Graphs and Combinatorics (2017) 33:913–927DOI 10.1007/s00373-017-1789-0

ORIGINAL PAPER

On 2-Step and Hop Dominating Sets in Graphs

Michael A. Henning1· Nader Jafari Rad2

Received: 8 November 2016 / Revised: 10 April 2017 / Published online: 25 May 2017

© Springer Japan 2017

Abstract Two vertices in a graph are said to 2-step dominate each other if they are

at distance 2 apart. A set S of vertices in a graph G is a 2-step dominating set of G

if every vertex is 2-step dominated by some vertex of S. A subset S of vertices of G

is a hop dominating set if every vertex outside S is 2-step dominated by some vertex

of S. The hop domination number, γh(G), of G is the minimum cardinality of a hop

dominating set of G. It is known that for a connected graph G, γh(G) = |V (G)| if

and only if G is a complete graph. We characterize the connected graphs G for which

γh(G) = |V (G)| − 1, which answers a question posed by Ayyaswamy and Natarajan

[An. Stt. Univ. Ovidius Constanta 23(2):187–199, 2015]. We present probabilistic

upper bounds for the hop domination number. We also prove that almost all graphs

G = G(n, p(n)) have a hop dominating set of cardinality at most the total domination

number if p(n) ≪ 1/n, and almost all graphs G = G(n, p(n)) have a hop dominating

set of cardinality at most 1 + np(1 + o(1)), if p is constant. We show that the decision

problems for the 2-step dominating set and hop dominating set problems are NP-

complete for planar bipartite graphs and planar chordal graphs.

Keywords 2-Step dominating set · Hop dominating set · NP-complete

Research supported in part by the South African National Research Foundation and the University of

Johannesburg.

B Michael A. Henning

mahenning@uj.ac.za

Nader Jafari Rad

n.jafarirad@gmail.com

1 Department of Pure and Applied Mathematics, University of Johannesburg, Auckland Park 2006,

South Africa

2 Department of Mathematics, Shahrood University of Technology, Shahrood, Iran

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914 Graphs and Combinatorics (2017) 33:913–927

AMS subject classification 05C69

1 Introduction

In this paper, we continue the study of 2-step domination and hop domination in

graphs. Two vertices are neighbors if they are adjacent. A subset S of vertices of a

graph G is a dominating set of G if every vertex in V (G)\S has a neighbor in S. The

domination number, γ (G), is the minimum cardinality of a dominating set of G. A

total dominating set of a graph G with no isolated vertex is a set S of vertices of G

such that every vertex in V (G) has a neighbor in S. The total domination number,

γt (G), is the minimum cardinality of a total dominating set of G. The literature on

the subject of domination parameters in graphs up to the year 1997 has been surveyed

and detailed in the two books [11,12], and a recent book on total dominating sets is

also available [14].

Let G be a graph with vertex set V (G) of order n(G) = |V (G)| and edge set E(G)

of size m(G) = |E(G)|. If the graph G is clear from context, we abbreviate V (G)

to V , E(G) to E , n(G) to n and m(G) to m. The distance between two vertices u

and v in G, denoted dG(u, v) or simply d(u, v) if the graph G is clear from context,

is the minimum length of a (u, v)-path in G. The diameter, diam(G), of G is the

maximum distance among all pairs of vertices in G. The subgraph induced by a set

S of vertices in a graph G is denoted by G[S]. A chordal graph is a graph that does

not contain an induced cycle of length greater than 3. We use the standard notation

[k] = {1, . . . , k}. For notation and graph theory terminology not defined here we

generally follow [11].

For an integer k ≥ 1, two vertices in a graph G are said to k-step dominate each

other if they are at distance exactly k apart in G. A set S of vertices in G is a k-step

dominating set of G if every vertex in V (G) is k-step dominated by some vertex of S.

The k-step domination number, γkstep(G), of G, is the minimum cardinality of a k-step

dominating set of G. The concept of 2-step domination in graphs was introduced by

Chartrand et al. [6] and further studied, for example in [5,7,16,21].

Recently, Ayyaswamy and Natarajan [2] introduced a parameter similar to the 2-

step domination number, namely the hop domination number of a graph. A subset S of

vertices of G is a hop dominating set if every vertex outside S is 2-step dominated by

some vertex of S. The hop domination number, γh(G), of G is the minimum cardinality

of a hop dominating set of G. The concept of hop domination was further studied, for

example, in [3,18].

We denote the degree of the vertex v in G by dG(v). The maximum (minimum)

degree among the vertices of G is denoted by �(G) (δ(G), respectively). The open

neighborhood of v is NG(v) = {u ∈ V (G) | uv ∈ E(G)} and the closed neighborhood

of v is NG [v] = {v} ∪ NG(v). The set of vertices at distance 2 from v in G is denoted

by N2(v; G). If the graph G is clear from context, we abbreviate NG(v) to N (v),

NG [v] to N [v], and N2(v; G) to N2(v). A set S of vertices in G is a 2-step dominating

set of G if ∪v∈S N2(v) = V (G). For a set S ⊆ V , its open neighborhood is the set

NG(S) =⋃

v∈S NG(v), and its closed neighborhood is the set NG [S] = NG(S) ∪ S.

