EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 16, No. 3, 2023, 1685-1694 ISSN 1307-5543 – ejpam.com Published by New York Business Global Global Stable Location-Domination in Graphs Marivir M. Ortega1,∗, Gina A. Malacas1,2, Sergio R. Canoy, Jr.1,2 1 Department of Mathematics and Statistics, College of Science and Mathematics, Mindanao State University-Iligan Institute of Technology, 9200 Iligan City, Philippines 2 Center for 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 introduce and investigate the concept of global stable location- domination in graphs. We also characterize the global stable locating-dominating sets in the join, edge corona, corona, and lexicographic product of graphs and determine the value of the corresponding global stable locating-domination number. 2020 Mathematics Subject Classifications: 05C69 Key Words and Phrases: Global stable location-domination, location-domination, stable domination, global domination, join, edge corona, corona, lexicographic product 1. Introduction Over the past decades, the study of dominating sets have been explored by many mathematicians. A lot of its variants have been introduced. An interesting variant of domination which was introduced by Slater is the location-domination [10, 11]. Locating- dominating sets were first introduced in order to identify the location of fires or intruders in a building [7]. In a building, a fire alarm system is placed on a wall or a ceiling. Each alarm will send a signal when it detects a fire in any adjacent vertices and the activated signal will determine the location of the fire. Suppose that exactly one of the alarm fails or will not function. When this situation happens, the fire alarm system may not function precisely. Moreover, if this fire alarm system fails, a backup placement of fire alarms will be useful in a more precise detection of fires in the building. To address these problems, some additional conditions can be imposed to the concept of location-domination. Some variations of location-domination that incorporate the concepts of globality and stability are introduced and investigated in [1–6, 8, 9]. ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v16i3.4748 Email addresses: marivir.ortega@g.msuiit.edu.ph (M. Ortega), gina.malacas@g.msuiit.edu.ph (G. Malacas), sergio.canoy@g.msuiit.edu.ph (S. Canoy, Jr.) https://www.ejpam.com 1685 © 2023 EJPAM All rights reserved. M. Ortega, G. Malacas, S. Canoy, Jr. / Eur. J. Pure Appl. Math, 16 (3) (2023), 1685-1694 1686 2. Terminology and Notation Let G = (V (G), E(G)) be a graph and v ∈ V (G). The open neighborhood of v in G is the set NG(v) = {u ∈ V (G) : uv ∈ E(G)} and the closed neighborhood of v is the set NG[v] = NG(v) ∪ {v}. The degree of v ∈ V (G), denoted by degG(v), is equal to the cardinality of NG(v) and the maximum degree of G is ∆(G) = max{degG(v) : v ∈ V (G)}. A vertex w of G is a leaf if degG(w) = 1. A vertex u is a support vertex if uw ∈ E(G) for some leaf w of G. The sets L(G) annd S(G) denote the sets of leaves and support vertices of G, respectively. A set D ⊆ V (G) is a dominating set in G if for every v ∈ V (G)\D, there exists u ∈ D such that uv ∈ E(G), that is, N [D] = V (G). The minimum cardinality of a dominating set in G, denoted by γ(G), is the domination number of G. A subset S of V (G) is a locating set in a graph G if every two vertices u and v of V (G)\S, NG(u) ∩ S ̸= NG(v) ∩ S. A subset S of V (G) is a strictly locating set if it is locating and NG(u) ∩ S ̸= S for all u ∈ V (G)\S. 