EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 17, No. 3, 2024, 1737-1750
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

On Minimal Geodetic Hop Domination In Graphs

Dyjay Catian1∗, Imelda S. Aniversario1, Ferdinand P. Jamil1

1 Department of Mathematics and Statistics, College of Science and Mathematics
Center for Mathematical and Theoretical Physical Science, PRISM
MSU-Iligan Institute of Technology, 9200 Iligan City, Philippines

Abstract. Let G be a nontrivial connected graph with vertex set V (G) and the edge set E(G). A set
S ⊆ V (G) is a geodetic hop dominating set of G if the following two conditions hold for each x ∈ V (G)\S:
(1) x lies in some u-v geodesic in G with u, v ∈ S, and (2) x is of distance 2 from a vertex in S. The
minimum cardinality γhg(G) of a geodetic hop dominating set of G is the geodetic hop domination
number of G.

A geodetic hop dominating set S is a minimal geodetic hop dominating set if S does not contain a proper
subset that is itself a geodetic hop dominating set. The maximum cardinality of a minimal geodetic hop
dominating set in G is the upper geodetic hop domination number of G, and is denoted by γ+

hg(G).
This paper initiates the study of the minimal geodetic hop dominating set and the corresponding upper
geodetic hop domination number of nontrivial connected graphs. Interestingly, every pair of positive
integers a and b with 2 ≤ a ≤ b is realizable as the geodetic domination number and the upper geodetic
hop domination number, respectively, of some graph. Furthermore, this paper investigates the concept
in the join, corona and lexicographic product of graphs.

2020 Mathematics Subject Classifications: 05C69

Key Words and Phrases: minimal geodetic hop dominating set, upper geodetic hop dominating
number, 2-path closure absorbing pointwise non-dominating set

1. Introduction

In 2021, D. Anusha et al. [17] introduced the concept of geodetic hop domination in
graphs, and initially investigated the concept in the complementary prism of graphs. Further
investigation of the concept was done by D. Anusha [3] in 2022, by C.J. Saromines and
S.R. Canoy Jr. [26, 27] in 2023, and just recently by D. Anusha et al. [4].

This present paper introduces and initiates the study of the concept of minimal geodetic
hop domination, a natural variation of the geodetic hop domination.

Throughout this paper, all graphs considered are simple and undirected. Common graph
terminologies and notations used here are adapted from [6, 11].

Given two graphs G and H with disjoint vertex sets, the join of G and H is the graph G+H
with V (G+H) = V (G) ∪ V (H) and E(G+H) = E(G) ∪ E(H) ∪ {uv : u ∈ V (G), v ∈ V (H)}.
The corona of G and H is the graph G ◦ H obtained by taking |V (G)| copies of H in every
vertex of V (G) and then joining the ith vertex of V (G) to every vertex of the ith copy of H.

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v17i3.5251

Email addresses: dyjaybill.catian@g.msuiit.edu.ph (D.B. Catian),
imelda.aniversario@g.msuiit.edu.ph (I. Aniversario), ferdinand.jamil@g.msuiit.edu.ph (F. Jamil)

https://www.ejpam.com 1737 © 2024 EJPAM All rights reserved.



D. Catian, I. S. Aniversario, F. P. Jamil / Eur. J. Pure Appl. Math, 17 (3) (2024), 1737-1750 1738

The lexicographic product of G and H is the graph G[H] with V (G[H]) = V (G) × V (H) and
(u, v)(w, z) ∈ E(G[H]) if and only if uw ∈ E(G) or u = w and vz ∈ E(H).

Let G be a connected graph. For any two distinct u, v ∈ E(G), a shortest path joining u
and v is called a u-v geodesic. The length of a u-v geodesic is called the distance between u
and v, which is denoted by dG(u, v). The diameter of the graph G, denoted diam(G), is the
maximum distance between any pair of vertices in G. If diam(G) = 1, then G is a complete
graph. A clique of a graph G is any complete subgraph of G. The clique number of G is the
maximum cardinality ω(G) of S ⊆ V (G) for which ⟨S⟩ is a clique.

By the (open) neighborhood NG(v) of a vertex v is meant the set of vertices that are adjacent
to v, that is, NG(v) = {u ∈ V (G) : uv ∈ E(G)}. The closed neighborhood of vertex v is
NG[v] = NG(v) ∪ {v}. For S ⊆ V (G), NG(S) = ∪v∈SNG(v), while NG[S] = NG(S) ∪ S. A
vertex v is an extreme vertex if the induced subgraph ⟨NG(v)⟩ is a complete graph. The symbol
Ext(G) denotes the set of all extreme vertices in G.

