EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 14, No. 4, 2021, 1367-1378 ISSN 1307-5543 – ejpam.com Published by New York Business Global On Restrained Strong Resolving Domination in Graphs Helyn C. Sumaoy1, 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 dominating set in G if S is a strong resolving dominating set in G and S = V (G) or ⟨V (G) \ S⟩ has no isolated vertex. The restrained strong resolving domination number of G, denoted by γrsR(G), is the smallest cardinality of a restrained strong resolving dominating set in G. In this paper, we present characterizations of the restrained strong resolving dominating sets in the join, corona and lexicographic product of two graphs and determine the exact value of the restrained strong resolving domination number of each of these graphs. 2020 Mathematics Subject Classifications: 05C69 Key Words and Phrases: restrained strong resolving dominating set, restrained strong resolving domination number, join, corona, lexicographic product 1. Introduction Domination in graphs was first introduced by C. Berge in 1958 [1]. There are many studies involving domination and its variations. Slater [8] introduced and studied the concept of resolving set. In 2003, Robert Brigham et al. [18] linked the concepts of resolving and domination. In their article, they defined a resolving dominating set as a set that is both resolving and dominating. Resolving sets and resolving dominating sets were further studied in [2, 3]. Oellermann, O. R, and Peters-Fransen, J. [6] introduced and studied the concept of strong resolving set. Domke et. al [7] introduced and investigated the concept of restrained domination in graphs. Khuller, et. al. [11] introduced the concept of metric dimension and this has grown to become an interesting topic in graph theory. In line with this, Sebo and Tannier [13] in- troduced the concept of strong metric dimension, a concept which is more restrictive than ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v14i4.4112 Email addresses: helyn.sumaoy@g.msuiit.edu.ph (H. Sumaoy), helen.rara@g.msuiit.edu.ph, helenrara@gmail.com (H. Rara) http://www.ejpam.com 1367 © 2021 EJPAM All rights reserved. H. Sumaoy, H. Rara / Eur. J. Pure Appl. Math, 14 (4) (2021), 1367-1378 1368 metric dimension. After its introduction, Slater in [8] studied it and discovered its useful- ness when working with the United States sonar and Coast Guard Loran (long range aids to navigation) stations. Its applications arise in many diverse fields including chemistry, for representing chemical compounds [10], the robot navigation [11] and geographical rout- ing protocols [12], to name a few. In [13], an invariant called the strong metric dimension, was presented where the authors illustrated its application to combinatorial search. Along with the increasing discovery of its applications, theoretical studies on this invariant also appear in several number of other papers including [14], [15], [16], [17]. This paper intends to generate additional theoretical results and help widen the pool of existing studies from where new researchers may draw new insights and directions for further investigation. In this study, we investigate the concept of restrained strong resolving domination in the join, corona, and lexicographic product of graphs. Readers are referred to [9] for elementary Graph Theory concepts. Let G = ( V (G), E(G) ) be a graph. The open neighborhood of v ∈ V (G) is NG(v) = {u ∈ V (G) : uv ∈ E(G)}. An element u ∈ NG(v) is called a neighbor of v. The closed neighborhood v ∈ V (G) is NG[v] = NG(v) ∪ {v}. Thus, the degree of v ∈ V (G) is given by degG(v) = |NG(v)|. For S ⊆ V (G), NG(S) = ⋃ v∈S NG(v) and NG[S] = ⋃ v∈S NG[v]. A clique in a graph G is a complete induced subgraph. 