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 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. A locating (resp. strictly locating) set S in G is a stable locating (resp. stable strictly locating) set in G if Sv = S\{v} is a locating (resp. strictly locating) set of G for each v ∈ S. The minimum cardinality of a stable strictly locating set, denoted by ηssls(G), is called the stable strictly location number of G. Any stable strictly locating set with cardinality ηssls is called a minimum stable strictly locating set or an ηssls-set. A locating (resp. strictly locating) set S in G is a stable locating (resp. stable strictly locating) set in G if Sv = S\{v} is a locating (resp. strictly locating) set of G for each v ∈ S. A locating-dominating (resp. stricty locating-dominating) set S of G is a stable locating-dominating (resp. stable strictly locating-dominating) set of G if Sv = S\{v} is a locating-dominating (resp. strictly locating-dominating) set of G for each v ∈ S. The minimum cardinality of a stable locating dominating (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 γsl -set (resp. γssl-set) of G. A locating-dominating set S of a graph G is a global locating-dominating set if it is a locating-dominating set of both G and its complement, G. The global locating-domination number λg(G) is the minimum cardinality of a global locating-dominating set of G. A set S in G is a global stable locating-dominating (resp. global stable strictly locating- dominating) set in G if S is a stable locating-dominating (resp. stable strictly locating- dominating) set inG and in its complement, G. The minimum cardinality of a global stable locating-dominating (resp. global stable strictly locating-dominating) set of G, denoted by λs gl(G) (resp. λs gsl(G)), is called the global stable locating-domination (resp. global stable strictly locating-domination) number of G. A global stable locating-dominating (resp. M. Ortega, G. Malacas, S. Canoy, Jr. / Eur. J. Pure Appl. Math, 16 (3) (2023), 1685-1694 1687 global stable strictly locating-dominating) set of G with cardinality λs gl(G) (resp. λs gsl(G)) is called a λs gl-set (resp. λs gsl-set) of G. Let G and H be two graphs. The join G+H of G and 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)}. The edge corona G ⋄H of G and H is the graph obtained by taking one copy of G and |E(G)| copies of H and joining each of end vertices u and v of every edge uv of G to every vertex of the copy Huv of H (that is, forming the join ⟨{u, v}⟩+Huv for each uv ∈ E(G)). The corona G ◦H of G and H is the graph obtained by taking one copy of G of order n and n copies of H, and then joining the i-th vertex of G to every vertex in the i-th copy of H. For every v ∈ V (G), we denote by Hv the copy of H whose vertices are joined or attached to the vertex v. For each v ∈ V (G), the subgraph ⟨v⟩ + Hv of G◦H will be denoted by v+Hv. The lexicographic product G[H] of G and H is the graph with vertex-set V (G[H]) = V (G) × V (H) and edge-set E(G[H]) satisfying the following conditions: (u1, v1)(u2, v2) ∈ E(G[H]) if and only if either u1u2 ∈ E(G) or u1 = u2 and v1v2 ∈ E(H). 