For u, v ∈ V (G), the set IG(u, v) refers to the set consisting of all the vertices lying in any
u − v geodesic of G and IG[u, v] = IG(u, v) ∪ {u, v}. For a subset S ⊆ V (G), the geodetic
closure IG[S] is defined by IG[S] = ∪{IG[u, v] : u, v ∈ S}. A geodetic set is any set S ⊆ V (G)
with IG[S] = V (G). The minimum cardinality g(G) of a geodetic set is the geodetic number
of G. A geodetic set of cardinality g(G) is called a geodetic basis. A geodetic set S in G is a
minimal geodetic set if S does not have a proper subset that is itself a geodetic set in G. The
maximum cardinality of a minimal geodetic set in G is denoted by g+(G). Geodetic sets and
geodetic numbers of graphs are, in fact, among the very well-studied concepts in graph theory
(see [1, 2, 7, 8, 10, 12, 15, 16, 18, 19, 28].

For S ⊆ V (G), the 2-path closure of S, denoted by P2[S]G, is the set

P2[S]G = S ∪ {w ∈ V (G) : w ∈ IG[u, v] ϶ u, v ∈ S with dG(u, v) = 2}

A set S is called 2-path closure absorbing if P2[S]G = V (G). The minimum cardinality of a
2-path closure absorbing set in G is denoted by ρ2(G). A 2-path closure absorbing set S is a
minimal 2-path closure absorbing set if S does not contain a proper subset that is itself a 2-path
closure absorbing. The maximum cardinality of a minimal 2-path closure absorbing set in G is
denoted by ρ+2 (G). The concept of 2-path closure absorbing sets was introduce in [15, 18, 19].

The (open) hop neighborhood of a vertex v refers to the set N2
G(v) = {u ∈ V (G) : dG(u, v) = 2}.

The closed hop neighborhood of a vertex v is N2
G[v] = N2

G(v) ∪ {v}. For S ⊆ V (G), the (open)
hop neighborhood S is the set N2

G(S) = ∪v∈SNG(v). The closed hop neighborhood of the set S
is N2

G[S] = N2
G(S) ∪ S.

A set S ⊆ V (G) is called a hop dominating set if N2
G[S] = V (G). The hop domination

number of G, denoted by γh(G), is the minimum cardinality among all hop dominating sets in
G. A hop dominating set with cardinality equal to γh(G) is called a γh-set of G. The authors
refer to [4, 5, 9, 13, 14, 20–25] for the definitions and results on hop domination which are
essential in this study.

A subset S ⊆ V (G) is a geodetic hop dominating set if S is both a geodetic and a hop
dominating set of G. The geodetic hop domination number γhg(G) of G is the minimum
cardinality among all geodetic hop dominating sets in G. Any geodetic hop dominating set
of G with cardinality γhg(G) is called a γhg-set.



D. Catian, I. S. Aniversario, F. P. Jamil / Eur. J. Pure Appl. Math, 17 (3) (2024), 1737-1750 1739

2. Main Results

2.1. The minimal geodetic hop domination in graphs

Let G be a graph. A geodetic hop dominating set S is a minimal geodetic hop
dominating set if S does not contain a proper subset that is itself a geodetic hop
dominating set. The maximum cardinality of a minimal geodetic hop dominating set of G is
the upper geodetic hop domination number of G denoted by γ+hg(G). A minimal geodetic

hop dominating set with cardinality γ+hg(G) is called a γ+hg-set .

Every dominating vertex and every extreme vertex in G is included in any minimal geodetic
hop dominating set of G.

Proposition 1. Let G be a nontrivial connected graph of order n. Then

2 ≤ γhg(G) ≤ γ+hg(G) ≤ n

In particular, γ+hg(G) < n if G does not have any dominating vertex.

Proof. Since a γhg-set is a minimal geodetic hop dominating set, γhg(G) ≤ γ+hg(G). Suppose

that γ+hg(G) = n and G has no dominating vertex . Since G is not a complete graph, G contains
a geodesic of the form [u,w, v]. Since w ∈ IG(u, v), S = V (G)\{w} is a geodetic set of G. Since
w is a non-dominating vertex, there exists z ∈ V (G) such that dG(w, z) = 2. Because z ∈ S, S
is a hop dominating set of G. Thus, S is a geodetic hop dominating set of G. Since V (G) is a
minimal geodetic hop dominating set, this is impossible.

The contrapositive of the second statement in Proposition 1 gives the following:

Corollary 1. If γ+hg(G) = n for a nontrivial connected graph, then G contains at least one
dominating vertex.

Theorem 1. Let G be a nontrivial connected graph of order n. Then γ+hg(G) = n if and only
if one of the following holds:

(i.) G = Kn

(ii.) G ̸= Kn and V (G) \ S induces a disconnected graph of complete components where
S ⊊ V (G) is the set of all dominating vertices of G.

Proof. Clearly, if G = Kn, then γ+hg(G) = n. Suppose G ̸= Kn with a nonempty set S of
dominating vertices of G and each of the components of ⟨V (G) \S⟩ is complete. Note first that
V (G) is a geodetic hop dominating set of G. Suppose T ⊆ V (G) is a geodetic hop dominating
set of G. By the above remark, S ⊆ T . Let C be a component of ⟨V (G) \ S⟩. We claim that
V (C) ⊆ Ext(G). Let x ∈ V (C), and let u, v ∈ NG(x) with u ̸= v. Then u, v ∈ S ∪ V (C).
If u, v ∈ V (C), then since C is complete, uv ∈ E(G). If u ∈ S or v ∈ S, then uv ∈ E(G).
Accordingly, x ∈ Ext(G). Since x is arbitrary, V (C) ⊆ Ext(G). Thus, V (C) ⊆ T . Since C is
arbitrary, V (G) \ S ⊆ T . Therefore, T = V (G). This means G does not have a proper subset
that is itself a geodetic hop dominating set. In other words, V (G) is a minimal geodetic hop
dominating set of G. Therefore, γ+hg(G) = n.

