EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 17, No. 1, 2024, 93-104
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Vertex Cover Hop Dominating Sets in Graphs

Vergel T. Bilar1,∗, Maria Andrea O. Bonsocan1, Javier A. Hassan2,
Susan C. Dagondon1

1Department of Mathematics and Statistics, College of Science and Mathematics,
MSU-Iligan Institute of Technology, 9200 Iligan City, Philippines
2Mathematics and Sciences Department, College of Arts and Sciences, MSU Tawi-Tawi
College of Technology and Oceanography, Bongao, Tawi-Tawi, Philippines

Abstract. Let G be a graph. Then a subset C of vertices of G is called a vertex cover hop domi-
nating if C is both a vertex cover and a hop dominating of G. The vertex cover hop domination
number of G, denoted by γvch(G), is the minimum cardinality among all vertex cover hop domi-
nating sets in G. In this paper, we initiate the study of vertex cover hop domination in a graph
and we determine its relations with other parameters in graph theory. We characterize the vertex
cover hop dominating sets in some special graphs, join, and corona of two graphs and we finally
obtain the exact values or bounds of the parameters of these graphs.
2020 Mathematics Subject Classifications: 05C69
Key Words and Phrases: Vertex cover, vertex cover hop dominating set, vertex cover hop
domination number

1. Introduction

One of the rapidly developing areas of research in graph theory is the study of dom-
ination. The idea was first introduced by Claude Berge [1] in 1958 and Ore [6] in 1962.
Moreover, graph theory has also witnessed a surge of interest in recent research, with hop
domination as an intriguing area of investigation which is one of the variations of domina-
tion in graphs. First introduced and investigated by Natarajan et al. [5], this concept can
be used in modelling social networks. Over time, researchers have extensively studied hop
domination and its numerous variants, as evidenced by a range of notable studies found
in [2–4, 7–11].

In this study, we will introduce the concept of vertex cover hop domination in a graph.
We will investigate and characterize these sets in some special graphs and graphs obtained
from some binary operations. Moreover, some bounds or exact values of these graphs will
be presented.

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v17i1.4978

Email addresses: vergel.bilar@g.msuiit.edu.ph (V.T. Bilar)
mariaandrea.bonsocan@g.msuiit.edu.ph (M.A. Bonsocan)
javierhassan@msutawi-tawi.edu.ph (J. Hassan)
susan.dagondon@g.msuiit.edu.ph (S.C. Dagondon)

https://www.ejpam.com 93 © 2024 EJPAM All rights reserved.



V. T. Bilar et al. / Eur. J. Pure Appl. Math, 17 (1) (2024), 93-104 94

2. Terminology and Notation

Let G be a graph. A subset B of V (G) is an independent if for every pair of distinct
vertices v, w ∈ B, dG(v, w) ̸= 1. The maximum cardinality of an independent set in G,
denoted by α(G), is called the independence number of G. Any independent set B with
cardinality equal to α(G) is called an α-set of G.

A vertex a in G is a hop neighbor of a vertex b in G if dG(a, b) = 2. The set N2
G(a) =

{b ∈ V (G) : dG(a, b) = 2} is called the open hop neighborhood of a. The closed hop
neighborhood of a in G is given by N2

G[a] = N2
G(a) ∪ {a}. The open hop neighborhood of

S ⊆ V (G) is the set N2
G(S) =

⋃
a∈S

N2
G(a). The closed hop neighborhood of S in G is the

set N2
G[S] = N2

G(S) ∪ S.
A subset S of V (G) is a hop dominating of G if for every a ∈ V (G)\S, there exists

b ∈ S such that dG(a, b) = 2. The minimum cardinality among all hop dominating sets of
G, denoted by γh(G), is called the hop domination number of G. Any hop dominating set
with cardinality equal to γh(G) is called a γh-set.

