EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 16, No. 3, 2023, 1464-1479 ISSN 1307-5543 – ejpam.com Published by New York Business Global 1-movable 2-Resolving Hop Domination in Graphs Angelica Mae Mahistrado1, Helen 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 set S of vertices in G is a 1-movable 2-resolving hop dominating set of G if S is a 2-resolving hop dominating set in G and for every v ∈ S, either S\{v} is a 2-resolving hop dominating set of G or there exists a vertex u ∈ ( (V (G)\S) ∩NG(v) ) such that ( S\{v} ) ∪ {u} is a 2-resolving hop dominating set of G. The 1-movable 2-resolving hop domination number of G, denoted by γ1 m2Rh(G) is the smallest cardinality of a 1-movable 2- resolving hop dominating set of G. In this paper, we investigate the concept and study it for graphs resulting from some binary operations. Specifically, we characterize the 1-movable 2-resolving hop dominating sets in the join, corona and lexicographic products of graphs, and determine the bounds of the 1-movable 2-resolving hop domination number of each of these graphs. 2020 Mathematics Subject Classifications: 05C69 Key Words and Phrases: 1-movable 2-resolving hop dominating set, 1-movable 2-resolving hop domination number, join, corona, edge corona, lexicographic product 1. Introduction The concept of domination was formally studied by Claude Berge [1] in 1958 and Oystein Ore in 1962. In 2015, Natarajan and Ayyaswamy introduced and studied the concept of hop domination [14]. On the other hand, in 1975 using the term locating set, the concept of resolving sets for a connected graph was first introduced by Slater [17]. These concepts were studied much earlier in the context of the coin-weighing problem. Later that year, Harary and Melter introduced independently these concepts, but with different terminologies [8]. The term metric dimension was used by Harary and Melter instead of locating number. Recently, 2-resolving hop dominating sets in graphs was studied in [9]. Other variations of 2-resolving hop dominating sets in graphs are found in [10, 11]. Moreover, other variations of resolving sets and hop dominating sets in graphs were also studied in [4–7, 12, 13]. ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v16i3.4770 Email addresses: angelicamae.mahistrado@g.msuiit.edu.ph (A.M. Mahistrado), helen.rara@g.msuiit.edu.ph (H. Rara) https://www.ejpam.com 1464 © 2023 EJPAM All rights reserved. A.M. Mahistrado, H. Rara / Eur. J. Pure Appl. Math, 16 (3) (2023), 1464-1479 1465 2. Terminology and Notation In this study, we consider finite, simple, connected, undirected graphs. For basic graph- theoretic concepts, we then refer readers to [2] and [3]. The following concepts are found in [2], [14], and [16], respectively. Let G be a connected graph. 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 neighborhood of X in G is the set NG[X, 2] = NG(X, 2) ∪X. A set S ⊆ V (G) is a hop dominating set of G if NG[S, 2] = V (G), that is, 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. For an ordered set of vertices W = {w1, w2, ..., wk} ⊆ V (G) and a vertex v in G, we refer to the k-vector (ordered k-tuple) rG(v/W ) = (dG(v, w1), dG(v, w2), ..., dG(v, wk)) as the (metric) representation of v with respect to W . The set W is called a resolving set for G if distinct vertices have distinct representations with respect to W . Hence, if W is a resolving set of cardinality k for a graph G of order n, then the set {rG(v/W ) : v ∈ V (G)} consists of n distinct k-vectors. A resolving set of minimum cardinality is called aminimum resolving set or a basis, and the cardinality of a basis for G is the dimension dim(G) of G. An ordered set of vertices W = {w1, ..., wk} is a k-resolving set for G if, for any distinct vertices u, v ∈ V (G), the (metric) representations rG(u/W ) and rG(v/W ) of u and v, respectively, differ in at least k positions. If k = 1, then the k-resolving set is called a resolving set for G. If