EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 14, No. 3, 2021, 638-649 ISSN 1307-5543 – ejpam.com Published by New York Business Global Stable Locating-Dominating Sets in Graphs Eman C. Ahmad1,∗, Gina A. Malacas1, Sergio R. Canoy, Jr.1 1 Department of Mathematics and Statistics, College of Science and Mathematics, Center for Graph Theory, Algebra and Analysis-PRISM, MSU-Iligan Institute of Technology, 9200 Iligan City, Philippines Abstract. A set S ⊆ V (G) of a (simple) undirected graph G is a locating-dominating set of G if for each v ∈ V (G) \ S, there exists w ∈ S such tha vw ∈ E(G) and NG(x) ∩ S 6= NG(y) ∩ S for any distinct vertices x and y in V (G) \ S. S is a stable locating-dominating set of G if it is a locating-dominating set of G and S \ {v} is a locating-dominating set of G for each v ∈ S. The minimum cardinality of a stable locating-dominating set of G, denoted by γsl (G), is called the stable locating-domination number of G. In this paper, we investigate this concept and the corresponding parameter for some graphs. Further, we introduce other related concepts and use them to characterize the stable locating-dominating sets in some graphs. 2020 Mathematics Subject Classifications: 05C69 Key Words and Phrases: locating, stable, domination, join, corona 1. Introduction The standard concept of domination in a graph has been continuously modified to give rise to new domination parameters. Indeed, a lot of variations of domination have been introduced and studied at different angles and in many ways. One variation of domination which finds an interesting application in the location-determination problem of monitoring devices in a system to ensure its safety was defined and studied by Slater in [6] and [7]. This variant is called locating-domination, a combination of the concepts of locating and domination, and can be used to model a protection strategy that determines locations of monitoring devices (e.g. fire alarms or surveilance cameras) in such a way that the exact location of an intruder (e.g. fire, burglar) can be singled out when a problem (presence of an intruder or fire) at a facility or system arises. The papers in [1], [2], [3], [4], and [5] also dealt with the concept of locating-domination and some of its related concepts. ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v14i3.3998 Email addresses: eman.ahmad@g.msuiit.edu.ph (E. Ahmad), gina.malacas@g.msuiit.edu.ph (G. Malacas), sergio.canoy@g.msuiit.edu.ph (S. Canoy, Jr.) http://www.ejpam.com 638 © 2021 EJPAM All rights reserved. E. Ahmad, G. Malacas, S. Canoy, Jr. / Eur. J. Pure Appl. Math, 14 (3) (2021), 638-649 639 In a system where monitoring devices are installed using the concept of locating- domination, an interesting issue to consider is when, at a given time, exactly one device unexpectedly becomes defective or non-functional. When this situation happens, the re- maining number of devices may not necessarily function as intended, that is, presence of an intruder at a certain location in a system may not be precisely detected or iden- tified. To address this specific problem, an additional condition can be imposed to the locating-domination concept to ensure stability and consistency of the devices installed using the modified concept. Thus, in this paper, we introduce the concept of stable locating-dominating set in a graph. Let G = (V (G), E(G)) be a simple graph. The distance between two vertices u and v of G, denoted by dG(u, v), is equal to the length of a shortest path connecting u and v. Any path connecting u and v of length dG(u, v) is called a u-v geodesic. The open neighbourhood of a vertex v of G is the set NG(v) = {u ∈ V (G) : uv ∈ E(G)} and its closed neighbourhood is the set NG[v] = NG(v) ∪ {v}. The open neighbourhood of a subset S of V (G) is the set NG(S) = ∪v∈SNG(v) and its closed neighbourhood is the set NG[S] = NG(S) ∪ S. The degree of v, denoted by degG(v), is equal to |NG(v)|. A vertex v of G is called isolated if degG(v) = 0. Vertex v is a leaf if degG(v) = 1 and the vertex u ∈ V (G) ∩NG(v) is called a support vertex. The minimum degree of G is δ(G) = min{degG(v) : v ∈ V (G)} and its maximum degree is ∆(G) = max{degG(v) : v ∈ V (G)}. A set S ⊆ V (G) is a dominating set (resp. total dominating set) of G if NG[S] = V (G) (resp. NG(S) = V (G)). The smallest cardinality of a dominating set of G, denoted by γ(G), is called the domination number of G. A dominating set of G with cardinality γ(G) is called a γ-set of G. A subset S of G is a locating set in G if NG(u)∩ S 6= NG(v)∩ S for every two distinct vertices u and v of V (G) \ S. Set S is said to be a strictly locating set if it is a locating set and NG(u) ∩ S 6= S for all u ∈ V (G) \ S. A locating (resp. strictly locating ) subset S of V (G) which is also a dominating set is called a locating-dominating (resp. strictly locating-dominating set in 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 locating-domination(resp. strictly locating-domination) number of G. A locating set (resp. strictly locating set) S in G is a stable locating set (resp. stable strictly locating set) in G if Sv = S \ {v} is a locating set (resp. strictly locating set) of G for each v ∈ S. A locating-dominating (strictly locating dominating) set S of G is a stable locating-dominating set (resp. stable strictly locating-dominating set of G if Sv = S \ {v} is a locating-dominating set (resp. strictly locating-dominating set) of G for each v ∈ S. The minimum cardinality of a stable locating dominating set (resp. stable strictly locating-dominating set) of G, denoted by γsl (G) (resp. γssl(G) ), is called the stable locating-domination (resp. stable strictly locating-domination) number of G. A stable locating-dominating (resp. stable strictly locating-dominating) set of G with cardinality γsl (G) (resp. γssl(G)) is called a γsl -set (resp. γssl-set) of G. Slater in [8] introduced and studied the concept of fault-tolerant locating-dominating set. He showed that every fault-tolerant locating-dominating set is also a stable locating-dominating set. E. Ahmad, G. Malacas, S. Canoy, Jr. / Eur. J. Pure Appl. Math, 14 (3) (2021), 638-649 640 2. Results Given a graph G, we denote by L(G) and S(G) the sets of leaves and support vertices of G, respectively. Proposition 1. Let G be a non-trivial graph. Then G admits a stable locating-dominating set if and only if G has no isolated vertices. If G has no isolated vertices, then 2 ≤ γsl (G) ≤ |V (G)|. Moreover, the following statements hold: (i) γsl (G) = 2 if and only if G = K2. (ii) If S is a stable locating-dominating set of G, then L(G) ∪ S(G) ⊆ S. (iii) γsl (G) = |V (G)| if and only if for every v ∈ V (G), v ∈ L(G) ∪ S(G) or there exists w ∈ V (G) \ {v} such that NG(v) = NG(w). Proof. Suppose G has no isolated vertices. Let S = V (G) and let v ∈ S. Clearly, S \ {v} is locating set. Since v is not an isolated vertex, there exists u ∈ S \ {v} such that uv ∈ E(G). Hence, S is a dominating set, showing that S is a stable locating-dominating set of G. For the converse, suppose that G admits a stable locating-dominating set, say S′. Suppose further that G has an isolated vertex, say z. Since S′ is a dominating set, z ∈ S′. This implies that S′ \ {z} is not a dominating set, contrary to the assumption that S is a stable locating-dominating set of G. Therefore, G has no isolated vertices. Next, suppose that G has no isolated vertices. Let S be a stable locating-dominating set of G. Clearly, |S| ≤ |V (G)|. Since an empty set is not a dominating set, |S| ≥ 2. Since S was an arbitrary stable locating-dominating set, it follows that 2 ≤ γsl (G) ≤ |V (G)|. (i) Clearly, γsl (K2) = 2. For the converse, suppose that γsl (G) = 2, say S = {x, y} is γsl -set of G. Since Sx = S \ {x} = {y} is a dominating set, xy ∈ E(G). Suppose there exists z ∈ V (G) \ S. Then NG(x) ∩ Sx = {y} = NG(z) ∩ Sx, showing that Sx