Conversely, suppose γ+hg(G) = n. If G = Kn, then we are done. Now, suppose G ̸= Kn. By
Corollary 1, the set S ⊆ V (G) consisting of the dominating vertices of G is nonempty. Suppose
that ⟨V (G) \ S⟩ has a component C that is not complete. Then C contains a geodesic of the
form [u,w, v]. We claim that W = V (G) \ {w} is a geodetic hop dominating set of G. Since



D. Catian, I. S. Aniversario, F. P. Jamil / Eur. J. Pure Appl. Math, 17 (3) (2024), 1737-1750 1740

w ∈ IG(u, v), W is a geodetic set of G. Since w /∈ S, there exists z ∈ V (G) such that wz /∈ E(G).
Necessarily, z /∈ S. Pick t ∈ S. Then [w, t, z] is a geodesic in G. Thus, dG(w, z) = 2. This means
thatW is a geodetic hop dominating set of G. Consequently, V (G) is not a minimal geodetic hop
dominating set, implying that γ+hg(G) < n, a contradiction. Therefore, C is complete. Finally,
suppose ⟨V (G) \ S⟩ = C, and let u ∈ V (C). Then uv ∈ E(G) for each v ∈ V (G) \ S = V (C)
with u ̸= v. Also, uv ∈ E(G) for each (dominating) vertex v ∈ S. Hence, u is a dominating
vertex of G, a contradiction. Therefore, ⟨V (G) \ S⟩ is a disconnected graph of at least two
complete components.

Theorem 2. Let G be a nontrivial graph connected graph. Then γ+hg(G) = 2 if and only if G
satisfies the following graphs:

(i) G = P2;

(ii) G = C6;

(iii) G has a geodetic set S = {u, v} such that dG(u, v) = 3 and u, v ∈ Ext(G).

Proof. It is easy to verify that if G = P2 or G = C6, then γ+hg(G) = 2. Suppose that
G satisfies condition (iii). Let w ∈ V (G) \ S. Then w lies on a u-v geodesic. Thus, either
dG(u,w) = 2 or dG(w, v) = 2. Since w is arbitrary, S is a hop dominating set of G. Therefore,
S is a (minimal) geodetic hop dominating set of G. Now, S ⊆ T for every geodetic hop
dominating set T of G. Hence, S is a γ+hg-set of G. Therefore, γ+hg(G) = 2.

Conversely, assume γ+hg(G) = 2. Suppose that G /∈ {P2, C6}. Let S = {u, v} be a γ+hg-set
of G. Since G ̸= P2, uv /∈ E(G). For each a ∈ V (G) \ S, a lies on a u-v geodesic. Choose
a ∈ V (G) \ S such that ua ∈ E(G). Since S is a hop dominating set of G, dG(a, v) = 2.
Necessarily, dG(u, v) = 3. Suppose that u /∈ Ext(G). Then G contains a geodesic [x, u, y]
containing u. Let [u, y, z, v] be a u-v geodesic containing y. We consider two cases:

Case 1: Suppose degG(u) ≥ 3, and let w ∈ NG(u) \ {x, y}. If xz ∈ E(G), then a minimal
geodetic hop dominating set T of G can be constructed containing x, y, z. Clearly, in this
case, |T | ≥ 3. On the other hand, if xz /∈ E(G), a minimal geodetic hop dominating set T
can be constructed containing x, z. Since w /∈ NG[{x, z}], |T | ≥ 3. Either subcase yields a
contradiction.

Case 2: Suppose that degG(u) = 2. If xz ∈ E(G), then T = {x, y, z, v} makes a minimal
geodetic hop dominating set of G. Suppose xz /∈ E(G). Then there exists a u-v geodetic in G
of the form [u, x, w, v] with w ̸= z. Consequently, |V (G)| ≥ 6. Suppose that |V (G)| ≥ 7. Then
a minimal geodetic hop dominating set can be constructed containing x and y. In this case,
|T | ≥ 3. Finally, suppose that |V (G)| = 6. Since G ̸= C6, G is obtained from C6 by adding
at least one edge to join a pair of nonadjacent vertices. In this case, a minimal geodetic hop
dominating set can be constructed with |T | ≥ 3. All possibilities yield to contradiction.

