EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 15, No. 4, 2022, 1705-1715 ISSN 1307-5543 – ejpam.com Published by New York Business Global Locating Hop Sets in a Graph Ethel Mae A. Pagcu1,∗, 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 Tibanga, Iligan City, Philippines 2 Center for Graph Theory, Algebra, and Analysis, Premier Research Institute in Science and Mathematics, Mindanao State University-Iligan Institute of Technology, 9200 Tibanga, Iligan City, Philippines Abstract. Let G be a connected graph with vertex set V (G) and edge set E(G). The open hop neighborhood of vertex v ∈ V (G) is the set NG(v, 2) = {w ∈ V (G) : dG(v, w) = 2}, where dG(v, w) denotes the distance between v and w. A non-empty set S ⊆ V (G) is a locating hop set of G if NG(u, 2) ∩ S ̸= NG(v, 2) ∩ S for every pair of distinct vertices u, v ∈ V (G) \ S. The smallest cardinality of a locating hop set of G, denoted by lhn(G) is called the locating hop number of G. This study focuses mainly on the concept of locating hop set in graphs. Characterizations of locating hop sets in the join and corona of two graphs are given and bounds for the corresponding locating hop numbers of these graphs are determined. 2020 Mathematics Subject Classifications: 05C69 Key Words and Phrases: Locating hop set, strictly locating hop set, join, corona 1. Introduction Let G = (V (G), E(G)) be a simple graph and v ∈ V (G). The set of neighbors of a vertex u in G, denoted by NG(u), is called the open neighborhood of u in G. The closed neighborhood of u in G is the set NG[u] = NG(u) ∪ {u}. The degree of a vertex v in a graph G, denoted by degG(v), is the number of edges incident with v in G and the minimum degree δ(G) of the vertices of G is the minimum degree of G. The open hop neighborhood of vertex v is the set NG(v, 2) = {w ∈ V (G) : dG(v, w) = 2}, where dG(v, w) denotes the distance between v and w. The closed hop neighborhood of vertex v is the set NG[v, 2] = NG(v, 2) ∪ {v}. The concept of hop neighborhood was used in [10] to define and investigate the concept of hop domination. Hop domination and some of its variants had been studied also in [6], [7], [9], [12], and [13]. ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v15i4.4500 Email addresses: ethelmae.pagcu@g.msuiit.edu.ph (EM. Pagcu), gina.malacas@g.msuiit.edu.ph (G. Malacas), sergio.canoy@g.msuiit.edu.ph (S. Canoy, Jr.) https://www.ejpam.com 1705 © 2022 EJPAM All rights reserved. E.M. A. Pagcu, G. A. Malacas, S. R. Canoy, Jr. / Eur. J. Pure Appl. Math, 15 (4) (2022), 1705-1715 1706 The concept of locating set was first introduced by Slater (for which a protection device can determine the distance to an intruder) in 1975 (see [16]). Omega and Canoy in [11] studied the locating sets in graphs and characterized the locating sets in the join and corona of graphs where they also determined the locating numbers of these graphs. A set S ⊆ V (G) is a locating set if for every two distinct vertices u, v ∈ V (G) \ S, NG(u) ∩ S ̸= NG(v) ∩ S. A set S ⊆ V (G) is strictly locating if it is locating and NG(u) ∩ S ̸= S for all u ∈ V (G) \ S. The minimum cardinality of a locating set in G, denoted by ln(G), is called the locating number of G. The minimum cardinality of a strictly locating set in G, denoted by sln(G), is the strict locating number of G. Any locating (resp. strictly locating) set with cardinality equal to ln(G) (resp. sln(G)), is called a minimum locating set or ln-set (resp. minimum strictly locating set or sln-set). In 1987, Slater in [17] further investigated locating set with another concept called domination. A set D ⊆ V (G) is a dominating set of G if ∪x∈DN [x] = V (G). The domination number of G, denoted by γ(G), is the minimum cardinality of a dominating set of G. Eventually, the concept of locating dominating set was introduced and is one of the widely studied topics nowadays (see [14], [15]). A locating subset S ⊆ V (G) which is also a dominating set is called locating-dominating set (LD-set) in a graph G. A strictly locating subset S of V (G) which is also a dominating set is called strictly locating-dominating set (SLD-set) in a graph G. The locating-domination number or L-domination number of G, denoted by γL(G), is the minimum cardinality of a locating-dominating set. The minimum cardinality of a strictly locating-dominating set of G, denoted by γSL(G), is called the SL- domination number of G. A locating-dominating (resp. strictly locating-dominating) set with cardinality equal to γL(G) (resp. γSL(G))is called a minimum locating-dominating set or γL-set (minimum strictly locating-dominating set or γSL-set). Canoy et al. [8] characterized the locating dominating sets in the corona and composition of graphs. They also determined the locating-domination number of these graphs. There are other studies involving the concept of locating set and locating dominating set (see [4], [5], [8], [10], and [11]). A non-empty set S ⊆ V (G) is a locating hop set of G if NG(u, 2)∩S ̸= NG(v, 2)∩S for every pair of distinct vertices u, v ∈ V (G) \S. A locating hop set is a strictly locating hop set if NG(v, 2)∩ S ̸= S for every v ∈ V (G) \ S. The smallest cardinality of a locating hop set (resp. strictly locating hop set) of G, denoted by lhn(G) (resp. slhn(G)) is called the locating hop (resp. strictly locating hop) number of G. Any locating hop set (resp. strictly locating hop set) with cardinality equal to lhn(G) (resp. slhn(G)) is called a minimum locating hop set or lhn-set (resp. minimum strictly locating hop set or shln-set). In this paper, we investigate the concept of locating hop set in the join and corona of two graphs. Investigation of several parameters in graphs under some binary operations had been done in many studies (see [1], [2], [3]). A point determining graph is defined in [18] as a graph in which distinct non-adjacent vertices have distinct neighborhoods. E.M. A. Pagcu, G. A. Malacas, S. R. Canoy, Jr. / Eur. J. Pure Appl. Math, 15 (4) (2022), 1705-1715 1707 2. Preliminary Results Proposition 1. For any graph G of order n ≥ 2, 1 ≤ lhn(G) ≤ n− 1. Proof : Let G be a connected non-trivial graph. By the