EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 15, No. 3, 2022, 1201-1210 ISSN 1307-5543 – ejpam.com Published by New York Business Global On Movable Strong Resolving Domination in Graphs Helyn C. Sumaoy1,∗, Helen M. Rara1 1 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 strong resolving dominating set S is a 1-movable strong resolving dominating set of G if for every v ∈ S, either S \ {v} is a strong resolving dominating set or there exists a vertex u ∈ (V (G) \ S) ∩NG(v) such that (S \ {v}) ∪ {u} is a strong resolving dominating set of G. The minimum cardinality of a 1-movable strong resolving dominating set of G, denoted by γ1 msR(G) is the 1-movable strong resolving domination number ofG. A 1-movable strong resolving dominating set with cardinality γ1 msR(G) is called a γ1 msR-set of G. In this paper, we study this concept and the corresponding parameter in graphs resulting from the join, corona and lexicographic product of two graphs. Specifically, we characterize the 1-movable strong resolving dominating sets in these types of graphs and determine the exact values of their 1-movable strong resolving domination numbers. 2020 Mathematics Subject Classifications: 05C69 Key Words and Phrases: Movable strong resolving dominating set, movable strong resolving domination number, join, corona, lexicographic product 1. Introduction Domination in graphs was first introduced by C. Berge in 1958 [4]. There are now many studies involving domination and its variations. Domke et. al [6] introduced and investigated the concept of restrained domination in graphs. Oellermann, O. R, and Peters-Fransen, J. [11] introduced and studied the concept of strong resolving set. Slater [14] introduced and studied the concept of resolving set. Resolving sets and resolving dominating sets were also studied in [1, 10]. The concept of metric dimension has grown to become an interesting topic in graph theory. In line with this, some researchers introduced another variant, more restricted than the metric dimension, called the strong metric dimension which is the cardinality ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v15i3.4440 Email addresses: helyn.sumaoy@g.msuiit.edu.ph (H. Sumaoy), helen.rara@g.msuiit.edu.ph (H. Rara) https://www.ejpam.com 1201 © 2022 EJPAM All rights reserved. H. Sumaoy, H. Rara / Eur. J. Pure Appl. Math, 15 (3) (2022), 1201-1210 1202 of a minimum strong resolving set. Furthermore, several remarkable studies are contin- uously appearing after its introduction by P.J. Slater who discovered its usefulness when working with the United States sonar and Coast Guard Loran (long range aids to naviga- tion) stations. Its applications have arisen in many diverse fields including chemistry, for representing chemical compounds [7], the robot navigation [12] and geographical routing protocols [9], to name a few. In [13], an variant 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 [3], [2], [5], [8]. 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. Let G = ( V (G), E(G) ) be a graph. NG(v) = {u ∈ V (G) : uv ∈ E(G)} is a neighborhood 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]. 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 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. 