We define the hop-degree of a vertex v in a graph G, denoted dh(v), to be the number of

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Graphs and Combinatorics (2017) 33:913–927 915

vertices at distance 2 from v in G; that is, dh(v) = |N2(v)|. The minimum hop-degree

among the vertices of G we denote by δh(G).

2 Preliminary Observations and Results

If G is a graph and k ≥ 1 an integer, then we denote by Dist(G; k) the graph consisting

of the vertex set V (G) and edge set {uv | dG(u, v) = k}. For example, Dist(G; 2) =G2 − E(G), where G2 is the square of G with the same set of vertices of G, but

in which two vertices are adjacent in G2 when their distance in G is at most 2. The

following observations follows readily from the definitions given in the introductory

section.

Observation 1 If G is a graph, then the following holds.

(a) A set S ⊆ V (G) is a hop dominating set of G if and only if S is a dominating set

in Dist(G; 2).

(b) γh(G) = γ (Dist(G; 2)).

(c) N2(v; G) = NDist(G;2)(v).

(d) δh(G) = δ(Dist(G; 2)).

Observation 2 For an integer k ≥ 1 and a graph G, the following holds.

(a) A set S ⊆ V (G) is a k-step dominating set of G if and only if S is a total dominating

set in Dist(G; k).

(b) γkstep(G) = γt (Dist(G; k)).

A hop dominating set S in a graph G is minimal if no proper subset of S is a

hop dominating set. For a graph G, a subset S of vertices of G, and a vertex v ∈ S,

we define the S-external hop neighborhood of v to be the set epn2(v, S) = {w ∈V (G)\S | N2(w; G) ∩ S = {v}}. We call each vertex in epn2(v, S) an S-external hop

neighbor of v. We say that the vertex v ∈ S is S-hop isolated if N2(v; G) ∩ S = ∅.

We note that a set S is a minimal dominating set in Dist(G; 2) if and only if S is a

minimal hop dominating set in G. From known properties of minimal dominating sets

in graphs, we therefore have the following two results. However as a gentle introduction

to the concept of hop domination in graphs, we give a direct proof of these properties

of minimal hop dominating sets.

Proposition 3 Let S be a hop dominating set in a graph G. Then, S is a minimal hop

dominating set in G if and only if each vertex in S is an S-hop isolated vertex or has

an S-external hop neighbor.

Proof Let S be a minimal hop dominating set in G and let v ∈ S. If v is not S-hop

isolated and epn2(v, S) = ∅, then every vertex x ∈ V (G)\(S\{v}) is at distance 2 in G

from some vertex in S\{v}. Hence, S\{v} is a hop dominating set of G, contradicting

the minimality of S. Therefore, v is S-hop isolated or |epn2(v, S)| ≥ 1 for every vertex

v ∈ S. Conversely, if v is S-hop isolated or |epn2(v, S)| ≥ 1 for each v ∈ S, then

S\{v} is not a hop dominating set of G, implying that S is a minimal hop dominating

set in G. ⊓⊔

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916 Graphs and Combinatorics (2017) 33:913–927

Proposition 4 Every graph G with δh(G) ≥ 1 contains a minimum hop dominating

set S such that each vertex in S has an S-external hop neighbor.

Proof Among all minimum hop dominating sets of G, let S be chosen so that the

number of S-hop isolated vertices in minimum. We show that for every vertex of S,

|epn2(v, S)| ≥ 1. Suppose, to the contrary, that epn2(v, S) = ∅ for some vertex v ∈ S.

By Proposition 3, the vertex v is an S-hop isolated vertex. Since δh(G) ≥ 1, there is a

vertex v′ at distance 2 from v in G. Since v is S-hop isolated, we note that v ∈ V (G)\S.

Since epn2(v, S) = ∅, every vertex at distance 2 from v in G is at distance 2 from

at least one vertex of S different from v. In particular, the vertex v′ is hop dominated

by at least one vertex of S different from v. These observations imply that the set

S′ = (S\{v}) ∪ {v′} is a hop dominating set of G. Further since |S′| = |S| = γh(G)

and v is an S-hop isolated vertex while v′ is not an S′-hop isolated vertex, the set S′ is

a minimum hop dominating set of G that contains fewer S-hop isolated vertices than

does S, a contradiction. Therefore, |epn2(v, S)| ≥ 1 for every vertex of S. ⊓⊔

Suppose that S is a hop dominating set in a graph G. For every vertex v in V (G)\S,

there is a vertex v′ ∈ S at distance 2 from v in G. Thus in the complement, G, of G

the vertices v and v′ are adjacent, implying that S is a dominating set of G. Therefore,

we have the following relationship between the hop domination number of a graph

and the domination number of its complement.