definition of the locating hop set, lhn(G) ≥ 1. Let v ∈ V (G) and set S = V (G) \ {v}. Then S is a locating hop set of G. Hence, lhn(G) ≤ |S| = n− 1. □ Lemma 1. Let G be a graph with n vertices. If S is a locating hop set of G, then n ≤ |S|+ 2|S|. In particular, n ≤ lhn(G) + 2lhn(G). Proof : Let G be a graph of order n and S is a locating hop set in G. By definition of locating hop set, the collection {NG(a, 2) ∩ S : a ∈ V (G) \ S} contains exactly |V (G) \ S| distinct subsets of S. Hence, |V (G)\S| = n−|S| ≤ 2|S|, i.e., n ≤ |S|+2|S|. In particular, if S is an lhn-set of G, then n ≤ lhn(G) + 2lhn(G). □ Theorem 1. Let G be a non-trivial graph. Then lhn(G) = n − 1 if and only if every component of G is complete. Proof : Suppose that lhn(G) = n − 1 and suppose further that G has a component H which is not complete. Then there exist x, y ∈ V (H) such that dH(x, y) = dG(x, y) = 2. Let z ∈ NG(x) ∩NG(y) and S = V (G) \ {x, z}. Since y ∈ NG(x, 2) \NG(z, 2), it follows that NG(x, 2)∩S ̸= NG(z, 2)∩S. Thus, S is a locating hop set and lhn(G) ≤ |S| = n−2, contrary to the assumption lhn(G) = n−1. Therefore, every component of G is complete. For the converse, suppose that every component of G is complete. Let S be an lhn-set of G. Since NG(u, 2) ∩ S = ∅ ∀u ∈ V (G), V (G) \ S cannot contain two distinct vertices. Consequently, S = V (G) \ {v} for some vertex v of G. Thus, lhn(G) = |S| = n− 1. □ Corollary 1. For any positive integer n ≥ 2, lhn(Kn) = lhn(Kn) = n− 1. Proposition 2. Let G be a graph on n vertices. Then lhn(G) = 1 if and only if G ∈ {K1,K2, P2, P3}. Proof : Suppose lhn(G) = 1. By Lemma 1, n ≤ 3. Clearly, G = K1 if n = 1 and G = K2 = P2 or G = K2 if n = 2. Suppose n = 3. By Theorem 1, lhn(K3) = lhn(K1 ∪ P2) = lhn(K3) = 2. It follows that G = P3. Thus, G ∈ {K1,K2, P2, P3}. The converse is clear. □ Proposition 3. Let G be a connected graph of order n. If lhn(G) = 2, then 3 ≤ |V (G)| ≤ 6. Proof : Suppose that lhn(G) = 2. By Lemma 1, n ≤ lhn(G) + 2lhn(G) = 2 + 22 = 6. By Proposition 2, it follows that 3 ≤ |V (G)| ≤ 6. □ Proposition 4. Let G be a connected graph of order n = 4. Then lhn(G) = 2 if and only if G ̸= K4. Proof : Let lhn(G) = 2. Then by Corollary 1, G ̸= K4. For the converse, suppose that G ̸= K4. Since n = 4, by Proposition 2, lhn(G) ≥ 2. Choose any u, v ∈ V (G) such that dG(u, v) = 2. Let w ∈ NG(u) ∩ NG(v) and let s ∈ V (G) \ {u, v, w}. Since u ∈ NG(v, 2) \NG(w, 2), it follows that S = {u, s} is a locating set of G. Consequently, lhn(G) = |S| = 2. □ E.M. A. Pagcu, G. A. Malacas, S. R. Canoy, Jr. / Eur. J. Pure Appl. Math, 15 (4) (2022), 1705-1715 1708 Proposition 5. Let G be a connected graph of order n = 5. Then lhn(G) = 2 if and only if there exist distinct vertices x and y of G satisfying one of the following properties: (i) |NG(x, 2) ∩NG(y, 2)| = 0 and |NG(x, 2) \ {y}| = |NG(y, 2) \ {x}| = 1. (ii) |NG(x, 2)∩NG(y, 2)| = 1 and [(|NG(x, 2)\{y}| = |NG(y, 2)\{x}| = 2) or (|NG(x, 2)\ {y}| = 2 and |NG(y, 2)\{x}| = 1) or (|NG(x, 2)\{y}| = 1 and |NG(y, 2)\{x}| = 2)]. Proof : Suppose that lhn(G) = 2. Then there exist distinct