EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 15, No. 3, 2022, 1047-1053 ISSN 1307-5543 – ejpam.com Published by New York Business Global Restrained 2-Resolving Dominating Sets in the Join, Corona and Lexicographic Product of two Graphs Jean Cabaro1,∗, Helen Rara2 1 Mathematics Department, College of Natural Sciences and Mathematics, Mindanao State University-Main Campus, 9700 Marawi City, Philippines 2 Department of Mathematics and Statistics, College of Science and Mathematics, Center of Graph Theory, Algebra, and Analysis-Premier Research Institute of Science and Mathematics, Mindanao State University-Iligan Institute of Technology, 9200 Iligan City, Philippines Abstract. Let G be a connected graph. An ordered set of vertices {v1, ..., vl} is a 2-resolving set for G if, for any distinct vertices u,w ∈ V (G), the lists of distances (dG(u, v1), ..., dG(u, vl)) and (dG(w, v1), ..., dG(w, vl)) differ in at least 2 positions. A set S ⊆ V (G) is a restrained 2-resolving dominating set in G if S is a 2-resolving dominating set in G and S = V (G) or ⟨V (G)\S⟩ has no isolated vertex. The restrained 2R-domination number of G, denoted by γr2R(G), is the smallest cardinality of a restrained 2-resolving dominating set in G. Any restrained 2-resolving dominating set of cardinality γr2R(G) is referred to as a γr2R-set in G. This study deals with the concept of restrained 2-resolving dominating set of a graph. It characterizes the restrained 2-resolving dominating set in the join, corona and lexicographic product of two graphs and determine the bounds or exact values of the restrained 2-resolving domination number of these graphs. 2020 Mathematics Subject Classifications: 05C62 Key Words and Phrases: Restrained 2-resolving set, restrained 2-resolving dominating set, join, corona, lexicographic product of two graphs 1. Introduction The problem of uniquely determining the location of an intruder in a network was the principal motivation of introducing the concept of metric dimension in graphs by Slater [9], where the metric generators were called locating sets. The concept of metric dimension of a graph was also introduced independently by Harary and Melter in [4] where metric generators were called resolving sets. Bailey and Yero in [1] demonstrated a construction of error-correcting codes from graphs by means of k-resolving sets, and present a decoding algorithm which makes use ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v15i3.4451 Email addresses: amerjean1228@gmail.com (J. Cabaro), helenrara@gmail.com (H. Rara) https://www.ejpam.com 1047 © 2022 EJPAM All rights reserved. J. Cabaro, H. Rara / Eur. J. Pure Appl. Math, 15 (3) (2022), 1047-1053 1048 of covering designs. The distance between two vertices u and v of a graph is the length of a shortest path between u and v, and we denote this by dG(u, v). In recent years, much attention has been paid to the metric dimension of graphs: this is the smallest size of a subset of vertices (called a resolving set) with the property that the list of distances from any vertex to those in the set uniquely identifies that vertex and is denoted by dim(G). According to the paper of Saenpholphat et al. [8], for an ordered set of vertices W = {w1, w2, ..., wk} ⊆ V (G) and a vertex v in G, the k-vector (ordered k-tuple) r(v/W ) = (dG(v, w1), dG(v, w2), ..., dG(v, wk)) is referred to 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 representation with respect to W . Hence, if W is a resolving set of cardinality k for a graph G of order n, then the set {r(v/W ) : v ∈ V (G)} consists of n distinct k-vectors. A resolving set of minimum cardi- nality is called a minimum resolving set or a basis, and the cardinality of a basis for G is the dimension dim(G) of G. In the paper of Rara and Cabaro [5], an ordered set of vertices W = {w1, ..., wl} is a 2-resolving set for G if, for any distinct vertices u, v ∈ V (G), the (metric) representations r(u/W ) and r(v/W ) of u and v, respectively differ in at least 2 positions. Then W is said to be a 2-resolving set for G. If G has a 2-resolving set, the minimum cardinality dim2(G) is called the 2-metric dimension of G. If k = 2 is the largest integer for which G has a 2-resolving set, then we say that G is a 2-metric dimensional graph. In the paper of Cabaro and Rara [6], the concept of restrained 2-resolving set in the join, corona and lexicographic product of two graphs was discussed. In this paper, the concept of restrained 2-resolving dominating set in the join, corona and lexicographic product of two graphs is discussed. 