EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 14, No. 3, 2021, 829-841 ISSN 1307-5543 – ejpam.com Published by New York Business Global Resolving Restrained Domination in Graphs 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. Let G be a connected graph. Brigham et al. [3] defined a resolving dominating set as a set S of vertices of a connected graph G that is both resolving and dominating. A set S ⊆ V (G) is a resolving restrained dominating set of G if S is a resolving dominating set of G and S = V (G) or 〈V (G) \ S〉 has no isolated vertex. In this paper, we characterize the resolving restrained dominating sets in the join, corona and lexicographic product of graphs and determine the resolving restrained domination number of these graphs. 2020 Mathematics Subject Classifications: 05C69 Key Words and Phrases: Dominating set, resolving set, resolving dominating set, resolving restrained dominating set, 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 [7]. 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 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 ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v14i3.3985 Email addresses: gerald.monsanto@vsu.edu.ph (G. Monsanto), helen.rara@g.msuiit.edu.ph (H. Rara) http://www.ejpam.com 829 © 2021 EJPAM All rights reserved. G. Monsanto, H. Rara / Eur. J. Pure Appl. Math, 14 (3) (2021), 829-841 830 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. A vertex x of a connected graph G is said to resolve two vertices u and v of G if dG(x, u) 6= 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. Brigham et al. [3] 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. Let G be a connected graph. A set S ⊆ V (G) is a strictly resolving dominating set of G if it is a resolving dominating set of G and NG(u) ∩ S 6= S for all 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. Let G = (V (G), E(G)) be a graph. A set S ⊆ V (G) is a restrained dominating set of G if S is a dominating set of G and for every v ∈ V (G)\S there exists u ∈ (V (G)\S)∩NG(v). Equivalently, a dominating subset S of V (G) is a restrained dominating set of G if S = V (G) or 〈V (G) \ S〉 has no isolated vertex. The restrained domination number of G, denoted by γr(G) is the minimum cardinality of a restrained dominating set of G. Any restrained dominating set of G of cardinality γr(G) is referred to as a γr-set of G. Let G be a connected graph. A set S ⊆ V (G) is a resolving restrained dominating set of G if S is a resolving dominating set of G and S = V (G) or 〈V (G) \ S〉 has no isolated vertex. The resolving restrained domination number of G, denoted by γRr(G) is the smallest cardinality of a resolving restrained dominating set of G. Any resolving restrained dominating set of cardinality γRr(G) is referred to as a γRr-set of G. Omega and Canoy [11] defined a locating set of G as a set S ⊆ V (G) if for every two distinct vertices u and v of V (G) \ S, NG(u) ∩ S 6= 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 ln-set of G. Canoy and Malacas [14] defined a locating (resp. strictly locating) subset S of V (G) which is also dominating is called a locating-dominating (resp. strictly locating-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(G)), 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 a strictly locating set of G if it is a locating set of G and NG(u)∩S 6= S for all u ∈ V (G) \S. The strictly locating number of G. Monsanto, H. Rara / Eur. J. Pure Appl. Math, 