EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 15, No. 3, 2022, 1417-1425 ISSN 1307-5543 – ejpam.com Published by New York Business Global On 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 2-resolving set S ⊆ V (G) which is dominating is called a 2-resolving dominating set or simply 2R-dominating set in G. The minimum cardinality of a 2-resolving dominating set in G, denoted by γ2R(G), is called the 2R-domination number of G. Any 2R-dominating set of cardinality γ2R(G) is then referred to as a γ2R-set in G. This study deals with the concept of 2-resolving dominating set of a graph. It characterizes the 2-resolving dominating 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: 05C62 Key Words and Phrases: 2-resolving set, 2-resolving dominating set, 2R-domination number, 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 [7], 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 [3] where metric generators were called resolving sets. Bailey and Yero in [6] demonstrated a construction of error-correcting codes from graphs by means of k-resolving sets, and present a decoding algorithm which makes use of covering designs. ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v15i3.4426 Email addresses: amerjean1228@gmail.com (J. Cabaro), helenrara@gmail.com (H. Rara) https://www.ejpam.com 1417 © 2022 EJPAM All rights reserved. J. Cabaro, H. Rara / Eur. J. Pure Appl. Math, 15 (3) (2022), 1417-1425 1418 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 [4], 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 2-resolving dominating set in the join, corona and lexi- cographic 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 [5]. Theorem 1. [2] Let G and H be two nontrivial graphs such that G is connected. Then the following assertions hold for any a, c ∈ V (G) and b, d ∈ V (H) such that a ̸= c. (i) NG[H](a, b) = ({a} ×NH {b}) ∪ {NG {a} × V (H)} (ii) dG[H]((a, b), (c, d)) = dG(a, c) (iii) dG[H](a, b), (a, d) = min {dH(b, d), 2}. Proposition 1. [1] 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. J. Cabaro, H. Rara / Eur. J. Pure Appl. Math, 15 (3) (2022), 1417-1425 1419 Remark 1. For any connected graph G of order n ≥ 2, 1 < γ2R(G) ≤ n. Remark 2. For any connected graph G of order n ≥ 2, dim2(G) ≤ γ2R(G). Remark 3. For n ≥ 2, γ2R(Kn) = n. Theorem 2. Let G be a nontrivial connected graph of order n ≥ 2. Then γ2R(G) = 2 if and only if G ∼= Pn, for 2 ≤ n ≤ 4. Proof. Suppose that γ2R(G) = 2. Let S = {x, y} be a γ2R-set in G. By Remark 2 and Proposition 1, G ∼= Pn. Moreover, x and y are the end vertices in G. Since S is a dominating set in G, 2 ≤ n ≤ 4. The converse follows immediately from Proposition 1. Theorem 3. Let G be a connected graph of order n ≥ 2. If G ∈ {Kn, F3, P2 + P3,Km,n−m}, where m ≥ 2, then γ2R(G) = n. Example 1. The sets S1 = {b, e, g} and S2 = {a, d, e, g} in Figure 1 are 2-resolving dominating sets in G. Moreover, S1 is a γ2R-set in G. Thus, γ2R(G) = 3. .................................... .................................... .................................... .................................... .................................... ........................................................................ ......... ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ .... ............................................................................................. ......... ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ .... ............................................................................................. ......... ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ......................................................................................................................................................................................................................................... G: a b c d e f g Figure 1: A graph G with γ2R(G) = 3 Example 2. Consider the graph G in Figure 2. The ordered set of vertices W = {u1, u2, u3} is a 2-resolving set for the graph G since the representations rG(u1/W3) = (0, 1, 2), rG(u2,W3) = (1, 0, 1), rG(u3/W3) = (2, 1, 0), rG(u4/W3) = (2, 2, 1), rG(u5/W3) = (1, 2, 2) and rG(u6/W3) = (3, 3, 2) differ in at least 2 positions. But W is not a dominating set of G. 3. 2-Resolving Dominating 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 J. Cabaro, H. Rara / Eur. J. Pure Appl. Math, 15 (3) (2022), 1417-1425 1420 .................................... .................................... .................................... .................................... .................................... .................................... ......... ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ .... ................... .................. .................. .................. .................. ............... ........................................................................................................................................ ............................................................................................. ............ ........... ........... ........... ........... ........... ........... ........... ........... ........... ........... ........... ... .......................................................................................................... G: u1 u2 u3 u4 u5 u6 • • • Figure 2: A graph G with dim2(G) = 3 (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 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. Theorem 4. [4] 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 of 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. [4] Let G and H 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 6. Let G be a connected non-trivial graph and let K1 = {v}. Then S ⊆ V (K1 +G) is a 2-resolving dominating set in K1 +G if and only if it is a 2-resolving set in K1 +G. Proof. Let S ⊆ V (K1 + G) be a 2-resolving dominating set in K1 + G. Then, S is a 2-resolving set in K1 +G by the definition of 2-resolving dominating set. Conversely, if S is a 2-resolving set in K1 + G, then by Theorem 4, S is a 2-locating set. Hence, S is a dominating set in K1 + G. Thus, S is a 2-resolving dominating set in K1 +G. Corollary 1. γ2R(K1 +G) = dim2(K1 +G). J. Cabaro, H. Rara / Eur. J. Pure Appl. Math, 15 (3) (2022), 1417-1425 1421 Theorem 7. Let G andH be nontrivial connected graphs. A proper subset S of V (G+H) is a 2-resolving dominating set in G+H if and only if it is a 2-resolving set in G+H. Proof. Let S ⊆ V (G + H) be a 2-resolving dominating set in G + H. Then, S is a 2-resolving set in G+H. Conversely, if S is a 2-resolving set in G + H, then by Theorem 5, S is a 2-locating set. Hence, S is a dominating set in G + H. Thus, S is a 2-resolving dominating set in G+H. Corollary 2. Let G and H be connected nontrivial graphs. Then, γ2R(G+H) = dim2(G+H). The set consisting of the shaded vertices in Figure 3 is a 2-resolving dominating set of the join P5 + P6. .................................... .................................... .................................... .................................... .................................... .................................... .................................... .................................... .................................... .................................... .................................... ......... ........ ........ ........ ........ ........ ........ ........ ........ ...... ......... ........ ........ ........ ........ ........ ........ ........ ........ ...... ......... ........ ........ ........ ........ ........ ........ ........ ........ ...... ......... ........ ........ ........ ........ ........ ........ ........ ........ ...... .......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... 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.............................................................................................................................................................................. • • • • • • • Figure 3: A graph P5 + P6 with γ2R(P5 + P6) = 7 4. 2-Resolving Dominating Sets in the Corona of Graphs Theorem 8. [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 9. Let G and H be nontrivial connected graphs. Then S ⊆ V (G ◦ H) is a 2-resolving dominating set in G ◦ H if and only if S = A ∪ ( ⋃ v∈V (G) Sv), where A ⊆ V (G), Sv is a 2-resolving set for each v ∈ A and Sv is a 2-resolving dominating set for each v ∈ V (G)\A. J. Cabaro, H. Rara / Eur. J. Pure Appl. Math, 15 (3) (2022), 1417-1425 1422 Proof. Suppose S is a 2-resolving dominating 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). By Theorem 8, Sv is a 2-resolving set in Hv for each v ∈ A. If v ∈ V (G)\A, then Sv is a 2-resolving dominating set in G ◦H. Conversely, let S = A ∪ ( ⋃ v∈V (G) Sv ) where A ⊆ V (G) and Sv ⊆ V (Hv) satisfying the given conditions. By Theorem 8, S is a 2-resolving set in G ◦ H. Let x ∈ V (G ◦ H)\S and let v ∈ A such that x ∈ V (v +Hv). Then xv ∈ E(G ◦H). If v ∈ V (G)\A, then there exists y ∈ Sv such that xy ∈ E(G ◦H). Therefore, S is a dominating set in G ◦H. Hence, S is a 2-resolving dominating set in G ◦H. Corollary 3. Let G and H be nontrivial connected graphs, where |V (G)| = n. Then γ2R(G ◦H) ≤ min {n(1 + dim2(H)), nγ2R(H)}. The set consisting of the shaded vertices in Figure 4 is a 2-resolving dominating set of the corona P4 ◦ C5. .................................... .................................... .................................... .................................... .................................... .................................... .................................... .................................... .................................... .................................... .................................... .................................... .................................... .................................... .................................... .................................... .................................... .................................... .................................... ........................................................................ .................................... .................................... .................................... 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.......... .......... ......... ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ............................................................................................................................................. • •• • •• • • • •• • Figure 4: A graph P4 ◦ C5 with γ2R(P4 ◦ C5) = 12 5. 2-Resolving Dominating 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 10. 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) J. Cabaro, H. Rara / Eur. J. Pure Appl. Math, 15 (3) (2022), 1417-1425 1423 (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). Proof. Suppose W = ⋃ x∈S [ {x} × Tx ] is a 2-resolving set in G[H]. Suppose there exists x ∈ V (G)\S. Pick a, b ∈ V (H), where a ̸= b. Then (x, a), (x, b) /∈ W and (x, a) ̸= (x, b). Since x /∈ S and dG[H]((x, a), (y, p)) = dG[H]((x, b), (y, p)) for all y ∈ V (G)\ {x} and for all p ∈ V (H), rG[H]((x, a)/W ) = rG[H]((x, b)/W ). This implies that W is not a 2-resolving set of G[H], a contradiction to the assumption on W . Therefore, S = V (G). To prove (ii), let x ∈ V (G) and p, q ∈ V (H) where p ̸= q. Then (x, p) ̸= (x, q). If p, q /∈ Tx or [p ∈ Tx and q /∈ Tx], then (x, p), (x, q) /∈ W or [(x, p) ∈ W and (x, q) /∈ W ]. Since W is a 2-resolving set in G[H], rG[H]((x, p)/W ) and rG[H]((x, q)/W ) differ in at least 2 positions. Hence, by Theorem 1(iii) and Definition 1, Tx is a 2-locating set in H. Thus, (ii) follows. To prove (iii), let x and y be adjacent vertices of G with dG(x, z) = dG(y, z), for all z ∈ V (G)\ {x, y}. Let a, b ∈ V (H), a ̸= b. Since W is 2-resolving, rG[H]((x, a)/W ) and rG[H]((y, b)/W ) differ in at least 2 positions. By as- sumption, it is not possible that NH(a) ∩ Tx = Tx and NH(b) ∩ Ty = Ty. If Tx or Ty is (2, 2)-locating, then we are done. Otherwise, Tx and Ty are (2, 1)-locating. To prove (iv), let x, y ∈ V (G) where dG(x, y) = 2 and dG(x, z) = dG(y, z), for all z ∈ V (G)\ {x, y}. Let a, b ∈ V (H), a ̸= b. Suppose one of Tx and Ty, say Tx is not a dominating set in H. Pick a ∈ V (H)\NH [Tx] and let b ∈ V (H)\Ty. Since dG[H]((x, a), (z, q)) = 2, for all (z, q), it follows that |NH(b)∩Ty| ≥ 2, i.e., Ty is a 2-dominating set. Conversely, suppose (i),(ii), (iii) and (iv) hold. Let (x, a), (y, b) ∈ V (G[H]), (x, a) ̸= (y, b). Consider the following cases. Case 1. x = y Supppose (x, a), (y, b) /∈ W . Then a ̸= b and a, b /∈ Tx = Ty. By (ii), Tx is a 2-locating set. Hence, by Theore 1(iii) and by Definition 1, rG[H]((x, a)/W ) and rG[H]((y, b)/W ) differ in at least two positions. On the other hand, if (x, a) ∈ W , (y, b) /∈ W , then a ∈ Tx, b /∈ Ty. Using similar argument as in above, rG[H]((x, a)/W ) and rG[H]((y, b)/W ) differ in at least 2 positions. Case 2. x ̸= y. Subcase 2.1 xy ∈ E(G). If dG(x, z) ̸= dG(y, z) for some z ∈ V (G)\ {x, y}, then rG[H]((x, a)/W ) and rG[H]((y, b)/W ) differ in at least 2 positions since H is nontrivial. Suppose dG(x, z) = dG(y, z), for all z ∈ V (G)\ {x, y}. Then by (iii), Tx and Ty are (2, 1)-locating sets in H or one of Tx and Ty is a (2, 2)-locating set in H. Hence, by Definition 1, rG[H]((x, a)/W ) and rG[H]((y, b)/W ) differ in at least 2 positions. J. Cabaro, H. Rara / Eur. J. Pure Appl. Math, 15 (3) (2022), 1417-1425 1424 Subcase 2.2 xy /∈ E(G) If dG(x, y) > 2, then it follows that rG[H]((x, a)/W ) and rG[H]((y, b)/W ) differ in at least 2 positions. If dG(x, y) = 2 and dG(x, z) ̸= dG(y, z) for some z ∈ V (G)\ {x, y}, then it follows that rG[H]((x, a)/W ) and rG[H]((y, b)/W ) differ in at least 2 positions. Suppose dG(x, y) = 2 and dG(x, z) = dG(y, z), for all z ∈ V (G)\ {x, y}. Suppose (x, a), (y, b) /∈ W . Then a /∈ Tx and y /∈ Ty. If Tx and Ty are both dominating, then rG[H]((x, a)/W ) and rG[H]((y, b)/W ) differ in at least 2 positions. If one, say Ty, is a 2-dominating set, then rG[H]((x, a)/W ) and rG[H]((y, b)/W ) differ in at least 2 positions. Similarly, if (x, a) ∈ W , (y, b) /∈ W , then rG[H]((x, a)/W ) and rG[H]((y, b)/W ) differ in at least 2 positions. Accordingly, W is a 2-resolving set of G[H]. Theorem 11. 