunction f V(G)⟶ 0 1 2 such that every vertex u forwhich f(u) 0 is adjacent to at least one vertex v for whichf(v) 2e weight w(f) of a Roman dominating functionf is the value w(f) 1113936uisinV(G)f(u) e minimumweight ofa Roman dominating function on a graph G is called theRoman domination number cR(G) of G An RDF f of G withw(f) cR(G) is called a cR(G) function

A double Roman dominating function (DRDF) on agraph G is a function f V(G)⟶ 0 1 2 3 satisfying thecondition that every vertex u for which f(u) 0 is adjacentto at least one vertex v for which f(v) 3 or two vertices v1and v2 for which f(v1) f(v2) 2 and every vertex u forwhich f(u) 1 is adjacent to at least one vertex v for whichf(v)ge 2 e weight w(f) of a double Roman dominatingfunction f is the value w(f) 1113936uisinV(G)f(u) e minimumweight of a double Roman dominating function on a graphG is called the double Roman domination number cdR(G) ofG A DRDF f of G with w(f) cdR(G) is called a cdR(G)

function We denote by wS(f) the weight of a doubleRoman dominating function f in SsubeV(G) iewS(f) 1113936xisinSf(x)

HindawiDiscrete Dynamics in Nature and SocietyVolume 2020 Article ID 6481092 5 pageshttpsdoiorg10115520206481092

Beeler et al [12] initiated the study of the double Romandomination in graphs ey showed that 2c(G)lecdR(G)le 3c(G) and defined a graph G to be double Romanif cdR(G) 3c(G) Moreover they suggest to find doubleRoman graphs

In this paper we study properties of double Romandomination in graphs and show that the double Romandomination problem is NP-complete for bipartite graphsMoreover we find a class of double Roman graphs and givecharacterizations of trees with cdR(T) cR(T) + k fork 1 2

2 Properties of Double Roman Domination

Proposition 1 (see [12]) In a double Roman dominatingfunction of weight cdR(G) no vertex needs to be assigned thevalue 1

By Proposition 1 when we consider a cdR(G) functionwe assume no vertex has been assigned the value 1

Proposition 2 (see [12])

(i) Let G be a graph and f (V0 V1 V2) be a cR(G)

function +en cdR(G)le 2|V1| + 3|V2|(ii) For any graph G cdR(G)le 2cR(G) with equality if

and only if G Kn

Proposition 3 (see [12])

(i) For every graph G cR(G)lt cdR(G)(ii) If f (V0empty V2 V3) is any cdR(G) function then

cR(G)le 2(|V2| + |V3|) cdR(G) minus |V3|

e following result is immediate

Proposition 4 For any graph G cdR(G)ge (3|V(G)|Δ(G) + 1)

Proof e desired inequality obviously holds if Δ(G)le 1 Inorder to prove the proposition for Δ(G)ge 2 we introducethe discharging approach Let f (V0empty V2 V3) be acdR(G) function e initial charge of every vertex v isin V(G)

is set to be s(v) f(v) We apply the discharging proceduredefined by applying the following rule

For each vertex v isin V3 we send 3(d(v) + 1) charge toeach adjacent vertex in V0 en the final charge of v issatisfying with s(v) (3(d(v) + 1))ge (3(Δ(G) + 1))

For each vertex v isin V2 we send (2(d(v) + 1))minus

(3(d(v)(Δ(G) + 1))) ((2Δ(G) minus 1)(d(v)(Δ(G) + 1)))ge((2Δ(G) minus 1)(Δ(G)(Δ(G) + 1))) charge to each adjacentvertex in V0 en the final charge of v is satisfying withs(v) (3(d(v) + 1))ge (3(Δ(G) + 1))

For each vertex v isin V0 by the definition of doubleRoman domination v has a neighbor assigned 3 or twovertices u1 and u2 assigned 2 Due to the discharging ruleabove if v has a neighbor u assigned 3 v receives chargefrom u We have s(v)ge (3(Δ(G) + 1))

If v has two vertices u1 and u2 assigned 2 v receivescharge from u1 and u2 We have s(v)ge ((4Δ(G) minus 2)(Δ(G)(Δ(G) + 1)))ge (3(Δ(G) + 1)) us cdR(G)

1113936visinV(G)f(v) 1113936visinV(G)s(v)ge ((3|V(G)|) (Δ(G) + 1)) eproof is complete

Proposition 5 Let G be a graph If cR(G) + 1 cdR(G) andf (V0empty V2 V3) is a cdR(G) function

(i) then |V3|le 1(ii) if |V3| 1 then |V2| 0 and there exists a vertex v

with degree |V(G)| minus 1

Proof

(i) By Proposition 3 we have cR(G)le cdR(G) minus |V3|

cR(G) + 1 minus |V3| So we have |V3|le 1(ii) If |V3| 1 let V3 v and H G[V(G) minus N[V3]]

We have the following claim

Claim 1 H is empty

Proof Otherwise any vertex in H assigned with 0 has atleast two vertices assigned with 2 Let w be a vertex in H

assigned with 2 we consider a function fprime with fprime(w) 1fprime(v) 2 and fprime(x) f(x) for any x isin V(G) minus v w en fprime is an RDF of G with weight cdR(G) minus 2 SocR(G)le cdR(G) minus 2 a contradiction

Since H is empty we have v as a vertex with degree|V(G)| minus 1 Now we have cdR(G) 3 and cR(G) 2 and sothe result holds

