EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 16, No. 1, 2023, 18-28 ISSN 1307-5543 – ejpam.com Published by New York Business Global Resolving Domination in Graphs Under Some Binary Operations Gerald B. Monsanto1,∗, Helen M. Rara2 1 College of Teacher Education, Arts and Sciences, Visayas State University Villaba, 6537 Villaba, Leyte, 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. In this paper, we investigate the concept of resolving dominating set in a graph. In particular, we characterize the resolving dominating sets in the join, corona and lexicographic product of two graphs and determine the resolving domination number of these graphs. 2020 Mathematics Subject Classifications: 05C69 Key Words and Phrases: Resolving dominating set, resolving domination number, join, corona, lexicographic product 1. Introduction All graphs considered in this study are finite, simple, and undirected connected graphs, that is, without loops and multiple edges. For some basic concepts in Graph Theory, we refer readers to [5]. Let G = ( V (G), E(G) ) be a connected graph. The open neighborhood of v ∈ V (G) is NG(v) = {u ∈ V (G) : uv ∈ E(G)}. Any element u of NG(v) is called a neighbor of v. The closed neighborhood of v ∈ V (G) is NG[v] = NG(v) ∪ {v}. Thus, the degree of v ∈ V (G) is given by degG(v) = |NG(v)|. Customarily, for S ⊆ V (G), NG(S) = ⋃ v∈S NG(v) and NG[S] = ⋃ v∈S NG[v]. A nonempty set S ⊆ V (G) is a dominating set in graph G if NG[S] = V (G). Otherwise, we say S is a non-dominating set of G. The domination number of a graph G, denoted by γ(G), is given by γ(G) = min|S| : S is a dominating set of G. If S is a dominating set of G and if |S| = γ(G), then S is called a minimum dominating set or a γ-set of G. ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v16i1.4643 Email addresses: gerald.monsanto@vsu.edu.ph (G. Monsanto), helen.rara@g.msuiit.edu.ph (H. Rara) https://www.ejpam.com 18 © 2023 EJPAM All rights reserved. G. Monsanto, H. Rara / Eur. J. Pure Appl. Math, 16 (1) (2023), 18-28 19 The distance dG(u, v) in G of two vertices u, v is the length of a shortest u-v path in G. A vertex x of a connected graph G is said to resolve two 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 distinct bvertices 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. Slater [10] brought in the notion of locating sets and its minimum cardinality as locating number. The same concept was also introduced by Harary and Melter [5] but using the terms resolving sets and metric dimension to refer to locating sets and locating number, respectively. Some variations of locating sets and resolving sets are studied in [2, 6– 9, 11, 12]. Let G be a connected graph. A set S ⊆ V (G) is a locating set of G if for every two distinct vertices u and v of V (G) \ S, NG(u)∩ S ̸= NG(v)∩ S. The locating number of G, denoted by ln(G), is the smallest cardinality of a locating set of G. A locating set of G of cardinality ln(G) is referred to as an ln-set of G. Let G be a connected graph. A set S ⊆ V (G) is a strictly locating set of G if it is a locating set of G and NG(u)∩S ̸= S, for all u ∈ V (G)\S. The strictly locating number of G, denoted by sln(G), is the smallest cardinality of a strictly locating set of G. A strictly locating set of G of cardinality sln(G) is referred to as sln-set of G. A connected graph G of order n ≥ 3 is point distinguishing if for any two distinct vertices u and v of G, NG(u) ̸= NG(v) [3]. It