Observation 5 If G is a graph, then γ (G) ≤ γh(G) and γ (G) ≤ γh(G).

We remark that strict inequality can occur in Observation 5. The 2-corona of a graph

H is the graph of order 3|V (H)| obtained from H by attaching a path of length 2 to

each vertex of H so that the resulting paths are vertex-disjoint. If G is the 2-corona

of a connected graph H where |V (H)| ≥ 3, then the set V (H) is a minimum hop

dominating set of G, and so γh(G) = |V (H)|. However, any two leaves of G form

a dominating set in its complement G, implying that γ (G) = 2. Thus the difference,

γh(G) − γ (G) can be arbitrarily large for connected graphs G.

3 Graphs with Large Hop Domination Number

If G is a complete graph, Kn , on n vertices, then V (G) is the unique hop dominating

set of G. If G is a connected graph that is not a complete graph and v is a vertex with

at least one non-neighbor in G, then V (G)\{v} is a hop dominating set of G. Thus,

as observed by Ayyaswamy and Natarajan [2], for a connected graph G, γh(G) = n

if and only if G is a complete graph of order n. In this section, we provide an answer

to the following problem posed by Natarajan et al. [18]: characterize the connected

graphs G of order n with γh(G) = n − 1. For this purpose, for n ≥ 3, let K −n be

the graph obtained from a complete graph Kn by deleting exactly one edge. Thus,

K −n = Kn − e, where e is an arbitrary edge of the clique.

Theorem 6 For a connected graph G of order n ≥ 3, γh(G) = n − 1 if and only if

G ∼= K −n .

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Proof By Observation 1(c), γh(G) = γ (Dist(G; 2)). Thus we wish to determine

which connected graphs G of order n ≥ 3 satisfy γ (Dist(G; 2)) = n − 1. We observe

that a graph H satisfies γ (H) = n(H) − 1 if and only if H ∼= K2 ∪ (n(H) − 2)K1.

For a connected graph G of order n, Dist(G; 2) ∼= K2 ∪ (n − 2)K1 if and only

if G ∼= K −n . Equivalently, γh(G) = γ (Dist(G; 2)) = n − 1 if and only if G ∼=

K −n . ⊓⊔

A classic result due to Ore [19] shows that if G is a graph of order n with δ(G) ≥ 1,

then γ (G) ≤ n/2. If δh(G) ≥ 1, then by Observation 1(d), δ(Dist(G; 2)) = δh(G) ≥1, implying by Observation 1(b) and Ore’s result that γh(G) = γ (Dist(G; 2)) ≤ n/2.

We state this result formally as follows, where we also give a short direct proof of this

result independent of Ore’s result.

Theorem 7 If G is a graph of order n with δh(G) ≥ 1, then γh(G) ≤ n2

.

Proof By Proposition 4, there exists a minimum hop dominating set S in G such that

|epn2(v, S)| ≥ 1 for each v ∈ S. Thus since⋃

v∈S epn2(v, S) ⊆ V (G)\S, we note

that

|S| ≤∑

v∈S

|epn2(v, S)| =

∣

∣

∣

∣

∣

⋃

v∈S

epn2(v, S)

∣

∣

∣

∣

∣

≤ |V (G)\S| = n − |S|,

and so, γh(G) ≤ |S| = n/2. ⊓⊔

The corona H ◦ K1 of a graph H is the graph obtained from H by adding for

each vertex v of H a new vertex v′ and the pendant edge vv′. The following result

characterizes isolate-free graphs with domination number one-half their order.

Theorem 8 [8,20] If G is a graph of order n with no isolated vertex, then γ (G) = n/2

if and only if the components of G are the cycle C4 or the corona H ◦ K1 for any

connected graph H.

For n ≥ 4, let Fn be the family of all graphs F of order n such that F ≇ C4 and

every component of F is a 4-cycle or the corona H ◦ K1 for some connected graph

H . Thus, Theorem 8 can be restated as: If G is a graph of order n with no isolated

vertex, then γ (G) = n/2 if and only if G ∈ {K2, C4} ∪ Fn .

Theorem 9 If G is a graph of order n with δh(G) ≥ 1, then the following holds.

γh(G) =n

2if and only if Dist(G; 2) ∈ Fn .

Proof By Observation 1(d), δ(Dist(G; 2)) = δh(G) ≥ 1. Suppose that γh(G) =n/2. By Observation 1(b), γ (Dist(G; 2)) = γh(G) = n/2. Thus, by Theorem 8,

Dist(G; 2) ∈ {K2, C4} ∪ Fn . However if n(G) = 2, then G ∼= K2 or G ∼= 2K1

and in both cases Dist(G; 2) ∼= 2K1, implying that Dist(G; 2) ∼= K2 is not pos-

sible. If Dist(G; 2) ∼= C4, then G ∼= 2K2, and so δh(G) = 0, a contradiction.