k = 2, then the k-resolving set is called a 2-resolving set for G. If G has a k-resolving set, the minimum cardinality dimk(G) of a k-resolving set is called the k-metric dimension of G. A set S of vertices in G is a 1-movable 2-resolving hop dominating set of G if S is a 2-resolving hop dominating set in G and for every v ∈ S, either S\{v} is a 2-resolving hop dominating set of G or there exists a vertex u ∈ ( (V (G)\S) ∩ NG(v) ) such that( S\{v} ) ∪ {u} is a 2-resolving hop dominating set of G. The 1-movable 2-resolving hop domination number of G, denoted by γ1m2Rh(G) is the smallest cardinality of a 1-movable 2-resolving hop dominating set of G. Any 1-movable 2-resolving hop dominating set of cardinality γ1m2Rh(G) is referred to as a γ1m2Rh -set of G. Definition 1. [6] LetG be any nontrivial connected graph and S ⊆ V (G). A set S ⊆ V (G) is a 2-locating set of G if it satisfies the following conditions: (i) ∣∣[(NG(x)\NG(y) ) ∩S]∪ [ ( NG(y)\NG(x) ) ∩S] ∣∣ ≥ 2, for all x, y ∈ V (G)\S with x ̸= y. (ii) ( NG(v)\NG(w) ) ∩ S ̸= ∅ or ( NG(w)\NG[v] ) ∩ S ̸= ∅, for all v ∈ S and for all w ∈ V (G)\S. A.M. Mahistrado, H. Rara / Eur. J. Pure Appl. Math, 16 (3) (2023), 1464-1479 1466 The 2-locating number of G, denoted by ln2(G), is the smallest cardinality of a 2-locating set of G. A 2-locating set of G of cardinality ln2(G) is referred to as an ln2-set of G. Definition 2. [15] A set D ⊆ V (G) is a point-wise non-dominating set of G if for each v ∈ V (G)\D, there exists u ∈ D such that v /∈ NG(u). The smallest cardinality of a point- wise non-dominating set of G, denoted by pnd(G), is called the point-wise non-domination number of G. Any point-wise non-dominating set D of G with |D| = pnd(G), is called a pnd-set of G. Definition 3. [9] A 2-locating set S ⊆ V (G) which is point-wise non-dominating is called a 2-locating point-wise non-dominating set in G. The minimum cardinality of a 2-locating point-wise non-dominating set in G, denoted by lnpnd 2 (G) is called the 2-locating point-wise non-domination number of G. Any 2-locating point-wise non-dominating set of cardinality lnpnd 2 (G) is then referred to as a lnpnd 2 (G)-set in G. Definition 4. [6] Let G be any nontrivial connected graph and S ⊆ V (G). S is a (2, 2)- locating ((2, 1)-locating, respectively) set in G if S is 2-locating and |NG(y)∩ S| ≤ |S| − 2 (|NG(y)∩S| ≤ |S|− 1, respectively), for all y ∈ V (G). The (2, 2)-locating ( (2, 1)-locating, respectively) number of G, denoted by ln(2,2)(G) (ln(2,1)(G), respectively), is the smallest cardinality of a (2, 2)-locating ((2, 1)-locating, respectively) set in G. A (2, 2)-locating ((2, 1)-locating, respectively) set in G of cardinality ln(2,2)(G) (ln(2,1)(G), respectively) is referred to as an ln(2,2)-set (ln(2,1)-set, respectively) in G. Definition 5. [9] A (2,2)-locating ((2,1)-locating, respectively) set S ⊆ V (G) which is a point-wise non-dominating is called a (2,2)-locating point-wise non-dominating ((2,1)- locating point-wise non-dominating, respectively) set in G. The minimum cardinality of a (2,2)-locating point-wise non-dominating ((2,1)-locating point-wise non-dominating, respectively) set in G, denoted by lnpnd (2,2)(G) (lnpnd (2,1)(G),respectively) is called the (2,2)- locating point-wise non-domination ((2,1)-locating point-wise non-domination) number of G. Any (2,2)-locating point-wise non-dominating ((2,1)-locating point-wise non-dominating, respectively) set of cardinality lnpnd (2,2)(G) (lnpnd (2,1)(G), respectively) is then referred to as a lnpnd (2,2)-set (ln pnd (2,1)-set) in G. Definition 6. A set S ⊆ V (G) is a 1-movable 2-locating point-wise non-dominating set in G if S is a 2-locating point-wise non-dominating set in G and for every v ∈ S, either S\{v} is a 2-locating point-wise non-dominating set or there exists a vertex u ∈ ( (V (G)\S) ∩ NG(v) ) such that ( S\{v} ) ∪ {u} is a 2-locating point-wise non-dominating set of G. The 1-movable 2-locating point-wise non-domination number