vertices x, y ∈ V (G) such that S = {x, y} is a minimum locating hop set of G. Hence, |NG(x, 2)∩NG(y, 2)| ≤ 1. Suppose |NG(x, 2) ∩ NG(y, 2)| = 0. Since S is a locating hop set, |NG(x, 2) \ {y}| ≤ 1. Suppose |NG(x, 2) \ {y}| = 0. Then |NG(y, 2) \ {x}| = 1 since S is a locating hop set. This implies that there exist at least two vertices say z and w such that z, w /∈ NG(x, 2) ∪ NG(y, 2). Consequently, NG(z, 2) = NG(w, 2) = ∅, contrary to our assumption that S is a locating hop set. Thus, |NG(x, 2) \ {y}| = 1. Similarly, |NG(y, 2) \ {x}| = 1. Hence, (i) holds. Suppose that |NG(x, 2) ∩ NG(y, 2)| = 1. Let a ∈ V (G) such that dG(x, a) = 2 and dG(y, a) = 2 and let b, c ∈ V (G) \ {x, y, a}. Then b, c /∈ NG(x, 2) ∩ NG(y, 2). Since the subset of S are ∅, {x, y}, {x}, {y} and since NG(a, 2) ∩ S is {x, y}, the remaining two sets NG(b, 2)∩S and NG(c, 2)∩S are {x} and {y} or {x} and ∅ or {y} and ∅, respectively. Thus, |NG(x, 2) \ {y}| = |NG(y, 2) \ {x}| = 2 or |NG(x, 2) \ {y}| = 2 and |NG(y, 2) \ {x}| = 1 or |NG(x, 2) \ {y}| = 1 and |NG(y, 2) \ {x}| = 2. Therefore, (ii) holds. For the converse, suppose there exist distinct vertices x, y ∈ V (G) satisfying (i) or (ii). Let S = {x, y}. Then S is a minimum locating hop set in G. Therefore, lhn(G) = 2. □ Proposition 6. Let G be a connected graph of order n ≥ 3. If lhn(G) < slhn(G), then 1 + lhn(G) = slhn(G). Proof : Let S be a minimum locating hop set in G. Then S is not a strictly locating hop set inG. Hence, there exists a vertex u ∈ V (G)\S such thatNG(u, 2)∩S = S. Let S∗ = S∪{u} and let z ∈ V (G) \ S∗. Then z ̸= u. Since S is a locating hop set and NG(u, 2) ∩ S = S, NG(z, 2) ∩ S ̸= S. This implies that there exists w ∈ S such that w /∈ NG(z, 2). Since u /∈ S, w ̸= u. Thus, NG(z, 2) ∩ S∗ ̸= S∗. This implies that S∗ is a strictly locating hop set in G. Hence, slhn(G) ≤ 1 + lhn(G). Since lhn(G) < slhn(G), 1 + lhn(G) ≤ slhn(G). Hence, 1 + lhn(G) = slhn(G). □ 3. Locating Hop Sets in the Join of Graphs 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 2. Let G and H be connected non-trivial graphs. A set S ⊆ V (G + H) is a locating hop set in G+H if and only if S1 = V (G) ∩ S and S2 = V (H) ∩ S are locating sets in G and H, respectively, and S1 or S2 is a strictly locating set. E.M. A. Pagcu, G. A. Malacas, S. R. Canoy, Jr. / Eur. J. Pure Appl. Math, 15 (4) (2022), 1705-1715 1709 Proof : Suppose that S is a locating hop set in G +H. Let S1 ⊆ V (G) and S2 ⊆ V (H). Suppose S1 = ∅. Then for any two distinct vertices x, y ∈ V (G), NG+H(x, 2) ∩ S = NG+H(y, 2) ∩ S = ∅, contrary to our assumption that S is a locating hop set. Thus, S1 ̸= ∅. Similarly, S2 ̸= ∅. Next, suppose S1 or S2, say S1 is not a locating set. Then there exist u, v ∈ V (G) such that NG(u) ∩ S1 = NG(v) ∩ S1. Thus, x ∈ [V (G) \ NG(u)] ∩ S1 if and only if x ∈ [V (G) \ NG(v)] ∩ S1. This implies that [V (G) \ NG(u)] ∩ S1 = [V (G) \ NG(v)] ∩ S1. Since S2 ∩NG+H(u, 2) = ∅ and S2 ∩NG+H(v, 2) = ∅, it follows that NG+H(u, 2) ∩ S = NG+H(u, 2) ∩ S1 = [V (G) \NG(u)] ∩ S1 = [V (G) \NG(v)] ∩ S1 = NG+H(v, 2) ∩ S1 = NG+H(v, 2) ∩ S. Thus, S is not a