EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 16, No. 1, 2023, 479-490 ISSN 1307-5543 – ejpam.com Published by New York Business Global Stable Locating-Dominating Sets in the Edge Corona and Lexicographic Product of Graphs Gina A. Malacas1,∗, Sergio R. Canoy, Jr.1, Emmy Chacon1 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 an 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 ̸= NG(y) ∩ S for any two 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 γSLD(G), is called the stable locating-domination number of G. In this paper, we investigate this concept and the corresponding parameter for edge corona and lexicographic product of graphs. 2020 Mathematics Subject Classifications: 05C69 Key Words and Phrases: Locating, stable, domination, edge corona, lexicographic product 1. Introduction Let G = (V (G), E(G)) be an undirected 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 neighborhood of v ∈ V (G) is the set NG(v)= {x ∈ V (G) : xv ∈ E(G)}. 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(x) : x ∈ E(G)}. A vertex v of G is a leaf if degG(v) = 1. A vertex u of G is a support if uv ∈ E(G) for some leaf v 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]. 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]. These concepts are defined and studied in [7] and [21]. A subset S of V (G) is a dominating set of G if for every v ∈ V (G) \ S, there exists u ∈ S such that xv ∈ E(G). S is a locating set in G if NG(u) ∩ S ̸= NG(v) ∩ S for every ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v16i1.4645 Email addresses: gina.malacas@g.msuiit.edu.ph (G. Malacas), sergio.canoy@g.msuiit.edu.ph (S. Canoy, Jr.), emmy.chacon@g.msuiit.edu.ph (E. Chacon) https://www.ejpam.com 479 © 2023 EJPAM All rights reserved. G. Malacas, S. Canoy, Jr., E. Chacon / Eur. J. Pure Appl. Math, 16 (1) (2023), 479-490 480 two distinct vertices u, v ∈ V (G) \ S. A locating set S is said to be a strictly locating set if NG(u) ∩ S ̸= S for all u ∈ V (G) \ S. A locating set (strictly locating set) S is a stable locating set (resp. stable strictly locating set) if S \ {v} is a locating (resp. strictly locating) set for each v ∈ S. A locating (resp. strictly locating ) set S of V (G) which is also a dominating set is called a locating-dominating (resp. strictly locating-dominating) set of G. A locating-dominating (strictly locating-dominating) set S is a stable locating- dominating (resp. stable strictly locating-dominating) set of G if S \ {v} is a locating- dominating (resp. strictly locating-dominating) set of G for each v ∈ S. The minimum cardinality of a locating (strictly locating, stable locating, stable strictly locating) set of G is denoted by ln(G) (resp. sln(G), sbln(G) , sbsln(G)). Any locating (strictly locating, stable locating, stable strictly locating) set of G with cardinality ln(G) (resp. sln(G), sbln(G), sbsln(G)) is called an ln-set (resp. sln-set, sbln-set, sbsln-set) of G. The minimum cardinality of a locating-dominating (resp. strictly locating-dominating, stable locating-dominating, stable strictly locating-dominating) set of G is denoted by