14 (3) (2021), 829-841 831 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 a sln-set of G. Let G be a connected graph. A set S ⊆ V (G) is a restrained locating set of G if S is a locating set of G and S = V (G) or 〈V (G) \ S〉 has no isolated vertex. The restrained locating number of G, denoted rln(G), is the smallest cardinality of a restrained locating set of G. A restrained locating set of cardinality rln(G) is then referred to as rln-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] 6= NG[v]. It is totally point determining if for any two distinct vertices u and v of G, NG(u) 6= NG(v) and NG[u] 6= NG[v]. In recent years, the concept of domination in graphs has been studied extensively and several research papers have been published on this topic. The said concept was not formally defined mathematically until the publications of the books by Claude Berge [2] in 1958 and Oystein Ore in 1962. In 1977, a survey paper by Cockayne and Hedetniemi [4] began to study the concept of domination. On the other hand, the problem of uniquely recognizing the possible position of an in- truder such as fault in a computer network and spoiled device was the principal motivation in introducing the concept of metric dimension in graphs. Slater [12] brought in the notion of locating sets and its minimum cardinality as lo- cating number. The same concept was also introduced by Harary and Melter [7] but using the terms resolving sets and metric dimension to refer to locating sets and locating number, respectively. However, in recent studies, locating sets and resolving sets are de- fined differently. In 2013, Canoy and Malacas [13], defined a locating set as a subset S of V (G) in a connected graph G satisfying the condition that NG(u) ∩ S 6= NG(v) ∩ S for all u, v ∈ V (G) \ S with u 6= v. Meanwhile, in the same year, Bailey et al. [1] defined a resolving set as a set of vertices S in a graph G such that for any two distinct vertices u, v, there exists x ∈ S such that the distances d(u, x) 6= d(v, x). In 1999, Domke et al. [5] introduced and investigated the concept of restrained domi- nation in graphs. In 2008, Hattingh et al. [8] investigated the same concept and obtained a Nordhaus-Gaddum results for restrained domination and total restrained domination in graphs. Moreover, in 2015, Omega et al. [10] introduced and characterized the restrained locating-dominating sets of some graphs and determined the restrained L-domination num- bers of these graphs. Inspired by the above works, this study aims to define and characterize the resolving restrained dominating sets and determine the resolving restrained domination number in the join, corona and lexicographic product of two graphs. 2. Preliminary Results Remark 1. For any connected graph G of order n ≥ 2, γRr(G) ∈ {2, 3, 4, . . . , n− 2, n} . G. Monsanto, H. Rara / Eur. J. Pure Appl. Math, 14 (3) (2021), 829-841 832 Remark 2. Every resolving restrained dominating set of a connected graph G of order n ≥ 2 is a restrained dominating set of G. Hence, γr(G) ≤ γRr(G). Theorem 1. Let G be a connected graph of order n ≥ 2. Then γRr(G) = n if and only if G ∼= Kn or G ∼= K1,n−1. Proof: Suppose that γRr(G) = n. If n = 2, then G = K2. If n = 3, then G ∼= K3 or G ∼= K1,2. Suppose that n ≥ 4 and suppose further that G � Kn. Let x ∈ V (G) with degG(x) = ∆(G). Suppose there exists y ∈ V (G) \ {x} such that xy /∈ E(G). Since G is connected, y can be chosen so that dG(x, y) = 2. Let z ∈ NG(x)∩NG(y). If