3. Results The first result gives the correct version of the one found in [1]. 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)\{w} = NG(w)\{v}. Theorem 1. Let G be a any graph. Then G admits a global stable locating-dominating set if and only if G and G have no isolated vertices. Proof. Suppose that G admits a global stable locating-dominating set. Then G and G admit a stable locating-dominating set. Thus, by Proposition 1, G and G have no isolated vertices. □ Corollary 1. Let G be a non-trivial connected graph. Then G admits a global stable locating-dominating set if and only if γ(G) ̸= 1. Remark 1. For any non-trivial connected graph G with γ(G) ̸= 1, λg(G) ≤ λs gl(G) ≤ |V (G)| and γsl (G) ≤ λs gl(G). Remark 2. Let G be a graph that admits a global stable locating-dominating set. Then λs gl(G) = λs gl(G). M. Ortega, G. Malacas, S. Canoy, Jr. / Eur. J. Pure Appl. Math, 16 (3) (2023), 1685-1694 1688 Proposition 2. For any non-trivial graph G of order n ≥ 4, 4 ≤ λs gl(G) ≤ n. Proof. Let S be λs gl-set. By Proposition 1(i), |S| ≥ 3. Now, suppose that λs gl(G) = 3 and let S = {a, b, c}. Pick any v ∈ V (G)\S. Suppose n = 4. Since S is a stable locating-dominating set, |NG(v)| ≥ 2. Suppose a, b ∈ NG(v). Since S\{c} is a dominating set, ac ∈ E(G) or bc ∈ E(G). If ac /∈ E(G), then bc ∈ E(G). Since S\{c} is a locating set, ab ∈ E(G). Hence, if c /∈ NG(v), then NG(a)∩S = NG(v)∩S. This implies that S\{a} is not a locating set. Similarly, if ac ∈ E(G), then S\{b} or S\{c} is not a locating set. Thus, NG(v) = {a, b, c}. This is not possible because v is an isolated vertex in G. If ⟨S⟩ = K3, then a and b are isolated vertices in G. Therefore, |V (G)| ≥ 5. Let v, w ∈ V (G)\S. Since S is a stable locating-dominating set, we may assume that |NG(v) ∩ S| = 2 and |NG(w) ∩ S| = 3. Assume that NG(v) ∩ S = {a, b}. Then NG(w) ∩ (S\{c}) = NG(v) ∩ (S\{c}), implying that S\{c} is not a locating set. Therefore, λs gl(G) ≥ 4. □ Lemma 1. [2] Let G be a graph and S ⊆ V (G). If x, y ∈ V (G)\S, then NG(x) ∩ S ̸= NG(y) ∩ S if and only if NG(x) ∩ S ̸= NG(y) ∩ S. Remark 3. Let G be a graph of order n ≥ 4 and suppose it admits a global stable locating-dominating set. If γsl (G) = n or γsl (G) = n, then λs gl(G) = n. Note that the converse of Remark 3 is not true. To see this, consider the graphs in Figure 1. It can be verified that S = V (G)\{4} and S′ = V (G)\{1} are γsl -sets in G and G, respectively. Hence, γsl (G) = γsl (G) = 4. However, λs gl(G) = 5. G : 2 4 1 5 3 G : 2 4 1 5 3 Figure 1: A graph G with γs l (G) = 4 and γs l (G) = 4 but λs gl(G) = 5 This particular example shows that the concept of stable locating-dominating set is not equivalent to the concept of global stable locating-dominating set. Theorem 2. Let G be a non-trivial graph of order n ≥ 5 such that ∆(G) ≤ n − 3. If G admits a global stable locating-dominating set, then λs gl(G) = n if and only if γsl (G) = n. Proof. Suppose that λs gl(G) = n. Suppose that γsl (G) ̸= n. By Proposition 1, there exists v ∈ V (G) such that v /∈ L(G) ∪ S(G) and NG(v)\{w} ̸= NG(w)\{v} for all w ∈ V (G)\{v}. Let S = V (G)\{v}. Then S is a locating-dominating set of G. Let z ∈ S and set Sz = S\{z} = V (G)\{v, z}. Suppose that vz /∈ E(G) and choose x, y ∈ V (G) such that xz, vy ∈ E(G). Then x, y ∈ Sz. Since v /∈ L(G) ∪ S(G), degG(v) ≥ 2. Hence there exists x ∈ V (G)\{v, z} = Sz