2. Preliminary Results In this study, we consider finite, simple and connected undirected graphs. For basic graph-theoretic concepts, we refer readers to [3]. Proposition 1. [2] Let G be a connected graph of order n ≥ 2. Then dim2(G) = 2 if and only if G ∼= Pn. Remark 1. For any connected graph G of order n ≥ 2, γ2R(G) ≤ γr2R(G). Remark 2. For n ≥ 2, γ2R(Kn) = n = γr2R(Kn). Remark 3. Let G be a connected graph of order n ≥ 2. Then γr2R(G) = n if and only if G ∼= Kn or G ∼= K1,n−1 or G ∼= C4. The following remark follows from Proposition 1. J. Cabaro, H. Rara / Eur. J. Pure Appl. Math, 15 (3) (2022), 1047-1053 1049 Remark 4. Let G be a connected graph of order n ≥ 2. Then γr2R(G) = 2 if and only if G ∼= K2 or G = P4. Proof. Proof follows immediately from Proposition 1. 3. Restrained 2-Resolving Dominating Sets in the Join of Graphs Theorem 1. Let G and H be nontrivial connected 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 are 2-locating sets in G and H, respectively where SG or SH is a (2, 2)-locating set or SG and SH are (2, 1)-locating sets 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 2. [7] Let G be a connected non-trivial graph and let K1 = {v}. Then S ⊆ V (K1 + G) is a restrained 2-resolving set of K1 + G if and only if either v /∈ S and S is a (2, 2)-locating set in G with V (G) ̸= S or S = {v} ∪ T , where T is a restrained (2, 1)-locating set in G. Theorem 3. Let G andH be nontrivial connected graphs. A proper subset S of V (G+H) is a restrained 2-resolving dominating set inG+H if and only if S is a restrained 2-resolving set in G+H. Proof. Let S ⊆ V (G+H) be a restrained 2-resolving dominating set in G+H. Then S is a restrained 2-resolving set in G+H. Conversely, if S is a restrained 2-resolving set in G+H, then by Theorem 1, conditions (i), (ii) and (iii) hold. Since S is a 2-resolving set, SG ̸= ∅ and SH ̸= ∅. Thus, S = SG∪SH is a dominating set in G +H. Therefore, S is a restrained 2-resolving dominating set in G+H. Corollary 1. Let G and H be a connected non-trivial graphs of order m and n, respec- tively. Then γr2R(G+H) = rdim2(G+H). The set consisting of the shaded vertices in Figure 1 is a restrained 2-resolving domi- nating set of the join P5 + P6. J. Cabaro, H. Rara / Eur. J. Pure Appl. Math, 15 (3) (2022), 1047-1053 1050 .................................... .................................... .................................... .................................... .................................... .................................... .................................... .................................... .................................... .................................... .................................... ......... ........ ........ ........ ........ ........ ........ ........ ........ ......... ........ ........ ........ ........ ........ ........ ........ ........ ......... ........ ........ ........ ........ ........ ........ ........ ........ ......... ........ ........ ........ ........ ........ ........ ........ ........ .......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... 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P5 + P6: • • • • • • • Figure 1: A graph P5 + P6 with γr2R(P5 + P6) = 7 Theorem 4. Let G be a connected non-trivial graph and let K1 = {v}. Then S ⊆ V (K1 + G) is a restrained 2-resolving dominating set in K1 + G if and only if it is a restrained 2-resolving set in K1 +G. Proof. Suppose S is a restrained 2-resolving dominating set in K1 + G. Then S is a restrained 2-resolving in K1 +G. Conversely, if S is a restrained 2-resolving set in K1 + G, then by Theorem 2 either v ∈ S and S is a (2, 2)-locating set in G with V (G) ̸= S or S = {v} ∪ T , where T is a restrained (2, 1)-locating set in G. Hence, S is a restrained 2-resolving dominating set in K1 +G. Corollary 2. Let G be a connected nontrivial graph of order m. Then γr2R(K1 +G) = { 1 +m, if ln(2,2)(G) = m and rln(2,1)(G) = m min { ln(2,2)(G), rln(2,1)(G) + 1 } , otherwise . 4. Restrained 2-Resolving Dominating Sets in the Corona of Graphs Theorem 5. [7] Let G and H be nontrivial connected graphs. A set S ⊆ V (G ◦ H) is a restrained 2-resolving set in G ◦ H if and only if S = A ∪ ( ⋃ v∈V (G) Sv) satisfying the following conditions. (i) A ⊆ V (G) (ii) Sv is a 2-resolving set for each v ∈ V (G)\A (iii) Sv is a restrained 2-resolving set for each v ∈ A (iv) w ∈ NG(V (G)\A) for each w ∈ V (G)\A with Sw = V (Hw). J. Cabaro, H. Rara / Eur. J. Pure Appl. Math, 15 (3) (2022), 1047-1053 1051 Theorem 6. Let G and H be nontrivial connected graphs. Then S ⊆ V (G ◦ H) is a restrained 2-resolving dominating set in G ◦ H if and only if S = A ∪ ( ⋃ v∈A Sv)( ⋃ w∈V (G)\A Sw) satisfying the following