EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 16, No. 2, 2023, 1180-1195 ISSN 1307-5543 – ejpam.com Published by New York Business Global Outer-Connected 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 ⊆ V (G) is an outer-connected 2-resolving hop dominating set of G if S is a 2-resolving hop dominating set of G and S = V (G) or the subgraph ⟨V (G)\S⟩ induced by V (G)\S is connected. The outer-connected 2-resolving hop domination number of G, denoted by γ̃c2Rh(G) is the smallest cardinality of an outer-connected 2-resolving hop dominating set of G. This study aims to combine the concept of outer-connected hop domination with the 2-resolving hop dominating sets of graphs. The main results generated in this study include the characterization of outer-connected 2-resolving hop dominating sets in the join, corona, edge corona and lexicographic product of graphs, as well as their corresponding bounds or exact values. 2020 Mathematics Subject Classifications: 05C69 Key Words and Phrases: Outer-connected 2-resolving hop dominating set, outer-connected 2-resolving hop domination number, join, corona, edge 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 2007, outer-connected domination, a variation of domination, was first introduced by Cyman [10]. In 2015, Natarajan and Ayyaswamy introduced and studied the concept of hop domination [16]. In 2022, Canoy and Saromines studied and published the outer-connect hop dominating sets in graphs [9]. On the other hand, in 1975 the term locating set, the concept of resolving sets for a connected graph was first introduced by Slater [19]. 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 [11]. The term ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v16i2.4771 Email addresses: angelicamae.mahistrado@g.msuiit.edu.ph (A.M. Mahistrado), helen.rara@g.msuiit.edu.ph (H. Rara) https://www.ejpam.com 1180 © 2023 EJPAM All rights reserved. A.M. Mahistrado, H. Rara / Eur. J. Pure Appl. Math, 16 (2) (2023), 1180-1195 1181 metric dimension was used by Harary and Melter instead of locating number. Recently, 2-resolving hop dominating sets in graphs was studied in [12]. Moreover, other variations of resolving sets and hop dominating sets in graphs were also studied in [4–6, 8, 13–15], respectively. Motivated by the 2-resolving hop domination concept and the introduction of the outer- connected hop domination concept by S.R. Canoy and C.J. Saromines [9], here authors introduced and studied the concept of outer-connected 2-resolving hop domination in graphs. 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], [16] and [18]. 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 ⊆ V (G) is an outer-connected 2-resolving hop dominating set of G if S is a 2-resolving hop dominating set of G and S = V (G) or the subgraph ⟨V (G)\S⟩ induced by V (G)\S is connected. The outer-connected 2-resolving hop domination number of G, denoted by γ̃c2Rh(G) is the smallest cardinality of a outer-connected 2-resolving hop A.M. Mahistrado, H. Rara / Eur. J. Pure Appl. Math, 16 (2) (2023), 1180-1195 1182 dominating 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. 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. [17] 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 pointwise 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. [12] 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 -set in G. Definition 4. A set S ⊆ V (G) is an outer-connected 2-locating point-wise non-dominating set in G if S is a 2-locating point-wise non-dominating set in G and S = V (G) or the subgraph ⟨V (G)\S⟩ induced by V (G)\S is connected. The outer-connected 2-locating point-wise non-dominating number of G, denoted by l̃npnd 2 (G), is the smallest cardinality of an outer-connected 2-locating point-wise non-dominating set in G. An outer-connected 2-locating point-wise non-dominating