Therefore, Dist(G; 2) ∈ Fn . Conversely, if Dist(G; 2) ∈ Fn , then by Theorem 8,

γ (Dist(G; 2)) = n/2. Thus, by Observation 1(b), γh(G) = γ (Dist(G; 2)) =n/2. ⊓⊔

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To illustrate Theorem 9, if G ∼= C8, then Dist(G; 2) ∼= 2C4 ∈ F8, implying by

Theorem 9 that γh(G) = n(G)/2 = 4. If G is the graph obtained by removing a

perfect matching from K2n , then Dist(G; 2) ∼= nK2 ∈ F2n , implying by Theorem 9

that γh(G; 2) = n(G)/2 = n.

Let Fn be the family of all graphs whose complement belongs to Fn ; that is,

Fn = {F | F ∈ Fn}. We note that if F ∈ Fn , then δh(F) ≥ 1. Further if F ∈ Fn ,

then F ∈ Fn and so, by Theorem 8, γ (F) = n/2. Thus, by Observation 5 and

Theorem 7, n/2 = γ (F) ≤ γh(F) ≤ n/2. Consequently, we must have equality

throughout this inequality chain. In particular, γh(F) = n/2. We state this formally

as follows.

Proposition 10 If G ∈ Fn , then G has order n ≥ 4, δh(G) ≥ 1 and γh(G) = n2

.

We note that the converse of Proposition 10 is not true. That is, if G is a graph of

order n ≥ 4 satisfying δh(G) ≥ 1 and γh(G) = n/2, then it is not necessary true

that G ∈ Fn . For example, if G ∼= C8, then δh(G) ≥ 1 and γh(G) = n(G)/2 = 4.

However, G /∈ F8.

4 Probabilistic Bounds

Let R be the set of real numbers and let Cn = {p = (p1, . . . , pn) | pi ∈ R, 0 ≤ pi ≤1, i ∈ [n]}. Let f : Cn → R be the function defined by

f (p) =n

∑

i=1

pi +

⎛

⎝

n∑

i=1

⎡

⎣(1 − pi ) ·∏

j∈N2(i)

(1 − p j )

⎤

⎦

⎞

⎠ .

As a consequence of Observation 1 and a result due to Harant, Pruchnewski, and

Voigt [10], we have the following probabilistic bound on the hop domination number

of a graph. For completeness, we give a direct proof of Theorem 11 in Appendix A.

Theorem 11 If G is a graph of order n, then γh(G) = minp ∈Cn

f (p).

As a consequence of a result due to Alon and Spencer [1], and using Observa-

tion 1(a), 1(c) and 1(f), we have the following upper bound on the hop domination

number of a graph in terms of its hop degree. For completeness, we deduce the result

of Theorem 12 directly from Theorem 11 in Appendix A.

Theorem 12 If G is a graph of order n with δh = δh(G) ≥ 1, then

γh(G) ≤(

ln(δh + 1) + 1

δh + 1

)

n.

If G is a triangle-free graph with minimum degree δ = δ(G) ≥ 2, then we note

that the hop degree is δh(G) ≥ δ − 1 ≥ 1. Hence, as a consequence of Theorem 12,

we have the following upper bound on the hop domination number of a triangle-free

graph in terms of its minimum degree.

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Graphs and Combinatorics (2017) 33:913–927 919

Corollary 13 If G is a triangle-free graph of order n with minimum degree δ =δ(G) ≥ 2, then

γh(G) ≤(

1 + ln δ

δ

)

n.

We remark that the upper bound of (1 + ln(δ))n/δ on the hop domination number

of a triangle-free graph with minimum degree δ ≥ 2 and order n is precisely the

same upper bound on the total domination number of a general graph with minimum

degree δ and order n established in [13]. It is therefore a natural question to ask if there

is a relationship between the hop domination and total domination numbers of a graph.

This is indeed the case for triangle-free graphs. Suppose that S is a total dominating

set in a triangle-free graph G. Each vertex u of G outside S has a neighbor, say v,

inside S; that is, v ∈ N (u)∩ S. Since S is a total dominating set in G, the vertex v has a

neighbor w that belongs to S. By the triangle-freeness of G, we note that d(u, w) = 2,

implying that u is hop dominated by S. Since u is an arbitrary vertex in V (G)\S, the

set S is therefore a hop dominating set of G. We state this formally as follows.

Observation 14 If G is a triangle-free graph, then γh(G) ≤ γt (G).

We note that for general graphs G, the relationship γh(G) ≤ γt (G) does not nec-

essarily hold. Indeed, there are connected graphs G with arbitrarily large minimum

hop degree for which the difference γh(G) − γt (G) can be made arbitrarily large. For

example, if G is the complete ℓ-partite graph Kk,...,k with each partite set of size k

for positive integers k and ℓ, then every hop dominating set of G contains at least one

vertex from each partite set, and so γh(G) ≥ ℓ. However, the set consisting of one

vertex from each partite set is a hop dominating set of G, and so γh(G) ≤ ℓ. Conse-

quently, γh(G) = ℓ. Since γt (G) = 2, we therefore note that γh(G) − γt (G) = ℓ − 2

which can be made arbitrarily large by making ℓ arbitrarily large. We also note that

δh(G) = k − 1 which can be made arbitrarily large by making k arbitrarily large.

