EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 15, No. 4, 2022, 1472-1481 ISSN 1307-5543 – ejpam.com Published by New York Business Global Restrained Strong Resolving Hop Domination in Graphs Armalene H. Abragan1,∗, Helen M. Rara2 1 Department of Mathematics and Statistics, College of Science and Mathematics, Center of Graph Theory, Algebra, Mindanao State University-Iligan Institute of Technology, 9200 Iligan City, Philippines 2 Analysis-Premier Research Institute of Science and Mathematics, Mindanao State University- Iligan Institute of Technology, 9200 Iligan City, Philippines Abstract. A set S ⊆ V (G) is a restrained strong resolving hop dominating set in G if for every v ∈ V (G)\S, there exists w ∈ S such that dG(v, w) = 2 and S = V (G) or V (G)\S has no isolated vertex. The smallest cardinality of such a set, denoted by γrsRh(G), is called the restrained strong resolving hop domination number of G. In this paper, we obtained the corresponding parameter in graphs resulting from the join, corona and lexicographic product of two graphs. Specifically, we characterize the restrained strong resolving hop dominating sets in these types of graphs and determine the bounds or exact values of their restrained strong resolving hop domination numbers. 2020 Mathematics Subject Classifications: 05C69 Key Words and Phrases: restrained strong resolving hop dominating set, restrained strong resolving hop domination number, join, corona, lexicographic product 1. Introduction The study of domination can be traced way back 1960. Since then numerous authors contribute several interesting domination parameters to nurture the growth of this research area. In 1977, E.J Cockayne and S.T Hedetniemi introduced the notation γ(G) for the domination number of graph G. Until the initiation of the concept of 2-step domination number by Chartrand et al [3] in 1995, which is closely related to hop domination number. Subsequently, Natarajan and Ayyaswamy (2015) introduced the Hop Domination concept. Some variation of domination can be seen in these papers [2], [5], [4]. In this study, the researcher defines and establishes a new concept of hop domination called a restrained strong resolving hop domination and generates some characterizations ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v15i4.4484 Email addresses: armalene.abragan@g.msuiit.edu.ph (A. Abragan), helen.rara@g.msuiit.edu.ph (H. M. Rara) https://www.ejpam.com 1472 © 2022 EJPAM All rights reserved. A. H. Abragan, H. M. Rara / Eur. J. Pure Appl. Math, 15 (4) (2022), 1472-1481 1473 of restrained strong resolving hop domination in graphs. For an application, in [8] Haynes and Henning considered a factory with large number of employees and a need to imple- ment a quality assurance checking system of their workers. The factory manager decides to designate an internal committee to do this, i.e, the manager will select a subset of the workers to form a quality assurance team to inspect the work of their co-workers. The manager desires to keep this team as small as possible in order to minimize costs (inspec- tors’ extra pay) and to protect privacy (keeping the identity of inspector secret).To avoid bias, an inspector should neither be close friends nor enemies with any of the workers he/she is responsible for inspecting. To model this situation, a social network graph can be constructed, where each worker is represented by a vertex and an edge between two workers represent possible bias, i.e if the two workers are either close friend or enemies. Ideally, an inspector should not be adjacent to any worker under his inspection. In hop domination, every worker will be inspected by the nearest non-biased inspector, that is, an inspector who is a close friend (or enemy) of the worker’s close friend (or enemy). This is to save time and effort locating a particular worker. If we desire a situation where every worker including the inspector has his/her work inspected, then restrained strong resolving hop domination numbers gives us the minimum number of inspectors needed. In this study, we only consider graphs that are finite, simple, undirected and connected. Readers are referred to [6] for elementary Graph Theory concepts. Let G be a connected graph. 