The above cases imply that u ∈ Ext(G). Similarly, v ∈ Ext(G)

Proposition 2. For the complete graph Kn, path Pn, cycle Cn and Petersen graph P , and for
k ≥ 1,

(i) γ+hg(Kn) = n;

(ii) γ+hg(Pn) =


n if n = 1, 2, 3

2k + 1 if n = 4k + 1

2
⌈n
4

⌉
otherwise



D. Catian, I. S. Aniversario, F. P. Jamil / Eur. J. Pure Appl. Math, 17 (3) (2024), 1737-1750 1741

(iii) γ+hg(Cn) =



3 if n = 3, 4, 5

2 if n = 6

2k + 1 if n = 4k + 3

2
⌈n
4

⌉
otherwise

(iv) γ+hg(P ) = 6.

Proof. Since γhg(Kn) = n, Proposition 1 yields γ+hg(Kn) = n.

Let Pn = [x1, x2, . . . , xn]. The case where n = 1, 2, 3 is trivial. Suppose n = 4k + 1,
where k ≥ 1. Since S = {x1, x2, x5, x6, . . . , xn} is a minimal geodetic hop dominating set,
2k+1 = |S| ≤ γ+hg(G). Conversely, let S be a γ+hg-set of G. Being a geodetic set, x1, xn ∈ S. For
every 1 ≤ j ≤ n− 3, S contains at most two vertices in xj , xj+1, xj+2, xj+3. Thus, |S| ≤ 2k+1.
Hence, γ+hg(Pn) = 2k + 1.

For the third case, assume n ̸= 4k+1. If n = 4k, then assume S = {x1, x2, x5, x6, . . . , xn−3, xn}.
By the same argument provided above, we have S be a minimal geodetic hop dominating
set so that S is a γ+hg-set. Hence, |S| = 4k

2 = 2k = 2
⌈
n
4

⌉
. If n > 4k + 1, choose the set

S = {x1, x2, x5, x6, . . . , xn−1, xn} to be minimal geodetic hop dominating set. By the same
argument, we have S to be a γ+hg-set with |S| = 2

⌈
n
4

⌉
. Therefore, γ+hg(G) = 2

⌈
n
4

⌉
.

Let G = Cn = [v1, v2, . . . , vn, v1]. The first and second case is trivial. Let k ≥ 1. For the
third case, assume n = 4k + 3. Suppose S = {v1, v2, v5, v6, . . . , vn−2}. Note that for every
vi, vi+1 /∈ S where i ≥ 3, there exist vi−1, vi+2 ∈ S such that vi, vi+1 ∈ IG(vi−1, vi+2) with
vi ∈ N2

G(vi+2) and vi+1 ∈ N2
G(vi−1) so that S is both a geodetic set and hop dominating set.

Thus, S is a minimal geodetic hop dominating set so that |S| = 4k
2 + 1 = 2k + 1 ≤ γ+hg(G).

Conversely, by the same argument provided in above, S is a minimal geodetic hop dominating
set and since S is arbitrary we have γ+hg(G) ≤ 2k+1. Therefore, γ+hg(G) = 2k+1. For the fourth
case, assume n < 4k + 3 with k ≥ 2. Suppose S = {v1, v2, v5, v6, . . . , vn−3, vn−2}. The same
argument would have S to be a minimal geodetic hop dominating set so that S is a γ+hg-set. It

follows that |S| = 4k
2 = 2k = 2

⌈
n
4

⌉
.

Let G be the Petersen graph shown in Figure 1.

a

e

dc

b
f

j

ih

g

Figure 1: A petersen graph

Then S = {a, c, d, f, g, j} is a γ+hg-set. Hence, γ
+
hg(P ) = 6.

Proposition 3. Let G = Km,n with partite sets U and W with |U | = m ≥ 2 and
|W | = n ≥ 2. Then S ⊆ V (G) is a minimal geodetic hop dominating set of G if and only
if S is one of the following:

(i) S = U ∪ {w} where w ∈ W



D. Catian, I. S. Aniversario, F. P. Jamil / Eur. J. Pure Appl. Math, 17 (3) (2024), 1737-1750 1742

(ii) S = W ∪ {u} where u ∈ U

(iii) S = {u, v, w, z} where u, v ∈ U and w, z ∈ W , in case where m,n ≥ 3.

Proof. It is easy to verify that if S is any of the sets described in (i), (ii) and (iii), then S
is a minimal geodetic hop dominating set of G.

Conversely, suppose S is a minimal geodetic hop dominating set of G. Being a hop
dominating set, S ∩ U ̸= ∅ and S ∩ W ̸= ∅. Suppose that U ⊆ S. Note that if w ∈ S ∩ W ,
then U ∪ {w} is a geodetic hop dominating set of G. By the minimality of S, S = U ∪ {w}.
Similarly, if W ⊆ S, then S = W ∪{u}, where u ∈ U . Now, suppose U \S ̸= ∅ and W \S ̸= ∅.
Since S is geodetic, |S ∩ U | ≥ 2 and |S ∩W | ≥ 2. Pick u, v ∈ S ∩ U and w, z ∈ S ∩W . Then,
{u, v, w, z} is a minimal geodetic hop dominating set of G. Thus, S = {u, v, w, z}.

Corollary 2. Let G = Km,n where m,n ≥ 2. Then γ+hg(G) = max{m,n}+ 1.

