EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 16, No. 1, 2023, 418-429 ISSN 1307-5543 – ejpam.com Published by New York Business Global 1-Movable Resolving Hop Domination in Graphs Jerson S. Mohamad1,∗, Helen M. Rara2 1 Department of Mathematics and Statistics, College of Science and Mathematics, Western Mindanao State University, 7000 Zamboanga City, Philippines 2 Department of Mathematics and Statistics, College of Science and Mathematics, Center of Graph Theory, Algebra, and Analysis-Premier Research Institute of Science and Mathematics, Mindanao State University-Iligan Institute of Technology, 9200 Iligan City, Philippines Abstract. Let G be a connected graph. A set W ⊆ V (G) is a resolving hop dominating set of G if W is a resolving set in G and for every vertex v ∈ V (G) \ W there exists u ∈ W such that dG(u, v) = 2. A set S ⊆ V (G) is a 1-movable resolving hop dominating set of G if S is a resolving hop dominating set of G and for every v ∈ S, either S \ {v} is a resolving hop dominating set of G or there exists a vertex u ∈ ((V (G) \ S) ∩NG(v)) such that (S \ {v}) ∪ {u} is a resolving hop dominating set of G. The 1-movable resolving hop domination number of G, denoted by γ1 mRh(G) is the smallest cardinality of a 1-movable resolving hop dominating set of G. This paper presents the characterization of the 1-movable resolving hop dominating sets in the join, corona and lexicographic product of graphs. Furthermore, this paper determines the exact value or bounds of their corresponding 1-movable resolving hop domination number. 2020 Mathematics Subject Classifications: 05C69 Key Words and Phrases: 1-movable resolving hop dominating set, 1-movable resolving hop domination number, hop dominated superclique, join, corona, lexicographic product 1. Introduction Dominating sets in graphs have been studied extensively and there have been many published studies that have introduced different variants of domination in graphs [7, 13]. In 2015, Natarajan and Ayyaswamy [12] studied the concept of hop domination in graphs and the hop domination number. Movable resolving domination in graphs was studied in [11] and the resolving hop domination sets in graphs was introduced in [10]. Other variations of resolving sets can be found in [2, 3, 6] and resolving dominating sets in [1, 4, 5, 9, 14]. This paper introduces and characterizes the concept of 1-movable resolving hop domination in graphs. ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v16i1.4671 Email addresses: mohamad.jerson@wmsu.edu.ph (J. Mohamad), helen.rara@g.msuiit.edu.ph (H. Rara) https://www.ejpam.com 418 © 2023 EJPAM All rights reserved. J. Mohamad, H. Rara / Eur. J. Pure Appl. Math, 16 (1) (2023), 418-429 419 We consider connected graphs that are finite, simple, and undirected. For elementary Graph Theory concepts, it is recommended that readers refer to [8]. Let G = ( V (G), E(G) ) be a graph. NG(v) = {u ∈ V (G) : uv ∈ E(G)} is a neighbor- hood of v. An element u ∈ NG(v) is called a neighbor of v. NG[v] = NG(v) ∪ {v} is a closed neighborhood of v. The degree of v, denoted by degG(v), is equal to |NG(v)|. For S ⊆ V (G), NG(S) = ⋃ v∈S NG(v) and NG[S] = ⋃ v∈S NG[v]. The distance dG(u, v) of two vertices u, v in G is the length of a shortest u-v path in G. The greatest distance between any two vertices in G, denoted by diam(G), is called the diameter of G. A set S ⊆ V (G) is a dominating set if every u ∈ V (G) \ S is adjacent to at least one vertex v ∈ S. The domination number of a graph G, denoted by γ(G), is given by γ(G) = min{|S| : S is a dominating set of G}. A set S ⊆ V (G) is a total dominating set if every vertex in graph G is adjacent to some vertex of S. The minimum cardinality of a total dominating set in G is the total domination number of G, denoted by γt(G), and we refer to such a set as γt-set of G. A set S ⊆ V (G) is a hop dominating set of G if for every v ∈ V (G)\S, there exists u ∈ S such that dG(u, v) = 2. The minimum cardinality of a hop dominating set of G, denoted by γh(G), is called the hop domination number of G. Any hop dominating set with cardinality equal to γh(G) is called a γh-set. A vertex v in G is a hop neighbor of vertex u in G if dG(u, v) = 2. The set NG(u, 2) = {v ∈ V (G) : dG(v, u) = 2} is called the open hop neighborhood of u. The closed hop neighborhood of u in G is given by NG[u, 2] = NG(u, 2) ∪ {u}. The open hop neighborhood of X ⊆ V (G) is the set NG(X, 2) = ⋃ u∈X NG(u, 2). The closed hop neighbor- hood of X in G is the set NG[X, 2] = NG(X, 2) ∪X. A set S ⊆ V (G) is a total hop dominating set of G if for every v ∈ V (G), there exists u ∈ S such that dG(u, v) = 2. That is, S is a hop dominating set of G and for all z ∈ S, NG(z, 2) ∩ S ̸= ∅. The smallest cardinality of a total hop dominating set of G, denoted by γth(G), is called the total hop domination number of G. Any total hop dominating set with cardinality equal to γth(G) is called a γth-set. A set S ⊆ V (G) is a locating set of G if for every two distinct vertices u and v of V (G)\S, NG(u) ∩ S ̸= NG(v) ∩ S. The locating number of G, denoted by ln(G), is the smallest cardinality of a locating set of G. A locating set of G of cardinality ln(G) is referred to as ln-set of G. A set S ⊆ V (G) is a strictly locating set of G if it is a locating set of G and NG(u)∩S ̸= S for all u ∈ V (G)\S. The strictly locating number of G, denoted by sln(G), is the smallest cardinality of a strictly locating set of G. A strictly locating set of G of cardinality sln(G) is referred to as a sln-set of G. A locating (resp. strictly locating) subset S of V (G) is a 1-movable locating (resp. 1-movable strictly locating) set of G if for every v ∈ S, either S \ {v} is a locating (resp. strictly locating) set of G or there exists a vertex u ∈ ((V (G) \ S) ∩NG(v)) such that (S \ {v})∪{u} is a locating (resp. strictly locating) set of G. The minimum cardinality of a 1-movable locating (resp. 1-movable strictly locating) set of G, denoted by mln(G)(resp. J. Mohamad, H. Rara / Eur. J. Pure Appl. Math, 16 (1) (2023), 418-429 420 msln(G)) is