such that xz ∈ E(G). Since v /∈ S(G), NG(z) ̸= {v}. Hence, there exists y ∈ Sz such that yz ∈ E(G). Thus, Sz is a dominating set of G. Since M. Ortega, G. Malacas, S. Canoy, Jr. / Eur. J. Pure Appl. Math, 16 (3) (2023), 1685-1694 1689 Sz is a locating set in G, Sz is a locating set in G by Lemma 1. This shows that S is a stable locating-dominating set in G. Therefore, S is a global stable locating-dominating set in G and λs gl(G) ≤ |S| = n− 1, a contradiction. Therefore, γsl (G) = n. Conversely, suppose that γsl (G) = n. By Remark 1, λs gl(G) = n. □ Remark 4. Theorem 2 may not hold if the condition ∆(G) ≤ n− 3 is removed. To see this, consider the graphs G and G shown in Figure 1. Note that ∆(G) = |V (G)| − 2 and γsl (G) ̸= 5 but λs gl(G) = 5 = |V (G)|. Corollary 2. Let G be a non-trivial connected graph of order n = 4. Then λs gl(G) = 4 if and only if G = C4 or G = P4. Theorem 3. [1] 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. Proposition 3. Let G and H be non-trivial graphs. Then G +H admits a global stable locating-dominating set if and only if G and H admit a stable strictly locating set. Moreover, S ⊆ V (G+H) is a global stable locating-dominating set in G+H if and only if S = A∪B, where A and B are stable strictly locating sets in G and H, respectively. Proof. Suppose that G + H admits a global stable locating-dominating set, say S. Then S is a stable locating-dominating set in G+H. By Theorem 3, A and B are stable locating sets in G and H, respectively. Now, suppose that A is not a stable strictly locating set. Let w ∈ A and set Aw = A\{w}. Since A is not a stable strictly locating set, there exists z ∈ V (G)\Aw such that NG(z) ∩ Aw = Aw. This implies that NG(z) ∩ Aw = ∅, a contradiction since G+H = G ∪H is a stable locating-dominating set. Thus, A is a stable strictly locating set in G. Similarly, B is a stable strictly locating set in H. Conversely, suppose that S = A′ ∪ B′, where A′ and B′ are stable strictly locating sets in G and H, respectively. Since A′ is stable strictly locating set in G, A′ is a stable locating set and a strictly locating set. Similarly, since B′ is stable strictly locating set in H, B′ is a stable locating set and a strictly locating set. By Theorem 3, S is a stable locating-dominating set in G + H. Since A′ and B′ are strictly locating sets, A′ and B′ are dominating sets in G and H, respectively. Hence, S is a dominating set in G+H. Let a, b ∈ V (G+H)\S with a ̸= b. Suppose that a, b ∈ V (G). Since A′ is a locating set, NG(a) ∩ A′ ̸= NG(b) ∩ A′. Hence, NG+H(a) ∩ S = NG(a) ∩ A′ ̸= NG(b) ∩ A′ = NG+H(b)∩ S by Lemma 1. Similarly, NG+H(a)∩ S ̸= NG+H(b)∩ S if a, b ∈ V (H). Thus, S is a locating set in G+H = G ∪H. Next, let w ∈ S and set Sw = S\{w}. Suppose w ∈ A′ and let Aw′ = A\{w}. Since A is a stable strictly locating set in G, Aw′ is a strictly locating set. Hence, Aw′ is a dominating set in G. Now, let a, b ∈ V (G+H)\Aw′ with a ̸= b. Suppose that M. Ortega, G. Malacas, S. Canoy, Jr. / Eur. J. Pure Appl. Math, 16 (3) (2023), 1685-1694 1690 a, b ∈ V (G)\Aw′ . Since Aw′ is a stable locating set, NG(a) ∩ Aw′ ̸= NG(b) ∩ Aw′ . Hence, NG+H(a)∩Sw = NG(a)∩Aw′ ̸= NG(b)∩Aw′ = NG+H(b)∩Sw by Lemma 1. Thus, Aw′ is a locating-dominating set in G. Similarly, Bw′ = B\{w} is a locating-dominating set in H if w ∈ B. Thus, S is a stable locating-dominating