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) 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, 15 (3) (2022), 1201-1210 1203 A non-empty set S ⊆ V (G) of a connected graph G is a 1-movable dominating set of G if S is a dominating set of G and for every v ∈ S, either S \ {v} is a dominating set of G or there exists a vertex u ∈ (V (G) \ S) ∩ NG(v) such that (S \ {v}) ∪ {u} is a dominating set of G. The 1-movable domination number of a graph G, denoted by γ1m(G) is the smallest cardinality of a 1-movable dominating set of G. A 1-movable dominating set of cardinality γ1m(G) is referred to as a γ1m-set of G. A resolving dominating set S of a graph G is a 1-movable resolving dominating set of G if for every v ∈ S, either S \ {v} is a resolving dominating set or there exists a vertex u ∈ (V (G) \ S)∩NG(v) such that (S \ {v})∪ {u} is a resolving dominating set of G. The minimum cardinality of a 1-movable resolving dominating set of G, denoted by γ1mR(G) is the 1-movable R-domination number of G. A 1-movable resolving dominating set with cardinality γ1mR(G) is called a γ1mR-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 introduces the movable strong resolving domination in some graphs. It also characterizes some graphs in terms of its movable strong resolving domination number. Remark 1. [1] Any superset of a strong resolving set is a strong resolving set. Lemma 1. [10] Let G be a nontrivial connected graph with diam(G) ≤ 2. Then S = V (G) \ C is a strong resolving dominating set of G if and only if C = ∅ or C is a dominated superclique in G. In particular, γsR(G) = |V (G)| − ωDS(G). Theorem 1. [1] 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), or S = V (K1 +G) \ C∗ or S = (V (G) \ C∗) ∪ {x ∈ C∗ : degG(x) = n− 1} where C∗ is a superclique in G. Theorem 2. [1] 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), or S = V (G) \ C, or S = V (K1 +G) \ C where C is a superclique in G. 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 H. Sumaoy, H. Rara / Eur. J. Pure Appl. Math, 15 (3) (2022), 1201-1210 1204 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}. Remark 2. Every movable strong resolving dominating set of a connected graph G is a strong resolving dominating set in G. Hence, γsR(G) ≤ γ1msR(G). Remark 3. The converse of Remark 2 does not hold. To see this, the set S = {v1, v2, v3} of the path P4 = [v1, v2, v3, v4] is a strong resolving dominating set of P4 but it is not movable strong resolving dominating set since S \ {v1} is not a strong resolving set of P4. Proposition 1. Any superset of a movable strong resolving dominating set is a movable strong resolving dominating set. Proof. Let S be a movable strong resolving dominating set of G and S ⊆ S′. Then by Remark 1, S′ is a strong resolving dominating set of G. We show that a S′ is movable strong resolving dominating set of G. Let x ∈ S′. If x ∈ S, then S \ {x} ⊆ S′ \ {x}. Since S is a movable strong resolving dominating set of G, either S \ {x} is strong resolving dominating set of G or ∃y ∈ (V (G) \ S) ∩ NG(x) such that (S \ {x}) ∪ {y} is strong resolving dominating set of G. If S \ {x} is a strong resolving dominating set of G, then S′\{x} is also strong resolving dominating set ofG by Remark 1. If ∃y ∈ (V (G)\S)∩NG(x) such that (S \ {x}) ∪ {y} is a strong resolving dominating set, then (S \ {x}) ∪ {y} ⊆ (S′ \ {x}) ∪ {y} (S′\{x})∪{y} is a strong resolving dominating set of G. Therefore, S′ is a movable strong resolving dominating set of G. Proposition 2. Let Pn = [v1, v2, . . . , vn] where n ≥ 1. If a set S ⊆ V (Pn) is a movable strong resolving dominating set of Pn, then S is a dominating set containing the vertices v1 and vn. Proof. Suppose S is a movable strong resolving dominating set of Pn and suppose that S does not contain v1 or vn, say v1. Since v1MMDvn, S ∩ {v1, vn} ≠ ∅. Hence vn ∈ S. This implies that S \{vn} and (S \{vn})∪{vn−1} if vn−1 /∈ S are not strong resolving sets of Pn, a contradiction. Therefore, S contains v1 and vn. Lemma 2. Let G be a nontrival connected graph with diam(G) ≤ 2. Then S = V (G)\C is a movable strong resolving dominating set of G if and only if C = ∅ or C is a dominated superclique in