the 1-movable location number (resp. 1-movable strictly location number) of G. Any 1-movable locating (resp. 1-movable strictly locating) set of cardinality mln(G) (resp. msln(G)) is referred to as mln-set (resp. msln-set) of 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 basis. A set S ⊆ V (G) is a resolving hop dominating set of G if S is both a resolving set and a hop dominating set. The minimum cardinality of a resolving hop dominating set of G, denoted by γRh(G), is called the resolving hop domination number of G. Any resolving hop dominating set with cardinality equal to γRh(G) is called a γRh-set. A set S ⊆ V (G) is a 1-movable resolving hop dominating set of G if S is a resolving hop dominating set of G and for every v ∈ S, either S \ {v} is a resolving hop dominating set of G or there exists a vertex u ∈ ((V (G) \ S) ∩NG(v)) such that (S \ {v}) ∪ {u} is a resolving hop dominating set of G. The 1-movable resolving hop domination number of G, denoted by γ1mRh(G) is the smallest cardinality of a 1-movable resolving hop dominating set of G. Any 1-movable resolving hop dominating set of cardinality γ1mRh(G) is referred to as a γ1mRh-set of G. 2. Preliminary Results Remark 1. Every 1-movable resolving hop dominating set of G is a resolving hop domi- nating set. Thus, 2 ≤ γRh(G) ≤ γ1mRh(G). Remark 2. Every 1-movable resolving hop dominating set of G is a hop dominating set. Thus, 2 ≤ γh(G) ≤ γ1mRh(G). Remark 3. Every 1-movable resolving hop dominating set of G is a resolving set. Thus, 1 ≤ dim(G) ≤ γ1mRh(G). Consider G = P5 where V (G) = {v1, v2, v3, v4, v5} with deg(v1) = deg(v5) = 1 and NG(v3) = {v2, v4}. Let S1 = {v1}, S2 = {v2, v3} and S3 = V (G). Then, S1 is a resolving set of G, S2 is a hop dominating set and a resolving set of G and S3 is a 1-movable resolving hop dominating set of G. It can be verified that dim(G) = 1, γh(G) = 2, γRh(G) = 2 and γ1mRh(G) = 5. Hence for G = P5, Remarks 1, 2 and 3 holds. J. Mohamad, H. Rara / Eur. J. Pure Appl. Math, 16 (1) (2023), 418-429 421 Proposition 1. Let G be a nontrivial connected graph. Then G admits a 1-movable resolving hop dominating set if and only if γ(G) ̸= 1. Proof: Suppose G has a 1-movable resolving hop dominating set S. Suppose further that γ(G) = 1. Let A = {x ∈ V (G) : {x} is a dominating set of G}. Then A ̸= ∅ since γ(G) = 1. Since S is a hop dominating set, A ⊆ S. Let x ∈ A. Then S \ {x} and (S \ {x}) ∪ {y} for each y ∈ V (G) \ S are not hop dominating sets of G. Thus, S is not a 1-movable resolving hop dominating set, a contradiction. Conversely, suppose that γ(G) ̸= 1. Let S = V (G). Then S is a resolving hop dominating set of G. For each x ∈ S, S \ {x} is a resolving set of G. Also, since {x} is not a dominating set, there exists y ∈ (S \ {x}) ∩ NG(x, 2). Hence, S \ {x} is a hop dominating set of G. Therefore, S \ {x} is a resolving hop dominating set of G for each x ∈ S. Accordingly, S is a 1-movable resolving hop dominating set of G. As a consequence of Proposition 1 the next result follows. Corollary 1. A