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 ωS(G) of G is the cardinality of a maximum superclique in G. A superclique C is called a dominated superclique if for every u ∈ C there exists v ∈ V (G) \ C such that uv ∈ E(G). The dominated superclique number ωDS(G), of G is the cardinality of a maximum dominated superclique in G. 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 a basis. 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 subset S ⊆ V (G) is a strong resolving dominating set of G if S is a dominating set and for every pair of vertices u, v ∈ V (G), there exists a vertex w ∈ S such that u ∈ IG[v, w] or v ∈ IG[u,w]. The smallest cardinality of a strong resolving dominating set of G is called the strong resolving domination number of G and is denoted by γsR(G). A strong resolving dominating set of cardinality γsR(G) is called a γsR-set of G. H. Sumaoy, H. Rara / Eur. J. Pure Appl. Math, 14 (4) (2021), 1367-1378 1369 A set S ⊆ V (G) is a restrained dominating set of G if S is a dominating set of G and for every v ∈ V (G) \ S there exists u ∈ (V (G) \ S) ∩NG(v). Equivalently, a dominating subset S of V (G) is a restrained dominating set of graph G if S = V (G) or ⟨V (G) \ S⟩ has no isolated vertex. The restrained domination number of G, denoted by γr(G) is the minimum cardinality of a restrained dominating set of G. Any restrained dominating set of G of cardinality γr(G) is referred to as a γr-set of G. The join of two graphs G and H is the graph G + H 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 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). 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). 2. Preliminary Results This section presents some of the important known results in strong resolving domi- nation of graphs and some properties of restrained strong resolving dominating set in a graph. Theorem 1. [3] 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 dominating set of K1 + G if and only if S = V (G); or S = V (K1 +G) \ C or S = V (G) \ C∗ where C is a superclique and C∗ is dominated superclique in G. Theorem 2. [3] 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 connected resolving dominating set of K1 +G if and only if S = V (G) or S = V (K1 +G) \C or S = (V (G) \ C∗) ∪ {x ∈ C∗ : deg(x) = n− 1} where C and C∗ are superclique and dominated superclique, respectively in G. Theorem 3. [3] 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 dominating set of K1 +G if and only if S = V (G) or S = V (G) \ C∗ i or S = V (K1 +G) \ Ci where Ci is a superclique in Gi, for i = 1, 2, . . . ,m and C∗ i is a dominated superclique of Gi. Theorem 4. [3] Let G be a nontrivial 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: H. Sumaoy, H. Rara / Eur. J. Pure Appl. Math, 14 (4) (2021), 1367-1378 1370 (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) \ {v} and Bv is a strong resolving dominating set of Hv if γ(H) = 1 or Bv is a strong resolving dominating set of ⟨v⟩+Hv if γ(H) ̸= 1. Theorem 5. [3] Let G and H be non-trivial connected graphs of orders m and n, respectively. A proper subset S of V (G+H) is a strong resolving 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 superclique of G. (ii) S = V (G+H) \ CH where CH is a superclique of G. (iii) If γ(G) = 1 and γ(H) = 1, S = [V (G+H) \ (CG ∪ CH)] ∪ {z ∈ CG : degG(z) = m− 1} or S = [V (G+H) \ (CG ∪ CH)] ∪ {w ∈ CH : degH(w) = n− 1} where CG and CH are 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 supercliques in G and H, respectively. Lemma 1. [2] 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). Theorem 6. [2] 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. Remark 1. Every restrained strong resolving dominating set of a connected graph G is a strong resolving dominating set. Hence, γsR(G) ≤ γrsR(G). Also, every restrained strong resolving dominating set of G is a restrained dominating set. Thus,γr(G) ≤ γrsR(G). Remark 2. For any connected graph G of order n, 1 ≤ γrsR(G) ≤ n. Moreover, γ(G) = 1 if and only if G is a non-trivial graph and γrsR(Kn) = n for n ≥ 1. Proposition 1. Let G be a nontrivial connected graph with diam(G) ≤ 2. Then S ⊆ V (G) is a restrained strong resolving dominating set of G if and only if S = V (G)\C where C = ∅ or C is a nonsingleton dominated superclique in G. In particular, γrsR(G) = |V (G)| − ωDS(G) . H. Sumaoy, H. Rara / Eur. J. Pure Appl. Math, 14 (4) (2021), 1367-1378 1371 Proof : Let S be a restrained strong resolving dominating set of G. Let C = V (G) \ S. Then, S = V (G) \ C. Since S is a strong resolving, by Lemma 1, C = ∅ or C is a superclique in G. Since S is restrained dominating, S = V (G) or V (G) \S has no isolated vertex. Thus, C = ∅ or C is a singleton dominated superclique in G. The converse follows immediately from Lemma 1, definition of dominated superclique and definition of restrained dominating set. Suppose S is a γrsR-set of G. Then, S = V (G)\C where C is a nonsingleton dominated superclique in G and |C| = ωDS(G). Thus, γrsR(G) = |S| = |V (G)| − |C| = |V (G)| − ωDS(G). 3. Restrained Strong Resolving Domination in the Join of Graphs Theorem 7. Let G be a nontrivial connected graph of order n with γ(G) ̸= 1 . Then S ⊆ V (K1 + G) is a restrained strong resolving dominating set of K1 + G if and only if S = V (G) \ C or S = V (K1 +G) \ C∗ where C is a non-singleton dominated superclique and C∗ = ∅ or C∗ is non-singleton dominated superclique of G. Proof : Let S be a restrained strong resolving set of K1 + G. Since S is strong resolving dominating set by Theorem 1, S = V (G) or S = V (G) \ C or S = V (K1 + G) \ C∗ where C is a dominated superclique and C∗ is a superclique in G. Since S is restrained dominating, S = V (G+K1) or V (G+K1) \ S has no isolated vertex. Hence, S ̸= V (G) and S = V (G) \ C or S = V (K1 +G) \ C∗ where C is dominated supeclique and C∗ = ∅ or C∗ is a nonsingleton superclique of G. The converse follows immediately from Theorem 1, definitions of dominated super- clique and restrained dominating set of a graph. Theorem 8. 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 dominating set of K1+G if and only if S = V (K1 +G) \C or S = (V (G) \C∗)∪ {x ∈ C∗ : degG(x) = n− 1} where C = ∅ or C is nonsingleton superclique of G and C∗ is a superclique of G. Proof : Let S be a restrained strong resolving dominating set of K1 +G. Then by Theorem 2, S = V (G) or S = V (K1 +G) \C or S = (V (G) \C∗)∪ {x ∈ C∗ : degG(x) = n − 1} where C and C∗ are superclique and dominated superclique of G, respectively. Since S is restrained dominating, S = V (K1+G) or V (K1+G)\S has no isolated vertex. Thus, S ̸= V (G), C = ∅ or C is non-singleton dominated superclique of G and C∗ is a superclique of G. The converse follows immediately from Theorem 2, definitions of superclique and restrained dominating set of a graph. The next results follow immediately from Theorem 7. Corollary 1. Let G be a nontrivial connected graph of order n. Then γrsR(K1 +G) = { n− ωS(G) + 