is not a locating set. Therefore, S is not a stable locating-dominating set, contrary to our assumption that S is γsl -set. Therefore, G = K2. (ii) Let S be a stable locating-dominating set. Let v ∈ L(G) and let u ∈ NG(v). Suppose v /∈ S. Since S is a dominating set, u ∈ S. This, however, would mean that Su = S \ {u} is not a dominating set (there is no vertex in Su that dominates v), contrary to the assumption that S is a stable locating-dominating set of G. Thus, v ∈ S and, consequently, L(G) ⊆ S. Next, let w ∈ S(G) and let z ∈ L(G) be such that zw ∈ E(G). Since z ∈ S (by the first part) and w /∈ S, Sz = S \ {z} is not a dominating set (because w /∈ Sz), a contradiction. This implies that w ∈ S, that is, S(G) ⊆ S. Therefore, L(G) ∪ S(G) ⊆ S. (iii) Suppose γsl (G) = |V (G)|. Let v be vertex ofG such that v /∈ L(G)∪S(G). Suppose further that NG(v) 6= NG(w) for all w ∈ V (G) \ {v}. Let S = V (G) \ {v}. Clearly, S is a locating-dominating set of G. Now let z ∈ V (G) \ {v} and set Sz = S \ {z}. Suppose vz /∈ E(G). Choose any x, y ∈ V (G) such that xz, vy ∈ E(G). Then x, y ∈ Sz. Next, suppose E. Ahmad, G. Malacas, S. Canoy, Jr. / Eur. J. Pure Appl. Math, 14 (3) (2021), 638-649 641 that vz ∈ E(G). Since v /∈ S(G), z /∈ L(G). Hence, there exists a vertex a ∈ Sz such that az ∈ E(G). Further, since v /∈ L(G), there exists b ∈ Sz such that bv ∈ E(G). Therefore, Sz is a dominating set of G. By the additional assumption that NG(v) 6= NG(w), it follows that Sz is a locating set in G. Since z was arbitrarily chosen from S, it follows that S is a stable locating-dominating set of G. Thus, γsl (G) ≤ |S| = |V (G) − 1, a contradiction. Consequently, there exists a w ∈ V (G) \ {v} such that NG(v) = NG(w). For the converse, suppose that the given conditions hold in G and let S be a γsl -set of G. Suppose there exists v ∈ V (G) \ S. By (ii), v /∈ L(G) ∪ S(G) ⊆ S. It follows from the assumption that there exists w ∈ V (G) \ {v} such that NG(v) = NG(w). Since S is a locating set, w ∈ S. However, the condition that NG(v) = NG(w) would imply that NG(v) ∩ Sw = NG(w) ∩ Sw, where Sw = S \ {w}. Hence, Sw is not a locating set of G, contrary to the assumption that S is a stable locating-dominating set of G. Therefore, S = V (G) and γsl (G) = |V (G)|. The join of two graphs G and H, denoted by G + H is the graph 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. Let G be a graph without isolated vertices and K1 = 〈v〉. A set S is a stable locating-dominating set of H = K1 +G if and only if (i) v ∈ S and SG = S \ {v} is both a strictly locating-dominating and stable locating set of G or (ii) S is a stable strictly locating-dominating set of G. Proof. Suppose that S is a stable locating-dominating set of H. Consider the following cases: Case 1. v ∈ S Since S is a stable locating-dominating set of H, SG = S \{v} is a locating-dominating set of H. Hence, NH(v) ∩ SG = SG 6= NH(w) ∩ SG = NG(w) ∩ SG for all w ∈ V (G) \ SG. Also, for x, y ∈ V (G)\SG with x 6= y, we have [NG(x)∩SG]∪{v} = NH(x)∩S 6= NH(y)∩ S = [NG(y) ∩ SG] ∪ {v}. Thus, NG(x) ∩ SG 6= NG(y) ∩ SG, showing that SG is a strictly locating-dominating set of G. Next, let z ∈ SG and let Sz G = SG \ {z}. Since Sz = S \ {z} is a locating-dominating set in H, NH(p) ∩ Sz 6= NH(q) ∩ Sz for all p, q ∈ V (G) \ Sz G with p 6= q. Since Sz = (SG ∪ {v}) \ {z} = Sz G ∪ {v}, NG(p) ∩ Sz G 6= NG(q) ∩ Sz G for all p, q ∈ V (G) \ Sz G with p 6= q. This implies that Sz G is a locating set in G. Therefore, SG is a stable locating set in G. This shows that (i) holds. Case 2. v /∈ S Clearly, S is a dominating set of G. Since S is a locating-dominating set of H. H and v /∈ S, NG(x) ∩ S = NH(x) ∩ S 6= NH(y) ∩ S = NG(y) ∩ S for all x, y ∈ V (G) \ S with x 6= y and S = NH(v) ∩ S 6= NH(z) ∩ S = NG(z) ∩ S for all z ∈ V (G) \ S. Hence, S is a strictly locating-dominating set in G. Next, let w ∈ S and let Sw = S \ {w}. Since S is a stable locating-dominating set of H, Sw is a locating-dominating