locating hop set in G + H, contrary to our assumption. Therefore, S1 and S2 are locating sets in G and H, respectively. Now, suppose that both are not strictly locating sets. Then there exist p ∈ V (G)\S1 and q ∈ V (H)\S2 such that NG(p)∩S1 = S1 and NH(q) ∩ S2 = S2. Consequently, NG(p, 2) ∩ S1 = ∅ and NH(q, 2) ∩ S2 = ∅. This implies that NG+H(p, 2)∩ S = NG+H(q, 2)∩ S = ∅, contrary to our assumption that S is a locating hop set. Therefore, S1 is a strictly locating set in G or S2 is a strictly locating set in H. For the converse, suppose that S1 and S2 are locating sets in G and H, respectively, and S1 or S2 is a strictly locating set. Let x, y ∈ V (G+H) \S with x ̸= y. If x, y ∈ V (G), then NG(x) ∩ S1 ̸= NG(y) ∩ S1. Moreover, NG+H(x, 2) ∩ S = [V (G) \ NG(x)] ∩ S1 ̸= [V (G) \NG(y)] ∩ S1 = NG+H(y, 2) ∩ S. Similarly, if x, y ∈ V (H), then NG+H(x, 2) ∩ S ̸= NG+H(y, 2) ∩ S. Suppose that x ∈ V (G) and y ∈ V (H) and suppose that S1 is a strictly locating set in G. Then NG(x)∩S1 ̸= S1. It follows that [V (G)\NG(x)]∩S1 = NG+H(x)∩ S ̸= ∅. Since S1 ∩NG+H(y, 2) = ∅, NG+H(x, 2) ∩ S ̸= NG+H(y, 2) ∩ S. Therefore, S is a locating hop set in G+H. □ Corollary 2. Let G and H be connected non-trivial graphs. Then lhn(G+H) = min{sln(H) + ln(G), sln(G) + ln(H)}. Proof : Let S be a minimum locating hop set in G + H. Let S1 = V (G) ∩ S and S2 = V (H) ∩ S. By Theorem 2, S1 and S2 are locating sets in G and H, respectively, where S1 or S2 is a strictly locating set. If S1 is strictly locating set, then sln(G)+ln(H) ≤ |S1| + |S2| ≤ |S| = lhn(G + H). If S2 is strictly locating set, then sln(H) + ln(G) ≤ |S2|+|S1| ≤ |S| = lhn(G+H). Thus, lhn(G+H) ≥ min{sln(H)+ln(G), sln(G)+ln(H)}. Next, suppose that sln(G) + ln(H) ≤ sln(H) + ln(G). Let S1 be a minimum strictly locating set in G and S2 be a minimum locating set in H. Then S = S1 ∪ S2 is a locating hop set by Theorem 2. Hence, lhn(G+H) ≤ |S| = |S1|+|S2| = sln(G)+ln(H). Therefore, lhn(G+H) = min{sln(H) + ln(G), sln(G) + ln(H)}. □ Theorem 3. ([5],[11]) Let G be a connected graph of order n ≥ 2. If ln(G) < sln(G), then 1 + ln(G) = sln(G). E.M. A. Pagcu, G. A. Malacas, S. R. Canoy, Jr. / Eur. J. Pure Appl. Math, 15 (4) (2022), 1705-1715 1710 Corollary 3. Let G be a connected non-trivial graph and let Kn be a complete graph of order n ≥ 2. Then lhn(G+Kn) = sln(G) + n− 1. Proof : Note that ln(Kn) = n − 1 and sln(Kn) = n. By Corollary 2, lhn(G + Kn) = min{sln(G) + n − 1, ln(G) + n} and by Theorem 3, sln(G) − 1 ≤ ln(G). Therefore, lhn(G+Kn) = min{sln(G) + n− 1, ln(G) + n} = sln(G) + n− 1. □ Theorem 4. Let G be a connected non-trivial graph and let K1 = ⟨v⟩. Then S ⊆ V (G+K1) is a locating hop set in G+K1 if and only if v /∈ S and S is a strictly locating set in G or S = {v} ∪ S1, where S1 is a locating set in G. Proof : Let S ⊆ V (G + K1) be a locating hop set in G + K1. If v /∈ S, then S ⊆ V (G). Let u, s ∈ V (G) \ S. Then NG+K1(u, 2) ∩ S ̸= NG+K1(s, 2) ∩ S. It follows that [V (G) \NG(u)] ∩ S ̸= [V (G) \NG(s)] ∩ S. Therefore, NG(u) ∩ S = [V (G) \NG+K1(u, 2)] ∩ S ̸= [V (G) \NG+K1(v, 2)] ∩ S = NG(v) ∩ S, showing that