EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 16, No. 1, 2023, 131-143 ISSN 1307-5543 – ejpam.com Published by New York Business Global Strong Resolving Hop Domination in Graphs Jerson S. Mohamad1, Helen M. Rara2 1 Department of Mathematics and Statistics, College of Science and Mathematics, Western Mindanao State University, 7000 Zamboanga City, Philippines 2 Department of Mathematics and Statistics, College of Science and Mathematics, Center of Graph Theory, Algebra, and Analysis-Premier Research Institute of Science and Mathematics, Mindanao State University-Iligan Institute of Technology, 9200 Iligan City, Philippines Abstract. A vertex w in a connected graph G strongly resolves two distinct vertices u and v in V (G) if v is in any shortest u-w path or if u is in any shortest v-w path. A set W of vertices in G is a strong resolving set G if every two vertices of G are strongly resolved by some vertex of W . A set S subset of V (G) is a strong resolving hop dominating set of G if S is a strong resolving set in G and for every vertex v ∈ V (G) \ S there exists u ∈ S such that dG(u, v) = 2. The smallest cardinality of such a set S is called the strong resolving hop domination number of G. This paper presents the characterization of the strong resolving hop dominating sets in the join, corona and lexicographic product of graphs. Furthermore, this paper determines the exact value or bounds of their corresponding strong resolving hop domination number. 2020 Mathematics Subject Classifications: 05C69 Key Words and Phrases: Strong resolving hop dominating set, strong resolving hop domination number, hop dominated superclique, join, corona, lexicographic product 1. Introduction Domination in graphs have been widely studied in Graph Theory. Over the years, many variations of domination have been studied. Hop domination in graphs was defined and characterized by Natarajan and Ayyaswamy [8], they also determined the hop domination number of some graphs. Variations of the domination and hop domination in graphs were studied in [4, 10, 11]. Resolving sets was studied by Slater in [12]. Variations of resolving sets and resolving dominating sets were studied in [2, 6, 7]. The strong resolving sets and strong metric dimension were introduced and characterized by Oellermann and Peters-Fransen [9]. Resolving hop dominating sets in graphs was studied and presented with an example of its application in minimization problems in [5]. This paper introduces and characterizes the concept of strong resolving hop domination in graphs. DOI: https://doi.org/10.29020/nybg.ejpam.v16i1.4578 Email addresses: mohamad.jerson@wmsu.edu.ph (J. Mohamad), helen.rara@g.msuiit.edu.ph (H. Rara) https://www.ejpam.com 131 © 2023 EJPAM All rights reserved. J. Mohamad, H. Rara / Eur. J. Pure Appl. Math, 16 (1) (2023), 131-143 132 We consider connected graphs that are finite, simple, and undirected. For elementary Graph Theory concepts, it is recommended that readers refer to [3]. Let G = ( V (G), E(G) ) be a graph. NG(v) = {u ∈ V (G) : uv ∈ E(G)} is a neighbor- hood of v. An element u ∈ NG(v) is called a neighbor of v. NG[v] = NG(v) ∪ {v} is a closed neighborhood of v. The degree of v, denoted by degG(v), is equal to |NG(v)|. For S ⊆ V (G), NG(S) = ⋃ v∈S NG(v) and NG[S] = ⋃ v∈S NG[v]. The distance dG(u, v) of two vertices u, v in G is the length of a shortest u-v path in G. The greatest distance between any two vertices in G, denoted by diam(G), is called the diameter of G. A set S ⊆ V (G) of vertices of G is a dominating