graph G does not admit a 1-movable resolving hop dominating set if and only if G = K1 +H for any graph H. Proposition 2. Let G be a connected graph and S a 1-movable resolving hop dominating set of G. Then for all z ∈ S, NG(z, 2)∩S ̸= ∅ and for each x ∈ V (G)\S, |NG(x, 2)∩S| ≥ 1 and there exists w ∈ (V (G) \ S) ∩NG(x, 2) ∩NG(v) whenever NG(x, 2) ∩ S = {v}. Proof: Let S be a 1-movable resolving hop dominating set of G and z ∈ S. Suppose NG(z, 2) ∩ S = ∅. Then S \ {z} and (S \ {z}) ∪ {u} where u ∈ (V (G) \ S) ∩ NG(z) are not hop dominating sets of G since z has no hop neighbor in both sets, a contra- diction. Thus, NG(z, 2) ∩ S ̸= ∅. Now, let x ∈ V (G) \ S. Since S is hop dominating, NG(x, 2)∩S ̸= ∅. Suppose |NG(x, 2)∩S| = 1. Let v ∈ NG(x, 2)∩S. Then S\{v} is not hop dominating, since x has no hop neighbor in S \{v}. It follows that (S \{v})∪{w} for some w ∈ (V (G) \ S) ∩NG(v) is a resolving hop dominating set of G. Hence, x must be a hop neighbor of w and so w ∈ (V (G) \ S) ∩NG(x, 2) ∩NG(v). As a consequence of Proposition 2, the next corollary follows. Corollary 2. Every 1-movable resolving hop dominating set is a total hop dominating set. Moreover, γth(G) ≤ γ1mRh(G). 3. On 1-Movable Resolving Hop Domination in the Join of Graphs Let A and B be sets which are not necessarily disjoint. The disjoint union of A and B, denoted by A • ∪ B, is the set obtained by taking the union of A and B treating each element in A as distinct from each element in B. The union G1 ∪ G2 of graphs G1 and G2 with disjoint vertex-sets V (G1) and V (G2), respectively, is the graph G with V (G) = V (G1) • ∪ V (G2) and E(G) = E(G1) • ∪ E(G2). The join of two graphs G and H, denoted by G+H, is the graph with vertex-set V (G+H) = V (G) • ∪ V (H) and edge-set E(G+H) = E(G) • ∪ E(H) ∪ {uv : u ∈ V (G), v ∈ V (H)}. J. Mohamad, H. Rara / Eur. J. Pure Appl. Math, 16 (1) (2023), 418-429 422 Theorem 1. [10] Let G and H be nontrivial connected graphs. A set W ⊆ V (G+H) is a resolving hop dominating set of G+H if and only if W = WG ∪WH where WG and WH are strictly locating sets of G and H, respectively. As an illustration, consider the graph P3 + P3 in Figure 1. It is easy to verify that sln(P3) = 2, and by Theorem 1, the set of shaded vertices is a resolving hop dominating set of P3 + P3. It follows that γRh(P3 + P3) = 4. P3 + P3: Figure 1: Graph P3 + P3 with γRh(P3 + P3) = 4 Theorem 2. Let G and H be connected graphs with γ(G) ̸= 1 and γ(H) ̸= 1. A set W ⊆ V (G +H) is a 1-movable resolving hop dominating set of G +H if and only if W = WG ∪WH where WG ⊆ V (G) and WH ⊆ V (H) are 1-movable strictly locating sets of G and H, respectively, and one of the following statements holds: (i) For each u ∈ WG, WG \ {u} and WH ∪ {v} are strictly locating sets of G and H, respectively, for some v ∈ V (H) \WH ; (ii) For each q ∈ WH , WH \ {q} and WG ∪ {b} are strictly locating sets of H and G, respectively, for some b ∈ V (G) \WG. Proof: Suppose that W ⊆ V (G + H) is a 1-movable resolving hop dominating set of G +H. Then W is resolving