Theorem 1 For every graph G on n vertices without isolatedvertex cdR(G)le 3n minus (3n(2(1 + δ(G))))e1δ(G)

Proof Clearly we have δ(G)ge 1 We select a subset S ofV(G) where each vertex is selected with probability p in-dependently Let T V(G) minus N[S] we consider a functionf V(G)⟶ 0 2 3 with f(x) 3 for x isin S f(x) 2 forx isin T and f(x) 0 for other vertex x en f is a DRDF ofG We have cdR(G)le 3|S| + 2|T| When we consider theexpectation we also have cdR(G)leE[3|S| + 2|T|]

3E[|S|] + 2E[|T|] First it is clear that E[|S|] np For eachvertex with degree d(x) if neither x nor any neighbor isselected then x isin T So we have P(x isin T) (1 minus p)1+d(x)and thus E[|T|]le n(1 minus p)1+δ(G) Consequently cdR(G)le3np + 2n(1 minus p)1+δ(G) Let F(p) 3np + 2n(1 minus p)1+δ(G) ByF(p) 3np minus 2n(1 + δ(G))(1 minus p)δ(G) 0 the maximumof F is given by Fmax 3n minus (3n(2(1 + δ(G))))e1δ(G) atp 1 minus (3(2(1 + δ(G))))e1δ(G) isin (0 1) us cdR(G)le3n minus (3n(2(1 + δ(G))))e1δ(G)

If G is a graph with some isolated vertices then δ(G) 0Let W be the set of isolated vertices of G and let Gprime G minus Werefore δ(Gprime)ge 1 Because all isolated vertices must beassigned 3 it is easy to prove that cdR(G)le 3n minus

(3nprime(2(1 + δ(Gprime))))e1δ(Gprime) where nprime |V(Gprime)|

2 Discrete Dynamics in Nature and Society

Proposition 6 If G is a connected graph of order n thencdR(G) + 1 2cR(G) if and only if there exists a vertex v ofdegree n + 1 minus ((cdR(G) + 1)2) in G

Proof

(rArr) Let f (V0 V1 V2) be a cR(G) function withminimum |V1| en we have V1 being independentTogether with Proposition 2 we have cdR(G)le 2|V1| +

3|V2|le 2|V1| +4|V2| 2cR(G) cdR(G) + 1 en wehave |V2|le 1 If |V2| 0 we have |V0| 0 and thuscR(G) n |V1| Since V1 is independent it is im-possible If |V2| 1 we have cdR(G) 2|V1| + 3|V2|

lt 2|V1| + 4|V2| 2cR(G) cdR(G) + 1 Let V2 v V0 N(v) and V1 V(G) minus V0 minus V2 en we havecR(G) n minus 1 minus d(v) + 2 and so d(v) n + 1 minus cR(G)

n + 1 minus ((cdR(G) + 1)2)(lArr) By Proposition 2 (ii) we have cdR(G) + 1le 2cR(G)

for a connected graph G Assume G contains a vertex v

of degree n + 1 minus (cdR(G) + 1)2 in G Let V2 v V0 N(v) and V1 V(G) minus V0 minus V2 enf (V0 V1 V2) is a cR(G) function and socR(G)le |V1| + 2|V2| (cdR(G) + 1)2 HencecdR(G) + 1ge 2cR(G)

Let F be the family of connected graphs G such that forany cdR(G) function f (V0empty V2 V3) we have |V3| 0

Proposition 7 Let G isin F then

(i) G contains no strong support vertex(ii) if f(u) f(v) 2 for an edge uv isin E(G) then G minus

e isinF and Ge isinF

Proof

(i) Suppose v be a strong support vertex then there existtwo leaves x y isin N(v) Since f(v)ne 3 we havef(x) f(y) 2 Now consider the function fprimewith fprime(z) 0 for any z isin L(v) fprime(v) 3 andfprime(z) f(z) for any z isin V(G)∖L[v] en fprime is aDRDF of G with fewer weight than f acontradiction

(ii) If f(u) f(v) 2 for an edge uv isin E(G) we have f

as also a DRDF of G minus e SocdR(G minus e)lew(f) cdr(G) Since for any graph Gwe have cdR(G minus e)ge cdR(G) socdR(G minus e) cdR(G) Suppose to the contrary thatthere exists a cdR(G minus e) function such that f(v) 3for a vertex v isin V(G minus e) en f is also a cdR(G)

function contradicting with |V3| 0 For the graphGe the proof is similar

Lemma 1 Let G be a graph on nge 4 vertices then cdR(G)

4 if and only if G contains a complete bipartite graph K2nminus 2 asa subgraph and Δ(G)le n minus 2

Proof

(rArr) If cdR(G) 4 then no vertex is assigned with 3and thus we have two vertices v andw assigned with 2and the others 0 Also each vertex 0 must be adjacent toboth v and w erefore G contains a complete bi-partite graph K2nminus 2 as a subgraph Since cdR(G)gt 3 wehave Δ(G)le n minus 2(lArr) If G contains a complete bipartite graph K2nminus 2with partitions X Y (|X| 2 |Y| n minus 2) as a subgraphand Δ(G)le n minus 2 en let f(x) 2 for any x isin X andf(x) 0 for x isin Y en f is a DRDF of G and socdR(G)le 4 Since G contains no vertex with degree|V(G)| minus 1 we have cdR(G)ge 4

Note that Δ(G)ge n minus 2 if G contains a complete bipartitegraph K2nminus 2 as a subgraph us Δ(G)le n minus 2 can bereplaced with Δ(G) n minus 2 in the lemma