set if every u ∈ V (G) \ S is adjacent to at least one vertex v ∈ S. The domination number of a graph G, denoted by γ(G), is given by γ(G) = min{|S| : S is a dominating set of G}. A set S ⊆ V (G) is a hop dominating set of G if for every v ∈ V (G)\S, there exists u ∈ S such that dG(u, v) = 2. The minimum cardinality of a hop dominating set 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. The floor function of a real number x, denoted by ⌊x⌋, is a function that returns the highest integer less than or equal to x. Formally, for all real number x, ⌊x⌋ = max{n ∈ Z |n ≤ x}. A vertex v in G is a hop neighbor of vertex u in G if dG(u, v) = 2. The set NG(u, 2) = {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 NG[u, 2] = NG(u, 2) ∪ {u}. The open hop neighborhood of X ⊆ V (G) is the set NG(X, 2) = ⋃ u∈X NG(u, 2). The closed hop neighbor- hood of X in G is the set NG[X, 2] = NG(X, 2) ∪X. A vertex x of a graph G is said to resolve two vertices u and v of G if dG(x, u) ̸= dG(x, v). For an ordered set W = {x1, ..., xk} ⊆ V (G) and a vertex v in G, the k − vector rG(v/W ) = (dG(v, x1), dG(v, x2), · · · , dG(v, xk)) is called the representation of v with respect to W . The set W is a resolving set for G if and only if no two vertices of G have the same representation with respect to W . The metric dimension of G, denoted by, dim(G), is the minimum cardinality over all resolving sets of G. A resolving set of cardinality dim(G) is called basis. A set S ⊆ V (G) is a resolving hop dominating set of G if S is both a resolving set and a hop dominating set. The minimum cardinality of a resolving hop dominating set of G, denoted by γRh(G), is called the resolving hop domination number of G. Any resolving hop dominating set with cardinality equal to γRh(G) is called a γRh-set. A u-v path of length dG(u, v) is called a u-v geodesic. The set IG[u, v] is defined as the set of all vertices of G lying on any u-v geodesic. J. Mohamad, H. Rara / Eur. J. Pure Appl. Math, 16 (1) (2023), 131-143 133 A vertex w ∈ V (G) strongly resolves two different vertices u, v ∈ V (G) if v ∈ IG[u,w] or if u ∈ IG[v, w]. A set W of vertices in G is a strong resolving set of G if every two vertices of G are strongly resolved by some vertex of W . The smallest cardinality of a strong resolving set of G is called the strong metric dimension of G and is denoted by sdim(G). A strong resolving set of cardinality sdim(G) is called a strong metric basis of G. A vertex u of G is maximally distant from vertex v of G, u ̸= v, if for every vertex w ∈ NG(u), dG(v, w) ≤ dG(u, v). If u is maximally distant from v, and v is maximally distant from u, then we say that u and v are mutually maximally distant, denoted by uMMDv. The boundary of a graph G, denoted by ∂(G), is defined as ∂(G) = {u ∈ V (G) : uMMDv for some v ∈ V (G)} . A set S ⊆ V (G) is a strong resolving hop dominating set of G if S is both a strong resolving set and a hop dominating set. The minimum cardinality of a strong resolving hop dominating set of G, denoted by γsRh(G), is called the strong resolving hop domination number of G. Any resolving hop dominating set with cardinality equal to γsRh(G) is called a γsRh-set. A clique in a graph G is a complete induced subgraph of G. A clique C in a connected graph G is called a superclique if for every pair of distinct vertices u, v ∈ C, there exists w ∈ V (G) \ C such that w ∈ NG(u) \ NG(v) or w ∈ NG(v) \ NG(u). A superclique C is maximum in G if |C| ≥ |C∗| for all supercliques C∗ in G. The superclique number, ωS(G), of G is