A subset U of vertices of a graph G is called a vertex cover of G if every edge in G is
incident with a vertex in U . The minimum cardinality of such set is the vertex covering
number of G and is denoted by β(G).

A subset C of V (G) is a pointwise non-dominating set if for every v ∈ V (G) \ C,
there exists u ∈ C such that v /∈ NG(u). The minimum cardinality of a pointwise non-
dominating set of G, denoted by pnd(G), is called a pointwise non-domination number of
G. Any pointwise non-dominating set of G with cardinality pnd(G) is called a pnd-set of
G.

Let G and H be any two graphs. The join G + H is the graph with vertex set
V (G+H) = V (G) ∪ V (H) and edge set

E(G+H) = E(G) ∪ E(H) ∪ {uv : u ∈ V (G), v ∈ V (H)}.

The corona G ◦H is the graph obtained by taking one copy of G and |V (G)| copies of
H, and then joining the ith vertex of G to every vertex of the ith copy of H. We denote
by Hv the copy of H in G ◦ H corresponding to the vertex v ∈ G and write v + Hv for
⟨{v}⟩+Hv.

3. Results

We shall define the concept of vertex cover hop domination in a graph as follows:

Definition 1. Let G be a graph. A subset C of vertices V (G) of G is said to be a vertex
cover hop dominating if C is both a vertex cover and a hop dominating set in G. The
vertex cover hop domination number of G, denoted by γvch(G), is the minimum cardinality
among all vertex cover hop dominating sets in G. Any vertex cover hop dominating set
with cardinality equal to γvch(G) is called a γvch-set of G.



V. T. Bilar et al. / Eur. J. Pure Appl. Math, 17 (1) (2024), 93-104 95

Example 1. Consider the graph G in Figure 1. Let C = {e, f, g, h}. Notice that every
edge of G is incident to atleast one vertex in C. Thus, C is a vertex cover set of G. Observe
that a, d ∈ N2

G(f), b ∈ N2
G(g) and c ∈ N2

G(e). It follows that N
2
G[C] = V (G), showing that

C is a hop dominating set in G. Therefore, C is a vertex cover hop dominating set of G.
Moreover, it can be verified that γvch(G) = 4.

a b c d

e f g h

G :

Figure 1: Graph G with γvch(G) = 4

Proposition 1. Let G be any graph. Then each of the following is true.

(i) A vertex cover may not be a hop dominating.

(ii) A hop dominating may not be a vertex cover.

Proof. (i) Consider the graph H given in Figure 2. Let C ′ = {b, d, e, f, g}. Observe
that every edge of H is incident to at least one vertex in C ′, and so C ′ is a vertex cover
set of H. However C ′ is not a hop dominating set of H since i /∈ N2

G[x] ∀x ∈ C ′.

h

H :

b d e

f ga

c

i

Figure 2: Graph H

(ii) Consider again the graph G in Figure 1. Let M = {c, f, g}. Notice that
a, d, h ∈ N2

G(f) and b, e ∈ N2
G(g). Thus, N2

G[M ] = V (G), showing that M is a hop



V. T. Bilar et al. / Eur. J. Pure Appl. Math, 17 (1) (2024), 93-104 96

dominating set of G. However, M is not a vertex cover of G since edges ae, be and dh are
not incident to any vertex in M .

Proposition 2. Let G be any graph. Then each of the following holds:

(i) γh(G) ≤ γvch(G), and this bound is sharp.

(ii) β(G) ≤ γvch(G), and this bound is sharp.

Proof. (i) Let C be a minimum vertex cover hop dominating set of G. Then C is a
hop dominating in G (by definition). Since γh(G) is the minimum cardinality among all
hop dominating sets in G, it follows that γh(G) ≤ |C| = γvch(G). To see that the bound
is sharp, consider G = K1,n. Then γh(G) = γvch(G) = 2.