is totally point determining if for any two distinct vertices u and v of G, NG(u) ̸= NG(v) and NG[u] ̸= NG[v] [13]. Brigham et al. [1] defined a resolving dominating set as a set S of vertices of a connected graph G that is both resolving and dominating. The cardinality of a minimum resolving dominating set is called the resolving domination number of G and is denoted by γR(G). A resolving dominating set of cardinality γR(G) is called a γR-set of G. Canoy and Malacas [11] defined a locating-dominating (resp. strictly locating-dominating) as a locating (resp. strictly locating) subset S of V (G) which is also dominating set in a connected graph G. The minimum cardinality of a locating-dominating (resp. strictly locating-dominating) set in G, denoted by γL(G) (resp. (γSL)), is called the L-domination (resp. SL-domination) number of G. Any L-dominating (resp. SL-dominating) set of cardinality γL(G) (resp. γSL(G)) is then referred to as a γL-set (γSL-set) of G. Let G be a connected graph. A set S ⊆ V (G) is strictly resolving dominating set of G if it is a resolving dominating set of G and NG ∩ S ̸= S for u ∈ V (G) \ S. The strictly resolving dominating number of G, denoted by γSR(G), is the smallest cardinality of a strictly resolving dominating set of G. A strictly resolving dominating set of G of cardinality γSR(G) is referred to as γSR-set of G. This study aims to define and characterize the resolving dominating sets in the join, corona and lexicographic product of graphs and determine their corresponding resolving domination number. G. Monsanto, H. Rara / Eur. J. Pure Appl. Math, 16 (1) (2023), 18-28 20 2. Preliminary Results Remark 1. Every locating-dominating set of a connected graph G is a resolving dominat- ing set and every resolving dominating set is a dominating set of G. Hence, γ(G) ≤ γR(G) ≤ γL(G). Example 1. Consider the graph G in Figure 1 and let S = {v1, v5}. Observe that NG[S] = V (G), that is, S is a dominating set in G. Moreover, S is a resolving set since the representation of each vertex in G, with respect to S is unique: rG(v1/S) = (0, 1), rG(v2/S) = (1, 2), rG(v3/S) = (2, 1), rG(v4/S) = (1, 1) and rG(v5/S) = (1, 0). Hence, γR(G) ≤ |S| = 2. Since any singleton is not a resolving set, γR(G) = 2. Also, S is a locating-dominating set of a graph G since NG(v2) ∩ S = {v1}, NG(v3) ∩ S = {v5}, and NG(v4) ∩ S = {v1, v5}. Thus, γL(G) = 2. Figure 1: A graph G with γ(G) = γR(G) = γL(G) = 2 Example 2. Consider the graph G in Figure 2. Let S1 = {x, y, z}. Observe that NG[S1] = V (G), that is, S1 is a dominating set in G. Hence, γ(G) = 3. Moreover, let S2 = {w1, w2, w3, w4}. Then rG(x/S2) = (1, 1, 1, 2), rG(y/S2) = (1, 1, 1, 4), rG(z/S2) = (2, 2, 2, 1), rG(w1/S2) = (0, 2, 2, 3), rG(w2/S2) = (2, 0, 2, 3), rG(w3/S2) = (2, 2, 0, 3), and rG(w4/S2) = (3, 2, 2, 0). Thus, S2 is a resolving dominating set of a graph G. Hence, γR(G) = 4. Furthermore, let S3 = {x,w1, w2, w3, w4}. Then S3 is a locating-dominating set of a graph G since NG(y)∩S3 = {w1, w2, w3} and NG(z)∩S3 = {w4}. Thus, γL(G) = 5. Figure 2: A graph G with γ(G) = 3, γR(G) = 4, and γL(G) = 5 Remark 2. Every resolving dominating set of a connected graph G is a resolving set of G. Thus, dim(G) ≤ γR(G). G. Monsanto, H. Rara / Eur. J. Pure Appl. Math, 16 (1) (2023), 18-28 21 Example 3. Consider the graph G in Figure 3. The set S = {v1, v2} is a resolving set. Since no single vertex constitutes a resolving set for G, it follows that W is a minimum