Proof. If n = m = 2, then γ+hg(G) = 3 = 1+max{m,n}. If n ≥ 3 or m ≥ 3, then γ+hg(G) ≥ 4.

Thus, γ+hg(G) = 1 +max{m,n}.

3. Realization problem

Theorem 3. For every pair of positive integers a and b with 2 ≤ a ≤ b, there exists a connected
graph G such that γhg(G) = a and γ+hg(G) = b.

Proof. If a = b, then take G = Ka. For this graph G,

γhg(G) = a = b = γ+hg(G)

Assume a < b. Write b = a + k for some k ≥ 1. If a = 2, then we consider the graph G = G1

in Figure 2 obtained by constructing k + 2 copies of P4 with common end-vertices.

u
w1 z1 v

w2 z2

...

wk+2 zk+2

Figure 2: G1: A connected graph complying with the specifications of Theorem 3 when a = 2

Let S = {u, v} and

T =

{
{w2i−1, z2i : i = 1, 2, 3, . . . , k2 + 1}, k is even;

{w1, w2i+1, z2i : i = 1, 2, 3, . . . , ⌊k+1
2 ⌋}, k is odd.

Then S and T are γhg-set and γ+hg-set of G, respectively. Suppose that a ≥ 3. Obtain the graph
G = G2 as in Figure 3 from G1 by adding to G1 (a− 2) pendant edges vxj , j = 1, 2, . . . , a− 2.



D. Catian, I. S. Aniversario, F. P. Jamil / Eur. J. Pure Appl. Math, 17 (3) (2024), 1737-1750 1743

u
w1 z1 v

w2 z2

...

wk+2 zk+2

x1
x2
...

xa−2

Figure 3: G2: A connected graph complying with the specifications of Theorem 3 when a ≥ 3

Put S = {u, v, xi : i = 1, 2, . . . , a− 2} and

T =

{
{w2i−1, z2i, xj : i = 1, 2, 3, . . . , k2 + 1; j = 1, 2, . . . , a− 2}, k is even;

{w1, w2i+1, z2i, xj : i = 1, 2, 3, . . . , ⌊k+1
2 ⌋; j = 1, 2, . . . , a− 2}, k is odd.

Then S and T are γhg-set and γ+hg-set of G, respectively. In any case,

γhg(G) = a and γ+hg(G) = a+ k = b

Corollary 3. The difference between γ+hg(G) and γhg(G) can be made arbitrarily large.

4. In the join of graphs

For the purposes of the remaining sections, we define the following variations of pointwise
non-dominating sets.

A pointwise non-dominating set S ⊆ V (G) is a minimal pointwise non-dominating set
of G if S does not contain a proper subset which is itself a pointwise non-dominating set in G.

A set S ⊆ V (G) is a 2-path closure absorbing pointwise non-dominating set if S is
both a 2-path closure absorbing set and a pointwise non-dominating set in G. A 2-path closure
absorbing pointwise non-dominating set S is said to be a minimal 2-path closure absorbing
pointwise non-dominating set whenever S does not contain a proper subset which is itself a
2-path closure absorbing pointwise non-dominating set. We denote by ρ+2pnd(G) the maximum
cardinality of a minimal 2-path closure absorbing pointwise non-dominating set in G. A minimal
2-path closure absorbing pointwise non-dominating set S is called a ρ+2pnd-set if |S| = ρ+2pnd(G).

Theorem 4. [26] Let G and H be any two graphs. A set S ⊆ V (G + H) is a geodetic 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, such that

i) SG is a 2-path closure absorbing set in G whenever ⟨SH⟩ is a complete subgraph of H and

(ii) SH is a 2-path closure absorbing set in H whenever ⟨SG⟩ is a complete subgraph of G.

Proposition 4. Let G and H be connected graphs, and let S ⊆ G + H. If S is a minimal
geodetic hop dominating set of G+H, then S = SG ∪ SH , where SG ⊆ V (G) and SH ⊆ V (H)
are pointwise non-dominating sets of G and H, respectively, such that

(i) SG is a minimal 2-path closure absorbing pointwise non-dominating set in G whenever
⟨SH⟩ is a complete subgraph of H;



D. Catian, I. S. Aniversario, F. P. Jamil / Eur. J. Pure Appl. Math, 17 (3) (2024), 1737-1750 1744

(ii) SH is a minimal 2-path closure absorbing pointwise non-dominating set in H whenever
⟨SG⟩ is a complete subgraph of G.

Proof. Let S be a minimal geodetic hop dominating set of G + H. By Theorem 4,
S = SG ∪ SH , where SG and SH are pointwise non-dominating sets in G and H,
respectively. Suppose that ⟨SH⟩ is a complete subgraph of H. By Theorem 4, SG is a 2-
path closure absorbing set in G. Let S∗

G ⊆ V (G) be a 2-path closure absorbing set in G with
S∗
G ⊆ SG. By Theorem 4, S∗

G∪SH is a geodetic hop dominating set of G+H. Since S∗
G∪SH ⊆ S,

the minimality of S implies that SG = S∗
G. Thus, SG is a minimal 2-path closure absorbing

pointwise non-dominating set in G and (i) holds. Similarly, if ⟨SG⟩ is complete, then (ii) holds.

