EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 15, No. 4, 2022, 1982-1997 ISSN 1307-5543 – ejpam.com Published by New York Business Global On 2-Resolving Hop Dominating Sets in the Join, Corona and Lexicographic Product of Graphs Angelica Mae Mahistrado1,∗, Helen Rara2 1 Department of Mathematics and Statistics, College of Science and Mathematics, Mindanao State University-Iligan Institute of Technology, 9200 Iligan 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 S of vertices in G is a 2-resolving hop dominating set of G if S is a 2-resolving set in G and for every vertex x ∈ V (G)\S, there exists y ∈ S such that dG(x, y) = 2. The minimum cardinality of a set S is called the 2-resolving hop domination number of G and is denoted by γ2Rh(G). This study aims to combine the concept of hop domination with the 2-resolving sets of graphs. The main results generated in this study include the characterization of 2-resolving hop dominating sets in the join, corona and lexicographic product of two graphs, as well as their corresponding bounds or exact values. 2020 Mathematics Subject Classifications: 05C69 Key Words and Phrases: 2-resolving hop dominating set, 2-resolving hop domination number, join, corona, lexicographic product 1. Introduction The concept of domination in graphs is one of the most studied problems and one of the fastest growing areas in graph theory. This 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 [13]. 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 [15]. 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 [10]. The term metric dimension was used by Harary and Melter instead of locating number. ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v15i4.4585 Email addresses: angelicamae.mahistrado@g.msuiit.edu.ph (A.M. Mahistrado), helen.rara@g.msuiit.edu.ph (H. Rara) https://www.ejpam.com 1982 © 2022 EJPAM All rights reserved. A.M. Mahistrado, H. Rara / Eur. J. Pure Appl. Math, 15 (4) (2022), 1982-1997 1983 Moreover, in the paper of Brigham et al. [9], the concept of resolving dominating in graphs was studied wherein they defined it as a set that is both resolving and dominating, and determined the resolving domination number γR(G) of a graph G [9]. In 2021, the concept of resolving hop domination in graphs was studied and published by Mohamad and Rara [11] wherein they characterized the resolving hop dominating sets in some binary operations namely, the join and corona of two graphs and determined the bounds or exact values of the resolving hop domination number of these mentioned graphs [11]. Also, in the same year, Cabaro and Rara published the concept of 2-resolving sets in the join and corona of graphs [4]. Their paper also presents some characterizations involving this concept and investigates the 2-resolving dominating sets in the join, corona and lexicorgraphic product of two graphs [7]. Other variations of 2-resolving sets in graphs were also studied in [4–6, 8]. 