γL(G) (resp. γSL(G), γsL(G), γsSL(G)). Any locating-dominating (strictly locating-dominating, stable locating-dominating, stable strictly locating-dominating) set of G with cardinality γL(G) (resp. γSL(G), γsL(G), γsSL(G)) is called an γL-set (resp. γSL-set, γsL-set, γsSL-set) of G. Domination and some variations of the concept are found in the book by Haynes et al. (see [9]). Other variations of domination can be found in [2], [3], [4], [5], [11], [12], [16], and [18]. The concepts of locating, stricly locating, locating-dominating, and strictly locating-dominating, and the associated parameters are studied in [6], [8], [10], [13], [14], [15], [17], [19], [20]. The concept of stable locating-dominating and related concepts are studied in [1]. Let G andH be any two graphs. The edge corona G⋄H is the graph obtained by taking one copy of G and |E(G)| copiesH and joining each end vertices u and v of every edge uv to every vertex of the copyHuv ofH (i.e. forming the join ⟨{u, v}⟩+Huv for each uv ∈ E(G)). The lexicographic product G[H] is the graph with vertex-set V (G[H]) = V (G)×V (H) and edge-set E(G[H]) satisfying the following conditions: (x, u)(y, v) ∈ E(G[H]) if and only if either xy ∈ E(G) or x = y and uv ∈ E(H). It is easily observed that for any non-empty subset C of V (G[H]) = V (G)× V (H), this set can be expressed as C = ∪x∈S({x} × Tx), where S ⊆ V (G) and Tx ⊆ V (H) for each x ∈ S. Set S = CG = {x ∈ V (G) : (x, a) ∈ C for some a ∈ V (H)} is called the G-projection of C. Moreover, for each x ∈ S, Tx = {a ∈ V (H) : (x, a) ∈ C}. 2. Results Throughout, we denote by L(G) the set containing all leaves of a graph G. The first result is found in [1]. Theorem 1. Let G be graph without isolated vertices. Then G has a stable strictly locating set if and only if γ(G) ̸= 1. G. Malacas, S. Canoy, Jr., E. Chacon / Eur. J. Pure Appl. Math, 16 (1) (2023), 479-490 481 Theorem 2. Let G be a connected graph of order m ≥ 3 and let H be any non-trivial connected graph. Then C is a 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 locating set of Huv; (b) Suv is a locating-dominating set of Huv whenever u, v /∈ A; (c) Suv is a strictly locating set of Huv for each v ∈ L(G) with v /∈ A; and (d) Suv is a strictly locating-dominating set of Huv whenever u, v /∈ A and {u, v}∩ L(G) ̸= ∅. (iii) For each uv ∈ E(G) with v ∈ A and u /∈ A, if x ∈ V (Huv)\Suv and NHuv(x)∩Suv = ∅, then for each w ∈ NG(v)\{u} and for each y ∈ V (Hwv)\Swv, it holds that w ∈ A or NHwv(y) ∩ Swv ̸= ∅. Proof. Suppose C is a locating-dominating set of G ⋄ H. Let A = C ∩ V (G) and Suv = C ∩ V (Huv) for each uv ∈ E(G). Then C = A ∪ [∪uv∈E(G)Suv] and (i) holds. Let uv ∈ E(G). Since C is a locating set of G ⋄ H, Suv ̸= ∅. Let x, y ∈ V (Huv) \ Suv with x ̸= y and let S = A ∩ {u, v}. Since C is a locating set, [NHuv(x) ∩ Suv] ∪ S = NG⋄H(x) ∩ C ̸= NG⋄H(y) ∩ C = [NHuv(y) ∩ Suv] ∪ S. This implies that NHuv(x) ∩ Suv ̸= NHuv(y) ∩ Suv, showing that Suv is a locating set of Huv. Suppose u, v /∈ A. Since C is a dominating set of G ⋄H, Suv is a dominating set of V (Huv. Hence, (a) and (b) hold. Next, suppose that uv ∈ E(G) and v ∈ L(G) \ A. Let S∗ = A ∩ {u} and