EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 13, No. 1, 2020, 170-179 ISSN 1307-5543 – www.ejpam.com Published by New York Business Global On Strong Resolving Domination in the Join and Corona of Graphs Gerald B. Monsanto1,∗, Penelyn L. Acal2, Helen M. Rara3 1 Department of Teacher Education, Visayas State University-Villaba, 6537 Villaba, Leyte, Philippines 2 Department of Mathematical Sciences, University of Science and Technology of Southern Philippines, 9023, Cagayan de Oro City, Philippines 3 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. Let G be a connected graph. 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. In this paper, we characterize the strong resolving dominating sets in the join and corona of graphs and determine the bounds or exact values of the strong resolving domination number of these graphs. 2020 Mathematics Subject Classifications: 05C69 Key Words and Phrases: Strong resolving dominating set, strong resolving domination number, join, corona 1. Introduction All graphs considered in this study are finite, simple, and undirected connected graphs, that is, without loops and multiple edges. For some basic concepts in Graph Theory, we refer readers to [4]. Let G = ( V (G), E(G) ) be a connected graph. The open neighborhood NG(v) = {u ∈ V (G) : uv ∈ E(G)}. Any element u of NG(v) is called a neighbor of v. The closed neighborhood NG[v] = NG(v) ∪ {v}. Thus, the degree of v is given by degG(v) = |NG(v)|. Customarily, for S ⊆ V (G), NG(S) = ⋃ v∈S NG(v) and NG[S] = ⋃ v∈S NG[v]. ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v13i1.3625 Email addresses: monger2006@yahoo.com (G. Monsanto), penelyn.acal@g.msuiit.edu.ph (P. Acal), helen.rara@g.msuiit.edu.ph (H. Rara) http://www.ejpam.com 170 c© 2020 EJPAM All rights reserved. G. Monsanto, P. Acal, H. Rara / Eur. J. Pure Appl. Math, 13 (1) (2020), 170-179 171 A nonempty set S ⊆ V (G) is a dominating set in graph G if NG[S] = V (G). Otherwise, we say S is a non-dominating set of G. 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}. If |S| = γ(G), then S is said to be a minimum dominating set or γ-set of G. A vertex w ∈ S 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 subset S ⊆ V (G) is a strong resolving dominating set of G if it is both strong resolving and dominating. 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. A clique in a graph G is a complete induced subgraph of G. A clique C in 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 in G is called a dominated superclique if for every u ∈ C, there exists v ∈ V (G) \ C such that uv ∈ E(G) [3]. A superclique (resp. dominated superclique) C is maximum in G if |C| ≥ |C∗| for all supercliques (resp. dominated supercliques) C∗ in G. The superclique (resp. domi- nated superclique) number, ωS(G) ( resp. ωDS(G) ) of G is the cardinality of a maximum superclique (resp. maximum dominated superclique) in G. In recent years, the concept of domination in graphs has been studied extensively and several research papers have been published on this topic. The said concept was not formally defined