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], [13], and [14], 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 a mini- mum 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.M. Mahistrado, H. Rara / Eur. J. Pure Appl. Math, 15 (4) (2022), 1982-1997 1984 A 2-resolving set S ⊆ V (G) which is hop dominating is called a 2-resolving hop domi- nating set or simply 2R-hop dominating set inG. The minimum cardinality of a 2-resolving hop dominating set in G, denoted by γ2Rh(G), is called the 2R-hop domination number of G. Any 2R-hop dominating set of cardinality γ2Rh(G) is then referred to as a γ2Rh-set in G. 3. Preliminary Results 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. 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. [12] 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 ofG. A dominating set D which is also a point-wise non-dominating set of G is called a dominating point-wise non-dominating set of G. The smallest cardinality of a dominating point-wise non-dominating set of G will be denoted by γpnd(G). Any dominating point-wise non-dominating set D of G with |D| = γpnd(G), is called a γpnd-set of G. Definition 3. 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. Example 1. For any graph G, (i) lnpnd 2 (Kn) = n; (ii) lnpnd 2 (K1,n) = n+ 1; (iii) lnpnd 2 (Km,n) = m+ n; (iv) lnpnd 2 (Pn) = { 3, n = 3 ⌈n+1 2 ⌉, n ≥ 4; A.M. Mahistrado, H. Rara / Eur. J. Pure Appl. Math, 15 (4) (2022), 1982-1997 1985 (v) lnpnd 2 (Cn) =  3, n = 3 4, n = 4 ⌈n2 ⌉, n ≥ 5. Remark 1. [7] Every 2-locating set in G is a 2-resolving set in G. However, a 2-resolving set in G need not be a 2-locating set in G. Thus, dim2(G) ≤ ln2(G). Example 2. For a path graph P6, dim2(P6) = 2 < 4 = ln2(P6). 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. 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. Example 3. (i) For all n ≥ 5, lnpnd (2,2)(Pn) =  5, n = 5 n 2 + 1, n is even ⌈n2 ⌉, n is odd. (ii) For all n ≥ 7, lnpnd (2,2)(Cn) = { n 2 , n is even ⌈n2 ⌉, n is odd. Example 4. (i) For all n ≥ 4, lnpnd (2,1)(Pn) = { n 2 + 1, n is even ⌈n2 ⌉, n is odd. (ii) For all n ≥ 5, lnpnd (2,1)(Cn) = { n 2 , n is even ⌈n2 ⌉, n is odd. Definition 6. A 2-resolving set S ⊆ V (G) which is point-wise non-dominating is called a 2-resolving point-wise non-dominating set in G. The minimum cardinality of a 2-resolving A.M. Mahistrado, H. Rara / Eur. J. Pure Appl. Math, 15 (4) (2022), 1982-1997 1986 point-wise non-dominating set in G, denoted by dim2pnd (G) is called the 2-resolving point- wise non-domination number of G. Any 2R-point-wise non-dominating set of cardinality dim2pnd (G) is then referred to as a dim2pnd (G)-set in G. Example 5. For any graph G, (i) dim2pnd (Kn) = n; (ii) dim2pnd (Km,n) = m+ n where m,n ≥ 1; (iii) dim2pnd (Pn) = { 3, n = 3 2, n ≥ 4. Remark 2. Every (2, 1)-locating point-wise non-dominating set in G is a 2-resolving point-wise non-dominating set in G. However, a 2-resolving point-wise non-dominating set in G need not be a (2, 1)-locating point-wise non-dominating set in G. Thus, dim2pnd (G) ≤ lnpnd (2,1)(G). Example 6. For a path graph P4, dim2pnd (P4) = 2 < 3 = lnpnd (2,1)(P4). Remark 