let z ∈ V (Huv) \ Suv. Again, since C is a locating set, [NHuv(z) ∩ Suv] ∪ S∗ = NG⋄H(z) ∩ C ̸= NG⋄H(v) ∩ C = Suv ∪ S∗. This implies that [NHuv(z)∩Suv] ̸= Suv, showing that Suv is a strictly locating set of Huv. If S∗ = ∅ (that is, u /∈ A), then Suv is a dominating set of V (Huv). Thus, (c) and (d) hold. Finally, let uv ∈ E(G) with v ∈ A and u /∈ A. Suppose x ∈ V (Huv) \ Suv and NHuv(x) ∩ Suv = ∅. Then NG⋄H(x) ∩ C = {v}. Let w ∈ NG(v) \ {u} and let y ∈ V (Hwv) \Swv. Suppose w /∈ A. Then NG⋄H(y)∩C = [NHwv(y)∩Swv]∪{v}. Since C is a locating set of G⋄H, NG⋄H(x)∩C ̸= NG⋄H(y)∩C. This implies that NHwv(y)∩Swv ̸= ∅, showing that (iii) holds. For the converse, suppose that C has the form described and satisfies (i), (ii), and (iii). Let z ∈ V (G ⋄ H) \ C and let uv ∈ E(G) such that z ∈ ⟨{u, v}⟩ + Huv. If z = u or z = v, then there exists t ∈ Suv ⊂ C such that z ∈ NG⋄H(t) by (ii)(a). Suppose z ∈ V (Huv) \ Suv. If u ∈ A or v ∈ A, then uz ∈ E(G ⋄H) or vz ∈ E(G ⋄H). If u, v /∈ A, then there exists s ∈ Suv ∩ NG⋄H(z) by (ii)(b). Hence, C is a dominating set of G ⋄ H. G. Malacas, S. Canoy, Jr., E. Chacon / Eur. J. Pure Appl. Math, 16 (1) (2023), 479-490 482 Next, let p, q ∈ V (G⋄H)\C with p ̸= q and let uv, xy ∈ E(G) such that p ∈ ⟨{u, v}⟩+Huv and q ∈ ⟨{x, y}⟩+Hxy. Consider the following cases: Case 1. The edges uv and xy are non-adjacent (i.e., they do not share a common vertex). Suppose that p ∈ {u, v} or q ∈ {x, y}. Since Suv ⊆ NG⋄H(p) and Sxy ⊆ NG⋄H(q), NG⋄H(p) ∩ C ̸= NG⋄H(q) ∩ C. Suppose that p /∈ {u, v} and q /∈ {x, y}. Then p ∈ V (Huv) \ Suv and q ∈ V (Hxy) \ Sxy. Since NG⋄H(p) ∩ C ⊆ V (⟨{u, v}⟩ + Huv) and NG⋄H(q) ∩ C ⊆ V (⟨{x, y}⟩+Hxy), it follows that NG⋄H(p) ∩ C ̸= NG⋄H(q) ∩ C. Case 2. The edges uv and xy are distinct and adjacent. We may assume that x = u. If p ∈ {u, v} or q ∈ {x, y}, then NG⋄H(p)∩C ̸= NG⋄H(q)∩C (as in Case 1). So suppose that p /∈ {u, v} and q /∈ {x, y}. If NHuv(p) ∩ Suv ̸= ∅ or NHxy(y) ∩ Sxy ̸= ∅, then NG⋄H(p) ∩C ̸= NG⋄H(q) ∩C. Suppose that NHuv(p) ∩ Suv = ∅ or NHxy(y)∩ Sxy = ∅. If u ∈ A, then y ∈ A or v ∈ A by (iii). Suppose that u /∈ A. Then by (ii)(b), y, v ∈ A. Since v ∈ NG⋄H(p) ∩ C, y ∈ NG⋄H(y) ∩ C, and y ̸= v, it follows that NG⋄H(p) ∩ C ̸= NG⋄H(q) ∩ C. Case 3. The edges uv and xy are the same. We may assume that x = u and y = v. Suppose first that p = u and q = v. Since G is connected and G ̸= K2, we may assume that there exists w ∈ V (G) \ {u, v} such that vw ∈ E(G). Because ∅ ̸= Svw ⊆ (NG⋄H(q) ∩C \ (NG⋄H(p) ∩C), we have NG⋄H(p) ∩C ̸= NG⋄H(q)∩C. Suppose that p, q ∈ V (Huv) \Suv. By (ii)(a), NG⋄H(p)∩C ̸= NG⋄H(q)∩C. Finally, suppose that p ∈ V (Huv) \ Suv (or q ∈ V (Huv) \ Suv) and q ∈ {u, v} (resp. p ∈ {u, v}). We may assume without loss of generality that q = u. Consider the following subcases: Subcase 1. u, v /∈ L(G). Then there exist a, b ∈ V (G) such that au, bv ∈ E(G). Since Sau ⊆ NG⋄H(q) ∩ C and Sau ∩NG⋄H(p) = ∅, NG⋄H(p) ∩ C ̸= NG⋄H(q) ∩ C. Subcase 2. u ∈ L(G) or v ∈ L(G). By (ii)(c) and (ii)(d), Suv is a strictly locating set of V (Huv). It follows that NHuv(p) ∩ Suv ̸= Suv. Since Suv ⊆ NG⋄H(q) ∩ C, NG⋄H(p) ∩ C ̸= NG⋄H(q) ∩ C. Accordingly, C is