set of cardinality l̃npnd 2 (G) is then referred to as an l̃npnd 2 -set in G. Definition 5. [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. A.M. Mahistrado, H. Rara / Eur. J. Pure Appl. Math, 16 (2) (2023), 1180-1195 1183 Definition 6. [12] 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 7. A set S ⊆ V (G) is an outer-connected (2, 2)-locating point-wise non- dominating ((2, 1)-locating point-wise non-dominating, respectively) set in G if S is a (2, 2)-locating point-wise non-dominating ((2, 1)-locating point-wise non-dominating, re- spectively) set in G and S = V (G) or the subgraph ⟨V (G)\S⟩ induced by V (G)\S is connected. The outer-connected (2, 2)-locating point-wise non-domination ((2, 1)-locating point-wise non-domination, respectively) number of G, denoted by l̃npnd (2,2)(G) (l̃npnd (2,1)(G), respectively), is the smallest cardinality of an outer-connected (2, 2)-locating point-wise non-dominating ((2, 1)-locating point-wise non-dominating, respectively) set in G. An outer-connected (2, 2)-locating point-wise non-dominating ((2, 1)-locating point-wise non- dominating, respectively) set of cardinality l̃npnd (2,2)(G) (l̃npnd (2,1)(G), respectively) is then re- ferred to as an l̃npnd (2,2)-set (l̃n pnd (2,1)- set) in G. 3. Preliminary Results Every nontrivial connected graph G admits an outer-connected 2-resolving hop dominating set. Indeed, the vertex set V (G) of G is an outer-connected 2-resolving hop dominating set. Remark 1. For any connected graph G of order n ≥ 2, 2 ≤ γ̃c2Rh(G) ≤ n. Moreover, γ̃c2Rh(P2) = 2 and γ̃c2Rh(Kn) = n. Proposition 1. (i) For a path Pn on n vertices γ̃c2Rh(Pn) =  n, if n = 2, 3; n− 2, if n = 4, 5, 6; n− 3, if n = 7; n− 4, if n ≥ 8. A.M. Mahistrado, H. Rara / Eur. J. Pure Appl. Math, 16 (2) (2023), 1180-1195 1184 (ii) For a cycle Cn on n vertices γ̃c2Rh(Cn) =  n, if n = 3, 4; n− 2, if n = 5; n− 3, if n = 6; n− 4, if n ≥ 7. Now, consider the following results of outer-connected 2-locating point-wise non-dominating sets which are used in characterizing the outer-connected 2-resolving hop dominating sets in the join of two graphs. Proposition 2. Let G be any nontrivial connected graph. Then for any positive integers n, we have (i) l̃npnd 2 (Pn) =  n, if n = 2, 3; n− 1, if 4 ≤ n ≤ 7; n− 2, if n ≥ 8. (ii) l̃npnd 2 (Cn) = { n, if n = 3, 4; n− 2, if n ≥ 5. (iii) For all n ≥ 5, l̃npnd (2,2)(Pn) = { n− 1, if 5 ≤ n ≤ 7; n− 2, if n ≥ 8; For all n ≥ 6, l̃npnd (2,2)(Cn) = n− 2. (iv) For all n ≥ 4, l̃npnd (2,1)(Pn) = { n− 1, if 4 ≤ n ≤ 7; n− 2, if n ≥ 8; For all n ≥ 4, l̃npnd (2,1)(Cn) = { n, if n = 4; n− 2, if n ≥ 5. Proof. (i) Let Pn = [v1, v2, v3, . . . , vn]. Clearly, l̃n pnd 2 (Pn) = n for n = 2, 3. Let n ≥ 4 and let S be an l̃npnd 2 -set in Pn. Since ⟨V (Pn) \ S⟩ is connected and S is a 2-locating point-wise non-dominating set, 1 ≤ |V (Pn)\S| ≤ 2. Clearly, at least one of v1 and vn is in S. Suppose that v1 ∈ S. Suppose further that |V (Pn) \ S| = 1. Then 4 ≤ n ≤ 7. Hence, l̃npnd 2 (Pn) = n− 1 for 4 ≤ n ≤ 7. Next, suppose that |V (Pn) \ S| = 2. If p is the smallest integer such that vp /∈ S, then p /∈ {1, 2, 3}. It follows that v1, v2, v3 ∈ S. In this case, for n ≥ 8, the set S ′ = V (Pn) \ {v4, v5} is clearly an outer-connected 2-locating point-wise non-dominating set. Thus, l̃npnd 2 (Pn) = n− 2 for all n ≥ 8. (ii) Let Cn = [v1, v2, v3, . . . , vn]. Clearly, l̃n pnd 2 (Cn) = n for n = 3, 4. Let n ≥ 5 and let S be an l̃npnd 2 -set of Cn. Since ⟨V (Cn) \ S⟩ is connected and S is a 2-locating point-wise A.M. Mahistrado, H. Rara / Eur. J. Pure Appl. Math, 16 (2) (2023), 1180-1195 1185 non-dominating set, |V (Cn) \ S| = 2. Therefore, l̃npnd 2 (Cn) = n− 2 for all n ≥ 5. The proofs of (iii) and (iv) are similar to (i) and (ii). Next, we show that every pair of positive integers are realizable as 2-resolving hop domination number and outer-connected 2-resolving