Theorem 2 Let G be a graph on nge 3 vertices then8le cdR(G) + cdR(G)le 2n + 3 Furthermore equality holds inthe upper bound if G or G is Kn

Proof If G is a graph on nge 3 vertices we have cdR(G)ge 3and if cdR(G) 3 then G has a vertex with degree n minus 1 Butits complement is neither a star nor a graph G with cdR(G)

4 (see the graph stated in Lemma 1) So we have cdR(G)ge 5and thus cdR(G) + cdR(G)ge 8 If G is a star then cdR(G) 5and so the lower bound is attainable

Let v be a vertex with maximum degree Δ(G) consider afunction f with f(v) 3 f(x) 0 for any x isin N(v) andf(x) 2 for x isin V(G) minus N[x] en f is a DRDF of G andso cdR(G)lew(f) 2n minus 2Δ(G) + 1 Since Δ(G) + δ(G)

n minus 1 we have cdR(G)le 2n minus 2Δ(G) + 1 2δ(G) + 3erefore cdR(G) + cdR(G)le 2n + 4 minus 2Δ(G) + 2δ(G)le2n + 4 It can be seen that if cdR(G) + cdR(G) 2n + 4 thenΔ(G) δ(G) Hence G is k-regular for some k By sym-metry we may assume that kle (n minus 1)2 en ifcdR(G) + cdR(G) 2n + 4 we have cdR(G) 2n minus 2k + 1and cdR(G) 2k + 3 Let v isin V(G) If |N(u)capN(v)|le k minus 2for some u isin V(G) minus N[v] then f (N(v)cupN(u)

empty V(G) minus N[u] minus N[v] u v ) is a DRDF of G with fewerweight than 2n minus 2k + 1 a contradiction erefore eachvertex not in N[v] has at least k minus 1 neighbors in N(v)Analogously each vertex in N[v] has at most 2 neighborsoutside N[v] en we have (k minus 1)(n minus k minus 1)le 2k Sincekle (n minus 1)2 we have kle 2 + 2(k minus 1) If k 3 then n 7is is impossible If k 2 then n isin 5 6 7 If G C5 wehave cdR(G) cdR(G) 6 and so cdR(G) + cdR(G)ne2n + 4 a contradiction If G C6 we have cdR(G)

cdR(G) 6 and so cdR(G) + cdR(G)ne 2n + 4 a contradic-tion If G C7 we have cdR(G) 8 and cdR(G) 6 and socdR(G) + cdR(G)ne 2n + 4 a contradiction If k 1 thenG (n2)K2 and we have cdR(G) 3n2 and cdR(G) 4and so cdR(G) + cdR(G)ne 2n + 4 a contradictionereforewe conclude that 8le cdR(G) + cdR(G)le 2n + 3 If G Knthen cdR(G) 3 and cdR(G) 2n and thus the upper boundis attainable

Discrete Dynamics in Nature and Society 3

3 Some Double Roman Graphs

e Cartesian product of graphs G and H is the graph GHwith vertex set G times H and (x1 x2)(y1 y2) isin E(GH)

whenever x1y1 isin E(G) and x2 y2 or x2y2 isin E(H) andx1 y1 e Cartesian product is commutative and asso-ciative having the trivial graph as a unit (cf [13])

Let f be a double Roman dominating function ofCmCn and we write V

fi v isin V(CmCn) | f(v) i1113864 1113865 for

i isin 0 1 2 3 When no confusion arise we simply write Vfi

as Vi We use f(i j) to denote the value f(v) forv (i j) isin V(CmCn) Let xi be the weight of Ci iexi 1113936xisinCi f(x)1113864 1113865

Theorem 3 Let m nge 1 +en the Cartesian product graphsC5mC5n are double Roman

Proof e lower bound follows from Proposition 4 LetV(C5mC5n) vij 0le ile 5m minus 1 0le jle 5n minus 11113966 1113967V1 V2 empty V3 v(5i)(5j+2)1113966 v(5i+1)(5j) v(5i+2)(5j+3)v(5i+3)(5j+1) v(5i+4)(5j+4) 0le ilem minus 1 0le jle n minus 1 andV0 N(V3) en N[V3] V(C5mC5n) and sof (V0 V1 V2 V3) is a DRDF of C5mC5n with weight15mn and we have cdR(C5mC5n)le 15mn Sincec(C5mC5n) 5mn we have C5mC5n is doubleRoman

4 Trees T with γdR(T)= γR(T)+ k

Theorem 4 If T is a tree then cR(T) + 1 cdR(T) if andonly if T is a star K1s for sge 1

Proof

(lArr) If T is a star K1s for sge 1 it is clear that cdR(T) 3and cR(T) 2 and the theorem holds(rArr) By Proposition 3 we have |V3|le 1 If |V3| 1 byProposition 5 we have T is a star K1s for some sge 1 If|V2| 0 then each vertex in T is assigned 2 or 0 SinceT is a tree T has a least two leaves v andw andf(v) f(w) 2 If v andw are adjacent to the samevertex x then we can obtain a DRDF of T with fewerweight by changing f(x) to 3 and f(v) andf(w) to 0and obtaining a contradiction If v andw are adjacent todifferent vertices we consider a function fprime withfprime(w) 1 fprime(v) 1 and fprime(x) f(x) for anyx isin V(T) minus v w en fprime is an RDF of T with weightcdR(T) minus 2 So cR(T)le cdR(T) minus 2 a contradiction