set in H. This implies that NG(a) ∩ Sw = NH(a) ∩ Sw 6= NH(b) ∩ Sw = NG(b) ∩ Sw for all a, b ∈ V (G) \ Sw E. Ahmad, G. Malacas, S. Canoy, Jr. / Eur. J. Pure Appl. Math, 14 (3) (2021), 638-649 642 with a 6= b. Since v /∈ Sw, Sw = NH(v) ∩ Sw 6= NH(z) ∩ Sw = NG(z) ∩ Sw for all z ∈ V (G) \ Sw. Thus, Sw is a strictly locating-dominating set in G. Therefore, S is a stable strictly locating-dominating set of G, showing that (ii) holds. For the converse, suppose first that (i) holds. Since SG = S \ {v} is a strictly locating- dominating set of G, S is a locating-dominating set of H. Let x ∈ S and let Sx = S \ {x}. If x = v, then Sx = SG is a strictly locating-dominating set of G by assumption. Hence, NH(a)∩Sx = NG(a)∩Sx 6= NG(b)∩Sw = NH(b)∩Sx for all a, b ∈ V (H)\ (Sx∪{v}) with a 6= b, and NH(x)∩Sx = Sx 6= NG(d)∩Sx = NH(d)∩Sx for all d ∈ V (H)\(Sx∪{v}). This implies that Sx is an locating-dominating set of H. Suppose x 6= v. Then x ∈ SG. Since SG is an stable locating set of G, Sx G = SG\{x} is a locating set of G. Since Sx = Sx G∪{v}, Sx is a locating set of H. Further, because v ∈ Sx, Sx is a locating-dominating set in H. Therefore, S is a stable locating -dominating set of H. Next, suppose that (ii) holds. Since S is strictly locating-dominating set in G, it is a locating-dominating set of H. Let z ∈ S. By assumption, Sz = S \ {z} is a strictly locating-dominating set G. Therefore, Sz is a locating-dominating set of H. This shows that S is a stable locating-dominating set of H. Corollary 1. Let G be a graph without isolated vertices. Then γsl (K1 +G) = { ρ(G), if ρ(G) = γssl(G) ρ(G) + 1, if ρ(G) < γssl(G), where ρ(G) = min{|S| : S is both a strictly locating-dominating and a stable locating set of G}. Proof. Let S be a γssl-set of G. Then S is a strictly locating-dominating set of G. In particular, S is a locating set of G. Since S is a a stable strictly locating-dominating set of G, S \ {z} is a strictly locating-dominating set of G for each z ∈ S. This implies that S \ {z} is a locating set of G for each z ∈ S. Thus, S is a stable locating set of G, showing that ρ(G) ≤ |S| = γssl(G). Now, suppose that S0 is a γsl -set of K1 + G. Suppose that ρ(G) = γssl(G). Then γssl(G) < ρ(G) + 1. By Theorem 1, S0 must be a γssl-set of G. Hence, γsl (K1 +G) = |S0| = γssl(G). If ρ(G) < γssl(G), then ρ(G)+1 ≤ γssl(G). By Theorem 1, v ∈ S0 and S0\{v} must be both a strictly locating-dominating and stable locating set of G and |S0 \ {v}| = ρ(G). Therefore, γsl (K1 +G) = |S0| = |S0 \ {v}|+ 1 = ρ(G) + 1. This proves the assertion. Theorem 2. Let G and H be non-trivial graphs. A set S is a stable locating-dominating set of G+H if and only if S = SG ∪ SH and SG and SH are stable locating sets of G and H, respectively, and at least one of them is a stable strictly locating set or both of them are strictly locating sets. Proof. Suppose S is a stable locating-dominating set of G + H. Let SG = S ∩ V (G) and SH = S ∩ V (H). Suppose SG = ∅. Then S = SH . Pick x, y ∈ V (G) ⊂ V (G+H) \ S with x 6= y (these vertices exist because G is non-trivial). Then NG+H(x) ∩ S = S = NG+H(y) ∩ S, a contradiction to the fact that S is a locating set. Therefore, SG 6= ∅. E. Ahmad, G. Malacas, S. Canoy, Jr. / Eur. J. Pure Appl. Math, 14 (3) (2021), 638-649 643 Similarly, SH 6= ∅. Now, let a, b ∈ V (G) \ SG with a 6= b. Since S is a locating set of G+H, [NG(a)∩SG)]∪SH = NG+H(a)∩S 6= NG+H(b)∩S = [NG(b)∩SG)]∪SH . Hence, NG(a)∩SG) 6= NG(b)∩SG), showing that SG is a locating set of G. Next, let z ∈ SG and let Sz G = SG \ {z}. Since S is a stable locating-dominating set of G+H, Sz = S \ {z} = Sz G ∪ SH is a locating-dominating in G + H. It follows that for any two distinct vertices p, q ∈ V (G)\Sz G, [NG(p)∩Sz G]∪SH = NG+H(p)∩Sz 6= NG+H(q)∩Sz = [NG(q)∩Sz G)]∪SH . This implies that NG(p) ∩ Sz G 6= NG(q) ∩ Sz G. Thus, Sz G is a locating set in G, showing that SG is a stable locating set of G. Similarly, SH is a stable locating set of