S is a locating set in G. Suppose S is not a strictly locating set in G. Then there exists z ∈ V (G) \ S such that NG(z) ∩ S = S. This implies that NG(z, 2) ∩ S = ∅ = NG(v, 2) ∩ S, contrary to our assumption that S is a locating hop set. Hence, S is a strictly locating set in G. Next, suppose that S = {v} ∪ S1, where S1 = V (G) ∩ S. Then S1 ̸= ∅ and is a locating set in G. For the converse, suppose v /∈ S and S is a strictly locating set in G. Let x, y ∈ V (G+K1) \ S. If x, y ∈ V (G), then NG+K1(x, 2) ∩ S = [V (G) \NG(x)] ∩ S ̸= [V (G) \NG(y)] ∩ S = NG+K1(y, 2) ∩ S. Suppose x ∈ V (G) and y = v. Then NG+K1(v, 2) ∩ S = ∅. Since S is a strictly locating set in G, NG(x) ∩ S ̸= S. Then NG+K1(x, 2) ∩ S = [V (G) \NG(x)] ∩ S ̸= [V (G) \NG(v)] ∩ S = NG+K1(v, 2) ∩ S. Therefore, S is a locating hop set in G+K1. Next, suppose that S = {v} ∪ S1, where S1 is a locating set of G. Let x, y ∈ V (G +K1) \ S with x ̸= y. Then x, y ∈ V (G) \ S1 and NG(x) ∩ S1 ̸= NG(y) ∩ S1. Thus, NG+K1(x, 2) ∩ S = [V (G) \NG(x)] ∩ S1 ̸= [V (G) \NG(y)] ∩ S1 = NG+K1(y, 2) ∩ S. Hence, S is a locating hop set in G+K1. □ Corollary 4. Let G be a connected non-trivial graph. Then lhn(G+K1) = sln(G). Proof : By Theorem 4, lhn(G+K1) = min{sln(G), ln(G)+1}. By Theorem 3, sln(G)−1 ≤ ln(G). Hence, sln(G) ≤ ln(G) + 1. Therefore, lhn(G+K1) = sln(G). □ E.M. A. Pagcu, G. A. Malacas, S. R. Canoy, Jr. / Eur. J. Pure Appl. Math, 15 (4) (2022), 1705-1715 1711 4. Locating Hop Sets in the Corona of Graphs The corona of two graphs G and H, denoted by G ◦ H, is the graph obtained by taking one copy of G of order n and n copies of H, and then joining the vertex vi of G to every vertex of the ith copy of H. For every v ∈ V (G), denote by Hv the copy of H whose vertices are joined or attached to the vertex v. Denote by v +Hv the subgraph of the corona G ◦H corresponding to the join ⟨{v}⟩+Hv. Theorem 5. Let G be a non-trivial connected graph and let H be any non-trivial graph. Then S ⊆ V (G ◦H) is a locating hop set of G ◦H if and only if S = A∪ [∪v∈V (G)Dv] and (i) A ⊆ V (G) such that for any two distinct vertices v, w ∈ V (G) \ A, NG(v) ̸= NG(w) or NG(v, 2) ∩A ̸= NG(w, 2) ∩A; (ii) Dv is a locating set in Hv for each v ∈ V (G); (iii) Dw is a dominating set of Hw for each w ∈ V (G) such that NG(v) = {w} for some v ∈ V (G) \A; and (iv) Dv or Dw is a strictly locating set for each pair of distinct vertices v and w of G with NG(v) ∩A = NG(w) ∩A. Proof : Suppose S is a locating hop set in G◦H. Let A = S∩V (G) and let Dv = S∩V (Hv) for each v ∈ V (G). Then S = A ∪ [∪v∈V (G)Dv]. Let v, w ∈ V (G) \A with v ̸= w. Since S is a locating hop set in G ◦H, [NG(v, 2) ∩A] ∪ [∪x∈NG(v)Dx] = NG◦H(v, 2) ∩ S ̸= NG◦H(w, 2) ∩ S = [NG(w, 2) ∩A] ∪ [∪y∈NG(w)Dy]. This implies that NG(v, 2)∩A ̸= NG(w, 2)∩A or NG(v) ̸= NG(w), showing that (i) holds. Next, let v ∈ V (G) and let a, b ∈ V (Hv) \Dv with a ̸= b. Since S is a locating hop set in G ◦H, ([V (Hv) \NHv(a)] ∩Dv) ∪ [NG(v) ∩A] = NG◦H(a, 2) ∩ S ̸= NG◦H(b, 2) ∩ S = ([V (Hv) \NHv(b)] ∩Dv) ∪ [NG(w) ∩A]. Hence, [V (Hv) \ NHv(a)] ∩ Dv ̸= [V (Hv) \ NHv(b)] ∩ Dv. This