For a positive integer t a wounded spider is a star K1t

with at most t minus 1 of its edges subdivided In a woundedspider a vertex of degree t will be called the head vertex andthe vertices at distance two from the head vertex will be thefoot vertices

Theorem 5 If T is a tree then cR(T) + 2 cdR(T) if andonly if T is a wounded spider with only one foot or T isobtained by adding an edge between two stars K1s and K1t

for s tge 2

Proof

(lArr) If T is a wounded spider with only one foot it isclear that cdR(T) 5 and cR(T) 3 and the theoremholds If T is a tree obtained by adding an edge betweentwo stars K1s and K1t for s tge 2 then cdR(T) 6 andcR(T) 4 and the theorem holds(rArr) By Proposition 3 we have |V3|le 2 If |V3| 2 letV3 v w and H G[V(G) minus N[w] minus N[v]] Similarto the proof of eorem 5 we have H as emptyOtherwise there exists a vertex u in H assigned with 2Now we change the function values of v w u from3 3 2 to 2 2 1 respectively and obtain an RDF of T

with weight cdR(T) minus 3 a contradiction In this case T

is a tree obtained by adding an edge between two starsK1s and K1t for s tge 2 If |V3| 1 let V3 v andH G[V(G) minus N[v]] en H has at most one con-nected component Otherwise we can make a vertex ineach connected component to change the functionvalues including the vertex v to obtain an RDF of T withweight cdR(T) minus 3 Since H contains no vertex assignedwith 3 then H is not a star with at least two leaves Weclaim that the leaves of H are at most two Otherwisewe change the function values of v and choose twoleaves to obtain an RDF of T with weight cdR(T) minus 3erefore H is a path on at least four vertices In thiscase we can obtain an RDF of T with weight at mostcdR(T) minus 3 a contradiction erefore T is a woundedspider with only one foot

Data Availability

No data were used to support this study

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

is work was supported by Sichuan Science and Tech-nology Program under grant 2018ZR0265 Sichuan Militaryand Civilian Integration Strategy Research Center undergrant JMRH-1818 and Sichuan Provincial Department ofEducation (Key Project) under grant 18ZA0118

References

[1] M R Garey and D S Johnson Computers and IntractabilityA Guide to the +eory of NP-Completeness W H FreemanSan Francisco CA USA 1979

[2] O Ore +eory of Graphs American Mathematical SocietyProvidence RI USA 1967

[3] TW Haynes S T Hedetniemi and P J Slater Fundamentalsof Domination in Graphs Marcel Dekker NewYork NYUSA 1998

[4] X Zhang Z Li H Jiang and Z Shao ldquoDouble romandomination in treesrdquo Information Processing Letters vol 134pp 31ndash34 2018

4 Discrete Dynamics in Nature and Society

[5] E Zhu and Z Shao ldquoExtremal problems on weak romandomination numberrdquo Information Processing Letters vol 138pp 12ndash18 2018

[6] Z Shao P Wu H Jiang Z Li J Zerovnik and X ZhangldquoDischarging approach for double roman domination ingraphsrdquo IEEE Access vol 6 pp 63345ndash63351 2018

[7] Z Shao J Amjadi S M Sheikholeslami and M ValinavazldquoOn the total double roman dominationrdquo IEEE Access vol 7pp 52035ndash52041 2019

[8] J-B Liu C Wang S Wang and B Wei ldquoZagreb indices andmultiplicative zagreb indices of Eulerian graphsrdquo Bulletin ofthe Malaysian Mathematical Sciences Society vol 42 no 1pp 67ndash78 2019

[9] J-B Liu X-F Pan F-T Hu and F-F Hu ldquoAsymptoticLaplacian-energy-like invariant of latticesrdquo Applied Mathe-matics and Computation vol 253 pp 205ndash214 2015

[10] J-B Liu and X-F Pan ldquoMinimizing Kirchhoff index amonggraphs with a given vertex bipartitenessrdquo Applied Mathe-matics and Computation vol 291 pp 84ndash88 2016

[11] H Yang P Wu S Nazari-Moghaddam et al ldquoBounds forsigned double roman k-domination in treesrdquo RAIRO-Oper-ations Research vol 53 no 2 pp 627ndash643 2019

[12] R A Beeler T W Haynes and S T Hedetniemi ldquoDoubleroman dominationrdquo Discrete Applied Mathematics vol 211pp 23ndash29 2016

[13] R HammackW Imrich and S KlavzarHandbook of ProductGraphs CRC Press Boca Raton FL USA Second edition2011

Discrete Dynamics in Nature and Society 5

Proposition 6 If G is a connected graph of order n thencdR(G) + 1 2cR(G) if and only if there exists a vertex v ofdegree n + 1 minus ((cdR(G) + 1)2) in G

Proof

(rArr) Let f (V0 V1 V2) be a cR(G) function withminimum |V1| en we have V1 being independentTogether with Proposition 2 we have cdR(G)le 2|V1| +

