EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 15, No. 4, 2022, 1623-1636 ISSN 1307-5543 – ejpam.com Published by New York Business Global Grundy Hop Domination in Graphs Javier A. Hassan1,∗, Sergio R. Canoy, Jr.2 1,2 Department of Mathematics and Statistics, College of Science and Mathematics, Center for Graph Theory, Premier Research Institute of Science and Mathematics, MSU-Iligan Institute of Technology, 9200 Iligan City, Philippines Abstract. Let G be an undirected graph with vertex and edge sets V (G) and E(G), respectively. Let S = (v1, v2, · · · , vk) be a sequence of distinct vertices of G and let Ŝ = {v1, v2, . . . , vk}. Then S is a legal closed hop neighborhood sequence of G if N2 G[vi]\∪ i−1 j=1N 2 G[vj ] ̸= ∅ for each i ∈ {2, · · · , k}. If, in addition, Ŝ is a hop dominating set of G, then S is called a Grundy hop dominating sequence. The maximum length of a Grundy hop dominating sequence in a graph G, denoted by γh gr(G), is called the Grundy hop domination number of G. In this paper, we determine some (extreme) values for the Grundy hop domination number. It is pointed out that the Grundy hop domination number is at least equal to the hop domination. Bounds for the Grundy hop domination numbers of some graphs resulting from some binary operations of two graphs are also obtained. 2020 Mathematics Subject Classifications: 05C69 Key Words and Phrases: Hop domination, hop domination number, closed hop neighborhood sequence, Grundy hop dominating sequence, Grundy hop domination number 1. Introduction One of the several considered variations of the standard domination concept is hop domination. This concept was introduced and initially studied by Natarajan and Ayyaswamy in [15]. Just like domination, hop domination and its variations find plenty of applications in various fields and networks. In fact, some real-life problems (including pro- tection strategies and facility location) that can be modeled by the concept of domination can be slighty modified for the concept of hop domination. Domination, hop domination, and some of their variations are also studied in [1], [2], [9], [10], [11], [12], [13], and [16]. In 2014, the concept of Grundy domination in graphs was introduced by Bresar et al. [6]. The newly defined parameter has subsequently attracted other researchers in the area who generated more interesting results (see [3], [4], [5], and [7]). Grundy domination was further studied in [5], where exact formulas for Grundy domination numbers of Sierpinski ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v15i4.4511 Email addresses: javier.hassan@g.msuiit.edu.ph (J. Hassan), sergio.canoy@g.msuiit.edu.ph (S. Canoy) https://www.ejpam.com 1623 © 2022 EJPAM All rights reserved. J. Hassan, S. Canoy / Eur. J. Pure Appl. Math, 15 (4) (2022), 1623-1636 1624 graphs were proven and a linear algorithm for determining these numbers in arbitrary interval graphs was given. Grundy domination number was also studied in Kneser graphs [7] and graph products in [3] and [14]. In this study, the concept of Grundy hop domination in a graph will be introduced and initially investigated. In particular, bounds for the parameter will be given for the join, corona, and lexicographic product of graphs. 