EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 16, No. 1, 2023, 286-303 ISSN 1307-5543 – ejpam.com Published by New York Business Global Restrained 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. LetG be a connected graph. A set S ⊆ V (G) is a restrained 2-resolving hop dominating set of G if S is a 2-resolving hop dominating set of G and S = V (G) or ⟨V (G)\S⟩ has no isolated vertex. The restrained 2-resolving hop domination number of G, denoted by γr2Rh(G) is the smallest cardinality of a restrained 2-resolving hop dominating set ofG. This study aims to combine the concept of hop domination with the restrained 2-resolving sets of graphs. The main results generated in this study include the characterization of restrained 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: Restrained 2-resolving hop dominating set, restrained 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 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 [10]. 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 [11]. Moreover, other variations of 2-resolving sets in graphs were also studied in [4–6, 8, 12, 13], respec- tively. ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v16i1.4665 Email addresses: angelicamae.mahistrado@g.msuiit.edu.ph (A.M. Mahistrado), helen.rara@g.msuiit.edu.ph (H. Rara) https://www.ejpam.com 286 © 2023 EJPAM All rights reserved. A.M. Mahistrado, H. Rara / Eur. J. Pure Appl. Math, 16 (1) (2023), 286-303 287 2. Terminology and Notation In this study, we consider finite, simple and connected graphs. For basic graph- theoretic concepts, we then refer readers to [2] and [3]. The following concepts are found in [2], [14] and [16]. 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 neigh- borhood of u. The closed hop neighborhood of u in G is given by NG[u, 2] = NG(u, 2)∪{u}. The open hop neighborhood ofX ⊆ 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 a restrained 2-resolving hop dominating set of G if S is a 2- resolving hop dominating set of G and S = V (G) or ⟨V (G)\S⟩ has no isolated vertex. The restrained 2-resolving hop domination number of G, denoted by γr2Rh(G) is the smallest cardinality of a restrained 2-resolving hop dominating set of G. Any restrained 2-resolving hop dominating set of cardinality γr2Rh(G) is referred to as a γr2Rh-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. A.M. Mahistrado, H. Rara / Eur. J. Pure Appl. Math, 16 (1) (2023), 286-303 288 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. [11] 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 a restrained 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 ⟨V (G)\S⟩ has no isolated vertex. The restrained 2-locating point-wise non-dominating number of G, denoted by rlnpnd 2 (G), is the smallest cardinality of a restrained 2-locating point-wise non- dominating set in G. A restrained 2-locating point-wise non-dominating set of cardinality rlnpnd 2 (G) is then referred to as an rlnpnd 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. Definition 6. [11] 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 a restrained (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 S = V (G) or ⟨V (G)\S⟩ has no isolated vertex. The restrained (2, 2)-locating point-wise non-domination ((2, 1)-locating point-wise non-domination, respectively) number of G, de- noted by rlnpnd (2,2)(G) (rlnpnd (2,1)(G), respectively), is the smallest cardinality of a