locating-dominating set of G ⋄H. A set S ⊆ V (G) is a vertex cover of G if for every uv ∈ E(G), u ∈ S or v ∈ S. A vertex cover S is a perfect vertex cover of G if for each v ∈ S and for each pair of distinct edges uv and wv of G, u ∈ S or w ∈ S. The smallest size of a perfect vertex cover of G. Malacas, S. Canoy, Jr., E. Chacon / Eur. J. Pure Appl. Math, 16 (1) (2023), 479-490 483 G, denoted by βp(G), is called the the perfect vertex covering number of G. Any perfect vertex cover of G of size βp(G) is called a βp-set or a minimum perfect vertex cover of G. Example 1. βp(Kn) = n− 1 for each n ≥ 2. Corollary 1. Let G be a connected graph of order m ≥ 3 and let H be any non-trivial connected graph. (i) If L(G) = ∅, then γL(G ⋄H) ≤ min{βp(G) + |E(G)|ln(H), |E(G)|γL(H)}. (ii) If L(G) ̸= ∅, then γL(G ⋄H) ≤ min{βp(G) + |E(G)|sln(H), |E(G)|γSL(H)}. Proof. (i) Suppose that L(G) = ∅. Let S1 be a βp-set of G and let Suv be a minimum locating set of Huv for each uv ∈ E(G). Then C1 = S ∪ [∪uv∈E(G)Suv] is a locating dominating set of G ⋄H by Theorem 2. Hence, γL(G ⋄H) ≤ |C1| = βp(G)+ |E(G)|ln(H). Now, let Luv be a γL-set of H uv for each uv ∈ E(G). Then C2 = ∪uv∈E(G)Luv is a locating dominating set of G⋄H by Theorem 2. This implies that γL(G⋄H) ≤ |C2| = |E(G)|γL(H). Therefore, (i) holds. (ii) Suppose that L(G) ̸= ∅. Let S be a βp-set of G and let S′ uv be a minimum strictly locating set of Huv for each uv ∈ E(G). Then C3 = S ∪ [∪uv∈E(G)S ′ uv is a locating domi- nating set of G⋄H by Theorem 2. Hence, γL(G⋄H) ≤ |C3| = βp(G)+ |E(G)|sln(H). Let Ruv be a γSL-set of H uv for each uv ∈ E(G). Then C4 = ∪uv∈E(G)Ruv is a locating dom- inating set of G ⋄H by Theorem 2. This implies that γL(G ⋄H) ≤ |C4| = |E(G)|γSL(H), showing that (ii) holds. Remark 1. The bounds in Corollary 2 are sharp. Indeed, it can be verified that γL(K3 ⋄ P5) = |E(K3)|γL(P5) = 6 < 8 = βp(K3) + |E(K3)|ln(P5), γL(K3 ⋄ P3) = βp(K3) + |E(K3)|ln(P3) = 5 < 6 = |E(K3)|γL(P3), γL(P3 ⋄ P3) = |E(P3)|γSL(P3) = 4 < 6 = βp(P3) + |E(P3)|sln(P3), and γL(P4 ⋄ P5) = βp(P4) + |E(P4)|sln(P5) = 8 < 9 = |E(P4)|γSL(P5). Theorem 3. 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). G. Malacas, S. Canoy, Jr., E. Chacon / Eur. J. Pure Appl. Math, 16 (1) (2023), 479-490 484 (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(x)∩ (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 ̸= ∅. Proof. Suppose C is a stable locating-dominating set of G ⋄ H. Let A = C ∩ V (G) and Suv = C ∩ V (Huv) for each uv ∈ E(G). Then C = A ∪ [∪uv∈E(G)Suv] and (i) holds. Let xy ∈ E(G). By Theorem 2(ii)(a), Sxy is a locating set of Hxy. Let p ∈ Sxy. Then by assumption, C \{p} = A∪ [∪uv∈[E(G)\{xy}]Suv]∪ (Sxy \{p}) is a locating-dominating set of G⋄H. It follows from Theorem 2(ii)(a) that Sxy \{p} is a locating set of Hxy. If x, y /∈ A, then Sxy \ {p} is a locating-dominating set of Hxy by Theorem 2(ii)(b). If one of x and y, say x ∈ L(G) \ A, then Sxy \ {p} is a strictly locating set of Hxy by Theorem 2(ii)(c). Morover, if x, y /∈ A and x ∈ L(G) or y ∈ L(G), then Sxy is a strictly locating-dominating set of Hxy by Theorem 2(ii)(d). Therefore, (a), (b), (c), and (d) hold. Next, let