hop dominating. By Theorem 1, W = WG ∪WH where WG ⊆ V (G) andWH ⊆ V (H) are strictly locating sets ofG andH, respectively. Moreover, since G and H are connected graphs with γ(G) ̸= 1 and γ(H) ̸= 1, WG ̸= ∅ and WH ̸= ∅. Let x ∈ WG. By assumption, W \ {x} = (WG \ {x}) ∪ WH or (W \ {x}) ∪ {w} = [(WG \ {x} ∪ {w})] ∪ WH for some w ∈ NG(x) ∩ (V (G) \ WG) or (W \ {x}) ∪ {z} = (WG \ {x}) ∪ (WH ∪ {z}) for some z ∈ V (H) \WH is a resolving hop dominating set of G+H. Thus, by Theorem 1, WG \ {x} or (WG \ {x})∪{w} is a strictly locating set of G. This implies that WG is a 1-movable-strictly locating set of G. Similarly, WH is a 1-movable strictly locating set of H. Now, let u ∈ WG. Since W is a 1-movable resolving hop dominating set, W \ {u} = (WG \ {u}) ∪ WH or (W \ {u}) ∪ {r} = [(WG \ {u}) ∪ {r}] ∪ WH for some r ∈ NG(u) ∩ (V (G) \ WG) or (W \ {u}) ∪ {v} = (WG \ {u}) ∪ (WH ∪ {v}) for some v ∈ V (H) \WH is a resolving hop dominating set of G +H. It follows from Theorem 1 that WG \ {u} and WH ∪{v} are strictly locating sets of G and H, respectively. Thus, (i) holds. Similarly, (ii) holds. For the converse, suppose that WG and WH are 1-movable strictly locating sets of G and H, respectively. Suppose (i) holds. Then W = WG ∪ WH is a resolving hop J. Mohamad, H. Rara / Eur. J. Pure Appl. Math, 16 (1) (2023), 418-429 423 dominating set of G+H by Theorem 1. Let u ∈ W . If u ∈ WG, then by assumption and Theorem 1, W \ {u} = (WG \ {u}) ∪WH or (W \ {u}) ∪ {w} = [(WG \ {u}) ∪ {w}] ∪WH for some w ∈ NG(u)∩ (V (G) \WG) or W \ {u}∪ {z} = (WG \ {u})∪ (WH ∪{z}) for some z ∈ V (H \WH) is a resolving hop dominating set of G+H. Now, suppose that u ∈ WH . Since WG and WH are 1-movable strictly locating sets of G and H, respectively, it follows from Theorem 1 thatW\{u} = (WH\{u})∪WG or (W\{u})∪{y} = [(WH\{u})∪{y}]∪WG for some y ∈ NH(u)∩ (V (H)\WH) is a resolving hop dominating set of G+H. Therefore, W is a 1-movable resolving hop dominating set of G + H. Similarly, W is a 1-movable resolving hop dominating set of G+H if (ii) holds. Corollary 3. Let G and H be nontrivial connected graphs with γ(G) ̸= 1 and γ(H) ̸= 1. If G and H have 1-movable strictly locating sets, then γ1mRh(G+H) ≤ msln(G) +msln(H). Proof: Suppose G and H have 1-movable strictly locating sets. Let WG and WH be msln-sets of G and H, respectively. Then W = WG ∪ WH is a 1-movable resolving hop dominating set of G+H by Theorem 2. Thus, γ1mRh(G+H) ≤ |W | = |WG|+ |WH | = msln(G) +msln(H). 4. On 1-Movable Resolving Hop Domination in the Corona of Graphs The corona of two graphs G and H, denoted by G◦H, is the graph obtained by taking one copy of G of order n and n copies of H, and then joining every vertex of the ith copy of H to the ith vertex of G. For v ∈ V (G), denote by Hv the copy of H whose vertices are attached one by one to the vertex v. Subsequently, denote by v+Hv the subgraph of the corona G ◦H corresponding to the join ⟨{v}⟩+Hv, v ∈ V (G). Theorem 3. [10] Let G and H be nontrivial connected graphs. Then W ⊆ V (G ◦ H) is a resolving hop dominating