mathematically until the publications of the books by Claude Berge [1] in 1958 and Oystein Ore in 1962. In 1977, a survey paper by Cockayne and Hedetniemi [2] began to study the concept of domination. On the other hand, the problem of uniquely recognizing the possible position of an in- truder such as fault in a computer network and spoiled device was the principal motivation in introducing the concept of metric dimension in graphs. Slater [6] brought in the notion of locating sets and its minimal cardinality as locating number. The same concept was also introduced by Harary and Melter [4] but using the terms resolving sets and metric dimension to refer to locating sets and locating number, respectively. In 2007, Oellerman and Peter-Fransen [5] introduced the strong resolving graph GSR of a connected graph G as a tool to study the strong metric dimension of G. This study aims to define and characterize the strong resolving dominating sets and determine the exact values or bounds in the join and corona of two graphs. 2. Preliminary Results Remark 1. Every strong resolving dominating set of a connected graph G is a dominating set. Hence, γ(G) ≤ γsr(G). Remark 2. Every strong resolving dominating set of a connected graph G is a strong resolving set. Thus, sdim(G) ≤ γsr(G). G. Monsanto, P. Acal, H. Rara / Eur. J. Pure Appl. Math, 13 (1) (2020), 170-179 172 Remark 3. For any connected graph G of order n, 1 ≤ γsr(G) ≤ n− 1. Remark 4. Any superset of a strong resolving dominating set is a strong resolving dom- inating set. Proposition 1. Let G be a connected graph of order n ≥ 2. Then, (i) γsr(Pn) = ⌈ n+1 3 ⌉ (ii) γsr(Kn) = n− 1 (iii) γsr(Cn) =  2 , if n = 3 n− 2 , if n > 3 and n is odd⌈ n 2 ⌉ , if n > 3 and n is even Proposition 2. Let G be a connected graph of order n and let A = {x ∈ V (G) : degG(x) = n− 1} . If A 6= ∅ and C is a superclique in G, then |C ∩ A| ≤ 1. Moreover, if C is a maximum superclique of G, then |C ∩A| = 1. Remark 5. Let G be a nontrivial connected graph with diam(G) ≤ 2. For distinct vertices u, v, w ∈ G, u ∈ IG[v, w] if and only if dG(v, w) = 2 and u ∈ NG(v) ∩NG(w). Proposition 3. 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). Proof: Assume that S is a strong resolving set of G. If S∩V (G) = V (G), then C = ∅. Suppose S ( V (G). Let C = V (G) \ S. Then S = V (G) \ C. Let x, y ∈ C, where x 6= y. Since S is a strong resolving set of G, x and y are strongly resolved by some z ∈ S. We may assume that x ∈ IG[y, z]. Then dG(y, z) = 2 and x ∈ NG(y) ∩ NG(z) by Remark 5. Thus, z ∈ NG(x) \NG(y), showing that C is a superclique in G. Conversely, assume that S = V (G) \ C, where C is a superclique in G. Let x, y /∈ S, where x 6= y. Then x, y ∈ C. Since C is a superclique in G, there exists z ∈ S such that z ∈ NG(x)\NG(y) or z ∈ NG(y)\NG(x). Since diam(G) = 2, dG(y, z) = 2 or dG(x, z) = 2. By Remark 5, x ∈ IG[y, z] or y ∈ IG[x, z]. Hence, S is a strong resolving set of G. Suppose S is a strong resolving set of G. Then S = V (G)\C, where C is a superclique in G and |C| = ωS(G). Thus, sdim(G) = |S| = |V (G)| − |C| = |V (G)| − ωS(G). 