of G, denoted by mlnpnd 2 (G) is the smallest cardinality of a 1-movable 2-locating point-wise non-dominating set of G. Any 1-movable 2-locating point-wise non-dominating set of cardinality mlnpnd 2 (G) is referred to as a mlnpnd 2 -set of G. Definition 7. A set S ⊆ V (G) is a 1-movable (2, 2)-locating point-wise non-dominating ((2, 1)-locating point-wise non-dominating, respectively) in G if S is a (2, 2)-locating point- wise non-dominating ((2, 1)-locating point-wise non-dominating, respectively) set in G and A.M. Mahistrado, H. Rara / Eur. J. Pure Appl. Math, 16 (3) (2023), 1464-1479 1467 for every v ∈ S, either S\{v} is a 2-locating point-wise non-dominating set or there exists a vertex u ∈ ( (V (G)\S) ∩NG(v) ) such that ( S\{v} ) ∪ {u} is (2, 2)-locating point- wise non-dominating ((2, 1)-locating point-wise non-dominating, respectively) set in G. The 1-movable (2,2)-locating point-wise non-domination ((2,1)-locating point-wise non- domination) number of G, denoted by mlnpnd (2,2)(G) (mlnpnd (2,1)(G),respectively) is s the smallest cardinality of a 1-movable called the (2,2)-locating point-wise non-dominating ((2,1)-locating point-wise non-dominating) number set of G. Any 1-movable (2,2)-locating point-wise non-dominating ((2,1)-locating point-wise non-dominating, respectively) set of cardinality mlnpnd (2,2)(G) (mlnpnd (2,1)(G), respectively) is then referred to as a mlnpnd (2,2)-set (mlnpnd (2,1)-set) in G. 3. Preliminary Results Remark 1. A 1-movable 2-resolving hop dominating set does not always exist in a graph G. Example 1. A complete bipartite graph Km,n and a complete graph Kn do not admit 1-movable 2-resolving hop dominating set. Remark 2. Let G be a nontrivial connected graph. If S is a 2-resolving set in G, then {x, y} ⊆ S for every x, y ∈ V (G) with x ̸= y and dG(x, z) = dG(y, z) for each z ∈ V (G)\{x, y}. Proposition 1. Let G be a nontrivial connected graph. Then G admits a 1 -movable 2-resolving hop dominating set if and only if γ(G) ̸= 1, dim2(G) ̸= V (G) and G is a free-equidistant graph. Proof. Suppose that G admits a 1-movable 2-resolving hop dominating set. Let S be a 1-movable 2-resolving hop dominating set of G. Suppose 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) ∩NG(x)) are not hop dominating sets of G. Thus, S is not a 1-movable 2-resolving hop dominating set. Therefore, γ(G) ̸= 1. If dim2(G) = V (G), then V (G)\{z} for each z ∈ V (G) is not a 2-resolving set. Thus, S is not a 1-movable 2-resolving hop dominating set. Therefore, dim2(G) ̸= V (G). If G is not a free-equidistant graph, then there exist a pair of vertices y, w ∈ V (G) with dG(w, z) = dG(y, z) for all z ∈ V (G)\{y, w}. By Remark 2, y, w ∈ S. Hence, rG(y/(S\{y})) and rG(w/(S\{y})) differ in at most one position. Thus, S is not a 1-movable 2-resolving hop dominating set. Therefore, G is a free-equidistant graph. Conversely, suppose that γ(G) ̸= 1, dim2(G) ̸= V (G) and G is a free-equidistant graph. Let S = V (G). Then S is a 2-resolving hop dominating set in G. For each x ∈ S, S \ {x} is a 2-resolving set in G since dim2(G) ̸= V (G) and G is a free-equidistant graph. 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 2-resolving hop dominating A.M. Mahistrado, H. Rara / Eur. J. Pure Appl. Math, 16 (3) (2023), 1464-1479 1468 set in G for each x ∈ S. It follows that S is a 1-movable 2-resolving hop dominating set in G. Accordingly, G admits 1-movable 2-resolving hop dominating set. Remark 3. Every 1-movable 2-resolving hop dominating set in G is a 2-resolving hop dominating set in G. Thus, γ2Rh(G) ≤ γ1m2Rh(G). Proposition 2. (i) For a path Pn on n vertices (n ≥ 4), γ1m2Rh(Pn) = n. (ii) For a cycle Cn on n vertices, γ1m2Rh(Cn) = 3, if n = 5; 2n+ k 3 , if n = k(mod 3), 0 ≤ k ≤ 2 and n > 5. Proof. Let Pn = [v1, v2, v3, . . . vn]. By Proposition 1, V (Pn) is a 1-movable 2-resolving hop dominating set. Suppose Si = V (Pn)\{vi1 , vi2 , . . . vik} is a 2-resolving hop dominating set in Pn for 1 ≤ k ≤ n− 