EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 15, No. 3, 2022, 1229-1236 ISSN 1307-5543 – ejpam.com Published by New York Business Global Restrained 2-Resolving 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 set in G if S is a 2-resolving set in G and S = V (G) or ⟨V (G)\S⟩ has no isolated vertex. The restrained 2-resolving number of G, denoted by rdim2(G), is the smallest cardinality of a restrained 2-resolving set in G. A restrained 2-resolving set of cardinality rdim2(G) is then referred to as an rdim2-set in G. This study deals with the concept of restrained 2-resolving set of a graph. It characterizes the restrained 2-resolving set in the join, corona and lexicographic product of two graphs and determine the bounds or exact values of the 2-resolving dominating number of these graphs. 2020 Mathematics Subject Classifications: 05C69 Key Words and Phrases: 2-resolving set, restrained 2-resolving set, restrained 2-resolving num- ber, 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 [8], 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 of ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v15i3.4427 Email addresses: amerjean1228@gmail.com (J. Cabaro), helenrara@gmail.com (H. Rara) https://www.ejpam.com 1229 © 2022 EJPAM All rights reserved. J. Cabaro, H. Rara / Eur. J. Pure Appl. Math, 15 (3) (2022), 1229-1236 1230 covering designs. In [6], the explicit interpretation for F-index of different forms of corona products involving Zagreb indices, graph size and order are obtained. 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. [7], 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 this paper, the concept of restrained 2-resolving set in the join, corona and lexico- graphic 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]. Remark 1. Let G be a connected graph. Then every restrained 2-resolving set in G is 2-resolving. Hence, dim2(G) ≤ rdim2(G). Proposition 1. [2] Let G be a connected graph of order n ≥ 2. Then dim2(G) = 2 if and only if G ∼= Pn. Proposition 2. dim2(Kn) = n for n ≥ 2. Proposition 3. Let G be any connected graph of order n ≥ 2. i. rdim2(G) = 2 if and only if G ∼= Pn, n ̸= 3. ii. rdim2(Kn) = n. J. Cabaro, H. Rara / Eur. J. Pure Appl. Math, 15 (3) (2022), 1229-1236 1231 Proof. i. Suppose rdim2(G) = 2. By Remark 1, dim2(G) = 2. Hence, by Proposition 1, G = Pn. Since rdim2(P3) = 3, G = Pn except n = 3. Conversely, if G = Pn = [v1, v2, ..., vn], then S = {v1, vn} is a restrained 2-resolving set of G. Hence, rdim2(G) = 2. ii. By Proposition 2, S = V (Kn) is the only 2-resolving set of Kn. Thus, rdim2(Kn) = n. 3. Restrained 2-Resolving Sets in the Join of Graphs Definition 1. Let G 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| ≥ 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. 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 ofG, 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. Theorem 1. Let G andH be nontrivial connected graphs. A proper subset S of V (G+H) is a 2-resolving set in G + H if and only if SG = V (G) ∩ S and SH = V (H) ∩ S are 2-locating sets in G and H, respectively, where SG or SH is a (2, 2)-locating set or SG and SH are (2, 1)-locating sets. Theorem 2. 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). J. Cabaro, H. Rara / Eur. J. Pure Appl. Math, 15 (3) (2022), 1229-1236 1232 Proof. Let S ⊆ V (G+H) be a restrained 2-resolving set in G+H. Then by Theorem 1, 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. To show that (i), (ii), and (iii) hold we consider the following cases: Case 1. SG = V (G). Suppose that SH ̸= V (H). Since ⟨V (G+H)\S⟩ = ⟨V (H)\SH⟩ and ⟨V (G+H)\S⟩ has no isolated vertex, it follows that ⟨V (H)\SH⟩ has no isolated vertex. Thus, SH is a restrained 2-locating set in H. Hence, (i) holds. Case 2. Suppose that SG ̸= V (G). If SH ̸= V (H), then (iii) holds. Suppose that SH = V (H). Then ⟨V (G+H)\S⟩ = ⟨V (G)\SG⟩ has no isolated vertex. Hence, SG is a restrained 2-locating set in G. Thus, (ii) holds. For the converse, suppose that SG and SH 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. Then by Theorem 1, S is a 2-resolving set in G + H. Suppose that SG = V (G). If SH = V (H), then S = V (G + H) is a restrained 2-resolving set in G + H. If SH ̸= V (H), then by (i) ⟨V (H)\SH⟩ has no isolated vertex. Since V (G+H)\S = V (H)\SH , S is a restrained 2-resolving set in G +H. Similarly, if (ii) holds, then S is a restrained 2-resolving set in G+H. Finally, suppose that SG ̸= V (G) and SH ̸= V (H). Then clearly, S is a restrained 2-resolving set in G+H. Corollary 1. Let G and H be connected non-trivial graphs of order m and n, respectively. Then rdim2(G+H) =  m+ n, if rln2(G) = m and rln2(H) = n min{rln(2,2)(G) + rln2(H), rln2(G) + rln(2,2)(H), rln(2,1)(G) + rln(2,1)(H)}, otherwise The set consisting of the shaded vertices in Figure 1 is a restrained 2-resolving set of the join P5 + P6. Theorem 3. 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. Corollary 2. Let G be a connected nontrivial graph of order m. Then rdim2(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. J. Cabaro, H. Rara / Eur. J. Pure Appl. Math, 15 (3) (2022), 1229-1236 1233 .................................... .................................... .................................... .................................... .................................... .................................... .................................... .................................... .................................... .................................... .................................... ......... ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ .... ......... ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ .... ......... ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ .... ......... ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ .... .......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... 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P5 + P6: • • • • • • • Figure 1: A graph P5 + P6 with rdim2(P5 + P6) = 7 4. Restrained 2-Resolving Sets in the Corona of Graphs Theorem 4. Let G and H be nontrivial connected graphs. A set S ⊆ V (G ◦H) is a 2-resolving set of G ◦H if and only if S = A∪B, where A ⊆ V (G) and B = ⋃ {Sv : Sv is a 2-resolving set of Hv, for all v ∈ V (G)}. Theorem 5. 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). Proof. Suppose S is a restrained 2-resolving set in G ◦ H. Let A = V (G) ∩ S and Sv = S ∩ V (Hv) for all v ∈ V (G). Then S = A ∪ ( ⋃ v∈V (G) Sv ) where A ⊆ V (G) and Sv ⊆ V (Hv) for each v ∈ V (G). By Theorem 4, Sv is a 2-resolving set in Hv for every v ∈ V (G). Since S is a restrained 2-resolving set in G◦H, S = V (G◦H) or ⟨V (G ◦H)\S⟩ has no isolated vertex. Let v ∈ A. If Sv = V (Hv), then Sv is a restrained 2-resolving set of Hv. Suppose Sv ̸= V (Hv). Since v ∈ A, ⟨V (Hv)\Sv⟩ must have no isolated vertex. Hence, Sv is a restrained 2-resolving set in Hv. Next, let w ∈ V (G)\A with Sw = V (Hw). Since V (G ◦H)\S has no isolated vertex, w ∈ NG(V (G)\A). Hence, (i)-(iv) hold. J. Cabaro, H. Rara / Eur. J. Pure Appl. Math, 15 (3) (2022), 1229-1236 1234 Conversely, let S = A ∪ ( ⋃ v∈V (G) Sv ) , where A ⊆ V (G) and Sv ⊆ V (Hv) for each v ∈ V (G) satisfying (i)-(iv). By Theorem 4, S is a 2-resolving set in G ◦ H. Moreover, because of (i)-(iv), S is a restrained 2-resolving set in G ◦H. Corollary 3. Let G and H be nontrivial connected graphs, where |V (G)| = n. Then rdim2(G ◦H) = n · dim2(H). The set consisting of the shaded vertices in Figure 2 is a restrained 2-resolving set of the corona P4 ◦ C5. .................................... .................................... .................................... .................................... .................................... .................................... .................................... .................................... .................................... .................................... .................................... .................................... .................................... .................................... .................................... .................................... .................................... .................................... .................................... ........................................................................ .................................... .................................... .................................... 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............................................................................................................................................. ............................................................................................. ............................................................................................. ............................................................................................. ............................................................................................. • •• • •• • • • •• • Figure 2: A graph P4 ◦ C5 with rdim2(P4 ◦ C5) = 12 5. Restrained 2-Resolving Sets in the Lexicographic Product of Graphs Definition 3. A vertex x is said to be 1-equidistant to y if xy ∈ E(G) and dG(x, z) = dG(y, z), for all z ∈ V (G)\ {x, y} and it is 2-equidistant to y if dG(x, y) = 2 and dG(x, z) = dG(w, z), for all z ∈ V (G)\ {x,w}. A vertex is called a free-vertex in G if it is neither 1-equidistant nor 2-equidistant to any vertex. The set containing all 1-equidistant, 2-equidistant, and free-vertices in G are denoted by EQ1(G), EQ2(G) and fr(G), respectively. Theorem 6. 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 2-resolving 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 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 (iv) Tx and Ty are (2-locating) dominating sets inH or one of Tx and Ty is a 2-dominating set whenever x, y ∈ EQ2(G). J. Cabaro, H. Rara / Eur. J. Pure Appl. Math, 15 (3) (2022), 1229-1236 1235 Theorem 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). Proof. Suppose W = ⋃ x∈S [ {x}×Tx ] where S ⊆ V (G) and Tx ⊆ V (H) is a restrained 2- resolving set in G[H]. Then by Theorem 6, (i), (ii), (iv) and (v) hold. Now, let x ∈ V (G) with Ty = V (H), for all y ∈ NG(x). Suppose that Tx is not restrained 2-locating set. Then ⟨V (H)\Tx⟩ has an isolated vertex, say u. Thus, (x, u) is an isolated vertex in ⟨V (G[H])\W ⟩, contrary to the assumption that W is a restrained 2-resolving set