implies that NHv(a) ∩ Dv ̸= NHv(b) ∩ Dv, showing Dv is a locating set of Hv. Hence, (ii) holds. To show that (iii) holds, suppose there exists w ∈ V (G) such that NG(v) = {w} for some v ∈ V (G) \ A. If Dw = V (Hw), then we are done. So suppose that Dw ̸= V (Hw) and let q ∈ V (Hw) \Dw. Then by assumption and the fact that S is a locating hop set in G ◦H, Dw ∪ (NG(w) ∩A) = NG◦H(v, 2) ∩ S E.M. A. Pagcu, G. A. Malacas, S. R. Canoy, Jr. / Eur. J. Pure Appl. Math, 15 (4) (2022), 1705-1715 1712 ̸= NG◦H(q, 2) ∩ S = ([V (Hw) \NHw(q)] ∩Dw) ∪ [NG(w) ∩A]. This implies that [(V (Hw) \ NHw(q)) ∩ Dw] ̸= Dw, that is, NHw(q) ∩ Dw ̸= ∅. This shows that Dw is a dominating set of Hw. Finally, let v, w ∈ V (G) with v ̸= w and NG(w) ∩ A = NG(v) ∩ A. Suppose Dv and Dw are not strictly locating sets of Hv and Hw, respectively. Then there exist x ∈ V (Hv) \ Dv and y ∈ V (Hw) \ Dw such that NHv(x) ∩Dv = Dv and NHw(y) ∩Dw = Dw. It follows that [V (Hv) \NHv(x)] ∩Dv = ∅ and [V (Hw) \NHw(y)] ∩Dw = ∅. This would imply that NG◦H(x, 2) ∩ S = [(V (Hv) \NHv(x)) ∩Dv] ∪ (NG(v) ∩A) = NG(v) ∩A = NG(w) ∩A = [(V (Hw) \NHw(y)) ∩Dw] ∪ (NG(w) ∩A) = NG◦H(y, 2) ∩ S, contrary to the assumption that S is a locating hop set of G ◦H. Thus, (iv) holds. For the converse, suppose that S is as described and satisfies properties (i)-(iv). Let a, b ∈ V (G ◦ H) \ S with a ̸= b and let v, w ∈ V (G) such that a ∈ V (v + Hv) and b ∈ V (w +Hw). Consider the following cases: Case 1: v = w. Suppose a, b ∈ V (Hv) \Dv. By (ii), NG◦H(a, 2) ∩ S ̸= NG◦H(b, 2) ∩ S. Suppose a = v and b ∈ V (Hv) \Dv. Pick any z ∈ NG(v). Since Dz ⊆ NG◦H(a, 2) \NG◦H(b, 2), it follows that NG◦H(a, 2) ∩ S ̸= NG◦H(b, 2) ∩ S. Case 2: v ̸= w. Suppose a = v and b = w. Then v, w ∈ V (G) \ A. By property (i), NG(v) ̸= NG(w) or NG(v, 2) ∩ A ̸= NG(w, 2) ∩ A. If NG(v, 2) ∩ A ̸= NG(w, 2) ∩ A, then NG◦H(a, 2) ∩ S ̸= NG◦H(b, 2) ∩ S. Suppose NG(v) ̸= NG(w). We may assume that there exists p ∈ NG(v) \ NG(w). Then Dp ⊆ NG◦H(a, 2) \NG◦H(b, 2). Hence, NG◦H(a, 2) ∩ S ̸= NG◦H(b, 2) ∩ S. Next, suppose that a = v and b ∈ V (Hw) \ Dw (or b = w and a ∈ V (Hv) \ Dv). If |NG(v)| > 1 or vw /∈ E(G), pick any z ∈ NG(v)\{w}. Then Dz ⊆ NG◦H(a, 2)\NG◦H(b, 2). It follows that NG◦H(a, 2)∩S ̸= NG◦H(b, 2)∩S. Suppose that NG(v) = {w}. Then Dw is a dominating set by (iii). Hence, [(V (Hw) \NHw(b)) ∩Dw] ̸= Dw. This implies that NG◦H(a, 2) ∩ S = Dw ∪ (NG(w) ∩A) ̸= [(V (Hw) \NHw(b)) ∩Dw] ∪ (NG(w) ∩A) = NG◦H(b, 2) ∩ S. Finally, suppose that a ∈ V (Hv)\Dv and b ∈ V (Hw)\Dw. If [V (Hv)\NHv(a)]∩Dv ̸= ∅ and [V (Hw) \ NHw(b)] ∩ Dw ̸= ∅, then NG◦H(a, 2) ∩ S ̸= NG◦H(b, 2) ∩ S. Suppose one, say [V (Hv) \ NHw(a)] ∩ Dv = ∅. If NG(v) ∩ A ̸= NG(w) ∩ A, then NG◦H(a, 2) ∩ S ̸= NG◦H(b, 2) ∩ S. If NG(v) ∩ A = NG(w) ∩ A, then [V (Hw) \ NHw(b)] ∩ Dw ̸= ∅ by (iv). Thus, NG◦H(a, 2) ∩ S ̸= NG◦H(b, 2) ∩ S. Accordingly, S is a locating hop set of G ◦H. □ The next result is an immediate consequence of Theorem 5. E.M. A. Pagcu, G. A. Malacas, S. R. Canoy, Jr. / Eur. J. Pure Appl. Math, 15 (4) (2022), 1705-1715 1713 Corollary 5. Let G be a non-trivial connected graph of order m and