set of G ◦ H if and only if W ∩ V (Hv) ̸= ∅ for every v ∈ V (G) and W = A ∪B ∪D where A ⊆ V (G), B = ∪ {Bv : v ∈ V (G) ∩NG(A) and Bv is a locating set of Hv} and D = ∪ {Du : u ∈ V (G) \NG(A) and Du is a strictly locating set of Hu} . As an illustration, consider the graph P3 ◦P4 in Figure 2 and let G = P3 and H = P4. It can be easily verified that ln(P4) = sln(P4) = 2 and by Theorem 3, the set of shaded vertices is a resolving hop dominating set of P3◦P4. It can be verified that γRh(P3◦P4) = 6. J. Mohamad, H. Rara / Eur. J. Pure Appl. Math, 16 (1) (2023), 418-429 424 Figure 2: Graph P3 ◦ P4 with γRh(P3 ◦ P4) = 6 Theorem 4. Let G and H be nontrivial connected graphs. Then W ⊆ V (G ◦ H) is a 1-movable resolving hop dominating set of G ◦ H if and only if W ∩ V (Hv) ̸= ∅ for every v ∈ V (G) and W = A ∪  ⋃ v∈NG(A) Bv  ∪  ⋃ u∈V (G)\NG(A) Du  where A ⊆ V (G), Bv ⊆ V (Hv) for all v ∈ V (G) ∩ NG(A) and Du ⊆ V (Hu) for all u ∈ V (G) \NG(A) are 1-movable locating and 1-movable strictly locating sets of Hv and Hu, respectively. Proof: Suppose that W ⊆ V (G ◦ H) is a 1-movable resolving hop dominating set of G ◦H. Then W is a resolving hop dominating set. By Theorem 3, W ∩ V (Hv) ̸= ∅ and W ∩V (Hv) is a locating set of Hv for all v ∈ V (G). Let A = W ∩V (G), Bv = W ∩V (Hv) for all v ∈ V (G) ∩NG(A) and Du = W ∩ V (Hu) for all u ∈ V (G) \NG(A). By Theorem 3, Bv is a locating set of Hv and Du is a strictly locating set of Hu. Let x ∈ Bv. Since W is a 1-movable resolving hop dominating set and x ∈ W , either W \ {x} is a resolving hop dominating set of G ◦ H or there exists y ∈ (V (G ◦ H) \ W ) ∩ NG◦H(x) such that (W \ {x}) ∪ {y} is a resolving hop dominating set of G ◦H. Note that W \ {x} = (Bv \ {x}) ∪  ⋃ u∈V (G)\{v} D∗ u  ∪A and (W\{x})∪{y} is equal to ((Bv \ {x}) ∪ {y})∪  ⋃ u∈V (G)\{v} D∗ u ∪A if y ∈ V (Hv)\Bv or equal to (Bv \ {x}) ∪  ⋃ u∈V (G)\{v} D∗ u  ∪ (A ∪ {y}) if y = v ∈ V (G) \ A. Hence, either Bv \ {x} is a locating set of Hv or (Bv \ {x}) ∪ {y} for some y ∈ (V (Hv) \ Bv) ∩ NHv(x) is a locating set of Hv. Thus, Bv is a movable locating set of Hv. The proof that Du is a 1-movable strictly locating set of Hu is similar. For the converse, suppose that W is a set described above. Then by Theorem 3, W is a resolving hop dominating set. Let x ∈ W and let v ∈ V (G) such that x ∈ V (⟨v⟩+Hv). Suppose that x ̸= v. Consider the following cases. J. Mohamad, H. Rara / Eur. J. Pure Appl. Math, 16 (1) (2023), 418-429 425 Case 1. v ∈ V (G) ∩NG(A) Then x ∈ Bv and W \ {x} = (Bv \ {x}) ∪  ⋃ u∈V (G)\{v} Du  ∪ A or (W \ {x}) ∪ {y} for some y ∈ (V (G ◦ H) \ W ) ∩ NG◦H(x) is a resolving hop dominat- ing set by Theorem 3. Case 2. v ∈ V (G) \NG(A) Then x ∈ Dv and W \ {x} = (Dv \ {x}) ∪  ⋃ u∈V (G)\{v} Bu  ∪ A or (W \ {x}) ∪ {y} is a resolving hop dominating set by Theorem 3. Therefore W is a 1-movable resolving hop dominating set of G ◦H. Corollary 4. Let G and H be nontrivial connected graphs where |V (G)| = p. Then γ1mrRh(G ◦H) ≤ min {p(msln(H)), γt(G) + p(mln(H))} . Proof: Let W ⊆ V (G ◦ H) be a 1-movable