resolving set. Hence, dim(G) = 2. Moreover, S is a resolving dominating set of a graph G since NG[S] = V (G). Thus, γR(G) = 2. Figure 3: A graph G with dim(G) = 2 = γR(G) Example 4. Consider the graph G in Figure 4. Let S1 = {v, y} is a resolving set of a graph G since the representation of each vertex in G, with respect to S1 is unique: rG(x/S1) = (2, 2), rG(u1/S1) = (2, 1), rG(u2/S1) = (1, 2), rG(u3/S1) = (1, 3), rG(z/S1) = (2, 3), rG(v/S1) = (0, 3), and rG(y/S1) = (3, 0). Hence, dim(G) = 2. Moreover, let S2 = {u1, u2, u3}. Then S2 is a resolving dominating set of a graph G. Thus, γR(G) = 3. Figure 4: A graph G with dim(G) = 2 and γR(G) = 3 Proposition 1. [1] Let G be a connected graph of order n ≥ 2, then (i) γR(P3) = 2 and for n ≥ 4, γR(Pn) = ⌈n3 ⌉ (ii) For n ≥ 3, γR(Cn) = ⌈n3 ⌉ if n ̸= 6 and γR(C6) = 3. Proposition 2. For any connected graph G of order n ≥ 2, 1 ≤ γR(G) ≤ n−1. Moreover, (i) γR(G) = 1 if and only if G = P2 and (ii) γR(G) = n− 1 if and only if G = Kn or K1,n−1. Proof: Suppose that γR(G) = 1, say W = {v} is a minimum resolving dominating set of G. Since G is connected and non-trivial, there exists x ∈ V (G) \ {v} such that xv ∈ E(G). If |V (G)| = 2, then G = K2 = P2. Therefore, (i) holds. G. Monsanto, H. Rara / Eur. J. Pure Appl. Math, 16 (1) (2023), 18-28 22 Suppose G ̸= Kn. Let V (G) \ {v} = W is γR-set. Let w ∈ NG(v). Consider that |NG(v)| = 1. Claim: G = ⟨w⟩+Kn−1 ∼= K1,n−1. Suppose there exists u ∈ V (G) \ {v, w} such that uv /∈ E(G). Let W = V (G) \ {u, v}. Clearly, W is a dominating set of G. Since v ∈ NG(w) and u ∈ NG(w), r(u/W ) ̸= r(v/W ). Hence, W is a resolving dominating set of G and γR(G) ≤ n − 2, a contradiction. Thus, w ∈ NG(z) for all z ∈ V (G) \ {w}. Next, suppose there exist distinct vertices a, b ∈ V (G) \ {w, v} such that ab ∈ E(G). Let W1 = V (G) \ {a, v}. Then W1 is a dominating set of G. Since w ∈ NG(v) and b /∈ NG(v), r(w/W1) ̸= r(b/W1). Hence, W1 is a resolving dominating set of G. Thus, γR(G) ≤ |W1| = n−2, a contradiction. Therefore, ab /∈ E(G). Accordingly, G = ⟨w⟩+Kn−1 ∼= K1,n−1. For the converse, suppose G = Kn or G = K1,n−1. Then γR(G) = n − 1. Hence, (ii) holds. 3. Resolving Domination in the Join of Graphs The join of two graphs G and H is the graph G + H with vertex set V (G + H) = V (G) • ∪ V (H) and edge set E(G+H) = E(G) • ∪ E(H) ∪ {uv : u ∈ V (G), v ∈ V (H)}. Theorem 1. [9] Let G and H be non-trivial connected graphs. A set W ⊆ V (G+H) is a resolving set of G+H if and only if W = WG ∪WH where WG ⊆ V (G) and WH ⊆ V (H) are locating sets of G and H, respectively, where WG or WH is a strictly locating set. Theorem 2. [4] Let G and H be connected graphs. Then C ⊆ V (G+H) is a dominating set in G+H if and only if at least one of the following is true: (i) C ∩ V (G) is a dominating set in G. (ii) C ∩ V (H) is a dominating set in H. (iii) C ∩ V (G) ̸= ∅ and C ∩ V (H) ̸= ∅. Theorem 3. Let G and H be non-trivial connected graphs. A set W ⊆ V (G + H) is a resolving dominating set of G+H if and only if W is a locating-dominating set of G+H. Proof: Suppose that W is a resolving dominating set of G+H. Then W is a resolving set of G +H. By Theorem 1, W = WG ∪WH where WG ⊆ V (G) and WH ⊆ V (H) are locating sets of G and H, respectively, where WG or WH is a strictly locating set. Since W is a dominating set of G + H, WG and WH are also dominating sets of G and H, respectively. By Theorem 1, W is a locating-dominating set of G+H. The converse follows immediately from