in G[H]. Hence, Tx is a restrained 2-locating set. For the converse, suppose that (i), (ii), (iii), and (iv) hold. Then by Theorem 6, W = ⋃ x∈S [ {x}×Tx ] is a 2-resolving set in G[H]. If W = V (G[H]), then W is a restrained 2-resolving set in G[H]. Suppose that W ̸= V (G[H]). Let (x, a) ∈ 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 by (iii), Tx is a restrained 2-locating 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 set in G[H]. The following corollaries are the direct consequences of Theorem 7. Corollary 4. Let G and H be nontrivial connected graphs such that G is not free- equidistant. Then, rdim2(G[H]) ≤ n · ln(2,1)(H) +m · γ2L(H) + p · rln2(H), where n+m+ p = |V (G)| with |EQ1(G)| = n, |EQ2(G)| = m and |fr(G)| = p. The following result follows from Theorem 7. Corollary 5. LetG andH be non-trivial connected graphs such thatG is free-equidistant. Then rdim2(G[H]) = { |V (G)| · ln2(H), if ln2(H) ̸= |V (H)| |V (G)| · rln2(H), otherwise. The set consisting of the shaded vertices in Figure 3 is a restrained 2-resolving set of the lexicographic product P4[P3]. REFERENCES 1236 .................................... .................................... .................................... .................................... .................................... .................................... .................................... .................................... .................................... .................................... .................................... .................................... ............................................................................................. ............................................................................................. ............................................................................................. ............................................................................................. ............................................................................................. 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........ ........ .... ......... ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ .... ............ ........... ........... ........... ........... ........... ........... ........... ........... ........... ........... ........... ... ......... ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ .... ........................................................................................................................................ ......................................................................................................................................................................................................................................... ........... ........... ........... ........... ........... ........... ........... ........... ........... ........... ........... ... ................... .................. .................. .................. .................. 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............................................................................................................................................................................................................................. ................... .................. .................. .................. .................. .................. .................. .................. .................. .................. .................. .................. .... ............ ........... ........... ........... ........... ........... ........... ........... ........... ........... ........... ........... ... ...................................................................................................................................................................................................... ........................................................................................................................................ ........................................................................................................................................ ........................................................................................................................................ ............ ........... ........... ........... ........... ........... ........... ........... ........... ........... ........... ........... ... ................... .................. .................. .................. .................. .................. .................. .................. .................. .................. .................. .................. .... ............ ........... ........... ........... ........... ........... ........... ........... ........... ........... ........... ........... ... • • • • • • • • Figure 3: A graph P4[P3] with rdim2 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] V Lokesha A S Cevik, Jain and I. Cangul. New results on the f-index of graphs based on corona-type of graphs. Proc. of the Jangjeon Mathematical Society, 23(2):141–147, 2020. [7] V Saenpholphat and P Zang. On connected resolvability of graphs. Australian Journal of Combinatorics, 28:25–37, 2003. [8] P. Slater. Congressus Numerantium,., 14:549–559, 1975.