Lemma 1. [1] Let G be a connected noncomplete graph and S ⊆ V (G). If S is a 2-path closure
absorbing set in G, then ⟨S⟩ is not complete.

Proposition 5. Let G and H be connected graphs, and let S ⊆ G + H. Suppose that
S = SG ∪ SH , where SG ⊆ V (G) and SH ⊆ V (H) are pointwise non-dominating sets of G
and H, respectively, such that one of the following holds:

(i) ⟨SH⟩ is a complete subgraph of H and SG is a minimal 2-path closure absorbing pointwise
non-dominating set in G.

(ii) ⟨SG⟩ is a complete subgraph of G and SH is a minimal 2-path closure absorbing pointwise
non-dominating set in H.

Then S is a minimal geodetic hop dominating set of G+H.

Proof. Assume that SG and SH are pointwise non-dominating sets of G and H,
respectively. Then S is a hop dominating set in G+H. Assume further that (i) holds for S. Let
x ∈ V (G + H) \ S. Suppose that x ∈ V (G) \ S. Since SG is a 2-path closure absorbing
set in G, there exists u, v ∈ SG for which dG(u, v) = 2 and x ∈ IG(u, v). Observe that
dG+H(u, v) = dG(u, v) = 2 and IG(u, v) ⊆ IG+H(u, v). Suppose that x ∈ V (H) \ S. By
Lemma 1, ⟨SG⟩ is noncomplete. Thus, there exists u, v ∈ SG such that dG(u, v) = 2. Then
dG+H(u, v) = 2 and x ∈ IG+H(u, v). Since x is arbitrary, S is a geodetic set in G+H. Therefore,
S is a geodetic hop dominating set in G+H.

Now, let T ⊆ V (G + H) be a geodetic hop dominating set in G + H with T ⊆ S. Write
T = TG ∪ TH , where TG = T ∩ V (G) and TH = T ∩ V (H). Then TG and TH are pointwise
non-dominating sets in G and H, respectively, by Theorem 4. Note that TG ⊆ SG and TH ⊆ SH .
Since ⟨SH⟩ is complete, ⟨TH⟩ is a complete subgraph of H. If SH \ TH ̸= ∅ and x ∈ SH \ TH ,
then xy ∈ E(G + H) for all y ∈ TH , a contradiction since TH is a pointwise non-dominating
set in H. Thus, SH = TH . By Theorem 4, TG is a 2-path closure absorbing set in G. Since
TG ⊆ SG, the minimality of SG implies that SG = TG. Therefore, S = T and S is a minimal
geodetic hop dominating set in G+H.

Similarly, if condition (ii) holds, then S is a minimal geodetic hop dominating set in G+H.

Corollary 4. Let G be a nontrivial connected graphs and p ≥ 1, and let S ⊆ V (G+Kp). Then
S is a minimal geodetic hop dominating set of G + Kp if and only if S = V (Kp) ∪ SG where
SG ⊆ V (G) is a minimal 2-path closure absorbing pointwise non-dominating set of G. More
precisely,

γ+hg(G+Kp) = p+ ρ+2pnd(G).



D. Catian, I. S. Aniversario, F. P. Jamil / Eur. J. Pure Appl. Math, 17 (3) (2024), 1737-1750 1745

Proof. If G is complete, then G + H is complete, and the assertion is obvious. Suppose
that G is noncomplete. Put H = Kp, and let SG = S ∩ V (G) and SH = S ∩ V (H) = V (Kp).
Assume S is a minimal geodetic hop dominating set in G + Kp. Note that ⟨SH⟩ is complete
and SH = V (Kp) is a pointwise non-dominating set in H. Thus, SG is a minimal 2-path closure
absorbing pointwise non-dominating set in G by Proposition 4.

Conversely, suppose that SG is a minimal 2-path closure absorbing pointwise non-dominating
set in G. By Proposition 5, S = SG ∪ SH is a minimal geodetic hop dominating set in G+H.

Proposition 6. Let G and H be connected noncomplete graphs. Then

γ+hg(G+H) ≥ max{ρ+2pnd(G) + ω(H), ρ+2pnd(H) + ω(G)}.

Proof. Let SG ⊆ V (G) be a ρ+2pnd-set of G and SH ⊆ V (H) be ω-set of H. Because H is
noncomplete, V (H) ̸= SH . Let x ∈ V (H) \SH . Since ⟨SH ∪{x}⟩ is a not complete, there exists
y ∈ SH for which xy /∈ E(H). Since x is arbitrary, SH is a pointwise non-dominating set in H.
By Proposition 4, S = SG ∪ SH is a minimal geodetic hop dominating set in G+H. Thus,

γ+hg(G+H) ≥ |S| = ρ+hg(G) + ω(H).

Similarly, γ+hg(G+H) ≥ |S| = ρ+hg(H) + ω(G).

Since γ+hg(P3 + C3) = 5 = ρ+2pnd(P3) + ω(P3), the lower bound in Proposition 6 is sharp.