the cardinality of a maximum superclique in G. A superclique C in G is called a hop dominated superclique if for every v ∈ C there exists u ∈ V (G)\C such that dG(u, v) = 2. A hop dominated superclique C is maximum in G if |C| ≥ |C∗| for all hop dominated supercliques C∗ in G. The hop dominated superclique number, ωhS(G), of G is the cardinality of a maximum hop dominated superclique in G. 2. Preliminary Results Remark 1. Every strong resolving hop dominating set of G is a resolving hop dominating set. Thus, 2 ≤ γRh(G) ≤ γsRh(G). Remark 2. Every strong resolving hop dominating set of G is a hop dominating set. Thus, 2 ≤ γh(G) ≤ γsRh(G). Remark 3. Every strong resolving hop dominating set of G is a resolving set. Thus, 1 ≤ dim(G) ≤ γsRh(G). Remark 4. Every strong resolving hop dominating set of a connected graph G is hop dominating and every strong resolving hop dominating set of G is strong resolving. Thus, γh(G) ≤ γsRh(G) and sdim(G) ≤ γsRh(G). J. Mohamad, H. Rara / Eur. J. Pure Appl. Math, 16 (1) (2023), 131-143 134 Remark 5. For any connected graph of order n, 2 ≤ γsRh(G) ≤ n. Moreover, γsRh(G) = 1 if and only if G is a trivial graph and γsRh(G) = n if G = Kn. Proposition 1. Let S be a hop dominating set of Pn for n ≥ 3. Then S is a strong resolving hop dominating set of Pn if and only if there exists u ∈ S with degPn(u) = 1. Proof: Suppose S is a strong resolving hop dominating set of Pn. Let u and v be distinct vertices of Pn such that deg(u) = deg(v) = 1. Then the pair of vertices u and v is only strongly resolved by u or by v, since u ∈ I[u, v] or v ∈ I[u, v] and u ∈ I[v, u] or v ∈ I[v, u]. Implying that u, v ∈ S. For the converse, let u ∈ S with degPn(u) = 1. Then every pair of distinct vertices x, y ∈ V (Pn) are strongly resolved by u, since x ∈ I[u, y] or y ∈ I[u, x]. Hence, S is a strong resolving set of Pn. Since S is hop dominating by assumption, S is a strong resolving hop dominating set of Pn. Proposition 2. Let S be a hop dominating set of Cn for n ≥ 4. If dCn(u, v) < ⌊n/2⌋ for each pair u, v ∈ V (Cn) \ S, then S is a strong resolving hop dominating set of Cn. Proof: Let S be a hop dominating set of Cn and let u, v ∈ V (Cn) \ S such that d(u, v) < ⌊n/2⌋. Note that diam(Cn) = ⌊n/2⌋. Then there exists, s ∈ S such that dCn(s, u) ≤ dCn(s, v) ≤ ⌊n/2⌋ or dCn(s, v) ≤ dCn(s, u) ≤ ⌊n/2⌋. It follows that u ∈ I[s, v] or v ∈ I[s, u]. Hence, s strongly resolves u and v. Therefore, S is a strong resolving hop dominating set of Cn. Remark 6. Every strong resolving hop dominating set of a connected graph G is a resolv- ing hop dominating set. Thus, γRh(G) ≤ γsRh(G). Remark 7. Any superset of a strong resolving hop dominating set is a strong resolving hop dominating set. Proposition 3. [1] Let G be a connected graph of order n and let A = {x ∈ V (G) : degG(x) = n− 1} . If A ̸= ∅ and ⟨C⟩ is a superclique in G, then |C ∩A| ≤ 1. Moreover, if ⟨C⟩ is a maximum superclique of G, then |C ∩A| = 1. Proposition 4. [1] Let G be a non-trivial connected graph with diam(G) ≤ 2. Then S = V (G) \ C is a strong resolving set of G if and only if C = ∅ or ⟨C⟩ is a superclique in G. In particular, sdim(G) = |V (G)| − ωS(G). Proposition 5. Let G be a connected graph of order n, and let A = {x ∈ V (G) : degG(x) = n − 1}. If A ̸= ∅ and C is a hop dominated superclique in G, then C ∩A = ∅. Proof: Suppose C is a hop dominated superclique in G. By Proposition 3, |C ∩A| ≤ 1. If |C ∩A| = 1 and x ∈ C ∩A, then dG(x, v) = 1 for each v ∈ V (G)\{x} since degG(x) = n − 1. This is a contradiction to the definition of hop