w ∈ A and let z ∈ NG(w). Since C is a stable locating-dominating set of G ⋄H, C \ {w} = (A \ {w}) ∪ [∪uv∈E(G)Suv] is a locating-dominating set of G⋄H. It follows from (c) and (d) of Theorem 2 that Szw is a strictly locating set of Hzw whenever w ∈ L(G) and Szw is a strictly locating-dominating set of Hzw whenever z /∈ A (hence, z /∈ A \ {w}) and {z, w} ∩L(G) ̸= ∅. This shows that (iii) holds. Finally, let zw ∈ E(G) with z ∈ A and w /∈ A. Let p ∈ Szw. Then, again, C \ {p} = A ∪ [∪uv∈[E(G)\{zw}]Suv] ∪ (Szw \ {p}) is a locating-dominating set of G ⋄H. Hence, by Theorem 2(iii), statement (iv) holds. For the converse, suppose that C has the given form and satisfies (i), (ii), (iii) and (iv). By (i) and (ii), it follows that (i) and (ii) of Theorem 2 are satisfied by C. Let zw ∈ E(G) with z ∈ A and w /∈ A. Let x ∈ V (Hzw) \Szw. Then x ∈ V (Hzw) \ [Szw \ {p}] for p ∈ Szw. Suppose NHzw(x) ∩ Szw = ∅. Then NHzw(x) ∩ (Szw \ {p}) = ∅. Hence, by (iv), for each y ∈ NG(z) \ {w} and for each q ∈ V (Hyz) \ Syz, it holds that y ∈ A or G. Malacas, S. Canoy, Jr., E. Chacon / Eur. J. Pure Appl. Math, 16 (1) (2023), 479-490 485 NHyz(q) ∩ Syz ̸= ∅. Thus, (iii) of Theorem 2 also holds for C. Therefore, C is a locating dominating set of G ⋄H. Let q ∈ C and let uv ∈ E(G) such that q ∈ V (⟨{u, v}⟩+Huv). Suppose first that p ∈ {u, v}. Then C∗ = C \ {p} = (A \ {p}) ∪ [∪uv∈E(G)Suv]. Accordingly, C is a stable locating-dominating set of G ⋄H. The next two results follow from Theorem 3. Corollary 2. Let G be a connected graph of orderm ≥ 3 with L(G) = ∅ and letH be any non-trivial connected graph. If C = ∪uv∈E(G)Suv and Suv is a stable locating-dominating set of Huv for each uv ∈ E(G), then C is a stable locating-dominating set of G ⋄H. In particular, γsL(G ⋄H) ≤ |E(G)|γsL(H). Proof. Since L(G) = ∅ and Suv is a stable locating-dominating set of Huv for each uv ∈ E(G), C = ∪uv∈E(G)Suv satisfies the conditions in Theorem 3. Thus, C is a stable locating-dominating set of G ⋄H and γsL(G ⋄H) ≤ |C| = |E(G)|γsL(H). Corollary 3. Let G be a connected graph of order m ≥ 3 with L(G) ̸= ∅ and let H be any non-trivial connected graph with γ(H) ̸= 1. If C = ∪uv∈E(G)Suv and Suv is a stable strictly locating-dominating set of Huv for each uv ∈ E(G), then C is a stable locating-dominating set of G ⋄H. Moreover, γsL(G ⋄H) ≤ |C| = |E(G)|γsSL(H). Proof. By Theorem 1 and the assumption that γ(H) ̸= 1, H admits a stable strictly locating-dominating set. Since Suv is a stable strictly locating-dominating set of Huv for each uv ∈ E(G) and L(G) ̸= ∅, C = ∪uv∈E(G)Suv satisfies the conditions in Theorem 3. Hence, C is a stable locating-dominating set of G ⋄ H and γsL(G ⋄ H) ≤ |C| = |E(G)|γsSL(H). The next result is found in [15]. Theorem 4. Let G and H be non-trivial connected graphs such that ∆(H) ≤ |V (H)|−2. Then C = ⋃ 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 the following hold. (i) S = V (G). (ii) Tx is a locating set in H for every x ∈ V (G). (iii) Tx or Ty is strictly locating in H whenever x and y are adjacent vertices of G with NG[x] = NG[y]. G. Malacas, S. Canoy, Jr., E. Chacon / Eur. J. Pure Appl. Math, 16 (1) (2023), 