EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 16, No. 2, 2023, 763-772 ISSN 1307-5543 – ejpam.com Published by New York Business Global On 1-Movable Strong Resolving Hop Domination in Graphs Armalene H. Abragan1,∗, Helen M. Rara2 1 Department of Mathematics and Statistics, College of Science and Mathematics, Center of Graph Theory, Algebra, Mindanao State University-Iligan Institute of Technology, 9200 Iligan City, Philippines 2 Analysis-Premier Research Institute of Science and Mathematics, Mindanao State University-Iligan Institute of Technology, 9200 Iligan City, Philippines Abstract. A set S is a 1-movable strong resolving hop dominating set of G if for every v ∈ S, either S\{v} is a strong resolving hop dominating set or there exists a vertex u ∈ (V (G)\S)∩NG(v) such that (S \ {v})∩ {u} is a strong resolving hop dominating set of G. The minimum cardinality of a 1-movable strong resolving hop dominating set of G is denoted by γ1 msRh(G). In this paper, we obtained the corresponding parameter in graphs resulting from the join, corona and lexicographic product of two graphs. Specifically, we characterize the 1-movable strong resolving hop dominating sets in these types of graphs and determine the bounds or exact values of their 1-movable strong resolving hop domination numbers. 2020 Mathematics Subject Classifications: 05C69 Key Words and Phrases: 1-movable strong resolving hop dominating set, 1-movable strong resolving hop domination number, join, corona, lexicographic product 1. Introduction The study of domination can be traced way back 1960. Since then numerous authors contribute several interesting domination parameters to nurture the growth of this research area. In 1977, E.J Cockayne and S.T Hedetniemi introduced the notation γ(G) for the domination number of graph G. Until the initiation of the concept of 2-step domination number by Chartrand et al. [1] in 1995, which is closely related to hop domination number. Subsequently, Natarajan and Ayyaswamy (2015) introduced the hop domination concept. Some variation of domination can be seen in these papers [7], [6]. Blair et al. [3] introduced and investigated a new variant of the standard domination parameter called 1-movable domination. In 2011, they established results on the 1-movable ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v16i2.4658 Email addresses: armalene.abragan@g.msuiit.edu.ph (A. Abragan), helen.rara@g.msuiit.edu.ph (H. M. Rara) https://www.ejpam.com 763 © 2023 EJPAM All rights reserved. A. H. Abragan, H. M. Rara / Eur. J. Pure Appl. Math, 16 (2) (2023), 763-772 764 dominating sets of some graphs and identified bounds on the 1-movable domination num- ber for certain classes of graphs. The concept of 1-movable dominating set was discussed in the paper of Hinampas and Canoy [4]. Their paper also presented some characterizations involving the concept and investigated the 1-movable dominating sets in the join and corona of graphs. Inspired by the above works, this present study investigates the concepts of restrained strong resolving hop dominating and 1- movable strong resolving hop dominating sets of some graphs. In this study, we only consider graphs that are finite, simple, undirected and connected. Readers are referred to [2] for elementary Graph Theory concepts. Let G be a connected graph. A set S ⊆ V (G) is a hop dominating set of G if for every v ∈ V (G)\S, there exists u ∈ S such that dG(u, v) = 2. The minimum cardinality of a hop dominating set of G, denoted by γh(G), is called the hop domination number of G. Any hop dominating set with cardinality