resolving hop dominating set of G ◦ H. Then W ∩ V (Hv) ̸= ∅ and W ∩ V (Hv) is a 1-movable locating set for each v ∈ V (G) and W = A ∪  ⋃ v∈NG(A) Bv  ∪  ⋃ u∈V (G)\NG(A) Du  where A ⊆ V (G) and Bv and Du satisfy the given properties in Theorem 4. Consider the following cases for set A. Case 1. A = ∅ Then NG(A) = ∅. Let Du = W ∩ V (Hu) be an msln-set of Hu for each u ∈ V (G). Thus, W =  ⋃ u∈V (G) Du  is a 1-movable resolving hop dominating set ofG◦H by Theorem 4. Implying that, γ1mRh(G ◦H) ≤ |W | = |V (G)||Du| ≤ p(msln(H)). Case 2. A is a γt-set of G Then NG(A) = V (G). Let Bv = W ∩ V (Hv) be an mln-set of Hv for each v ∈ V (G). Hence, W = A ∪  ⋃ v∈V (G) Bv  is a 1-movable resolving hop dominating set of G ◦H by Theorem 4. It follows that γ1mRh(G ◦H) ≤ |W | = |A|+ |V (G)||Bv| = γt(G) + p(mln(H)). Therefore, γ1mrRh(G ◦H) ≤ min {p(msln(H)), γt(G) + p(mln(H))} . J. Mohamad, H. Rara / Eur. J. Pure Appl. Math, 16 (1) (2023), 418-429 426 5. On 1-Movable Resolving Hop Domination in the Lexicographic Product of Graphs The lexicographic product of two graphs G and H, denoted by G[H], is the graph with vertex-set V (G[H]) = V (G) × V (H) such that (u1, u2)(v1, v2) ∈ E(G[H]) if either u1v1 ∈ E(G) or u1 = v1 and u2v2 ∈ E(H). Theorem 5. [10] Let G and H be nontrivial connected graphs with △(H) ≤ |V (H)| − 2. Then W = ⋃ x∈S [{x} × Tx], where S ⊆ V (G) and Tx ⊆ V (H) for each x ∈ S, is a resolving hop dominating set of G[H] if and only if (i) S = V (G); (ii) Tx is a locating set for every x ∈ V (G); (iii) Tx or Ty is a strictly locating set of H whenever x and y are adjacent vertices of G with NG[x] = NG[y]; (iv) Tx or Ty is a (locating) dominating set of H whenever x and y are nonadjacent vertices of G with NG(x) = NG(y); and (v) Tx is a strictly locating set of H for each x ∈ S \NG(S, 2). The set of shaded vertices in the lexicographic product P3[P4] in Figure 3 where G = P3 and H = P4 satisfies the conditions in Theorem 5 and thus it is a resolving hop dominating set of G[H]. In fact, the set of vertices that are not shaded is also a resolving hop dominating set of G[H]. Figure 3: Resolving hop dominating sets of P3[P4] Theorem 6. Let G and H be nontrivial connected graphs with △(H) ≤ |V (H)| − 2. Then W = ⋃ x∈S ({x} × Tx) where S ⊆ V (G) and Tx ⊆ V (H) for each x ∈ S, is a 1-movable resolving hop dominating set of G[H] if and only if the follow- ing conditions hold: (i) S = V (G). (ii) Tx is a 1-movable locating set for each x ∈ S. J. Mohamad, H. Rara / Eur. J. Pure Appl. Math, 16 (1) (2023), 418-429 427 (iii) Tx \ {a} or Ty is a strictly locating set of H whenever x and y are adjacent vertices of G with NG[x] = NG[y] and for each a ∈ Tx. (iv) Tx \ {a} or Tx \ {a} ∪ {b} or Ty is a (locating) dominating set of H whenever x and y are nonadjacent vertices of G with NG(x) = NG(y) and for each a ∈ Tx and for some b ∈ NH(a). (v) Tx \ {a} or Tx \ {a} ∪ {b} is a strictly locating set of H for each x ∈ S \NG(S, 2) and for each a ∈ Tx and for some b ∈ NH(a). Proof: Suppose W is a 1-movable resolving hop dominating