restrained (2, 2)-locating point-wise non-dominating ((2, 1)-locating point-wise non-dominating, re- spectively) set in G. A restrained (2, 2)-locating point-wise non-dominating ((2, 1)-locating A.M. Mahistrado, H. Rara / Eur. J. Pure Appl. Math, 16 (1) (2023), 286-303 289 point-wise non-dominating, respectively) set of cardinality rlnpnd (2,2)(G) (rlnpnd (2,1)(G), respectively) is then referred to as an rlnpnd (2,2)(G) (rlnpnd (2,1)(G), respectively)-set in G. Definition 8. A restrained 2-resolving set S ⊆ V (G) which is point-wise non-dominating is called a restrained 2-resolving point-wise non-dominating set in G. The minimum cardinality of a restrained 2-resolving point-wise non-dominating set in G, denoted by rdim2pnd (G) is called the restrained 2-resolving point-wise non-domination number of G. Any r2R-pointwise non-dominating set of cardinality rdim2pnd (G) is then referred to as a rdim2pnd -set in G. Proposition 1. [9] Let G be a connected graph of order n ≥ 2. Then dim2(G) = 2 if and only if G ∼= Pn. Remark 1. [11] For a path Pn on n vertices, lnpnd 2 (Pn) = { 3, n = 3 ⌈n+1 2 ⌉, n ≥ 4 3. Preliminary Results Remark 2. Every nontrivial connected graph G admits a restrained 2-resolving hop dom- inating set. Indeed, the vertex set V (G) of G is a restrained 2-resolving hop dominating set. Theorem 1. If S ⊆ V (G) is a restrained 2-resolving hop dominating set in G, then S is a restrained 2-resolving point-wise non-dominating set in G. Proof. Suppose S is a restrained 2-resolving hop dominating set in G. Let v ∈ V (G)\S. Since S is 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, S is a restrained 2- resolving point-wise non-dominating set in G. The next result follows from [5]. Remark 3. Let G be any nontrivial connected graph. Then 2 ≤ rlnpnd 2 (G) ≤ |V (G)|. Moreover, (i) rlnpnd 2 (G) = 2 if and only if G = K2. (ii) If G is a connected graph with 2 ≤ |V (G)| ≤ 4, then rlnpnd 2 (G) = |V (G)|. A.M. Mahistrado, H. Rara / Eur. J. Pure Appl. Math, 16 (1) (2023), 286-303 290 Proposition 2. Let G be any nontrivial connected graph. Then for any positive integers n and k, we have (i) rlnpnd 2 (Pn) = n, if 2 ≤ n ≤ 7; 3n+ 2k 5 , if n = k(mod 5), 3 ≤ k ≤ 7. (ii) rlnpnd 2 (Cn) = n, if n = 3, 4; 3n+ 2k 5 , if n = k(mod 5), 0 ≤ k ≤ 4. (iii) For all n ≥ 4, rlnpnd (2,2)(Pn) = n, if 4 ≤ n ≤ 7; 3n+ 2k 5 , if n = k(mod 5), 3 ≤ k ≤ 7. For all n ≥ 6, rlnpnd (2,2)(Cn) = n, if n = 4; 3n+ 2k 5 , if n = k(mod 5), 0 ≤ k ≤ 4. (iv) For all n ≥ 2, rlnpnd (2,1)(Pn) = n, if 2 ≤ n ≤ 7; 3n+ 2k 5 , if n = k(mod 5), 3 ≤ k ≤ 7. For all n ≥ 3, rlnpnd (2,1)(Cn) = n, if n = 3, 4; 3n+ 2k 5 , if n = k(mod 5), 0 ≤ k ≤ 4. Proof. (i) Let Pn = [v1, v2, . . . , vn] and S be an rlnpnd 2 - set of Pn. The case where n ≤ 7 can be easily verified by Remark 1. Next, let n ≥ 8 and n ≡ k(mod 5) where 3 ≤ k ≤ 7. Then n = 5r + k. Hence, r = n− k 5 . Then the set S = {v1, v2, v3, v6, v7, v8, v11, v12, v13, . . . , v5r+1, v5r+2, . . . , v5r+k} is an rlnpnd 2 - set of Pn. Therefore, |S| = 5r + k − 2r = 3n+ 2k 5 . The proofs of (ii), (iii) and (iv) are similar to (i). Theorem 2. Let G be a connected graph. Then 2 ≤ rdim2pnd (G) ≤ |V (G)|. Moreover, (i) rdim2pnd (G) = 2 if and only if G is a path Pn except n = 3. (ii) If G is a cycle Cn for n ̸= 4, then rdim2pnd (Cn) = 3. Proof. (i) Suppose rdim2pnd (G) = 2. Note that every restrained 2-resolving point- wise non-dominating set is a 2-resolving point-wise non-dominating