3. Every nontrivial connected graph G admits a 2-resolving hop dominating set. Indeed, the vertex set V (G) of G is a 2-resolving hop dominating set. Remark 4. [4] Let S ⊆ G. Since any pair of vertices u and v where u and v are in the 2-resolving set S of a graph G, rG(u/S) and rG(v/S) differ in at least 2 positions. Hence, to show that S is a 2-resolving set of G, we only need to show that every pair u, v of distinct vertices in V (G)\S or u ∈ S and v ∈ V (G)\S, rG(u/S) and rG(v/S) differ in at least 2 positions. Theorem 1. Let G be any nontrivial connected graph. Then S ⊆ V (G) is a 2-resolving hop dominating set in G if it satisfies the following conditions: (i) For every pair of vertices x, y ∈ V (G) where x ∈ S and y ∈ V (G)\S or both x, y ∈ V (G)\S, rG(x/S) and rG(y/S) differ in at least 2 positions; (ii) S is a point-wise non-dominating set in G. Proof. Suppose S ⊆ V (G) is a 2-resolving hop dominating set in G. Then by Remark 4, (i) holds. To prove (ii), let v ∈ V (G)\S. Since S is a hop dominating set, there exists z ∈ S such that dG(v, z) = 2. Hence, v /∈ NG(z). This shows that S is a point-wise non-dominating set of G. Thus, (ii) holds. A.M. Mahistrado, H. Rara / Eur. J. Pure Appl. Math, 15 (4) (2022), 1982-1997 1987 Remark 5. Every 2-resolving hop dominating set in G is a 2-resolving point-wise non- dominating set in G. However, a 2-resolving point-wise non-dominating set in G need not be a 2-resolving hop dominating set in G. Thus, dim2pnd (G) ≤ γ2Rh(G). Remark 6. Let G be a connected graph. Then every 2-resolving hop dominating set in G is a 2-resolving set in G. However, a 2-resolving set in G need not be a 2-resolving hop dominating set in G. Thus, dim2(G) ≤ γ2Rh(G). Proposition 1. For a path Pn on n vertices γ2Rh(Pn) =  2, if n = 2, 4; 3, if n = 3, 5, 6; 2s, if n = 6s, s ≥ 2; 2s+ 1, if n = 6s+ 1, s ≥ 1; 2s+ 2, if n = 6s+ x; 2 ≤ x ≤ 5, s ≥ 1. Proposition 2. For a cycle Cn on n vertices γ2Rh(Cn) =  3, if n = 3, 5, 6; 2s if n = 6s, s ≥ 2; 2s+ 1, if n = 6s+ 1, s ≥ 1; 2s+ 2, if n = 6s+ x; 2 ≤ x ≤ 5, s ≥ 1. Proposition 3. Let G be a connected graph of order 4. Then γ2Rh(P4) = 2 and γ2Rh(C4) = 4, respectively. Theorem 2. For a complete graph Kn on n vertices, γ2Rh(Kn) = n. Proof. Suppose that γ2Rh(Kn) < n. Let S be the minimum 2-resolving hop dominating set of Kn. Let x ∈ V (Kn)\S and y ∈ S. Then x and y differ at exactly one position, that is a contradiction to the assumption that S is a 2-resolving hop dominating set of Kn. Therefore, γ2Rh(Kn) = n. Theorem 3. A set S ⊆ V (Km,n) is a 2-resolving hop dominating set in Km,n if and only if S = V (Km,n). Proof. Let S be a subset of V (Km,n). Let U and V be partite sets with |U | = m and |V | = n; m,n ≥ 1. Let U = {u1, u2, . . . um} and V = {v1, v2, . . . vn}. Suppose there exists uk ∈ U\S. Then ∀ i ∈ {1, . . . ,m}\{k}, rKm,n(ui/S) and rKm,n(uk/S) differ in at most one position. Similarly, suppose there exists vk ∈ V \S ∀ j ∈ {1, . . . , n}\{k}, rKm,n(vj/S) and rKm,n(vk/S) differ in at most one position. Thus, it follows that S = V (Km,n). The converse is clear. Corollary 1. A set S ⊆ V (K1,n) is a 2-resolving hop dominating set in K1,n if and only if S = V (K1,n). Corollary 2. For a complete bipartite graph Km,n, γ2Rh(Km,n) = m+ n. A.M. Mahistrado, H. Rara / Eur. J. Pure Appl. Math, 15 (4) (2022), 1982-1997 1988 4. 