H. Suppose that SG and SH are not stable strictly locating sets. Suppose that SG is not a strictly locating set of G. Then there exists v ∈ V (G) \ SG such that NG(v)∩ SG = SG. Suppose SH is not a strictly locating set. Then there exists w ∈ V (H) \ SH such that NH(w) ∩ SH = SH . Consequently, NG+H(v) ∩ S = S = NG+H(w) ∩ S, contrary to the fact that S is a locating set in G+H. Thus, SH is a strictly locating set of H. Now, let y ∈ SH and set Sy H = SH \ {y}. Since SH is a stable locating set of H, it follows that Sy H is a locating set of H. From the assumption that S is a stable locating-dominating set of G + H, the set Sy = S \ {y} = SG ∪ Sy H is a locating set of G + H. This implies that [NG(v) ∩ SG] ∪ Sy H = SG ∪ Sy H = NG+H(v) ∩ Sy 6= NG+H(u) ∩ Sy = SG ∪ [NH(u) ∩ Sy H)] for all u ∈ V (H) \ Sy H . Hence, NH(u) ∩ Sy H) 6= Sy H for all u ∈ V (H) \ Sy H . This shows that Sy H is a strictly locating set of H. Therefore, SH is a stable strictly locating set of H, a contradiction. Therefore, SG is a strictly locating set of G. Similarly, SH is a strictly locating set of H. Conversely, suppose that S = SG∪SH and SG and SH are stable locating sets of G and H, respectively, such that at least one of them is a stable strictly locating set or both of them are strictly locating sets. Then S is a dominating set of G+H. Suppose first that one of SG or SH , say SG is a stable strictly locating set. Let a, b ∈ V (G+H) \S where a 6= b. Since SG and SH are locating sets, NG+H(a) ∩ S 6= NG+H(b) ∩ S if a, b ∈ V (G) \ SG or a, b ∈ V (H)\SH . Suppose a ∈ V (G)\SG and b ∈ V (H)\SH . Since SG is strictly locating, NG(a) ∩ SG 6= SG. Hence, NG+H(a) ∩ S = [NG(a) ∩ SG] ∪ SH 6= SG ∪ [NH(b) ∩ SH ] = NG+H(b)∩S. Therefore, S is a locating set of G+H. Next, let w ∈ S and let Sw = S\{w}. Suppose w ∈ SG and let Sw G = SG \ {w}. Then Sw G is strictly locating set because SG is a stable strictly locating set of G. Following an earlier argument, we find that Sw = Sw G∪SH is a locating-dominating set of G+H. If w ∈ SH , then Sw = S \ {w} = SG ∪ Sw H , where Sw H = SH \ {w} is a locating set of H by an assumption. These and the fact that SG is strictly locating will imply that Sw is a locating-dominating set of G+H. Therefore, S is a stable locating-dominating set of G. Next, suppose that SG and SH are both strictly locating sets. Then S is a locating- dominating set of G + H. Let Let v ∈ S. Suppose that v ∈ SG and let Sv = S \ {v} = (SG \ {v}) ∪ SH . Since SG is a stable locating set of G, Sv G = SG \ {v} is a locating set of G. Clearly, Sv is a dominating set of G. Let x, y ∈ V (G+H) \ Sv with x 6= y. Consider the following cases: Case 1. x, y ∈ V (G) Then x, y ∈ V (G)\Sv G, where Sv G = SG\{v}. Since Sv G is a locating set in G, NG+H(x) ∩ Svs = [NG(x) ∩ Sv G] ∪ SH 6= [NG(y) ∩ Sv G)] ∪ SH = NG+H(y) ∩ Sv. Case 2. x, y ∈ V (H) Then x, y ∈ V (H) \ SH . Since SH is a locating set in H, E. Ahmad, G. Malacas, S. Canoy, Jr. / Eur. J. Pure Appl. Math, 14 (3) (2021), 638-649 644 NG+H(x) ∩ Sv = Sv G ∪ (NH(x) ∪ SH) 6= Sz G ∪ (NH(y) ∩ SH)] = NG+H(y) ∩ Sv. Case 3. x ∈ V (G) and y ∈ V (H) (or y ∈ V (G) and x ∈ V (H)) Then x ∈ V (G) \ Sv G, y ∈ V (H) \ SH , NG+H(x) ∩ Sv = [NG(x) ∩ Sv G] ∪ SH , and NG+H(y)∩Sv = Sv G∪[NH(y)∩SH)]. Since SH is a strictly locating set of H, NH(y)∩SH) 6= SH . Therefore, NG+H(x) ∩ Sv 6= NG+H(y) ∩ Sv. Since SG is also strictly locating, it follows that NG+H(x)∩ Sv 6= NG+H(y)∩ Sv when v ∈ SH and Sv = S \{v} = SG∪ (SH \{v}. Accordingly, S is a stable locating-dominating set of G+H. Lemma 1. Let G be a non-trivial connected graph. If γ(G) = 1, then every nonempty set S ⊆ V (G) is not a stable strictly locating set of G. In particular, G does not admit a stable strictly locating set. Proof. Suppose ∅ 6= S ⊆ V (G) and let {v} be a dominating set of G. If v /∈ S, then NG(v) ∩ S = S. Hence, S is not a strictly locating set. Suppose v ∈ S. Then NG(v) ∩ Sv = Sv, where Sv = S \ {v}. This implies that Sv cannot be a