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. A set C ⊆ V (G) is called a superclique in G if ⟨C⟩ is a clique and 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 of G, denoted by ωS(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 super- clique number denoted by ωhS(G), of G is the cardinality of a maximum hop dominated superclique in G. A superclique C ⊆ V (G) is called a point-wise non-dominated superclique of G if for every x ∈ C there exists y ∈ V (G) \C such that y /∈ NG(x). A maximum cardinality of a point-wise non-dominated superclique in G is denoted by ωpndS(G). A vertex x of a connected graph G is said to resolve 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 A. H. Abragan, H. M. Rara / Eur. J. Pure Appl. Math, 15 (4) (2022), 1472-1481 1474 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 a basis. For two vertices u, v ∈ V (G), the interval IG[u, v] between u and v is the collection of all vertices that belong to some shortest u-v path. A vertex w strongly resolves two vertices u and v 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 subset 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 set S ⊆ V (G) is a restrained strong resolving hop dominating set on G if S is a strong resolving hop dominating set in G and S = V (G) or ⟨V (G) \ S⟩ has no isolated vertex. The restrained strong resolving hop domination number of G, denoted by γrsRh(G), is the smallest cardinality of a restrained strong resolving dominating set in G. A restrained strong resolving hop dominating set of cardinality γrsRh(G) is then referred to as γrsRh-set of G. 2. Preliminary Results Lemma 1. [7] Let G be a nontrivial 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 1. Let G be a connected graph of order n and A = {x ∈ G : degG(x) = n− 1}. If A ̸= ∅ and C is a hop dominated superclique in G, then C ∩A = ∅. Theorem 1. [7] Let G be a nontrivial 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), S = V (G) \ C, or S = V (K1 +G) \ C where C is a superclique in G. Theorem 2. [7] Let G and H be nontrivial connected graphs of orders m and n, respec- tively. A proper subset S of V (G+H) is a strong resolving 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 superclique in G. (ii) S = V (G+H) \ CH where CH is a superclique in H. (iii) If γ(G) ̸= 1 or γ(H) ̸= 1, S = V (G+H) \ (CG ∪ CH) = (V (G) \ CG) ∪ (V (H) \ CH), A. H. Abragan, H. M. Rara / Eur. J. Pure Appl. Math, 15 (4) (2022), 1472-1481 1475 where CG and CH are supercliques in G and H, respectively. Lemma 2. [1] Let G = Kn for n > 1 and H a nontrivial 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. Theorem 3. [1] Let G = Kn for n > 1 and H a nontrivial connected graph with γ(H) ̸= 1. A subset S of V (G[H]) is a strong resolving set of G[H] if and only S = V (G[H])\(A×C), where A is a subset of V (G) and C = ∅ or C is a superclique in H. Lemma 3. [1] Let G = Kn for n > 1 and H a nontrivial 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 C∗ of H. Theorem 4. [7] Let G be a nontrivial connected graph and H a connected graph. A proper subset S of V (G ◦H) is a strong resolving 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 = ∪ ( ∪ u∈V (G)\{v} V (Hu ) ) ∪Bv for a unique v in V (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. Remark 1. Every restrained strong resolving hop dominating set of a connected graph G is a strong resolving set. Hence, sdim(G) ≤ γrsRh(G). Also, every restrained strong resolving hop dominating set of G is a hop dominating set. Thus, γh(G) ≤ γrsRh(G). Remark 2. For any connected graph G of order n, 1 ≤ γrsRh(G) ≤ n. Moreover, γrsRh(G) = 1 if G is a trivial graph and γrsRh(Kn) = n for n ≥ 1. The next result follows immediately from Lemma 1. Proposition 2. Let G be a nontrivial connected graph with diam(G) ≤ 2. Then S ⊆ V (G) is a restrained strong resolving hop dominating set of G if and only if S = V (G) \ C where C = ∅ or C is a nonsingleton hop dominated superclique in G. In particular, γrsRh(G) = |V (G)| − ωhS(G). 