hop domination number. Remark 2. Every outer-connected 2-resolving hop dominating set of G is a 2-resolving hop dominating set of G. Thus, γ2Rh(G) ≤ γ̃c2Rh(G). Theorem 1. Let a and b be positive integers such that 2 ≤ a ≤ b. Then there exists a nontrivial connected graph G such that γ2Rh(G) = a and γ̃c2Rh(G) = b. Proof. Suppose 2 ≤ a = b. Consider Figure 1. Then S = {u1, u2, u3, u4, . . . un} is both a γ2Rh-set and γ̃c2Rh-set of G1. Hence, 2 ≤ γ2Rh(G1) = γ̃c2Rh(G1) = a = b. ................................................................................................................ .................................................................................................................................................... .................................................................................................................................................... .................................... ............................................................................................................... .................................... .................................... ...................................................................................................................................................................................... .................................... .................................... ............ ........... ........... ........... ........... ........... ........... ........... ........... ........... .................................... .................................... .......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... . .................................... .................................... u1 u2 u3 u4 u5 un • • • • • • ... G1 : Figure 1 Suppose 2 < a < b. Consider the graph G2 in Figure 2. Then S = {x1, x2, . . . , xa} is a γ2Rh-set of G2 and X = S ∪ {y1, y2, . . . , yb−a} is a γ̃c2Rh-set of G2. Hence γ2Rh(G2) = a and γ̃c2Rh(G2) = |X| = |S|+ (b− a) = a+ b− a = b. ........... .......... .......... .......... .......... .......... . .................................... .................................... ........... .......... .......... .......... .......... .......... . .................................... .................................... .............................................................. .................................... .................................... .............................................................. .................................... .................................... ......... ........ ........ ........ ........ ...... .................................... .................................... ......... ........ ........ ........ ........ ...... .................................... .................................... .......................... ......................... ......................... .... .................................... .................................... .................... ................... .............. .................................... .................................... ...................................................................................... .................................... .................................... ................................................... .................................... .................................... ............................................................................................................................ ........................................................................................................................................................................................................................................ .................................... .............................................................. .................................