dominated superclique. It follows that |C ∩A| = 0. Therefore C ∩A = ∅. J. Mohamad, H. Rara / Eur. J. Pure Appl. Math, 16 (1) (2023), 131-143 135 Lemma 1. Let G be a nontrivial connected graph with diam(G) ≤ 2. Then S = V (G)\C is a strong resolving hop dominating set of G if and only if C = ∅ or C is a hop dominated superclique in G. In particular, γsRh(G) = |V (G)| − ωhS(G). Proof: Suppose that S = V (G) \ C is a strong resolving hop dominating set of G. Then S is a strong resolving set. By Proposition 4, C = ∅ or C is a superclique in G. Let x ∈ C. Then x ∈ V (G) \ S. Since S is hop dominating, a vertex y ∈ S ∩NG(x, 2) exists. Hence, y ∈ V (G) \ C and dG(x, y) = 2. Thus, C is a hop dominated superclique in G. Conversely, suppose C = ∅ or C is a hop dominated superclique in G. Then C ̸= ∅ or C is a superclique in G. By Proposition 4, S = V (G) \ C is a strong resolving set of G. Let x ∈ V (G) \ S. Then x ∈ C. Since C is a hop dominated superclique, there exists y ∈ V (G) \ C ∩ NG(x, 2). Since S = V (G) \ C, y ∈ S ∩ NG(x, 2). Therefore, S is a hop dominating set of G. Accordingly, S is a strong resolving hop dominating set of G. Suppose S is a γsRh-set of G. Then S = V (G) \ C, where C is a hop dominated superclique in G and |C| = ωhS(G). Hence, γsRh(G) = |S| = |V (G)| − |C| = |V (G)| − ωhS(G). 3. On Strong Resolving Hop Domination in the Join of Graphs Let A and B be sets which are not necessarily disjoint. The disjoint union of A and B, denoted by A • ∪ B, is the set obtained by taking the union of A and B treating each element in A as distinct from each element in B. The union G1 ∪ G2 of graphs G1 and G2 with disjoint vertex-sets V (G1) and V (G2), respectively, is the graph G with V (G) = V (G1) • ∪ V (G2) and E(G) = E(G1) • ∪ E(G2). The join of two graphs G and H, denoted by 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)}. Remark 8. For the joins ⟨v⟩+ Pn and ⟨w⟩+ Cn, γsRh(⟨v⟩+ Pn) = n− 1 for n ≥ 3 and γsRh(⟨w⟩+ Cn) = n− 1 for n ≥ 4. Theorem 1. [1] Let G be a non-trivial connected graph of order n with γ(G) ̸= 1 and K1 = ⟨v⟩. Then S ⊆ V (K1+G) is a strong resolving set of K1+G if and only if S = V (G), or S = V (G) \ C, or S = V (K1 +G) \ C where ⟨C⟩ is a superclique in G. Theorem 2. [1] Let G be a non-trivial connected graph of order n with γ(G) = 1 and K1 = ⟨v⟩. Then S ⊆ V (K1+G) is a strong resolving set of K1+G if and only if S = V (G), or S = V (K1 +G) \ C∗, or S = (V (G) \ C∗) ∪ {x ∈ C∗ : degG(x) = n− 1} where ⟨C∗⟩ is a superclique in G. J. Mohamad, H. Rara / Eur. J. Pure Appl. Math, 16 (1) (2023), 131-143 136 Theorem 3. [1] Let K1 = ⟨v⟩ and G be a disconnected graph whose components are Gi for i = 1, 2, . . . ,m. A proper subset S of V (K1 +G) is a strong resolving set of K1 +G if and only if S = V (G) or S = V (G)\Ci or S = V (K1+G)\Ci where ⟨Ci⟩ is a superclique in Gi, for some i ∈ {1, 2, . . . ,m}. Lemma 2. Let G be a connected graph of order n and let K1 = ⟨v⟩. Then C ⊆ V (K1+G) is a hop dominated superclique in K1 +G if and only if C is a hop dominated superclique in G. Proof: Suppose C is a hop dominated superclique in K1 +G. Since degK1+G(v) = n, by Proposition 5, v /∈ C. Thus, C is a hop dominated superclique in G. The converse follows immediately from Proposition 5 and definition of hop dominated superclique. Theorem 4. Let G be a connected graph of order n > 1, and let K1 = ⟨v⟩. Then S ⊆ V (K1 + G) is a strong resolving hop dominating set of K1 + G if and only if S = V (K1 +G) \ C where C = ∅ or C is a hop dominated superclique in G. Proof: Suppose S is