479-490 486 (iv) Tx or Ty is a dominating set in H whenever x and y are distinct non-adjacent vertices of G with NG(x) = NG(y). Theorem 5. Let G and H be 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 stable locating-dominating set of G [H] if and only if each of the following conditions hold. (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 of G 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. Proof. Suppose C is a stable locating-dominating set of G [H]. By Theorem 4, S = V (G) and Tx is a locating set of H for each x ∈ V (G). Let z ∈ S and let a ∈ Tz. By assumption, C \{(z, a)} = [ ⋃ x∈S\{z} ({x}×Tx)]∪ [{z}× (Tz \{a})] is a locating-dominating set of G[H]. By Theorem 4(ii), Tz \ {a} is a locating set of H. This implies that Tz is a stable locating set of H. Thus, (ii) holds. Suppose now that x and y are adjacent vertices of G with NG[x] = NG[y]. Suppose that one, say Tx is not a strictly locating set of H. By Theorem 4(iii), Ty is a strictly locating set of H. Let p ∈ Ty. Since C \ {(y, p)} = [ ⋃ z∈S\{y} ({z} × Tz)] ∪ [{y} × (Ty \ {p})] is a locating-dominating set of G[H] and Tx is not strictly locating, it follows from Theorem 4(iii) that Ty \ {p} is strictly locating. Therefore, Ty is a stable strictly locating set of H, showing that (iii) holds. Next, suppose that x and y are distinct non-adjacent vertices of G with NG(x) = NG(y). Suppose that one, say Tx is not a dominating set of H. Then Ty is a dominating set of G by Theorem 4(iv). Let q ∈ Ty. Since C \ {(y, q)} = [ ⋃ z∈S\{y} ({z} × Tz)] ∪ [{y} × (Ty \ {q})] is a locating-dominating set of G[H] and Tx is not a dominating set of G, Ty \ {q} is a dominating set of H by Theorem 4(iv). This shows that Ty is a stable dominating set of H. Thus, (iv) holds. For the converse, suppose that C satisfies (i), (ii), (iii) and (iv). Then C satisfies the conditions (i), (ii), (iii) and (iv) of Theorem 4. Hence, C is a locating-dominating set of G. Malacas, S. Canoy, Jr., E. Chacon / Eur. J. Pure Appl. Math, 16 (1) (2023), 479-490 487 G[H]. Let (y, a) ∈ C. Then C∗ = C \ {(y, a)} = [ ⋃ x∈S\{y} ({x} × Tx)] ∪ [{y} × (Ty \ {a})]. By (ii), Ty \ {a} is a locating set and Tx are stable locating sets of H for each x ∈ S \ {y}. This would also imply that C∗ G = S∗ = S = V (G). Let x and z be adjacent vertices of G with NG[x] = NG[z]. If Tx is strictly locating, then we are done. So suppose that Tx is not strictly locating. Then by (iii), Tz is a stable strictly locating set. Hence, if z ̸= y, then Ty is strictly locating and, if z = y, then Ty \ {a} is strictly locating. Finally, let u and w be distinct non-adjacent vertices of G with NG(u) = NG(w). If Tu is dominating is a dominating set, then we are done. Suppose Tu is not a dominating set in H. Then by (iv), Tw is a stable dominating set of H. This implies that Tw is a dominating set if w ̸= y and Ty \ {a} is a dominating set if w = y. Therefore, C∗ is a locating-dominating set of G[H]. Accordingly, C is a stable locating-dominating set of G[H]. Given a non-trivial connected graph H, we denote by γsL(H) the smallest size of a dominating stable locating set of