set in G+H = G∪H. Therefore, S is a global stable locating-dominating set in G+H. □ Corollary 3. Let G and H be non-trivial connected graphs. A subset S of V (G+H) is a minimum global stable locating-dominating set in G+H if and only if S = A ∪B, where A and B is a minimum stable strictly locating set in G and H, respectively. In particular, λs gl(G+H) = ηssls(G) + ηssls(H). Corollary 4. Let G be a graph and let Km = ⟨{v1, v2, ..., vm}⟩ where m ≥ 2. A subset S of V (Km + G) is a global stable locating-dominating set in Km + G if and only if S = {v1, v2, ..., vm} ∪ SG, where SG is a stable strictly locating set in G. In particular, λs gl(Km +G) = m+ ηssls(G). Proof. The only stable strictly locating set in Km is the V (Km). Thus, by Proposition 3, S is a global stable locating-dominating set in Km +G. □ Theorem 4. [4] Let G be a connected graph of order m ≥ 3 and let H be any non-trivial connected graph. Then C is a stable locating-dominating set of G ⋄ H if and only if C = A ∪ ( ∪uv∈E(G)Suv ) and satisfies the following conditions: (i) A ⊆ V (G), (ii) For each uv ∈ E(G), (a) Suv is a stable locating set of Huv; (b) Suv is a stable locating-dominating set of Huv whenever u, v /∈ A; (c) Suv is a stable strictly locating set of Huv for each v ∈ L(G) with v /∈ A; and (d) Suv is a stable strictly locating-dominating set of Huv whenever u, v /∈ A and {u, v} ∩ L(G) ̸= ∅. (iii) For each w ∈ A and for each z ∈ NG(w), we have: (a) Szw is a strictly locating set of Hzw whenever w ∈ L(G) and (b) Szw is a strictly locating-dominating set of Hzw whenever z /∈ A and {z, w} ∩ L(G) ̸= ∅. (iv) For each zw ∈ E(G) with z ∈ A and w /∈ A, if x ∈ V (Hzw)\[Szw\{p}] for p ∈ Szw and NHzw ∩ (Szw\{p}) = ∅, then for each y ∈ NG(z)\{w} and for each q ∈ V (Hyz)\Syz, it holds that y ∈ A or NHyz(q) ∩ Syz ̸= ∅. M. Ortega, G. Malacas, S. Canoy, Jr. / Eur. J. Pure Appl. Math, 16 (3) (2023), 1685-1694 1691 Lemma 2. Let G be a connected graph such that G ̸= K2 and H be any non-trivial connected graph. Then γ(G ⋄H) = 1 if and only if γ(G) = 1. Proof. Suppose γ(G ⋄ H) = 1, say {p} is a dominating set in G ⋄ H. If p ∈ V (G), then γ(G) = 1. Suppose that p ∈ V (Huv) for some uv ∈ E(G). If G ̸= K2, there exists an xy ∈ E(G)\{uv}. Then pq /∈ E(G ⋄ H) for all q ∈ V (Hxy), a contradiction. Thus, G = K2. Therefore, γ(G) = 1. Conversely, if γ(G) = 1, say {q} is a dominating set in G. Then for every vertex u ∈ V (G ⋄H), qu ∈ E(G ⋄H). Thus, γ(G ⋄H) = 1. □ Theorem 5. Let G be a connected graph of order n ≥ 4 and ∆(G) ≤ n− 2 and H be any non-trivial connected graph. Then G ⋄ H admits a global stable locating-dominating set. Moreover, S ⊆ V (G ⋄H) is a global stable locating-dominating set in G ⋄H if and only if S is a stable locating-dominating set in G ⋄H. In particular, λs gl(G ⋄H) = γsl (G ⋄H). Proof. Since γ(G) ̸= 1, γ(G ⋄H) ̸= 1 by Lemma 2. Therefore since G ⋄H is connected and γ(G ◦ H) ̸= 1, G ◦ H admits a global stable locating-dominating set by Corollary 1. Let S be a global stable locating-dominating set in G ◦ H. Then S is a stable locating-dominating set in G ◦H. Conversely, suppose that S is a stable locating-dominating set in G ⋄ H. Let A ⊆ V (G) and Duv = V (Huv) ∩ S. By Theorem 4, S = A ∪ ( ∪uv∈E(G)Duv ) is a stable locating-dominating set in G ⋄ H. Let x ∈ V (G ⋄H)\S. If x ∈ V (G)\A, then pick v ∈ V (G)\NG(x). Since G