2-Resolving Hop Dominating Sets in the Join of Graphs Definition 7. [2] The join G + H of two graphs G and 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)} . Note that the star K1,n can be expressed as the join of the trivial graph K1 and the empty graph Kn of order n, that is, K1,n = K1 + Kn. The graphs Fn = K1 + Pn and Wn = K1 + Cn of orders n+ 1 are called fan and wheel, respectively. Theorem 4. [4] Let G be a connected graph of order greater than 3 and let K1 = {v}. Then S ⊆ V (K1 +G) is a 2-resolving set in K1 +G if and only if either v /∈ S and S is a (2, 2)-locating set in G or S = {v} ∪ T where T is a (2, 1)-locating set in G. Theorem 5. Let G be a connected graph and let K1 = {x}. Then S ⊆ V (K1 + G) is a 2-resolving hop dominating set in K1 + G if and only if S = {x} ∪ T where T is a (2, 1)-locating point-wise non-dominating set in G. Proof. Let S ⊆ V (K1 +G) be a 2-resolving hop dominating set in K1 +G. Since S is a hop dominating set, x ∈ S. Hence, S = {x} ∪ T for T ⊆ V (G). By Theorem 4, T is a (2, 1)-locating set in G. Now, since S is a hop dominating set ∀ v ∈ V (G)\T, there exists z ∈ T such that dK1+G(v, z) = 2. This follows that v /∈ NG(z). Thus, T is a point-wise non-dominating set in G. Therefore, T is a (2, 1)-locating point-wise non-dominating set in G. Conversely, suppose S = {x}∪T where T is a (2, 1)-locating point-wise non-dominating set in G. Then by Theorem 4, S is a 2-resolving set in K1+G. Since T is a point-wise non- dominating set in G, ∀ y ∈ V (G)\T , there exists v ∈ T such that vy /∈ E(G). This follows that dK1+G(v, y) = 2. Thus, S is a hop dominating set. Therefore, S is a 2-resolving hop dominating set in K1 +G. Corollary 3. Let G be connected nontrivial graph. Then γ2Rh(K1 +G) = lnpnd (2,1)(G) + 1. Proof. Suppose S is a γ2Rh-set in K1 +G. Then by Theorem 5, S = {x} ∪ T where T is a (2, 1)- locating point-wise non-dominating set in G. Thus, γ2Rh(K1 +G) = |S| = |T |+ 1 ≥ lnpnd (2,1)(G) + 1. On the other hand, let T ′ be a lnpnd (2,1)-set of G. Then by Theorem 5, S = {x} ∪ T ′ is a 2-resolving hop dominating set of K1 +G. Thus, γ2Rh(K1 +G) ≤ |S| = |T ′ |+ 1 = lnpnd (2,1)(G) + 1. Therefore, γ2Rh(K1 +G) = lnpnd (2,1)(G) + 1. A.M. Mahistrado, H. Rara / Eur. J. Pure Appl. Math, 15 (4) (2022), 1982-1997 1989 Example 7. For a fan Fn = Pn +K1 on n+ 1 (n ≥ 3) vertices γ2Rh(Fn) = lnpnd (2,1)(Pn) + 1 =  4, if n = 3 n 2 + 2, if n is even ⌈n2 ⌉+ 1, if n is odd. Example 8. For a wheel Wn = Cn + 1 on n+ 1 (n ≥ 3) vertices γ2Rh(Wn) = lnpnd (2,1)(Cn) + 1 =  4, if n = 3 5, if n = 4 n 2 + 1, if n is even ⌈n2 ⌉+ 1, if n is odd. Theorem 6. [4] Let G and H be nontrivial connected graphs. A proper subset S of V (G+H) is a 2-resolving set in G+H if and only if SG = V (G)∩S and SH = V (H)∩S are 2-locating sets in G and H , respectively, where SG or SH is a (2, 2)-locating set or SG and SH are (2, 1)-locating sets. Theorem 7. [12] Let G and H be any two graphs. A set S ⊆ V (G + H) is a hop dominating set of G + H if and only if S = SG ∪ SH , where SG and SH are point-wise non-dominating sets of G and H, respectively. Theorem 8. Let G and H be any two graphs. A set S ⊆ V (G+H) is a 2-resolving hop dominating set in 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 in G and H, respectively, where SG or SH is a (2, 2)-locating point-wise non-dominating set or SG and SH (2, 1)- locating point-wise non-dominating sets. Proof. Suppose that S is a 2-resolving hop dominating set of G + H. Then S is a 2-resolving set of G + H. By Theorem 6, S = SG ∪ SH where SG = V (G) ∩ S and SH = V (H) ∩ S are 2-locating sets in G and H, respectively where SG or SH is a (2, 2)- locating set or SG and SH are (2, 1)-locating sets. Also, since S is a hop dominating set in G+H, it follows by Theorem 7 that SG and SH are point-wise non-dominating sets of G and H, respectively. Therefore, 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 of G and H. Conversely, suppose 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 of G and H, respectively. Then by Theorem 6, S is a 2-resolving set in G + H. Similarly, by Theorem 7 it follows that S is a hop dominating set of G+H. Therefore, S is a 2-resolving hop dominating set in G+H. Corollary 4. Let G and H be connected nontrivial graphs. Then, γ2Rh(G+H) = min{lnpnd (2,2)(G) + lnpnd (2) (H), lnpnd (2) (G)lnpnd (2,2)(H), lnpnd (2,1)(G) + lnpnd (2,1)(H)}. A.M. Mahistrado, H. Rara / Eur. J. Pure Appl. Math, 15 (4) (2022), 1982-1997 1990 Proof. Let S be a minimum 2-resolving hop dominating set inG+H. Since S = SG∪SH where SG = V (G)∩S and SH = V (H)∩S. By Theorem 8, SG and SH are 2-locating point- wise non-dominating sets in 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. If SG is (2, 2)-locating point-wise non-dominating set in G, then lnpnd (2,2)(G) + lnpnd 2 (H) ≤ |SG|+ |SH | = |S| = γ2Rh(G+H). If SH is a (2, 2)-locating point-wise non-dominating set in H, then lnpnd (2,2)(H) + lnpnd 2 (G) ≤ |SH |+ |SG| = |S| = γ2Rh(G+H). If SG and SH are both (2, 1)-locating point-wise non-dominating sets, then lnpnd (2,1)(G) + lnpnd (2,1)(H) ≤ |SG|+ |SH | = |S| = γ2Rh(G+H). Thus, γ2Rh(G+H) ≥ min{lnpnd (2,2)(G) + lnpnd (2) (H), lnpnd (2) (G)lnpnd (2,2)(H), lnpnd (2,1)(G) + lnpnd (2,1)(H)}. Next, suppose that lnpnd (2,1)(G) + lnpnd (2,1)(H) ≤ lnpnd (2,2)(G) + lnpnd 2 (H) and lnpnd 2,1 (G) + lnpnd (2,1)(H) ≤ lnpnd 2 (G) + lnpnd (2,2)(H). Let SG be a minimum (2, 1)-locating point-wise non-dominating set in G and SH be a minimum (2, 1)-locating point-wise non- dominating set in H. Then S = SG∪SH is a 2-resolving hop dominating set in G+H, by Theorem 8. Hence, γ2Rh(G+H) ≤ |S| = |SG|+ |SH | = lnpnd (2,1)(G)+ lnpnd (2,1)(H). Therefore, γ2Rh(G+H) ≤ lnpnd (2,1)(G) + lnpnd (2,1)(H). Similarly, if lnpnd (2,2)(G) + lnpnd 2 (H) ≤ lnpnd (2,1)(G) + lnpnd (2,1)(H) and lnpnd (2,2)(G) + lnpnd 2 (H) ≤ lnpnd 2 (G) + lnpnd (2,2)(H), then