a strong resolving hop dominating set of K1 + G. By Theorem 1 and Theorem 2, S = V (G), or S = (V (G) \ C) ∪ {x ∈ C : degG(x) = n − 1}, or S = V (K1 + G) \ C where C is a superclique in G. Since S is a hop dominating set of K1 +G, v ∈ S. Thus, S ̸= V (G) and S ̸= (V (G) \ C) ∪ {x ∈ C : degG(x) = n− 1}. Since diam(K1 +G) ≤ 2, by Lemma 1 and Proposition 5, S = V (K1 +G) \ C where C = ∅ or C is a hop dominated superclique in G. The converse follows immediately from Lemma 1 and Proposition 5. The next result is a consequence of Theorem 4. Corollary 1. Let G be a connected graph of order n. Then γsRh(K1 +G) = n+ 1− ωhS(G). The next result follows immediately from Theorem 3, and definitions of strong resolving hop domination and K1 +G. Theorem 5. Let K1 = ⟨v⟩ and G be a disconnected graph whose components are Gi for i = 1, 2, · · · ,m. A proper subset S of V (K1 +G) is a strong resolving hop dominating set of K1 + G if and only if S = V (K1 + G) \ Ci where Ci is a superclique in Gi for some i ∈ {1, 2, · · · ,m}. Proof: Suppose S is a strong resolving hop dominating proper subset of K1 + G. By Theorem 3, S = V (G), or S = V (G) \Ci or S = V (K1+G) \Ci where Ci is a superclique in Gi for some i ∈ {1, 2, · · · ,m}. By definition of strong resolving hop domination, v ∈ S. Hence, S ̸= V (G) and S ̸= V (G)\Ci. Thus, S = V (K1+G)\Ci where Ci is a superclique in Gi for some i ∈ {1, 2, · · · ,m}. Conversely, suppose S = V (K1 + G) \ Ci where Ci is a superclique in Gi for some i ∈ {1, 2, · · · ,m}. Let x ∈ V (K1 + G) \ S. Then x ∈ Ci for some i ∈ {1, 2, · · · ,m}. Since m ≥ 2, there exists Gj , j ̸= i and vertex y ∈ V (Gj) ∩ NK1+G(x, 2). Since J. Mohamad, H. Rara / Eur. J. Pure Appl. Math, 16 (1) (2023), 131-143 137 S = V (K1 + G) \ Ci, y ∈ S ∩ NK1+G(x, 2). Therefore, S is a hop dominating set of K1 +G. As a consequence of Theorem 5, the next result follows. Corollary 2. Let Gi be connected graphs of orders ni and G be a disconnected graph whose components are Gi for i = 1, 2, · · · ,m. Then γsRh(K1 +G) = m∑ i=1 ni −max {ωhS(Gi)(i = 1, 2, · · · ,m)}. Theorem 6. Let G and H be nontrivial connected graphs. A proper subset S of V (G+H) is a strong resolving hop dominating set of G+H if and only if at least one of the following is satisfied: (i) S = V (G+H) \ CG where CG is a hop dominated superclique in G. (ii) S = V (G+H) \ CH where CH is a hop dominated superclique in H. (iii) S = V (G+H) \ (CG ∪CH) where CG and CH are hop dominated supercliques of G and H, respectively. Proof: Note that diam(G +H) ≤ 2 and CG, CH , and (CG ∪ CH) are hop dominated supercliques in G+H. Then by Lemma 1, the theorem holds. The next result is a consequence of Theorem 6. Corollary 3. Let G and H be nontrivial connected graphs of orders m and n, respectively. Then γsRh(G+H) = m+ n− (ωhS(G) + ωhS(H)). 4. On Strong Resolving Hop Domination in the Corona of Graphs The corona of two graphs G and H, denoted by G◦H, is the graph obtained by taking one copy of G of order n and n copies of H, and then joining every vertex of the ith copy of H to the ith vertex of G. For v ∈ V (G), denote by Hv the copy of H whose vertices are attached one by one to the vertex v. Subsequently, denote by v+Hv the subgraph of the corona G ◦H corresponding to the join ⟨{v}⟩+Hv, v ∈ V (G). Theorem 7. [1] Let G be a non-trivial connected graph and H a connected graph. A proper subset S of V (G ◦H) is a strong resolving dominating set of G ◦H if and only if one of the following holds: (i) S = A ⋃( ⋃ u∈V (G) V (Hu) ) where A ⊆ V (G); (ii) S = A ⋃( ⋃ u∈V (G)\{v} V (Hu) )⋃ Bv for a unique vertex v in G, where A ⊆ V (G) and Bv is a strong resolving set of Hv if γ(H) = 1 or Bv is a resolving set of ⟨v⟩ +Hv if γ(H) ̸= 1. J. Mohamad, H. Rara / Eur. J. Pure Appl. Math, 16 (1) (2023), 131-143 138 Theorem 8. Let G be a nontrivial connected graph and H a connected graph. A proper subset S of V (G ◦H) is a strong resolving hop dominating set of G ◦H if and only if one of the following holds: (i) S = A ∪  ⋃ v∈V (G) V (Hv)  where A ⊆ V (G); (ii) S = A ∪  ⋃ v∈V (G)\{u} V (Hv)  ∪ Bu for a unique vertex u in G, where A ⊆ V (G), Bu is a strong resolving set of Hu and Bu is strong resolving hop dominating of Hu if NG(u) ∩A = ∅. Proof: Suppose S is a strong resolving hop dominating set of G◦H. By Theorem 7, (i) holds or S = A∪  ⋃ v∈V (G)\{u} V (Hv) ∪Bu for a unique vertex u in G, where A ⊆ V (G), Bu is a strong resolving set of Hu if γ(H) = 1 or Bu is a strong resolving set of {u}+Hu if γ(H) ̸= 1. Let NG(u) ∩ A = ∅ and x ∈ V (Hu) \ Bu. Since S is hop dominating and x ∈ V (G ◦H) \ S, there exists y ∈ NG◦H(x, 2) ∩ S. Thus, y ∈ NHu(x, 2) ∩ Bu. It follows that Bu is strong resolving hop dominating set of Hu. Hence, (ii) holds. For the converse, suppose (i) and (ii) hold. Then by Theorem 7, S is a strong resolving set of G ◦H. Let x ∈ V (G ◦H) \ S. Consider the following cases: Case 1. x ∈ V (G) \A Since G is nontrivial connected, there exists a vertex y ∈ V (G)∩NG(x). By (i) or (ii), vertex z ∈ NG◦H(x, 2) ∩ V (Hy) or z ∈ NG◦H(x, 2) ∩By exists. Case 2. x ∈ V (Hu) \Bu for a unique vertex u in G. If NG(u) ∩ A ̸= ∅, then w ∈ ((NG(u) ∩ A) ∩NG◦H(x, 2) exists. On the other hand, if NG(u) ∩A = ∅, by (ii), Bu is a strong resolving hop dominating set of Hu. Hence, there exists p ∈ NHu(x, 2) ∩Bu. This implies that p ∈ NG◦H(x, 2) ∩ S. In any case, S is a hop dominating set of G ◦H. Accordingly, S is a strong resolving hop dominating set of G ◦H. Corollary 4. Let G and H be connected graphs of orders m > 1 and n, respectively. Then γsRh(G ◦H) = (m− 1)n+ γsRh(H). Moreover, if diam(H) ≤ 2, then γsRh(G ◦H) = (m− 1)n+ |V (H)| − ωhS(H) = mn− ωhS(H). Proof: Let S be a γsRh-set of G ◦H. Then, S = A ∪  ⋃ v∈V (G)\{u} V (Hv)  ∪ Bu for a unique vertex u in G, where A ⊆ V (G), and Bu satisfies (ii) of Theorem 8. Let A = ∅ J. Mohamad, H. Rara / Eur. J. Pure Appl. Math, 16 (1) (2023), 131-143 139 and Bu be γsRh-set of H u. Then γsRh(G ◦H) =|S| =|A|+ ∣∣∣∣∣∣ ⋃ v∈V (G)\{u} V (Hv) ∣∣∣∣∣∣+ |Bu| =0 + (m− 1)n+ γsRh(H) =(m− 1)n+ γsRh(H). Suppose that diam(H) ≤ 2. Then by Lemma 1, γsRh(H) = |V (H)| − ωhS(H) = n− ωhS(H). Hence, γsRh(G ◦H) =(m− 1)n+ n− ωhS(H) =mn− n+ n− ωhS(H) =mn− ωhS(H). Therefore, γsRh(G ◦H) = (m− 1)n+ |V (H)| − ωhS(H) = mn− ωhS(H). 5. On Strong Resolving Hop Domination in the Lexicographic Product of Graphs The lexicographic product of two graphs G and H, denoted by G[H], is the graph with vertex-set V (G[H]) = V (G) × V (H) such that (u1, u2)(v1, v2) ∈ E(G[H]) if either u1v1 ∈ E(G) or u1 = v1 and u2v2 ∈ E(H). Lemma 3. [1] Let G = Kn for n > 1 and H a non-trivial connected graph with γ(H) ̸= 1. Then A × C ⊆ V (G[H]) induces a superclique in G[H] if and only if A is a nonempty subset of V (G) and ⟨C⟩ is a superclique in H. Lemma 4. [1] Let G = Kn for n > 1 and H a non-trivial connected graph with γ(H) = 1. Then A×C ⊆ V ( G[H] ) is a superclique in G[H] if and only if A is a nonempty subset of V (G) and C is a superclique in H such that |A| = 1 whenever C ∩C∗ ̸= ∅ for some γ-set of H. Lemma 5. Let G = Kn for n > 1 and H a connected graph with |V (H)| ≥ 3. Then A×C ⊆ V (G[H]) is