is connected, we may choose u ∈ NG(v). It follows that xp ∈ E(G ⋄H) for all p ∈ Duv. By Theorem 4 (ii)(a), Duv ̸= ∅ and xp ∈ E(G ⋄H) for all p ∈ Suv. Suppose x ∈ V (Hyz)\Dyz. Choose wt ∈ E(G)\{yz}. Then xq ∈ E(G ⋄H) for all q ∈ Dwt. Thus, S is a dominating set in G ⋄H. Now, let a, b ∈ V (G ⋄H)\S where a ̸= b. Since S is a locating set in G ⋄ H, NG⋄H(a) ∩ S ̸= NG⋄H(b) ∩ S. Hence, NG⋄H(a) ∩ S ̸= NG⋄H(b) ∩ S by Lemma 1. Thus, S is a locating set in G ⋄H. Finally, let w ∈ S and set Sw = S\{w}. Again, since Duv ̸= ∅, for every uv ∈ E(G), Sw is a dominating set in G ⋄H. Since S is a stable locating set in G⋄H, Sw is a locating set in G ⋄H. Thus, Sw is a locating set in G ⋄H by Lemma 1. Therefore, S is a stable locating-dominating set in G ⋄H. Accordingly, S is a global stable locating-dominating in G ⋄ H. Consequently, λs gl(G ⋄H) = γsl (G ⋄H). □ Theorem 6. [1] 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). (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. M. Ortega, G. Malacas, S. Canoy, Jr. / Eur. J. Pure Appl. Math, 16 (3) (2023), 1685-1694 1692 (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 = ∅. Proposition 4. Let G be a connected non-trivial graph and let H be any graph. Then G ◦H admits a global stable locating-dominating set. Moreover, S ⊆ V (G ◦H) is a global stable locating-dominating set in G ◦H if and only if S is a stable locating-dominating set in G ◦H. In particular, λs gl(G ◦H) = γsl (G ◦H). Proof. Since G ◦ H is connected and γ(G ◦ H) ̸= 1, G ◦ H admits a global stable locating-dominating set by Corollary 1. Let S be a global stable locating-dominating set in G ◦H. Then S is a stable locating-dominating set in G ◦H. Conversely, suppose that S is stable locating-dominating set in G ◦H. Let A ⊆ V (G) and Dv = V (Hv) ∩ S. Then S = A ∪ ( ∪v∈V (G)Dv ) and Dv ̸= ∅ by Theorem 6. If v ∈ V (G)\A, then we may choose w ∈ V (G)\NG(v). Since Dw ̸= ∅, vp ∈ E(G ◦H) for all p ∈ Dw. Suppose v ∈ V (Hu)\Du. Then vs ∈ E(G ◦H) for all s ∈ Dz\{u}. Thus, S is a dominating set in G ◦H. Since S is a locating set in G ◦ H, NG◦H(a) ∩ S ̸= NG◦H(b) ∩ S. Hence, NG◦H(a) ∩ S ̸= NG◦H(b) ∩ S by Lemma 1. Thus, S is a locating-dominating set in G ◦H. Finally, let y ∈ S and set Sy = S\{y}. Again, since Dv ̸= ∅ for each v ∈ V (G), Sy is a dominating set in G ◦H. Since S is a stable locating-dominating set in G ◦H, Sy is a locating set in G ◦H. Thus, Sy is a locating-dominating set in G ◦H. Therefore, S is a stable locating-dominating set in G ◦H. Accordingly, S is a global stable locating-dominating in G ◦ H. Consequently, λs gl(G ◦H) = γsl (G ◦H). □ Theorem 7. [4] Let G and H be a non-trivial connected graphs with ∆(H) ≤ |V (H)|− 2. Then C = ⋃ x∈S ({x} × Tx), where S ⊆ V (G) and Tx ⊆ V (H) for each x ∈ S, is a global stable locating-dominating set of G[H] if and only if each of the following holds: (i) S = V (G). (ii) Tx is a stable locating set in H for every x ∈ V (G). (iii) If x and y are adjacent vertices with NG[x] = NG[y] and one, say Tx is not strictly locating, then Ty is a stable strictly locating set of H. (iv) If x and y are distinct non-adjacent vertices of G with NG(x) = NG(y) and one, say Tx is not a dominating set, then Ty is a stable dominating set of H. Proposition 5. Let G and H be a non-trivial connected graphs with ∆(H) ≤ |V (H)| − 2. Then G[H] admits a global stable