EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 15, No. 4, 2022, 1783-1796 ISSN 1307-5543 – ejpam.com Published by New York Business Global Hop Independent Hop Domination in Graphs Javier A. Hassan 1, Sergio R. Canoy, Jr.1,∗ 1 Department of Mathematics and Statistics, College of Science and Mathematics, Center for Graph Theory, Premier Research Institute of Science and Mathematics, MSU-Iligan Institute of Technology, 9200 Iligan City, Philippines Abstract. Let G be an undirected graph with vertex and edge sets V (G) and E(G), respectively. A set S ⊆ V (G) is called a hop independent hop dominating set of G if S is both hop independent and hop dominating set of G. The minimum cardinality of hop independent hop dominating set of G, denoted by γhih(G), is called the hop independent hop domination number of G. In this paper, we show that the hop independent hop domination number of a graph G lies between the hop domination number and the hop independence number of graph G. We characterize these types of sets in the shadow graph, join, corona, and lexicographic product of two graphs. Moreover, either exact values or bounds of the hop independent hop domination numbers of these graphs are given. 2020 Mathematics Subject Classifications: 05C69 Key Words and Phrases: Hop domination, hop domination number, hop independent set, hop independence number, hop independent hop domination number 1. Introduction Since its introduction in 2015 (see [10]), hop domination has been one of the topics of research or investigation in the area. To date, a number of variants of hop domination have been introduced and studied. The newly defined variations and parameters are studied in many classes of graphs including those which result from some binary operations (see [1], [2], [4], [5], [6], [7], [9], and [11]). Recently, Hassan et al. [3] introduced the concept of hop independent set in a graph and defined the parameter called hop independence number of a graph. The hop independence number may be equal or less or greater than the independence number of graph. A result in [3] shows that the absolute difference between the independence number and the hop independence number of a graph can be made arbitrarily large. Moreover, it was pointed out that the maximum hop independent set in a graph G forms a hop dominating set, that is, the hop independence number of a graph G is at least equal to the hop domination ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v15i4.4577 Email addresses: javier.hassan@g.msuiit.edu.ph (J. Hassan), sergio.canoy@g.msuiit.edu.ph (S. Canoy) https://www.ejpam.com 1783 © 2022 EJPAM All rights reserved. J. Hassan, S. Canoy / Eur. J. Pure Appl. Math, 15 (4) (2022), 1783-1796 1784 number of G. This observation, the way the variation “independent domination” for the standard domination concept was introduced, and the possible future applications it may offer researchers in the field, are some of the motivations the authors have for introducing and studying hop independent hop domination in a graph. The newly defined concept and the corresponding parameter will be studied initially for some graphs including those that result from some binary operations. 2. Terminology and Notation Two vertices u, v of a graph G are adjacent, or neighbors, if uv is an edge of G. Moreover, an edge uv of G is incident to two vertices u, v of G. The set of neighbors of a vertex u in G, denoted by NG(u), is called the open neighborhood of u in G. The closed neighborhood of u in G is the set NG[u] = NG(u) ∪ {u}. If X ⊆ V (G), the open neighborhood of X in G is the set NG(X) = ⋃ u∈X NG(u). The closed neighborhood of X in G is the set NG[X] = NG(X) ∪X. Let G be a graph. A set D ⊆ V (G) is a dominating set of G if for every v ∈ V (G) \D, there exists u ∈ D such that uv ∈ E(G), that is, NG[D] = V (G). The domination number of