G and either for each x ∈ S, C ∪ {x} is a dominated superclique or there exists y ∈ C ∩NG(x) such that (C \ {y}) ∪ {x} is a dominated superclique in G. Proof. Suppose S = V (G)\C is a movable strong resolving dominating set of G. Then S is a strong resolving dominating set of G . By Lemma 1, C = ∅ or C is a dominated superclique in G. Let x ∈ S. Since S is a movable strong resolving dominating set, either S \ {x} is a strong resolving dominating or there exists y ∈ (V (G) \ S)∩NG(x) such that (S\{x})∪{y} is a strong resolving dominating set of G. Since S\{x} = V (G)\(C∪{x}) H. Sumaoy, H. Rara / Eur. J. Pure Appl. Math, 15 (3) (2022), 1201-1210 1205 and (S \ {x}) ∪ {y} = V (G) \ ((C \ {x}) ∪ {y}), by Lemma 1 C ∪ {x} is a dominated superclique or (C \ y) ∪ {x} is a dominated superclique in G. For the converse, suppose C = ∅. Then, S = V (G) is strong resolving dominating set of G. Thus, S \ {x} = V (G) \ {x} is strong resolving dominating since {x} is a dominated superclique for each x ∈ V (G). So, suppose C is a dominated superclique in G and for each x ∈ S either C ∪ {x} is a dominated superclique or there exists y ∈ C ∩ NG(x) such that (C \ {y}) ∪ {x} is a dominated superclique. Hence, for each x ∈ S , (S \ {x})∪ {y} = V (G) \ (C \ {y})∪ {x} is a strong resolving dominating set of G. Therefore, S is a movable strong resolving set of G. Lemma 3. Let G be a nontrivial connected graph. A set C ⊆ V (G) is a superclique in G and for each x ∈ V (G)\C either C∪{x} is a superclique or there exists y ∈ C∩NG(x) such that (C \ {y} ∪ {x}) is a superclique in G if and only if |C| = 1 and degG(v) = |V (G)| − 1 for v ∈ C. Proof. Suppose C is a superclique in G satisfying the given condition and let |C| > 1. Let u, v ∈ C. Then uv ∈ E(G). Let x ∈ V (G) \ C. If C ∪ {x} is a superclique, then x ∈ NG(u)∩NG(v), a contradiction since C is a superclique. On the other hand, suppose there exists y ∈ C∩NG(x) such that (C\{y})∪{x} is superclique. If y = u, then x ∈ NG(v) and there exists z ∈ (NG(x)\NG(v))∩(V (G)\C). This is a contradiction since z ∈ C∪{x} or z ∈ (C \{w})∪{x} for some w ∈ C∩NG(x), that is z ∈ NG(x)∩NG(v). Hence, |C| = 1. Now, suppose degG(v) < |V (G)|−1 where v ∈ C. Then there exists y ∈ V (G)\NG(v) and C ∪ {y} is not a superclique in G, a contradiction. Thus, degG(v) = |V (G)| − 1, showing that C is a γ-set of G. For the converse, suppose C is γ-set of G. Then C is a superclique in G and for every x ∈ V (G) \ C, C ∪ {x} is a superclique in G since ⟨C ∪ {x}⟩ is a path in G. As a consequence of Lemma 2 and Lemma 3, the next result follows. Theorem 4. Let G be a nontrivial connected graph with diam(G) ≤ 2. Then S = V (G) \ C is a movable strong resolving dominating set of G if and only if C = ∅ or C is a γ-set of G if γ(G) = 1. 3. γ1 msR(G+H) in the Join of Graphs This section gives characterization of the movable strong resolving dominating sets in the join of graphs as well as its movable strong resolving domination number. 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 movable strong resolving dominating set of K1 +G if and only if S = V (G), or S = V (K1 +G) \ C where C is a γ-set of G. Proof. Suppose S is a movable strong resolving dominating set of K1 +G. Then, by Theorem 1, S = V (G), or S = V (K1 +G) \ C∗, or S = (V (G) \ C) ∪ {x ∈ C : degG(x) = n− 1} H. Sumaoy, H. Rara / Eur. J. Pure Appl. Math, 15 (3) (2022), 1201-1210 1206 where C is a superclique in G. Since S is a dominating set, C is a dominated superclique in G and C∗ is a superclique in G. By Lemma 2 and Lemma 3, C is a γ-set of G. Hence, S = V (G) or S = V (K1 +G) \ C where C is a γ-set of G. Conversely, suppose S = V (G) or S = V (K1 +G) \ C where C is a γ-set of G. Then, by Theorem 1, S is a strong