yp ∈ E(G) for all p ∈ NG(x), then choose W = V (G) \ {y, z}. Since degG(x) ≥ degG(z) ≥ 2, there exists u ∈ NG(x) \ {z}. Let W = V (G) \ {x, z}. Since xz, xu, zy ∈ E(G), W is a restrained dominating set of G. Moreover, because y ∈ NG(z)\NG(x), rG(x/W ) 6= rG(z/W ). Thus, W is a resolving set of G and we have γRr(G) ≤ |W | = n− 2, a contradiction. Therefore, xy ∈ E(G) for all y ∈ V (G) \ {x}. It remains to show that uv /∈ E(G) for every distinct vertices u, v ∈ V (G) \ {x}. To this end, suppose there exist distinct vertices u and v in V (G) \ {x} such that uv ∈ E(G). Since G 6= Kn, there exist a, b ∈ V (G) such that ab /∈ E(G). If av ∈ E(G) or bv ∈ E(G), say av ∈ E(G), then consider W = V (G) \ {x, a}. Note that xv, av ∈ E(G). Thus, W is a dominating set of G. Since b ∈ NG(x) \ NG(a), rG(a/W ) 6= rG(x/W ). Thus, W is a resolving dominating set of G. Since V (G) \ W = {x, a} and xa ∈ E(G), it follows that W is a resolving restrained dominating set of G. If av, bv /∈ E(G), then take W = V (G) \ {x, v}. Again, W is a resolving restrained dominating set of G. In either case, γR(G) ≤ |W | = n− 2, a contradiction to the asssumption. Therefore, uv /∈ E(G) for every two distinct vertices u, v ∈ V (G) \ {x}. Thus, G ∼= K1,n−1. The converse is easy. Theorem 2. Let G be a connected graph of order n = 4. Then γRr = 2 if and only if G /∈ {K4,K1,3}. Proof: Suppose that γRr(G) = 2. Then by Theorem 1, G /∈ {K4,K1,3}. For the converse, suppose that G /∈ {K4,K1,3}. Since n = 4, γR(G) ≥ 2. Choose x, y ∈ V (G) such that dG(x, y) = 2. Let z ∈ NG(x) ∩ NG(y) and p ∈ V (G) \ {x, y, z}. Consider the following cases: Case 1. Suppose that p ∈ NG(x) ∩NG(y). Let W = {x, z}. Then W is a resolving restrained dominating set of G. Case 2. Suppose that p ∈ NG(x) \NG(y). Then W = {y, p}, thus W is a resolving restrained dominating set of G. Case 3. Suppose that p ∈ NG(y) \NG(x). Then W = {x, p}, hence W is a resolving restrained dominating set of G. Therefore, in all cases, γRr(G) = 2. Theorem 3. Let G be a connected graph of order n = 5. Then γRr(G) = 2 if and only if there exist distinct vertices x and y that dominate G such that |NG(x) ∩ NG(y)| = 1, |NG(x) \ {y} | = |NG(y) \ {x} | = 2 and 〈V (G) \ {x, y}〉 has no isolated vertex. G. Monsanto, H. Rara / Eur. J. Pure Appl. Math, 14 (3) (2021), 829-841 833 Proof: Suppose that γRr(G) = 2. Then there exist distinct vertices x and y such that W = {x, y} is a minimum restrained resolving dominating set of G. Hence, |NG(x) ∩ NG(y)| ≤ 1. Suppose |NG(x) ∩NG(y)| = 0. Then one of x and y, say x, has at least two neighbors, u1, u2 ∈ (V (G) \W ) \ NG(y). Thus, rG(u1/W ) = rG(u2/W ), a contradiction to the assumption. Hence, |NG(x) ∩ NG(y)| = 1. Next, let z ∈ NG(x) ∩ NG(y) and let b, c ∈ V (G) \ {x, y, z}. Then b, c /∈ NG(x) ∩NG(y). Since W is dominating, b ∈ NG(x) or b ∈ NG(y). Suppose b ∈ NG(x). Then W is a resolving dominating set of G implies that c ∈ NG(y) \ NG(x). Hence, |NG(x) \ {y} | = |NG(y) \ {x} | = 2. Since W is a restrained resolving dominating set, 〈V (G) \W 〉 has no isolated vertex. For the converse, suppose there exist distinct vertices x, y ∈ V (G) satisfying the given properties. Let W = {x, y}. Then W is a resolving restrained dominating set of G. Hence, γRr(G) = 2. 