2. Note that 0 < |NPn(vim , 2)| ≤ 2 for each m ∈ {1, 2, . . . k}. If |NPn(vim , 2)| = 1, then Si \{vj} is not a hop dominating set for vj ∈ NPn(vim , 2). Suppose |NPn(vim , 2)| = 2. Let vj , vl ∈ NPn(vim , 2). If |NPn(vj , 2)| = 1 or |NPn(vl, 2)| = 1, then Si\{vj} or Si\{vl} is not a hop dominating set in Pn. On the other hand, if |NPn(vj , 2)| = 2 or |NPn(vl, 2)| = 2, then Si \ {vp} or Si \ {vq} is not a hop dominating set where vp ∈ NPn(vj , 2) and vq ∈ NPn(vl, 2). Thus, Si is not a 1-movable 2-resolving hop dominat- ing set in Pn. Therefore, γ 1 m2Rh(Pn) = n. (ii) Let Cn = [v1, v2, . . . , vn] and S be a γ1m2Rh- set of Cn. The case when n = 5 can be verified. Next, let n > 5 and n ≡ k(mod 3) where 0 ≤ k ≤ 2. Then n = 3r + k. Hence, r = n− k 3 . Then the set S = {v1, v3, v4, v6, v7, v9, v10, v12, v13, . . . , v3r+k−3, v3r+k−2, . . . , v3r+k} is a γ1m2Rh- set of Cn. Therefore, |S| = 3r + k − r = 2n+ k 3 . Now, consider the following results of 1-movable 2-locating point-wise non-dominating sets which are used in characterizing the 1-movable 2-resolving hop dominating sets in some binary operations. Remark 4. A 1-movable 2-locating point-wise non-dominating set does not always exist in a graph G. Example 2. A complete bipartite graph Km,n and a complete graph Kn do not admit 1-movable 2-locating point-wise non-dominating set. Remark 5. Let G be a nontrivial connected graph. If S is a 2-locating set in G, then {x, y} ⊆ S for every x, y ∈ V (G) with x ̸= y and NG(x) = NG(y). A.M. Mahistrado, H. Rara / Eur. J. Pure Appl. Math, 16 (3) (2023), 1464-1479 1469 Proposition 3. Let G be a nontrivial connected graph. Then G admits a 1-movable 2-locating set if and only if G is a point determining graph. Proof. Let S be a 1-movable 2-locating set of G. Suppose G is not a point determining graph. Then there exist x, y ∈ V (G) with x ̸= y and NG(x) = NG(y). This implies that x, y ∈ S by Remark 5. Thus, S \ {x} and (S \ {x}) ∪ {z} are not 2-locating sets where z ∈ (V (G) \ S) ∩NG(x). Thus, G is a point determining graph. Conversely, suppose G is a point determining graph. Let S = V (G). Then S is a 2-locating set of G. Since G is a point determining graph, S \ {x} is a 2-locating set for all x ∈ S. Therefore, G admits a 1-movable 2-locating set. Proposition 4. Let G be a nontrivial connected graph. Then G admits a 1-movable 2-locating point-wise non-dominating set if and only if G is a point determining graph where γ(G) ̸= 1. Proof. Let S be a 1-movable 2-locating point-wise non-dominating set of G. Suppose γ(G) = 1. Set A = {x ∈ V (G) : {x} is a dominating set of G}. Then A ̸= ∅ since γ(G) = 1. Since S is a point-wise non-dominating set, A ⊆ S. Let x ∈ A. Then S\{x} and ( S\{x} ) ∪ {y} for each y ∈ V (G)\S ∩ NG(x) are not point-wise non-dominating sets of G. Thus, S is not a 1-movable 2-locating point-wise non-dominating set. Therefore, γ(G) ̸= 1. By Proposition 3, G is a point determining graph. Conversely, suppose G is a point determining graph where γ(G) ̸= 1. Then by Proposition 3, S = V (G) is a 1-movable 2-locating set. Thus, S is a 1-movable 2-locating point-wise non-dominating set of G. Accordingly, G admits a 1-movable 2-locating point- wise non-dominating set. Proposition 5. Let G be a nontrivial connected graph of order n ≥ 4. Then (i) mlnpnd 2 (Pn) = n, if n = 4, 5; 2n+ k 3 , if n = k(mod 3), 0 ≤ k ≤ 2 and n > 5; (ii) mlnpnd (2,1)(Pn) = mlnpnd (2,2)(Pn) = n, if n = 5; 2n+ k 3 , if n = k(mod 3), 0 ≤ k ≤ 2 and n > 5; (iii) mlnpnd 2 (Cn) = 3, if n = 5; 2n+ k 3 , if n = k(mod 3), 0 ≤ k ≤ 2 and n > 5; A.M. Mahistrado, H. Rara / Eur. J. Pure Appl. Math, 16 (3) (2023), 1464-1479 1470 (iv) mlnpnd (2,1)(Cn) = 3, if n = 5; 2n+ k 3 , if n = k(mod 3), 0 ≤ k ≤ 2 and n > 5; (v) mlnpnd (2,2)(Cn) = 5, if n = 5; 2n+ k 3 , if n = k(mod 3), 0 ≤ k ≤ 2 and n > 5. Proof. (i) Let Pn = [v1, v2, . . . , vn] and S be an mlnpnd 2 - set of Pn. The case where n = 4, 5 can be verified. Next, let n > 5 and n ≡ k(mod 3) where 0 ≤ k ≤ 