G, denoted by γ(G), is the minimum cardinality of a dominating set of G. Any dominating set D of G with cardinality γ(G), is called a γ-set of G. Let G be a graph. A set D ⊆ V (G) is a total dominating set of G if for every v ∈ V (G), there exists u ∈ D such that uv ∈ E(G), that is, NG(D) = V (G). The total domination number of G, denoted by γt(G), is the minimum cardinality of a total dominating set of G. Any total dominating set with cardinality equal to γt(G) is called a γt-set. A vertex v in G is a hop neighbor of vertex u in G if dG(u, v) = 2. The set N2 G(u) = {v ∈ V (G) : dG(v, u) = 2} is called the open hop neighborhood of u. The closed hop neighborhood of u in G is given by N2 G[u] = N2 G(u) ∪ {u}. The open hop neighborhood of X ⊆ V (G) is the set N2 G(X) = ⋃ u∈X N2 G(u). The closed hop neighborhood of X in G is the set N2 G[X] = N2 G(X) ∪X. A set S ⊆ V (G) is a hop dominating set of G if N2 G[S] = V (G), that is, for every v ∈ V (G)\S, there exists u ∈ S such that dG(u, v) = 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 nonempty set S ⊆ V (G) is a hop independent set of G if dG(x, y) ̸= 2 for any two vertices x, y ∈ S. The hop independence number of G, denoted by αh(G), is the largest cardinality of a hop independent set of G. Any hop independent set of G with cardinality αh(G) is called an αh-set of G. A set S ⊆ V (G) is called a hop independent hop dominating set of G if S is both hop independent and hop dominating set of G. The minimum cardinality of a hop independent hop dominating set of G, denoted by γhih(G), is called the hop independent hop domination number of G. Any hop independent hop dominating set of G with cardinality γhih(G) is called a γhih-set of G. J. Hassan, S. Canoy / Eur. J. Pure Appl. Math, 15 (4) (2022), 1783-1796 1785 A set S ⊆ V (G) is a clique if the subgraph ⟨S⟩ induced by S is a complete graph. The maximum cardinality of a clique of G, denoted by ω(G), is called the clique number of G. A set C ⊆ 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. A set S ⊆ V (G) is a clique pointwise non-dominating set if S is both a clique and a pointwise non-dominating set of G. The smallest cardinality of a clique pointwise non- dominating set of G, denoted by cpnd(G), is called the clique pointwise non-domination number of G. Any clique pointwise non-dominating set of G with cardinality cpnd(G) is called a cpnd-set of G. A total dominating set D ⊆ V (G) is called a total dominating hop independent hop dominating set if it is a hop independent hop dominating set. The minimum cardinality of a total dominating hop independent hop dominating set of G is denoted by γhiht (G). Any total dominating hop independent hop dominating set D with cardinality γhiht (G) is referred to as a γhiht -set. The shadow graph S(G) of graph G is constructed by taking two copies of G, say G1 and G2, and then joining each vertex u ∈ G1 to the neighbors of its corresponding vertex u′ ∈ G2. 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. The lexicographic product G[H] is the graph with vertex set V (G[H]) = V (G) × V (H) and (v, a)(u, b) ∈ E(G[H]) if and only if either uv ∈ E(G) or u = v and ab ∈ E(H). Note that any non-empty set C ⊆ V (G)× V (H) can be written as C = ⋃ x∈S [{x}× Tx], where S ⊆ V (G) and Tx ⊆ V (H) for each x ∈ S. Specifically, Tx = {a ∈ V (H) : (x, a) ∈ C} for each x ∈ S. 3. Results Proposition 1. Let G be any graph on n vertices. Then γh(G) ≤ γhih(G) ≤ αh(G). Proof. Since every hop independent hop dominating set S is a hop dominating set, it follows that γh(G) ≤ γhih(G). Also, since every αh-set is a hop independent hop dominating set, the right inequality holds. Remark 1. The bounds given in Proposition 1 are tight. Moreover, strict inequalities can also be attained. J. Hassan, S. Canoy / Eur. J. Pure Appl. Math, 15 (4) (2022), 1783-1796 1786 To see this, consider G = C4. Then