3. Resolving Restrained Domination in the Join of Graphs Theorem 4. [6] 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) 6= ∅ and C ∩ V (H) 6= ∅. Theorem 5. [14] Let G and H be non-trivial connected graphs. A set W ⊆ V (G+H) is a locating-dominating set of G+H if and only if W = WG ∪WH where WG ⊆ V (G) and WH ⊆ V (H) are locating-dominating sets of G and H, respectively, where WG or WH is a strictly locating set. Theorem 6. [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 7. 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 6, 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 dominatings sets of G and H, respectively. By Theorem 5, W is a locating-dominating set of G+H. The converse immediately follows from Theorem 5 and Theorem 4(iii). The next result follows immediately from Theorem 5. Theorem 8. 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, 14 (3) (2021), 829-841 834 Theorem 9. Let G and H be nontrivial connected graphs. A set W ⊆ V (G + H) is a resolving restrained dominating set of G + H if and only if W = WG ∪WH and satisfies the following conditions: (i) WG and WH are locating sets of G and H, respectively; (ii) WG (WH) is a restrained locating set of G (resp. H) whenever WG = V (G) (resp. WH = V (H)); and (iii) WG or WH is a strictly locating set. Proof: Let W ⊆ V (G+H) be a resolving restrained dominating set of G+H. Then by Theorem 8, 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. Hence, (i) and (iii) hold. Suppose WG = V (G) or WH = V (H). Since W = V (G + H) or V (G + H) \W has no isolated vertex, WH = V (H) or V (H) \WH has no isolated vertex. Hence, WH is a restrained locating set of H. Similarly, WG is a restrained locating set of G if WH = V (H). Thus, (ii) holds. For the converse, suppose that W = WG ∪WH and satisfies (i), (ii), and (iii), where WG and WH are locating sets of G and H, respectively, and WG or WH is a strictly locating set. Then by Theorem 8, W is a resolving dominating set of G+H. By (ii) and the fact that WG and WH are non-empty, W is a restrained dominating set of G+H. Lemma 1. Let G and H be non-trivial connected graphs such that sln(G) = |V (G)| = m and sln(H) 6= |V (H)|. Then m+ rln(H) ≥ sln(H) + ln(G). Proof: If ln(H) = sln(H), then m + rln(H) ≥ ln(G) + sln(H). Suppose ln(H) < sln(H). Then ln(H) = sln(H)− 1 ≤ rln(H). Hence, m+ rln(H) ≥ m+ sln(H)− 1 = sln(H) + (m− 1) = sln(H) + ln(G). Theorem 10. [11] Let G be a connected graph of order n ≥ 2. If ln(G) < sln(G), then 1 + ln(G) = sln(G). Corollary 1. Let G and H be non-trivial connected graphs of order m and n, respectively. Then γRr(G+H) =  m+ n , if sln(G) = m and sln(H) = n sln(H) + ln(G) , if sln(G) = m and sln(H) 6= n sln(G) + ln(H) , if sln(G) 6= m and sln(H) = n min {sln(H) + ln(G), sln(G) + ln(H)} , if sln(G) 6= m and sln(H) 6= n. G. Monsanto, H. Rara / Eur. J. Pure Appl. Math, 14 (3) (2021), 829-841 835 Proof: Consider the following cases: Case 1. Suppose sln(G) = m and sln(H) = n. Then V (G) and V (H) are the only strictly locating sets of G and H, respectively. Since ln(G) ≤ m − 1, with 1 + ln(G) = sln(G) by Theorem 10. Thus, ln(G) = m − 1. Since ln(G) ≤ rln(G) and rln(G) cannot be equal to m − 1, it follows that rln(G) = m. Similarly, rln(H) = n. Thus, V (G) and V (H) are the only restrained locating sets of G and H, respectively. Hence, by Theorem 10, if W is a minimum restrained resolving dominating set of G+H, then W = V (G) ∪ V (H). Therefore, γRr(G+H) = m+ n. Case 2. Suppose sln(G) = m and sln(H) 6= n. Suppose first that sln(G) = m and sln(H) 6= n. Let WG and WH be minimum locating set and minimum strictly locating set of G and H, respectively. Then by Theorem 9, W = WG ∪WH is a restrained resolving dominating