We establish next a relationship between the hop domination and total domination

numbers in a random graph. For this purpose, let n be a positive integer and 0 <

p(n) < 1. The random graph G(n, p(n)) is a probability space over the set of graphs

on the vertex set [n] determined by Pr[{i, j} ∈ E(G)] = p(n) with these events

mutually independent. We say that an event holds asymptotically almost surely (or

almost always) if the probability that it holds tends to 1 as n tends to infinity. If x and

y are real numbers, we write x ≪ y (respectively, x ≫ y) to mean that x is much

less (respectively, much greater) than y . We say that a function r(n) is a threshold

function for an event Q, if the probability that Q holds tends to 1 as n tends to infinity

whenever p(n) ≫ r(n), and the probability that Q holds tends to 0 as n tends to

infinity whenever p(n) ≪ r(n) (see [1,4]). We shall make use of the following two

results.

Theorem 15 ([1]) r(n) = 1n

is the threshold function for the property “having a

triangle”.

Theorem 16 ([22]) If p is constant then almost every graph G(n, p) has diameter

two.

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920 Graphs and Combinatorics (2017) 33:913–927

We note that if G is a graph of diameter two and v is a vertex of G of minimum degree

δ(G), then NG [v] is a hop dominating set in G, implying that γh(G) ≤ |NG[v]| =1 + δ(G). From [4] we know that if G = G(n, p), where p is constant, then δ(G) =np(1+o(1)). As a consequence of these observations and Observation 14, Theorems 15

and 16 imply the following result on the hop domination number.

Theorem 17 The following holds.

(a) For almost all graphs G = G(n, p(n)), γh(G) ≤ γt (G), if p(n) ≪ 1/n.

(b) For almost all graphs G = G(n, p), γh(G) ≤ 1 + np(1 + o(1)), if p is constant.

Let

cn =

√

1

2+

1

ln(n)+

ln(ln(n))

2 ln(n).

The constant, cn , was first defined in [15] where it was shown that cn < 1 when

n ≥ 24. Further, it was shown in [15] that if G is a graph of diameter two and of

order n ≥ 24 with δ(G) ≥ cn

√n ln(n), then γt (G) < cn

√n ln(n). As observed

earlier, the relationship γh(G) ≤ γt (G) does not necessarily hold in general. However

using analogous ideas as in the proof in [15], we can prove that this upper bound

established in [15] on the total domination number also holds for the hop domination

number of a graph of diameter two.

Theorem 18 Let G be a graph of diameter two and of order n ≥ 24 with �(G) <

n − 1. If �(G) ≤ n − cn

√n ln(n), then γh(G) < cn

√n ln(n).

Proof Let n ≥ 24, and let G be defined as in the statement of the theorem. Let

δh(G) = δh and let �(G) = �. Since � < n − 1, we note that δh ≥ 1. Since G is a

graph of diameter two, we note that the hop degree of a vertex v in G is the number of

vertices at distance 2 from v in G, that is, dh(v) = n − d(v) − 1. Thus a vertex with

minimum hop degree in G is a vertex of maximum (ordinary) degree in G. Hence,

δh = n − � − 1, implying that δh + 1 = n − � ≥ cn

√n ln(n). Since n ≥ 24, we

note that cn < 1 (see [15]), implying that ln(cn) < 0. Since (ln(δh + 1) + 1)/(δh + 1)

is a decreasing function for all δh ≥ 1, and since δh + 1 ≥ cn

√n ln(n), the following

now holds by Theorem 12.

γh(G) ≤(

1 + ln(δh + 1)

δh + 1

)

n

≤(

1 + ln(cn

√n ln(n))

cn

√n ln(n)

)

n

=(

1 + ln(cn) +ln(n)

2+

ln(ln(n))

2

)√

n ln(n)

cn ln(n)

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Graphs and Combinatorics (2017) 33:913–927 921

<1

cn

(

1

ln(n)+

1

2+

ln(ln(n))

2 ln(n)

)

√

n ln(n)

=1

cn

· c2n ·

√

n ln(n)

= cn

√

n ln(n).

⊓⊔

5 Complexity

In this section, we consider the complexity issue related to computing the 2-step

domination and hop domination numbers of a graph. We will state the correspond-

ing decision problems in the standard Instance Question form [9] and indicate the

polynomial-time reduction used to prove that it is NP-complete. A graph is non-empty

if it contains at least one edge. Consider the following decision problems:

2-Step Dominating Set Problem (2SDP).

Instance: A non-empty graph G, and a positive integer k.

Question: Does G have a 2-step dominating set of size at most k?

Hop Dominating Set Problem (HDP).