(ii) Let C be a minimum vertex cover hop dominating set of G. Then C is a vertex cover
in G (by definition). Since β(G) is the minimum cardinality of a vertex cover set in G, it
follows that β(G) ≤ |C| = γvch(G). To see that sharpness is attainable, consider the graph
G in Figure 3. Let T = {b, d, e}. Then T is a vertex cover set of G, and so β(G) ≤ 3. Since
{x, y} is not a vertex cover of G for any pair of distinct vertices x, y ∈ V (G), it follows that
β(G) ≥ 3. Thus, β(G) = 3. Now, observe that N2

G[b] = {b, c, f, g}, N2
G[d] = {a, d, f, g}

and N2
G[e] = {a, c, e}. Thus, N2

G[T ] = V (G), showing that T is a vertex cover hop
dominating set of G. Since T is a minimum vertex cover of G, it follows that T is a
minimum vertex cover hop dominating set of G, and so γvch(G) = 3. Consequently,
γvch(G) = 3 = β(G).

a

b

c d

e

f

g

G :

Figure 3: Graph G with β(G) = γvch(G)

Theorem 1. Let G be any graph on n ≥ 1 vertices. Then 1 ≤ γvch(G) ≤ n. Moreover,

(i) γvch(G) = 1 if and only if G is trivial.

(ii) γvch(G) = 2 if and only if γh(G) = 2 = |T | such that V (G) \T is an independent set
of G.



V. T. Bilar et al. / Eur. J. Pure Appl. Math, 17 (1) (2024), 93-104 97

(iii) γvch(G) = n if and only if every component of G is complete.

Proof. Clearly, 1 ≤ γvch(G) ≤ n.

(i) Suppose γvch(G) = 1. If G is not trivial graph, then γh(G) ≥ 2. By Proposition 2,
γvch(G) ≥ 2, a contradiction. Hence, G is trivial.

Conversely, suppose G is trivial graph. Clearly, γvch = 1.

(ii) γvch(G) = 2, say T = {a, b} is a minimum vertex cover hop dominating set of G.
Then by (i), G is non-trivial and so γh(G) ≥ 2. By assumption and Proposition 2, we
have γh(G) = 2. Now, suppose that V (G) \ T is not an independent set in G. Then there
exist a, b ∈ V (G) \ T such that dG(a, b) = 1, a contradiction to the fact that T is a vertex
cover of G. Hence, V (G) \ T is an independent set in G.

Conversely, suppose γh(G) = 2 = |T |, say T = {a, b} is a γh-set of G such that V (G)\T
is an independent set in G. Then γvch(G) ≥ 2 by Proposition 2. Now, suppose T is not
a vertex cover of G. Then there exist an edge e = xy such that x, y /∈ T . It follows
that x, y ∈ V (G) \ T , a contradiction to the fact that V (G) \ T is an independent set
in G. Hence, T is a vertex cover of G, and so T is a vertex cover hop dominating of G.
Consequently, γvch(G) = 2.

(iii) Suppose that γvch(G) = n. Suppose there is a component H of G which is non-
complete. Then dH(a, b) = 2 = dG(a, b) for some a, b ∈ V (H). Let C ′ = V (G) \ {a}.
Then C ′ is a vertex cover hop dominating set of G. It follows that γvch(G) ≤ n − 1, a
contradiction. Thus, every component of G is complete.

Conversely, suppose that every component of G is complete. Then γh(G) = n. By
Proposition 2, γvch(G) = n.

The next result is a direct consequence of Theorem 1.

Corollary 1. Let G be any graph of order n ≥ 1. Then each of the following statements
holds.

(i) γvch(G) ≥ 2 if and only if G is non-trivial.

(ii) γvch(G) = n if G = Kn.

(iii) γvch(G) ≤ n− 1 if G has non-complete component.

(iv) γvch(G) + γvch(G) = 2n if G = Kn or G = Kn.

(v) γvch(G) · γvch(G) = n2 if G = Kn or G = Kn.