set of G + H. Thus, γRr(G + H) ≤ |W | = sln(H) + ln(G). Case 3. Suppose sln(G) 6= m and sln(H) = n. Now, suppose that W ′ is a minimum resolving restrained dominating set of G + H. Then by Theorem 9, W ′ = W ′G ∪ W ′H , where W ′G 6= V (G) is a locating set of G and W ′H 6= V (H) is a strictly locating set of H. Hence, γRr(G+H) = |W ′| ≥ ln(G) + sln(H). Therefore, γRr(G+H) = ln(G) + sln(H). Similarly, if sln(G) 6= m and sln(H) = n, then γRr(G+H) = sln(G) + ln(H). Case 4. Suppose sln(G) 6= m and sln(H) 6= n. Let W be a minimum resolving restrained dominating set of G + H. Let WG = V (G) ∩W and WH = V (H) ∩W . Then by Theorem 9, WG and WH are locating sets of G and H, respectively, where WG or WH is a strictly locating set. If WG = V (G), then by Theorem 10(i), WH is a restrained locating set of H. Thus, γRr(G+H) = m+ rln(H) ≥ sln(G) + ln(H) by Lemma 1. Similarly, by Theorem 9(i), if WH = V (H), then γRr(G+H) = n+rln(G) ≥ sln(H) + ln(G) by Lemma 1. Suppose that WG 6= V (G) and WH 6= V (H). Assume first that WG is a strictly locating set of G. Then sln(G) + ln(H) ≤ |WG|+ |WH | = |W | = γRr(G+H). If WH is a strictly locating set of H, then sln(H) + ln(G) ≤ |WH |+ |WG| = |W | = γRr(G+H). Thus, γRr(G+H) ≥ min {sln(G) + ln(H), sln(H) + ln(G)}. Let WG and W ′H be minimum strictly locating sets of G and H, respectively, and let WH and W ′G be minimum locating sets of H and G, respectively. Then W = WG ∪WH and W ′ = W ′G ∪W ′H are resolving restrained dominating sets of G + H by Theorem 8. Thus, γRr(G+H) ≤ |W | = |WG|+ |WH | = sln(G) + ln(H) and γRr(G+H) ≤ |W ′| = ln(G) + sln(H). Therefore, γRr(G+H) = min {sln(G) + ln(H), sln(H) + ln(G)} . G. Monsanto, H. Rara / Eur. J. Pure Appl. Math, 14 (3) (2021), 829-841 836 Lemma 2. Let H be a non-trivial connected graph of order m and let Kn be a complete graph of order n ≥ 3. Then n− 1 + sln(H) ≤ n+ rln(H). Proof: Suppose ln(H) < sln(H). Then ln(H) = sln(H)−1. If ln(H) = rln(H). Then rln(H) = sln(H) − 1. Hence, sln(H) + n − 1 = n + rln(H). If ln(H) = sln(H), then sln(H) ≤ rln(H). Hence, sln(H)− 1 < rln(H), that is, n− 1 + sln(H) < n+ rln(H). The next result follows immediately from Corollary 1. Corollary 2. Let H be a non-trivial connected graph of order m and let Kn be a complete graph of order n ≥ 3. Then γRr(H +Kn) = { sln(H) + n− 1 , if sln(H) 6= m m+ n , otherwise. Theorem 11. [14] 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}∪W1, where W1 is a locating set of H. The next result follows immediately from Theorem 11. Theorem 12. 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. Theorem 13. Let H be a non-trivial connected graph and let K1 = 〈v〉. Then W ⊆ V (H +K1) is a resolving restrained dominating set of H +K1 if and only if either v /∈W and W is a strictly resolving dominating set of H with V (H) 6= W or W = {v} ∪WH , where WH is a restrained locating set of H. Proof: Suppose that W is a resolving restrained dominating set of H + K1 and let WH = V (H) ∩ W . Then by Theorem 12, W = WH ∪ {v}, where WH is a lo- cating set of H or v /∈ W and W is a strictly resolving dominating set of H. Sup- pose that v /∈ W . Since 〈V (H +K1) \W 〉 = 〈{v} ∪ (V (H) \W )〉 has no isolated ver- tex, it follows that W 6= V (H). Next, suppose that W = WH ∪ {v}. Again, since 〈V (H +K1) \W 〉 = 〈V (H) \WH〉, and W is a restrained locating set, it follows that WH = V (H) or 〈V (H) \WH〉 has no isolated vertex. Hence, WH is a restrained locating set of H. Conversely, assume first that v /∈ W and