3. Join of Graphs Definition 1. [6] The join G+H of graphs G and 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) and v ∈ V (H)}. A. H. Abragan, H. M. Rara / Eur. J. Pure Appl. Math, 15 (4) (2022), 1472-1481 1476 Theorem 5. Let G be a nontrivial connected graph of order n with γ(G) ̸= 1 and K1 = ⟨v⟩. Then S ⊆ V (K1+G) is a restrained 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 of G. Proof : Let S be a restrained strong resolving set of K1+G. Since S is strong resolving, by Theorem 1, S = V (G) or S = V (G)\C∗ or S = V (K1+G)\C∗ where C∗ is a superclique in G. Since S is restrained hop dominating set in K1 +G, so S = V (G+K1) or ⟨V (G+K1) \S⟩ has no isolated vertex and v ∈ S. Hence , S ̸= V (G) and S = V (G) \C∗. Hence, S = V (K1 + G) \ C∗ where C∗ = ∅ or C∗ is a nonsingleton hop dominated superclique of G. The converse follows immediately from Theorem 1. Theorem 6. Let G be a nontrivial connected graph of order n with γ(G) = 1 and K1 = ⟨v⟩. Then S ⊆ V (K1+G) is a restrained strong resolving hop dominating set of K1+G if and only if S = (V (K1 + G) \ C) ∪ {x ∈ C : degG(x) = n − 1} where C = ∅ or C is a hop dominated superclique in G. Proof : Let S be a restrained strong resolving hop dominating set of K1 + G. Then by Theorem 1, S = V (G)orS = V (K1 +G) \ C∗orS = (V (G) \ C∗) ∪ {x ∈ C∗ : degG(x) = n− 1} where C∗ is a superclique of G. Since S is a restrained hop dominating set, S = V (K1+G) or ⟨V (K1 + G) \ S⟩ has no isolated vertex and v, x ∈ S where degG(x) = n − 1. Thus, S = (V (G) \ C∗) ∪ {x ∈ C∗ : degG(x) = n− 1} where C∗ = ∅ or C∗ is nonsingleton hop dominated superclique of G. The converse follows immediately from Theorem 1. Corollary 1. Let G be a nontrivial connected graph of order n. Then γrsRh(K1 +G) = n− ωhS(G) + 1. Corollary 2. Let Pn = [v1, v2, . . . , vn] and Cm = [c1, c2, . . . cm, c1] where n,m ≥ 4. (i) The set [ V (Pn) ∪ {v} ] \ {vk, vk+1} for k = 1, 2, . . . , n− 1 are the restrained strong resolving hop dominating sets of ⟨v⟩+ Pn. (ii) The sets [(V (Cm)∪{v})\{ci, ci+1}] and (V (Cm)∪{v}\{c1, cm}) for i = 1, 2, . . . ,m−1, are the restrained strong resolving hop dominating sets of ⟨v⟩+ Cn. Theorem 7. Let G be a disconnected graph whose components are Gi for i = 1, 2, . . . , n. A subset S of V (K1+G) is a restrained strong resolving hop dominating set of K1+G if and only if S = V (K1 +G) \ Ci where Ci = ∅ or Ci is a nonsingleton superclique of Gi. Proof : Let S be a restrained strong resolving hop dominating set of K1 + G. Then by Theorem 1, S = V (G) or S = V (G)\C∗ i or S = V (K1+G)\C∗ i where C∗ i A. H. Abragan, H. M. Rara / Eur. J. Pure Appl. Math, 15 (4) (2022), 1472-1481 1477 is a superclique of Gi and C∗ i = ∅ or C∗ i is a nonsingleton superclique in Gi. Since S is a restrained and hop dominating, S = V (K1+G) or ⟨V (K1+G)\S⟩ has no isolated vertex. Hence, S ̸= V (G) and S ̸= V (G) \ C∗ i where C∗ i = ∅ or C∗ i is a nonsingleton superclique in Gi. The converse follows immediately from Theorem 1. Corollary 3. Let Gi be connected graphs of order ni and G be a disconnected graph whose components are Gi for i = 1, 2, . . . ,m. Then, γrsRh(K1 +G) = m∑ i=1 ni −max{ωS(Gi) : i = 1, · · · ,m}. In the join of two graphs G and H, the results of Theorem 5 and Theorem 6 have already considered the case when G or H is trivial. Hence, the next result considers the characterizations of the restrained strong resolving hop dominating sets of nontrivial con- nected graphs G and H. The next result follows from Theorem 2. Theorem 8. Let G and H be nontrivial connected graphs of orders m and n, respectively. A subset S of V (G +H) is a restrained 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 nonsingleton hop dominated superclique of G. (ii) S = V (G+H) \ CH where CH is a nonsingleton hop dominated superclique of H. (iii) If γ(G) = 1 and γ(H) = 1, S = [V (G+H)\(CG∪CH)]∪{z ∈ CG : degG(z) = m−1} ∪{w ∈ CH : degH(w) = n−1} where CG and CH are hop dominated supercliques in G and H, respectively. (iv) If γ(G) ̸= 1 and γ(H) ̸= 1, S = [V (G+H) \ (CG ∪ CH)] = (V (G) \ CG) ∪ (V (H) \ CH) where CG and CH are hop dominated supercliques in G and H, respectively. Corollary 4. Let G and H be nontrivial connected graphs of orders m and n, respectively. Then γsrRh(G+H) = { (m− ωhS(G)) + (n− ωhS(H)) + 1, if γ(G) = 1 or γ(H) = 1 (m− ωhS(G)) + (n− ωhS(H)) , if γ(G) ̸= 1 and γ(H) ̸= 1. Example 1. Consider the graphs ⟨w⟩+ C4. Then γrsRh(⟨w⟩+ C6) = 3. A. H. Abragan, H. M. Rara / Eur. J. Pure Appl. Math, 15 (4) (2022), 1472-1481 1478 4. Corona of Graphs Definition 2. [8] The corona G ◦H of graphs G and H, is the graph obtained by taking one copy of G of order n and n copies of H, and then joining the ith vertex of G to every vertex of the ith copy of H. For every 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 9. Let G be a nontrivial connected graph and H a connected graph. A proper subset S ⊆ V (G ◦H) is a restrained strong resolving hop 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) and ⟨V (G) \A⟩ has no isolated vertex. (ii) S = A∪ ( ⋃ u∈V (G)\{v} V (Hu) ) ∪Bv for a unique vertex v in G, where A = V (G) \ {v} or ⟨V (G) \ (A∪{v})⟩ has no isolated vertex and Bv is a strong resolving hop dominating set of Hv + ⟨v⟩ if NG(v) ∩ A = ∅ and Bv is a strong resolving set where ⟨V (Hv) \ Bv⟩ has no isolated vertex if v ∈ A and NG(v) ∩A ̸= ∅. Proof : Suppose S is a restrained resolving hop dominating set of G ◦ H. Then S is a strong resolving hop dominating set and by Theorem 4, one of the following holds: (a) S = A ∪ ( ⋃ u∈V (G) V (Hu) ) , where A ⊆ V (G); (b) S = A∪ ( ⋃ u∈V (G)\{v} V (Hu) ) ∪Bv for a unique vertex v in G, where A ⊆ V (G)\{v} and Bv is a strong resolving set of Hv if γ(H) = 1 or Bv is a strong resolving set of ⟨v⟩+Hv if γ(H) ̸= 1. Suppose (a) holds. Since S is a proper restrained hop dominating subset of G ◦ H, ⟨V (G ◦H) \ S = V (G) \A⟩ has no isolated vertex. Thus, (i) holds. On the other hand, suppose (b) holds. Since S is a restrained hop dominating set of G ◦ H, S = A ∪ ( ⋃ u∈V (G)\{v} V (Hu) ) ∪ Bv for a unique vertex v ∈ V (G), where A = V (G) \ {v} or ⟨V (G) \ (A ∪ {v})⟩ has no isolated vertex and Bv is a strong resolving hop dominating set of Hv + ⟨v⟩ if NG(v) ∩A = ∅ and Bv is a strong resolving set, where ⟨V (Hv)\Bv⟩ has no isolated vertex if v ∈ A and NG(v)∩A ̸= ∅. Since v ∈ S, ⟨V (Hv)\Bv⟩ has no isolated vertex and Bv is a strong hop dominating set of Hv + ⟨v⟩, if v ∈ S and Bv is strong resolving set of Hv and ⟨V (Hv) \ Bv⟩ has no isolated vertex if v ∈ S. Thus (ii) holds. Conversely, suppose (i) and (ii) hold. By Theorem 4, S is a strong resolving set of G ◦H. If (i) holds, then ⟨V (G ◦H) \ S = V (G) \ A⟩ has no isolated vertex. If (ii) holds A. H. Abragan, H. M. Rara / Eur. J. Pure Appl. Math, 15 (4) (2022), 1472-1481 1479 then V (G ◦H) \ S = (V (G) \A) ∪ (V (Hv + ⟨v⟩) \Bv). Since A = V (G) \ {v} or ⟨V (G) \ (A ∪ {v})⟩ has no isolated vertex, ⟨V (G +H) \ S⟩ has no isolated vertex. In either case, ⟨V (G ◦H) \ S⟩ has no isolated vertex. Therefore, S is a restrained strong resolving hop dominating set of G ◦H. Corollary 5. Let G andH be nontrivial connected graphs of ordersm and n, respectively. Then, γrsRh(G ◦H) = (m− 1)n+ γsR(H +K1). Proof : Let S be a γrsRh-set of G ◦H. Then by Theorem 9 (ii), S = A ⋃ u∈V (G)\{v} V (Hu) ⋃ Bv for a unique vertex v in G and Bv is a strong resolving hop dominating set of Hv. Hence, γrsRh(G ◦H) = |S| = |V (H)||V (G) \ {v}|+ |Bv| ≥ (m− 1)n+ γsRh(H). Let Cv be a minimum strong resolving hop dominating set ofK1+Hv. For a unique ver- tex v ∈ V (G), let ⟨Bv⟩ ∼= ⟨Cv⟩. Then by Theorem 9, S = A ⋃( ⋃ u∈V (G)\{v} V (Hu) )⋃ Bv is a restrained strong resolving hop dominating set of G ◦H. Thus, γrsRh(G ◦H) ≤ |S| = ∣∣∣∣ ⋃ u∈V (G)\{v} V (Hu) ∣∣∣∣+ |Bv| = (m− 1)(n) + |Cv| = (m− 1)(n) + γsRh(K1 +H). Therefore, γrsRh(G ◦H) = (m− 1)n+ γsRh(K1 +H). Example 2. Consider the graph P3 ◦ P3. Then the minimum restrained strong resolving hop dominating set is γrsRh(P3 ◦ P3) = 8. 