set in G, that is dim2pnd (G) = 2. Hence, by Proposition 1, G = Pn. Since rdim2pnd (P3) = 3, G = Pn except n = 3. Conversely, if G = Pn = [v1, v2, . . . , vn], then S = {v1, vn} is a restrained 2-resolving A.M. Mahistrado, H. Rara / Eur. J. Pure Appl. Math, 16 (1) (2023), 286-303 291 point-wise non-dominating set of G. Hence, rdim2pnd (G) = 2. (ii) Suppose G = Cn = [v1, v2, . . . , vn]. Let S be the rdim2pnd -set of Cn. By (i), rdim2pnd (Cn) > 2. Thus, S = {v1, v2, v3} is a restrained 2-resolving point-wise non- dominating set of G. Hence, rdim2pnd (Cn) = 3. Remark 4. For any connected graph G of order n ≥ 2, 2 ≤ γr2Rh(G) ≤ n. Moreover, γr2Rh(P2) = 2 and γr2Rh(Kn) = n. Example 1. (i) For complete graph Kn on n ≥ 2 vertices, γr2Rh(Kn) = n. (ii) For complete bipartite graph Km,n on m+ n vertices where m,n ≥ 1, γr2Rh(Km,n) = m+ n. (iii) For star graph K1,n on n+ 1 vertices where n ≥ 1, γr2Rh(K1,n) = n+ 1. The next results follow from [14] and by definition of restrained 2-resolving hop dom- inating set. Proposition 3. (i) For a path Pn on n vertices γr2Rh(Pn) =  2, if n = 2, 4; 3, if n = 3, 5; 4, if n = 6; n+ 2s 3 , if n ≡ s(mod 6) where 0 ≤ s ≤ 2 and n > 6; n+ 6− s 3 , if n ≡ s(mod 6) where s = 3, 4 and n > 8; n+ 4 3 , if n ≡ 5(mod 6)where n > 10. (ii) For a cycle Cn on n vertices γr2Rh(Cn) =  3, if n = 3, 5, 6; 4, if n = 4; n+ 2s 3 , if n ≡ s(mod 6) where 0 ≤ s ≤ 2 and n > 6; n+ 6− s 3 , if n ≡ s(mod 6) where 3 ≤ s ≤ 5 and n > 8. Next, we show that every pair of positive integers are realizable as 2-resolving hop domination number and restrained 2-resolving hop domination number. Thus, as a con- sequence, the difference γr2Rh − γ2Rh can be made arbitrarily large. Remark 5. Every restrained 2-resolving hop dominating set of G is a 2-resolving hop dominating set of G. Thus, γ2Rh(G) ≤ γr2Rh(G). A.M. Mahistrado, H. Rara / Eur. J. Pure Appl. Math, 16 (1) (2023), 286-303 292 Theorem 3. Let a and b be positive integers such that 2 ≤ a ≤ b. Then there exists a nontrivial connected graph H such that γ2Rh(H) = a and γr2Rh(H) = b. Proof. Suppose 2 ≤ a = b. Consider graphH1 in Figure 1. Hence, S = {x1, x2, x3 . . . , xa} is both γ2Rh and a γr2Rh-set of H1. Thus, 2 ≤ γ2Rh(H1) = a = b = γr2Rh(H1). ......... ........ ........ ........ ........ ........ ........ ........ ........ ... .................................... .................................... ......... ........ ........ ........ ........ ........ ........ ........ ........ ... .................................... .................................... ......... ........ ........ ........ ........ ........ ........ ........ ........ ... .................................... .................................... ......... ........ ........ ........ ........ ........ ........ ........ ........ ... .................................... .................................... ................................................................................................................ ................................................................................................................ .................................... ......... ........ ........ ........ ........ ........ ........ ........ ........ ... .................................... .................................... ................................................................................................................ .................................... x1 x2 x3 xa−1 xa=b • • • • • . . . H1 : Figure 1 Suppose 2 < a < b. Consider the graph H2 in Figure 2. Then S = {x1, x2, . . . , xa} is a γ2Rh-set of H2 and X = S ∪ {y1, y2, . . . , yb−a} is a γr2Rh-set of H2. Hence γ2Rh(H2) = a and γr2Rh(H2) = |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 H2 : We now characterize the restrained 2-resolving hop dominating sets in some graphs under some binary operations. 4. Restrained 2-Resolving Hop Dominating Sets in the Join of Graphs This section presents characterizations on the restrained 2-resolving hop dominating sets in the join of graphs. Theorem 4. [7] 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. [11] 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 6. Let G be a connected graph and let K1 = {x}. Then S ⊆ V (K1 + G) is a restrained 2-resolving hop dominating set in K1 +G if and only if S = {x} ∪ T where T is a restrained (2, 1)-locating point-wise non-dominating set in G. A.M. Mahistrado, H. Rara / Eur. J. Pure Appl. Math, 16 (1) (2023), 286-303 293 Proof. Let S ⊆ V (K1 +G) be a restrained 2-resolving hop dominating set in K1 +G. Then S is a restrained 2-resolving set in K1 +G. Since S is a hop dominating set, x ∈ S. Hence, S = {x} ∪ T for T ⊆ V (G). Then by Theorem 5, T is a (2,1)-locating point-wise non-dominating set in G. Now, since ⟨V (K1 +G)\S⟩ = ⟨V (G)\T ⟩, and S is a restrained 2-resolving hop dominating set in K1+G, then it follows that T = V (G) or ⟨V (G)\T ⟩ has no isolated vertex. Therefore, T is a restrained (2, 1)-locating point-wise non-dominating set in G. Conversely, assume that S = {x}∪T , where T is a restrained (2,1)-locating point-wise non-dominating set in G. By Theorem 5, S is a 2-resolving hop dominating set in K1+G. Next, since ⟨V (K1 +G)\S⟩ = ⟨V (G)\T ⟩ and T is a restrained (2,1)-locating point-wise non-dominating set in G, it follows that S is a restrained 2-resolving hop dominating set in K1 +G. As a consequence of Theorem 6 the next result follows. Corollary 1. Let G be connected nontrivial graph. Then γr2Rh(K1+G) = rlnpnd (2,1)(G)+1. Example 2. For a fan Fn = Pn +K1 on n+ 1 vertices γr2Rh(Fn) = rlnpnd (2,1)(Pn) + 1 = n+ 1, if 2 ≤ n ≤ 7; 3n+ 2k 5 + 1, if n = k(mod 5), 3 ≤ k ≤ 7. Example 3. For a wheel Wn = Cn + 1 on n+ 1 vertices γr2Rh(Wn) = rlnpnd (2,1)(Cn) + 1 = n+ 1, if n = 3, 4; 3n+ 2k 5 + 1, if n = k(mod 5), 0 ≤ k ≤ 4. Theorem 7. [11] 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. Theorem 8. [8] Let G and H be any two graphs. A set S ⊆ V (G+H) is a restrained 2- resolving set in G+H if and only if SG = V (G)∩S and SH = V (H)∩S where S = SG∪SH are 2-locating set in G and H, respectively where SG or SH is a (2, 2)-locating or SG and SH are (2, 1)-locating sets and one of the following holds: (i) SG = V (G) and SH is a restrained 2-locating set in H; (ii) SH = V (H) and SG is a restrained 2-locating set in G; (iii) SG ̸= V (G) and SH ̸= V (H). Theorem 9. Let G and H be any two graphs. A set S ⊆ V (G + H) is a restrained 2-resolving hop dominating set in G+H if and only if SG = V (G)∩S and SH = V (H)∩S A.M. Mahistrado, H. Rara / Eur. J. Pure Appl. Math, 16 (1) (2023), 286-303 294 are 2-locating pointwise 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 and one of the following holds: (i) SG = V (G) and SH is a restrained 2-locating point-wise non-dominating set in H; (ii) SH = V (H) and SG is a restrained 2-locating point-wise non-dominating set in