H, i.e., γsL(H) = min{|S| : S is a dominating stable locating set of H}. Any dominating stable locating set of H of size γsL(H) is called a γsL-set of H. Note that since V (H) is a dominating stable locating set, it follows that H admits a dominating stable locating set. Consider the graph H in Figure 1 below. Clearly, S = {c, d, e} is a dominating stable locating set of H and γsL(H) = |S| = 3. ............................................................................................................................................ .................................... ............................................................................................................................................ ................................................................................................................................................................................ .................................... ............ ........... ........... ........... ........... ........... ........... ........... ........... ........... ........... ........... ........... ........ .................................... .................................... ......... ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ....... .................................... .................................... • • • a b c d e Figure 1 Corollary 4. Let G and H be non-trivial connected graphs. If G is point determining, then γsL(G[H]) ≤ |V (G)|γsL(H). Proof. Let D be a γsL-set of H and let Tx = D for each x ∈ V (G). Then C =⋃ x∈V (G) ({x} × Tx) is a stable locating-dominating set of G[H] by Theorem 5. Thus, γsL(G[H]) ≤ |C| = |V (G)|γsL(H). REFERENCES 488 Note that the bound in Corollary 4 is tight. To see this, consider the graph H in Figure 1. It can easily be verified that γsL(P3[H]) = 9 = 3.3 = |V (P3)|γsL(H). The next result follows from Theorem 5. Corollary 5. Let G be a connected totally point determining graph and let H be any non-trivial connected graph. Then C = ⋃ x∈S ({x} × Tx), where S ⊆ V (G) and Tx ⊆ V (H) for each x ∈ S, is a γsL-set of G [H] if and only if S = V (G) and Tx is an sbln-set of H for every x ∈ V (G). In particular, γsL(G[H]) = |V (G)|sbln(H). Proof. By Theorem 5, C = ⋃ x∈S ({x} × Tx), where is a γsL-set of G [H] if and only if S = V (G) and Tx is an sbln-set of H for every x ∈ V (G). Now, let D be an sbln-set of H and let Tx = D for each x ∈ V (G). Then C0 = ⋃ x∈V (G) ({x} × Tx) is a γsL-set of G [H]. Therefore, γsL(G[H]) = |C| = |V (G)|sbln(H). 3. Conclusion The locating dominating sets in the edge corona of graphs were characterized and bounds for its locating-domination number were obtained. The stable locating-dominating sets in the edge corona and lexicographic products of graphs were also characterized. Tight bounds for their stable locating-domination numbers were determined. A further study of stable locating-domination in other graphs is highly recommended. It is not yet known if the stable locating dominating set problem is NP-complete. 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] E. Ahmad, G. Malacas, and S. 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