conditions. (i) A ⊆ V (G) (ii) Sw is a 2-resolving dominating set in Hw for each w ∈ V (G)\A (iii) Sv is a restrained 2-resolving set in Hv for each v ∈ A (iv) w ∈ NG(V (G)\A) for each w ∈ V (G)\A with Sw = V (Hw). Proof. Suppose S is a restrained 2-resolving dominating set in G ◦H. By Theorem 5, conditions (i), (ii), (iii) and (iv) hold. Conversely, suppose S = A∪( ⋃ v∈A Sv)( ⋃ w∈V (G)\A Sw) satisfying the conditions (i),(ii),(iii) and (iv). Then by Theorem 5, S is a restrained 2-resolving set inG◦H. Let x ∈ V (G◦H)\S and let w ∈ V (G) such that x ∈ V (w +Hw). If w ∈ S, then xw ∈ E(G ◦H). If w /∈ S, then x /∈ Sw = S ∩ V (Hw), where Sw is a 2-resolving dominating set in Hw. Thus, there exists z ∈ V (Hw) ∩ Sw such that xz ∈ E(G ◦ H). Therefore, S is a dominating set in G ◦H. Thus, S is a restrained 2-resolving dominating set in G ◦H. Corollary 3. Let G and H be nontrivial connected graphs, where |V (G)| = n. Then γr2R(G ◦H) ≤ min {|V (G)| · γ2R(H), |V (G)|(1 + rdim2(H))} . The set consisting of the shaded vertices in Figure 2 is a restrained 2-resolving domi- nating set of the corona P4 ◦ C5. .................................... .................................... .................................... .................................... .................................... .................................... .................................... .................................... .................................... .................................... .................................... .................................... .................................... .................................... .................................... .................................... .................................... .................................... .................................... ........................................................................ .................................... .................................... 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............................................................................................................................................. ............................................................................................. ............................................................................................. ............................................................................................. ............................................................................................. • •• • •• • • • •• • Figure 2: A graph P4 ◦ C5 with γr2R(P4 ◦ C5) = 12 J. Cabaro, H. Rara / Eur. J. Pure Appl. Math, 15 (3) (2022), 1047-1053 1052 5. Restrained 2-Resolving Dominating Sets in the Lexicographic Product of Graphs Theorem 7. [7] Let G and H be non-trivial 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 set in G[H] if and only if (i) S = V (G) (ii) Tx is a 2-locating set in H for all x ∈ V (G); (iii) Tx is a restrained 2-locating set for each x with Ty = V (H), for all y ∈ NG(x); (iv) Tx and Ty are (2, 1)-locating sets or one of Tx and Ty is a (2, 2)-locating set in H whenever x, y ∈ EQ1(G); and (v) Tx and Ty are (2-locating) dominating sets in H or if one of Tx and Ty, say Tx is not dominating, then Ty is 2-dominating whenever x, y ∈ EQ2(G). Theorem 8. Let G and H be non-trivial 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 dominating set in G[H] if and only if it is a restrained 2-resolving set in G[H]. Proof. Suppose W = ⋃ x∈S [ {x} × Tx ] where S ⊆ V (G) and Tx ⊆ V (H) is a restrained 2-resolving dominating set in G[H]. Then W is a restrained 2-resolving set in G[H]. For the converse, suppose that W is a restrained 2-resolving set in G[H]. Then by Theorem 7, (i)-(v) hold. Since W is a 2-resolving set, Tx ̸= ∅ for every x ∈ V (G). Thus, W is a dominating set in G[H]. Therefore, W is a restrained 2-resolving dominating set in G[H]. The following are the direct consequences of Theorem 8. Corollary 4. Let G and H be nontrivial connected graphs such that G is not free- equidistant. Then, γr2R(G[H]) = rdim2(G[H]). The following result follows from Theorem 8. Corollary 5. LetG andH be non-trivial connected graphs such thatG is free-equidistant. Then γr2R(G[H]) = rdim2(G[H]). The set consisting of the shaded vertices in Figure 3 is a restrained 2-resolving domi- nating set of the lexicographic product P4[P3]. REFERENCES 1053 .................................... .................................... .................................... .................................... .................................... .................................... .................................... .................................... .................................... .................................... .................................... .................................... ............................................................................................. ............................................................................................. ............................................................................................. ............................................................................................. ............................................................................................. ............................................................................................. ............................................................................................. ............................................................................................. ......... ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ .... ......... ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ .... ......... ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ .... ......... ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ .... ......... ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ .... ......... ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ .... ......... ........ ........ ........ ........ ........ ........ ........ ........ 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............................................................................................................................................................................................................................. ................... .................. .................. .................. .................. .................. .................. .................. .................. .................. .................. .................. .... ............ ........... ........... ........... ........... ........... ........... ........... ........... ........... ........... ........... ... ...................................................................................................................................................................................................... ........................................................................................................................................ ........................................................................................................................................ ........................................................................................................................................ ............ ........... ........... ........... ........... ........... ........... ........... ........... ........... ........... ........... ... ................... .................. .................. .................. .................. .................. .................. .................. .................. .................. .................. .................. .... ............ ........... ........... ........... ........... ........... ........... ........... ........... ........... ........... ........... ... • • • • • • • • Figure 3: A graph P4[P3] with γr2R(P4[P3]) = 8 Acknowledgements The authors would like to thank the Commission on Higher Education (CHED) and Mindanao State University-Marawi and MSU-Iligan Institute of Technology, Philippines. References [1] R Bailey and I Yero. Error-correcting codes from k-resolving sets. Discussiones Math- ematicae, Graph Theory, 39:341–355, 2019. [2] J Estrada-Moreno, A Rodriguez-Velasquez and I Yero. The k-metric dimension of a graph. Applied Math Information Science., 9:2829–2840, 2015. [3] F Harary. Graph Theory. Addison-Wesley Publishing Company, USA, 1969. [4] F Harary and R Melter. On the metric dimension of a graph. Ars Combinatoria, 2, 1976. [5] H Rara and J Cabaro. On 2-resolving sets in the join and corona of graphs. European journal of pure and applied mathematics, 14:773–782, 2021. [6] H Rara and J Cabaro. Restrained 2-resolving sets in the join, corona and lexicographic product of two graphs. European journal of pure and applied mathematics, Accepted:To publish, 2022. [7] H Rara and J Cabaro. Restrained 2-resolving sets in the join, corona and lexicographic product of two graphs. European journal of pure and applied mathematics, Accepted:To publish, 2022. [8] V Saenpholphat and P Zang. On connected resolvability of graphs. Australian Journal of Combinatorics, 28:25–37, 2003. [9] P. Slater. Congressus Numerantium,., 14:549–559, 1975.