equal to γh(G) is called a γh-set. A set C ⊆ V (G) is called a superclique in G if ⟨C⟩ is a clique and for every pair of distinct vertices u, v ∈ C, there exists w ∈ V (G) \ C such that w ∈ NG(u) \ NG(v) or w ∈ NG(v)\NG(u). A superclique C is maximum in G if |C| ≥ |C∗| for all supercliques C∗ in G. The superclique number of G, denoted by ωS(G), is the cardinality of a maximum superclique in G. A superclique C in G is called a hop dominated superclique if for every v ∈ C there exists u ∈ V (G)\C such that dG(u, v) = 2. A hop dominated superclique C is maximum in G if |C| ≥ |C∗| for all hop dominated supercliques C∗ in G. The hop dominated super- clique number denoted by ωhS(G), of G is the cardinality of a maximum hop dominated superclique in G. A superclique C ⊆ V (G) is called a point-wise non-dominated superclique of G if for every x ∈ C there exists y ∈ V (G) \C such that y /∈ NG(x). A maximum cardinality of a point-wise non-dominated superclique in G is denoted by ωpndS(G). A vertex u of G is maximally distant from vertex v of G, u ̸= v, if for every vertex w ∈ NG(u), dG(v, w) ≤ dG(u, v). If u is maximally distant from v and v is maximally distant from u, then we say that u and v are mutually maximally distant, denoted by uMMDv. A vertex x of a connected graph G is said to resolve vertices u and v of G if dG(x, u) ̸= dG(x, v). For an ordered set W = {x1, . . . , xk} ⊆ V (G) and a vertex v in G, the k-vector rG(v/W ) = (dG(v, x1), dG(v, x2), . . . dG(v, xk)) is called the representation of v with respect to W . The set W is a resolving set for G if and only if no two vertices of G have the same representation with respect to W . The metric dimension of G, denoted by dim(G), is the minimum cardinality over all resolving sets of G. A resolving set of cardinality dim(G) is called a basis. For two vertices u, v ∈ V (G), the interval IG[u, v] between u and v is the collection of all vertices that belong to some shortest u-v path. A vertex w strongly resolves two vertices u and v if v ∈ IG[u,w] or if u ∈ IG[v, w]. A set W of vertices in G is a strong A. H. Abragan, H. M. Rara / Eur. J. Pure Appl. Math, 16 (2) (2023), 763-772 765 resolving set of G if every two vertices of G are strongly resolved by some vertex of W . The smallest cardinality of a strong resolving set of G is called the strong metric dimension of G and is denoted by sdim(G). A strong resolving set of cardinality sdim(G) is called a strong metric basis of G. A subset S ⊆ V (G) is a strong resolving hop dominating set of G if S is both a strong resolving set and a hop dominating set. The minimum cardinality of a strong resolving hop dominating set of G, denoted by γsRh(G), is called the strong resolving hop domination number of G. Any resolving hop dominating set with cardinality equal to γsRh(G) is called a γsRh-set. A strong resolving hop dominating set S is a 1-movable strong resolving hop dominating set of G if for every v ∈ S, either S \ {v} is a strong resolving hop dominating set or there exists a vertex u ∈ (V (G)\S)∩NG(v) such that (S \{v})∩{u} is a strong resolving hop dominating set of G. The minimum cardinality of a 1-movable strong resolving hop dominating set of G is denoted by γ1msRh(G). 