G; and (iii) SG ̸= V (G) and SH ̸= V (H). Proof. Suppose that S ⊆ V (G+H) is a restrained 2-resolving hop dominating set in G+H. Let SG = V (G) ∩ S and SH = V (H) ∩ S where S = SG ∪ SH . Now, since S is a 2-resolving hop dominating set by Theorem 7, 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 (2, 1)-locating point-wise non-dominating sets of G and H, respectively. Suppose SG = V (G). Let SH ̸= V (H). Since S is restrained 2-resolving hop dominating, S = V (G +H) or ⟨V (G+H)\S⟩ = ⟨V (H)\SH⟩ has no isolated vertex. Hence, SH = V (H) or ⟨V (H)\SH⟩ has no isolated vertex. Thus, it follows that SH is a restrained 2-locating point-wise non-dominating set of H and so (i) holds. Next, suppose that SG ̸= V (G). If SH ̸= V (H), then (iii) holds. On the other hand, if SH = V (H), then ⟨V (G)\SG⟩ has no isolated vertex and so (ii) holds. Conversely, suppose that S = SG ∪ SH where SG ⊆ V (G) and SH ⊆ V (H) are 2- locating point-wise non-dominating sets of G and H, respectively, and (i), (ii) and (iii) hold. By Theorem 7, S is a 2-resolving hop dominating set of G + H. If (i) holds, then S = V (G +H) or ⟨V (G+H)\S⟩ = ⟨V (H)\SH⟩ has no isolated vertex since SH is restrained 2-resolving hop dominating. Similarly, if (ii) holds, then S = V (G + H) or ⟨V (G+H)\S⟩ = ⟨V (G)\SG⟩ has no isolated vertex since SG is restrained 2-resolving hop dominating set. Therefore, it follows that S is a restrained 2-resolving hop dominating set of G+H. As a consequence of Theorem 9 the next result follows. Corollary 2. Let G and H be nontrivial connected graphs. Then γr2Rh(G+H) =  m+ n, if rlnpnd 2 (G) = m and rlnpnd 2 (H) = n min{lnpnd (2,2)(G) + lnpnd 2 (H), lnpnd 2 (G) + lnpnd (2,2)(H), lnpnd (2,1)(G) + lnpnd (2,1)(H)}, otherwise. Example 4. For any nontrivial connected graph G and H of order n and m, respectively; (i) γr2Rh(G+H) = m+ n if G and H are complete; A.M. Mahistrado, H. Rara / Eur. J. Pure Appl. Math, 16 (1) (2023), 286-303 295 (ii) γr2Rh(G+H) =  ( 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. where G = Pn and H = Pm and n,m ≥ 4. (iii) γr2Rh(G+H) =  ( n 2 ) + ( m 2 ) , if n,m are even( n 2 ) + ⌈m2 ⌉, if n is even,m is odd ⌈n2 ⌉+ ( m 2 ) , if n is odd,m is even ⌈n2 ⌉+ ⌈m2 ⌉, if n,m are odd. where G = Cn and H = Cm and n,m ≥ 5. 5. Restrained 2-Resolving Hop Dominating Sets in the Corona of Graphs This section presents characterizations on the restrained 2-resolving hop dominating sets in the corona of graphs. Remark 6. [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 10. [11] 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). A.M. Mahistrado, H. Rara / Eur. J. Pure Appl. Math, 16 (1) (2023), 286-303 296 Theorem 11. Let G and H be nontrivial connected graphs. A set S ⊆ V (G ◦ H) is a restrained 2-resolving hop dominating set of G ◦H if and only if S = A∪  ⋃ v∈(V (G)\A)∩NG(A) Sv ∪  ⋃ w∈(V (G)\A)\NG(A) Dw ∪  ⋃ u∈A∩NG(A) Eu ∪  ⋃ j∈A\NG(A) Fj  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 is a 2-locating set of Hv for all v ∈ (V (G)\A) ∩NG(A); (iii) Dw is a 2-locating point-wise non- dominating set ofHw for all w ∈ (V (G)\A)\NG(A); (iv) Eu is a restrained 2-locating set of Hu for all u ∈ A ∩NG(A); (v) Fj is a restrained 2-locating point-wise non-dominating set ofHj for all j ∈ A\NG(A). Proof. Suppose S ⊆ V (G◦H) be a restrained 2-resolving hop dominating set of G◦H. Let A = S∩V (G), Sv = S∩V (Hv) for each v ∈ (V (G)\A)∩NG(A), Dw = S∩V (Hw) for each w ∈ (V (G)\A)\NG(A), Eu = S∩V (Hu) for each u ∈ A∩NG(A) and Fj = S∩V (Hj) for each j ∈ A\NG(A). Then S = A∪  ⋃ v∈(V (G)\A)∩NG(A) Sv ∪  ⋃ w∈(V (G)\A)\NG(A) Dw ∪  ⋃ u∈A∩NG(A) Eu ∪  ⋃ j∈A\NG(A) Fj  . Since S is a 2-resolving hop dominating set, (i), (ii) and (iii) follow immediately from Theorem 10. Next, let u ∈ A∩NG(A). If Eu = V (Hu), then Eu is a restrained 2-locating. Suppose that Eu ̸= V (Hu). Then V (G ◦H) ̸= S. Now, since V (Hu)\Eu ⊆ V (G ◦H)\S and S is a restrained 2-resolving, it follows that ⟨V (Hu)\Eu⟩ has no isolated vertex. Thus, Eu is a restrained 2-locating set of Hu. Hence, (iv) follows. Finally, suppose j ∈ A\NG(A). Since S is a restrained 2-resolving hop dominating set and Fj ⊆ S, Fj is a restrained 2-locating point-wise non-dominating set of Hj . Thus, (v) follows. Conversely, let S be the set as described and satisfies the given conditions. By Theorem 10, S is 2-resolving hop dominating set. Furthermore, because (i), (ii), (iii), (iv) and (v) hold, S is a restrained 2-resolving hop dominating set in G ◦H. As a consequence of Theorem 11 the next results follow. Corollary 3. Let G and H be nontrivial connected graphs and |V (G)| = n. Then (i) γr2Rh(G ◦H) ≤ n(1 + rln2(H)). A.M. Mahistrado, H. Rara / Eur. J. Pure Appl. Math, 16 (1) (2023), 286-303 297 (ii) γr2Rh(G ◦H) ≤ n(lnpnd 2 (H)). Proof. (i) Let A = V (G), E be an rln2-set of H and Eu ⊆ V (Hu) be an rln2-set of Hu with ⟨Eu⟩ ∼= ⟨E⟩ for each u ∈ V (G). Then S = A ∪ ( ⋃ u∈V (G) Ew ) is a restrained 2-resolving hop dominating set of G ◦H by Theorem 11. Hence, γr2Rh(G ◦H) ≤ |S| = |V (G)|+ ∑ w∈V (G) |Eu| = |V (G)|+ |V (G)| · |E| = n(1 + rln2(H)). (ii) Let A = ∅, D be a lnpnd 2 -set of H and Dw ⊆ V (Hw) be a lnpnd 2 -set of Hw with ⟨Dw⟩ ∼= ⟨D⟩ for each w ∈ V (G). Then S = A ∪ ( ⋃ w∈V (G) Dw ) is a restrained 2-resolving hop dominating set of G ◦H by Theorem 11. Hence, γr2Rh(G ◦H) ≤ |S| = |A|+ ∑ w∈V (G) |Dw| = |V (G)| · |D| = n(lnpnd 2 (H)). Corollary 4. Let G and H be nontrivial connected graphs where |V (G)| = n and lnpnd 2 (H) = ln2(H). Then γr2Rh(G ◦H) = n(lnpnd 2 (H)). Proof. We have γr2Rh(G ◦ H) ≤ n(lnpnd 2 (H)) by Corollary 3 (ii). Since lnpnd 2 (H) = ln2(H), then by Remark 5 and Corollary 5 in [11], we have γr2Rh(G◦H) ≥ γ2Rh(G◦H) = n(lnpnd 2 (H)). Therefore, γr2Rh(G ◦H) = n(lnpnd 2 (H)). Example 5. For any nontrivial connected graph G of order n, (i) γr2Rh(G ◦H) ≤ 4n if H = P3; (ii) γr2Rh(G ◦H) = n · (⌈ m+1 2 ⌉) if H = Pm and m ≥ 4; (iiii) γr2Rh(G ◦H) = n · (⌈ m 2 ⌉) if H = Cm and m ≥ 5. 6. Restrained 2-Resolving Hop Dominating Sets in the Edge Corona of Graphs This section presents characterizations on the 2-resolving hop dominating sets and restrained 2-resolving hop dominating sets in the edge corona of graphs. Remark 7. 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). A.M. Mahistrado, H. Rara / Eur. J. Pure Appl. Math, 16 (1) (2023), 286-303 298 Remark 8. 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 9. A leaf l(G) of a graph G is a set of vertices v in G with degG(v) = 1. Theorem 12. 