Theorem 2. Let G be a graph of order n ≥ 1. Then each of the following statements
holds.

(i) Let G be a connected graph. Then γvch(G) = n if and only if G = Kn.



V. T. Bilar et al. / Eur. J. Pure Appl. Math, 17 (1) (2024), 93-104 98

(ii) 4 ≤ γvch(G) + γvch(G) ≤ 2n− 1 if G has one non-complete component.

(iii) 4 ≤ γvch(G) · γvch(G) ≤ n2 − n if G has one non-complete component.

Proof. (i) Suppose γvch(G) = n. Then by Theorem 1(iii), every component of G is
complete. Since G is connected, it follows that G = Kn.

The converse follows from Corollary 1(ii).

(ii), (iii) Suppose that G has one non-complete component. Then by Corollary 1(iii),
γvch(G) ≤ n − 1. By Theorem 1, γvch(G) ≤ n. Thus, γvch(G) + γvch(G) ≤ 2n − 1 and
γvch(G) · γvch(G) ≤ n2 − n. Since G is non-trivial, γvch(G) ≥ 2 and γvch(G) ≥ 2 by
Corollary 1(i). Therefore, γvch(G)+γvch(G) ≥ 4 and γvch(G) ·γvch(G) ≥ 4. Consequently,
4 ≤ γvch(G) + γvch(G) ≤ 2n− 1 and 4 ≤ γvch(G) · γvch(G) ≤ n2 − n.

Proposition 3. Let G be any graph on n ≥ 2 vertices. If γvch(G) = 2, then γh(G) = 2.
However, the converse of is not always true.

Proof. Suppose γvch(G) = 2. Then γh(G) ≤ 2 by Proposition 2. Since γh(G) ≥ 2 for
any graph of order n ≥ 2, it follows that γh(G) = 2.

To see the converse is not necessarily true, consider P5 = [v1, v2, v3, v4, v5]. Let
C = {v2, v3}. Then C is a γh-set of P5 and so γh(P5) = 2. However, γvch(P5) = 3.

Definition 2. A pointwise non-dominating set C ⊆ V (G) is called a vertex cover pointwise
non-dominating set of G if it is a vertex cover of G. The minimum cardinality of a vertex
cover pointwise non-dominating set of G, denoted by vcpnd(G), is called a vertex cover
pointwise non-domination number of G. Any vertex cover pointwise non-dominating set
of G with cardinality equal to vcpnd(G) is called a vcpnd-set of G.

Example 2. Consider the graph G in Figure 4. Let C = {a, b, f}. Notice that every
edge of G is incident to atleast one vertex in C. Thus, C is a vertex cover set of G. Now,
since c, d, e /∈ NG(a), it follows that C is a pointwise non-dominating set of G. Hence,
C is a vertex cover pointwise non-dominating set of G. Moreover, it can be verified that
vcpnd(G) = 3.



V. T. Bilar et al. / Eur. J. Pure Appl. Math, 17 (1) (2024), 93-104 99

b

a

c

d

e

f
G :

Figure 4: Graph G with vcpnd(G) = 3

Theorem 3. Let G be any graph of order n. Then each of the following is true.

(i) 1 ≤ vcpnd(G) ≤ n

(ii) pnd(G) ≤ vcpnd(G).

(iii) vcpnd(G) = 1 if and only if every component of G is trivial.

(iv) vcpnd(G) = n if and only if every component of G is a non-trivial complete graph.

Proof. (i) Since any vertex cover pointwise non-dominating set in G is a non-empty, it
follows that vcpnd(G) ≥ 1. Also, since every vertex cover pointwise non-dominating set
C is contained in V (G), we have vcpnd(G) ≤ n. Therefore, 1 ≤ vcpnd(G) ≤ n.

(ii) Let C be a minimum vertex cover pointwise non-dominating set of G. Then C is
a pointwise non-dominating (by definition). Since pnd(G) is the minimum cardinality
among all pointwise non-dominating sets in G, we have vcpnd(G) = |C| ≥ pnd(G).