1 , if γ(G) = 1 n− ωDS(G) , if γ(G) ̸= 1. H. Sumaoy, H. Rara / Eur. J. Pure Appl. Math, 14 (4) (2021), 1367-1378 1372 Corollary 2. Let G be a connected graph with diam(G) ≤ 2. Then γrsR(K1 +G) = { sdim(G) + 1 , if γ(G) = 1 γsR(G) , if γ(G) ̸= 1. Corollary 3. Let Pn = [v1, v2, . . . , vn] and Cm = [c1, c2, . . . cm, c1] where n,m ≥ 3. (i) The sets V (Pn) \ {vi, vi+1} for i = 2, 3, . . . , n− 2, V (Pn) \ {vj} for j = 1, 2, . . . , n and V (Pn)∪{v} \ {vk, vk+1} for k = 1, 2, . . . , n− 1 are the restrained strong resolving dominating sets of ⟨v⟩+ Pn. (ii) The sets V (Cm) \ {ci, ci+1}, (V (Cm)∪ {v}) \ {ci, ci+1}, V (Cm) \ {vi}, V (Cm) \ {vm}, V (Cm) \ {ci, cm} and (V (Cm)∪ {v} \ {c1, cm}) for i = 1, 2, . . . ,m− 1 , are the restrained strong resolving dominating sets of ⟨v⟩+ Cn. Theorem 9. 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 dominating set of K1 +G if and only if S = V (G) \ Ci or S = V (K1 + G) \ C∗ i where Ci is a dominated superclique of Gi and C∗ i = ∅ or C∗ i is a nonsingleton superclique of G. Proof : Let S be a restrained strong resolving dominating set of K1 +G. Then by Theorem 3 S = V (G) or S = V (G) \ Ci or S = V (K1 + G) \ C∗ i where Ci is a dominated superclique of Gi and C∗ i = ∅ or C∗ i is a nonsingleton superclique in Gi. Since S is restrained dominating, S = V (K1 + G) or V (K1 + G) \ S has no isolated vertex. Hence, S ̸= V (G) and C∗ i = ∅ or C∗ i is nonsingleton. Therefore, S = V (G)\C1 or S = V (K1 +G) \C∗ i where Ci is a dominated superclique in Gi and C∗ i is a nonsingleton superclique in Gi. The converse follows immediately from Theorem 3, definitions of dominated super- clique and restrained dominating set of a graph. Corollary 4. Let Gi be connected graphs of order ni and G be a disconnected graph whose components are Gi for i = 1, 2, . . . ,m. Then, γrsR(K1 + G) = ∑m i=1 ni − rG where rG = max {max{ωDS(Gi)},max{ωS(Gi) + 1} : i = 1, 2, . . . ,m} . In the join of two graphs G and H, the results Theorem 7 and Theorem 8 have already considered the case when G or H is trivial. Hence, the next results, considered the characterizations of the restrained strong resolving dominating sets of nontrivial connected graphs G and H. Theorem 10. LetG andH be nontrivial connected graphs of ordersm and n, respectively. A subset S of V (G +H) is a restrained strong resolving 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 superclique of G. (ii) S = V (G+H) \ CH where CH is a nonsingleton superclique of G. H. Sumaoy, H. Rara / Eur. J. Pure Appl. Math, 14 (4) (2021), 1367-1378 1373 (iii) If γ(G) = 1 and γ(H) = 1, S = [V (G+H) \ (CG ∪ CH)] ∪ {z ∈ CG : degG(z) = m− 1} or S = [V (G+H) \ (CG ∪ CH)] ∪ {w ∈ CH : degH(w) = n− 1} where CG and CH are 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 supercliques in G and H, respectively. Proof : Let S be a restrained strong resolving dominating set of G+H. By Theorem 5 (a) S = V (G+H) \ CG where CG is a superclique of G. (b) S = V (G+H) \ CH where CH is a superclique of G. (c) If γ(G) = 1 and γ(H) = 1, S = [V (G+H) \ (CG ∪ CH)] ∪ {z ∈ CG : degG(z) = m− 1} or S = [V (G+H) \ (CG ∪ CH)] ∪ {w ∈ CH : degH(w) = n− 1} where ⟨CG⟩ and CH are supercliques in G and H, respectively. (d) 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 supercliques in G and H, respectively. Since S is a restrained dominating set of G+H, (i), (ii), (iii), (iv) hold. The converse immediately follows from Theorem 5 and from definition of restrained dominating set of a graph. Corollary 5. Let G andH be nontrivial connected graphs of ordersm and n, respectively. Then sdim(G+H) =  (m− ωs(G)) + (n− ωs(H)) + 1, if γ(G) = 1 or γ(H) = 1 (m− ωs(G)) + (n− ωs(H)) , if γ(G) ̸= 1 and γ(H) ̸= 1. 