Instance: A non-empty graph G, and a positive integer k.

Question: Does G have a hop dominating set of size at most k?

A vertex cover of a graph is a set of vertices such that each edge of the graph is

incident to at least one vertex of the set. The Vertex Cover Problem is the following

decision problem.

Vertex Cover Problem (VCP).

Instance: A non-empty graph G, and a positive integer k.

Question: Does G have a vertex cover of size at most k?

The Vertex Cover Problem was one of Karp’s 21 NP-complete problems [17] and

is therefore a classical NP-complete problem in computational complexity theory. We

show that the two decision problems, 2-Step Dominating Set and Hop Dominating

Set, are NP-complete by reducing the Vertex Cover Problem to them.

Theorem 19 2SDP is NP-complete for planar bipartite graphs.

Proof Clearly, the 2SDP is in NP, since it is easy to verify a “yes” instance of 2SDP

in polynomial time. Now let us show how to transform the vertex cover problem to

the 2SDP so that one of them has a solution if and only if the other has a solution.

Let G be a connected planar graph of order nG

and size mG

≥ 2. Let H be the graph

obtained from G as follows. For each edge e = uv ∈ E(G) we subdivide the edge e

three times, and add a path ve1v

e2v

e3v

e4, and join ve

1 to both u and v. The resulting graph

H has order nH

= nG

+ 7mG

and size mH

= 9mG

. The transformation can clearly be

performed in polynomial time. We note that since G is connected and planar, so too

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922 Graphs and Combinatorics (2017) 33:913–927

Fig. 1 The graphs G and H in

the proof of Theorem 19

a

b

c

e

f

G H

ve4

ve3

ve2

ve1

vf4

vf3

vf2

vf1

c

b

aeab

ebc

(a) (b)

is H . Further, by construction, H is bipartite. Thus, H is a connected planar bipartite

graph (Fig. 1).

We show that G has a vertex cover of size at most k if and only if H has a 2-step

dominating set of size at most k + 2mG

.

Suppose firstly that G has a vertex cover, SG , of size at most k. We now consider

the set

SH = SG ∪⋃

e∈E(G)

{

ve1, v

e2

}

.

We note that since mG

≥ 2, the set SG �= ∅. For every edge e = uv ∈ E(G), the vertex

ve2 2-step dominates the vertices ve

4, u and v in H , while the vertex ve1 2-step dominates

the vertex ve3 and the neighbors of u and v in H that belong to the (u, v)-path in H

that resulted from subdividing the edge e = uv of G. Since SG is a vertex cover in G,

every subdivided vertex that is not a neighbor of a vertex in V (G) is 2-step dominated

by the set SG in H . Further, the set SG 2-step dominates the vertex ve2 for every edge

e ∈ E(G). Since G is connected and mG

≥ 2, for every two adjacent edges e and f in

G the vertices ve1 and v

f1 2-step dominate each other. Therefore, the set SH is a 2-step

dominating set of size at most k + 2mG

in H .

Suppose next that H has a 2-step dominating set, DH , of size at most k + 2mG

.

Let e = uv ∈ E(G) and let euv be the subdivided vertex at distance 2 from both u

and v in H that resulted from subdividing the edge e three times. In order to 2-step

dominate ve3 in H , the vertex ve

1 must belong to the set DH . Analogously, in order

to 2-step dominate ve4 in H , the vertex ve

2 belongs to the set DH . In order to 2-step

dominate the vertex euv in H , we note that u or v or both u and v belong to DH . Thus,

DG = DH ∩ V (G) is a vertex cover of G. Further, since both ve1 and ve

2 belong to

DH for every edge e ∈ E(G), we note that |DG | ≤ |DH | − 2mG

= k. Thus, G has a

vertex cover of size at most k. ⊓⊔

Theorem 20 2SDP is NP-complete for planar chordal graphs.

Proof Clearly, the 2SDP is in NP, since it is easy to verify a “yes” instance of 2SDP

in polynomial time. Now let us show how to transform the vertex cover problem to

the 2SDP so that one of them has a solution if and only if the other has a solution.

Let G be a connected planar chordal graph of order nG

and size mG

≥ 2. Let H be

the graph obtained from G as follows. For each edge e = uv ∈ E(G) we add a new

vertex euv adjacent to both u and v in H and we add a pendant vertex e′uv to euv , and

so e′uv has degree 1 and is adjacent to euv . Further, we add a path ve

1ve2v

e3v

e4, and join

123

Graphs and Combinatorics (2017) 33:913–927 923

e

f

h

a

b

c

G H

ve4

ve3

ve2

vf4

vf3

vf2

ve1

vf1

vh1

vh2

vh3

vh4

c

b

a

eab

ebc

eab

ebc

eaceac

(a) (b)

Fig. 2 The graphs G and H in the proof of Theorem 20

ve1 to u and v. The resulting graph H has order n

H= n

G+ 6m

Gand size m

H= 9m

G.