2. Terminology and Notation Two vertices u, v of a graph G are adjacent, or neighbors, if uv is an edge of G. Moreover, an edge uv of G is incident to two vertices u, v of 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}. If X ⊆ V (G), the open neighborhood of X in G is the set NG(X) = ⋃ u∈X NG(u). The closed neighborhood of X in G is the set NG[X] = NG(X) ∪X. Let G be a graph. A set D ⊆ V (G) is a dominating set of G if for every v ∈ V (G) \D, there exists u ∈ D such that uv ∈ E(G), that is, NG[D] = V (G). The domination number of G, denoted by γ(G), is the minimum cardinality of a dominating set of G. Let S = (v1, v2, · · · , vk) be a sequence of distinct vertices of a graph G, and let Ŝ = {v1, v2, · · · , vk}. Then S is a legal closed neighborhood sequence if NG[vi]\ ⋃i−1 j=1NG[vj ] ̸= ∅ for every i ∈ {2, · · · , k}. If, in addition, Ŝ is a dominating set of G, then S is called a Grundy dominating sequence. The maximum length of a Grundy dominating sequence in a graph G is called the Grundy domination number of G, and is denoted by γgr(G). A vertex v in G is a hop neighbor of vertex u in G if dG(u, v) = 2. The set N2 G(u) = {v ∈ V (G) : dG(v, u) = 2} is called the open hop neighborhood of u. The closed hop neighborhood of u in G is given by N2 G[u] = N2 G(u) ∪ {u}. The open hop neighborhood of X ⊆ V (G) is the set N2 G(X) = ⋃ u∈X N2 G(u). The closed hop neighborhood of X in G is the set N2 G[X] = N2 G(X) ∪X. A set S ⊆ V (G) is a hop dominating set of G if N2 G[S] = V (G), that is, for every v ∈ V (G)\S, there exists u ∈ S such that dG(u, v) = 2. The minimum cardinality among all hop dominating sets of G, denoted by γh(G), is called the hop domination number of G. Any hop dominating set with cardinality equal to γh(G) is called a γh-set. Let S = (v1, v2, · · · , vk) be a sequence of distinct vertices of G and let Ŝ = {v1, · · · , vk}. Then S is a legal closed hop neighborhood sequence of G if N2 G[vi]\∪ i−1 j=1N 2 G[vj ] ̸= ∅ for each i ∈ {2, · · · , k}. If, in addition, Ŝ is a hop dominating set of G, then S is called a Grundy hop dominating sequence. The maximum length of a Grundy hop dominating sequence in a graph G, denoted by γhgr(G), is called the Grundy hop domination number of G. We say that vertex vi hop-footprints the vertices from N2 G[vi]\∪i j=1N 2 G[vj ], and that vi is their hop- footprinter. A legal closed hop neighborhood sequence S = (v1, v2, · · · , vk) with maximum length, i.e., k = max{p ∈ N : ∃ a legal closed hop neighborhood sequence (x1, · · · , xp) ofG}, will be referred to as a maximum legal closed hop neighborhood sequence. J. Hassan, S. Canoy / Eur. J. Pure Appl. Math, 15 (4) (2022), 1623-1636 1625 Let S1 = (v1, · · · , vn) and S2 = (u1, · · · , um), n,m ≥ 1 be two sequences of distinct vertices of G. The concatenation of S1 and S2, denoted by S1 ⊕ S2, is the sequence given by S1 ⊕ S2 = (v1, · · · , vn, u1, · · · , um). A sequence S = (v1, v2, · · · , vk) of distinct vertices of a graph G is a co-legal closed neighborhood sequence in G if [V (G) \NG(vi)] \ ∪i−1 j=1[V (G) \NG(vj)] ̸= ∅ for each i ∈ {2, . . . , k}. A co-legal sequence S = (v1, v2, . . . , vk) is a co-Grundy dominating sequence if V (G) = ∪k i=1[V (G)\NG(vi)]. The maximum length of a co-Grundy dominating sequence in a graph G is called the co-Grundy domination number of G, and is denoted by γcogr(G). A set D ⊆ V (G) is hop independent if for every pair of distinct vertices v, w ∈ D, we have dG(v, w) ̸= 2. This concept was introduced and studied in [8]. Let S = (v1, v2, · · · , vk) be a sequence of distinct vertices of a graph G and let Ŝ = {v1, v2, . . . , vk}. Then S is a legal closed hop independent neighborhood sequence in G if it is a legal closed hop neighborhood sequence and Ŝ is a hop independent set. A legal closed hop independent neighborhood sequence S = (v1, v2, . . . , vk) is a Grundy hop independent hop dominating sequence if Ŝ is a hop independent hop dominating set of