strictly locating set. Therefore, S is not a stable strictly locating set of G. Theorem 3. Let G be a graph without isolated vertices. Then G has a stable strictly locating set if and only if γ(G) 6= 1. Proof. Suppose G has a stable strictly locating set. Then γ(G) 6= 1 by Lemma 1. For the converse, suppose that γ(G) 6= 1. Let S = V (G). Then clearly, S is a strictly locating set. Let w ∈ S and let Sw = S \ {w}. Then Sw is a locating set of G. Moreover, since {w} is not a dominating set of G, NG(w) ∩ Sw 6= Sw. This implies that Sw is a strictly locating set of G. Therefore, S = V (G) is a stable strictly locating set of G. Theorem 4. Let G be a connected graph with ∆(G) = n− 2, where n is the order G. If S is a stable strictly locating set, then v, w ∈ S for every pair of vertices v and w with degG(v) = n− 2 and vw /∈ E(G). Proof. Let v, w ∈ V (G) with degG(v) = n − 2 and vw /∈ E(G). Suppose v /∈ V (G). Since S is a strictly locating set of G, w ∈ S. By the assumption that S is a stable strictly locating set, it follows that Sw = S \ {w} is a strictly locating set. This, however, is not possible because NG(v) ∩ Sw = Sw. Therefore, v ∈ S. Next, suppose that w /∈ S. Then NG(v) ∩ Sv = Sv = S \ {v}. Hence, S is not a stable strictly locating set of G, a contradiction. Thus, w ∈ S, proving our assertion. Given a graph G without isolated vertices, we will use the following notations: ηsls(G) = min{|S| : S is a stable locating set of G} ηssls(G) = min{|S| : S is a stable strictly locating set of G} ηslsls(G) = min{|S| : S is a strictly locating set and a stable locating set of G} E. Ahmad, G. Malacas, S. Canoy, Jr. / Eur. J. Pure Appl. Math, 14 (3) (2021), 638-649 645 Corollary 2. Let G and H be graphs without isolated vertices. (i) If γ(G) = γ(H) = 1, then γsl (G+H) = ηslsls(G) + ηslsls(H). (ii) If γ(G) 6= 1 and γ(H) = 1, then γsl (G+H) = min{ηslsls(G) + ηslsls(H), ηssls(G) + ηsls(H)}. (iii) If γ(G) = 1 and γ(H) 6= 1, then γsl (G+H) = min{ηslsls(G) + ηslsls(H), ηsls(G) + ηssls(H)}. (iv) If γ(G) 6= 1 and γ(H) 6= 1, then γsl (G+H) = min{ηslsls(G) + ηslsls(H), ηsls(G) + ηssls(H), ηssls(G) + ηsls(H)}. Proof. Let S be a γsl -set of G+H. By Theorem 2, S = SG ∪ SH and SG and SH are stable locating sets of G and H, respectively. (i) If γ(G) = γ(H) = 1, then G and H do not admit stable strictly locating sets by Theorem 3. By Theorem 2, SG and SH are both strictly locating sets. Thus, γsl (G+H) = ηslsls(G) + ηslsls(H). (ii) Suppose γ(G) 6= 1 and γ(G) = 1. Then only H does not admit a stable strictly locating set. By Theorem 2 and by combining all possible pairings, we have γsl (G+H) = min{ηslsls(G) + ηslsls(H), ηssls(G) + ηsls(H)}. (iii) This is similar to (ii). (iv) Since each of the graphs G and H admits a stable strictly locating set, it follows from Theorem 2 that γsl (G+H) = min{ηslsls(G) + ηslsls(H), ηssls(G) + ηsls(H), ηssls(G) + ηsls(H)}. These prove our assertions. The corona G ◦H of two graphs G and H is the graph obtained by taking one copy of G and |V (G)| copies of H, and then forming the join 〈{v}〉+Hv = v +Hv, where Hv is a copy of H, for each v ∈ V (G). Theorem 5. Let G be a connected non-trivial graph and let H be any graph without isolated vertices. Then S ⊆ V (G ◦H) is a stable locating-dominating set of G ◦H if and only if S = A ∪ [∪v∈V (G)Dv] and satisfies the following properties: (i) A ⊆ V (G). E. Ahmad, G. Malacas, S. Canoy, Jr. / Eur. J. Pure Appl. Math, 14 (3) (2021), 638-649 646 (ii) Dv is a stable locating-dominating set of Hv for each v ∈ (V (G) \ A), and, in addition, strictly locating when |NG(v) ∩A| = 1. (iii) Dv is a stable strictly locating-dominating set of Hv for each v ∈ V (G) \NG(A). (iv) Dv is a dominating stable locating set for each v ∈ A and, in addition, strictly locating when NG(v) ∩A = ∅. Proof. Suppose that S is a stable locating-dominating set of G ◦H. Let A = S ∩V (G) and let Dv = S ∩ V (Hv) for each v ∈ V (G). Then (i) holds and S = A ∪ [∪v∈V (G)Dv]. Suppose