5. In the corona of graphs

Theorem 5. [27] Let G and H be any two graphs. A set S ⊆ V (G ◦ H) is a geodetic hop
dominating set of G ◦H if and only if

S = A ∪
(
∪v∈V (G)Sv

)
,

where A ⊆ V (G) and Sv ⊆ V (Hv) for each v ∈ V (G), and satisfies the following conditions:

(i) Sv is a pointwise non-dominating set in Hv for each v ∈ V (G) \NG(A);

(ii) For each w ∈ V (G) \A, one of the following holds:

(a) ∃a, b ∈ Sw with dHw(a, b) ̸= 1;

(b) ∃x, y ∈ V (G) with w ∈ IG(x, y);

(c) ∃s ∈ Sw and t ∈ A.

(iii) Sv is a 2-path closure absorbing set in Hv for all v ∈ V (G).

Observe that if G is a nontrivial connected graph, then condition (ii) in Theorem 5 may be
removed.

Corollary 5. Let G and H be two graphs, where G is connected and nontrivial. A set
S ⊆ V (G ◦H) is a geodetic hop dominating set of G ◦H if and only if

S = A ∪
(
∪v∈V (G)Sv

)
,

where A ⊆ V (G) and Sv ⊆ V (Hv) for each v ∈ V (G), and satisfies the following conditions:



D. Catian, I. S. Aniversario, F. P. Jamil / Eur. J. Pure Appl. Math, 17 (3) (2024), 1737-1750 1746

(i) Sv is a pointwise non-dominating set in Hv for each v ∈ V (G) \NG(A);

(ii) Sv is a 2-path closure absorbing set in Hv for all v ∈ V (G).

Proof. The necessity part follows from Theorem 5. Suppose that (i) and (ii) hold for S.
Following the same proof as one given in [9] for Theorem 5, S is a hop dominating set in G ◦H
and for every w ∈ V (Hw) \ Sw, there exist u, v ∈ S such that w ∈ IG+H(u, v). Now, suppose
that w ∈ V (G) \A. Since G is nontrivial and connected, NG(w) ̸= ∅, say z ∈ NG(w). By (ii),
Sz ̸= ∅ and Sw ̸= ∅. Pick u ∈ Sz and v ∈ Sw. Then w ∈ IG◦H(u, v).

Proposition 7. Let G and H be two graphs, where G is connected of order n ≥ 2. Then

γ+hg(G ◦H) ≥ n · ρ+2pnd(H),

and this bound is sharp.

Proof. For each v ∈ V (G), let Sv ⊆ V (Hv) be a ρ+2pnd-set of H
v. By Corollary 5, ∪v∈V (G)Sv

is a geodetic hop dominating set in G ◦H. Let T ⊆ V (G ◦H) be a geodetic hop dominating
set in G ◦ H with T ⊆ S. By Corollary 5 and since T ⊆ S, T = ∪v∈V (G)Tv, where Tv ⊆ Sv

is a 2-path closure absorbing pointwise non-dominating set in Hv for each v ∈ V (G). By the
minimality of Sv, Tv = Sv for each v ∈ V (G). Therefore, S = T and S is a minimal geodetic
hop dominating set in G ◦H. Thus,

γ+hg(G ◦H) ≥ |S| = n · ρ+2pnd(H).

Further, since γ+hg(P2 ◦ P2) = 4 = 2ρ+2pnd(P2), the given bound is sharp.

Proposition 8. Let G and H be any two graphs, where G is connected and nontrivial. If
S ⊆ V (G ◦H) is a minimal geodetic hop dominating set in G ◦H, then

S = A ∪
(
∪v∈V (G)Sv

)
,

where A ⊆ V (G) and Sv ⊆ V (Hv) for each v ∈ V (G), and satisfies the following conditions:

(i) Sv is a minimal 2-path closure absorbing pointwise non-dominating set in Hv for each
v ∈ V (G) \NG(A);

(ii) Sv is a minimal 2-path closure absorbing set in Hv for all v ∈ V (G) ∩NG(A).

Proof. In view of Corollary 5, we are left to work only on the minimality part. Let S be a
minimal geodetic hop dominating set in G ◦ H. Let v ∈ V (G) \ NG(A). Let
D ⊆ V (Hv) be a 2-path closure absorbing pointwise non-dominating set in Hv with D ⊆ Sv.
Then T = A∪

(
∪u∈V (G)\{v}Su

)
∪D is a geodetic hop dominating set of G◦H. Since T ⊆ S, the

minimality of S implies that T = S. Necessarily, Sv = D, showing that Sv is a minimal 2-path
closure absorbing pointwise non-dominating set in Hv, and (i) holds. Similarly, (ii) holds.

Corollary 6. Let G and H be two graphs, where G is connected of order n ≥ 2 and
ρ+2pnd(H) = ρ+2 (H). Then

γ+hg(G ◦H) = n · ρ+2pnd(H).

In particular, for p ≥ 1,
γ+hg(G ◦Kp) = np.