... .................................... ..................... .................... .................... . ....... ............................. .................................... ................................................... .................................... ....................................................................................... .................................... ..................................................................................................... .................................... .................................... ............ ........... ........... ........... ...... . ................................... .................................... ............ ........... ........... ........... ...... . ................................... .................................... ............ ........... ........... ........... ........... ......... .................................... .................................... ......... ........ ........ ........ .................................... .................................... ......... ........ ........ ........ .................................... .................................... ......... ........ ........ ........ ........ ...... .................................... .................................... ................................................................................ .................................... .............................................................................. .................................... ...................................................................................................................... .................................... .................................... .......................... ......................... ......................... .... .................................... .................................... ............... .............. ............. .................................... .................................... .............. ............. ............. ............. ............. ............. ... ... ................................. .................................... x1 x2 x3 x5 x4 y1 y2 y3 y4 yb−ax6 x7 x8 x9 xa xa−1 • ••• • • . . . • ••• • • •• • • • • . . . . . . Figure 2 G2 : Corollary 1. For each positive integer n, there exists a connected graph G such that γ̃c2Rh(G)− γ2Rh(G) = n, that is, γ̃c2Rh − γ2Rh can be made arbitrarily large. We now characterize the outer-connected 2-resolving hop dominating sets in some graphs under some binary operations. A.M. Mahistrado, H. Rara / Eur. J. Pure Appl. Math, 16 (2) (2023), 1180-1195 1186 4. Join of Graphs This section presents characterizations in the outer-connected 2-resolving hop domi- nating sets in the join of graphs. Theorem 2. [12] 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. Theorem 3. Let G be a connected graph and let K1 = {x}. Then S ⊆ V (K1 + G) is an outer-connected 2-resolving hop dominating set in K1 +G if and only if S = {x} ∪ T where T is an outer-connected (2, 1)-locating point-wise non-dominating set in G. Proof. Let S ⊆ V (K1 + G) be an outer-connected 2-resolving hop dominating set in K1 + G. Then S is a 2-resolving hop dominating set in K1 + G. Then by Theorem 2, S = {x} ∪ T where T is a (2,1)-locating point-wise non-dominating set in G. Now, since S is an outer-connected 2-resolving hop dominating set in K1 + G, it follows that S = V (K1 + G) or ⟨V (K1 +G)\S⟩ = ⟨V (G)\T ⟩ is connected. Thus, T = V (G) or the subgraph ⟨V (G)\T ⟩ induced by V (G)\T is connected. Therefore, T is an outer-connected (2, 1)-locating point-wise non-dominating set in G. Conversely, assume that S = {x} ∪ T , where T is an outer-connected (2,1)-locating point-wise non-dominating set in G. By Theorem 2, S is a 2-resolving hop dominating set in K1 + G. Next, since ⟨V (K1 +G)\S⟩ = ⟨V (G)\T ⟩ and T is an outer-connected (2,1)-locating point-wise non-dominating set in G, it follows that S is a outer-connected 2-resolving hop dominating set in K1 +G. Corollary 2. Let G be connected nontrivial graph. Then γ̃c2Rh(K1+G) = l̃npnd (2,1)(G)+1. Example 1. For a fan Fn = Pn + 1 on n+ 1 vertices γ̃c2Rh(Fn) = l̃npnd (2,1)(Pn) + 1 = { n, if 4 ≤ n ≤ 7; n− 1, if n ≥ 8. Example 2. For a wheel Wn = Cn + 1 on n+ 1 vertices γ̃c2Rh(Wn) = l̃npnd (2,1)(Cn) + 1 = { n+ 1, if n = 4; n− 1, if n ≥ 5. Theorem 4. [12] 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 are (2, 1)-locating point-wise non-dominating sets of G and H, respectively. A.M. Mahistrado, H. Rara / Eur. J. Pure Appl. Math, 16 (2) (2023), 1180-1195 1187 Theorem 5. [9] Let G and H be any two graphs. A set S ⊆ V (G + H) is an outer- connected hop dominating set in G+H if and only if S = SG ∪SH , where SG and SH are pointwise non-dominating subsets of G and H, respectively, such that (i) ⟨V (H)\SH⟩ is connected whenever SH ̸= V (H) and SG = V (G) and (ii) ⟨V (G)\SG⟩ is connected whenever SG ̸= V (G) and SH = V (H). Theorem 6. Let G and H be any two graphs. A set S ⊆ V (G + H) is an outer- connected 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 are (2, 1)-locating point-wise non-dominating sets of G and H, respectively, such that (i) ⟨V (H)\SH⟩ is connected whenever SH ̸= V (H) and SG = V (G) and (ii) ⟨V (G)\SG⟩ is connected whenever SG ̸= V (G) and SH = V (H). Proof. Suppose that S ⊆ V (G+H) is an outer-connected 2-resolving hop dominating set in G+H. Let SG = V (G) ∩ S and SH = V (H) ∩ S then S = SG ∪ SH . Now, since S is a 2-resolving hop dominating set, by Theorem 4, 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. Suppose SG = V (G) and SH ̸= V (H). Since S is an outer-connected hop dominating set, by Theorem 5, ⟨V (H)\SH⟩ is connected. Hence, (i) holds. Similarly, suppose that SG ̸= V (G) and SH = V (H). By Theorem 5, ⟨V (G)\SG⟩ is connected and so (ii) holds. Conversely, suppose that S = SG ∪ SH where SG ⊆ V (G) and SH ⊆ V (H) are sets as described and satisfying (i) and (ii). By Theorem 4, S is a 2-resolving hop dominating set of G+H. If SG = V (G) and SH = V (H), then S = V (G+H) is an outer-connected 2-resolving hop dominating set. Suppose, S ̸= V (G+H). Consider the following cases: Case 1: SG ̸= V (G) and SH ̸= V (H) Then ⟨V (G+H)\S⟩ = ⟨V (G)\SG⟩+ ⟨V (H)\SH⟩ is connected. Case 2: SG = V (G) and SH ̸= V (H) Then ⟨V (G+H)\S⟩ = ⟨V (H)\SH⟩ is connected by (i). Case 3: SH = V (H) and SG ̸= V (G) Then ⟨V (G+H)\S⟩ = ⟨V (G)\SG⟩ is connected by (ii). Accordingly, S is an outer-connected 2-resolving hop dominating set of G+H. As a consequence of Theorem 6 the next result follows. Corollary 3. Let G and H be nontrivial connected graphs. Then γ̃c2Rh(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, 16 (2) (2023), 1180-1195 1188 5. Corona of Graphs This section presents characterizations in the outer-connected 2-resolving hop domi- nating sets in the corona of graphs. Remark 3. [7] 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 7. [12] 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 8. [9] Let G be a connected graph and let H be any graph. Then a subset C of V (G ◦H) is an outer-connected hop dominating set of G ◦H if and only if C = A ∪  ⋃ v∈V (G) Sv  where Sv ⊆ V (Hv) for each v ∈ V (G) and satisfies each of the following statements: (i) A = V (G) or ⟨V (G)\A⟩ is connected; (ii) If A = V (G), then ⟨V (Hv)\Sv⟩ is a connected proper subgraph of Hv for at most one vertex v ∈ A. Otherwise, Sv = V (Hv) for all v ∈ A. (iii) For all v ∈ (V (G)\NG[A, 2], there exists w ∈ NG(v) such that Sw ̸= ∅; (iv) Sv is a point-wise non-dominating set of Hv for all v ∈ (V (G)\NG[A]). Theorem 9. Let G and H be nontrivial connected graphs. A set S ⊆ V (G ◦ H) is an outer-connected 2-resolving hop