γ2Rh(G+H) ≤ lnpnd (2,2)(G) + lnpnd 2 (H). Also, if lnpnd 2 (G) + lnpnd (2,2)(H) ≤ ln(2,2)( pndG) + lnpnd 2 (H) and lnpnd 2 (G) + lnpnd (2,2)(H) ≤ lnpnd (2,1)(G) + lnpnd (2,1)(H), then γ2Rh(G+H) ≤ lnpnd 2 (G) + lnpnd (2,2)(H). Thus, γ2Rh(G+H) ≤ min{lnpnd (2,2)(G) + lnpnd (2) (H), lnpnd (2) (G)lnpnd (2,2)(H), lnpnd (2,1)(G) + lnpnd (2,1)(H)}. A.M. Mahistrado, H. Rara / Eur. J. Pure Appl. Math, 15 (4) (2022), 1982-1997 1991 Therefore, γ2Rh(G+H) = min { lnpnd (2,2)(G) + lnpnd 2 (H), lnpnd 2 (G) + lnpnd (2,2)(H), lnpnd (2,1)(G) + lnpnd (2,1)(H) } . Example 9. For paths Pn and Pm on n and m vertices (n,m ≥ 4), we have γ2Rh(Pn + Pm) =  ( n 2 + 1 ) + ( m 2 + 1 ) , if n,m are even( n 2 + 1 ) + ⌈m2 ⌉, if n is even,m is odd ⌈n2 ⌉+ ( m 2 + 1 ) , if n is odd,m is even ⌈n2 ⌉+ ⌈m2 ⌉, if n,m are odd. In particular, for n,m = 2, 3, γ2Rh(Pn + Pm) = n+m. 4.1. 2-Resolving Hop Dominating Sets in the Corona of Graphs Let the following notations be described as follows: For k ∈ V (G ◦H), v ∈ V (G) and Sv ⊂ V (Hv). V k Sv is a vector whose components are the distances of k from the elements of Sv. Definition 8. [2] The corona G ◦ H of two graphs G and H is the graph obtained by taking one copy of G of order n and n copies of H, and then joining the ith vertex of G to every vertex in the ith copy of H. For every 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). Remark 7. [4] Let v ∈ V (G). For every x, y ∈ V (Hv), dG◦H(x,w) = dG◦H(y, w) and dG◦H(v, w) + 1 = dG◦H(x,w) for every w ∈ V (G ◦H)\V (Hv). Theorem 9. [12] Let G and H be any two graphs. A set C ⊆ V (G ◦ H) is a hop dominating set of G ◦H if and only if C = A ∪  ⋃ v∈V (G)∩NG(A) Sv  ∪  ⋃ w∈V (G)\NG(A) Ew  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) ∩ C ̸= ∅, (ii) Sv ⊆ V (Hv) for each v ∈ V (G) ∩NG(A), and (iii) Ew ⊆ V (Hw) is a point-wise non-dominating set of Hw for each w ∈ V (G)\NG(A). A.M. Mahistrado, H. Rara / Eur. J. Pure Appl. Math, 15 (4) (2022), 1982-1997 1992 Theorem 10. 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). Proof. Suppose that S ⊆ V (G ◦H) is a 2-resolving hop dominating set of G ◦H. Let A = V (G) ∩ S, Sv = S ∩ V (Hv) for all v ∈ V (G) ∩NG(A) and Dw = S ∩ V (Hw) for all w ∈ V (G)\NG(A) . Then S = A ∪  ⋃ v∈V (G)∩NG(A) Sv  ∪  ⋃ w∈V (G)\NG(A) Dw  where A ⊆ V (G), Sv ⊆ V (Hv) and Dw ⊆ V (Hw). Now, since S is 2-resolving hop dominating set of G ◦ H. By Theorem 9, (i) holds. Next, suppose that Sv = ∅ for some v ∈ V (G) ∩ NG(A). Let x, y ∈ V (Hv). Then rG◦H(x/S) = rG◦H(y/S) which is a contradiction to the assumption of S. Thus, Sv ̸= ∅. Similarly, Dw ̸= ∅ for some w ∈ V (G)\NG(A). Next, claim that Sv is a 2-locating set inHv for each v ∈ V (G)∩NG(A). Let p, q ∈ V (Hv)\Sv where p ̸= q. Since S is a 2-resolving set in G ◦H, rG◦H(p/S) and rG◦H(q/S) differ in at least 2 positions and by Remark 7, rHv(p/Sv) and rHv(q/Sv) must differ in at least 2 positions. Thus, it follows that Sv is a 2-locating set of Hv and so (ii) holds. Similarly, Dw is a 2-locating set of Hw. Now, since S is a hop dominating set of G ◦H, Theorem 9 follows, showing Dw