3|V2|le 2|V1| +4|V2| 2cR(G) cdR(G) + 1 en wehave |V2|le 1 If |V2| 0 we have |V0| 0 and thuscR(G) n |V1| Since V1 is independent it is im-possible If |V2| 1 we have cdR(G) 2|V1| + 3|V2|

lt 2|V1| + 4|V2| 2cR(G) cdR(G) + 1 Let V2 v V0 N(v) and V1 V(G) minus V0 minus V2 en we havecR(G) n minus 1 minus d(v) + 2 and so d(v) n + 1 minus cR(G)

n + 1 minus ((cdR(G) + 1)2)(lArr) By Proposition 2 (ii) we have cdR(G) + 1le 2cR(G)

for a connected graph G Assume G contains a vertex v

of degree n + 1 minus (cdR(G) + 1)2 in G Let V2 v V0 N(v) and V1 V(G) minus V0 minus V2 enf (V0 V1 V2) is a cR(G) function and socR(G)le |V1| + 2|V2| (cdR(G) + 1)2 HencecdR(G) + 1ge 2cR(G)

Let F be the family of connected graphs G such that forany cdR(G) function f (V0empty V2 V3) we have |V3| 0

Proposition 7 Let G isin F then

(i) G contains no strong support vertex(ii) if f(u) f(v) 2 for an edge uv isin E(G) then G minus

e isinF and Ge isinF

Proof

(i) Suppose v be a strong support vertex then there existtwo leaves x y isin N(v) Since f(v)ne 3 we havef(x) f(y) 2 Now consider the function fprimewith fprime(z) 0 for any z isin L(v) fprime(v) 3 andfprime(z) f(z) for any z isin V(G)∖L[v] en fprime is aDRDF of G with fewer weight than f acontradiction

(ii) If f(u) f(v) 2 for an edge uv isin E(G) we have f

as also a DRDF of G minus e SocdR(G minus e)lew(f) cdr(G) Since for any graph Gwe have cdR(G minus e)ge cdR(G) socdR(G minus e) cdR(G) Suppose to the contrary thatthere exists a cdR(G minus e) function such that f(v) 3for a vertex v isin V(G minus e) en f is also a cdR(G)

function contradicting with |V3| 0 For the graphGe the proof is similar

Lemma 1 Let G be a graph on nge 4 vertices then cdR(G)

4 if and only if G contains a complete bipartite graph K2nminus 2 asa subgraph and Δ(G)le n minus 2

Proof

(rArr) If cdR(G) 4 then no vertex is assigned with 3and thus we have two vertices v andw assigned with 2and the others 0 Also each vertex 0 must be adjacent toboth v and w erefore G contains a complete bi-partite graph K2nminus 2 as a subgraph Since cdR(G)gt 3 wehave Δ(G)le n minus 2(lArr) If G contains a complete bipartite graph K2nminus 2with partitions X Y (|X| 2 |Y| n minus 2) as a subgraphand Δ(G)le n minus 2 en let f(x) 2 for any x isin X andf(x) 0 for x isin Y en f is a DRDF of G and socdR(G)le 4 Since G contains no vertex with degree|V(G)| minus 1 we have cdR(G)ge 4

Note that Δ(G)ge n minus 2 if G contains a complete bipartitegraph K2nminus 2 as a subgraph us Δ(G)le n minus 2 can bereplaced with Δ(G) n minus 2 in the lemma

Theorem 2 Let G be a graph on nge 3 vertices then8le cdR(G) + cdR(G)le 2n + 3 Furthermore equality holds inthe upper bound if G or G is Kn

Proof If G is a graph on nge 3 vertices we have cdR(G)ge 3and if cdR(G) 3 then G has a vertex with degree n minus 1 Butits complement is neither a star nor a graph G with cdR(G)

4 (see the graph stated in Lemma 1) So we have cdR(G)ge 5and thus cdR(G) + cdR(G)ge 8 If G is a star then cdR(G) 5and so the lower bound is attainable

Let v be a vertex with maximum degree Δ(G) consider afunction f with f(v) 3 f(x) 0 for any x isin N(v) andf(x) 2 for x isin V(G) minus N[x] en f is a DRDF of G andso cdR(G)lew(f) 2n minus 2Δ(G) + 1 Since Δ(G) + δ(G)

n minus 1 we have cdR(G)le 2n minus 2Δ(G) + 1 2δ(G) + 3erefore cdR(G) + cdR(G)le 2n + 4 minus 2Δ(G) + 2δ(G)le2n + 4 It can be seen that if cdR(G) + cdR(G) 2n + 4 thenΔ(G) δ(G) Hence G is k-regular for some k By sym-metry we may assume that kle (n minus 1)2 en ifcdR(G) + cdR(G) 2n + 4 we have cdR(G) 2n minus 2k + 1and cdR(G) 2k + 3 Let v isin V(G) If |N(u)capN(v)|le k minus 2for some u isin V(G) minus N[v] then f (N(v)cupN(u)

empty V(G) minus N[u] minus N[v] u v ) is a DRDF of G with fewerweight than 2n minus 2k + 1 a contradiction erefore eachvertex not in N[v] has at least k minus 1 neighbors in N(v)Analogously each vertex in N[v] has at most 2 neighborsoutside N[v] en we have (k minus 1)(n minus k minus 1)le 2k Sincekle (n minus 1)2 we have kle 2 + 2(k minus 1) If k 3 then n 7is is impossible If k 2 then n isin 5 6 7 If G C5 wehave cdR(G) cdR(G) 6 and so cdR(G) + cdR(G)ne2n + 4 a contradiction If G C6 we have cdR(G)

cdR(G) 6 and so cdR(G) + cdR(G)ne 2n + 4 a contradic-tion If G C7 we have cdR(G) 8 and cdR(G) 6 and socdR(G) + cdR(G)ne 2n + 4 a contradiction If k 1 thenG (n2)K2 and we have cdR(G) 3n2 and cdR(G) 4and so cdR(G) + cdR(G)ne 2n + 4 a contradictionereforewe conclude that 8le cdR(G) + cdR(G)le 2n + 3 If G Knthen cdR(G) 3 and cdR(G) 2n and thus the upper boundis attainable