G. The maximum length of a Grundy hop independent hop dominating sequence in a graph G is called the Grundy hop independent hop domination number of G, and is denoted by γhihgr (G). Let G and H be any two graphs. The join of 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)}. The corona G and H, denoted by G ◦H, the graph obtained by taking one copy of G and |V (G)| copies of H, and then joining the ith vertex of G to every vertex of the ith copy of H. We denote by Hv the copy of H in G ◦H corresponding to the vertex v ∈ G and write v +Hv for ⟨{v} +Hv⟩. The lexicographic product of graphs G and H, denoted by G[H], is the graph with vertex set V (G[H]) = V (G) × V (H) and (v, a)(u, b) ∈ E(G[H]) if and only if either uv ∈ E(G) or u = v and ab ∈ E(H). We note that any non-empty set C ⊆ V (G) × V (H) can be written as C = ⋃ x∈S [{x} × Tx], where S ⊆ V (G) and Tx ⊆ V (H) for each x ∈ S. Specifically, Tx = {a ∈ V (H) : (x, a) ∈ C} for each x ∈ S. 3. Results Remark 1. The vertex set of a graph need not form a legal closed hop neighborhood sequence (a Grundy hop dominating sequence). To see this, consider the graph G = C5 in Figure 1. Let S = (v1, v2, v3, v4, v5). Notice that N2 G[v5] = {v2, v3, v5} ⊆ N2 G[v1] ∪N2 G[v2] = {v1, v2, v3, v4, v5}. J. Hassan, S. Canoy / Eur. J. Pure Appl. Math, 15 (4) (2022), 1623-1636 1626 Hence, N2 G[v5] \ ∪4 j=1N 2 G[vj ] = ∅. Thus, S is not a legal closed hop neighborhood sequence (hence, not a Grundy hop dominating sequence). It is routine to show that any rearrangement of the terms of S is not legal closed hop neighborhood sequence of G. G : v1 v2 v3v4 v5 Figure 1: A graph G such that V (G) does not form a legal closed hop neighborhood sequence Remark 2. A proper hop dominating set need not form a legal closed hop neighborhood sequence (a Grundy hop dominating sequence). Consider the graph G in Figure 2. Let S = (v1, v2, v3). Clearly, Ŝ is a proper hop dominating set of G. Observe that N2 G[v3] = {v2, v3, v5} = N2 G[v2]. Hence, N2 G[v3] \ ∪2 j=1N 2 G[vj ] = ∅. Thus, S is not a legal closed hop neighborhood sequence (not a Grundy hop dominating sequence). v1 v2 v4 v3 v5 G : Figure 2: A graph G with a proper hop dominating set which does not form a legal closed hop neighborhood sequence Our first result shows that every graph G admits a Grundy hop dominating sequence. Theorem 1. Let G be any graph on n vertices. Then the following statements hold. (i) If γh(G) = k and D = {v1, v2, . . . , vk} is a minimum hop dominating set of G, then S = (v1, v2, · · · , vk) is a Grundy hop dominating sequence. In particular, γh(G) ≤ γhgr(G). (ii) If S = (v1, v2, · · · , vm) is a minimum Grundy hop dominating sequence, then γh(G) = |Ŝ|. J. Hassan, S. Canoy / Eur. J. Pure Appl. Math, 15 (4) (2022), 1623-1636 1627 Proof. (i) Suppose there exists i ∈ {2, 3, . . . , k} such that N2 G[vi] \ ∪ i−1 j=1N 2 G[vj ] = ∅. Then N2 G[vi] ⊆ ∪i−1 j=1N 2 G[vj ]. It follows that D \ {vi} is a hop dominating set of G, contra- dicting the minimality of D. Therefore, N2 G[vi]\∪ i−1 j=1N 2 G[vj ] ̸= ∅ for each i ∈ {2, 3, . . . , k}. Consequently, γh(G) ≤ γhgr(G). (ii) Note that from (i), every γh-set of G forms a Grundy hop dominating sequence. Since S is a minimum Grundy hop dominating sequence, it follows that |Ŝ| ≤ γh(G). On the other hand, since every Grundy hop dominating sequence forms a hop dominating set (by definition), it follows that γh(G) ≤ |Ŝ|. This establishes the desired equality. Theorem 2. Let G be any graph. Then S = (v1, v2, · · · , vk) is a maximum legal closed hop neighborhood sequence of G if and only if S is a Grundy hop