γh(G) = γhih(G) = αh(G) = 2. For the strict inequalities, let S1 = {c, e, g} (see Figure 1). Then S1 is a minimum hop dominating set of G. Since dG(c, g) = 2, it follows that S1 is not a hop independent set. Next, let S2 = {a, b, g, h}(see Figure 2). Then S2 is a minimum hop independent hop dominating set of G. However, S2 is not a maximum hop independent set of G. In fact, S3 = {a, b, h, i, j} (see Figure 3 ) is a maximum hop independent set of G. Consequently, γh(G) = 3 < 4 = γhih(G) < 5 = αh(G). G : c d b ea f g h i j Figure 1: A graph G with γh(G) < γhih(G) < αh(G) G : c d b ea f g h i j Figure 2: A graph G with γh(G) < γhih(G) < αh(G) G : c d b ea f g h i j Figure 3: A graph G with γh(G) < γhih(G) < αh(G) Theorem 1. [3] Let G be any graph on n vertices. Then S is a hop independent set of G if and only if every component of ⟨S⟩ is complete. Moreover, αh(G) = n if and only if every component of G is complete. Theorem 2. Let G be a graph. Then γh(G) = γhih(G) if and only if G has a γh-set S such that every component of ⟨S⟩ is complete. Proof. Suppose γh(G) = γhih(G) and let S be a γhih-set of G. Then S is a hop independent set of G. Hence, by Theorem 1, every component of ⟨S⟩ is complete. By assumption, S is a γh-set of G. Conversely, suppose G has a γh-set S such that every component of ⟨S⟩ is complete. Then by Theorem 1, S is a hop independent set. Since S is a hop dominating set, it follows that S is a hop independent hop dominating set. Hence, γhih(G) ≤ |S| = γh(G). By Proposition 1, γh(G) = γhih(G). The next result is immediate from Theorem 2. J. Hassan, S. Canoy / Eur. J. Pure Appl. Math, 15 (4) (2022), 1783-1796 1787 Corollary 1. Let n ≥ 3 be any positive integer. Then (i) γhih(Pn) = γh(Pn) and (ii) γhih(Cn) = γh(Cn). Theorem 3. Let G be any graph on n ≥ 2 vertices. Then 2 ≤ γhih(G) ≤ n. Moreover, the following statements hold. (i) γhih(G) = 2 if and only if γh(G) = 2. (ii) γhih(G) = n if and only if every component C of G is complete. Proof. Clearly, 2 ≤ γhih(G) ≤ n. (i) Suppose γhih(G) = 2. By Proposition 1, γh(G) ≤ γhih(G) = 2. Since γh(G) ≥ 2 for any graph of order n ≥ 2, it follows that γh(G) = 2. Conversely, suppose γh(G) = 2, say S = {a, b} is a γh-set. Suppose, on the contrary, that S is not a hop independent set, that is dG(a, b) = 2. Then there exists x ∈ V (G) such that x ∈ NG(a)∩NG(b). This implies that x /∈ N2 G(a)∪N2 G(b), contrary to the assumption that S is a hop dominating set. Therefore, S is a hop independent set and γhih(G) ≤ 2. Consequently, γhih(G) = 2. (ii) Suppose γhih(G) = n. Then S = V (G) = {v1, . . . , vn} is the only hop independent hop dominating set of G. Thus, by Theorem 1, every component of ⟨S⟩ = G is complete. For the converse, suppose that every component C of G is complete. Then by Proposition 1, n = γh(G) ≤ γhih(G) ≤ αh(G) = n. Hence, γhih(G) = n. The next result is a direct consequence of Theorem 3. Corollary 2. Let n be a positive integer. Then each of the following statements holds. (i) γhih(Kn) = n. (ii) γhih(Kn) = n. (iii) γhih(G) + γhih(G) = 2n if G = Kn. (iv) γhih(G) ≤ n − 1 if G is a graph on n vertices and has at least one non-complete component. Proposition 2. Let G be a graph on n ≥ 3 vertices. If G has at least one non-complete component, then (i) 4 ≤ γhih(G) + γhih(G) ≤ 2n− 1, and (ii) 4 ≤ γhih(G) · γhih(G) ≤ n2 − n. J. Hassan, S. Canoy / Eur. J. Pure Appl. Math, 15 (4) (2022), 1783-1796 1788 Proof. By Corollary 2(iv), γhih(G) ≤ n − 1 and by Theorem 3, γhih(G) ≤ n. These imply that γhih(G)+γhih(G) ≤ n−1+n = 2n−1 and γhih(G)·γhih(G) ≤ (n−1)n = n2−n. Since γhih(G) ≥ 2 for any graph of order at least 2, the left inequalities follow. Theorem 4. Let G be any graph of order n ≥ 3. Then γhih(G) = n− 1 if and only if all but one component H of G are cliques, where H = K|V (H)| \e (H is obtained from K|V (H)| by deleting an edge e). Proof. Suppose