locating-dominating set. Moreover, C = ⋃ u∈S ({u} × Tu), where S ⊆ V (G) and Tu ⊆ V (H) for each u ∈ S, is a global stable locating-dominating set of G[H] if and only if each of the following holds: M. Ortega, G. Malacas, S. Canoy, Jr. / Eur. J. Pure Appl. Math, 16 (3) (2023), 1685-1694 1693 (i) S = V (G). (ii) Tu is a stable locating set in H for every u ∈ V (G). (iii) If u and v are adjacent vertices with NG[u] = NG[v] and one, say Tu is not strictly locating, then Tv is a stable strictly locating set of H. (iv) If u and v are distinct non-adjacent vertices of G with NG(u) = NG(v) and one, say Tu is not a dominating set, then Tv is a stable dominating set of H. (v) For each u ∈ V (G) with degG(u) = |V (G)| − 1, it hold that Tu\NH(x) ̸= ∅ for all x ∈ V (H)\Tu, that is, Tu is dominating in H. Proof. Since γ(H) ̸= 1, γ(G[H]) ̸= 1. By Corollary 1, G[H] admits a global stable locating-dominating set. Let C be a global stable locating-dominating set in G[H]. Then C is a stable locating-dominating set in G[H]. By Theorem 7, (i), (ii), (iii), and (iv) hold. Now, let u ∈ V (G) with degG(u) = |V (G)|−1. Suppose that Tu is not a dominating set of H. Then there exists w ∈ V (H)\NH [Tu] and NH(u)∩Tu = Tu. Thus, there exists (u,w) ∈ V (G[H])\C such that NG[H]((u,w)) ∩ C = C which implies that N G[H] ((u,w)) ∩ C = ∅. Hence, C is not a dominating set of G[H], a contradiction. Therefore, Tu is a dominating set of H. Conversely, suppose that (i), (ii), (iii), (iv), and (v) hold. Then by Proposition 7, C is a stable locating-dominating set of G[H]. By (i) and (v), C is a dominating set of G[H]. By Lemma 1 and (ii), C is a stable locating-dominating set of G[H]. Thus, C is a stable locating-dominating set of G[H]. Therefore, C is a global stable locating-dominating set of G[H]. □ The following result is a consequence of Proposition 5. Corollary 5. Let G and H be a non-trivial connected graphs with ∆(H) ≤ |V (H)| − 2. If γ(G) ̸= 1, then λs gl(G[H]) = γsl (H). Conclusion This study introduced and initially investigated the concept of global stable locating- dominating set. This concept was shown to be different from the previously defined concept of stable locating-dominating set. Graph which attains the order as its global stable locating-domination number was characterized. Global stable locating-dominating sets in the join, edge corona, corona, and lexicographic product of graphs were characterized and global stable locating-domination number of each of these graphs were determined. In this paper, the authors did not provide any realization result involving global stable locating-domination number and any locating-related parameters. Thus, it is recommended that a realization result involving global stable locating-domination number and stable locating-domination number be constructed. Further, the concept can be studied for other graphs not considered in this study. REFERENCES 1694 Acknowledgements The authors would like to thank the reviewers for their invaluable comments and suggestions that led to this improved version of the paper. This research is funded by the Department of Science and Technology-Accelerated Science and Technology Human Resource Development Program (DOST-ASTHRDP) and the Mindanao State University- Iligan Institute of Technology. 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