2. Then n = 3r + k. Hence, r = n− k 3 . Then the set S = {v1, v3, v4, v6, v7, v9, v10, v12, v13, . . . , v3r+k−3, v3r+k−2, . . . , v3r+k} is an mlnpnd 2 - set of Pn. Therefore, |S| = 3r + k − r = 2n+ k 3 . The proofs of (ii), (iii), (iv) and (v) are similar to (i). Remark 6. Let G be a nontrivial connected graph. Then G admits a 1-movable (2, 1)- locating point-wise non-dominating set if and only if ∆(G) ≤ |V (G)|| − 2. Remark 7. Let G be a nontrivial connected graph. Then G admits a 1-movable (2, 2)- locating point-wise non-dominating set if and only if ∆(G) ≤ |V (G)|| − 3. We now characterize the 1-movable 2-resolving hop dominating sets in some graphs under some binary operations. 4. Join of Graphs As a consequence of Proposition 1 the next result follows. Corollary 1. A graph G does not admit a 1-movable 2-resolving hop dominating set if and only if G = K1 +H for any nontrivial connected graph H. Theorem 1. [9] Let G and H be nontrivial connected graphs with γ(G) ̸= 1 and γ(H) ̸= 1. A set S ⊆ V (G + H) is a 2-resolving hop dominating set of G + H if and only if S = SG ∪ SH where SG = V (G) ∩ S and SH = V (H) ∩ S are 2-locating point-wise non-dominating sets of G and H, respectively where SG or SH is a (2, 2)-locating point-wise non-dominating set or SG and SH are (2, 1)-locating point-wise non-dominating sets. A.M. Mahistrado, H. Rara / Eur. J. Pure Appl. Math, 16 (3) (2023), 1464-1479 1471 Theorem 2. Let G and H be nontrivial connected graphs with γ(G) ̸= 1 and γ(H) ̸= 1. A set S ⊆ V (G + H) is a 1-movable 2-resolving hop domi- nating set of G+H if and only if S = SG∪SH where SG = V (G)∩S and SH = V (H)∩S are 1-movable 2-locating point-wise non-dominating sets of G and H, respectively where SG or SH is a 1-movable (2, 2)-locating point-wise non-dominating set or SG and SH are 1-movable (2, 1)-locating point-wise non-dominating sets. Proof. Suppose that S ⊆ V (G + H) is a 1-movable 2-resolving hop dominating set of G + H. Since S is 2-resolving hop dominating set by Theorem 1, S = SG ∪ SH where SG and SH are 2-locating point-wise non-dominating sets of G and H, respectively where SG or SH is a (2, 2)-locating (point-wise non-dominating) set or SG and SH are (2, 1)-locating (point-wise non-dominating) sets. Now, let p ∈ SG. Then p ∈ S. Thus, S\{p} = ( SG\{p} ) ∪ SH or ( S\{p} ) ∪ {z} = [( SG\{p} ∪ {z} )] ∪ SH for some z ∈ NG(p)∩(V (G)\SG) or (S\{p})∪{q} = (SG\{p})∪(SH∪{q}) for some q ∈ V (H)\SH is a 2-resolving hop dominating set inG+H. Hence, by Theorem 1, SG\{p} or (SG\{p})∪{z} is a 2-locating point-wise non-dominating set of G. This shows that SG is a 1-movable 2-locating point-wise non-dominating set of G. Similarly, SH is a 1-movable 2-locating point-wise non-dominating set of H. Therefore, SG and SH are 1-movable 2-locating point-wise non-dominating sets of G and H, respectively where SG or SH is a 1-movable (2, 2)-locating (point-wise non-dominating) set or SG and SH are 1-movable (2, 1)-locating (point-wise non-dominating) sets. Conversely, suppose that SG and SH satisfy the given conditions. Then by Theorem 1, S = SG ∪ SH is a 2-resolving hop dominating set in G + H. Let p ∈ S. If p ∈ SG, then S\p = (SG\{p})∪SH or (S\{p})∪{w} = [ (SG\{p})∪{w} ] ∪SH for some w ∈ NG(p)∩ (V (G)\SG) is a 2-resolving hop dominating set in G+H. Similarly, suppose that p ∈ SH . Then S\{p} = (SH\{p})∪SG or (S\{p})∪{w} = [ (SH\{p})∪{w} ] ∪SG for some w ∈ NH(p) ∩ (V (H)\SH) is a 2-resolving hop dominating set in G+H. Therefore, S is a 1-movable 2-resolving hop dominating set in G+H. Corollary 2. Let G and H be nontrivial connected graphs with γ(G) ̸= 1 and γ(H) ̸= 1. Then γ1m2Rh(G+H) =min{mlnpnd (2,2)(G) +mlnpnd (2) (H),mlnpnd (2) (G) +mlnpnd (2,2)(H), mlnpnd (2,1)(G) +mlnpnd (2,1)(H)}. A.M. Mahistrado, H. Rara / Eur. J. Pure Appl. Math, 16 (3) (2023), 1464-1479 1472 5. Corona of Graphs Theorem 3. [9] Let G and H be nontrivial connected graphs. A set S ⊆ V (G ◦H) is a 2-resolving hop dominating set of G ◦H if and only