is a point-wise non-dominating set of Hw that is (iii) holds. Conversely, let S = A ∪ ( ⋃ v∈V (G)∩NG(A) Sv ) ∪ ( ⋃ w∈V (G)\NG(A) Dw ) where A ⊆ V (G), Sv ⊆ V (Hv) and Dw ⊆ V (Hw) satisfying the given conditions. Let x ∈ V (G ◦H)\S and let v ∈ V (G) such that x ∈ V (v+Hv). Suppose x = v. Then x /∈ A. From the assumption that (i) holds, it follows that there exists y ∈ S such that dG◦H(x, y) = 2. Next, suppose x ̸= v. If v ∈ NG(A), then there exists a vertex say z ∈ A ∩ NG(v) such that z ∈ S and dG◦H(x, z) = 2. Suppose v /∈ NG(A). Then x ∈ V (Hv)\Dv. Since Dv is point-wise non-dominating by Theorem 9, it follows that there exists y ∈ Dv, that is y ∈ S such that dG◦H(x, y) = 2. This shows that S is a hop dominating set of G ◦H. Since Sv or Dv is a A.M. Mahistrado, H. Rara / Eur. J. Pure Appl. Math, 15 (4) (2022), 1982-1997 1993 2-locating set of G ◦H, by our assumption. By Remark 1, it then follows that Sv or Dv is a 2-resolving set of G ◦H. Accordingly, S is a 2-resolving hop dominating set of G ◦H. Corollary 5. Let G be a nontrivial graph of order n and H be any graph. Then the following statements hold. (i) γ2Rh(G ◦H) ≤ n(1 + ln2(H)). (ii) If lnpnd 2 (H) = ln2(H), then γ2Rh(G ◦H) = n(lnpnd 2 (H)). Proof. (i) Let A = V (G) and let Sv be a 2-locating set of Hv for all v ∈ V (G). Then S = A∪ ( ⋃ v∈V (G) Sv ) is a 2-resolving hop dominating set of G ◦H by Theorem 10. Hence, γ2Rh(G ◦H) ≤ |S| = |V (G)|+ |V (G)||Sv| = n(1 + ln2(H)). (ii) Suppose that lnpnd 2 (H) = ln2(H). Now, set A = ∅ and let Dw be a 2-locating point-wise non-dominating set of Hw for all w ∈ V (G). Then S = A ∪ ( ⋃ w∈V (G) Dw ) is a 2-resolving hop dominating set by Theorem 10. Hence, γ2Rh(G ◦H) ≤ |S| = |A|+ |V (G)||Dw| = n(lnpnd 2 (H)). Next,let S0 = A0 ∪ ( ⋃ v∈V (G)\C0 Sv ) ∪ ( ⋃ w∈C0 Dw ) be a 2-resolving hop dominating set of G ◦ H. By Theorem 10, A0 ⊆ V (G) where C0 = {x ∈ V (G) : x /∈ NG(A0)}, Sv is a 2- locating set of Hv for all v ∈ V (G)\C0 and Dw is a 2-locating point-wise non-dominating set of Hw for all w ∈ C0. Thus, we can write this as γ2Rh(G ◦H) = |S| (1) = |A|+ |V (G)\C0||Sv|+ |C0||Dw| ≥ |V (G)\C0|ln2(H) + |C0|lnpnd2 (H) = (|V (G)| − |C0|)ln2(H) + |C0|lnpnd2 (H) = (|V (G)| − |C0|)lnpnd2 (H) + |C0|lnpnd2 (H) = n · lnpnd2 (H) Therefore, γ2Rh(G ◦H) = n(lnpnd 2 (H)). Example 10. (i) For n = 4 and m = 3, γ2Rh(P4 ◦ P3) = 10 < 12 = 4(1 + 2). A.M. Mahistrado, H. Rara / Eur. J. Pure Appl. Math, 15 (4) (2022), 1982-1997 1994 (ii) For any integer n ≥ 2 and m ≥ 4, γ2Rh(G ◦ Pm) = n · lnpnd 2 (Pm) = n (⌈m+ 1 2 ⌉) . Example 11. For any integer n ≥ 2 and m ≥ 5, γ2Rh(G ◦ Cm) = n · lnpnd 2 (Cm) = n (⌈m 2 ⌉) . 4.2. 