5. Lexicographic of Graphs Definition 3. [6] The lexicographic product of 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]) satisfying the following conditions: (u1, v1)(u2, v2) ∈ E(G[H]) if and only if either u1u2 ∈ E(G) or u1 = u2 and v1v2 ∈ E(H). Lemma 4. Let G = Kn for n > 1 and H a nontrivial connected graph with γ(H) ̸= 1. 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 superclique in H. A. H. Abragan, H. M. Rara / Eur. J. Pure Appl. Math, 15 (4) (2022), 1472-1481 1480 Proof : Suppose that A × C ⊆ V (G[H]) is a hop dominated superclique in G[H]. By Lemma 2 and Lemma 3, 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 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 1, 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 2, Lemma 3 and Proposition 1, 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 1, 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 10. Let G = Kn for n > 1 and H a nontrivial connected graph with γ(H) ̸= 1. A subset S of V (G[H]) is a restrained strong resolving dominating set of G[H] if and only if S = V (G[H]) \ (A× C) and one of the following is satisfied: (i) A ⊆ V (G) and C = ∅. (ii) A is a singleton subset of V (G) and C is a nonsingleton superclique in H. (iii) A is a nonempty nonsingleton subset of V (G) and C is a hop dominated superclique in H. Proof : Let S be a restrained strong resolving hop dominating set of G[H]. By Theorem 3, S = V (G[H]) \ (A× C) where A is a subset of V (G) and C = ∅ or C is a superclique in H. Since S is a restrained strong resolving set, S = V (G[H]) or ⟨V (G[H]) \ S⟩ has no isolated vertex. If S = V (G[H]) then A × C = ∅, showing that A ⊆ V (G) and C = ∅. Thus, (i) holds. If ⟨V (G[H]) \S⟩ has no isolated vertex, then A×C is a nonsingleton hop dominated superclique in G[H]. This implies that A is a singleton subset of V (G) and C is a nonsingleton superclique in H or A is nonempty nonsingleton subset of V (G) and C is hop dominated superclique in H. Hence (ii) or (iii) holds. For the converse, suppose S = V (G[H]) \ (A × C), where A and C satisfy (i),(ii) or (iii). Then, either A × C = ∅ or by Lemma 4, A × C is a nonsingleton hop dominated superclique in G[H]. By Theorem 3, S is a strong resolving set in G[H]. Since A × C is hop dominated superclique, S is a strong resolving hop dominating set of G[H]. If (i) is true, then A ⊆ V (G) and C = ∅, that is, A× C = ∅ and S = V (G[H]). If (ii) or (iii) is satisfied, then ⟨V (G[H]) \ S⟩ has no isolated vertex. Therefore, S is a restrained strong resolving hop dominating set G[H]. Example 3. Consider the graph of K3[P5] then the γrsRh(K3[P5]) = 13. REFERENCES 1481 Acknowledgments 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] P. Acal and H. Rara. The strong connected metric dimension in the join and corona of graphs. Advances and Applications in Discrete Mathematics, 21(1):91–101, 2019. [2] Sergio Canoy, Jr, Reynaldo Villarobe Mollejon, and John Gabriel E. Canoy. Hop dominating sets in graphs under binary operations. European Journal of Pure and Applied Mathematics, 12(4):1455–1463, Oct. 2019. [3] G. Chartrand, L. Eroh, M. Johnson, and O.R. Oellermann. Resolvability in graphs and the metric dimension of a graph the metric dimension of a graph. Discrete Applied Mathematics, 105:99–113, 2000. [4] Natarajan Chidambaram and S.K. Ayyaswamy. Hop domination in graphs-ii. Analele Stiintifice ale Universitatii Ovidius Constanta, Seria Matematica, 23:187–199, 06 2015. [5] Sumaoy H. and H. Rara. On restrained strong resolving domination in graphs. Euro- pean Journal of Pure and Applied Mathematics, 14(4):1367–1378, 2021. [6] F. Harary and R.A. Melter. On the metric dimension of a graph. Ars Combinatoria, 2:191–195, 1976. [7] Gerald Bacon Monsanto, Penelyn L. Acal, and Helen M. Rara. On strong resolving domination in the join and corona of graphs. European Journal of Pure and Applied Mathematics, 13(1):170–179, Jan. 2020. [8] Haynes T.W., Hedetniemi S., and P. Slater. (Fundamentals of Domination in Graphs (1st ed.)). CRC Press, 1998.