3. On Strong Resolving Domination in the Join of Graphs 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)}. G. Monsanto, P. Acal, H. Rara / Eur. J. Pure Appl. Math, 13 (1) (2020), 170-179 173 Remark 6. For the joins 〈v〉+Pn and 〈w〉+Cn, it can be verified that γsr ( 〈v〉+Pn ) = n−1 for n ≥ 3 and γsr ( 〈w〉+ Cn ) = n− 2 for n ≥ 4. Proposition 4. Let G be a connected graph with γ(G) 6= 1 and let K1 = 〈v〉 . Then C ⊆ V (K1 +G) is a superclique of K1 +G if and only if |C| = 1 or |C| ≥ 2 and C \ {v} is a superclique of G. Proof: The conditions follow immediately if C ⊆ V (K1+G) is a superclique of K1+G. For the converse, the case when |C| = 1 is obvious. Suppose |C| ≥ 2. Since C \ {v} is a superclique of G, we only need to consider the pair of distinct vertices z, v ∈ C. Since γ(G) 6= 1, there exists w ∈ V (G) such that zw /∈ E(G). Since diam(K1 + G) = 2, dG(z, w) = 2. Hence, z ∈ NG(v)\NG(w), showing that C is a superclique of K1+G. Theorem 1. Let G be a nontrivial connected graph of order n with γ(G) 6= 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 and C∗ are superclique and dominated superclique, respectively, in G. Proof: Let S be a strong resolving dominating set of K1 + G. Suppose γ(G) 6= 1. If v /∈ S, then S ( V (G). By Proposition 3, S = V (K1 +G)\ (C ∪{v}) = V (G)\C, where S is a dominating set in K1 +G and C∪{v} is a superclique in K1 +G. By Proposition 4, C is a superclique in G. Since {v} is a superclique in K1 +G,S = V (K1 +G) \ {v} = V (G). On the other hand, if v ∈ S and C = V (K1 +G) \S, then S = V (K1 +G) \C where C is a superclique in K1 +G by Proposition 3. By Proposition 4, C \ {v} = C is a superclique in G. Conversely, the case when S = V (G) and S = V (K1 + G) \ C follows immediately from Proposition 3. Suppose S = V (G) \ C∗, where C∗ is a dominated superclique in G. By Proposition 4, C ∪ {v} is a superclique of K1 + G. Since v /∈ S, S = V (G) \ C∗ = V (K1 + G) \ ( C ∪ {v} ) . By Proposition 3, S is a strong resolving dominating set of K1 +G. Theorem 2. 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∗) ∪ {x ∈ C∗ : deg(x) = n− 1} where C and C∗ are superclique and dominated superclique, respectively, in G. Proof: Let S be a strong resolving dominating set of K1 + G. Suppose γ(G) = 1. If v ∈ S and C = V (K1+G)\S, then S = V (K1+G)\C = {v}∪(V (G)\C). By Proposition 3, C is a superclique in K1 + G. Hence for x, y ∈ C, x 6= y, there exists w ∈ V (G) \ C such that w ∈ NG(x) \ NG(y) or w ∈ NG(y) \ NG(x), showing that C is a superclique in G. On the other hand, if v /∈ S, then S ( V (G). Let C = V (K1 + G) \ S. Hence, S = V (K1 +G)\C = V (G)\C. By Proposition 3, C is a superclique in K1 +G. Hence, C is also a superclique in G. Since γ(G) = 1, AG = {z ∈ V (G); degG(z) = n− 1} 6= ∅. By Proposition 2, |C ∩ AG| = 1. Let z ∈ C ∩ AG. Since dK1+G(z, v) = 1 and dK1+G(v) = n, none of the elements in S strongly resolves z and v, a contradiction. Hence, z ∈ S. Thus, S = (V (G) \ C) ∪ {z} . In addition, since {v} is a superclique in K1 + G, S = G. Monsanto, P. Acal, H. Rara / Eur. J. Pure Appl. Math, 13 (1) (2020), 170-179 174 V (K1 +G) \ {v} = V (G). Similarly, if C∗ = V (G) \S, then C∗ is a dominated superclique in G. For the converse, the case when S = V (G) is trivial. Suppose S = V (K1 + G) \ C, where C is a superclique in G. Let x, y /∈ S, x 6= y. Then x, y ∈ C and there exists w ∈ V (G) \ C such that xy ∈ E(G) and w ∈ NG(x) \ NG(y) or w ∈ NG(y) \ NG(x). By Remark 5, x ∈ IK1+G[y, w] or y ∈ IK1+G[x,w], showing that S is a strong resolving dominating set of K1 +G. Suppose S = (V (G) \ C∗) ∪ {z ∈ C∗; degG(z) = n− 1}, where C∗ is a dominated su- perclique in G. Let x, y /∈ S, x 6= y. Then x, y ∈ C∗. By the same argument above, there exists w ∈ S that strongly resolves x and y. Now, consider the vertices x and v. Since x /∈ S, then degG(x) < n − 1. Hence, there exists z ∈ V (G) such that xz /∈ E(G). It follows that v ∈ IK1+G[x, z]. Thus, S is a strong resolving dominating set of K1+G. Corollary 1. 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 . . . , n− 2 are the strong resolving dominating sets of 〈v〉+ Pn. (ii) The sets V (Cm)\{ci, ci+1} and V (Cm)\{c1, cm}, for i = 1, 2 . . . ,m−1 are the strong resolving dominating sets of 〈v〉+ Cm. Corollary 2. Let G be a nontrivial connected graph of order n. Then (i) for γ(G) = 1, we have γsr(K1 +G) = n− ωS(G) + 1; (ii) for γ(G) 6= 1, we have γsr(K1 +G) = min { γsr(G), n− ωS(G) } . The next result follows from Proposition 3, Theorem 1 and Theorem 2. Corollary 3. Let G be nontrivial connected graph with diam(G) ≤ 2. Then (i) for γ(G) = 1, we have γsr(K1 +G) = sdim(G) + 1; (ii) for γ(G) 6= 1, we have γsr(K1 +G) = min { γsr(G), sdim(G) + 1 } . The following theorem gives a characterization of the strong resolving dominating sets in the join of K1 and a disconnected graph G. Theorem 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. Proof: Let S be a strong resolving dominating set of K1+G. Suppose v /∈ S. Then S ( V (G). Let Ci = V (K1 +G)\S, for i = 1, 2, . . . ,m. Then S = V (K1 +G)\Ci = V (G)\Ci. Let x, y ∈ Ci, x 6= y. Since dK1+G(w, x) = dK1+G(w, y), for all w ∈ V (G) \ V (Gi), there exists z ∈ V (Gi) \ Ci such that x ∈ IGi [y, z] or y ∈ IGi [x, z]. By Remark 5, G. Monsanto, P. Acal, H. Rara / Eur. J. Pure Appl. Math, 13 (1) (2020), 170-179 175 x ∈ NGi(y) \NGi(z) or y ∈ NGi(x) \NGi(z). Thus, Ci is a superclique in Gi. Since {v} is a superclique in K1 +G, by Proposition 3, S = ( V (K1 +G) \ {v} ) = V (G). On the other hand, if v ∈ S and Ci = V (K1 + G) \ S, for i = 1, 2, . . . ,m, then S = V (K1 + G) \ Ci, where Ci is a superclique in K1 + G, by Proposition 3. Hence, Ci is a superclique in Gi. Similarly, if C∗i = V (G) \ S for i = 1, 2, . . . ,m where C∗i is a dominated superclique of Gi and since S is dominating, then V (Gi) \ C∗i is a dominating set of Gi. For the converse, if S = V (G), then we are done. Suppose S = V (G) \ C∗i , or S = V (K1 +G)\Ci, where Ci and C∗i are superclique and dominated superclique, respectively, in Gi for i = 1, 2, . . . ,m. Then by Theorem 1 and Theorem 2, V (Gi) \ C∗i is a strong resolving dominating set of K1 + Gi. By Remark 4, S = V (K1 + G) \ Ci is a strong resolving dominating set of