γhih(G) = n−1. Let S = V (G)\{v} be a γhih-set. Let G1, G2, . . . , Gk be the components of ⟨S⟩. Then each Gi is a clique. Since v /∈ S and S is a hop dominating set, there exists w ∈ S such that dG(v, w) = 2. Assume that w ∈ V (G1). Let z ∈ NG(v)∩NG(w). Then z ∈ V (G1). Suppose there exists j ̸= 1 such that vp ∈ E(G) for some p ∈ V (Gj). Then S∗ = V (G) \ {v, p} is a hop independent hop dominating set in G, contrary to the assumption that γhih(G) = n−1. This implies that the components ofG are ⟨V (G1)∪{v}⟩, G2, . . . , Gk. Now, we will show that xv ∈ E(G) for every x ∈ V (G1) \ {w}. Suppose there exists q ∈ V (G1) \ {w} such that qv /∈ E(G). Then dG(q, v) = 2. Let D = {t ∈ V (G1) : vt /∈ E(G)}. Then |D| ≥ 2 and ⟨V (G1) \ D⟩ is a clique. Notice that dG(v, t) = 2 for every t ∈ D. Now, let S′ = V (G) \ D. Then S′ is a hop dominating set. Since the components of ⟨S′⟩ are ⟨V (G1) \ D⟩, G2, . . . , Gk, it follows that S′ is a hop independent set. Hence, γhih(G) ≤ |S′| = n − |D| ≤ n − 2, a contradiction. Hence, xv ∈ E(G) for every x ∈ V (G1) \ {w}. This implies that H = ⟨V (G1)∪ {v}⟩ = KV (H) \ e, where e = vw. The converse is clear. The next result follows from Theorem 4. Corollary 3. Let G be a connected graph of order n ≥ 3. Then γhih(G) = n − 1 if and only if G = Kn \ e for some e ∈ E(Kn). Theorem 5. Let G be a non-trivial connected graph. Then S is a hop independent hop dominating set of S(G) if and only if one of the following conditions holds: (i) S is a hop independent hop dominating set in G1. (ii) S is a hop independent hop dominating set in G2. (iii) S = SG1 ∪ SG2 such that SG1 ∪ S′ G2 and S′ G1 ∪ SG2 are hop independent hop dominating sets in G1 and G2, respectively, where S ′ G2 = {a ∈ V (G1) : a ′ ∈ SG2} and S′ G1 = {a ∈ V (G2) : a ′ ∈ SG1}. Proof. Let S be a hop independent hop dominating set of S(G). Set SG1 = S ∩V (G1) and SG2 = S ∩ V (G2). If SG2 = ∅, then S = SG1 is a hop independent hop dominating set of G1. If SG1 = ∅, then S = SG2 is a hop independent hop dominating set of G2. Hence, (i) or (ii) holds. Next, suppose SG1 and SG2 are both non-empty. Let S′ G2 = {a ∈ V (G1) : a′ ∈ SG2}. Suppose SG1 ∪ S′ G2 is not a hop independent set in G1. Then there exist p, q ∈ SG1 ∪ S′ G2 such that dG1(p, q) = 2 = dS(G)(p, q). Since SG1 J. Hassan, S. Canoy / Eur. J. Pure Appl. Math, 15 (4) (2022), 1783-1796 1789 and SG2 are hop independent sets, we may assume that p ∈ SG1 and q ∈ S′ G2 . Then q′ ∈ SG2 and dS(G)(p, q ′) = 2, contrary to the assumption that S is a hop independent set. Thus, SG1 ∪ S′ G2 is a hop independent set. Next, let x ∈ V (G1) \ SG1 ∪ S′ G2 . Then x ∈ V (S(G)) \ S. Since S is a hop dominating of S(G), there exists y ∈ S such that dS(G)(x, y) = 2. If y ∈ SG1 , then we are done. Suppose y ∈ SG2 , say y = z′, where z ∈ V (G1). Then z ∈ S′ G2 and dS(G)(x, z) = dG1(x, z) = 2. Therefore, SG1 ∪ S′ G2 is a hop dominating set. Consequently, SG1 ∪ S′ G2 is a hop independent hop dominating set of G1. Similarly, S′ G1 ∪ SG2 is a hop independent hop dominating set of G2. Hence, (iii) holds. For the converse, suppose (i) holds. Then S is a hop independent set of S(G). Let a ∈ V (S(G)) \ S. If a ∈ V (G1) \ S, then there exists b ∈ S such that dG1(a, b) = dS(G)(a, b) = 2. Suppose a ∈ V (G2), say a = v′, where v ∈ V (G1). If v ∈ S, then dG1(a, v) = dS(G)(a, v) = 2. If v /∈ S, then there exists w ∈ S such that dG1(v, w) = 2. It follows that dS(G)(a,w) = dS(G)(v ′, w) = 2. Therefore, S is a hop independent hop dominating set of S(G). Similarly, if (ii) holds, then S is a hop independent hop dominating set of S(G). Now, suppose (iii) holds. Suppose further that S = SG1 ∪SG2 is not a hop independent set in S(G). Then there exist a, b ∈ S such that dS(G)(a, b) = 2. Since SG1 and SG2 are hop independent sets, we may assume that a ∈ SG1 and b ∈ SG2 . Then dG2(a ′, b) = 2, contrary to the assumption that S′ G1 ∪SG2 is a hop independent set in G2. Therefore, S is a hop independent set S(G). Next, let y ∈ V (S(G)) \ S. Then y /∈ SG1 ∪ SG2 . Suppose y ∈ V (G2) \ SG2 , say y = z′, where z ∈ V (G1). Then z /∈ S′ G2 . If z ∈ SG1 , then dS(G)(y, z) = dS(G)(z ′, z) = 2. Suppose z /∈ SG1 . Since SG1 ∪ S′ G2 is a hop dominating set of G1, there exists p ∈ SG1 ∪ S′ G2 such that dG1(p, z) = 2 = dS(G)(p, z). If p ∈ SG1 , then p ∈ S and dS(G)(p, z ′) = 2. If p ∈ S′ G2 , then p′ ∈ SG2 ⊆ S and dG2(p ′, z′) = dS(G)(p ′, z′) = 2. Therefore, S is a hop dominating set of S(G). Consequently, S is a hop independent hop dominating set of S(G). Corollary 4. Let G be a non-trivial connected graph. Then γhih(S(G)) = γhih(G). In particular, we have (i) γhih(S(Kn)) = γhih(Kn) = n for every n ≥ 2, (ii) γhih(S(Pn)) = γhih(Pn) = γh(Pn) for every n ≥ 2, and (iii) γhih(S(Cn)) = γhih(Cn) = γh(Cn) for evey n ≥ 3. Proof. Let S be a γhih-set of G. Then by Theorem 5, S is a hop independent hop dominating set of S(G). Thus, γhih(S(G)) ≤ |S| = γhih(G). J. Hassan, S. Canoy / Eur. J. Pure Appl. Math, 15 (4) (2022), 1783-1796 1790 On the other hand, suppose S∗ is a γhih-set of S(G). If S∗ is of type (i) or (ii), then S∗ is a hop independent hop dominating set of G by (i) and (ii) of Theorem 5. Hence, γhih(S(G)) = |S∗| ≥ γhih(G). Suppose S∗ is of type (iii), say S∗ = SG1 ∪ SG2 . Then S∗ G = SG1 ∪ S′ G2 is a hop independent hop dominating set of G1 by Theorem 5(iii). This implies that γhih(S(G)) = |S∗| = |S∗ G| ≥ γhih(G). Consequently, γhih(S(G)) = γhih(G). Statements (i), (ii) and (iii) follow from Corollary 1, Theorem 3 and Theorem 5. Lemma 1. [8] Let G be a graph of order n. Then every component of S(G) is a complete graph if and only if G = Kn. Theorem 6. Let G be a graph on n vertices. Then 2 ≤ γhih(S(G)) ≤ 2n. Moreover, γhih(S(G)) = 2n if and only if G = Kn. Proof. Since 2 ≤ |V (S(G))| ≤ 2n, it follows that 2 ≤ γhih(S(G)) ≤ 2n by Theorem 3. Suppose γhih(S(G)) = 2n. Then every component of S(G) is a complete graph by Theorem 3. Hence, G = Kn by Lemma 1. For the converse, suppose G = Kn. Then S(G) = K2n. Hence, γhih(S(G)) = 2n by Theorem 3. Theorem 7. Let G be a graph of order n. Then 1 ≤ cpnd(G) ≤ n. Moreover, (i) cpnd(G) = 1 if and only if G has an isolated vertex. (ii) cpnd(G) = 2 if and only if G has no isolated vertex and there exist adjacent vertices x and y of G such that NG(x) ∩NG(y) = ∅. (iii) cpnd(G) = n if and only if G is a complete graph. Proof. Celarly, 1 ≤ cpnd(G) ≤ n. (i) Suppose cpnd(G) = 1, say {p} is a cpnd-set of G. Clearly, p is an isolated vertex of G. Conversely, suppose G has an isolated vertex, say a. Then {a} is a cpnd-set of G. Hence, cpnd(G) = 1. (ii) Suppose cpnd(G) = 2, say C = {x, y} is a clique pointwise non-dominating set of G. Then x and y are adjacent. By (i), G has no isolated vertex. Now, suppose that NG(x) ∩ NG(y) ̸= ∅. Let a ∈ NG(x) ∩ NG(y). Then a ∈ V (G) \ C. Since a is adjacent to both x and y, it follows that C is not a pointwise non-dominating set, a contradiction. Hence, NG(x) ∩NG(y) = ∅. Conversely, suppose G has no isolated vertex and there exist adjacent vertices x and y of G such that NG(x) ∩NG(y) = ∅. Let C = {x, y} and a ∈ V (G) \C. By assumption, a is not adjacent to x or y. This implies that C is a clique pointwise non-dominating set, and so cpnd(G) ≤ 2. Since G has no isolated vertex, cpnd(G) ≥ 2. Consequently, cpnd(G) = 2. (iii) Suppose cpnd(G) = n. Then C = V (G) is the only clique pointwise non- dominating set of G. It follows that G is a complete graph. The converse is clear. The next result follows immediately from Theorem 7(ii). J. Hassan, S. Canoy / Eur. J. Pure Appl. Math, 15 (4) (2022), 1783-1796 1791 Corollary 5. Let n be any positive integer. Then (i) cpnd(Pn) = 2 for any n ≥ 2. (ii) cpnd(Cn) = 2 for any n ≥ 4. Theorem 8. [7] 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 9. [3] Let