W is a strictly resolving dominating set of H with W 6= V (H). By Theorem 12, W is a resolving dominating set of H + K1. Since 〈V (H +K1) \W 〉 has no isolated vertex. Thus, W is a restrained dominating set of H + K1. Finally, suppose that W = WH ∪ {v}, where WH is a restrained locating set of H. By Theorem 12, W is a resolving dominating set of H +K1. Consequently, W is a restrained dominating set of H + K1. Therefore, W is a resolving restrained dominating set of H +K1. G. Monsanto, H. Rara / Eur. J. Pure Appl. Math, 14 (3) (2021), 829-841 837 Corollary 3. Let G be a non-trivial connected graph of order m. Then γRr(G+K1) =  m , if γSR(G) = m and rln(G) = m; rln(G) + 1 , if γSR(G) = m and rln(G) 6= m; min {γSR(G), rln(G) + 1} , if γSR(G) 6= m and rln(G) 6= m. 4. Resolving Restrained Domination in the Corona of Graphs Theorem 14. [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) 6= ∅ for every v ∈ V (G) and W = A∪B, where A ⊆ V (G) and B = ∪{Bv : v ∈ A and Bv is a locating set of Hv} . Theorem 15. 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) 6= ∅ 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 Hu} . Proof: Let W be a resolving dominating set of G ◦ H. Then by Theorem 14, W ∩ V (Hv) 6= ∅ 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 14. 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 14, 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. Theorem 16. Let G and H be non-trivial connected graphs. Then W ⊆ V (G ◦ H) is a resolving restrained dominating set of G ◦ H if and only if W ∩ V (Hv) 6= ∅ for every v ∈ V (G) and W = A ∪B ∪D, where A ⊆ V (G), B = ∪{Bv : v ∈ A and Bv is a restrained locating set of Hv} and D = ∪{Du : u /∈ A and Du is a locating-dominating set of Hu} where u ∈ NG(V (G) \A) if Du = V (Hu)}. Proof: Suppose W is a resolving restrained set of G◦H. By Theorem 15, W ∩V (Hv) 6= ∅ for all v ∈ V (G) and W = A∪B ∪D where A,B and D are sets described in Theorem 15. Since W is a restrained dominating set of G ◦ H, Bv is a restrained dominating set of Hv for each v ∈ A, and u ∈ NG(V (G) \ A) for each u ∈ V (G) \ A with Du = V (Hu). Therefore, B = ∪{Bv : v ∈ A and Bv is a restrained locating set of Hv} and G. Monsanto, H. Rara / Eur. J. Pure Appl. Math, 14 (3) (2021), 829-841 838 D = ∪{Du : u /∈ A and Du is a locating dominating set of Hu} where u ∈ NG ( V (G) \A ) if Du = V (Hu)}. For the converse, suppose that W ∩V (Hv) 6= ∅ for every v ∈ V (G) and W = A∪B∪D where A,B and D are as described above. By Theorem 15, W is a resolving dominating set of G ◦H. Further, because of the additional properties of the sets in B and D, W is a restrained dominating set of G ◦H. Remark 3. [11] For any connected graph G, ln(G) ≤ γL(G) ≤ γSL(G). Theorem 17. [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 4. Let G and H be non-trivial connected graphs with |V (G)| = n. Then γRr(G ◦H) = n · γL(H). Proof: Let W be a minimum resolving restrained dominating of G ◦ H. Then W = A∪B ∪D are the sets described in Theorem 16. By Remark 3 and Theorem 17, it follows that γRr(G ◦H) = |W | = |A|+ |B|+ |D| ≥ |A|+ |A| · rln(H) + ( n− |A| ) · γL(H) ≥ |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 restrained dominating set of G ◦H by Theorem 16. Hence, γRr(G ◦H) ≤ |W | = n · γL(H). Therefore, γRr(G ◦H) = n · γL(H). 