dominating set of G ◦H if and only if S = A ∪  ⋃ v∈V (G) Sv  A.M. Mahistrado, H. Rara / Eur. J. Pure Appl. Math, 16 (2) (2023), 1180-1195 1189 where Sv ⊆ V (Hv) for each v ∈ V (G) and satisfies each of the following statements: (i) A = V (G) or ⟨V (G)\A⟩ is connected; (ii) If A = V (G), then ⟨V (Hv)\Sv⟩ is a connected proper subgraph of Hv for at most one vertex v ∈ A. Otherwise, Sv = V (Hv) for all v ∈ A; (iii) Sv is a 2-locating set for all v ∈ V (G) where Sv is a (2-locating) point-wise non-dominating set of Hv if v ∈ (V (G)\NG[A]). Proof. Suppose S ⊆ V (G ◦H) is an outer-connected 2-resolving hop dominating set of G◦H. Let A = S∩V (G), Sv = S∩V (Hv) for each v ∈ V (G). Then S = A∪ ( ⋃ v∈V (G) Sv ) Since S is an outer-connected hop dominating set, (i) and (ii) follow immediately from Theorem 8. Now, since S is a 2-resolving hop dominating set, by Theorem 7, (iii) holds. Conversely, let S be the set as described and satisfies the given conditions. By Theorem 7, S is 2-resolving hop dominating set. Furthermore, because (i) and (ii) hold, S is an outer-connected hop dominating set. Accordingly, S is an outer-connected 2-resolving hop dominating set in G ◦H. Corollary 4. Let G and H be connected graphs of orders n and m, respectively. Then γ̃c2Rh(G ◦H) ≤ min{γ̃c(G)(m+ 1) + (n− γ̃c(G))ln2(H), nlnpnd 2 }. Proof. Let A be a γ̃c-set of G and Sv be an ln2-set of Hv for each v ∈ V (G) \ A. Thus, by Theorem 9 S = A ∪ ( ⋃ v∈V (G) V (Hv) ) ∪ ( ⋃ v∈V (G)\A Sv ) is an outer-connected 2-resolving hop dominating set. Hence, γ̃c2Rh(G ◦H) ≤ |S| = |A|+ ∑ v∈V (G) |V (Hv)|+ ∑ v∈V (G)\A |Sv| = γ̃c(G)(m+ 1) + (n− γ̃c(G))ln2(H). Let A = ∅, Sw be a lnpnd 2 -set of Hw. Then S = A ∪ ( ⋃ w∈V (G) Sw ) is an outer-connected 2-resolving hop dominating set in G ◦H by Theorem 9. Hence, γ̃c2Rh(G ◦H) ≤ |S| = |A|+ ∑ w∈V (G) |Sw| = |V (G)| · |Sw| = n(lnpnd 2 (H)). Accordingly, γ̃c2Rh(G ◦H) ≤ min{γ̃c(G)(m+ 1) + (n− γ̃c(G))ln2(H), nlnpnd 2 }. A.M. Mahistrado, H. Rara / Eur. J. Pure Appl. Math, 16 (2) (2023), 1180-1195 1190 6. Edge Corona of Graphs This section presents characterizations in the outer-connected 2-resolving hop domi- nating sets in the edge corona of graphs. Remark 4. [12] Let uv ∈ E(G). For every x, y ∈ V (Huv), dG⋄H(x,w) = dG⋄H(y, w), dG⋄H(u,w) = dG⋄H(x,w), and dG⋄H(v, w)+1 = dG⋄H(x,w) for every w ∈ V (G⋄H)\V (Huv). Remark 5. [12] Let G and H be nontrivial connected graphs, C ⊆ V (G ⋄H) and Suv = V (Huv) ∩ C where uv ∈ E(G). For each x ∈ V (Huv)\Suv and z ∈ Suv, dG⋄H(x, z) = { 1 if z ∈ NHuv(x) 2 otherwise. Definition 8. A leaf l(G) of a graph G is a set of vertices v in G with degG(v) = 1. Theorem 10. [12] Let γ(G) ̸= 1 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 11. Let γ(G) ̸= 1 and H be any nontrivial connected graphs. A set S ⊆ V (G ⋄H) is an outer-connected 2-resolving hop dominating set of G ⋄H if and only if C = A ∪  ⋃ uv∈E(G) Suv  where Suv ⊆ V (Huv) for each uv ∈ E(G) and satisfies each of the following statements: (i) 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. (ii) A = V (G) or ⟨V (G)\A⟩ is connected; (iii) If A = V (G), then ⟨V (Huv)\Suv⟩ is a connected proper subgraph of Huv for at most one edge uv ∈ E(G). Otherwise, Suv = V (Huv) for all uv ∈ E(G); A.M. Mahistrado, H. Rara / Eur. J. Pure Appl. Math, 16 (2) (2023), 1180-1195 1191 Proof. Suppose C is an outer-connected 2-resolving hop dominating set in G ⋄H. Let A = V (G) ∩ C 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 each uv ∈ E(G). Then C is a 2-resolving hop dominating set in G ⋄ H. By Theorem 10, (i) holds. Now, suppose A ̸= V (G). Then C ̸= V (G ⋄H). Since C is an outer-connected 2-resolving hop dominating set, it follows that ⟨V (G ⋄H)\C⟩ = ⟨V (Huv)\Suv⟩ ∪ ⟨V (G)\A⟩ is connected. Hence, ⟨V (G)\A⟩ is connected. Hence, (ii) holds. Suppose A = V (G). If V (G ⋄ H) ̸= C, then ⟨V (G ⋄ H)\C⟩ = ⟨V (Huv)\Suv⟩. Since C is