Discrete Dynamics in Nature and Society 3

3 Some Double Roman Graphs

e Cartesian product of graphs G and H is the graph GHwith vertex set G times H and (x1 x2)(y1 y2) isin E(GH)

whenever x1y1 isin E(G) and x2 y2 or x2y2 isin E(H) andx1 y1 e Cartesian product is commutative and asso-ciative having the trivial graph as a unit (cf [13])

Let f be a double Roman dominating function ofCmCn and we write V

fi v isin V(CmCn) | f(v) i1113864 1113865 for

i isin 0 1 2 3 When no confusion arise we simply write Vfi

as Vi We use f(i j) to denote the value f(v) forv (i j) isin V(CmCn) Let xi be the weight of Ci iexi 1113936xisinCi f(x)1113864 1113865

Theorem 3 Let m nge 1 +en the Cartesian product graphsC5mC5n are double Roman

Proof e lower bound follows from Proposition 4 LetV(C5mC5n) vij 0le ile 5m minus 1 0le jle 5n minus 11113966 1113967V1 V2 empty V3 v(5i)(5j+2)1113966 v(5i+1)(5j) v(5i+2)(5j+3)v(5i+3)(5j+1) v(5i+4)(5j+4) 0le ilem minus 1 0le jle n minus 1 andV0 N(V3) en N[V3] V(C5mC5n) and sof (V0 V1 V2 V3) is a DRDF of C5mC5n with weight15mn and we have cdR(C5mC5n)le 15mn Sincec(C5mC5n) 5mn we have C5mC5n is doubleRoman

4 Trees T with γdR(T)= γR(T)+ k

Theorem 4 If T is a tree then cR(T) + 1 cdR(T) if andonly if T is a star K1s for sge 1

Proof

(lArr) If T is a star K1s for sge 1 it is clear that cdR(T) 3and cR(T) 2 and the theorem holds(rArr) By Proposition 3 we have |V3|le 1 If |V3| 1 byProposition 5 we have T is a star K1s for some sge 1 If|V2| 0 then each vertex in T is assigned 2 or 0 SinceT is a tree T has a least two leaves v andw andf(v) f(w) 2 If v andw are adjacent to the samevertex x then we can obtain a DRDF of T with fewerweight by changing f(x) to 3 and f(v) andf(w) to 0and obtaining a contradiction If v andw are adjacent todifferent vertices we consider a function fprime withfprime(w) 1 fprime(v) 1 and fprime(x) f(x) for anyx isin V(T) minus v w en fprime is an RDF of T with weightcdR(T) minus 2 So cR(T)le cdR(T) minus 2 a contradiction

For a positive integer t a wounded spider is a star K1t

with at most t minus 1 of its edges subdivided In a woundedspider a vertex of degree t will be called the head vertex andthe vertices at distance two from the head vertex will be thefoot vertices

Theorem 5 If T is a tree then cR(T) + 2 cdR(T) if andonly if T is a wounded spider with only one foot or T isobtained by adding an edge between two stars K1s and K1t

for s tge 2

Proof

(lArr) If T is a wounded spider with only one foot it isclear that cdR(T) 5 and cR(T) 3 and the theoremholds If T is a tree obtained by adding an edge betweentwo stars K1s and K1t for s tge 2 then cdR(T) 6 andcR(T) 4 and the theorem holds(rArr) By Proposition 3 we have |V3|le 2 If |V3| 2 letV3 v w and H G[V(G) minus N[w] minus N[v]] Similarto the proof of eorem 5 we have H as emptyOtherwise there exists a vertex u in H assigned with 2Now we change the function values of v w u from3 3 2 to 2 2 1 respectively and obtain an RDF of T

with weight cdR(T) minus 3 a contradiction In this case T

is a tree obtained by adding an edge between two starsK1s and K1t for s tge 2 If |V3| 1 let V3 v andH G[V(G) minus N[v]] en H has at most one con-nected component Otherwise we can make a vertex ineach connected component to change the functionvalues including the vertex v to obtain an RDF of T withweight cdR(T) minus 3 Since H contains no vertex assignedwith 3 then H is not a star with at least two leaves Weclaim that the leaves of H are at most two Otherwisewe change the function values of v and choose twoleaves to obtain an RDF of T with weight cdR(T) minus 3erefore H is a path on at least four vertices In thiscase we can obtain an RDF of T with weight at mostcdR(T) minus 3 a contradiction erefore T is a woundedspider with only one foot

Data Availability

No data were used to support this study

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

is work was supported by Sichuan Science and Tech-nology Program under grant 2018ZR0265 Sichuan Militaryand Civilian Integration Strategy Research Center undergrant JMRH-1818 and Sichuan Provincial Department ofEducation (Key Project) under grant 18ZA0118

References

[1] M R Garey and D S Johnson Computers and IntractabilityA Guide to the +eory of NP-Completeness W H FreemanSan Francisco CA USA 1979

[2] O Ore +eory of Graphs American Mathematical SocietyProvidence RI USA 1967

[3] TW Haynes S T Hedetniemi and P J Slater Fundamentalsof Domination in Graphs Marcel Dekker NewYork NYUSA 1998