5. Resolving Restrained Domination in the Lexicographic Product of Graphs Theorem 18. [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]. G. Monsanto, H. Rara / Eur. J. Pure Appl. Math, 14 (3) (2021), 829-841 839 Theorem 19. [6] Let G and H be non-trivial connected graphs. Then C ⊆ V ( G[H] ) is a dominating set in G[H] if and only if C = ⋃ 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 20. [14] 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 set of H whenever x and y are adjacent vertices of G with NG[x] = NG[y]; and (iv) Tx or Ty is (locating) dominating set of H whenever x and y are nonadjacent vertices of G with NG(x) = NG(y). Theorem 21. 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 20, W is a locating-dominating set of G[H]. The converse follows from Theorem 20 and Theorem 19. The next result follows immediately from Theorem 20 and Theorem 21. Theorem 22. 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); (ii) Tx is a locating set of H for every x ∈ V (G); (iii) Tx or Ty is strictly locating set of H whenever x and y are adjacent vertices of G with NG[x] = NG[y]; and (iv) Tx or Ty is locating-dominating set of H whenever x and y are nonadjacent vertices of G with NG(x) = NG(y). Theorem 23. 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 restrained dominating set of G[H] if and only if it is a resolving dominating set. REFERENCES 840 Proof: Let W = ⋃ x∈S [ {x} × Tx ] where S ⊆ V (G) and Tx ⊆ V (H) for each x ∈ S be a resolving restrained dominating set of G[H]. Then by Theorem 22, (i), (iii) and (iv) hold and Tx is a locating set of H for every x ∈ V (G). Now, let S′ = {y ∈ V (G) : Ty 6= V (H)} and let x ∈ S′\NG(S′). Suppose that Tx is not a restrained locating set. Then 〈V (H) \ Tx〉 has an isolated vertex, say u. Then (x, u) is an isolated vertex in V ( G[H] ) \W , contrary to the assumption that W is a resolving restrained dominating set of G[H]. Hence, Tx is a restrained locating set of H for all x ∈ S′\NG(S′). Therefore, W is a resolving dominating set of G[H]. For the converse, suppose that W = ⋃ x∈S [ {x} × Tx ] is a resolving dominating set of G[H]. Suppose that V ( G[H] ) = W . Then W is a resolving restrained dominating set of G[H]. Suppose that V ( G[H] ) 6= W . Let (y, u) ∈ V ( G[H] ) \W . Then u /∈ Ty. Hence, y ∈ S′. If y ∈ NG(S′), then there exists z ∈ S′ ∩NG(y). Pick any v ∈ V (H) \ Tz. Then (y, u)(z, v) ∈ E ( G[H] ) . If y /∈ NG(S′), then by (ii), there exists p ∈ ( V (H) \Ty ) ∩NG(u). Thus, (y, u)(y, p) ∈ E ( G[H] ) . Hence 〈 V ( G[H] ) \W 〉 has no isolated vertex. Therefore, W is a resolving restrained dominating set of G[H]. Corollary 5. Let G be a connected totally point determining graph of order n ≥ 3 and let H be any non-trivial connected graph. Then γRr ( G[H] ) ≤ |V (G)| · ln(H). Proof: Let D be a minimum locating set of H and let Tx = D for each x ∈ V (G). Then Tx 6= V (H) for all x ∈ V (G), that is, S′ = V (G). By Theorem 23 and hypothesis, W =⋃ x∈V (G) [ {x} × Tx ] is a resolving restrained dominating set of G[H]. Thus, γRr ( G[H] ) ≤ |W | = |V (G)| · ln(H). Acknowledgements This research is funded by the Commission on Higher Education (CHED) and Min- danao State University-Iligan Institute of Technology, Philippines. References [1] R.F. Bailey, J.Cáceres, A. González D. Garijo, A. Márquez. K. Meagher, and M.L. Puertas. Resolving sets for johnson and kneser graphs. European Journal of Combi- natorics, 34:736–751, 2013. [2] C. Berge. Theorie des graphes et ses applications. Dunod, Paris, 1958. [3] R.C. Brigham, G. Chartrand, R.D. Dutton, and P. Zhang. Resolving domination in graphs. 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