Theorem 1 and Theorem 2(iii). Theorem 4. Let G and H be non-trivial connected graphs. A set W ⊆ V (G + H) is a resolving dominating set of G+H if and only if W = WG ∪WH where WG = V (G) ∩W and WH = V (H) ∩W are locating sets of G and H, respectively, where WG or WH is a strictly locating set. G. Monsanto, H. Rara / Eur. J. Pure Appl. Math, 16 (1) (2023), 18-28 23 Proof: Suppose that W is a resolving dominating set of G+H. Then W is a resolving set of G + H. By Theorem 1, WG = W ∪ V (G) where WG ⊆ V (G) and WH ⊆ V (H) are locating sets of G and H, respectively, where WG or WH is a strictly locating set. Since W is a dominating set of G+H, WG and WH are also dominating sets of G and H, respectively. By Theorem 3, W is a locating-dominating set of G+H. Conversely, let WG = V (G) ∩W and WH = V (H) ∩W be locating sets of G and H, respectively, and WG or WH is a strictly locating set of G + H. By Theorem 2, W is a dominating set of G +H. Let u, v ∈ V (G +H) \W with u ̸= v. Consider the following cases: Case 1. u, v ∈ V (G) Since WG is a locating set of G, NG(u) ∩WG ̸= NG(v) ∩WG. Hence, rG+H(u/W ) ̸= rG+H(v/W ). Case 2. u, v ∈ V (H) The proof is similar to case 1. Case 3. u ∈ V (G) and v ∈ V (H) rG+H(u/W ) = ( dG+H(u,w1), . . . , dG+H(u,wn), 1, 1, . . . , 1 ) and rG+H(v/W ) = ( 1, 1, . . . , 1, dG+H(v, u1), . . . , dG+H(v, un) ) Suppose there exists j ∈ {1, 2, . . . , n} such that dG+H(u,wj) ̸= 1 or there exists k ∈ {1, 2, . . . ,m} such that dG+H(v, uk) ̸= 1. Hence, rG+H(u/W ) ̸= rG+H(v/W ). Therefore, W is a resolving set of G +H. Accordingly, W is a resolving dominating set of G+H. Corollary 1. Let G and H be non-trivial connected graphs. Then γR(G+H) = min {sln(H) + ln(G), sln(G) + ln(H)} . Proof: Let W be a minimum resolving dominating set in G+H. Let WG = V (G)∩W and WH = V (H) ∩ W . By Theorem 4, WG and WH are locating sets in G and H, respectively, where WG or WH is a strictly locating set. If WG is strictly locating, then sln(G) + ln(H) ≤ |WG|+ |WH | = |W | = γR(G+H). If WH is strictly locating, then sln(G) + ln(H) ≤ |WH |+ |WG| = |W | = γR(G+H). Thus, γR(G+H) = min {sln(H) + ln(G), sln(G) + ln(H)} . Next, suppose that sln(G)+ ln(H) ≤ sln(H) + ln(G). G. Monsanto, H. Rara / Eur. J. Pure Appl. Math, 16 (1) (2023), 18-28 24 Let WG be a minimum strictly locating set of G and WH be a minimum locating set of H. Then W = WG ∪WH is a resolving dominating set of G+H by Theorem 4. Hence, γR(G+H) ≤ |W | = |WG|+ |WH | = sln(G) + ln(H). Therefore, γR(G+H) = min {sln(H) + ln(G), sln(G) + ln(H)} . Theorem 5. [11] Let G be a connected graph of order n ≥ 2. (i) If ln(G) < sln(G), then 1 + ln(G) = sln(G). (ii) If ln(G) < γL(G), then 1 + ln(G) = γL(G). (iii) If sln(G) < γSL(G), then 1 + sln(G) = γSL(G). Corollary 2. Let G be a non-trivial connected graph and let Kn be a complete graph of order n ≥ 2. Then γR(G+Kn) = sln(G) + n− 1. Proof: Note that γR(Kn) = n− 1 and sln(Kn) = n. From Corollary 1, γR(G+Kn) = min {sln(H) + ln(G), sln(G) + ln(H)} . By Theorem 5, sln(G)− 1 ≤ ln(G). Therefore, γR(G+Kn) = min {sln(G) + n− 1, ln(G) + n} = sln(G) + n− 1. Theorem 6. [11] Let H be a non-trivial connected graph and let K1 = ⟨v⟩. Then W ⊆ V (H) is a locating-dominating set of H + K1 if and only if either v /∈ W and W is a strictly locating dominating set of H or W = {v} ∪WH , where WH is a locating set of H. Theorem 7. [9] Let H be a non-trivial connected graph and let K1 = ⟨v⟩. Then W ⊆ V (H) is a resolving set of H +K1 if and only if either v /∈ W and W is a strictly locating set of H or W = {v} ∪WH , where WH is a locating set of H. The next result follows immediately from Theorem 6 and Theorem 7. Theorem 8. Let H be a non-trivial connected graph and let K1 = ⟨v⟩. Then W ⊆ V (H) is a resolving dominating set of H +K1 if and only if either v /∈ W and W is a strictly resolving dominating set of H or W = {v} ∪WH , where WH is a locating set of H. Corollary 3. Let H be a non-trivial connected graph. Then γR(H +K1) = min {γSR(H), ln(H) + 1} . 