(iii) Suppose vcpnd(G) = 1, say, {v} is a vcpnd-set of G. Suppose there is a component W
of G which is non-trivial. Then there exist a, b ∈ V (W ) such that ab ∈ E(W ) ⊆ E(G). If
v ̸= a, b, then vcpnd(G) ≥ 2, a contradiction. Suppose v = a. Then b must also be in the
vertex cover pointwise non-dominating set S of G. Hence, vcpnd(G) ≥ 2, a contradiction.
Therefore, every component of G is trivial.

The converse is clear.

(iv) Suppose vcpnd(G) = n, say, C = V (G) is the vcpnd-set of G. Then by (iii), every
component of G is a non-trivial graph. Now, suppose there is a component C of G
which is non-complete. Then there exist u, v ∈ V (C) ⊆ V (G) such that dC(u, v) = 2 =
dG(u, v) = 2. Clearly, V (G) \ {u} is a vertex cover pointwise non-dominating set in G.
Hence, vcpnd(G) ≤ n− 1, a contradiction. Therefore, every component of G is non-trivial
complete graph.

Conversely, suppose that every component of G is a non-trivial complete graph. Then
pnd(G) = n. By (i) and (ii), we have vcpnd(G) = n.



V. T. Bilar et al. / Eur. J. Pure Appl. Math, 17 (1) (2024), 93-104 100

Observation 1. Let n be any positive integer. Then

(i) vcpnd(Pn) =


2 if n = 2, 3

3 if n = 5
n
2 if n ≥ 4 and even

⌊n2 ⌋ if n ≥ 7 and odd.

(ii) vcpnd(Cn) =


3 if n = 3
n
2 if n ≥ 4 and even

⌈n2 ⌉ if n ≥ 5 and odd.

Theorem 4. [10] Let G and H be any two graphs. A set S ⊆ V (G+H) is hop dominating
set of G+H if and only if S = SG ∪SH , where SG and SH are pointwise non-dominating
sets of G and H, respectively.

Theorem 5. Let G and H be two non-complete graphs. A subset C = CG ∪ CH of
V (G+H) is a vertex cover hop dominating set of G+H if and only if C satisfies one of
the following conditions:

(i) CG = V (G) and CH is a vertex cover pointwise non-dominating set of H.

(ii) CH = V (H) and CG is a vertex cover pointwise non-dominating set of G.

Proof. Suppose C is a vertex cover hop dominating set of G +H. Suppose CG = ∅.
Observe that V (G) ⊈ N2

G[C]. Since C is a hop dominating set of G +H, it follows that
V (G) ⊆ N2

G[C], a contradiction. Thus, CG ̸= ∅. Similarly, CH ̸= ∅. If CG = V (G)
and CH = V (H), then we are done. Suppose that CH ̸= V (H). Then there exists
a ∈ V (H)\CH such that a /∈ C. If CG ̸= V (G), then there exists b ∈ V (G)\CG such that
ab ∈ E(G+H). However, a, b /∈ C, a contradiction to the fact that C is a vertex cover of
G+H. Hence, CG = V (G). Since C is a hop dominating set in G+H, it follows that CH

is a pointwise non-dominating set of H by Theorem 4. Also, since C is a vertex cover of
G+H, CH is a vertex cover of H. Thus, (i) holds. Similarly, (ii) holds.

Conversely, suppose that (i) holds. Then C = V (G) ∪ CH is a vertex cover of G+H.
Since CH is a pointwise non-dominating set, it follows that C = V (G) ∪ CH is a hop
dominating set in G +H by Theorem 4. Therefore, C = CG ∪ CH is a vertex cover hop
dominating set of G+H. Similarly, if (ii) holds, then C = CG ∪CH is a vertex cover hop
dominating set of G+H.