Dv = ∅ for some v ∈ V (G). Since S is a dominating set, v ∈ A. As S is a stable locating-dominating set, this would imply that Sv = S \ {v} is a locating-dominating set of G ◦H. This, however, is impossible because the V (Hv)∩NG◦H [Sv] = ∅. Thus, Sv 6= ∅ for each v ∈ V (G). Let v ∈ V (G) \ A. Since S is a stable locating-dominating set of G ◦ H and v /∈ A, Dv is must be a stable locating dominating set of Hv. Suppose |NG(v) ∩ A| = 1, say w ∈ NG(v)∩A. Again, since S is a stable locating-dominating set of G ◦H, Sw = S \ {w} is a locating-dominating set of G ◦H. We also find that NG◦H(v) ∩ Sw = Dv. Therefore, since Sw is a locating set, NG◦H(a) ∩ Sw = NHv(a) ∩ Dv 6= Dv for all a ∈ V (Hv) \ Dv. Thus, Dv is a strictly locating set of Hv. This shows that (ii) holds. Suppose now that v ∈ V (G) \ NG[A]. Then, by Theorem 1(ii), Dv is a stable strictly locating-dominating set of Hv, showing that (iii) holds. Next, suppose that v ∈ A. Since S is a locating set of G ◦ H and v ∈ A, [NHv(p) ∩ Dv] ∪ {v} = NG◦H(p) ∩ S 6= NG◦H(q) ∩ S = NHv(p) ∩Dv] ∪ {v} for all p, q ∈ V (Hv) \Dv with p 6= q. Therefore, NHv(p) ∩Dv 6= NHv(q) ∩Dv for all p, q ∈ V (Hv) \Dv with p 6= q, showing that Dv is a locating set of Hv. Now, since S is a stable locating-dominating set, it follows that S \ {v} is a locating-dominating set of G ◦ H. This implies that Dv is a dominating set of Hv. Let x ∈ Dv and set Dx v = Dv \{x}. Since Sx = S \{x} is a locating set of G ◦H, it follows that Dx v is a locating set of Hv. Therefore, Dv is a stable locating set of Hv. If NG(v)∩A = ∅, then Dv ∪{v} is a stable locating-dominating set of v+Hv. By Theorem 1(i), Dv is a strictly locating set of Hv. This shows that (iv) holds. For the converse, suppose that S has the given form and satisfies properties (i)-(iv). Then clearly, S is a dominating set of G ◦ H. Let x, y ∈ V (G ◦ H) with x 6= y and let v, w ∈ V (G) such that x ∈ V (v +Hv) and y ∈ V (w +Hw). Consider the following cases: Case 1. v = w Suppose first that x = v and y ∈ V (Hv) \Dv. If NG(x)∩A 6= ∅, then NG◦H(x)∩S = (NG(x)∩A)∪Dv 6= NHv(y)∩Dv = NG◦H(y)∩S. SupposeNG(x)∩A = ∅. Then by (iii), Dv is a strictly locating set of Hv. Hence, NG◦H(y)∩S = NHv(y)∩Dv 6= Dv = NG◦H(x)∩S. Next, suppose that x, y ∈ V (Hv) \ Dv. Since Dv is a locating set by assumption, it follows that NHv(x) ∩ Dv 6= NHv(x) ∩ Dv. Hence, whether or not v is in A, we have NG◦H(x) ∩ S 6= NG◦H(y) ∩ S. Case 2. v 6= w Suppose x = v or y = w). Since Dv ⊆ NG◦H(x) ∩ S and Dw ⊆ NG◦H(y) ∩ S, NG◦H(x) ∩ S 6= NG◦H(y) ∩ S. Suppose x ∈ V (Hv) \ Dv and y ∈ V (Hw) \ Dw. Since E. Ahmad, G. Malacas, S. Canoy, Jr. / Eur. J. Pure Appl. Math, 14 (3) (2021), 638-649 647 Dv and Dw are dominating sets of Hv and Hw, respectively, NHv(x) ∩ Dv 6= ∅ and NHw(y) ∩Dw 6= ∅. Hence, NG◦H(x) ∩ S 6= NG◦H(y) ∩ S. Therefore, S is a locating set of G ◦H. Accordingly, S is a locating-dominating set of G ◦H. Next, let z ∈ S and let u ∈ V (G) such that z ∈ V (u+Hu). Let Sz = S \ {z} Suppose z = u ∈ A. Since Du is a dominating set of Hu according to (iv), it follows that Sz is a dominating set of G◦H. By assumption, Du is a locating set of Hu and strictly locating if NG(u)∩A = ∅. Using this and the assumption, it is routine to show that Sz is a locating set of G ◦H. Lastly, suppose that z 6= u. Then z ∈ Du. Note that Du \{z} is a locating-dominating set of Hu if u ∈ (V (G) \ A) by (ii); a strictly locating set if u ∈ (V (G) \ A) \ NG(A) by (iii); and a locating set if v ∈ A by (iv). Using this and the assumption, it can be shown that Sz is a locating set of G ◦H. Accordingly, S is a stable locating-dominating set of G ◦H. Let H be a graph without isolated vertices. We shall be using the following notations in our next results. γslsl (H) = min{|D| : D is a strictly locating and stable locating-dominating set of H}, γsSLl(H) = min{|D| : D is a strictly locating-dominating and stable locating set of H}. Any strictly locating and stable locating-dominating set (strictly locating-dominating and stable locating set) of H with cardinality γslsl (H) (resp. γsSLl(H)) is called a γslsl -set (resp. γsSLl-set) of H. Note that every graph without isolated vertices admits a strictly locating stable locating-dominating set and a strictly locating-dominating and stable locating set. Indeed, if H is a graph without isolated vertices, then V (H) is both a strictly locating stable locating-dominating set and a dominating and stable locating set of H. Also, one can easily verify that if H = K1 + P4, where V (K1) = {a} and P4 = [b, c, d, e], then S = {a, b, d, e} is a strictly locating stable locating-dominating set of H. Corollary 3. Let G be a connected non-trivial graph of order m and let H be any graph without isolated vertices. (i) If γ(H) = 1, then γsl (G) ≤ γ(G) + γ(G)γsSLl(H) + (m− γ(G))γssls(H). (ii) If γ(H) 6= 1, then γsl (G) ≤ min{γ(G)+γ(G)γsSLl(H)+(m−γ(G))γssls(H),m.γssl(H)}. Proof. (i) Suppose γ(H) = 1. Let A be a γ-set of G. Let Dv be a γsSLl-set for each v ∈ A and let it be a γslsl -set of H for each v ∈ V (G) \ A. By Theorem 5, S = A∪ [∪v∈ADv]∪ [∪v∈V (G)\ADv] is a stable locating-dominating set of G ◦H. It follows that γsl (G ◦H) ≤ |S| = γ(G) + γ(G)γsSLl(H) + (m− γ(G))γssls(H) = [1 + γsSLl(H)− γslsl (H)]γ(G) +m.γslsl (H). REFERENCES 648 (ii) Suppose γ(H) 6= 1. Let A = ∅ and letDv be a γssl-set ofH. Then S′ = [∪v∈V (G)Dv] is a stable locating-dominating set of G by Theorem 5. Hence, γsl (G ◦ H) ≤ |S′| = m.γssl(H). This and (i) imply that γsl (G ◦H) ≤ min{[1 + γsSLl(H)− γslsl (H)]γ(G) +m.γslsl (H),m.γssl(H)}. This establishes the desired results. Consider G = P3 and H = P4. One can easily verify that m = 3, γ(G) = 1, γslsl (H) = 4, γsSLl(H) = 3, and γsl (G ◦H) = 12 = [1 + γsSLl(H) − γslsl (H)]γ(G) + m.γslsl (H). Thus, the given bound in Corollary 3(i) is tight. Next, consider G = P4 and H = P3. Then m = 4, γ(G) = 2 and γssl(H)} = γslsl (H) = γsSLl(H) = 3. Also, γsl (G ◦H) = 12 = m.γssl(H)} < 14 = [1 + γsSLl(H)− γslsl (H)]γ(G) +m.γslsl (H). Hence, the bound in Corollary 3(ii) is also tight. Conclusion: The concept of stable locating-dominating set, when used to place or install monitoring devices at designated locations in a given system for safeguard by iden- tifying the exact location of an intruder when a problem in a facility occurs, ensures that the remaining devices in a system can still function as expected in an event when exactly one monitor becomes non-functional. Results generated in this study were obtained using other related concepts. It may be interesting and worthwhile to study these concepts and continue this initial investigation on the concept of stable locating-domination. Acknowledgements The authors would like to thank the Department of Science and Technology - Acceler- ated Science and Technology Human Resource Development Program (DOST-ASTHRDP)- Philippines, and MSU-Iligan Institute of Technology for funding this research. References [1] S. Canoy and G. Malacas. Determining the intruder’s location in a given network: Locating-dominating sets in a graph. NRCP Research Journal., 13(1):1–8, 2013. [2] S. Canoy and G. Malacas. Differentiating-dominating sets in graphs under binary operations. Tamkang Journal of Mathematics, 46(1):51–60, 2015. [3] A. Finbow and B.L. Hartnell. On locating-dominating sets and well-covered graphs. Congr. Numer., 65:191–200, 1988. [4] J. Gimbel, B. van Gorden, M. Nicolescu, C. Umstead, and N. Vaiana. Location with dominating sets. Congr. Numer., 151:129–144, 2001. REFERENCES 649 [5] T.W. Haynes, M.A. Henning, and J. Howard. Locating and total dominating sets in trees. Discrete Applied Mathematics, 154(8):1293–1300, 2006. [6] P.J. Slater. Dominating and location in acyclic graphs. Networks, 17:55–64, 1987. [7] P.J. Slater. Dominating and reference sets in graphs. J. Math. Phys. Sci., 22:445–455, 1988. [8] P.J. Slater. Fault-tolerant locating-dominating sets. Discrete Mathematics, 249:179– 189, 2002.