4. Resolving Domination in the Corona of Graphs The corona of two graphs G and H, denoted by G◦H, is the graph obtained by taking one copy of G of order n and n copies of H, and then joining every vertex of the ith copy of H to the ith vertex of G. For v ∈ V (G), denoted by Hv the copy of H whose vertices are attached one by one to the vertex v. Subsequently, denote by v+Hv the subgraph of the corona G ◦H corresponding to the join ⟨{v}⟩+Hv, v ∈ V (G). G. Monsanto, H. Rara / Eur. J. Pure Appl. Math, 16 (1) (2023), 18-28 25 Theorem 9. [9] Let G and H be non-trivial connected graphs. Then W ⊆ V (G ◦H) is a resolving set of G ◦H if and only if W ∩ V (Hv) ̸= ∅ for all v ∈ V (G) and W = A ∪ B, where A ⊆ V (G) and B = ∪{Bv : v ∈ V (G) and Bv is a locating set of Hv} . Theorem 10. Let G and H be non-trivial connected graphs. Then W ⊆ V (G ◦H) is a resolving dominating set of G ◦H if and only if W ∩ V (Hv) ̸= ∅ for every v ∈ V (G) and W = A ∪B ∪D, where A ⊆ V (G), B = ∪{Bv : v ∈ A and Bv is a locating set of Hv} and D = ∪{Du : u /∈ A and Du is a locating-dominating set of Hv} . Proof: LetW be a resolving dominating set ofG◦H. Then by Theorem 9,W∩V (Hv) ̸= ∅ for any v ∈ V (G). Since W is a resolving set, G = A ∪B∗, where A ⊆ V (G) and B∗ = ∪{Bv : v ∈ V (G) and Bv is a locating set of Hv} by Theorem 9. Let B = ∪{Bv : v ∈ A} and D = ∪{Bu : u ∈ V (G) \A}. Since W is a dominating set, it follows that Bu is a dominating set for each u ∈ V (G) \A. For the converse, suppose W = A ∪ B ∪D, where A,B and D are the sets possesing the properties described. Then by Theorem 9, W is a resolving set of G ◦H. Since Du is a dominating set of Hu for each u /∈ W , W is a resolving dominating set of G ◦H. Remark 3. [11] For any connected graph G, ln(G) ≤ γL(G) ≤ γSL(G). Corollary 4. Let G and H be non-trivial connected graphs with |V (G)| = n. Then γR(G ◦H) = n · γL(H). Proof: Let W be a minimum resolving dominating set in G◦H. Then W = A∪B ∪D where A,B and D are the sets described in Theorem 10. By Remark 3 and Theorem 5(ii), it follows that γR(G ◦H) = |W | = |A|+ |B|+ |D| ≥ |A|+ |A| · ln(H) + ( n− |A| ) · γL(H) = |A| ( 1 + ln(H) ) + ( n− |A| ) · γL(H) = |A| · γL(H) + ( n− |A| ) · γL(H) = n · γL(H). Now, let F be a minimum locating dominating set of H. For each v ∈ V (G), pick Fv ⊆ V (Hv) with ⟨Fv⟩ ∼= ⟨F ⟩. Then W = ⋃ v∈V (G) Fv is a resolving dominating set of G ◦H by Theorem 10. Hence, γR(G ◦H) ≤ |W | = n · γL(H). Therefore, γR(G ◦H) = n · γL(H). G. Monsanto, H. Rara / Eur. J. Pure Appl. Math, 16 (1) (2023), 18-28 26 5. Resolving Domination in the Lexicographic Product of Graphs The lexicographic product of graphs G and H, denoted by G[H], is the graph with vertex set V (G[H]) = V (G)× V (H) such that (v, a)(u, b) ∈ E(G[H]) if and only if either uv ∈ E(G) or u = v and ab ∈ E(H). Theorem 11. [9] Let G and H be non-trivial connected graphs with ∆(H) ≤ |V (H)| − 2. Then W = ⋃ x∈S [ {x}×Tx ] , where S ⊆ V (G) and Tx ⊆ V (H) for