a hop dominated superclique in G[H] if and only if A is a nonempty subset of V (G) and C is a hop dominated superclique in H. J. Mohamad, H. Rara / Eur. J. Pure Appl. Math, 16 (1) (2023), 131-143 140 Proof: Suppose that A × C ⊆ V (G[H]) is a hop dominated superclique in G[H]. By Lemma 3 and Lemma 4, A is a nonempty subset of V (G) and C is a superclique in H. Let x ∈ C. Then (a, x) ∈ A× C for any a ∈ A. Since A× C is a hop dominated superclique, there exists (b, y) ∈ [V (G[H]) \ (A × C)] ∩ NG[H]((a, x), 2). Suppose γ(H) ̸= 1. Since G = Kn for n > 1, a = b and y ∈ (V (H)\C)∩NH(x, 2). If γ(H) = 1, then by Proposition 5, C ∩ C∗ = ∅ for all γ-sets C∗ of H. Thus, x ∈ C \ C∗ and y ∈ NH(x, 2) exists. Hence, C is a hop dominated superclique in H. For the converse, suppose that A is a nonempty subset of V (G) and C is a hop dominated superclique in H. By Lemma 3, Lemma 4 and Proposition 5, A × C is a superclique in G[H]. Let (a, x) ∈ A × C and γ(H) ̸= 1. Since C is a hop dominated superclique in H, there exists y ∈ (V (H) \ C) ∩NH(x, 2). Hence, (a, y) ∈ [V (G[H]) \ (A× C)] ∩NG[H]((a, x), 2). Suppose γ(H) = 1. Then by Proposition 5, C ∩ C∗ = ∅ for all γ-sets C∗ of H. Thus, x ∈ C \ C∗. This implies that a vertex z ∈ NH(x, 2) exists. Since C is a superclique, z ∈ V (H) \ C. Hence, (a, z) ∈ [V (G[H]) \ (A× C)] ∩NG[H]((a, x), 2). Therefore A× C is a hop dominated superclique in G[H]. Theorem 9. Let G = Kn for n > 1 and H a connected graph with |V (H)| ≥ 3. A subset S of V (G[H]) is a strong resolving hop dominating set of G[H] if and only if S = V (G[H]) \ (A × C), where A is a subset of V (G) and C = ∅ or A is a nonempty subset of V (G) and C is a hop dominated superclique in H. Proof: Let S be a strong resolving hop dominating set of G[H]. Since diam(G[H]) = 2, by Lemma 1, S = V (G[H]) \ (A × C) where A × C = ∅ or A × C is a hop dominated superclique in G[H]. If A× C = ∅, then A = ∅ or C = ∅. On the other hand, if A× C is a hop dominated superclique in G[H], A is a nonempty subset of V (G) and C is a hop dominated superclique in H, by Lemma 5. For the converse, suppose S = V (G[H]) \ (A× C), where A ⊆ V (G) and C = ∅ or A is a nonempty subset of V (G) and C is a hop dominated superclique in H. If C = ∅, then A × C = ∅ and S = V (G[H]) is a strong resolving hop dominating set of G[H]. On the other hand, if A is a nonempty subset of V (G) and C is a hop dominated superclique in H, then by Lemma 5 and Lemma 1, S is a strong resolving hop dominating set of G[H]. Corollary 5. Let G = Kn for n > 1 and H a connected graph of order m ≥ 3. Then γsRh(G[H]) = mn− ωhS(G[H]). Proof: Let S be a γsRh-set of G[H]. Then by Theorem 9, S = V (G[H])\(A×C) where A × C is a hop dominated superclique in G[H]. Since S is a minimum strong resolving hop dominating set of G[H]. Hence, γsRh(G[H]) = |S| = |V (G[H])| − |A× C| = mn− ωhS(G[H]). By Lemma 5 and Corollary 5 the next result follows. J. Mohamad, H. Rara / Eur. J. Pure Appl. Math, 16 (1) (2023), 131-143 141 Corollary 6. Let G = Kn for n > 1 and H a connected graph of order m. Then γsRh(G[H]) = n(m− ωhS(H)). Theorem 10. [1] Let G be a non-trivial connected graph and H be non-trivial complete graph. A subset C = (⋃ x∈S { {x} × Tx })⋃ ⋃ x∈WG { {x} × (V (H) \ Tx) } of V ( G[H] ) where WG ⊆ S and Tx ⊆ V (H), ∀x ∈ S, is a strong resolving set of G[H] if and only if (i) S = V (G). (ii) V (H) \ Tx is a superclique of H. (iii) WG is a strong resolving set of G. Theorem 11. Let G be a nontrivial connected graph and H be a nontrivial complete graph. A subset C = ⋃ x∈S ({x}×Tx) of