Theorem 6. Let G and H be two non-complete graphs. Then

γvch(G+H) = min{|V (G)|+ vcpnd(H), |V (H)|+ vcpnd(G)}.

Proof. Let C = CG ∪ CH be a minimum vertex cover hop dominating set of G +H.
Then by Theorem 5, C satisfies one of the following:



V. T. Bilar et al. / Eur. J. Pure Appl. Math, 17 (1) (2024), 93-104 101

(i) CG = V (G) and CH is a vertex cover pointwise non-dominating set of H.

(ii) CH = V (H) and CG is a vertex cover pointwise non-dominating set of G.

Thus,
γvch(G+H) = |C| = |CG|+ |CH | ≥ V (G)|+ vcpnd(H)

and
γvch(G+H) = |C| = |CG|+ |CH | ≥ V (H)|+ vcpnd(G).

Consequently,

γvch(G+H) ≥ min{|V (G)|+ vcpnd(H), |V (H)|+ vcpnd(G)}.

On the other hand, suppose that C = V (G) ∪ CH , where CH is a minimum vertex
cover pointwise non-dominating set of H. Then C is a vertex cover hop dominating set
in G+H by Theorem 5. Thus, |V (G)|+ vcpnd(H) = |C| ≥ γvch(G+H). Next, suppose
that C = V (H) ∪ CG, where CG is a minimum vertex cover pointwise non-dominating
set of G. Then C is a vertex cover hop dominating set in G + H by Theorem 5. Thus,
|V (H)|+ vcpnd(G) = |C| ≥ γvch(G+H). Therefore,

γvch(G+H) = min{|V (G)|+ vcpnd(H), |V (H)|+ vcpnd(G)}.

The following result follows from Observation 1 and Theorem 6.

Corollary 2. Let n be any positive integer. Then each of the following is true.

(i) γvch(Pn + Pn) = n+ vcpnd(Pn) =


n+ 2 if n = 3

n+ 3 if n = 5

n+ n
2 if n ≥ 4 and even

n+ ⌊n2 ⌋ if n ≥ 7 and odd.

(ii) γvch(Cn + Cn) = n+ vcpnd(Cn) =

{
n+ n

2 if n ≥ 4 and even

n+ ⌈n2 ⌉ if n ≥ 5 and odd.

Theorem 7. Let G be any complete graph and H be any graph. A set C ⊆ V (G+H) is
vertex cover hop dominating set of G+H if and only if C = V (G) ∪ CH , CH is a vertex
cover pointwise non-dominating set in H.

Proof. Suppose C is a vertex cover hop dominating set of G +H. Suppose CG = ∅.
Observe that V (G) ⊈ N2

G[C]. Since C is a hop dominating set of G +H, it follows that
V (G) ⊆ N2

G[C], a contradiction. Thus, CG ̸= ∅. Similarly, CH ̸= ∅. Since C is a
hop dominating set in G + H, it follows that CH is a pointwise non-dominating set of
H by Theorem 4. Also, since C is a vertex cover of G + H, CH is a vertex cover of H.
Now, suppose CG ̸= V (G). Then there exists a ∈ V (G) \ CG such that a /∈ C. Since



V. T. Bilar et al. / Eur. J. Pure Appl. Math, 17 (1) (2024), 93-104 102

G is complete, it follows that a /∈ N2
G+H [C], a contradiction to the fact that C is a hop

dominating set in G+H. Hence, CG = V (G).
Conversely, suppose that C = V (G) ∪ CH , where CH is a vertex cover pointwise non-

dominating set in H. Then C is a vertex cover set of G+H. By Theorem 4, C is a hop
dominating of G+H. Thus, C is a vertex cover hop dominating set of G+H.

Theorem 8. Let G be complete graph and H be any graph. Then.

γvch(G+H) = |V (G)|+ vcpnd(H).