2. Some Known Results The following known results are taken from [5]. Theorem 1. Let G and H be nontrivial connected graphs of orders m and n, respectively. A proper subset S of V (G +H) is a strong resolving set of G +H if and only if at least one of the following is satisfied: (i) S = V (G+H) \ CG where CG is a superclique in G. (ii) S = V (G+H) \ CH where CH is a superclique in H. (iii) If γ(G) ̸= 1 or γ(H) ̸= 1, S = V (G+H) \ (CG ∪ CH) = (V(G) \ CG) ∪ (V (H) \ CH), where CG and CH are supercliques in G and H respectively. Lemma 1. Let G be a nontrivial connected graph with diam(G) ≤ 2. Then S = V (G)\C is a strong resolving set of G if and only if C = ∅ or C is a superclique in G. In particular, sdim(G) = |V (G)| − ωS(G). Theorem 2. Let G be a nontrivial connected graph and H a connected graph. A proper subset S of V (G ◦H) is a strong resolving set of G ◦H if and only if one of the following holds: (i) S = A ∪ ( ∪ u∈V (G) V (Hu)) where A ⊆ V (G). A. H. Abragan, H. M. Rara / Eur. J. Pure Appl. Math, 16 (2) (2023), 763-772 766 (ii) S = ∪( ∪ u∈V (G)\{v} V (Hu)) ∪Bv for a unique v in V (G), where A ⊆ V (G) and Bv is a strong resolving set of Hv if γ(H) = 1 or Bv is a resolving set of {v}+Hv if γ(H) ̸= 1. Remark 1. Any superset of a strong resolving set is a strong resolving set. Theorem 3. Let G = Kn for n > 1 and H a nontrivial connected graph with γ(H) ̸= 1 . A subset S of V (G[H]) is a strong resolving set of G[H] if and only S = V (G[H])\(A×C), where A is a subset of V (G) and C = ∅ or C is a superclique in H. 3. Preliminary Results This section introduces the 1-movable strong resolving hop domination in some graphs. It also characterizes some graphs in terms of its 1-movable strong resolving hop domination number. Remark 2. Every 1-movable strong resolving hop dominating set of a connected graph G is a strong resolving hop dominating set in G. Hence, γsRh(G) ≤ γ1msRh(G). Remark 3. The converse of Remark 2 does not hold. To see this, the set S = {v1, v2, v3} of the path P4 = [v1, v2, v3, v4] is a strong resolving dominating set of P4 but it is not a 1-movable strong resolving hop dominating set since S \ {v1} is not a strong resolving set of P4. Proposition 1. Any superset of a 1-movable strong resolving hop dominating set is a 1-movable strong resolving dominating set. Proof : Let S be a 1-movable strong resolving hop dominating set of G and S ⊆ S′. Then S is a strong resolving set. By Remark 1, S′ is a strong resolving set of G. We show that S′ is a 1-movable strong resolving hop dominating set of G. Let x ∈ S′. If x ∈ S then S \ {x} ⊆ S′ \ {x}. Since S is a 1-movable strong resolving hop dominating set of G either S \ {x} is strong resolving hop dominating set of G or ∃y ∈ (V (G) \ S) ∩NG(x) such that (S \ {x}) ∪ {y} is a strong resolving hop dominating set of G. If S \ {x} is a strong resolving hop dominating set of G, then S′ \ {x} is also a strong resolving set of G by Remark 1. If there exists y ∈ (V (G) \ S)∩NG(x) such that (S \ {x})∪ {y} is a strong resolving hop dominating set of G, then (S \ {x}) ∪ {y} ⊆ (S′ \ {x}) ∪ {y}. It follows that (S′ \ {x}) ∪ {y} is strong resolving set of G. It can be verified that every superset of hop dominating set is hop dominating. Therefore, S′ is a 1-movable strong resolving hop dominating set of G. Proposition 2. Let Pn = [v1, v2, . . . , vn] where n ≥ 1. If a set S ⊆ V (Pn) is a 1-movable strong resolving hop dominating set of Pn, then S contains the vertices v1 and vn. A. H. Abragan, H. M. Rara / Eur. J. Pure Appl. Math, 16 (2) (2023), 763-772 767 Proof : Suppose S is a 1-movable strong resolving hop dominating set of Pn and suppose that S does not contain v1 or vn, say v1. Since v1MMDvn, S ∩ {v1, vn} ̸= ∅. Hence, vn ∈ S. This implies that S \ {vn} and (S \ {vn}) ∪ {vn−1} are not strong resolving sets of Pn, a contradiction. Therefore, S contains v1 and vn. Proposition 3. Let G be a nontrival connected graph with diam(G) ≤ 2 and γ(G) ̸= 1. Then S = V (G) \ C is a 1-movable