K1 +G. Corollary 4. Let Gi be connected graphs of orders ni and G be a disconnected graph whose components are Gi for i = 1, 2, . . . ,m and Si = V (Gi) \Ci where Ci is a maximum dominated superclique of Gi. Then γsr(K1 +G) = m∑ i=1 ni −max { γsr(Gi), ωDS(Gi) + 1 ∣∣i = 1, 2, . . . ,m } . In the join of two graphs G and H, the previous results have already considered the case when G or H is trivial. Hence, in the following theorem, a characterization of the strong resolving dominating sets in the join of nontrivial connected graphs G and H is considered. Theorem 4. Let G and H be nontrivial 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 in G. (ii) S = V (G+H) \ CH where CH is a superclique in H. (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) 6= 1 and γ(H) 6= 1, S = [V (G+H) \ (CG ∪ CH)] = (V (G) \ CG) ∪ (V (H) \ CH), where CG and CH are supercliques in G and H, respectively. G. Monsanto, P. Acal, H. Rara / Eur. J. Pure Appl. Math, 13 (1) (2020), 170-179 176 Proof: Let S be a strong resolving dominating set of G+H. Since dG+H(x, y) = 1, for each x ∈ V (G) and y ∈ V (H), none of the vertices in V (G) and V (H) strongly resolves any pair of distinct vertices in V (H) and V (G), respectively. Thus, S ∩ V (G) 6= ∅ and S ∩ V (H) 6= ∅. If S ∩ V (H) = V (H), then S 6= V (G). Let CG = V (G) \ S. Hence, S = V (G + H) \ CG. Let u, v ∈ CG, u 6= v. Then there exists w ∈ S ∩ V (G) such that u ∈ IG+H [v, w] or v ∈ IG+H [u,w]. By Remark 5, w ∈ NG(u)\NG(v) or w ∈ NG(v)\NG(u). Thus, CG is a superclique in G. Similarly, S ∩ V (G) = V (G). On the other hand, if S ∩ V (G) 6= V (G), S ∩ V (H) 6= V (H) , CG = V (G) \ S and CH = V (H) \ S, then S = V (G+H)\ (CG∪CH). Hence, CG and CH are supercliques in G and H, respectively. Suppose γ(G) = 1 and γ(H) = 1. Then AG = {zG ∈ V (G); degG(z) = m− 1} 6= ∅ and AH = {zH ∈ V (H); degH(z) = m− 1} 6= ∅. By Proposition 2, |CG ∩ AG| ≤ 1 and |CH ∩ AH | ≤ 1. Hence, we may assume that there exists zG ∈ CG ∩ AG and zH ∈ CH ∩ AH . Then none of the vertices in S ∩ V (G) and S ∩ V (H) strongly resolves zG and zH , a contradiction. Thus zG ∈ S or zH ∈ S so that 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}. Suppose γ(G) 6= 1 or γ(H) = 1. Then AG = ∅ or AH = ∅. Hence, S = V (G+H) \ (CG ∪ CH) = (V (G) \ CG) ∪ (V (G) \ CG). Conversely, suppose S satisfies condition (i). Since CG is a superclique in G, there exists w ∈ (V (G) \CG) ⊆ S such that u ∈ IG[v, w] or v ∈ IG[u,w], for any u, v /∈ S, u 6= v, showing that w strongly resolves u, v. A similar argument applies if S satisfies condition (ii). Suppose S satisfies condition (iii) or (iv). Let u, v /∈ S, u 6= v. If u, v ∈ CG or u, v ∈ CH , then we are done. Consider the pair u ∈ V (G) \ S and v ∈ V (H) \ S. Since CG is a superclique in G, there exists z ∈ (V (G) \ CG) ⊆ S such that z ∈ NG(u) \NG(v) or z ∈ NG(v) \ NG(u). Hence, u ∈ IG[v, z] or v ∈ IG[u, z]. Thus, S is a strong resolving dominating set of G+H. Corollary 5. Let G and H be nontrivial connected graphs of orders m and n, respectively. Then γsr(G+H) = { (m− ωS(G)) + (n− ωS(H)) + 1, if γ(G) = 1 and γ(H) = 1 (m− ωS(G)) + (n− ωS(H)), if γ(G) 6= 1 or γ(H) 6= 1. Remark 7. If G is a nontrivial connected graph with γ(G) = 1, then diam(G) ≤ 2. Corollary 6. Let G and H be nontrivial connected graphs with γ(G) = 1 and γ(H) = 1. Then γsr(G+H) = sdim(G) + sdim(H) + 1. In particular, G. Monsanto, P. Acal, H. Rara / Eur. J. Pure Appl. Math, 13 (1) (2020), 170-179 177 (i) γsr(G+H) = 3 for G = Pm and H = Pn (m ≥ 2, n ≥ 2); (ii) γsr(G+H) = ⌈ n 2 ⌉ + 2 for G = Pm and H = Cn (m ≥ 2, n ≥ 3); (iii) γsr(G+H) = 4 for G = Cm and H = Cn (m = n = 3) (iv) γsr(G+H) = ⌈ m 2 ⌉ + ⌈ n 2 ⌉ + 1 for G = Cm and H = Cn (m,n ≥ 4) Theorem 5. Let G be a disconnected graph with components G1, . . . , Gn and H a dis- connected graph with components H1, . . . ,Hm. A proper subset S of V (G+H) is a strong resolving dominating set of G+H if and only if S satisfies any of the following: (i) S = SG ∪ V (H) where V (G) \ SG is a superclique of Gi for some i ∈ {1, 2, . . . , n} ; (ii) S = SH ∪ V (G) where V (H) \ SH is a superclique of Hj for some j ∈ {1, 2, . . . ,m} ; (iii) S = SG ∪ SH , where V (G) \ SG and V (H) \ SH are supercliques of Gi and Hj , for some i ∈ {1, 2, . . . , n} and some j ∈ {1, 2, . . . ,m}. Proof: Let S be a strong resolving dominating set of G+H and x ∈ Gi and y ∈ Gk, i 6= k. Since dG+H(x, y) = 2 and dG+H(x, h) = dG+H(y, h) = 1, for all h ∈ V (H), then x ∈ S or y ∈ S. Hence, S ∩ V (G) = ∅. Similarly, S ∩ V (H) 6= ∅. Let SG = S ∩ V (G) and SH = S ∩ V (H). Suppose S ∩ V (H) = V (H). Then SG ⊆ V (G). Let CG = V (G) \ SG. Then S = SG ∪ V (H). Let u, v /∈ SG, u 6= v. Hence, u, v ∈ CG. Since S is a strong resolving dominating set of G + H, CG ⊆ V (Gi) for some i ∈ {1, 2, . . . , n} . Then there exists z ∈ S ∩ V (Gi) such that u ∈ IG+H [v, z] or v ∈ IG+H [u, z]. It follows from Remark 5 that CG is a superclique of Gi. Similarly, S ∩ V (G) = V (G). On the other hand, if S ∩ V (G) 6= V (G), S ∩ V (H) 6= V (H), CG = V (G) \ SG and CH = V (H) \ SH , then S = SG ∪SH . Hence, CG and CH are supercliques of Gi and Hj for some i ∈ {1, 2, . . . , n} and j ∈ {1, 2, . . . ,m} . Conversely, suppose S satisfies condition (i). Let u, v /∈ S, u 6= v. Then u, v ∈ CG = V (G) \ SG. Hence, there exists w ∈ V (Gi) \ CG such that w ∈ NG+H(u) \ NG+H(v) or w ∈ NG+H(v) \ NG+H(u). By Remark 5, u ∈ IG+H [v, w] or v ∈ IG+H [u,w]. Thus, S is a strong resolving dominating set of G + H. Similarly, the same conclusion holds if S satisfies condition (ii). Suppose S satisfies condition (iii). Let u, v /∈ S, u 6= v. If u, v ∈ CG or u, v ∈ CH , then we are done. Assume u ∈ CG and v ∈ CH . Since dG+H(u, u′) = 2 for u′ ∈ Gk, k 6= i and dG+H(v, v′) = 2 for v′ ∈ Hp, p 6= j, v ∈ IG+H [u, u′] or u ∈ IG+H [v, v′]. Thus, S is a strong resolving dominating set of G+H. 4. On Strong Resolving 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). G. Monsanto, P. Acal, H. Rara / Eur. J. Pure Appl. Math, 13 (1) (2020), 170-179 178 Remark 8. For the coronas Pn ◦K1 and Cn ◦K1, it can be verified easily that γsr(Pn ◦K1) = γsr(Cn ◦K1) = n, ∀n ≥ 3. Theorem 6. 