G and H be graphs. Then S is a non-empty hop independent set of G+H if and only if one of the following statements holds: (i) S is a clique in G. (ii) S is a clique in H (iii) S ∩ V (G) and S ∩ V (G) are cliques in G and H, respectively. Theorem 10. Let G and H be two graphs. A set S ⊆ V (G + H) is a hop independent hop dominating set of G + H if and only if S = SG ∪ SH , where SG and SH are clique pointwise non-dominating sets of G and H, respectively. Proof. Suppose S is a hop independent hop dominating set of G +H. Then SG and SH are both non-empty. Since S is a hop dominating set, SG and SH are pointwise non- dominating sets of G and H, respectively by Theorem 8. Since S is a hop independent set, SG and SH are cliques in G and H, respectively, by Theorem 9. Therefore, SG and SH are clique pointwise non-dominating sets of G and H, respectively. Conversely, suppose that S = SG ∪ SH , where SG and SH are clique pointwise non- dominating sets of G and H, respectively. Since SG and SH are pointwise non-dominating sets, S = SG ∪ SH is a hop dominating set of G+H by Theorem 8. Since SG and SH are cliques, it follows that S = SG ∪ SH is a hop independent set of G + H by Theorem 9. Consequently, S = SG ∪ SH is a hop independent hop dominating set of G+H. The next result follows from Theorem 7, Corollary 5 and Theorem 10. Corollary 6. Let G and H be graphs. Then γhih(G + H) = cpnd(G) + cpnd(H). In particular, we have (i) γhih(Kn +H) = n+ cpnd(H) for all n ≥ 1, (ii) γhih(G+H) = 2 if G and H contain isolated vertices, (iii) γhih(Wn) = γhih(K1 + Cn) = 3 for all n ≥ 4, (iv) γhih(Fn) = γhih(K1 + Pn) = 3 for all n ≥ 2, (v) γhih(Kn +Km) = n+m for all n,m ≥ 1, J. Hassan, S. Canoy / Eur. J. Pure Appl. Math, 15 (4) (2022), 1783-1796 1792 (vi) γhih(Pn + Pm) = 4 for all n,m ≥ 2, (vii) γhih(Cn + Cm) = 4 for all n,m ≥ 4, and (viii) γhih(K1,n) = γhih(K1 +Kn) = 2 for all n ≥ 1. Theorem 11. [3] Let G be a non-trivial connected graph and let H be any graph. Then S is a hop independent set in G ◦H if and only if S = A ∪ (∪v∈V (G)Sv) and satisfies the following conditions: (i) A is a hop independent set of G. (ii) Sv is empty or a clique in Hv for each v ∈ V (G) \NG(A). (iii) Sv = ∅ for each v ∈ NG(A). Theorem 12. [7] Let G and H be any two graphs. A set C ⊆ V (G) 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) and satisfies the following conditions: (i) A ⊆ V (G) such that for each w ∈ V (G) \A, there exists x ∈ A with dG(w, x) = 2 or there exists y ∈ V (G) ∩NG(w) with V (Hy) ∩ C ̸= ∅. (ii) Sv ⊆ V (Hv) for each v ∈ V (G) ∩NG(A). (iii) Ew ⊆ V (Hw) is a pointwise non-dominating set of Hw for each w ∈ V (G) \NG(A). Theorem 13. Let G be a non-trivial connected graph and let H be any graph. Then C is a hop independent hop dominating set of G ◦ H if and only if C = A ∪ (∪v∈V (G)Cv), where A ⊆ V (G), Cv ⊆ V (Hv) for each v ∈ V (G), and satisfies the following conditions: (i) A = ∅ or A is a hop independent set of G. (ii) Cy = ∅ for each y ∈ NG(A). (iii) For each v ∈ V (G)\N2 G[A], there exists w ∈ NG(v) such that Cw is a hop independent set of Hw. (iv) For each v ∈ V (G) \NG(A), Cv is a clique pointwise non-dominating set. Proof. Suppose C is a hop independent hop dominating set of G◦H. Let A = C∩V (G) and Cv = C ∩ V (Hv) for each v ∈ V (G). Suppose further that A ̸= ∅. Since C is a hop independent set, A is a hop independent set of G by Theorem 11(i). This shows that (i) holds. Clearly, (ii) holds by Theorem 11(iii). Next, let v ∈ V (G) \ N2 G[A]. Then v /∈ A and dG(u, v) ̸= 2 for every u ∈ A. Since C is a hop dominating set of G ◦H, there exists x ∈ C such that dG◦H(x, v) = 2. It follows that x ∈ Cw for some w ∈ NG(v). Since C is a hop independent set, it follows that Cw is a hop independent set of Hw, showing that (iii) holds. Property (iv) follows immediately from Theorem 11(ii) and Theorem 12(iii). J. Hassan, S. Canoy / Eur. J. Pure Appl. Math, 15 (4) (2022), 1783-1796 1793 For the converse, suppose that C has the given form and satisfies conditions (i), (ii), (iii) and (iv). Then by Theorem 11, C is a hop independent set of G ◦ H. Next, let a ∈ V (G ◦H) \ C and v ∈ V (G) such that a ∈ V (v +Hv). Consider the following cases: Case 1: a = v. Then a ∈ V (G)\A. If a ∈ N2 G(A), then dG(a, b) = dG◦H(a, b) = 2 for some b ∈ A ⊆ C. If a /∈ N2 G(A), then there exists w ∈ NG(a) such that Cw is a hop independent set by (iii). Choose any p ∈ Cw. Then p ∈ C and dG◦H(a, p) = 2. Case 2: a ̸= v. Then a ∈ V (Hv) \ Cv. If v ∈ NG(A), say cv ∈ E(G ◦ H) for some c ∈ A ⊆ C, then dG(c, a) = 2. Suppose v /∈ NG(A). By (iv), there exists q ∈ Cv ⊆ C such that dG◦H(q, a) = 2. Thus, C is a hop dominating set. Therefore, C is a hop independent hop dominating set in G ◦H. Consider the following families of graphs: G1 = {G : G admits a dominating hop independent hop dominating set} and G2 = {G : G admits a total dominating hop independent hop dominating set}. Let G be a graph. Denote by H = {A : A is a hop independent hop dominating set of G}, H1 G = {B : B is a dominating hop independent hop dominating set of G} ifG ∈ G1, and H2 G = {T : T is a total dominating hop independent hop dominating set of G} ifG ∈ G2. Corollary 7. Let G be a non-trivial connected graph and let H be any graph. Then γhih(G ◦H) ≤ min{|A|+ |V (G) \NG(A)|cpnd(H) : A ∈ H}. In particular, we have (i) γhih(G ◦H) ≤ min{|B|+ |B \NG(B)|cpnd(H) : B ∈ H1 G} if G ∈ G1, and (ii) γhih(G ◦H) ≤ γhiht (G) if G ∈ G2 G. Proof. Let A ∈ H. For each v ∈ V (G) \ NG(A), let Cv be a cpnd-set of Hv. Then C = A∪(∪v∈V (G)\NG(A)Cv) is a hop independent hop dominating set of G◦H by Theorem 13. Thus, γhih(G ◦H) ≤ |C| = |A|+ ∑ v∈V (G)\NG(A) |Cv| = |A|+ |V (G) \NG(A)|cpnd(H). Since A is arbitrary, it follows that γhih(G ◦H) ≤ min{|A|+ |V (G) \NG(A)|cpnd(H) : A ∈ H}. J. Hassan, S. Canoy / Eur. J. Pure Appl. Math, 15 (4) (2022), 1783-1796 1794 Let G ∈ G1 and let B ∈ H1 G. Since B is a dominating set, it follows that V (G) \NG(B) = B \NG(B). Hence, (i) follows from the first inequality. Next, let G ∈ G2 and let T ∈ H2 G. Since T is a total dominating set of G, it follows that V (G) = NG(T ). Thus, γhih(G ◦H) ≤ min{|T | : T ∈ H2 G} = γhiht (G) by the first inequality. Hence, (ii) holds. Theorem 14. [3] Let G and H be non-trivial connected graphs. Then C = ⋃ x∈S [{x}×Tx], where S ⊆ V (G) and Tx ⊆ V (H) for each x ∈ S, is a hop independent set of G[H] if and only if the following conditions hold. (i) S is a hop independent set of G. (ii) Tx is a clique in H for each x ∈ S. Theorem 15. [7] Let G and H be connected non-trivial 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 pointwise non-dominating set of H for each x ∈ S with |N2 G(x) ∩ S| = 0. Theorem 16. Let G and H be non-trivial connected graphs. Then C = ⋃ x∈A [{x} × Tx], where A ⊆ V (G) and Tx ⊆ V (H) for each x ∈ A, is a hop independent hop dominating set of G[H] if and only if the following conditions hold. (i) A is a hop independent hop dominating set of G. (ii) Tx is a clique pointwise non-dominating set in H for each x ∈ A. Proof. Suppose C = ⋃ x∈A [{x} × Tx] is a hop independent hop dominating set of G[H]. Since C is a hop independent set, it follows that A is a hop independent set of G by Theorem 14. Moreover, since C is a hop dominating set of G, A is a hop dominating set by Theorem 15. Hence, (i) holds. Also, by Theorem 14 and Theorem 15, Tx is a clique pointwise non-dominating set of H for each x ∈ A, respectively. Hence, (ii) holds. Conversely, suppose that C = ⋃ x∈B [{x}×Tx] and satisfies (i) and (ii). Then by Theorem 14 and Theorem 15, C is a hop independent hop dominating set of G[H]. J. Hassan, S. Canoy / Eur. J. Pure Appl. Math, 15 (4) (2022), 1783-1796 1795 Corollary 8. Let G and H be non-trivial connected graphs. Then γhih(G[H]) = γhih(G)cpnd(H). In