if S = A ∪  ⋃ v∈V (G)∩NG(A) Sv  ∪  ⋃ w∈V (G)\NG(A) Dw  where (i) A ⊆ V (G) such that for each w ∈ V (G)\A, there exists x ∈ A with dG(w, x) = 2 or there exists y ∈ V (G) ∩NG(w) with V (Hy) ∩ S ̸= ∅; (ii) Sv ⊆ V (Hv) is a 2-locating set of Hv for all v ∈ V (G) ∩NG(A); and (iii) Dw ⊆ V (Hw) is a 2-locating point-wise non-dominating set of Hw for all w ∈ V (G)\NG(A). Theorem 4. Let G and H be nontrivial connected graphs. Then S ⊆ V (G ◦ H) is a 1-movable 2-resolving hop dominating set of G ◦H if and only if S ∩ V (Hv) ̸= ∅ and S = A ∪  ⋃ v∈V (G)∩NG(A) Sv  ∪  ⋃ w∈V (G)\NG(A) Dw  where (i) A ⊆ V (G) (ii) Sv ⊆ V (Hv) is a 1-movable 2-locating set of Hv for all v ∈ V (G) ∩NG(A). (iii) Dw ⊆ V (Hw) is a 1-movable 2-locating point-wise non-dominating set of Hw for all w ∈ V (G)\NG(A). Proof. Suppose that S ⊆ V (G ◦H) is a 1-movable 2-resolving hop dominating set of G◦H. Then S is a 2-resolving hop dominating set. Let A = S∩V (G) and Sv = S∩V (Hv) for all v ∈ V (G) ∩NG(A). By Theorem 3, Sv is a 2-locating set of Hv. Let p ∈ Sv. Since S is a 1-movable 2-resolving hop dominating set and p ∈ S , either S\{p} or (S\{p})∪{q} is a 2-resolving hop dominating set in G ◦ H for some q ∈ (V (G ◦ H)\S) ∩ NG◦H(p). Now, note that S\{p} = A ∪ ( Sv\{p} ) and (S\{p}) ∪ {q} = A ∪ ( (Sv\{p}) ∪ {q} ) or (S\{p}) ∪ {q} = ( A ∪ {q} ) ∪ (Sv\{p}). Hence, either Sv\{p} or (Sv\{p}) ∪ {q} for some q ∈ (V (Hv)\Sv)∩NHv(p) is a 2-locating set of Hv. Thus, Sv is a 1-movable 2-locating set of Hv. Finally, suppose w ∈ V (G)\NG(A). Then by similar argument, Dw is a 1-movable 2-locating point-wise non-dominating set of Hw. Thus, (ii) follows. Conversely, suppose that S is a set as described and satisfies the given conditions. Then by Theorem 3, S is a 2-resolving hop dominating set. Let x ∈ S and let v ∈ V (G) A.M. Mahistrado, H. Rara / Eur. J. Pure Appl. Math, 16 (3) (2023), 1464-1479 1473 such that x ∈ V (⟨v⟩+Hv). If x = v, then x ∈ A. By Theorem 3, S\{x} or ( S\{x} ) ∪{y} for some y ∈ ( V (G◦H)\S ) ∩NG◦H(x) is a 2-resolving hop dominating set. Next, suppose that x ̸= v. Consider the following cases. Case 1: v ∈ V (G) ∩NG(A) Then x ∈ Sv and S\{x} = (Sv\{x})∪ ( ⋃ u∈V (G)\{v} Du ) ∪A or (S\{x})∪ {y} for some y ∈ (V (G ◦H)\S) ∩NG◦H(x) is a 2-resolving hop dominating set, by Theorem 3. Case 2: v ∈ V (G)\NG(A) Then x ∈ Dv and S\{x} = (Dv\{x})∪ ( ⋃ u∈V (G)\{v} Su ) ∪A or (S\{x})∪{y} for some y ∈ (V (G ◦H)\S) ∩NG◦H(x) is a 2-resolving hop dominating set, by Theorem 3. Accordingly, S is a 1-movable 2-resolving hop dominating set in G ◦H. Corollary 3. Let G and H be nontrivial connected graphs where |V (G)| = n. Then γ1m2Rh(G ◦H) ≤ min{n ·mlnpnd 2 (H), γt(G) + n ·mln2(H)}. Proof. Let S ⊆ V (G ◦ H) be a 1-movable 2-resolving hop dominating set of G ◦ H. Then S ∩ V (Hv) ̸= ∅ and S ∩ V (Hv) is a 1-movable 2-locating set for each v ∈ V (G) and S = A ∪  ⋃ v∈V (G)∩NG(A) Sv  ∪  ⋃ w∈V (G)\NG(A) Dw  where A ⊆ V (G) and Sv and Dw satisfy the given properties in Theorem 4. Consider the following cases for set A. Case 1: A = ∅ Let Dw = S∩V (Hw) be an mlnpnd 2 -set of Hw for each w ∈ V (G). Thus, S = ( ⋃ v∈V (G) Dw ) s a 1-movable 2-resolving hop dominating set of G ◦H by Theorem 4. Implying that, γ1m2Rh(G ◦H) ≤ |S| = |V (G)||Dw| ≤ n · (mlnpnd 2 (H)). Case 2: A is a γt-set of G Let NG(A) = V (G). Sv = S ∩ V (Hv) be an mln2-set of Hv for each v ∈ V (G). Thus, S = A ∪ ( ⋃ v∈V (G) Sv ) s a 1-movable 2-resolving hop dominating set of G ◦H by Theorem 4. Implying that, γ1m2Rh(G ◦H) ≤ |S| = |A|+ |V (G)||Sv| ≤ γt(G) + n · (mln2(H)). A.M. Mahistrado, H. Rara / Eur. J. Pure Appl. Math, 16 (3) (2023), 1464-1479 1474 6. Edge Corona of Graphs Theorem 5. Let G ̸= P2 and H be any nontrivial connected graphs. A set C ⊆ V (G ⋄H) is a 2-resolving hop dominating set of G ⋄H if and only if C = A ∪  ⋃ uv∈E(G) Suv  where (i) A ⊆ V (G); (ii) Suv ⊆ V (Huv) is a 2-locating set of Huv for all uv ∈ E(G) or if uv is a pendant edge, then Suv is a (2, 1)-locating set of Huv whenever l(⟨{u, v}⟩) ⊆ A and Suv is a (2, 2)-locating set of Huv otherwise. Theorem 6. Let G and H be any nontrivial connected graphs where γ(G) ̸= 1 and ∆(H) ≤ |V (H)| − 3. A set C ⊆ V (G ⋄H) is a 1-movable 2-resolving hop dominating set of G ⋄H if and only if C = A ∪  ⋃ uv∈E(G) Suv  where (i) A ⊆ V (G); (ii) Suv ⊆ V (Huv) is a 1-movable 2-locating set of Huv for all uv ∈ E(G) or if uv is a pendant edge, then Suv is a 1-movable (2, 1)-locating set of Huv whenever l(⟨{u, v}⟩) ⊆ A and Suv is a 1-movable (2, 2)-locating set of Huv otherwise. Proof. Suppose that C ⊆ V (G ⋄ H) is a 1-movable 2-resolving hop dominating set of G ⋄ H. Then C is a 2-resolving hop dominating set. Let A = C ∩ V (G) and Suv = C ∩ V (Huv) for all uv ∈ E(G). Then C = A ∪ ( ⋃ uv∈E(G) Suv ) where A ⊆ V (G) and Suv ⊆ V (Huv) for all uv ∈ E(G). By Theorem 5, Suv is a 2-locating set of Huv for all uv ∈ E(G). Let p ∈ Suv. Since C is a 1-movable 2-resolving hop dominating set and p ∈ C, either C\{p} or (C\{p}) ∪ {q} is a 2-resolving hop dominating set of G ⋄ H for some q ∈ (V (G ⋄ H)\C) ∩ NG⋄H(p). Now, note that C\{p} = A ∪ ( Suv\{p} ) and (C\{p})∪ {q} = A∪ ( (Suv\{p})∪ {q} ) or (C\{p})∪ {q} = ( A∪ {q})∪ (Suv\{p}). Hence, either Suv\{p} or (Suv\{p}) ∪ {q} for some q ∈ (V (Huv)\Suv) ∩ NHuv(p) is a 2-locating set of Huv. Thus, Suv is a 1-movable 2-locating set of Huv. Next, suppose that uv is a pendant edge and suppose u is an end-vertex where u ∈ C. Since Suv = C ∩ V (Huv) ⊆ C and C is a 1-movable 2-resolving set it follows by Theorem 5, Suv is a 1- movable (2, 1)- locating set of Huv whenever l(⟨{u, v}⟩) ⊆ A and Suv is a 1-movable (2, 2)-locating set of A.M. Mahistrado, H. Rara / Eur. J. Pure Appl. Math, 16 (3) (2023), 1464-1479 1475 Huv otherwise. Thus, (ii) holds. Conversely, suppose that C is a set as described and satisfies the given conditions. By Theorem 5, C is 2-resolving hop dominating set of G ⋄ H. Let p ∈ C. If p ∈ Suv, then by assumption and Theorem 5, either C\{p} = A ∪ ( Suv\{p} ) or (C\{p}) ∪ {q} = A ∪ ( (Suv\{p}) ∪ {q} ) is a 2-resolving hop dominating set of G ⋄H for some q ∈ (V (G ⋄ H)\C)∩NG⋄H(p). Therefore, C is a 1-movable 2-resolving hop dominating set ofG⋄H. Corollary 4. Let γ(G) ̸= 1 and H a nontrivial connected graph with |E(G)| = p. Then the following statements hold. (i) If G is a graph with no pendant edges, then γ1m2Rh(G ⋄H) = p ·mln2(H). (ii) If G is a graph with k ≥ 1 pendant edges, then γ1m2Rh(G ⋄H) =min {( p− k ) mln2(H) + k ·mln(2,1)(H) + k,( p− k ) mln2(H) + k ·mln(2,2)(H) } and γ1m2Rh(G ⋄H) = ( p− k ) mln2(H) + k ·mln(2,2)(H) whenever mln(2,2)(H) = mln(2,1)(H). 7. Lexicographic Product of Graphs Theorem 7. [9] Let G and H be nontrivial connected graphs. Then W = ⋃ x∈S [{x} × Tx], where S ⊆ V (G) and Tx ⊆ V (H) for each x ∈ S, is a 2-resolving hop dominating set in G[H] if and only if (i) S = V (G); (ii) Tx is a 2-locating set in H for every x ∈ V (G) ; (iii) Tx or Ty is a (2, 1)-locating set or one of Tx and Ty is a (2, 2)-locating set in H whenever x, y ∈ EQ1(G); (iv) Tx and Ty are (2 − locating) dominating sets in H or one of Tx and Ty is a 2-dominating set whenever x, y ∈ EQ2(G). (v) Tx is a 2-locating point-wise non-dominating set in H for every x ∈ S with |NG(x, 2) ∩ S| = 0. Theorem 8. Let G and H be nontrivial connected graphs with △(H) ≤ |V (H)| − 3. Then W = ⋃ x∈S [{x} × Tx], where S ⊆ V (G) and Tx ⊆ V (H) for each x ∈ S, is a 1-movable 2-resolving hop dominating set in G[H] if and only if (i) S = V (G); (ii) Tx is a 1-movable 2-locating set of H for every x ∈ V (G) ; A.M. Mahistrado, H. Rara / Eur. J. Pure Appl. Math, 16 (3) (2023), 1464-1479 1476 (iii) Tx\{p} or Tx\{p}∪ {q} is a 2-locating point-wise non-dominating set of H for every x ∈ S with |NG(x, 2) ∩ S| = 0 and p ∈ Tx and for some q ∈ NH(p). (iv) Tx\{p} and Ty are (2, 1)-locating set or one of Tx\{p} and Ty is a (2, 2)-locating set of H whenever x, y ∈ EQ1(G) and for each p ∈ Tx; (v) Tx\{p} or Tx\{p}∪{q} or Ty is (2− locating) dominating sets in H or one of