The transformation can clearly be performed in polynomial time. We note that since

G is a connected planar chordal graph, so too is H (Fig. 2).

We show that G has a vertex cover of size at most k if and only if H has a 2-step

dominating set of size at most k + 2mG .

Suppose firstly that G has a vertex cover, SG , of size at most k. We now consider

the set

SH = SG ∪⋃

e∈E(G)

{

ve1, v

e2

}

.

We note that since mG

≥ 2, the set SG �= ∅. For every edge e = uv ∈ E(G), the vertex

ve2 2-step dominates the vertices ve

4, u and v in H , while the vertex ve1 2-step dominates

the vertices ve3 and euv . Since SG is a vertex cover in G, every vertex e′

uv of degree 1

in H associated with the edge uv ∈ E(G) is 2-step dominated by the set SG in H .

Further, the set SG 2-step dominates the vertex ve2 for every edge e ∈ E(G). Since G

is connected and mG

≥ 2, for every two adjacent edges e and f in G the vertices ve1

and vf

1 2-step dominate each other. Therefore, the set SH is a 2-step dominating set

of size at most k + 2mG

in H .

Suppose next that H has a 2-step dominating set, DH , of size at most k + 2mG

. Let

e = uv ∈ E(G). In order to 2-step dominate ve3 in H , the vertex ve

1 must belong to

the set DH . Analogously, in order to 2-step dominate ve4 in H , the vertex ve

2 belongs

to the set DH . In order to 2-step dominate the vertex e′uv in H , we note that u or v

or both u and v belong to DH . Thus, DG = DH ∩ V (G) is a vertex cover of G.

Further, since both ve1 and ve

2 belong to DH for every edge e ∈ E(G), we note that

|DG | ≤ |DH | − 2mG

= k. Thus, G has a vertex cover of size at most k. ⊓⊔

We consider next the decision problem HDP. The proof of the following results are

analogous, but simpler, to those of Theorems 19 and 20.

Theorem 21 HDP is NP-complete for planar bipartite graphs.

Proof Let G be a graph of order nG

and size mG

, and let H be the connected planar

bipartite graph constructed in the proof of Theorem 19. We show that G has a vertex

cover of size at most k if and only if H has a hop dominating set of size at most k+2mG

.

If G has a vertex cover, SG , of size at most k, then this is immediate since the 2-step

dominating set SH constructed in the proof of Theorem 19 is also a hop dominating

set in H of size |SH | ≤ k + 2mG

.

123

924 Graphs and Combinatorics (2017) 33:913–927

Suppose next that H has a hop dominating set, DH , of size at most k + 2mG

.

If |DH ∩ {ve1, v

e2, v

e3, v

e4}| ≤ 1 for some edge e ∈ E(G), then ve

3 or ve4 is not hop

dominated by DH , a contradiction. Therefore, |DH ∩ {ve1, v

e2, v

e3, v

e4}| ≥ 2 for every

edge e ∈ E(G). Let e = uv be an arbitrary edge of G, and let euv be the subdivided

vertex at distance 2 from both u and v in H that resulted from subdividing the edge

e three times. If euv /∈ DH , then in order to hop dominate the vertex euv in H , we

note that u or v or both u and v belong to DH . We now consider the set DG obtained

from DH ∩ V (G) as follows. For each subdivided vertex euv associated with an edge

uv ∈ E(G), if euv ∈ DH , then we add u or v to the set DG . The resulting set DG is

a vertex cover of G of size at most |DH | − 2mG

≤ k. Thus, G has a vertex cover of

size at most k. ⊓⊔

Theorem 22 HDP is NP-complete for planar chordal graphs.

Proof Let G be a graph of order nG

and size mG

, and let H be the connected planar

chordal graph constructed in the proof of Theorem 20. We show that G has a vertex

cover of size at most k if and only if H has a hop dominating set of size at most k+2mG

.

If G has a vertex cover, SG , of size at most k, then this is immediate since the 2-step

dominating set SH constructed in the proof of Theorem 20 is also a hop dominating

set in H of size |SH | ≤ k + 2mG

.

Suppose next that H has a hop dominating set, DH , of size at most k + 2mG

.

If |DH ∩ {ve1, v

e2, v

e3, v

e4}| ≤ 1 for some edge e ∈ E(G), then ve

3 or ve4 is not hop

dominated by DH , a contradiction. Therefore, |DH ∩ {ve1, v

e2, v

e3, v

e4}| ≥ 2 for every

edge e ∈ E(G). Let e = uv be an arbitrary edge of G. If e′uv /∈ DH , then in order to

hop dominate the vertex e′uv in H , we note that u or v or both u and v belong to DH .

We now consider the set DG obtained from DH ∩ V (G) as follows. For each vertex

e′uv associated with an edge uv ∈ E(G), if e′

uv ∈ DH , then we add u or v to the set

DG . The resulting set DG is a vertex cover of G of size at most |DH | − 2mG

≤ k.