[4] X Zhang Z Li H Jiang and Z Shao ldquoDouble romandomination in treesrdquo Information Processing Letters vol 134pp 31ndash34 2018

4 Discrete Dynamics in Nature and Society

[5] E Zhu and Z Shao ldquoExtremal problems on weak romandomination numberrdquo Information Processing Letters vol 138pp 12ndash18 2018

[6] Z Shao P Wu H Jiang Z Li J Zerovnik and X ZhangldquoDischarging approach for double roman domination ingraphsrdquo IEEE Access vol 6 pp 63345ndash63351 2018

[7] Z Shao J Amjadi S M Sheikholeslami and M ValinavazldquoOn the total double roman dominationrdquo IEEE Access vol 7pp 52035ndash52041 2019

[8] J-B Liu C Wang S Wang and B Wei ldquoZagreb indices andmultiplicative zagreb indices of Eulerian graphsrdquo Bulletin ofthe Malaysian Mathematical Sciences Society vol 42 no 1pp 67ndash78 2019

[9] J-B Liu X-F Pan F-T Hu and F-F Hu ldquoAsymptoticLaplacian-energy-like invariant of latticesrdquo Applied Mathe-matics and Computation vol 253 pp 205ndash214 2015

[10] J-B Liu and X-F Pan ldquoMinimizing Kirchhoff index amonggraphs with a given vertex bipartitenessrdquo Applied Mathe-matics and Computation vol 291 pp 84ndash88 2016

[11] H Yang P Wu S Nazari-Moghaddam et al ldquoBounds forsigned double roman k-domination in treesrdquo RAIRO-Oper-ations Research vol 53 no 2 pp 627ndash643 2019

[12] R A Beeler T W Haynes and S T Hedetniemi ldquoDoubleroman dominationrdquo Discrete Applied Mathematics vol 211pp 23ndash29 2016

[13] R HammackW Imrich and S KlavzarHandbook of ProductGraphs CRC Press Boca Raton FL USA Second edition2011

Discrete Dynamics in Nature and Society 5

3 Some Double Roman Graphs

e Cartesian product of graphs G and H is the graph GHwith vertex set G times H and (x1 x2)(y1 y2) isin E(GH)

whenever x1y1 isin E(G) and x2 y2 or x2y2 isin E(H) andx1 y1 e Cartesian product is commutative and asso-ciative having the trivial graph as a unit (cf [13])

Let f be a double Roman dominating function ofCmCn and we write V

fi v isin V(CmCn) | f(v) i1113864 1113865 for

i isin 0 1 2 3 When no confusion arise we simply write Vfi

as Vi We use f(i j) to denote the value f(v) forv (i j) isin V(CmCn) Let xi be the weight of Ci iexi 1113936xisinCi f(x)1113864 1113865

Theorem 3 Let m nge 1 +en the Cartesian product graphsC5mC5n are double Roman

Proof e lower bound follows from Proposition 4 LetV(C5mC5n) vij 0le ile 5m minus 1 0le jle 5n minus 11113966 1113967V1 V2 empty V3 v(5i)(5j+2)1113966 v(5i+1)(5j) v(5i+2)(5j+3)v(5i+3)(5j+1) v(5i+4)(5j+4) 0le ilem minus 1 0le jle n minus 1 andV0 N(V3) en N[V3] V(C5mC5n) and sof (V0 V1 V2 V3) is a DRDF of C5mC5n with weight15mn and we have cdR(C5mC5n)le 15mn Sincec(C5mC5n) 5mn we have C5mC5n is doubleRoman

4 Trees T with γdR(T)= γR(T)+ k

Theorem 4 If T is a tree then cR(T) + 1 cdR(T) if andonly if T is a star K1s for sge 1

Proof

(lArr) If T is a star K1s for sge 1 it is clear that cdR(T) 3and cR(T) 2 and the theorem holds(rArr) By Proposition 3 we have |V3|le 1 If |V3| 1 byProposition 5 we have T is a star K1s for some sge 1 If|V2| 0 then each vertex in T is assigned 2 or 0 SinceT is a tree T has a least two leaves v andw andf(v) f(w) 2 If v andw are adjacent to the samevertex x then we can obtain a DRDF of T with fewerweight by changing f(x) to 3 and f(v) andf(w) to 0and obtaining a contradiction If v andw are adjacent todifferent vertices we consider a function fprime withfprime(w) 1 fprime(v) 1 and fprime(x) f(x) for anyx isin V(T) minus v w en fprime is an RDF of T with weightcdR(T) minus 2 So cR(T)le cdR(T) minus 2 a contradiction

For a positive integer t a wounded spider is a star K1t

with at most t minus 1 of its edges subdivided In a woundedspider a vertex of degree t will be called the head vertex andthe vertices at distance two from the head vertex will be thefoot vertices