D. Catian, I. S. Aniversario, F. P. Jamil / Eur. J. Pure Appl. Math, 17 (3) (2024), 1737-1750 1747

Proof. Let S be a γ+hg-set of G◦H. By Proposition 8, S = A∪
(
∪v∈V (G)Sv

)
, where A ⊆ V (G)

and Sv ⊆ V (Hv) for each v ∈ V (G)such that Sv is a minimal 2-path closure absorbing pointwise
non-dominating set in Hv for each v ∈ V (G) \ NG(A) and Sv is a minimal 2-path closure
absorbing set in Hv for all v ∈ V (G) ∩ NG(A). Since S is a γ+hg-set, |Sv| = ρ+2pnd(H) for all
v ∈ V (G). Consequently, the minimality of S implies that A = ∅. Thus,

γ+hg(G ◦H) = |S| = |V (G)| · ρ+2pnd(H).

6. In the lexicographic product of graphs

Theorem 6. [27] Let G and H be connected nontrivial graphs. A subset
C = ∪x∈S ({x} × Sx) ⊆ V (G[H]), where S ⊆ V (G) and Sx ⊆ V (H), is a geodetic hop
dominating set in G[H] if and only if the following conditions hold:

(i) S is a geodetic hop dominating set in G;

(ii) Sx is a pointwise non-dominating set in H for each x ∈ S \N2
G(S);

(iii) Sx is a 2-path closure absorbing set in H for each x ∈ S \ IG(S).

The following follows from Theorem 6.

Corollary 7. Let G and H be connected nontrivial graphs. Let
C = ∪x∈S ({x} × Sx) ⊆ V (G[H]), where S ⊆ V (G) and Sx ⊆ V (H), such that the
following conditions hold:

(i) S is a minimal geodetic hop dominating set in G;

(ii) Sx is a minimal pointwise non-dominating set in H for each x ∈ S \N2
G(S);

(iii) Sx is a minimal 2-path closure absorbing set in H for each x ∈ S \ IG(S).

Then C is minimal geodetic hop dominating set in G[H].

Proof. By Theorem 6, C is a geodetic hop dominating set in G[H]. Let C∗ ⊆ C be a
geodetic hop dominating set in G[H]. By Theorem 6, there exist T ⊆ V (G) and Tx ⊆ V (H) for
each x ∈ T such that

C∗ = ∪x∈T ({x} × Tx) .

Moreover, T is a geodetic hop dominating set in G, Tx is a pointwise non-dominating set in H
for each x ∈ T \N2

G(T ), and Tx is a 2-path closure absorbing set in H for each x ∈ T \ IG(T ).
Since C∗ ⊆ C, T ⊆ S and Tx ⊆ Sx for each x ∈ T . The minimality of S implies that T = S.
Consequently, the minimality of Sx implies that Sx = Tx for all x ∈ T \N2

G(T ) and Sx = Tx for
all x ∈ T \ IG(T ). Hence, C = C∗, showing that C is a minimal geodetic hop dominating set in
G[H].

Corollary 8. For nontrivial connected graphs G and p ≥ 2,

(i) γ+hg(G[Kp]) = p · γ+hg(G).

(ii) γ+hg(Kp[G]) = p · ρ+2pnd(H).



REFERENCES 1748

Proof. Let S ⊆ V (G) be a γ+hg-set of G, and let D = V (Kp). Then C = ∪x∈S ({x} ×D) is
a minimal geodetic hop dominating set in G[Kp] by Corollary 7. Thus,

γ+hg(G[Kp]) ≥ |C| = |S| · |D| = p · γ+hg(G).

To get the other inequality, let C = ∪x∈S ({x} × Sx) ⊆ V (G[H]) be a γ+hg-set of G[Kp]. By
Theorem 6 and the minimality of C, S is a minimal geodetic hop dominating set in G and
Sx = V (Kp) for all x ∈ S \N2

G(S) and for all x ∈ S \ IG(S). Thus,

γ+hg(G[Kp]) = |C| =
∑
x∈S

|Sx| ≤
∑
x∈S

|V (Kp)| = p · |S| ≤ p · γ+hg(G).

This proves (i).

To prove (ii), let C = ∪x∈S ({x} × Tx) ⊆ V (Kp[G]) be a γ+hg-set of Kp[G]. By Theorem 6,
S = V (Kp) and Sx is a 2-path closure absorbing pointwise non-dominating set in H. Moreover,
by the minimality of C, Sx is a minimal 2-path closure absorbing pointwise non-dominating set
in H. Thus,

γ+hg(Kp[G]) = |C| =
∑

x∈V (G)

|Sx| ≤ p · ρ+2pnd(H).

To get the other inequality, D ⊆ V (H) be a ρ+2pnd-set of H. Then

C = ∪v∈V (Kp) ({x} ×D) = V (Kp)×D

is a minimal geodetic hop dominating set in Kp[G]. Therefore,

γ+hg(Kp[G]) ≥ |C| = p · ρ+2pnd(H).

Acknowledgements

The first author would like to thank the DOST-ASTHRD program of the Department of
Science and Technology, Philippines, for fully supporting his research activities for this project.
Moreover, the first author is thankful and

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