strong resolving hop dominating set of G if and only if C = ∅ or C is a hop dominated superclique in G and either for each x ∈ S, C ∪ {x} is a hop dominated superclique or there exists y ∈ [C ∩NG(x)] such that (C \ {y}) ∪ {x} is a hop dominated superclique in G. Proof : Suppose S = V (G) \ C is a 1-movable strong resolving hop dominating set of G. Then S is strong resolving set in G. By Lemma 1, C = ∅ or C is a dominated superclique in G. We claim that C is a hop dominated superclique. Let z ∈ C. Then z /∈ S. Since S is hop dominating, there exists y ∈ (S \C) such that dG(z, y) = 2. Hence, C is a hop dominated superclique. Let x ∈ S. Since S is a 1-movable strong resolving hop dominating set, either S \ {x} is strong resolving hop dominating or there exists y ∈ [(V (G) \ S) ∩NG(x)] such that (S \ {x}) ∪ {y} is a strong resolving hop dominating set of G. Since S \ {x} = V (G) \ (C ∪ {x}) and (S \ {x})∪ {y} = [V (G) \C \ {x}]∪ {y}, C ∪ {x} is a hop dominated superclique or (C \ y) ∪ {x} is a hop dominated superclique in G. For the converse, suppose C = ∅. Then S = V (G) is a strong resolving hop dominating set of G. Thus, S \ {x} = V (G) \ {x} is a strong resolving hop dominating since {x} is a superclique for each x ∈ V (G). Since γ(G) ̸= 1, a vertex y ∈ S \ {x} exists such that dG(x, y) = 2. Hence, S is a hop dominating. So, suppose C is a hop dominated superclique in G and for each x ∈ S either C ∪ {x} is a hop dominated superclique or there exists y ∈ [C ∩NG(x)] such that (C \{y})∪{x} is a hop dominated superclique. Hence, for each x ∈ S. (S \ {x}) ∪ {y} = [V (G) \ (C \ {y})] ∪ {x} is a strong resolving hop dominating set of G. Therefore, S is a 1-movable strong resolving hop dominating set of G. 4. Join of Graphs Theorem 4. Let G be a connected graph of order n and γ(G) = 1. Then a 1-movable strong resolving hop dominating set of G does not exist. Proof : Suppose G has a 1-movable strong resolving hop dominating set S. Let D = {x ∈ V (G) : degG(x) = n − 1}. Since S is hop dominating, D ⊆ S. Let x ∈ D. Then S \{x} is not hop dominating and for each y ∈ (V (G)\S)∩NG(x), (S \{x})∪{y} is also not hop dominating. Hence, S is not a 1-movable strong resolving hop dominating. As a consequence of Theorem 4 the next result follows. A. H. Abragan, H. M. Rara / Eur. J. Pure Appl. Math, 16 (2) (2023), 763-772 768 Corollary 1. Let G be a graph, then the 1-movable strong resolving hop dominating set of K1 +G does not exist. Theorem 5. Let G and H be graphs where γ(G) ̸= 1 and γ(H) ̸= 1. A proper subset S of V (G +H) is a 1-movable strong resolving hop dominating set of G +H if and only if at least one of the following is satisfied. (i) S = V (G+H) \ CG where CG and CG ∪ {x} or (CG ∪ {x}) \ {y} are point-wise non-dominated superclique in G for each x ∈ S. (ii) S = V (G+H) \ CH where CH and CH ∪ {z} or (CH ∪ {z}) \ {w} are point-wise non-dominated superclique in H for each z ∈ S and w ∈ (V (H) \ SH) ∩NH(z). (iii) S = V (G+H) \ (CH ∪ CG) where CH , CG, CH ∪ {x}, (CH ∪ {x}) \ {y}, CG ∪ {z}, (CG ∪ {z}) \ {w} are point-wise non-dominated supercliques in H and G, respectively for all x ∈ SH . Proof : Suppose S ⊆ V (G+H) is a 1-movable strong resolving hop dominating set of G. Then S is a strong resolving set of G+H. By Theorem 1, at least one of the following is satisfied: (a) S = V (G+H) \ CG where CG is a superclique in G. (b) S = V (G+H) \ CH where CH is a superclique in H. (c) If γ(G) ̸= 1 or γ(H) ̸= 1, S = V (G+H) \ (CG ∪ CH) = (V(G) \ CG) ∪ (V (H) \ CH). We claim that CG is a point-wise non-dominated set of G. Let x ∈ CG. Since S is hop dominating and