2-Resolving Hop Dominating Sets in the Lexicographic Product of Graphs Definition 9. [2] The lexicographic product of graphs G and H, denoted by G[H], is the graph with vertex set V (G[H]) = V (G) × V (H) such that (v, a)(u, b) ∈ E(G[H]) if and only if either uv ∈ E(G) or u = v and ab ∈ E(H). Note that every non-empty subset C of V (G)×V (H) can be expressed as C = ⋃ x∈S [{x}×Tx], where S ⊆ V (G) and Tx ⊆ V (H). Definition 10. [7] A vertex x is said to be 1-equidistant to y if xy ∈ E(G) and dG(x, z) = dG(y, z), for all z ∈ V (G)\{x, y} and it is 2-equidistant to y if dG(x, y) = 2 and dG(x, z) = dG(y, z), for all z ∈ V (G)\{x, y}. A vertex is called a free-vertex in G if it is neither 1-equidistant nor 2-equidistant to any vertex. The set containing all 1- equidistant, 2-equidistant, and free-vertices in G are denoted by EQ1(G), EQ2(G) and fr(G), respectively. Definition 11. [7] A graph is called free-equidistant if all of its vertices are free-vertices. Remark 8. [16] Let G and H be two nontrivial graphs such that G is connected. Then the following assertions hold for any a, c ∈ V (G) and b, d ∈ V (H) such that a ̸= c. (i) NG[H](a, b) = ({a} ×NH{b}) ∪ {NG{a} × V (H)} (ii) dG[H]((a, b), (c, d)) = dG(a, c) (iii) dG[H]((a, b), (a, d)) = min{dH(b, d), 2}. Theorem 11. [7] 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); and (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). A.M. Mahistrado, H. Rara / Eur. J. Pure Appl. Math, 15 (4) (2022), 1982-1997 1995 Theorem 12. [12] Let G and H be nontrivial connected graphs. A subset C = ⋃ x∈S [{x}×Tx] of V (G[H]) is a hop dominating set of G[H] if and only if the following conditions hold: (i) S is a hop dominating set of G; (ii) Tx is a point-wise non-dominating set of H for each x ∈ S with |NG(x, 2) ∩ S| = 0. Theorem 13. 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 is a hop dominating set of G, that is 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); and (v) Tx is a 2-locating point-wise non-dominating set inH for every x ∈ S with |NG(x, 2)∩ S| = 0. Proof. SupposeW = ⋃ x∈S [{x}×Tx] is a 2-resolving hop dominating set of G[H]. Then W is a 2-resolving set. By Theorem 11, (i) to (iv) hold. Since W is a hop dominating set then by Theorem 12, it follows that (i) and (v) hold. Now, let x ∈ V (G) and p, q ∈ S\NG(S, 2) where p ̸= q. Then (x, p) ̸= (x, q). If p, q /∈ Tx or [p ∈ Tx and q /∈ Tx], then (x, p), (x, q) /∈ W or [(x, p) ∈ W and (x, q) /∈ W ]. Since W is a 2-resolving hop dominating set in G[H], rG[H]((x, p)/W ) and rG[H]((x, q)/W ) differ in at least 2 positions. Hence, by Remark 8 and Definition 1, Tx is a 2-locating set in H. Thus, (v) holds. Conversely, suppose (i) to (iv) hold. By Theorem 11, W is a 2-resolving set. Also, since (i) and (v) hold. By Theorem 12, W is a hop dominating set. Accordingly, W is 2-resolving hop dominating set of G[H]. The following corollaries are the direct consequences of Theorem 13. Corollary 6. Let G and H be nontrivial connected graphs. Then, γ2Rh(G[H]) ≤ n · ln(2,1)(H) +m · γ2L(H) + p · lnpnd 2 (H), where n + m + p = |V (G)| with |EQ1(G)| = n, |EQ2(G)| = m and |fr(G)| = p. In particular, if n = m = 0, then γ2Rh(G[H]) = |V (G)| · lnpnd 2 (H). Corollary 7 follows from Theorem 13. Corollary 7. Let G and H be nontrivial connected graphs such that G is free-equidistant. Then γ2Rh(G[H]) = |V (G)| · lnpnd 2 (H). REFERENCES 1996 Proof. Since G is free-equidistant, then |EQ1(G)| = |EQ2(G)| = 0. 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