Thus, G has a vertex cover of size at most k. ⊓⊔

6 Open Problems

By Theorem 14, every triangle-free (isolate-free) graph G satisfies γh(G) ≤ γt (G).

It would be interesting to find other classes of graphs G that satisfy γh(G) ≤ γt (G).

More generally, we pose the following problem.

Problem 1 Characterize the (isolate-free) graphs G for which γh(G) ≤ γt (G).

By Observation 5, for every graph G it holds that γ (G) ≤ γh(G). We have yet to

characterize the graphs G achieving equality in this bound.

Problem 2 Characterize the graphs G for which γ (G) = γh(G); that is, characterize

the graphs achieving equality in the upper bounds of Observation 5.

Acknowledgements The authors express their sincere thanks to three anonymous reviewers. In particular,

we wish to thank one of the reviewers for pointing out to us Observations 1 and 2 which noticeably

simplified our original proofs. The very helpful and insightful comments of the reviewers greatly improved

the exposition and clarity of the paper.

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Graphs and Combinatorics (2017) 33:913–927 925

Appendix

In this appendix, we provide a direct proof of Theorem 11. Recall its statement.

Theorem 11 If G is a graph of order n, then γh(G) = minp ∈Cn

f (p).

Proof of Theorem 11 Let G be a graph with vertex set V = {1, 2, . . . , n}. We pick

randomly and independently each vertex i ∈ [n] with probability pi , where 0 ≤ pi ≤1, to form a set X ⊆ V . Thus, pi denotes the probability that the vertex i belongs to

X ; that is, Pr(i ∈ X) = pi for i ∈ [n]. Let Z be the set of vertices outside X that are

not at distance 2 in G from any vertex of X ; that is,

Z = {i /∈ X | N2(i) ∩ X = ∅}.

The set D = X ∪ Z is a hop dominating set of G. Clearly, |X | =∑n

i=1 X i , where

X i is a random variable with X i = 1 if i ∈ X , and X i = 0 otherwise. Similarly, Z can

be written as a sum of n indicator random variables, say Z1, . . . , Zn . By the linearity

of expectation,

E(|D|) ≤ E(|X | + |Z |) = E(|X |) + E(|Z |).

Hence using the well-known fact that for a random subset M of a given finite set

N ,

E(|M |) =∑

n∈N

Pr(n ∈ M),

we have

E(|X |) =n

∑

i=1

E(X i ) =n

∑

i=1

pi ,

and

E(|Z |) =n

∑

i=1

E(Zi ) =n

∑

i=1

(1 − pi )∏

j∈N2(i)

(1 − p j ).

Thus,

E(|D|) ≤ E(|X |) + E(|Z |) ≤n

∑

i=1

pi +

⎛

⎝

n∑

i=1

(1 − pi ) ·∏

j∈N2(i)

(1 − p j )

⎞

⎠ = f (p).

The expectation being an average value, there is consequently a hop dominating

set of G of cardinality at most E(|D|). Hence,

γh(G) ≤ minp ∈Cn

f (p).

123

926 Graphs and Combinatorics (2017) 33:913–927

Now let D∗ be a hop dominating set of G of minimum cardinality γh(G). Then for

p∗ = (p∗1, . . . , p∗

n) where p∗i = 1 if i ∈ D∗ and p∗

i = 0 otherwise,

f (p∗) =n

∑

i=1

pi = |D∗| = γh(G),

whence γh(G) = minp∈Cn

f (p). ⊓⊔

We show next that the result of Theorem 12 can be deduced directly from Theo-

rem 11. Recall the statement of Theorem 12.

Theorem 12 If G is a graph of order n with δh = δh(G) ≥ 1, then

γh(G) ≤(

ln(δh + 1) + 1

δh + 1

)

n.

Proof of Theorem 12 Following the notation introduced in the proof of Theorem 11,

we let p = (p1, . . . , pn) and we set pi = p for all i ∈ [n], where 0 ≤ p ≤ 1. Further,

for i ∈ [n], we let di denote the hop-degree of the vertex i in G, and so, di = dh(i).

Then,

f (p) =n

∑

i=1

p +n

∑

i=1

(1 − p)di +1

≤ np + n(1 − p)δh+1 (since δh ≤ di and 0 ≤ 1 − p ≤ 1)

≤ np + ne−(δh+1)p (since for x ∈ R, 1 − x ≤ e−x ).

The function g(p) = np + ne−(δh+1)p is minimized when p = p∗ where

e−(δh+1)p∗=

1

δh + 1,

i.e., where

p∗ =ln(δh + 1)

δh + 1.

We note that 0 < p∗ < 1. Let p∗ = (p∗, . . . , p∗) be the n-vector each entry of

which is equal to p∗. By Theorem 11,

γh(G) ≤ f (p∗) ≤ np∗ + ne−(δh+1)p∗ =(

ln(δh + 1) + 1

δh + 1

)

n.

which is the desired upper bound. ⊓⊔

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Graphs and Combinatorics (2017) 33:913–927 927

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