x ∈ V (G + H) \ S, there exists y ∈ S such that dG+H(x, y) = 2. By Definition of point-wise non-dominated superclique, y ∈ V (G)\CG and y /∈ NG(x). Thus, CG is a point-wise non-dominated superclique of G. Let x ∈ S. Since S is a 1-movable strong resolving hop dominating set of G + H, either S \ {x} or (S \ {x}) ∪ {y} is a strong resolving hop dominating set of G + H where y ∈ [V (G + H) \ S] ∩ NG+H(x) . Since S = V (G + H) \ CG, S \ {x} = V (G + H) \ (CG ∪ {x}) and (S \ {x}) ∪ {y} = [V (G+H)] \ [(CG ∪ {x}) \ {y}] by Lemma 1, CG ∪ {x} or (CG ∪ {x}) \ {y} is a superclique. By similar argument above, CG∪{x} or (CG ∪ {x})\{y} is a point-wise non-dominated superclique in G. This proves (i). Statements (i) and (iii) are proved similarly. For the converse, suppose (i) holds. By Theorem 1, S is a strong resolving set. Let u ∈ V (G+H) \ S. Then u ∈ CG. Since CG is a point-wise non-dominated superclique of G, there exists v ∈ V (G)\CG such that v /∈ NG(u). Hence, v ∈ S and dG+H(u, v) = 2. Let x ∈ S. Since S \ {x} = V (G+H) \ (CG ∪ {x}), by (i) of Theorem 1 and the Definition of point-wise non-dominated superclique, S is a 1-movable strong resolving hop dominating A. H. Abragan, H. M. Rara / Eur. J. Pure Appl. Math, 16 (2) (2023), 763-772 769 set of G + H. Similarly if (ii) and (iii) holds, S is a 1-movable strong resolving hop dominating set of G+H. The next result follows from Theorem 5. Corollary 2. Let G andH be nontrivial connected graphs of ordersm and n, respectively. Then, γ1msRh(G+H) = m− ωpndS}(G) + n− ωpndS(H). 5. Corona of Graphs This section gives characterization of the 1-movable strong resolving hop dominating sets in the corona of graphs as well as its 1-movable strong resolving hop domination number. Theorem 6. Let G be a nontrivial connected graph and H a connected graph with γ(H) ̸= 1 . A proper subset S of V (G◦H) is a 1-movable strong resolving hop dominating set of G ◦H if and only if S = A ⋃( ⋃ u∈V (G) V (Hu) ) where A ⊆ V (G). Proof : Suppose that a proper subset S of V (G ◦H) is a 1-movable strong resolving hop dominating set of G ◦H. Since S is strong resolving set of G ◦H, (i) or (ii) of Theorem 2 holds. If (i) holds, then S = A ⋃( ⋃ u∈V (G) V (Hu) ) , where A ⊆ V (G). Suppose (ii) holds. Let x ∈ V (Hw) for some w ∈ V (G) with w ̸= v. Then S \ {x} = A ⋃( ⋃ u∈V (G)\{w,v} V (Hu) )⋃( V (Hw) \ {x} )⋃ Bv is not a strong resolving set by Theorem 2. Hence, S = A ⋃( ⋃ u∈V (G) V (Hu) ) where A ⊆ V (G). For the converse, suppose S = A ⋃( ⋃ u∈V (G) V (Hu) ) where A ⊆ V (G). By Theorem 2, S is a strong resolving set of G ◦H. It can be seen that S is a hop dominating set also. Let p ∈ S. If p ∈ A , then S \ {p} = (A \ {p}) ∪ ( ⋃ u∈V (G) V (Hu) ) is a strong resolving hop dominating set. If p ∈ V (Hu) for each u ∈ V (G), then S \ {p} = A∪ ( ⋃ u∈V (G) V (Hu)\{p} ) ∪ ( ⋃ v∈V (G)\{u} V (Hv) ) is a strong resolving set by Theorem 2 and hop dominating since γ(H) ̸= 1. A. H. Abragan, H. M. Rara / Eur. J. Pure Appl. Math, 16 (2) (2023), 763-772 770 Accordingly S is a 1-movable strong resolving hop dominating set in G◦H. Corollary 3. Let G be a connected graph of order m > 1 and H be any graph of order n with γ(H) ̸= 1. Then γ1msRh(G ◦H) = mn. Proof : Let S be a γ1msRh-set of G ◦H. Then by Theorem 6, S = A ⋃( ⋃ u∈V (G) V (Hu) ) where A ⊆ V (G). Thus, γ1msRh(G ◦H) = |S| = |A|+ ∣∣∣∣ ⋃ u∈V (G) V (Hu) ∣∣∣∣ ≥ |V (G)||V (H)| = mn. Let A = ∅. Then S∗ = A ⋃( ⋃ u∈V (G) V (Hu) ) is a 1-movable strong resolving hop dominating set of G ◦H by Theorem 6. Hence, γ1msRh(G ◦H) ≤ |S∗| = ∣∣∣∣ ⋃ u∈V (G) V (Hu) ∣∣∣∣ = mn. Therefore, γ1msRh(G ◦H) = mn. Example 1. For the graph of P3 ◦P4, the minimum 1-movable strong resolving hop dom- inating set is γ1msRh(P3 ◦ P4) = 3(4) = 12. 