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: (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 v ∈ V (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) 6= 1; (iii) S = A ∪ ( ∪u∈V (G)\{v}V (Hu) ) ∪Bv for a unique v ∈ V (G) where v ∈ A ⊆ V (G) and Bv is a strong resolving set of Hv if γ(H) = 1 and Bv is a strong resolving set of 〈v〉+Hv if γ(H) 6= 1. Proof: Suppose S is a strong resolving dominating set of G ◦ H. Let A = S ∩ V (G) and Bv = S ∩ V (Hv), where v ∈ V (G). Consider the following cases: Case 1. S ∩ V (Hv) = V (Hv) Then Bv = Hv. Thus, S = A ∪ ( ∪u∈V (G) V (Hu) ) . Case 2. S ∩ V (Hv) 6= V (Hv) Let u, v ∈ V (G), u 6= v such that S ∩ V (Hu) 6= V (Hu) and S ∩ V (Hv) 6= V (Hv). Pick pu ∈ V (Hu) \ S and pv ∈ V (Hv) \ S. Then, none of the vertices in S strongly resolves pu and pv, a contradiction. Thus, the vertex v ∈ V (G) such that S ∩ V (Hv) 6= V (Hv) must be unique. Hence, S = A ∪ ( ∪u∈V (G)\{v}V (Hu) ) ∪Bv. Subcase 2.1 v ∈ S Let Cv = V (Hv) \ Bv. Hence, Bv = V (Hv) \ Cv. Then it can be verified that Cv is a superclique in Hv. If γ(H) 6= 1, by Theorem 1, Bv is a strong resolving set of {v}+Hv. If γ(H) = 1, then by Remark 7 and Theorem 2, Bv is a strong resolving set of Hv. Subcase 2.2 v /∈ S Since S is a dominating set of G◦H, Bv is a dominating set of Hv. By similar argument in the proof of subcase 2.1, 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) 6= 1. Conversely, suppose (i), (ii) and (iii) hold. Consider the following cases: Case 1. p, q ∈ V (G) \A Let p, q ∈ V (G ◦ H) \ S where p 6= q and p = v or q = v, but not both, then p ∈ IG◦H [q, z] or q ∈ IG◦H [p, z] for some z ∈ Bv. On the other hand, if p 6= v and q 6= v, then q ∈ IG◦H [p, w] for some w ∈ V (Hq) ⊂ S or p ∈ IG◦H [q, r] for some r ∈ V (Hp) \Bv. Case 2. p, q ∈ V (Hv) \Bv Since Bv is a strong resolving set of Hv, there exists t ∈ Bv ⊂ S that strongly resolves p and q. Case 3. p ∈ V (G) \ (A ∪ {v}) and q ∈ V (Hv) \Bv Since p 6= v, then V (Hp) ⊂ S and p ∈ IG◦H [q, z] for all z ∈ V (Hp). REFERENCES 179 Case 4. p = v, q ∈ V (Hv) \Bv Let t ∈ NG(v). Then V (Ht) ⊂ S and p ∈ IG◦H [q, u], for some u ∈ V (Ht). Cases 1 to 4 imply that S is a strong resolving dominating set of G◦H and (i), (ii), (iii) imply that S is a dominating set of G◦H. Accordingly, S is a strong resolving dominating set of G ◦H. Corollary 7. Let G and H be connected graphs of orders m and n, respectively γsr(G ◦H) = { (m− 1)n+ γsr(H), if γ(H) = 1 (m− 1)n+ γsr(K1 +H), if γ(H) 6= 1 Acknowledgements This research is funded by the Commission on Higher Education (CHED) and Min- danao State University-Iligan Institute of Technology, Philippines. References [1] C. Berge. Theorie des graphes et ses applications. Metheun and Wiley, London and New York, 1962. [2] E. Cockayne and S. Hedetniemi. Towards a theory of domination in graphs. Networks, 7(3):247–261, 1977. [3] A. Cuivillas and Jr. S. Canoy. Restrained double domination in the join and corona of graphs. International Journal of Math. Analysis, 8(27):1339–1347, 2014. [4] F. Harary. Graph Theory. Addison-Wesley Publishing Company, USA, 1969. [5] O. Oellermann and J. 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