EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 3, Article Number 6536 ISSN 1307-5543 – ejpam.com Published by New York Business Global On Pairs of Disjoint Hop Dominating Sets in Graphs Viralou Abrille B. Besana1,2,∗, Ferdinand P. Jamil1,2, Sergio R. Canoy, Jr.1,2 1 Department of Mathematics and Statistics, College of Science and Mathematics, Mindanao State University - Iligan Institute of Technology, 9200 Iligan City, Philippines 2 Center for Mathematical and Theoretical Physical Sciences, Premier Research Institute of Science and Mathematics, Mindanao State University - Iligan Institute of Technology, 9200 Iligan City, Philippines Abstract. A set S of vertices of a graph G is a hop dominating set of G if for every v ∈ V (G) \ S, v is at distance 2 from a vertex in S. The minimum cardinality γh(G) of a hop dominating set is the hop domination number of G. Any hop dominating set of cardinality γh(G) is a γh-set. A pair (S, T ) of sets of vertices of G is a disjoint hop dominating pair if S ∩ T = ∅ and both S and T are hop dominating sets of G. In particular, if S is a γh-set, then T is an inverse hop dominating set of G. The minimum sum |S| + |T | among all pairs (S, T ) of disjoint hop dominating sets of G is the disjoint hop domination number, denoted by γhh(G). The minimum cardinality of an inverse hop dominating set of G is the inverse hop domination number of G, denoted by γ̃h(G). In this paper, we initiate the study of inverse hop domination and disjoint hop domination. Interestingly, for every pair of positive integers m and n with 2 ≤ m ≤ n, there exists a connected graph G for which γh(G) = m and γ̃h(G) = n. Also, for each positive integer n ≥ 4, there exists a connected graph G for which γh(G) + γ̃h(G) − γhh(G) = n. Here we investigate these new concepts for some specific graphs including the join, corona and lexicographic product of graphs. 2020 Mathematics Subject Classifications: 05C69 Key Words and Phrases: Hop domination, inverse hop domination, disjoint hop domination 1. Introduction All throughout this paper, we consider only graphs which are simple, finite and undirected. Given a graph G = (V (G), E(G)), we call V (G) the vertex set of G and E(G) its edge set. The cardinality |V (G)| of V (G) is the order of G. All terminologies used here which are not defined are adapted from [1]. Let G and H be disjoint graphs. The join G + H of G and H is the graph with vertex set V (G) ∪ V (H) and edge set E(G) ∪ E(H) ∪ {uv : u ∈ V (G), v ∈ V (H)}. The corona G ◦ H of G and H is the graph obtained by taking one copy of G and |V (G)| copies of H, ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i3.6536 Email addresses: viralouabrille.besana@g.msuiit.edu.ph (V.A Besana), ferdinand.jamil@g.msuiit.edu.ph (F. Jamil),sergio.canoy@g.msuiit.edu.ph (S. Canoy Jr.) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) V. A Besana, F. Jamil, S. Canoy Jr. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6536 2 of 17 and then joining each ith vertex of G to every vertex in the ith copy of H. In particular, we call G ◦ K1 the corona of G, and write cor(G) = G ◦ K1. The lexicographic product G[H] of G and H is the graph with V (G[H]) = V (G) × V (H) and (u, v)(u′, v′) ∈ E(G[H]) if and only if either uu′ ∈ E(G) or u = u′ and vv′ ∈ E(H). In any of these graphs, G and H are referred to as their basic component graphs. Vertices u and v of a graph G are neighbors if uv ∈ E(G). The open neighborhood of v refers to the set NG(v) consisting of all neighbors of v. The degree of v refers to the cardinality |NG(v)| of the open neighborhood of v. Vertex v is isolated if the degree of v is 0. The closed neighborhood of v is the set NG[v] = NG(v) ∪ {v}. Customarily, for S ⊆ V (G), NG(S) = ∪v∈SNG(v) and NG[S] = ∪v∈SNG[v]. A subset S ⊆ V (G) is a dominating set of G if NG[S] = V (G). In case NG(S) = V (G), then S is a total dominating set of G. The minimum cardinality γ(G) of a dominating set of G is the domination number of G, and the minimum cardinality γt(G) of a total dominating set is the total domination number of G. A dominating set of cardinality γ(G) is called a γ-set of G. Similarly, a γt-set is a total dominating set of cardinality γt(G). The reader is referred to [2–6] for the history, fundamental concepts and some of the recent developments in domination in graphs as well as its various applications. A set S ⊆ V (G) is a (1, 2)∗-dominating set (resp. (1, 2)∗-total dominating set) of G if it is a dominating (resp. total dominating) set of G and for each x ∈ V (G)\S there exists z ∈ S such that dG(x, z) = 2. The smallest cardinality of a (1, 2)∗-dominating (resp. (1, 2)∗-total dominating) set of G, denoted by γ∗ 1,2(G) (resp. γ∗t 1,2(G)), is the (1, 2)∗-domination number (resp. (1, 2)∗-total domination number) of G. Any (1, 2)∗-dominating (resp. (1, 2)∗-total dominating) set of G of cardinality γ∗ 1,2(G) (resp. γ∗t 1,2(G)) is a γ∗ 1,2-set (resp. γ∗t 1,2-set) of G. Both (1, 2)∗-domination and (1, 2)∗-total domination are introduced and studied in [7]. A set S ⊆ V (G) is a point-wise non-dominating set of G if for each v ∈ V (G) \ S, there exists u ∈ S such that v /∈ NG(u). The smallest cardinality of a point-wise non-dominating set of G, denoted by pnd(G), is called the point-wise non-domination number of G. A dominating set S which is also a point-wise non-dominating set of G is called a dominating point-wise non-dominating set of G. The smallest cardinality of a dominating point-wise non-dominating set of G will be denoted by γpnd(G). Any point-wise non- dominating (resp. dominating point-wise non-dominating) set S of G of cardinality |S| = pnd(G) (resp. |S| = γpnd(G)), is called a pnd-set (resp. γpnd-set) of G. Point-wise non-dominating sets and dominating point-wise non-dominating sets are discussed in [7]. Let G be a graph without isolated vertices. A subset S ⊆ V (G) is an inverse dominating set of G if V (G) \ S contains a γ-set of G. A minimum cardinality of an inverse dominating set of G is the inverse domination number of G, and is denoted by γ̃(G). Motivated by C. Berge[2], inverse domination of a graph was introduced by V.R. Kulli and S.C. Sigarkanti [8] in 1991, and studied further in [9–12]. It may be noted that P.G. Bhat and S.R. Bhat in [9] made mention of its application in an Information Retrieval System. For a graph G with no isolated vertex, any pair of subsets S and D of V (G) is called dd-pair if S and D are disjoint dominating sets of G. The symbol γγ(G) is the smallest sum |S| + |D| for all dd-pairs (S, D) of G. Since an inverse dominating set together with its associated γ-set constitute a dd-pair, γγ(G) ≤ γ(G) + γ̃(G). Disjoint dominating sets V. A Besana, F. Jamil, S. Canoy Jr. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6536 3 of 17 are studied extensively in [11–13]. A subset S ⊆ V (G) of a connected graph G is a hop dominating set (resp. total hop dominating set) of G if for each v ∈ V (G) \ S (resp. v ∈ V (G)), there exists u ∈ S for which dG(u, v) = 2. The minimum cardinality of a hop dominating set (resp. total hop dominating set) is called the hop domination number (resp. total hop domination number) of G, and is denoted by γh(G) (resp. γth(G)). Any hop dominating set (resp. total hop dominating set) of cardinality γh(G) (resp. γth(G)) is called γh-set (resp. γth-set) of G. Using the symbol HD(G) to denote the family of all hop dominating sets of G, more precisely, γh(G) = min{|S| : S ∈ HD(G)}. Hop domination was introduced by C. Natarajan C and S.K. Ayyaswamy [14] in 2015, and is investigated further in [7, 15–19]. For a vertex v of a connected graph G, NG(v, 2) = {u ∈ V (G) : dG(u, v) = 2}, and for S ⊆ V (G), NG(S, 2) = ∪v∈SNG(v, 2) and NG[S, 2] = NG(S, 2) ∪ S. Precisely, S is a hop dominating set (resp. total hop dominating set) if and only if NG[S, 2] = V (G) (resp. NG(S, 2) = V (G)). The relevance of hop domination is very well illustrated by the relatively well-known application cited in [20] which, for our purpose, can be rephrased as follows : A factory wants to set up a quality assurance team where some employees evaluate their co-workers. To keep costs low and evaluators anonymous, the number of evaluators is kept as small as possible and evaluators should not be direct friends or enemies of the workers they assess to avoid bias. A social network can be modelled by a graph G with vertices representing the workers where two workers are adjacent in G whenever they are either friends or enemies of each other. In this graph, evaluators are not connected to the people they evaluate, but instead are connected to the friends or enemies of those people. In hop domination, every worker is evaluated by someone who is two steps away in the social network. This method ensures privacy, fairness and efficient evaluation. The present study is motivated by the situation where the management considers the possibility that the quality assurance team might fail to deliver the desired output, and reserves another separate team (composed of evaluators who are not members of the first team) that can perform the same evaluation job. It deals mainly with the following two more likely approaches: • The second team will proceed only after the management found that the first team’s evaluation result is a failure; or • the second team will perform its task simultaneously with the first team. 2. Some existing results in hop domination The following existing results are useful in the present study. Proposition 1. [14] (i) For a complete graph Kn, γh(Kn) = n. (ii) For a complete bipartite graph Km,n, γh(Km,n) = 2. V. A Besana, F. Jamil, S. Canoy Jr. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6536 4 of 17 (iii) For a path Pn on n vertices, γh(Pn) =  2r, if n = 6r; 2r + 1, if n = 6r + 1; 2r + 2, if n = 6r + s; 2 ≤ s ≤ 5. (iv) For a cycle Cn of length n, γh(Cn) =  2r, if n = 6r; 2r + 1, if n = 6r + 1; 2r + 2, if n = 6r + s; 2 ≤ s ≤ 5. (v) For the Petersen graph P , γh(P ) = 2. Proposition 2. [7] Let G be a graph of order n. Then 1 ≤ pnd(G) ≤ n. Moreover, (i) pnd(G) = n if and only if G = Kn. (ii) pnd(G) = 1 if and only if G has an isolated vertex. (iii) pnd(G) = 2 if and only if G has no isolated vertex and there exist distinct vertices a and b of G such that NG(a) ∩ NG(b) = ∅. Theorem 1. [7] Let G and H be any two graphs. A set S ⊆ V (G+H) is a hop dominating set of G + H if and only if S = SG ∪ SH , where SG and SH are point-wise non-dominating sets of G and H, respectively. Theorem 2. [7] Let G and H be any two graphs. A set C ⊆ V (G◦H) is a hop dominating set of G ◦ H if and only if C = A ∪ ( ∪v∈V (G)∩NG(A)Sv ) ∪ ( ∪w∈V (G)\NG(A)Ew ) , where (i) A ⊆ V (G) such that for each w ∈ V (G) \ A, there exists x ∈ A with dG(x, w) = 2 or there exists y ∈ V (G) ∩ NG(w) with V (Hy) ∩ C ̸= ∅; (ii) Sv ⊆ V (Hv) for each v ∈ V (G) ∩ NG(A); and (iii) Ew ⊆ V (Hw) is a point-wise non-dominating set of Hw for each w ∈ V (G) \ NG(A). Theorem 3. [7]Let G be a nontrivial connected graph and let H be any graph. Then (i) γh(G ◦ H) ≤ min{γ∗t 1,2(G), [1 + pnd(H)]γ(G)}. (ii) γh(G ◦ H) = 2 if γ∗t 1,2(G) = 2. V. A Besana, F. Jamil, S. Canoy Jr. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6536 5 of 17 (iii) γh(G ◦ H) = 2 if γ(G) = 1 and H has an isolated vertex. Theorem 4. [7] Let G and H be non-trivial connected graphs. A subset C = ∪x∈S ({x} × Tx) of V (G[H]) is a hop dominating set of G[H] if and only if the following conditions hold: (i) S is a hop dominating set of G; (ii) Tx is a point-wise non-dominating set of H for each x ∈ S with |NG(x, 2) ∩ S| = 0. Corollary 1. [7] Let G and H be non-trivial connected graphs of orders m and n, respectively. Then (i) γh(G[H]) = ρH(G) if γ(G) = 1, where ρH(G) = min{|S ∩ NG(S, 2)| + pnd(H)|S \ NG(S, 2)| : S is a hop dominating set of G}; (ii) γh(G[H]) = γth(G) if γ(G) ̸= 1; and (iii) γh(G[H]) = m[pnd(H)] if G = Km. 3. Results By an ntc graph we mean a nontrivial connected graph. For vertices u and v of an ntc graph G, a u-v geodesic is any shortest path in G joining u and v. The length of a u-v geodesic is the distance between u and v, and is denoted by dG(u, v). The eccentricity of v refers to the quantity e(v) = max{dG(u, v) : v ∈ V (G)}. The diameter and radius of G are defined, respectively, as diam(G) = max{e(v) : v ∈ V (G)} and r(G) = min{e(v) : v ∈ V (G)}. Proposition 3. Let G be an ntc graph with r(G) ≥ 2. For each γh-set S ⊆ V (G), V (G)\S is a hop dominating set of G. Proof : Let S ⊆ V (G) be a γh-set of G. Suppose, in the contrary, that there exists u ∈ S for which dG(u, v) ̸= 2 for all v ∈ V (G) \ S. Since r(G) ≥ 2, there exists v ∈ V (G) such that dG(u, v) = 2. The previous statement implies that v ∈ S. Put S∗ = S \ {u}. Then S∗ is a hop dominating set of G, a contradiction since |S∗| < |S| = γh(G). ■ In what follows, G is the family of all ntc graphs G such that r(G) ≥ 2. 3.1. Inverse hop domination Let G ∈ G . A subset S ⊆ V (G) is an inverse hop dominating set provided S is a hop dominating set and V (G)\S contains a γh-set of G. The minimum cardinality of an inverse hop dominating set is called the inverse hop domination number of G, and is denoted by γ̃h(G). Clearly, for G ∈ G of order n, 2 ≤ γh(G) ≤ γ̃h(G) ≤ n − γh(G) ≤ n − 2. (1) V. A Besana, F. Jamil, S. Canoy Jr. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6536 6 of 17 Theorem 5. Let G ∈ G of order n. Then (i) γ̃h(G) = 2 if and only if γh(G) = 2 and G has two disjoint γh-sets. (ii) γ̃h(G) = n − 2 if and only if γh(G) = 2 and for every γh-set {u, v} of G, dG(x, y) ̸= 2 for all x, y ∈ V (G) \ {u, v}. Proof : For (i), from Inequality 1, if γ̃h(G) = 2, then γh(G) = 2 and the conclusion follows. The converse is clear. Suppose that γ̃h(G) = n − 2. Then Inequality 1 implies that γh(G) = 2. Let {u, v} be a γh-set of G. Then S = V (G) \ {u, v} is a γ̃h-set of G. Let x, y ∈ S with dG(x, y) = 2. First, if u, v /∈ NG(x, 2), then S \ {x} is a hop dominating set of G, a contradiction. Next, if u, v ∈ NG(x, 2), then S \ {y} is a hop dominating set of G, a contradiction. Now assume that u ∈ NG(x, 2) and v /∈ NG(x, 2), and let [x, z, u] be a x-u geodesic in G. Suppose that z ̸= v. Then dG(z, v) = 2 and S\{y} is a hop dominating set of G, a contradiction. Suppose that z = v. If [x, v, y] is a x-y geodesic in G, then S \ {y} is a hop dominating set of G, a contradiction. Suppose not, and let [x, w, y] be a geodesic in G. If wv ∈ E(G), then S \{w} is a hop dominating set of G. If wv /∈ E(G), then S \ {y} is a hop dominating set of G, a contradiction. The above contradictions imply that dG(x, y) ̸= 2 for all x, y ∈ V (G)\{u, v}. Conversely, suppose that γh(G) = 2, and let {u, v} be a γh-set of G. If S = V (G)\{u, v} is not a γ̃h-set of G, then there exists x ∈ S such that S \ {x} is a hop dominating set of G. This means that, in particular, there exists y ∈ S \ {x} such that dG(x, y) = 2, contrary to the hypothesis. ■ Observe that for graph G1 in Figure 1, {u, v} in particular, is a γh-set and x, z ∈ V (G1) \ {u, v} with dG(x, z) = 2. By Theorem 5(ii), γ̃h(G1) < 3. Since {x, y} is a γh-set, γ̃h(G1) = 2 as also affirmed by statement (i). For G2 in Figure 1, {u, v} and {v, w} are the only γh-sets of G2. Both γh-sets satisfy the conditions in Theorem 5(ii). Thus, γ̃h(G2) = n − 2 = 6 − 2 = 4. ....................................................................................................................................... ....................................................................................................................................... ....................................................................................................................................... .................................... ........... .......... .......... .......... .......... .......... .......... .......... ....... .................................... ........................................................................................ .................................... ....................................• • • • • u vx y z G1 : ....................................................................................................................................... ....................................................................................................................................... ....................................................................................................................................... ....................................................................................................................................... .................................... ........... .......... .......... .......... .......... .......... .......... .......... ....... .................................... ........................................................................................ .................................... ....................................• • • • • • u v w G2 : Figure 1: Examples of graphs described in Theorem 5 In view of Proposition 1, the following observations hold. Observation 1. (i) For a complete multipartite graph G = Kr1,r2,...,rn with 2 ≤ r1 ≤ r2 ≤ · · · ≤ rn, γ̃h(G) = n. (ii) For a path Pn on n ≥ 4 vertices, γ̃h(Pn) = { 2r + 3, if n = 6r + 5; 2r + 2, if n = 6r + s; 0 ≤ s ≤ 4. V. A Besana, F. Jamil, S. Canoy Jr. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6536 7 of 17 (iii) For a cycle Cn of length n ≥ 4, γ̃h(Cn) =  r, if n = 3r; 2r + 1, if n = 6r + 1; 2r + 2, if n = 6r + s, s = 2, 4, 5. (iv) For the Petersen graph P , γ̃h(P ) = 2. Theorem 6. For every pair of positive integers m and n with 2 ≤ m ≤ n, there exists G ∈ G for which γh(G) = m and γ̃h(G) = n. Proof : If m = n, then we take G = Kr1,r2,...,rn with 2 ≤ r1 ≤ r2 ≤ · · · ≤ rn. Assume that m < n, and write n = m + k, where k ≥ 1. If m = 2, then take the graph G = (K1 ∪ Kk+1) + K2 (see graph G1 in Figure 2 when k = 2). If V (K1) = {v} and V (K2) = {y1, y2}, then {v, y1} and V (Kk+1) ∪ {y2} are, respectively, a γh-set and a γ̃h-set of G. Suppose that m ≥ 3. Then we take the graph G = (K1 ∪ Kk+1) + Kr1,r2,...,rm−1 , where r1 = r2 = · · · = rm−1 = 2 (see graph G2 in Figure 2 for m = 3 and k = 2). Put V (K1) = {v} and let Urj = {y1 rj , y2 rj } (j = 1, 2, . . . , m−1) be the partite sets of Kr1,r2,...,rm−1 . Then {v, y1 rj : j = 1, 2, . . . , m−1} and V (Kk+1)∪{y2 rj : j = 1, 2, . . . , m−1} are, respectively, a γh-set and a γ̃h-set of G. In any case, γh(G) = m and γ̃h(G) = (m − 1) + (k + 1) = n.■ .................................... .................................... .................................... .................................... .................................... .................................... • • • • • • ......... ........ ........ ........ ........ ........ ........ ........ ........ ... .................................... ................................................... .................................... ............ ........... ........... ........... ...... . ................................... .................................... ......................................... ........................................ ........................................ ........................................ ........................................ ........ ...................... ..................... ..................... ..................... ..................... ..................... ..................... ..................... ..................... ..................... .......... .......................... ......................... ......................... ......................... ......................... ......................... ......................... ......................... ......................... ......................... ........... ................... .................. .................. .................. .................. .................. .................. .................. .................. .................. .................. .................. .................. .................. .................. ........ ......................................................................................................................................................................................................................................................................................................... ................................................ ................................................ ................................................ ................................................ ....................................................................................................................................................................................................................................................................... ........................................................................................................................................................................................................................................................ G1 = (K1 ∪ K3) + K2 v x1 x2 x3 y1 y2 .................................... .................................... .................................... .................................... .................................... .................................... .................................... .................................... • • • • • • • • ......... ........ ........ ........ ........ ........ ........ ........ ........ ... .................................... ................................................... .................................... ............ ........... ........... ........... ...... . ................................... .................................... ......................................... ........................................ ........................................ ........................................ ........................................ ........ ...................... ..................... ..................... ..................... ..................... ..................... ..................... ..................... ..................... ..................... .......... .......................... ......................... ......................... ......................... ......................... ......................... ......................... ......................... ......................... ......................... ........... ................... .................. .................. .................. .................. .................. .................. .................. .................. .................. .................. .................. .................. .................. .................. ........ ......................................................................................................................................................................................................................................................................................................... ................................................ ................................................ ................................................ ................................................ ....................................................................................................................................................................................................................................................................... ........................................................................................................................................................................................................................................................ ...................................................................................................................................................................................................................................................................................................... .............................................................................................................................................................................................................................................................................................................. ...................................................................................................................................................................................................................................................................... ....................................................................................................................................................................................................................................................................................... ........................................................................................................................................................................................................................................................ ............................................................................................................................................................................................................................................................................................................. ................................................ ................................................ ................................................ ................................................ ................................................................................................................................................................................................................................................................... ............... ................ ................ ................. .................. .................... ...................... ......................... .............................. ............................................ ................................................................................................................................................................................................................................................................................................................................. ................... .................... ..................... ....................... .......................... ............................... ........................................ ............................................................................................................................................................................................................................................................................................................................................................................ ................ ................. .................. .................... ...................... ......................... ................................ ................................................................................................................................................................................................................................................................................................................................................................................................................... ............... ................ ................ ................. .................. ................... ..................... ....................... .......................... ................................. .................................................... ............................................................................................................................................................................................................................................................................ G2 = (K1 ∪ K3) + K2,2 v x1 x2 x3 y1 y2 z1 z2 Figure 2: Examples of graphs described in the proof of Theorem 6 Corollary 2. The difference γ̃h(G) − γh(G) can be made arbitrarily large. 3.2. Disjoint hop domination For G ∈ G , Proposition 3 guarantees the existence in G of hop dominating sets A and B with A ∩ B = ∅. Denote by PHD(G) the family of all pairs (A, B) where A and B are disjoint hop dominating sets of G. We define γhh(G) = min{|A| + |B| : (A, B) ∈ PHD(G)}. Any pair (A, B) ∈ PHD(G) with |A| + |B| = γhh(G) is called γhh-pair of G. V. A Besana, F. Jamil, S. Canoy Jr. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6536 8 of 17 It should be noted that for (A, B) ∈ PHD(G), any of A and B need not be a γh-set of G. For all G ∈ G of order n, 2γh(G) ≤ γhh(G) ≤ γh(G) + γ̃h(G) ≤ n. (2) If G is any of the graphs G1 and G2 in Figure 2, then γhh(G) = γh(G) + γ̃h(G) = |V (G)|. Consider the graph G in Figure 3, the sets {a1, b1, c1} and {a2, a3, b2, b3, c2, c3} are a γh-set and a γ̃h-set, respectively, of G. While the sets {a1, a2, b3, c3} and {b1, c1, b2, a3} constitute a γhh-pair of G. For this G, 2γh(G) < γhh(G) < γh(G) + γ̃h(G). ........... .......... .......... .......... .......... .......... .......... .......... .......... .......... .......... .......... ..... .................................... ......................................................................................................................................................... .................................................................................................................................................................. .................................... .................................... ............ ........... ......... .................................... ............ ........... ......... .................................... .................................... ................................ .................................... ................................ .................................... .................................... ......... ........ ........ ........ ........ ... .................................... ......... ........ ........ ........ ........ ... .................................... .................................... • •• • • • • • • • • • • a1 a2 a3 b1 b2 b3 c1 c2 c3 G: Figure 3: Graph G with 2γh(G) < γhh(G) < γh(G) + γ̃h(G) Proposition 4. For all G ∈ G , if γ̃h(G) ≤ 1 + γh(G), then γhh(G) = γh(G) + γ̃h(G), (3) but not conversely. In particular, (i) γhh(G) = 2γh(G) for any of the following graphs G: the complete multipartite graph, cycle Cn and the Petersen graph described in Proposition 1. (ii) For a path Pn on n ≥ 4 vertices, γhh(Pn) =  4r + 4, if n = 6r + s, 2 ≤ s ≤ 4; 4r + 2, if n = 6r; 4r + 3, if n = 6r + 1; 4r + 5, if n = 6r + 5. Proof : Equation 3 is clear if γ̃h(G) = γh(G). Assume γ̃h(G) = 1 + γh(G), and let (A, B) ∈ PHD(G). If |A|+|B| < 1+2γh(G), then |A| = |B| = γh(G). Consequently, γ̃h(G) = γh(G), a contradiction. Since (A, B) is arbitrary, γh(G)+ γ̃h(G) = 1+2γh(G) ≤ γhh(G). Equation 2 yields the desired equality. V. A Besana, F. Jamil, S. Canoy Jr. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6536 9 of 17 Revisit the graph G = (K1 ∪ Kk+1) + Kr1,r2,...,rm−1 , where r1 = r2 = · · · = rm−1 = 2, in Theorem 6. As shown, γhh(G) = 2m + k = γh(G) + γ̃h(G). However, if k ≥ 2, then γ̃h(G) > 1 + γh(G). The rest of the proof follows from Proposition 1 and Observation 1. ■ Proposition 5. For each positive integer n ≥ 4, there exists G ∈ G for which γh(G) + γ̃h(G) − γhh(G) = n. Proof : Let G be the graph given in Figure 4 which is obtained from the complete graph K4 (with vertices {u, v, w, z}) by adding to K4 three copies of the join K1 + C4 through the vertices u, v and w and then adding the join ⟨z⟩+Kn−2. Let V (Kn−2) = {x1, x2, . . . , xn−2} .......................... ......................... ......................... ......................... ......................... ......................... ........... .................................... ............................................................................................................................................................................................................................................................................................................................. .............................................................. ......................... ......................... ......................... ......................... ......................... ........... .................................... ......... ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ .. . ................................... .................................................................................................................................................................. .................................... .................................... • •• • • • • • •• • • • • • • .................................................................................................................................................................. ............................................................................................................ ................... .................. .............. .................................... ........................................................................ .................................... ................... .................. .............. .................................... .................................... ......... ........ ........ ........ ........ ........ ...... .............. ............. ............. ............. ............. ............. ............. .. ........................................................ ........................................................ .......... .......... .......... ....... ......... ........ ........ ........ ....................................b b1 b2 b3 b4 .................................... .................................... .................................... ..................................... . . u v w z x1 x2 x3 xn−2 ......... ......... .......... ........... ............. ................. ............................................................................................................................................................................................................................ ......... ......... ........... ................. ................................................................................................................................................................................ .......... ............... .................................................................................................................................................................... ............................................ ................. ......................................................................................................................... ............... ............ ............................................................................................................................................ ............. .. ............................................................................... .................. .. ........... .......... .......... .......... .......... .......... .......... .......... .......... .......... .......... .......... ..... ......... ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........................................................................................................... ........................................................................................................................................................................................... ............................................................................................................ ................... .................. .............. .................................... ........................................................................ .................................... ................... .................. .............. .................................... ....................................• • •• • •........... ........ ........ ........ ........ ........ .... .............. ............. ............. ............. ............. ............. ............. .. ........................................................ ........................................................ .......... .......... .......... ....... ......... ........ ........ ........ ....................................c c1 c2 c3 c4 ............................................................................................................ ................... .................. .............. .................................... ........................................................................ .................................... ................... .................. .............. .................................... ....................................• • •• • •........... ........ ........ ........ ........ ........ .... .............. ............. ............. ............. ............. ............. ............. .. ........................................................ ........................................................ .......... .......... .......... ....... ......... ........ ........ ........ ....................................a a1 a2 a3 a4 Figure 4: A graph G with γhh(G) < γh(G) + γ̃h(G) and let the copies of K1 +C4 be given by the vertices {a, a1, a2, a3, a4}, {b, b1, b2, b3, b4} and {c, c1, c2, c3, c4} with au, vb, wc ∈ E(G). Then {u, v, w, z} is a γh-set of G and {a, a1, a2} ∪ {b, b1, b2} ∪ {c, c1, c2} ∪ {x1, x2, . . . , xn−2} is a γ̃h-set of G. On the other hand, the sets {w, c, b3, b4, c3, c4} and {u, v, z, c1, c2} constitute a γhh-pair of G. Thus, γh(G) + γ̃h(G) − γhh(G) = 4 + (7 + n) − 11 = n. ■ Corollary 3. The quantity γh(G) + γ̃h(G) − γhh(G) can be made arbitrarily large. 3.3. In the join of graphs A proof similar to that of Proposition 3 establishes the following lemma. Lemma 1. Let G ∈ G . If S ⊆ V (G) is a pnd-set of G, then V (G)\S contains a point-wise non-dominating set of G. Lemma 1 makes sense to the following definition. Let G ∈ G . A subset S ⊆ V (G) is an inverse point-wise non-dominating set of G if there exists a pnd-set D of G for which S ∩ D = ∅. The minimum cardinality of an inverse point-wise non-dominating set of G is denoted by ipnd(G). Any inverse point-wise non-dominating set of G of cardinality ipnd(G) is called ipnd-set of G. V. A Besana, F. Jamil, S. Canoy Jr. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6536 10 of 17 Theorem 7. Let G, H ∈ G and S ⊆ V (G+H). Then S is an inverse hop dominating set of G+H if and only if S = SG ∪SH , where SG and SH are inverse point-wise non-dominating sets of G and H, respectively. Proof : Note first that G + H ∈ G . Assume that S is an inverse hop dominating set of G + H, and let D ⊆ V (G + H) be a γh-set of G + H such that S ∩ D = ∅. By Theorem 1, S = SG ∪ SH and D = DG ∪ DH , where SG and DG are point-wise non-dominating sets of G and SH and DH are point-wise non-dominating sets of H. Moreover, DG and DH are pnd-sets of G and H, respectively. Thus, SG and SH are inverse point-wise non-dominating sets of G and H, respectively. Conversely, suppose that S = SG ∪ SH , where SG ⊆ V (G) and SH ⊆ V (H) are inverse point-wise non-dominating sets of G and H, respectively. Then, there exist pnd-sets DG ⊆ V (G) and DH ⊆ V (H), such that SG ∩ DG = ∅ and SH ∩ DH = ∅. By Theorem 1, both S and D = DG ∪ DH are hop dominating sets of G + H. Using the same theorem, it is straightforward to show that D is a γh-set of G + H. Since S ∩ D = ∅, S is an inverse hop dominating set of G + H. ■ Corollary 4. For all G, H ∈ G , γ̃h(G + H) = ipnd(G) + ipnd(H). (4) Given G ∈ G , we use the symbol PPND(G) to denote the family of all pairs (A, B), where A, B ⊆ V (G) are disjoint point-wise non-dominating sets of G. By Lemma 1, PPND(G) ̸= ∅. We define ppnd(G) = min{|A| + |B| : (A, B) ∈ PPND(G)}. Any pair (A, B) ∈ PPND(G) for which |A| + |B| = ppnd(G) is called ppnd-pair of G. Theorem 8. Let G, H ∈ G , and let A, B ⊆ V (G + H). Then (A, B) ∈ PHD(G + H) if and only if A = AG ∪ AH and B = BG ∪ BH , where (AG, BG) ∈ PPND(G) and (AH , BH) ∈ PPND(H). Proof : Assume (A, B) ∈ PHD(G+H). By Theorem 1 and since A∩B = ∅, A = AG ∪AH and B = BG ∪ BH , where (AG, BG) ∈ PPND(G) and (AH , BH) ∈ PPND(H). Conversely, if A = AG ∪ AH and B = BG ∪ BH , where (AG, BG) ∈ PPND(G) and (AH , BH) ∈ PPND(H), then A and B are hop dominating sets of G + H by Theorem 1. Moreover, since A ∩ B = ∅, (A, B) ∈ PHD(G + H). ■ Corollary 5. For all G, H ∈ G , γhh(G + H) = ppnd(G) + ppnd(H). (5) V. A Besana, F. Jamil, S. Canoy Jr. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6536 11 of 17 3.4. In the corona of graphs Statements (ii) and (iii) of Theorem 3 assert that under some conditions, the value of γh(G ◦ H) is attainable by the value of γ∗t 1,2(G) or [1 + pnd(H)]γ(G). Moreover, It is shown in [7] that a strict inequality in statement (i) is also attainable. For a (1, 2)-total dominating set A of a graph G, we write Γ(A) = {v ∈ A : NG(v) \ A ̸= ∅}. For each v ∈ Γ(A), choose exactly one uv ∈ NG(v) \ A, and define A◦ = {uv : v ∈ Γ(A)}. Clearly, A ∩ A◦ = ∅. Proposition 6. Let G be an ntc graph of order n and let H be any graph. Then (i) γhh(G ◦ H) ≤ [1 + pnd(H)] γγ(G), and this bound is sharp. (ii) If γh(G ◦ H) = γ∗t 1,2(G), then γ̃h(G ◦ H) ≤ min{|A◦| + |A◦ ∩ NG(A◦)| + [n − |NG(A◦)|]pnd(H) : A is a γ∗t 1,2- set of G}, and equality is attained for star graphs G on n ≥ 3 vertices. (iii) If γh(G ◦ H) = [1 + pnd(H)]γ(G), then γ̃h(G ◦ H) ≤ [1 + pnd(H)]γ̃(G). In particular, if γ(G) = γ̃(G), then γh(G ◦ H) = γ̃h(G ◦ H). Proof : Let (A, B) be a γγ-pair of G. For each v ∈ A, let Sv ⊆ V (Hv) be a pnd-set of Hv. Similarly, for each v ∈ B, let Tv ⊆ V (Hv) be a pnd-set of Hv. Define S = A ∪ (∪v∈ASv) and T = B ∪ (∪v∈BTv). Let x ∈ V (G◦H)\S and let v ∈ V (G) for which x ∈ V (Hv +v). If x = v, then since A is a dominating set and v /∈ A, there exists u ∈ A such that uv ∈ E(G). Pick y ∈ Su. Then y ∈ S and dG◦H(x, y) = 2. On the other hand, if x ̸= v, then since Sv is a pnd-set of Hv and x ∈ V (Hv) \ Sv, there exists y ∈ Sv for which xy /∈ E(Hv). This means that y ∈ S and dG◦H(x, y) = 2. Accordingly, S is a hop dominating set of G ◦ H. Similarly, T is a hop dominating set of G ◦ H. Since S ∩ T = ∅, (S, T ) ∈ PHD(G ◦ H). Therefore, γγ(G ◦ H) ≤ |S| + |T | = [1 + pnd(H)] γγ(G). In particular, if H has an isolated vertex, then γhh(P4 ◦ H) = 4 = 2γγ(G). This proves (i). To prove (ii), let A ⊆ V (G ◦ H) be a γ∗t 1,2-set of G. Then A is a γh-set of G ◦ H. For each v ∈ A◦ ∩ NG(A◦), let Sv ⊆ V (Hv) be singleton. For each v ∈ V (G) \ NG(A◦), let Tv ⊆ V (Hv) be a pnd-set of Hv. Define C = A◦ ∪ ( ∪v∈A◦∩NG(A◦)Sv ) ∪ ( ∪v∈V (G)\NG(A◦)Tv ) . V. A Besana, F. Jamil, S. Canoy Jr. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6536 12 of 17 Clearly, C ∩ A = ∅. We claim that C is a hop dominating set of G ◦ H. Let v ∈ V (G) \ A◦. We consider the following cases: Case 1: v ∈ A Since A is a total dominating set of G, NG(v) ̸= ∅. Moreover, because v /∈ A◦, NG(v) ⊆ A. Pick w ∈ A ∩ NG(v). First, suppose that w ∈ Γ(A) and y = uw ∈ A◦. Since y /∈ NG(v), dG(v, y) = 2. Next, suppose that w /∈ Γ(A). Then w /∈ NG(A◦) and Tw is a pnd-set of Hw. Pick y ∈ Tw. Then y ∈ V (Hw) ∩ C. Case 2: v /∈ A Since A is a dominating set of G, there exists w ∈ A ∩ NG(v). Since v ∈ NG(w) \ A, w ∈ Γ(A) and there exists y = uw ∈ A◦. If vy /∈ E(G), then dG(v, y) = 2. Suppose that vy ∈ E(G). If y ∈ NG(A◦), then V (Hy) ∩ C = Sy ̸= ∅. If y /∈ NG(A◦), then V (Hy) ∩ C = Ty ̸= ∅. By Theorem 2, C is a (inverse) hop dominating set of G ◦ H. Thus, γ̃h(G ◦ H) ≤ |C| = |A◦| + |A◦ ∩ NG(A◦)| + [n − |NG(A◦)|]pnd(H). In particular, if G is the star graph K1,n−1 (n ≥ 3), then γh(G◦H) = 2 and γ̃h(G◦H) = 1+(n−1)pnd(H), for any graph H. Any γ∗t 1,2-set A contains the central vertex, |A◦| = 1 and A◦ ∩NG(A◦) = ∅. Thus, |A◦|+ |A◦ ∩NG(A◦)|+[n−|NG(A◦)|]pnd(H) = 1+(n−1)pnd(H). Finally, to prove (iii), let B ⊆ V (G) be γ̃-set of G and let A ⊆ V (G) be a γ-set of G for which A ∩ B = ∅. For each v ∈ A, let Sv ⊆ V (Hv) be a pnd-set of Hv. Similarly, for each v ∈ B, let Tv ⊆ V (Hv) be a pnd-set of Hv. Define S = A ∪ (∪v∈ASv) and T = B ∪ (∪v∈BTv). As shown in the proof of statement (i), (S, T ) ∈ PHD(G ◦ H). Moreover, since |S| = [1 + pnd(H)]γ(G), T is an inverse hop dominating set of G ◦ H. Therefore, γ̃h(G ◦ H) ≤ |T | = [1 + pnd(H)]γ̃(G). ■ The corona G ◦ H, where pnd(H) ≥ 2 and G is the graph in Figure 5, shows that strict inequality may be attained in Proposition 6(ii). Here A = {z, w} is the unique γh-set of G ◦ H and γ̃h(G ◦ H) = 2 + pnd(H). Now, choose A◦ = {y}. Then |A◦| + |A◦ ∩ NG(A◦)| + [4 − |NG(A◦)|]pnd(H) = 1 + 2pnd(H) > γ̃h(G ◦ H). .................................... .................................... .................................... .................................... .............................................................................................................................. ....................................................................................................................... .......................... ......................... ......................... ......................... ......................... ......... ........ ........ ........ ........ ........ ........ ........ ........ ... G x y z w • • Figure 5: Graph G for illustration of Proposition 6(ii) Let H be a graph with isolated vertex. For n ≥ 2, γ̃h(K1,n ◦ H) = n + 1 < 2n = [1 + pnd(H)]γ̃(K1,n). This means that inequality in Proposition 6(iii) is also attainable. V. A Besana, F. Jamil, S. Canoy Jr. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6536 13 of 17 3.5. In the lexicographic product of graphs A subset S ⊆ V (G) is a ρH-set of G if S is a hop dominating set of G with ρH(G) = |S ∩ NG(S, 2)| + pnd(H)|S \ NG(S, 2)|. Theorem 9. Let G and H be ntc graphs with γ(G) ̸= 1. Then γ̃h(G[H]) = γth(G). Proof : Since γ(G) ̸= 1, G admits a total hop dominating set. Let S ⊆ V (G) be a γth-set of G, and let u, v ∈ V (H) with u ̸= v. By Theorem 4, C1 = S × {u} and C2 = S × {v} are hop dominating sets of G[H]. Since |C1| = γth(G), C1 is a γh-set of G[H] by Corollary 1. Consequently, C2 is an inverse hop dominating set of G[H]. Thus, γth(G) = γh(G[H]) ≤ γ̃(G[H]) ≤ |C2| = γth(G). ■ Theorem 10. Let G and H be ntc graphs with γ(G) = 1 and H ∈ G . Let C = ∪x∈S ({x} × Tx) ⊆ V (G[H]) with Tx ̸= V (H) for x ∈ S. Then C is an inverse hop dominating set of G[H] if and only if each of the following holds: (i) S is a hop dominating set of G; (ii) Tx is a point-wise non-dominating set of H for all x ∈ S \ NG(S, 2); (iii) There exists a ρH-set S∗ of G such that for each x ∈ (S ∩ S∗) \ NG(S∗, 2), V (H) \ Tx admits a pnd-set of H. More particularly, for each x ∈ (S ∩ S∗)\(NG(S, 2) ∪ NG(S∗, 2)), Tx is an inverse point-wise non-dominating set of H. Proof : First, assume that C is an inverse hop dominating set of G[H]. By Theorem 4, both (i) and (ii) hold for S. Since C is an inverse hop dominating set, there exists a γh-set C∗ = ∪x∈S∗ ({x} × T ∗ x ) of G[H] for which C ⊆ V (G[H]) \ C∗. By Corollary 1, S∗ is a ρH -set of G and T ∗ x is a pnd-set of H for each x ∈ S∗ \ NG(S∗, 2). Let x ∈ (S ∩ S∗) \ NG(S∗, 2). Because C ∩ C∗ = ∅, T ∗ x ⊆ V (H) \ Tx. More particularly, if x ∈ (S ∩ S∗) \ (NG(S, 2) ∪ NG(S∗, 2)), then Tx is a point-wise non-dominating set of H. Further, since Tx ⊆ V (H) \ T ∗ x , Tx is an inverse point-wise non-dominating set of H. This proves (iii). Conversely, suppose that C satisfies all conditions (i), (ii) and (iii). Then, by Theorem 4, C is a hop dominating set of G[H]. We construct a γh-set C∗ = ∪x∈S∗ ({x} × T ∗ x ) for which C ⊆ V (G[H]) \ C∗ as follows: Let x ∈ S∗. Case 1: Suppose that x ∈ S. If x ∈ NG(S∗, 2), then we take T ∗ x = {y}, where y ∈ V (H)\Tx. If x /∈ NG(S∗, 2), then as provided by condition (iii), we take a pnd-set T ∗ x of H with which Tx ⊆ V (H) \ T ∗ x . Case 2: Suppose that x /∈ S. If x ∈ NG(S∗, 2), then choose T ∗ x = {y} for any y ∈ V (H). If x /∈ NG(S∗, 2), then we choose any pnd-set T ∗ x of H. V. A Besana, F. Jamil, S. Canoy Jr. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6536 14 of 17 Define C∗ = ∪x∈S∗ ({x} × T ∗ x ). Then C∗ is a hop dominating set of G[H] by Theorem 4. Moreover, C ∩ C∗ = ∅ and |C∗| = ∑ x∈S∗∩NG(S∗,2) |T ∗ x | + ∑ x∈S∗\NG(S∗,2) |T ∗ x | = |S∗ ∩ NG(S∗, 2)| + pnd(H)|S∗ \ NG(S∗, 2)| = ρH(G). Therefore, C is an inverse hop dominating set of G[H]. ■ Corollary 6. If G and H are ntc graphs with γ(G) = 1 and H ∈ G , then γ̃h(G[H]) ≤ min{|S ∩ NG(S, 2)| + ipnd(H)|S \ NG(S, 2)| : S ∈ HD(G)}. Proof : Put ρ̃H(G) = min{|S ∩ NG(S, 2)| + ipnd(H)|S \ NG(S, 2)| : S ∈ HD(G)}. Let S ⊆ V (G) be a ρH -set of G, y ∈ V (H) and A ⊆ V (H) an inverse point-wise non-dominating sets of H. Define C = ∪x∈S ({x} × Tx), where Tx = {y} for all x ∈ S ∩ NG(S, 2) and Tx = A for all x ∈ S \ NG(S, 2). By Theorem 10, C is an inverse hop dominating set of G[H]. Thus, γ̃h(G[H]) ≤ |C| = |S ∩ NG(S, 2)| + ipnd(H)|S \ NG(S, 2)|. Since S is arbitrary, γ̃h(G[H]) ≤ ρ̃H(G). ■ Corollary 7. For all H ∈ G and m ≥ 2, γ̃h(Km[H]) = m · ipnd(H). Proof : Note first that S = V (Km) is the unique hop dominating set of Km and NKm(S, 2) = ∅. Thus, Corollary 6 yields γh(Km[H]) ≤ m · ipnd(H). Now, let C ⊆ V (Km) be a γ̃h-set of Km[H]. By Theorem 10, C = ∪x∈V (Km) ({x} × Tx), where Tx ⊆ V (H) is an inverse point-wise non-dominating set of H for each x ∈ V (Km). Thus, γ̃h(Km[H]) = ∑ x∈V (Km) |Tx| ≥ m · ipnd(H). ■ Equality in Corollary 6 can be attained even with a noncomplete G. Consider, for example, G = P3 = [x1, x2, x3]. Then G has only three distinct hop dominating sets, namely S1 = {x1, x2}, S2 = {x2, x3} and S3 = V (G). In view of Proposition 2, for any graph H ∈ G , S3 is the unique ρH -set of G. Thus, γ̃h(G[H]) = ρ̃H(G) = 2 + ipnd(H). Proposition 7. Let G and H be ntc graphs with γ(G) ̸= 1. Then γhh(G[H]) = 2γth(G). V. A Besana, F. Jamil, S. Canoy Jr. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6536 15 of 17 Proof : Applying Corollary 1 and Theorem 9, we have 2γth(G) = 2γh(G[H]) ≤ γhh(G[H]) ≤ γh(G[H]) + γ̃h(G[H]) = 2γth(G). ■ The following follows immediately from Theorem 4. Theorem 11. Let G and H be ntc graphs with γ(G) = 1 and H ∈ G . Let C = ∪x∈S ({x} × Tx) , C∗ = ∪x∈S∗ ({x} × T ∗ x ) ⊆ V (G[H]) with Tx ̸= V (H) for x ∈ S and T ∗ x ̸= V (H) for all x ∈ S∗. Then (C, C∗) ∈ PHD(G[H]) if and only if each of the following holds: (i) Both S and S∗ satisfy the conditions (i) and (ii) of Theorem 4; and (ii) Tx ∩ T ∗ x = ∅ for all x ∈ S ∩ S∗. More particularly, (Tx, T ∗ x ) ∈ PPND(H) for all x ∈ (S ∩ S∗) \ (NG(S, 2) ∪ NG(S∗, 2)). Corollary 8. For all graphs H ∈ G and m ≥ 2, γhh(Km[H]) = m · ppnd(H). Proof : Put S = V (Km) and let C = ∪x∈S ({x} × Tx) , C∗ = ∪x∈S ({x} × T ∗ x ) ∈ V (Km[H]) such that (Tx, T ∗ x ) is a ppnd-pair of H for each x ∈ S. Then (C, C∗) ∈ PHD(Km[H]) by Theorem 4. Thus, γhh(Km[H]) ≤ |C| + |C∗| = ∑ x∈V (Km) (|Tx| + |T ∗ x |) = m · ppnd(H). Now let (C, C∗) be a γhh-pair of Km[H]. By Theorem 11(i), C = ∪x∈S ({x} × Tx) and C∗ = ∪x∈S∗ ({x} × T ∗ x ) for some hop dominating sets S and S∗ of Km with Tx a point-wise non-dominating sets of H for each x ∈ S \ NKm(S, 2) and T ∗ x a point-wise non-dominating set of H fo all x ∈ S∗ \ NKm(S∗, 2). Since V (Km) is the unique hop dominating set of Km, S = S∗ = V (Km) and NKm(S, 2) = NKm(S∗, 2) = ∅. Further, by Theorem 11(ii), (Tx, T ∗ x ) ∈ PPND(H). Thus, γhh(Km[H]) = |C| + |C∗| = ∑ x∈V (Km) (|Tx| + |T ∗ x |) ≥ m · ppnd(H). ■ Acknowledgements This research project is fully supported by the DOST-ASTHRDP, Philippines, and the Office of the Vice Chancellor for Research and Enterprise of the MSU-IIT, Philippines. The authors would like to thank the reviewers for reading and recommending invaluable suggestions to the improvement of the paper. V. A Besana, F. Jamil, S. Canoy Jr. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6536 16 of 17 References [1] F. Buckley and F. Harary. Distance in Graphs. Addison-Wesley, Redwood City, CA, 1990. [2] C. Berge. The theory of Graphs and its Applications. Wiley, New York, 1962. [3] E. Cockayne and S. Hedetniemi. Towards a theory of domination in graphs. Networks, 7(3):247–261, 1977. [4] T.W. Haynes, S.T. Hedetniemi, and P.J. Slater. Fundamentals of Domination in Graphs. Marcel Dekker, Inc., New York, 1998. [5] F.P. Jamil and H.N. Maglanque. Cost effective domination in the join, corona and composition of graphs. European Journal of Pure and Applied Mathematics, 12(3):978–998, 2019. [6] O. Ore. Theory of Graphs, volume 38 of American Mathematical Society Colloquium Publications. American Mathematical Society, Providence, RI, 1962. [7] S.R Canoy Jr., R.V. Mollejon, and J.G. Canoy. Hop dominating sets in graphs under binary operations. European Journal of Pure and Applied Mathematics, 12(4):1455–1463, 2019. [8] V.R. Kulli and S.C. Sigarkanti. Inverse domination in graphs. National Academy Science Letters, 14:473–475, 1991. [9] P.G. Bhat and S.R. Bhat. Inverse independence number of a graph. International Journal of Computer Applications, 42(5), 2012. [10] G.S. Domke, J.E. Dunbar, and L.R. Markus. The inverse domination number of a graph. Ars Combinatoria, 72:149–160, 2004. [11] E.M. Kiunisala and F.P. Jamil. Inverse domination numbers and disjoint domination numbers of graphs under some binary operations. Applied Mathematical Sciences, 8(107):5303–5315, 2014. [12] E.M. Kiunisala and F.P. Jamil. On pairs of disjoint dominating sets in a graph. International Journal of Mathematical Analysis, 10(13):623–637, 2016. [13] S.M. Hedetniemi, S.T. Hedetniemi, R.C. Laskar, L. Markus, and P.J. Slater. Disjoint dominating sets in graphs. In Proceedings of the International Conference on Discrete Mathematics, volume 7 of Ramanujan Mathematical Society Lecture Notes Series, pages 87–100. 2008. [14] C. Natarajan and S.K. Ayyaswamy. Hop domination in graphs – ii. Versita, 23(2):187–199, 2015. [15] M.A. Bonsocan and F.P. Jamil. Transversal hop domination in graphs. European Journal of Pure and Applied Mathematics, 16(1):192–206, 2023. [16] M.A. Henning, S. Pal, and D. Pradhan. Algorithm and hardness results on hop domination in graphs. Information Processing Letters, 153:105872, 2020. [17] M.A. Henning and N.J. Rad. On 2-step and hop dominating sets in graphs. Graphs and Combinatorics, 33:913–927, 2017. [18] C. Natarajan, S.K. Ayyaswamy, and G. Sathiamoorthy. A note on hop domination number of some special families of graphs. International Journal of Pure and Applied Mathematics, 119(12):14165–14171, 2018. V. A Besana, F. Jamil, S. Canoy Jr. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6536 17 of 17 [19] S. Shanmugavelan and C. Natarajan. On hop domination number of some generalized graph structures. Ural Mathematical Journal, 7(2):121–135, 2021. [20] W. Desormeaux, T. W. Haynes, and M. A. Henning. A note on non-dominating set partitions in graphs. Discussiones Mathematicae Graph Theory, 36:1043–1050, 2016.