6. Lexicographic of Graphs This section gives characterization of a 1-movable strong resolving hop dominating sets in the lexicographic product of graphs as well as its 1-movable strong resolving hop domination number. Theorem 7. Let G = Kn for n > 1 and H is a connected graph with γ(H) ̸= 1. A subset S of V (G[H]) is a 1-movable strong resolving hop dominating set of G[H] if and only if S = V (G[H]) \ (A× C), where A is a subset of V (G) and C = ∅. REFERENCES 771 Proof : Suppose S is a 1-movable strong resolving hop dominating set G[H]. By Theorem 3, S = V (G[H]) \ (A × C) where A ⊆ V (G) and C = ∅ or C is a superclique in H. Suppose C ̸= ∅ and C is a superclique in H. Let (x, y) ∈ S. Then y /∈ C. Hence, S \ {x, y} = V (G[H]) \ ((A × C) ∪ {x, y}). Since γ(H) ̸= 1, C ∪ {y} is not a superclique in H. Therefore A ⊆ V (G)and C = ∅. The converse follows immediately from Theorem 3. As a consequence of Theorem 7 the next result follows. Corollary 4. Let G = Kn for n > 1 and H is a connected graph of order m and γ(H) ̸= 1. Then γ1msRh(G[H]) = mn. Example 2. The sets of shaded vertices in K4[P4] and K4[P3] in Figure 1 represent 1- movable strong resolving hop dominating sets. ......... ........ ........ ........ ........ ........ ........ ..... ............ ........... ........... ........... ........... ........... ........... ........... .. ................... .................. .................. .................. .................. .................. .................. .................. ........................................................................................................ ........ ........ ........ ........ ........ ........ ...... ............ ........... ........... ........... ........... ........... ........... ........... ........................................................................................................................................................ ........................................................................................... 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................................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................... • • • • • • • • • • • K4[P3] Figure 1: A 1-movable strong resolving hop dominating sets of K4[P4] and K4[P3] References [1] G. Chartrand, L. Eroh, M. Johnson, and O.R. Oellermann. Resolvability in graphs and the metric dimension of a graph the metric dimension of a graph. Discrete Applied Mathematics, 105:99–113, 2000. [2] F. Harary and R.A. Melter. On the metric dimension of a graph. Ars Combinatoria, 2:191–195, 1976. [3] Blair J., Gera R., and Horton S. Movable dominating sets in networks. Journal of Combinatorial Mathematics and Combinatorial Computing, (77):102–123, 2011. [4] Renario G. Hinampas Jr. and Sergio R. Canoy Jr. 1-movable domination in graphs. Applied Mathematical Sciences, 8(172):8565 – 8571, 2014. [5] Gerald Bacon Monsanto, Penelyn L. Acal, and Helen M. Rara. On strong resolving domination in the join and corona of graphs. European Journal of Pure and Applied Mathematics, 13(1):170–179, Jan. 2020. REFERENCES 772 [6] Chidambaram Natarajan and Ayyaswamy S.K. Hop domination in graphs-ii. Analele Stiintifice ale Universitatii Ovidius Constanta, Seria Matematica, 23:187–199, 06 2015. [7] Canoy Jr Sergio, Mollejon Reynaldo Villarobe, and Canoy John Gabriel E. Hop dom- inating sets in graphs under binary operations. European Journal of Pure and Applied Mathematics, 12(4):1455–1463, Oct. 2019.