4. Restrained Strong Resolving Domination in the Corona of Graphs This section gives characterization of the restrained strong resolving dominating sets in the corona of graphs as well as its restrained strong resolving domination number. Theorem 11. Let G be a nontrivial connected graph and H a connected graph. A proper subset S ⊆ V (G ◦H) of a restrained 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) and V (G) \A has no isolated vertex. H. Sumaoy, H. Rara / Eur. J. Pure Appl. Math, 14 (4) (2021), 1367-1378 1374 (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 dominating set of Hv + ⟨v⟩ if v ∈ S and Bv is a strong resolving set where V (Hv) \ Bv has no isolated vertex if v ∈ S. Proof : Suppose S is a restrained resolving dominating set of G ◦H. Then S is a strong resolving dominating 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 dominating set of Hv if γ(H) = 1 or Bv is a strong resolving dominating set of ⟨v⟩+Hv if γ(H) ̸= 1. Suppose (a) holds. Since S is a proper restrained dominating subset of G ◦H, V (G ◦ H) \ S = V (G) \A has no isolated vertex. Thus, (i) holds. On the other hand if (b) holds, then since S is a restrained dominating set and V (G ◦ H) \ S = (V (G)\A)∪ (V (Hv+ ⟨v⟩)\Bv), A = V (G)\{v} or V (G)\ (A∪{v}) has no isolated vertex and Bv is a strong resolving dominating set of Hv + ⟨v⟩. Since v ∈ S, V (Hv) \Bv has no isolated vertex and Bv is a strong resolving 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. Hence, (ii) holds. Conversely, suppose (i) and (ii) hold. By Theorem 4, S is a strong resolving dominating set of G ◦H. If (i) holds, then V (G ◦H) \ S = V (G) \ A has no isolated vertex. If (ii) holds 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 dominating set of G ◦H. Corollary 6. Let G andH be nontrivial connected graphs of ordersm and n, respectively. Then, γrsR(G ◦H) = (m− 1)n+ γsR(H +K1). Proof : Let S be a γrsR-set of G ◦H. Then by Theorem 11 (ii), S = ⋃ u∈V (G)\{v} V (Hu) ⋃ Bv for a unique vertex v in G and Bv is a strong resolving dominating set of Hv. Hence, γrsR(G ◦H) = |S| = |V (H)||V (G) \ {v}|+ |Bv| ≥ (m− 1)n+ γsR(H) H. Sumaoy, H. Rara / Eur. J. Pure Appl. Math, 14 (4) (2021), 1367-1378 1375 Let Cv be a minimum strong resolving dominating set of Hv. For a unique vertex v ∈ V (G), let ⟨Bv⟩ ∼= ⟨Cv⟩. Then by Theorem 11 S = ( ⋃ u∈V (G)\{v} V (Hu) )⋃ Bv is a restrained strong resolving dominating set of G ◦H. Thus, γrsR(G ◦H) ≤ |S| = ∣∣∣∣ ⋃ u∈V (G)\{v} V (Hu) ∣∣∣∣+ |Bv| = (m− 1)(n) + |Cv| = (m− 1)(n) + γsR(H) Therefore, γrsR(G ◦H) = (m− 1)n+ γsR(H). 5. Restrained Strong Resolving Domination in the Lexicographic Product of Graphs Lemma 2. Let G = Kn for n > 1 and H a nontrivial connected graph with γ(H) ̸= 1 . Then A×C ⊆ V (G[H]) is a dominated superclique in G[H] if and only if A is a nonempty subset of V (G) and C is a superclique in H. Proof : Let G = Kn for n > 1 and H a nontrivial connected graph with γ(H) ̸= 1. Suppose A× C ⊆ V (G[H]) is a dominated superclique in G[H]. By a Lemma in [3], A is a nonempty subset of V (G) and C is a superclique in H. Conversely, suppose A ⊆ V (G), A ̸= ∅ and C is a superclique in H. Then A× C ⊆ V (G[H]) is a superclique in G[H] by a Lemma in [3]. We show that A× C is a dominated superclique . Let (a, b) ∈ A×C. Then a ∈ A and b ∈ C. Since γ(H) ̸= 1, there exists d ∈ V (H)\NH(b). Hence, d /∈ C. Since G = Kn for n > 1, a vertex v ∈ V (Kn)\{a} exists where av ∈ E(Kn). Thus, (v, d) /∈ A × C and (a, b)(v, d) ∈ E(G[H]). Therefore, A× C is a dominated superclique in G[H]. Theorem 12. 