Tx\{p} and Ty is a 2-dominating set whenever x, y ∈ EQ2(G) and for each p ∈ Tx and for some q ∈ NH(p). Proof. Suppose W is a 1-movable 2-resolving hop dominating set in G[H]. Then by Theorem 7, S = V (G) and Tx is a 2-locating set of H for each x ∈ V (G). Let p ∈ Tx. Then (x, p) ∈ W . Since W is a 1-movable 2-resolving hop dominating set, either W\{(x, p)} =  ⋃ v∈S\{x} ({v} × Tv)  ∪ [{x} × (Tx\{p})] or (W\{(x, p)}) ∪ {(x, q)} =  ⋃ z∈S\{x} ({z} × Tz)  ∪ [{x} × (Tx\{p} ∪ {q})] for some q ∈ (V (H)\Tx) ∩NH(p) or (W\{(x, p)}) ∪ {(y, w)} =  ⋃ a∈S\{(x,y)} ({a} × Ta)  ∪ [{x} × (Tx\{p})] ∪ [{y} × (Ty\{w})] for some y ∈ V (G)∩NG(x) and w ∈ V (H)\Ty is a 2-resolving hop dominating set of G[H]. By Theorem 7, Tx\{p} or (Tx\p) ∪ {q} is a 2-locating set of H for each p ∈ Tx and for some q ∈ (V (H)\Tx) ∩NH(p). Hence, Tx is a 1-movable 2-locating set of H for each x ∈ V (G) or Tx\{p} is 2-locating and (ii) holds. If (iii) does not hold, then W\{(x, p)} and (W\{(x, p)} ∪ {(y, q)}) are not hop domi- nating sets of G[H] for all y ∈ NG(x) and q ∈ V (H)\Tx or x = y and q ∈ NH(p). This is a contradiction to W being a 1-movable 2-resolving hop dominating set of G[H]. Hence, (iii) holds. To prove (iv), let x and y be adjacent vertices of G with dG(x, z) = dG(y, z) for all z ∈ V (G)\{x, y}. Let p, w ∈ V (H), p ̸= w. Suppose (iii) does not hold. Then there exist a ∈ V (H)\(Tx\{p}) and w ∈ V (H)\Ty such that NH(a) ∩ (Tx\{p}) = Tx\{p} and NH(w)∩Ty = Ty for some adjacent vertices x and y of G and for some p ∈ Tx. Hence, both W\{(x, p)} and (W\{(x, p)}) ∪ {(y, w)} are not 2-resolving sets, a contradiction. Thus, (iv) holds. To prove (v), let x, y ∈ V (G) where dG(x, y) = 2 and dG(x, z) = dG(y, z) for all z ∈ V (G)\{x, y}. Let p, w ∈ V (H), p ̸= w. Suppose one of Tx\{p} and Ty, say Tx\{p} A.M. Mahistrado, H. Rara / Eur. J. Pure Appl. Math, 16 (3) (2023), 1464-1479 1477 is not a dominating set in H. Pick p ∈ V (H)\NH [Tx] and let w ∈ V (H)\Ty. Since dG[H]((x, a), (y, w)) = 2, for all (y, w), it follows that |NH(b) ∩ Ty| ≥ 2, that is, Ty is a 2-dominating set. Thus, (v) holds. Conversely, suppose that W satisfies properties (i) to (v). By Theorem 7, W is a 2-resolving hop dominating set of G[H]. Let x ∈ V (G) and p ∈ Tx. Then (x, p) ∈ W and W\{(x, p)} = ( ⋃ v∈S\{x} ({v} × Tv) ) ∪ [{x} × (Tx\{p})] and (W\{(x, p)}) ∪ {(x, q)} =  ⋃ z∈S\{x} ({z} × Tz)  ∪ [{x} × (Tx\{p} ∪ {q})] for some q ∈ (V (H)\Tx) ∩NH(p) and (W\{(x, p)}) ∪ {(y, w)} =  ⋃ a∈S\{(x,y)} ({a} × Ta)  ∪ [{x} × (Tx\{p})] ∪ [{y} × (Ty\{w})] for some y ∈ V (G) ∩NG(x) and w ∈ V (H)\Ty. By (i) to (v), for every (x, p) ∈ W either W\{(x, p)} is a 2-resolving hop dominating set in G[H] or there exists (y, q) ∈ NG[H]((x, p)) ∩ (V (G[H])\W ) such that (W\{(x, p)}) ∪ {(y, q)} is a 2-resolving hop dominating set in G[H]. Accordingly, W is a 1-movable 2-resolving hop dominating set in G[H]. Corollary 5. Let G and H be nontrivial connected graph with γ(G) ̸= 1 and G is free- equidistant. Then γ1m2Rh(G[H]) = |V (G)| ·mln2(H). Proof. Let S = V (G) and let Rx be an mln2-set of H for each x ∈ S. Since γ(G) ̸= 1, x ∈ NG(S, 2) for each x ∈ S. By Theorem 8, W = ⋃ x∈S [{x} ×Rx] is a 1-movable 2-resolving hop dominating set in G[H]. Thus, γ1m2Rh(G[H]) ≤ |W | = |V (G)||Rx| = |V (G)|mln2(H). If W0 = ⋃ x∈S({x} × T ) is a γ1m2Rh -set of G[H], then S0 = V (G) and Tx is a 1-movable 2-locating set of H for each x ∈ V (G) by Theorem 8. Hence, γ1m2Rh(G[H]) = |W0| = |V (G)||Tx| ≥ |V (G)|mln2(H). Therefore, γ1m2Rh(G[H]) = |V (G)| ·mln2(H). 8. Conclusion 1-movable 2-resolving hop domination, a variant of 2-resolving hop domination, has been introduced and studied for some graphs and graphs resulting from the join, corona REFERENCES 1478 and lexicographic product of two graphs. It is recommended that some bounds on the 1-movable 2-resolving hop domination be determined and that the parameter can be in- vestigated further for graphs under other binary operations. 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