set of G[H]. Then by Theorem 5, S = V (G) and Tx is a locating set of H for each x ∈ V (G). Let a ∈ Tx. Then (x, a) ∈ W . Since W is a 1-movable resolving hop dominating set, either W \ {(x, a)} =  ⋃ v∈S\{x} ({v} × Tv)  ∪ [{x} × (Tx \ {a})] or (W \ {(x, a)}) ∪ {(x, b)} =  ⋃ z∈S\{x} ({z} × Tz)  ∪ [{x} × (Tx \ {a} ∪ {b})] for some b ∈ NH(a) ∩ (V (H) \ Tx) or (W \ {(x, a)}) ∪ {(y, u)} =  ⋃ p∈S\{(x,y)} ({p} × Tp)  ∪ [{x} × (Tx \ {a})] ∪ [{y} × (Ty ∪ {u})] for some y ∈ V (G) ∩ NG(x) and u ∈ V (H) \ Ty is a resolving hop dominating set of G[H]. By Theorem 5, Tx \ {a} or (Tx \ {a}) ∪ {b} is a locating set of H for each a ∈ Tx and for some b ∈ NH(a) ∩ (V (H) \ Tx). Hence, Tx is a 1-movable locating set of H for each x ∈ V (G) or Tx \ {a} is locating and (ii) holds. Suppose (iii) does not hold. Then there exist p ∈ V (H) \ (Tx \ {a}) and q ∈ V (H) \ Ty such that NH(p) ∩ (Tx \ {a}) = Tx \ {a} and NH(q)∩Ty = Ty for some adjacent vertices x and y of G with NG[x] = NG[y] and for some a ∈ Tx. Hence, both W \ {(x, a)} and (W \ {(x, a)})∪{(y, b)} are not resolving sets, a contradiction. Thus, (iii) holds. Statement (iv) is proved similarly. If (v) does not hold, then W \ {(x, a)} and (W \ {(x, a)} ∪ {(y, b)}) are not hop dominating sets of G[H] for all y ∈ NG(x) and b ∈ V (H) \ Tx or x = y and b ∈ NH(a). This is a contradiction to W being a 1-movable resolving hop dominating set of G[H]. Hence, (v) holds. For the converse, suppose that W satisfies properties (i) to (v). By Theorem 5, W is a resolving hop dominating set of G[H]. Let x ∈ V (G) and a ∈ Tx. Then (x, a) ∈ W and W \ {(x, a)} =  ⋃ v∈S\{x} ({v} × Tv)  ∪ [{x} × (Tx \ {a})] J. Mohamad, H. Rara / Eur. J. Pure Appl. Math, 16 (1) (2023), 418-429 428 and for some b ∈ NH(a) ∩ (V (H) \ Tx), (W \ {(x, a)}) ∪ {(x, b)} =  ⋃ z∈S\{x} ({z} × Tz)  ∪ [{x} × ((Tx \ {a}) ∪ {b})] and (W \ {(x, a)}) ∪ {(y, q)} =  ⋃ p∈S\{(x,y)} ({p} × Tp)  ∪ [{x} × (Tx \ {a})] ∪ [{y} × (Ty ∪ {q})] for some y ∈ V (G) ∩NG(x) and q ∈ V (H) \ Ty. By (i) to (v) and Theorem 5, for every (x, a) ∈ W either W \{(x, a)} is a resolving hop dominating set of G[H] or there exists (y, b) ∈ NG[H]((x, a)) ∩ (V (G[H]) \ W ) such that (W \ {(x, a)}) ∪ {(y, b)} is a resolv- ing hop dominating set of G[H]. Therefore, W is a 1-movable resolving hop dominating set of G[H]. Corollary 5. Let G be a nontrivial connected totally point determining graph with γ(G) ̸= 1 and H be a nontrivial connected graph with △(H) ≤ |V (H)| − 2. Then γ1mRh(G[H]) = |V (G)|mln(H). Proof: Let S = V (G) and let Rx be an mln-set of H for each x ∈ S. Since γG ̸= 1, x ∈ NG(S, 2) for each x ∈ S. By Theorem 6, W = ⋃ x∈S [{x} ×Rx] is a 1-movable resolving hop dominating set of G[H]. Thus, γ1mRh(G[H]) ≤ |W | = |V (G)||Rx| = |V (G)|mln(H). Now, if W0 = ⋃ x∈S0 ({x} × Tx) is a γ1mRh-set of G[H] then S0 = V (G) and Tx is a 1-movable locating set of H for each x ∈ V (G) by Theorem 6. Hence, γ1mRh(G[H]) = |W0| = |V (G)||Tx| ≥ |V (G)|mln(H). Therefore, γ1mRh(G[H]) = |V (G)|mln(H). 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