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 singleton subset of V (G) and C is a nonsingleton suerclique in H. (iii) A is nonempty nonsingleton subset of V (G) and C is a superclique in H. Proof : Let S be a restrained strong resolving dominating set of G[H]. By Theorem 6, 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 restrained strong resolving dominating, S = V (G[H]) or V (G[H]) \ S has H. Sumaoy, H. Rara / Eur. J. Pure Appl. Math, 14 (4) (2021), 1367-1378 1376 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 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 a 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 2, A × C is a nonsingleton dominated superclique in G[H]. By Theorem 6, S is a strong resolving set in G[H]. Since A×C is a dominated superclique, S is a strong resolving 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 dominating set G[H]. Lemma 3. Let G = Kn for n > 1 and H a nontrivial connected graph with γ(H) = 1. Then A×C ⊆ V (G[H]) is a dominated superclique of 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. Proof : Suppose A×C ⊆ V (G[H]) is a dominated superclique in G[H]. Then A×C ̸= ∅. By a Lemma in [3], 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. Conversely, suppose A ⊆ V (G) and A ̸= ∅ and C is a superclique in H such that |A| = 1 whenever C ∩ C∗ ̸= ∅ for some γ-set C∗ of H. Then by a Lemma in [3], A × C is a superclique in G[H]. We show that A × C is a dominated superclique in G[H]. Let (a, b) ∈ A × C. Then a ∈ A and b ∈ C. If C ∩ C∗ = ∅ for all γ-set C∗ of H, then there exists d ∈ V (H) \NH(b). Thus, d /∈ C. Since G = Kn for n > 1, a vertex y ∈ (V (Kn) \ {a}) ∪NG(a) exists. This implies that (y, d) ∈ [V (G[H]) \ (A× C)] ∩NG[H]((a, b)). Suppose, that C ∩ C∗ ̸= ∅ for some γ-set C∗ of H. Then |A| = 1. Let w ∈ A and p ∈ C ∩ C∗. Since G = Kn for n > 1, there exists u ∈ (V (Kn) \ {w}) ∩NG(w) and (u, v) /∈ (A× C) for all v ∈ V (H). Thus, (w, p)(u, v) ∈ E(G[H]). Therefore A× C is a dominated superclique in G[H]. Using Lemma 3, the proof of the next result will just be similar to that of Theorem 12. Theorem 13. 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 nonempty nonsingleton subset of V (G) and C is a superclique in H if C∩C∗ = ∅ for all γ-set C∗ in H. REFERENCES 1377 (iii) A is a singleton subset of V (G) and C is a nonsingleton superclique inH if C∩C∗ ̸= ∅ for some γ-set C∗ of H. Corollary 7. Let G = Kn for n > 1 and H a non-trivial connected graph of order m. Then γrsR(G[H]) = mn− ωDS(G[H]). Proof : Let S be a γrsR-set of G[H]. Then S is a restrained strong resolving dominating set of G[H]. By Theorem 12 or Theorem 13, S = V (G[H]) \ (A × C) where A × C is a dominated superclique in G[H]. Since S is a γrsR-set, A × C is a maximum dominated superclique of G[H]. Hence, γrsR(G[H]) = |S| = |V (G[H])| − |A× C| = mn− ωDS(G[H]). By Lemma 2 and Lemma 3, the dominated superclique A× C of Kn[H] is maximum if A = V (G) and C is a maximum superclique in H. Thus, the next result follows. Corollary 8. Let G = Kn for n > 1 and H be a connected graph of order m > 1. Then γrsR(G[H]) = n(m− ωS(H)). Acknowledgements This research is funded by the Department of Science and Technology - Accelerated Sci- ence and Technology Human Resource Development Program (DOST-ASTHRDP), Philip- pines. References [1] C. Berge. Theorie des graphes et ses applications. Metheun and Wiley, London and New York, 1962. [2] Acal P.L. and Rara H.M. 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