let H be any graph. Then the following statements hold: (i) m · ln(H) ≤ lhn(G ◦H) ≤ lhn(G) +m · γSL(H). (ii) If δ(G) ≥ 2, then lhn(G ◦H) ≤ lhn(G) +m · sln(H). (iii) If G is point determining, then lhn(G ◦H) ≤ m · γSL(H). (iv) If G is point determining and δ(G) ≥ 2, then lhn(G ◦H) ≤ m · sln(H). Moreover, if in addition, ln(H) = sln(H), then lhn(G ◦H) = m · ln(H) = m · sln(H). Proof : Let S be a minimum locating hop set (lhn-set) in G. Then S = A ∪ [∪v∈V (G)Dv] and satisfies the conditions in Theorem 5. In particular, Dv is a (minimum) locating set in Hv for each v ∈ V (G). Hence, m · ln(H) ≤ |A|+ ∑ v∈V (G) |Dv| = |S| = lhn(G ◦H). Now, let A1 be a locating hop set in G and let Lv be a strictly locating-dominating set (γSL-set) in Hv for each v ∈ V (G). Then S = A1 ∪ [∪v∈V (G)Lv] is a locating hop set in G ◦ H by Theorem 5. This implies that lhn(G ◦ H) ≤ |S| = lhn(G) + m · γSL(H), showing that (i) holds. If δ(G) ≥ 2 and each Lv is a minimum strictly locating set (sln- set) in Hv, then S is a locating hop set in G ◦H by Theorem 5. Thus, (ii) holds, that is, lhn(G ◦H) ≤ |S| = lhn(G) +m · sln(H). Suppose G is a point determining graph. For each v ∈ V (G), let Tv be a minimum strictly locating-dominating set (γSL-set) in Hv. Then S1 = ∪v∈V (G)Tv is a locating hop set in G ◦H by Theorem 5. This implies that lhn(G ◦H) ≤ |S| = m · γSL(H), showing that (iii) holds. Moreover, if we impose that δ(G) ≥ 2, then each set Tv can be taken as a strictly locating set of Hv. Now, S1 is still a locating hop set in G ◦H by Theorem 5. Thus, lhn(G ◦H) ≤ m · sln(H). Suppose now that, in addition, ln(H) = sln(H). Then lhn(G ◦H) ≤ m · sln(H) = m · ln(H). Combining this with an inequality in (i), it follows that lhn(G ◦H) = m · ln(H) = m · sln(H). □ Corollary 6. Let G be a cycle of order m = 4 and H be a non-trivial graph. Then lhn(G ◦H) = { m · sln(H) + 2 if ln(H) = sln(H) 2sln(H) + 2ln(H) + 2 if ln(H) < sln(H). Proof : Let C4 = [a, b, c, d, a] and let S be a minimum locating hop set in C4 ◦ H. Put Dv = S ∩ V (Hv) for each v ∈ C4. By Theorem 5(i), A = S ∩ V (C4) ̸= ∅. Without loss of generality, we suppose that a ∈ A. Suppose further that b, d /∈ A. Since NC4(b) = NC4(d) and NC4(b, 2) ∩A = NC4(d, 2) ∩A = ∅, A does not satisfy Theorem 5(i), contrary to our assumption that S is a locating hop set in C4 ◦H. Hence, either b ∈ A or d ∈ A, say b ∈ A. Since NC4(a) ∩ A = NC4(c) ∩ A, Da or Dc, say Da must be a minimum strictly locating REFERENCES 1714 set in Ha. It follows that Dc is a minimum locatng set in Hc by Theorem 5(iv) and the fact that S is an lhn-set. Similarly, one of Db and Dd is a minimum strictly locating set and the other a minimum locating set. Since S is a minimum locating hop set in C4 ◦H, |A| = 2 (increasing the number of elements of A will not change the above requirement for the sets Dv), and two subsets of S in copies of H are strictly locating sets. Therefore, lhn(C4 ◦H) = |A|+ 2ln(H) + 2sln(H) = 2ln(H) + 2sln(H) + 2. If ln(H) = sln(H), then lhn(C4 ◦H) = 4ln(H) + 2 = 4sln(H) + 2. □ 5. 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