outer-connected 2- resolving hop dominating set, ⟨V (Huv)\Suv⟩ is a connected proper subgraph of Huv for at most one edge uv ∈ E(G). Otherwise, if V (G ⋄ H) = C, then Suv = V (Huv) for all uv ∈ E(G). Hence, (iii) holds. Conversely, let C be a set as described and satisfies the given conditions. By (i), C is a 2-resolving hop dominating set. If V (G⋄H) = C, then we are done. Now, if V (G⋄H) ̸= C. Consider the following cases: Case 1: A = V (G) Then ⟨V (G ⋄ H)\C⟩ = ⟨V (Huv)\Suv⟩ and by (iii), ⟨V (Huv)\Suv⟩ is a connected proper subgraph of Huv for at most one edge uv ∈ E(G). Thus, ⟨V (G ⋄ H)\C⟩ is connected. Case 2: A ̸= V (G) Then V (Huv) = Suv for all uv ∈ E(G). Hence, ⟨V (G ⋄H)\C⟩ = ⟨V (Huv)\Suv⟩ ∪ ⟨V (G)\A⟩ = ⟨V (G)\A⟩. Thus, ⟨V (G ⋄H)\C⟩ is connected since ⟨V (G)\A⟩ is connected by (ii). Accordingly, C is an outer-connected 2-resolving hop dominating set in G ⋄H. Corollary 5. Let γ(G) ̸= 1 be any nontrivial connected graph of sizem andH a nontrivial connected graph. Then the following statements hold. (i) If G is a graph with no pendant edges, then γ̃c2Rh(G ⋄H) = m · ln2(H). (ii) If G is a graph with k ≥ 1 pendant edges, then γ̃c2Rh(G⋄H) = min {( m−k ) ln2(H)+k ·ln(2,1)(H)+k, ( m−k ) ln2(H)+k ·ln(2,2)(H) } and γ̃c2Rh(G ⋄H) = ( m− k ) ln2(G)+ k · ln(2,2)(G) whenever ln(2,2)(H) = ln(2,1)(H). 7. Lexicographic Product of Graphs This section presents characterizations on the outer-connected 2-resolving hop domi- nating sets in the lexicographic product of graphs. A.M. Mahistrado, H. Rara / Eur. J. Pure Appl. Math, 16 (2) (2023), 1180-1195 1192 Theorem 12. [12] 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 inH for every x ∈ S with |NG(x, 2)∩ S| = 0. Theorem 13. [9] Let G and H be connected nontrivial graphs. A subset C = ⋃ x∈S [{x}× Tx] of V (G[H]) is an outer-connected hop dominating set of G[H] if and only if (i) S is a hop dominating set of G; and (ii) Tx is a point-wise non-dominating set of H for every x ∈ S with |NG(x, 2) ∩ S| = 0; (iii) ⟨(V (G)\S) ∪ {v ∈ S : Tv ̸= V (H)}⟩ is a connected graph in G. Theorem 14. 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 an outer-connected 2-resolving hop dominating set in G[H] if and only if (i) S = V (G); (ii) Tx is a 2-locating set of H for every x ∈ V (G) ; (iii) Tx and Ty are (2, 1)-locating set or one of Tx and Ty is a (2, 2)-locating set of 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 of H for every x ∈ S with |NG(x, 2) ∩ S| = 0. (vi) ⟨∪{v ∈ V (G) : Tv ̸= V (H)}⟩ is a connected graph in G. REFERENCES 1193 Proof. Let W = ⋃ x∈S [{x} × Tx], where S ⊆ V (G) and Tx ⊆ V (H) for each x ∈ S, be an outer-connected 2-resolving hop dominating set in G[H]. Then W is a 2-resolving hop dominating set in G[H]. Since W is an outer-connected hop dominating set and S = V (G), by Theorem 13 (iii), ⟨ ⋃ {v ∈ V (G) : Tv ̸= V (H)}⟩ is a connected graph in G. For the converse, let W be a 2-resolving hop dominating set and satisfies the given condition. If V (G[H]) = W , then we are done. On the other hand, suppose V (G[H]) ̸= W . Since S = V (G), ⟨(V (G)\S) ∪ {v ∈ S : Tv ̸= V (H)}⟩ = ⟨∪{v ∈ V (G) : Tv ̸= V (H)}⟩ which is connected. By Theorem 13, Theorem 12(i), and by Theorem 12 (iii) Therefore, W is an outer-connected hop dominating set in G[H]. Accordingly, W is an outer-connected 2-resolving hop dominating set in G[H]. Corollary 6. Let G and H be any nontrivial connected graph with γ(G) ̸= 1 and G is a free-equidistant. Then γ̃c2Rh(G[H]) = |V (G)| · ln2(H). Proof. Let S = V (G) and let Rx be an ln2-set of H for each x ∈ S. By Theorem 14, W = ⋃ x∈S [{x} × Rx] is an outer-connected 2-resolving hop dominating set in G[H]. Thus, γ̃c2Rh(G[H]) ≤ |W | = |V (G)||Rx| = |V (G)|ln2(H). If W0 = ⋃ x∈S({x}×T ) is a γ̃c2Rh -set of G[H], then S0 = V (G) and Tx is a 2-locating set in H for each x ∈ V (G) by Theorem 14. 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