Proof. Let C = V (G)∪CH be a minimum vertex cover hop dominating set of G+H.
Then by Theorem 7, C = V (G)∪CH , where CH is a vertex cover pointwise non-dominating
set of H. Thus,

γvch(G+H) = |C| = |V (G)|+ |CH | ≥ |V (G)|+ vcpnd(H).

Next, suppose that C = V (G) ∪ CH , where CH is a minimum vertex cover pointwise
non-dominating set of H. Then C is a vertex cover hop dominating set in G + H by
Theorem 7. Hence, |V (G)|+ vcpnd(H) = |C| ≥ γvch(G+H). Therefore,

γvch(G+H) = |V (G)|+ vcpnd(H).

The following result follows that from Observation 1 and Theorem 7.

Corollary 3. Let n and m be positive integers. Then each of the following is true.

(i) γvch(Kn + Pm) = n+ vcpnd(Pm) =


n+ 2 if m = 3

n+ 3 if m = 5

n+ m
2 if m ≥ 4 and even

n+ ⌊m2 ⌋ if m ≥ 7 and odd.

(ii) γvch(Kn + Cm) = n+ vcpnd(Cm) =

{
n+ m

2 if m ≥ 4 and even

n+ ⌈m2 ⌉ if m ≥ 5 and odd.

(iv) γvch(Wn) = 1 + vcpnd(Cm) =

{
1 + m

2 if m ≥ 4 and even

1 + ⌈m2 ⌉ if m ≥ 5 and odd.

(v) γvch(Fn) = 1 + vcpnd(Pm) =


3 if m = 3

4 if m = 5

1 + m
2 if m ≥ 4 and even

1 + ⌊m2 ⌋ if m ≥ 7 and odd.



REFERENCES 103

Theorem 9. Let G be a non-trivial connected graph and let H be any graph. If
C = V (G) ∪ (∪v∈V (G)Cv), where Cv ⊆ V (Hv) is a vertex cover pointwise non-dominating
set of H for each v ∈ V (G), then C is a vertex cover hop dominating set of G ◦H.

Proof. Let C = V (G) ∪ (∪Cv), where Cv is a vertex cover pointwise non-dominating
set Hv for each v ∈ V (G). Since Cv is a pointwise non-dominating set in Hv for each
v ∈ V (G), it follows that N2

⟨v+Hv⟩[Cv] = V (Hv) for each v ∈ V (G). Thus,

N2
G◦H [C] = N2

G◦H [V (G) ∪ (
⋃

v∈V (G)

Cv)] = V (G ◦H),

and so C is a hop dominating set of G ◦ H. Since Cv is a vertex cover of Hv for each
v ∈ V (G), C = V (G) ∪ (∪v∈V (G)Cv) is a vertex cover of G ◦ H. Consequently, C is a
vertex cover hop dominating set of G ◦H.

Corollary 4. Let G be a non-trivial connected graph and let H be any graph. Then

γvch(G ◦H) ≤ |V (G)|+ |V (G)| · vcpnd(H).

Proof. Let C = V (G) ∪ (
⋃

v∈V (G)Cv), where Cv is a minimum vertex cover pointwise
non-dominating set of H. By Theorem 9, C is a vertex cover hop dominating set G ◦H.
Thus,

γvch(G ◦H) ≤ |C| = |V (G)|+ |V (G)| · vcpnd(H).

4. Conclusion

The concept of vertex cover hop domination has been introduced and initially inves-
tigated in this study. Graphs that attained some specific vertex cover hop domination
number have been characterized. The vertex cover hop domination number of the join
and corona of two graphs have been obtained. These characterizations have been used to
obtain bounds or exact values of the vertex cover hop domination number of each of these
graphs. Exploring bounds for this newly introduced parameter in relation to other known
parameters possibly provides insightful information.

Acknowledgements

The authors would like to thank MSU-Iligan Institute of Technology, MSU Tawi-Tawi
College of Technology and Oceanography and DOST-ASTHRDP for funding this research.

References

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