resolving dominating set of K1+G. If S = V (G), then S \{x} is a strong resolving dominating set of K1 + G since {x} is a dominated superclique in G by Lemma 2 and Lemma 3 and Theorem 1. Hence, S is a movable strong resolving dominating set of K1 +G. Similarly, if S = V (K1 +G) \C where C is a γ-set of G, then S is a movable strong resolving set of K1+G by Lemma 2, Lemma 3 and Theorem 1. Theorem 6. Let G be a nontrivial connected graph with γ(G) ̸= 1. Then, S ⊆ V (K1+G) is a movable strong resolving dominating set of K1 +G if and only if S = V (G). Proof. Suppose S is a movable strong resolving dominating set of K1+G. By Theorem 2 S = V (G) or S = V (G) \C, or S = V (K1+G) \C where C is a superclique in G. Since γ(H) ̸= 1, by Lemma 2, Lemma 3 and Theorem 2, S ̸= V (G) \C and S ̸= V (K1+G) \C. Hence, S = V (G). The converse is clear. Corollary 1. Let G be a nontrivial connected graph of order n. Then γ1msR(K1+G) = n. Proof. Let S be a γ1msR-set ofK1+G. If γ(G) = 1, then S = V (G) or S = V (K1+G)\C where C is a γ-set of G. Hence, γ1msR(K1 +G) = |S| = |V (G)| = |V (K1 +G)| − |C| = n+ 1− 1 = n. On the other hand, if γ(G) ̸= 1, then S = V (G). Thus, γmsR(K1 +G) = |S| = |V (G)| = n. Theorem 7. 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 movable strong resolving dominating set of K1 + G if and only if S = V (G) or S = V (K1 + G) \ Ci where Ci is γ-set of Gi if γ(Gi) = 1 for all i ∈ {1, 2, . . . ,m}. Proof. Suppose S is a movable strong resolving dominating set of K1 + G. Then, 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}. If γ(Gi) = 1, then by Lemma 2, Lemma 3 and Theorem 5, Ci is a γ-set of Gi. Thus , S ̸= V (G)\Ci showing that S = V (G) or S = V (K1+G)\Ci where Ci is a γ-set of Gi for some i ∈ {1, 2, . . . ,m}. If γ(Gi) ̸= 1 for all i ∈ {1.2. . . . ,m}, then S = V (G) by Lemma 2, Lemma 3, and Theorem 6. H. Sumaoy, H. Rara / Eur. J. Pure Appl. Math, 15 (3) (2022), 1201-1210 1207 Conversely, the case when S = V (G is trivial. Suppose S = V (K1 +G) \Ci where Ci is a γ-set of Gi if γ(Gi) = 1 for some i ∈ {1, 2, . . . ,m}. Then by Theorem 3, S is a strong resolving dominating set of K1 + G. By Lemma 2, Lemma 3 and Theorem 5, S is a movable strong resolving set of K1 +G. Corollary 2. Let Gi be connected graphs of order ni and G be a disconnected graph whose components are Gi for i ∈ {1, 2, . . . ,m}. Then, γ1msR(K1 +G) = |V (G)|. Theorem 8. Let G and H be nontrivial connected graphs of orders m and n, respectively. A proper subset S of V (G +H) is a movable strong resolving dominatiing 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 γ-set of G if γ(G) = 1. (ii) S = V (G+H) \ CH where CH is a γ-set of H if γ(H) = 1. Proof. Suppose S is a movable strong resolving dominating set of G+H. By Theorem [10], Lemma 2, and Lemma 3, (i) or (ii) holds. The converse is clear by Theorem [10], Lemma 2, and Lemma 3. Corollary 3. Let G andH be nontrivial connected graphs of ordersm and n, respectively. Then, γ1msR(G+H) = m+ n− 1. 4. γ1 msR(G ◦H) in the Corona of Graphs This section gives characterization of the movable strong resolving dominating sets in the corona of graphs as well as its movable strong resolving domination number. Theorem 9. Let G be a nontrivial connected graph and H a connected graph. A proper subset S of V (G ◦H) is a movable strong resolving dominating set of G ◦H if and only if S = A ⋃( ⋃ u∈V (G) V (Hu) ) where A ⊆ V (G). Proof. Suppose that a proper subset S of V (G ◦ H) is a movable strong resolving dominating set of G ◦ H. Since S is a strong resolving dominating set of G ◦ H (i) or (ii) of Theorem [10] holds. If (i) holds, then S = A ⋃( ⋃ u∈V (G) V (Hu) ) , where A ⊆ V (G). Suppose (ii) holds. Let x ∈ V (Hw) for some w ∈ V (G). Then S \ {x} = A ⋃( ⋃ u∈V (G)\{w,v} V (Hu) )⋃( V (Hw) \ {x} )⋃ Bv is not a strong resolving set by Theorem [10]. Hence, S = A ⋃( ⋃ u∈V (G) V (Hu) ) where A ⊆ V (G). H. Sumaoy, H. Rara / Eur. J. Pure Appl. Math, 15 (3) (2022), 1201-1210 1208 For the converse, suppose S = A ⋃( ⋃ u∈V (G) V (Hu) ) . By Theorem [10], S is a strong resolving dominating set. Let p ∈ S. If p ∈ A , then S \{p} = (A\{p})∪ ( ⋃ u∈V (G) V (Hu) ) is a strong resolving dominating set. If p ∈ V (Hu) for each u ∈ V (G), then S \ {p} = A ∪ ( ⋃ u∈V (G) V (Hu) \ {p} ) is a strong resolving dominating set since {p} is a dominated superclique in Hu. Accordingly, S is a movable strong resolving dominating set in G ◦H. Corollary 4. Let G be a connected graph of order m > 1 and H be any graph of order n. Then γ1msR(G ◦H) = mn. Proof. Let C be a γ1msR-set of G ◦H. Then by Theorem 9, S = A ⋃( ⋃ u∈V (G) V (Hu) ) where A ⊆ V (G). Thus, γ1msR(G ◦H) = |C| = |A|+ ∣∣∣∣ ⋃ u∈V (G) V (Hu) ∣∣∣∣ ≥ |V (G)||V (H)| = mn. Let A = ∅. Then S∗ = A ⋃( ⋃ u∈V (G) V (Hu) ) is a movable strong resolving dominat- ing set of G ◦H by Theorem 9. Hence, γ1msR(G ◦H) ≤ |S∗| = ∣∣∣∣ ⋃ u∈V (G) V (Hu) ∣∣∣∣ = mn. Therefore, γ1msR(G ◦H) = mn. 5. γ1 msR(G[H]) in the Lexicographic Product of Graphs This section gives characterization of the movable strong resolving dominating sets in the lexicographic product of graphs as well as its movable strong resolving domination number. H. Sumaoy, H. Rara / Eur. J. Pure Appl. Math, 15 (3) (2022), 1201-1210 1209 Lemma 4. Let G = Kn for n > 1 and H be a nontrivial connected graph with γ(H) = 1. Then A × C is a γ-set of G[H] if and only if A is a singleton subset of V (G) and C is a γ-set of H. Proof. Suppose A × C is a γ-set of G[H]. Since G = Kn for n > 1 and γ(H) = 1, γ(G[H]) = 1. Hence, A is a singleton subset of V (G) and C is a singleton subset of H. We claim that C is a γ-set of H. Suppose C is not a γ-set of H. Then there exists y ∈ (V (H) \ C) such that NH(y) ∩ C = ∅. Hence, NG[H]((a, y)) ∩ ( {a} × C ) = ∅ for all a ∈ A. This contradicts the assumption that A×C is a γ-set of G[H]. Thus, C is a γ-set of H. Conversely, suppose that A is a singleton subset of V (G) and C is a γ-set of H. We claim that A×C is a γ-set of G[H]. Let (a, x) ∈ V (G[H]) \ (A×C). Then (a ∈ A and x /∈ C) or (a /∈ A and x ∈ C) or (a /∈ A and x /∈ C). If a ∈ A and x /∈ C, then there exists y ∈ C∩NH(x) since C is a γ-set of H. Hence, (a, y) ∈ (A×C)∩NG[H]((a, x)). Suppose a /∈ A and x ∈ C. Since A is a singleton subset of G = Kn for n > 1, there exists b ∈ A ∩ NG(a). Hence, (b, x) ∈ (A × C) ∩ NG[H]((a, x)). Also, if a /∈ A and x /∈ C, then there exists (b, y) ∈ (A× C) ∩NG[H]((a, x)). Therefore, A× C is a γ-set of G[H]. Theorem 10. Let G = Kn for n > 1 and H a nontrivial connected graph. A subset S of V (G[H]) is a movable strong resolving 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 singleton subset of V (G) and C is a γ-set of H if γ(H) = 1. Proof. Suppose S is a movable strong resolving dominating set of G[H]. Since diam(G[H]) = 2, by Theorem 4, S = V (G[H]) \ (A× C) where A× C = ∅ or A× C is a γ-set of G[H] if γ(G[H]) = 1. By Lemma 4, A is a singleton subset of V (G) and C is a γ-set of H if γ(H) = 1. For the converse, suppose S = V (G[H])\(A×C), where A ⊆ V (G) and C = ∅ or A is a singleton subset of V (G) and C is a γ-set of H if γ(H) = 1. If A ⊆ V (G) and C = ∅, then A × C = ∅. Hence, S = V (G[H]) is a movable strong resolving dominating set of G[H]. On the other hand, if A is a singleton subset of V (G) and C is γ-set of H if γ(H) = 1, then A× C is a γ-set of G[H] if γ(G[H]) = 1. By Theorem 4, S = V (G[H]) \ (A× C) is a movable strong resolving dominating set of G[H]. 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