each x ∈ S, is a resolving set of G[H] if and only if W is a locating set of G[H]. Theorem 12. [4] Let G and H be non-trivial be connected graphs. Then C ⊆ V (G [H]) is a dominating set in G[H] if and only if W = ⋃ x∈S [ {x} × Tx ] and either (i) S is a total dominating set in G or (ii) S is a dominating set in G and Tx is a dominating set in H for every x ∈ S \NG(s). Theorem 13. Let G and H be non-trivial connected graphs with ∆(H) ≤ |V (H)| − 2. Then W = ⋃ x∈S [ {x}×Tx ] , where S ⊆ V (G) and Tx ⊆ V (H) for each x ∈ S, is a resolving dominating set of G[H] if and only if W is a locating-dominating set of G[H]. Proof: Suppose W is a resolving dominating set of G[H]. Then by Theorem 11, W is a locating-dominating set of G[H]. The converse follows from Theorem 11 and Theorem 12. Theorem 14. [11] Let G and H be non-trivial connected graphs with ∆(H) ≤ |V (H)|−2. Then W = ⋃ x∈S [ {x} × Tx ] , where S ⊆ V (G) and Tx ⊆ V (H) for each x ∈ S, is a locating-dominating set of G[H] if and only if (i) S = V (G); (ii) Tx is a locating set of H for every x ∈ V (G); (iii) Tx or Ty is strictly locating of H whenever x and y are adjacent vertices of G with NG[x] = NG[y]; and (iv) Tx or Ty is (locating) dominating of H whenever x and y are nonadjacent vertices of G with NG(x) = NG(y). The next result follows immediately from Theorem 13 and Theorem 14. Theorem 15. Let G and H be non-trivial connected graphs with ∆(H) ≤ |V (H)| − 2. Then W = ⋃ x∈S [ {x}×Tx ] , where S ⊆ V (G) and Tx ⊆ V (H) for each x ∈ S, is a resolving dominating set of G[H] if and only if (i) S = V (G); REFERENCES 27 (ii) Tx is a locating set of H for every x ∈ V (G); (iii) Tx or Ty is strictly locating of H whenever x and y are adjacent vertices of G with NG[x] = NG[y]; and (iv) Tx or Ty is (locating) dominating of H whenever x and y are nonadjacent vertices of G with NG(x) = NG(y). The following is a direct consquence of Theorem 15. Corollary 5. Let G be a connected totally point determining graph and let H be a non- trivial connected graph. Then W = ⋃ x∈S [ {x} × Tx ] is a minimum resolving dominating set of G[H] if and only if S = V (G) and Tx is a minimum locating set of H for every x ∈ V (G). Corollary 6. Let G be a connected totally point determining graph and let H be a non- trivial connected graph. Then γR ( G[H] ) = |V (G)| · ln(H). Proof: Let W = ⋃ x∈S ({x} x Tx) be a minimum resolving dominating set of G[H]. Then S = V (G) and Tx is a minimum locating set in H for every x ∈ V (G), by Corollary 5. Therefore, γR(G[H]) = |V (G)| · ln(H). 6. Conclusion This study did introduce the concept of resolving domination under some binary oper- ations. Let G and H be non-trivial connected graphs. It is shown that the resolving dom- ination number in the corona of two graphs is n ·γL(H), the resolving domination number in the join of two graphs is the min {sln(H) + ln(G), sln(G) + ln(H)}, and the resolving domination number of the lexicographic product of graphs G and H is |V (G)| · ln(H). The parameter can be investigated further for graphs under other binary operations. Acknowledgements This research is funded by the Commission on Higher Education (CHED) and Min- danao State University-Iligan Institute of Technology, Philippines. Also, the authors would like to recognize the efforts of the anonymous reviewers whose suggestions and recommen- dations contributed to the big improvement of the paper. 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