Theorem 5 If T is a tree then cR(T) + 2 cdR(T) if andonly if T is a wounded spider with only one foot or T isobtained by adding an edge between two stars K1s and K1t

for s tge 2

Proof

(lArr) If T is a wounded spider with only one foot it isclear that cdR(T) 5 and cR(T) 3 and the theoremholds If T is a tree obtained by adding an edge betweentwo stars K1s and K1t for s tge 2 then cdR(T) 6 andcR(T) 4 and the theorem holds(rArr) By Proposition 3 we have |V3|le 2 If |V3| 2 letV3 v w and H G[V(G) minus N[w] minus N[v]] Similarto the proof of eorem 5 we have H as emptyOtherwise there exists a vertex u in H assigned with 2Now we change the function values of v w u from3 3 2 to 2 2 1 respectively and obtain an RDF of T

with weight cdR(T) minus 3 a contradiction In this case T

is a tree obtained by adding an edge between two starsK1s and K1t for s tge 2 If |V3| 1 let V3 v andH G[V(G) minus N[v]] en H has at most one con-nected component Otherwise we can make a vertex ineach connected component to change the functionvalues including the vertex v to obtain an RDF of T withweight cdR(T) minus 3 Since H contains no vertex assignedwith 3 then H is not a star with at least two leaves Weclaim that the leaves of H are at most two Otherwisewe change the function values of v and choose twoleaves to obtain an RDF of T with weight cdR(T) minus 3erefore H is a path on at least four vertices In thiscase we can obtain an RDF of T with weight at mostcdR(T) minus 3 a contradiction erefore T is a woundedspider with only one foot

Data Availability

No data were used to support this study

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

is work was supported by Sichuan Science and Tech-nology Program under grant 2018ZR0265 Sichuan Militaryand Civilian Integration Strategy Research Center undergrant JMRH-1818 and Sichuan Provincial Department ofEducation (Key Project) under grant 18ZA0118

References

[1] M R Garey and D S Johnson Computers and IntractabilityA Guide to the +eory of NP-Completeness W H FreemanSan Francisco CA USA 1979

[2] O Ore +eory of Graphs American Mathematical SocietyProvidence RI USA 1967

[3] TW Haynes S T Hedetniemi and P J Slater Fundamentalsof Domination in Graphs Marcel Dekker NewYork NYUSA 1998

[4] X Zhang Z Li H Jiang and Z Shao ldquoDouble romandomination in treesrdquo Information Processing Letters vol 134pp 31ndash34 2018

4 Discrete Dynamics in Nature and Society

[5] E Zhu and Z Shao ldquoExtremal problems on weak romandomination numberrdquo Information Processing Letters vol 138pp 12ndash18 2018

[6] Z Shao P Wu H Jiang Z Li J Zerovnik and X ZhangldquoDischarging approach for double roman domination ingraphsrdquo IEEE Access vol 6 pp 63345ndash63351 2018

[7] Z Shao J Amjadi S M Sheikholeslami and M ValinavazldquoOn the total double roman dominationrdquo IEEE Access vol 7pp 52035ndash52041 2019

[8] J-B Liu C Wang S Wang and B Wei ldquoZagreb indices andmultiplicative zagreb indices of Eulerian graphsrdquo Bulletin ofthe Malaysian Mathematical Sciences Society vol 42 no 1pp 67ndash78 2019

[9] J-B Liu X-F Pan F-T Hu and F-F Hu ldquoAsymptoticLaplacian-energy-like invariant of latticesrdquo Applied Mathe-matics and Computation vol 253 pp 205ndash214 2015

[10] J-B Liu and X-F Pan ldquoMinimizing Kirchhoff index amonggraphs with a given vertex bipartitenessrdquo Applied Mathe-matics and Computation vol 291 pp 84ndash88 2016

[11] H Yang P Wu S Nazari-Moghaddam et al ldquoBounds forsigned double roman k-domination in treesrdquo RAIRO-Oper-ations Research vol 53 no 2 pp 627ndash643 2019

[12] R A Beeler T W Haynes and S T Hedetniemi ldquoDoubleroman dominationrdquo Discrete Applied Mathematics vol 211pp 23ndash29 2016

[13] R HammackW Imrich and S KlavzarHandbook of ProductGraphs CRC Press Boca Raton FL USA Second edition2011

Discrete Dynamics in Nature and Society 5

[5] E Zhu and Z Shao ldquoExtremal problems on weak romandomination numberrdquo Information Processing Letters vol 138pp 12ndash18 2018

[6] Z Shao P Wu H Jiang Z Li J Zerovnik and X ZhangldquoDischarging approach for double roman domination ingraphsrdquo IEEE Access vol 6 pp 63345ndash63351 2018

[7] Z Shao J Amjadi S M Sheikholeslami and M ValinavazldquoOn the total double roman dominationrdquo IEEE Access vol 7pp 52035ndash52041 2019

[8] J-B Liu C Wang S Wang and B Wei ldquoZagreb indices andmultiplicative zagreb indices of Eulerian graphsrdquo Bulletin ofthe Malaysian Mathematical Sciences Society vol 42 no 1pp 67ndash78 2019

[9] J-B Liu X-F Pan F-T Hu and F-F Hu ldquoAsymptoticLaplacian-energy-like invariant of latticesrdquo Applied Mathe-matics and Computation vol 253 pp 205ndash214 2015

[10] J-B Liu and X-F Pan ldquoMinimizing Kirchhoff index amonggraphs with a given vertex bipartitenessrdquo Applied Mathe-matics and Computation vol 291 pp 84ndash88 2016

[11] H Yang P Wu S Nazari-Moghaddam et al ldquoBounds forsigned double roman k-domination in treesrdquo RAIRO-Oper-ations Research vol 53 no 2 pp 627ndash643 2019

[12] R A Beeler T W Haynes and S T Hedetniemi ldquoDoubleroman dominationrdquo Discrete Applied Mathematics vol 211pp 23ndash29 2016

[13] R HammackW Imrich and S KlavzarHandbook of ProductGraphs CRC Press Boca Raton FL USA Second edition2011

Discrete Dynamics in Nature and Society 5