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. Proof. Suppose that C ⊆ V (G ⋄H) is a 2-resolving hop dominating set of 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). Now, suppose that Suv = ∅ for some uv ∈ E(G) where v ∈ V (G) ∩NG(A) or u ∈ V (G) ∩NG(A). Let x, y ∈ V (Huv). Then rG⋄H(x/C) = rG⋄H(y/C) which is a contradiction to the assumption of C. Thus, Suv ̸= ∅. Next, we claim that Suv is a 2-locating set in Huv for each uv ∈ E(G). Let a, b ∈ V (Huv)\Suv where a ̸= b or [a ∈ Suv and b /∈ Suv]. Since C is a 2-resolving set in G ⋄ H, rG⋄H(a/C) and rG⋄H(b/C) differ in at least 2 positions. By Remark 7, rHuv(a/Suv) and rHuv(b/Suv) must differ in at least 2 positions. By definition of G ⋄H, there exists at least two vertices say p, q ∈ V (Huv) ∩ Suv such that either p, q ∈ NHuv(a)\NHuv(b) or p, q ∈ NHuv(b)\NHuv(a) or p ∈ NHuv(a)\NHuv(b) and q ∈ NHuv(b)\NHuv(a). Similarly, if a ∈ Suv and b ∈ V (Huv)\Suv, then there exists a vertex s ∈ V (Huv)∩Suv such that s ∈ NHuv(a)\NHuv(b) or s ∈ NHuv(b)\NHuv(a). Thus, it follows that Suv is a 2-locating set of Huv. Next, suppose that uv is a pendant edge and suppose u is an end-vertex. Then ⟨v⟩ +Huv is a subgraph G ⋄ H. Since Suv = C ∩ V (Huv) ⊆ C and C is a 2-resolving set it follows by Theorem 4, Suv is a (2, 1)-locating set of Huv whenever u ∈ C and Suv is a (2, 2)-locating set of Huv otherwise. Conversely, let C be the set as described and satisfies the given conditions. Let x, y ∈ V (G ⋄H) with x ̸= y. Then it can be easily verify that rG⋄H(x/C) and rG⋄H(y/C) differ in at least two positions for all x, y ∈ V (G) or x ∈ V (Huv) and y ∈ V (G) for all edge uv ∈ E(G) or x ∈ V (Hpq) and y ∈ V (Hab) such that pq ̸= ab for some pq, ab ∈ E(G). A.M. Mahistrado, H. Rara / Eur. J. Pure Appl. Math, 16 (1) (2023), 286-303 299 Hence, consider only the following cases: Case 1: x, y ∈ V (Huv)\Suv or x ∈ V (Huv)\Suv and y ∈ Suv for some edge uv ∈ E(G). Now, since Suv is a 2-locating set, rHuv(x/Suv) and rHuv(y/Suv) differ in at least two positions. Then by definition of G ⋄H, rG⋄H(x/C) and rG⋄H(y/C) differ in at least two positions. Case 2: x ∈ V (Huv)\Suv or x ∈ Suv and y = u for some pendant edge uv ∈ E(G) and u is an end-vertex Since Suv is a (2, 2)-locating set, there exists a, b ∈ Suv\NHuv(x) but a, b ∈ NG⋄H(y). Thus, it follows that rG⋄H(x/C) and rG⋄H(y/C) differ in ath and bth positions. Therefore, C is a 2-resolving set in G ⋄H. Next, we claim that C is a hop dominating set. Let x ∈ V (G)\A. Since G is a connected graph and G ̸= P2, there exist y, q ∈ V (G) such that y ∈ NG(x) ∩ NG(q). Now, since Syq ̸= ∅, a vertex z ∈ Syq∩NG⋄H(x, 2) exists. On the other hand, if x ∈ V (Huv)\Suv, then there exists y ∈ NG(u)∪NG(v) such that NG⋄H(x, 2)∩Svy ̸= ∅ or NG⋄H(x, 2)∩Suy ̸= ∅. Thus, C is a hop dominating set in G ⋄H. Accordingly, C is a 2-resolving hop dominating set in G ⋄H. As a consequence of Theorem 12 the next result follows. Corollary 5. Let G ̸= P2 be any nontrivial connected graph of size m and H a nontrivial connected graph. Then the following statements hold. (i) If G is a graph with no pendant edges, then γ2Rh(G ⋄H) = m · ln2(H). (ii) If G is a graph with k ≥ 1 pendant edges, then γ2Rh(G⋄H) = min {( m−k ) ln2(H)+k · ln(2,1)(H)+k, ( m−k ) ln2(H)+k · ln(2,2)(H) } and γ2Rh(G ⋄H) = ( m− k ) ln2(H) + k · ln(2,2)(H) whenever ln(2,2)(H) = ln(2,1)(H). Theorem 13. Let G ̸= P2 and H be any nontrivial connected graphs. A set S ⊆ V (G⋄H) is a restrained 2-resolving hop dominating set of G ⋄H if and only if C = A ∪  ⋃ uv∈E(G) Suv  is a 2-resolving hop dominating set and (i) ⟨V (G)\A⟩ has no isolated vertex whenever Suv = V (Huv); and (ii) Suv is a restrained 2-locating set of Huv for all uv ∈ E(G) if u ∈ A and v ∈ A. Proof. Suppose C is a restrained 2-resolving hop dominating set in G ⋄ H. Then C is a 2-resolving hop dominating set in G ⋄H. By Theorem 12, Suv is a 2-locating set in Huv for all uv ∈ E(G). 