V (G[H]) where S ⊂ V (G) and Tx ⊆ V (H) for each x ∈ S, is a strong resolving hop dominating set of G[H] if and only if (i) S = V (G); (ii) 0 ≤ |V (H) \ Tx| ≤ 1 for every x ∈ S; (iii) Tx = V (H) or Ty = V (H) for every pair of vertices x, y ∈ V (G) such that x ∈ WG where WG is a strong resolving set of G; (iv) Tx = V (H) for every x ∈ S with degG(x) = |V (G)| − 1. Proof: Suppose C is a strong resolving hop dominating set of G[H]. By Theorem 10 (i) and (ii) hold. Now, let x ∈ S with degG(x) = |V (G)| − 1. Suppose Tx ̸= V (H). Let a ∈ V (H) \ Tx. Then (x, a) /∈ C. Since C is a hop dominating set of G[H], there exists (y, b) ∈ C ∩ NG[H]((x, a), 2). Since H is complete, y ∈ S ∩ NG(x, 2). This implies that y ∈ (V (G) \ {x}) \ NG(x), showing that degG(x) < |V (G)| − 1, a contradiction. Hence, Tx = V (H) and (iv) is true. For (iii), let x, y ∈ V (G) such that xMMDy. Suppose Tx ̸= V (H) and Ty ̸= V (H). Let p ∈ V (H) \ Tx and q ∈ V (H) \ Ty. Then (x, p), (y, q) /∈ C. Since C is a strong resolving set of G[H], there exists (z, w) ∈ C such that (z, w) strongly resolves (x, p) and (y, q). Suppose (x, p) ∈ IG[H][(y, q), (z, w)]. Then dG[H]((z, w), (x, p)) + dG[H]((x, p), (y, q)) = dG[H]((z, w), (y, q)). If z = x, then w ∈ Tx and dG[H]((z, w), (x, p)) = 1. Thus 1 + dG[H]((x, p), (y, q)) = dG[H]((z, w), (y, q)), J. Mohamad, H. Rara / Eur. J. Pure Appl. Math, 16 (1) (2023), 131-143 142 that is, 1 + dG[H]((z, p), (y, q)) = dG[H]((z, w), (y, q)) or 1 + dG(z, y) = dG(z, y), which is a contradiction. Thus, z ̸= x. Consequently, dG(z, x) + dG(x, y) = dG(z, y) implying that x ∈ IG[z, y]. This is a contradiction to the assumption that xMMDy. Similarly, if (y, q) ∈ IG[H][(x, p), (z, w)], then a contradiction can be obtained. Hence Tx = V (H) or Ty = V (H). For the converse, suppose C satisfies conditions (i) to (iv). Let ((x1, y1), (x2, y2)) /∈ C with (x1, y1) ̸= (x2, y2). Then y1 /∈ Tx1 and y2 /∈ Tx2 . Hence, x1 is not mutually maximally distant with x2. Thus, there exists u ∈ NG(x1) such that dG(u, x2) > dG(x1, x2), that is, x1 ∈ IG[x2, u]. Let v ∈ Tu. Then (u, v) ∈ C strongly resolves (x1, y1) and (x2, y2). Hence, C is a strong resolving set of G[H]. Now, we show that C is a hop dominating set of G[H]. Let (x, y) ∈ C. Then y ∈ V (H) \ Tx. By (iv), degG(x) < |V (G)| − 1. Thus, there exists z ∈ V (G\{x})\NG(x). Let z ∈ NG(x, 2) and w ∈ Tz. Then, (z, w) ∈ C ∩NG[H]((x, y), 2). This shows that C is a hop dominating set of G[H]. Accordingly, C is a strong resolving hop dominating set of G[H]. Corollary 7. Let G be a connected graph of order m > 1 and γ(G) ̸= 1 and H = Kn for n > 1. Then γsRh(G[H]) = m(n− 1) + |∂(G)| − 1. Proof: Let C be a γsRh-set of G[H]. Then C = ⋃ x∈S ({x} × Tx) satisfying conditions of Theorem 11. Thus, S = V (G) = [V (G) \ ∂(G)] ∪ (∂(G)). Hence, γsRh(G[H]) =|C| = ∣∣∣∣∣∣ ⋃ x∈V (G)\∂(G) ({x} × Tx) ∣∣∣∣∣∣+ ∣∣∣∣∣∣ ⋃ y∈∂(G) ({y} × Ty) ∣∣∣∣∣∣ =|V (G) \ ∂(G)||Tx|+ |∂(G)− 1||Ty|+ n− 1 =(m− |∂(G)|)(n− 1) + |∂(G)− 1|n+ n− 1 =m(n− 1)− |∂(G)|n+ |∂(G)|+ |∂(G)|n− n+ n− 1 =m(n− 1) + |∂(G)| − 1. Therefore, γsRh(G[H]) = m(n− 1) + |∂(G)| − 1. Acknowledgements This research is funded by the Department of Science and Technology - Accelerated Sci- ence and Technology Human Resource Development Program (DOST-ASTHRDP), Min- danao State University - Iligan Institute of Technology, and Western Mindanao State University, Philippines. REFERENCES 143 References [1] G. Monsanto, P. Acal and H. Rara. On Strong Resolving Domination in the Join and Corona of Graphs. 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