particular, we have (i) γhih(Kn[Km]) = γhih(Kn)cpnd(Km) = nm for any n,m ≥ 1, (ii) γhih(Kn[Pm]) = γhih(Kn)cpnd(Pm) = 2n for any n ≥ 1,m ≥ 3, (iii) γhih(Kn[Cm]) = γhih(Kn)cpnd(Cm) = 2n for any n ≥ 1,m ≥ 4, (iv) γhih(Pn[Pm]) = γhih(Pn)cpnd(Pm) = 2γh(Pn) for any n,m ≥ 3, and (v) γhih(Cn[Cm]) = γhih(Cn)cpnd(Cm) = 2γh(Cn) for any n,m ≥ 4. Proof. Let A be a γhih-set of G and let T be a cpnd-set of H. For each x ∈ A, set Tx = T . Then C = ⋃ x∈A [{x} × Tx] = A × T is a hop independent hop dominating set of G[H] by Theorem 16. Hence, γhih(G[H]) ≤ |C| = γhih(G)cpnd(H). On the other hand, if C ′ = ⋃ x∈A′ [{x} × Rx] is a γhih-set of G[H], then A′ is a hop independent hop dominating set of G and Rx is a clique pointwise non-dominating set of H for every x ∈ A′ by Theorem 16. Hence, γhih(G[H]) = |C ′| = ∑ x∈A′ |Rx| ≥ |A′|cpnd(H) ≥ γhih(G)cpnd(H). Consequently, γhih(G[H]) = γhih(G)cpnd(H). The assertions in (i), (ii), (iii), (iv) and (v) follow from Theorem 2, Corollary 1, Theorem 7, Corollary 5, and the first part. 4. Conclusion The concept of hop independent hop domination has been introduced and initially investigated in this study. As pointed out, every graph admits a hop independent hop dominating set and the hop independent hop domination number of a graph lies between the hop domination number and the hop independence number of the graph. Graphs which attained some specific values for their hop independent hop domination number have been characterized. Necessary and sufficient conditions for a subset in the shadow graph, join, corona, and lexicographic product of two graphs have been obtained. These characterizations have been used to obtain bounds or exact value of the hop independent hop domination number of each of these graphs. It may be interesting to find bounds for this newly defined parameter in terms of other known parameters, investigate the concept for trees and other interesting graphs, and determine the complexity of the hop independent hop domination problem. REFERENCES 1796 Acknowledgements The authors would like to thank the Department of Science and Technology - Acceler- ated Science and Technology Human Resource Development Program (DOST-ASTHRDP)- Philippines and MSU-Iligan Institute of Technology for funding this research. References [1] S. Ayyaswamy, B. Krishnakumari, B. Natarjan, and Y. Venkatakrishnan. Bounds on the hop domination number of a tree. Proceedings-Mathematical Sciences., 125(4):449–455, 2015. [2] S. Ayyaswamy, C. Natarajan, and G. Sathiamoorphy. A note on hop domination number of some special families of graphs. International Journal of Pure and Applied Mathematics., 119(12):11465–14171, 2018. [3] J. Hassan, S. Canoy Jr., and A. Aradais. Hop independent sets in graphs. Eur. J. Pure Appl. Math., 15(2):467–477, 2022. [4] M. Henning and N. Rad. On 2-step and hop dominating sets in graphs. Graphs and Combinatorics., 33(4):913–927, 2017. [5] S. Canoy Jr. and G. Malacas. Determining the intruder’s location in a given network: Locating-dominating sets in a graph. NRCP Research Journal., 13(1):1–8, 2013. [6] S. Canoy Jr. and G. Malacas. Differentiating-dominating sets in graphs under binary operations. Tamkang J. Math., 46(1):51–60, 2015. [7] S. Canoy Jr., R. Mollejon, and J. G. Canoy. Hop dominating sets in graphs under binary operations. Eur. J. Pure Appl. Math., 12(4):1455–1463, 2019. [8] S. Canoy Jr. and G. Salasalan. Revisiting domination, hop domination, and global hop domination in graphs. Eur. J. Pure Appl. Math., 14:1415–1428, 2021. [9] S. Canoy Jr. and G. Salasalan. Locating-hop domination in graphs. Kyungpook Mathematical Journal., 62:193–204, 2022. [10] C. Natarajan and S. Ayyaswamy. Hop domination in graphs ii. Versita, 23(2):187– 199, 2015. [11] G. Salasalan and S. Canoy Jr. Global hop domination numbers of graphs. Eur. J. Pure Appl. Math., 14(1):112–125, 2021.