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 dominating set in G[H] if and only if it is a 2-resolving set in G[H]. Proof. The proof is similar to that of Theorem 10. Corollary 4. Let G and H be nontrivial connected graphs such that G is not free- equidistant. Then, γ2R(G[H]) = dim2(G[H]). The set consisting of the shaded vertices in Figure 5 is a 2-resolving dominating set of the lexicographic product P4[P3]. .................................... .................................... .................................... .................................... .................................... .................................... .................................... .................................... .................................... .................................... .................................... .................................... ............................................................................................. ............................................................................................. ............................................................................................. ............................................................................................. ............................................................................................. ............................................................................................. ............................................................................................. ............................................................................................. ......... ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ .... ......... ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ .... ......... ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ .... ......... ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ .... ......... ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ .... ......... ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ .... ......... ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ .... ......... ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ .... ............ ........... ........... ........... ........... ........... ........... ........... ........... ........... ........... ........... ... ......... ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ .... ........................................................................................................................................ ......................................................................................................................................................................................................................................... ........... ........... ........... ........... ........... ........... ........... ........... ........... ........... ........... ... ................... .................. .................. .................. .................. .................. .................. .................. .................. .................. .................. .................. .... ........................................................................................................................................ ......................................................................................................................................................................................................................................... ........... ........... ........... ........... ........... ........... ........... ........... ........... ........... ........... ... ........................................................................................................................................ ............................................................................................................................................................................................................................. ................... .................. .................. .................. .................. .................. .................. .................. .................. .................. .................. .................. .... ............ ........... ........... ........... ........... ........... ........... ........... ........... ........... ........... ........... ... ...................................................................................................................................................................................................... ........................................................................................................................................ ........................................................................................................................................ ........................................................................................................................................ ............ ........... ........... ........... ........... ........... ........... ........... ........... ........... ........... ........... ... ................... .................. .................. .................. .................. .................. .................. .................. .................. .................. .................. .................. .... ............ ........... ........... ........... ........... ........... ........... ........... ........... ........... ........... ........... ... • • • • • • • • Figure 5: A graph P4[P3] with γ2RP4[P3] = 8 REFERENCES 1425 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. 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