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). A.M. Mahistrado, H. Rara / Eur. J. Pure Appl. Math, 16 (1) (2023), 286-303 300 Now, suppose Suv = V (Huv). Since C is a restrained 2-resolving hop dominating set, then ⟨V (G)\A⟩ must contain no isolated vertex. Thus, (i) holds. Next, let u, v ∈ A. If Suv = V (Huv), then Suv is a restrained 2-locating set of Huv. Suppose Suv ̸= V (Huv). Since V (Huv)\Suv ⊆ V (G ⋄H)\C and C is a restrained 2- resolving hop dominating set in G⋄H, it follows ⟨V (Huv)\Suv⟩ must have no isolated vertex. Hence, Suv is a restrained 2-locating set in Huv. Hence, (ii) holds. Conversely, let C be a 2-resolving hop dominating set as described and satisfies the given conditions. Suppose V (Huv) = Suv for all uv ∈ E(G). Then ⟨V (G ⋄ H)\C⟩ = ⟨V (G)\A⟩. By (i), ⟨V (G⋄H)\C⟩ has no isolated vertex. Next, suppose V (Huv) ̸= Suv for some uv ∈ E(G). If u or v is not an element of A, then ⟨V (Huv)\Suv⟩ + ⟨{u, v}⟩ has no isolated vertex. On the other hand, if u, v ∈ A, then V (Huv)\Suv has no isolated vertex by (ii). Thus, it follows that ⟨V (G ⋄ H)\C⟩ has no isolated vertex. Therefore, C is a restrained 2-resolving hop dominating set in G ⋄H. Corollary 6. Let G and H be a nontrivial connected graph. Then γr2Rh(G ⋄H) = γ2Rh(G ⋄H). 7. Restrained 2-Resolving Hop Dominating Sets in the Lexicographic Product of Graphs This section presents characterizations on the restrained 2-resolving hop dominating sets in the lexicographic product of graphs. Theorem 14. [11] 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 15. 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 restrained 2-resolving hop dominating set in G[H] if and only if it is a 2-resolving hop dominating set and Tx is a restrained 2-locating point-wise non-dominating set for each x with Ty = V (H) for all y ∈ NG(x). A.M. Mahistrado, H. Rara / Eur. J. Pure Appl. Math, 16 (1) (2023), 286-303 301 Proof. Let W = ⋃ x∈S [{x} × Tx], where S ⊆ V (G) and Tx ⊆ V (H) for each x ∈ S, be a restrained 2-resolving hop dominating set in G[H]. Then W is a 2-resolving hop domi- nating set in G[H]. By Theorem 14, (i)-(iv) hold and Tx is a 2-locating point-wise non- dominating set inH for every x ∈ S with |NG(x, 2)∩S| = 0. Since V (H)\Tx ⊆ V (G[H])\W and W is a restrained 2-resolving hop dominating set, it follows that ⟨V (H)\Tx⟩ has no isolated vertex. Hence, Tx is a restrained 2-locating point-wise non-dominating set of H. For the converse, let W be a 2-resolving hop dominating set as described and satisfies the given conditions. Suppose that V (G[H]) = W . Then W is a restrained 2-resolving hop dominating set of G[H]. Suppose that V (G[H]) ̸= W . Let (x, v) ∈ V (G[H])\W . If Ty ̸= V (H), for all y ∈ NG(x), then ⟨V (G[H])\W ⟩ has no isolated vertex. If Ty = V (H), for some y ∈ NG(x), then Tx is a restrained 2-locating point-wise non-dominating set. Thus, ⟨V (H)\Tx⟩ has no isolated vertex. Hence, ⟨V (G[H])\W ⟩ has no isolated vertex. Therefore, W is a restrained 2-resolving hop dominating set in G[H]. The following results follow from Theorem 15. Corollary 7. Let G and H be nontrivial connected graphs such that G is not free- equidistant.. Then, γr2Rh(G[H]) ≤ n · ln(2,1)(H) +m · γ2L(H) + p · rlnpnd 2 (H), where n+m+ p = |V (G)| with |EQ1(G)| = n, |EQ2(G)| = m and |fr(G)| = p. Corollary 8. 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