dominating sequence of G and γhgr(G) = k. Proof. Let S = (v1, · · · , vk) be a maximum legal closed hop neighborhood sequence of G. Suppose Ŝ is not a hop dominating set of G. Then there exists v ∈ V (G)\N2 G[Ŝ]. This implies that v /∈ N2 G[u] for every u ∈ Ŝ. Let S′ = (v1, · · · , vk, v). Since S is a legal closed hop neighborhood sequence, N2 G[vi] \ ∪i−1 j=1N 2 G[vj ] ̸= ∅ for each i ∈ {2, 3, . . . , k}. Also, since v ∈ N2 G[v] and v /∈ N2 G[u] for every u ∈ Ŝ, it follows that N2 G[v] \ ∪k j=1N 2 G[vi] ̸= ∅. Hence, S′ is a legal closed hop neighborhood sequence of G, contradicting the maximality of S. Therefore, Ŝ is a hop dominating set of G. Since S is a maximum legal closed hop neighborhood sequence of G, it is a Grundy hop dominating sequence and γhgr(G) = k. For the converse, suppose that S is a Grundy hop dominating sequence and γhgr(G) = k. Then S is a maximum legal closed hop neighborhood sequence of G. Corollary 1. Let G be a graph and let D = (x1, · · · , xt) be a legal closed hop neighborhood sequence of G. Then |D̂| = t ≤ γhgr(G). Proof. Let k be the length of a maximum legal closed hop neighborhood sequence of G. Then t ≤ k. By Theorem 2, |D̂| = t ≤ γhgr(G). Throughout, [n] = {1, 2, . . . , n} for each positive integer n. Theorem 3. Let G be any graph on n ≥ 2 vertices. Then 2 ≤ γhgr(G) ≤ n. Moreover, each of the following statements holds. (i) γhgr(G) = 2 if and only if for each pair of distinct vertices x, y ∈ V (G) such that N2 G[x] ̸= N2 G[y], we have V (G) = N2 G[x]∪N2 G[y] (i.e., {x, y} is a hop dominating set of G). (ii) γhgr(G) = n if and only if every component C of G is complete. Proof. Clearly, 2 ≤ γhgr(G) ≤ n. J. Hassan, S. Canoy / Eur. J. Pure Appl. Math, 15 (4) (2022), 1623-1636 1628 (i) Suppose γhgr(G) = 2. Then by Theorem 2, the maximum length of a legal closed hop neighborhood sequence of G is 2. Let x and y be distinct vertices of G such that N2 G[x] ̸= N2 G[y]. We may assume that N2 G[y] \ N2 G[x] ̸= ∅. Then (x, y) is a legal closed hop neighborhood sequence of G. Suppose there exists z ∈ V (G) \ (N2 G[x]∪N2 G[y]). Since z ∈ N2 G[z], it follows that N2 G[z] \ (N2 G[x] ∪ N2 G[y]) ̸= ∅. This implies that (x, y, z) is a legal closed hop neighborhood sequence of G, contrary to the assumption that γhgr(G) = 2. Hence, V (G) = N2 G[x] ∪N2 G[y]. For the converse, suppose that for each pair of distinct vertices x, y ∈ V (G) such that N2 G[x] ̸= N2 G[y], we have V (G) = N2 G[x] ∪ N2 G[y]. Since G is a non-trivial graph, the assumption implies that γh(G) = 2. Hence, γhgr(G) = k ≥ 2. Let (v1, v2, · · · , vk) be a Grundy hop dominating sequence of G. Because N2 G[v2] \ N2 G[v1] ̸= ∅, V (G) = N2 G[v1] ∪N2 G[v2], by assumption. Therefore, γhgr(G) = k = 2. (ii) Suppose γhgr(G) = n and let S = (v1, v2, · · · , vn) be a Grundy hop dominating sequence of G. Note that since N2 G[vn] \ ∪n−1 j=1N 2 G[vj ] ̸= ∅ and vj ∈ N2 G[vj ] for each j ∈ [n], N2 G[vn] \ ∪n−1 j=1N 2 G[vj ] = {vn}. This implies that vn /∈ N2 G[vj ] for all j ∈ [n − 1], i.e., N2 G[vn] = {vn}. This would imply that N2 G[vn−1] \ ∪n−2 j=1N 2 G[vj ] = {vn−1}. Using the same argument as earlier, N2 G[vn−1] = {vn−1}. Continuing in this fashion, we find that N2 G[vi] = {vi} for each i ∈ [n], i.e., degG(vi) = 0 or vi is adjacent to every other vertex in the component it belongs. Therefore, every component of G is complete. Conversely, if every component of G is complete, then N2 G[v] = {v} for each v ∈ V (G). Hence, if V (G) = {v1, v2, . . . , vn}, then N2 G[vi] \ ∪i−1 j=1N 2 G[vj ] = {vi} \ {vj : j ̸= i} = {vi} ≠ ∅. for each i ∈ {2, 3, . . . , n}. This shows that (v1, v2, · · · , vn) is a Grundy hop dominating sequence of G. Therefore γhgr(G) = n. The next result is immediate from the Theorem 3(ii). Corollary 2. Let G be a connected graph on n vertices. Then each of the following statements holds. (i) γhgr(G) = n if and only if G = Kn. (ii) If G is non-complete, then γhgr(G) ≤ n− 1. Theorem 4. Let G be a graph on n vertices. (i) If G is complete, then γhgr(G) + γhgr(G) = 2n. (ii) If G is non-complete, then (a) 4 ≤ γhgr(G) + γhgr(G) ≤ 2n− 1, and (b) 4 ≤ γhgr(G) · γhgr(G) ≤ n2 − n. J. Hassan, S. Canoy / Eur. J. Pure Appl. Math, 15 (4) (2022), 1623-1636 1629 Proof. (i) The equality follows from Theorem 3(ii). (ii) By Corollary 2(ii), γhgr(G) ≤ n− 1 and by Theorem 3, γhgr(G) ≤ n. These imply that γhgr(G) + γhgr(G) ≤ n− 1 + n = 2n− 1, and γhgr(G) · γhgr(G) ≤ (n− 1)n = n2 − n. The left inequalities follow from Theorem 3. The bounds in Theorem 4(ii) are tight. Indeed, one can easily verify that γhgr(P4) + γhgr(P 4) = γhgr(P4) · γhgr(P 4) = 4, γhgr(P3) + γhgr(P 3) = 5 = 2(3)− 1, and γhgr(P3) · γhgr(P 3) = 6 = 32 − 3. Proposition 1. For any positive integer n ≥ 2, γhgr(Pn) = { 2 if n = 2, 3 n− 2 if n ≥ 4. Proof. Let G = Pn = [v1, v2, · · · , vn]. Clearly, γhgr(Pn) = 2 for n = 2, 3. So sup- pose that n ≥ 4. Let S′ = (v1, · · · , vn−2). Clearly, S′ is a Grundy hop dominating sequence in G. Since G is not a complete graph, γhgr(G) ≤ n − 1 by Corollary 2(ii). Thus, n − 2 ≤ γhgr(G) ≤ n − 1. Suppose γhgr(G) = n − 1, say, S = (w1, · · · , wn−1) is a Grundy hop dominating sequence. Then N2 G[wn−1] \ ⋃n−2 j=1 N 2 G[wj ] ̸= ∅. Notice that N2 G[wn−1] \ ⋃n−2 j=1 N 2 G[wj ] ⊆ {wn−1, vr} for some r ∈ {1, . . . , n}, vr ̸= wn−1. Consider the following two cases: Case 1: N2 G[wn−1] \ ⋃n−2 j=1 N 2 G[wj ] = {wn−1, vr} for some vr ∈ V (G) \ {w1, . . . , wn−1}. Let wn−1 = vq. Then vq and vr are not hop dominated by each wj , where j ∈ {1, 2, . . . , n − 2}. If q < r, then vq is v1 or v2. If vq = v1, then vr = v3 and vn = v4. This is not possible because N2 G[v2] = N2 G[v4] = {v2, v4} where v2, v4 ∈ Ŝ. If vq = v2, then vr = v4 and n = 4 or 5. Again, this is not possible. A similar situation happens when q > r. Case 2: N2 G[wn−1] \ ⋃n−2 j=1 N 2 G[wj ] = {wn−1}. Then wn−1 is not hop dominated by any of the vertices w1, · · · , wn−2. Moreover, since n ≥ 4, dG(wn−1, vt) = 2 for some vertex vt ∈ V (G) \ Ŝ. Now, because N2 G[wn−1] \⋃n−2 j=1 N 2 G[wj ] = {wn−1}, vt must be hop-dominated by some vertex wj where 1 ≤ j ≤ n−2. This is not possible when n = 4, and so n ≥ 5. This would imply that wn−1 is either v1, v2, vn−1, or vn. It is routine to show that any of these vertices will contradict the assumption that S is a legal closed hop neighborhood sequence. Therefore, γhgr(Pn) = n− 2 when n ≥ 4. Lemma 1. Let G be a graph on n vertices. If |N2 G[v]| = 3 for every v ∈ V (G), then γhgr(G) ≤ n− 2. J. Hassan, S. Canoy / Eur. J. Pure Appl. Math, 15 (4) (2022), 1623-1636 1630 Proof. Suppose that |N2 G[v]| = 3 for every v ∈ V (G). Then G ̸= Kn. Thus, γhgr(G) ≤ n− 1. Suppose that γhgr(G) = n− 1, say, S = (v1, · · · , vn−1) is a Grundy hop dominating sequence of G. Then N2 G[vi] \ ⋃i−1 j=1N 2 G[vj ] ̸= ∅ for each i ∈ {2, . . . , n − 1}. Let p, q ∈ N2 G(vn−1). Since N 2 G[vn−1]\ ⋃n−2 j=1 N 2 G[vj ] ̸= ∅, p /∈ S or q /∈ S, say p /∈ S. Then q = vk ∈ Ŝ for some k ̸= n− 1 and vk, vn−1 ∈ N2 G[vk]. Now, since |N2 G[p]| = 3, there exists j ̸= n− 1 such that p ∈ N2(vj). It follows that vk, p, vn−1 ∈ N2 G[vk] ∪ N2 G[vj ], a contradiction. Therefore, γhgr(G) ≤ n− 2. The next result shows that the bound in Lemma 1 is tight. Proposition 2. For any positive integer n ≥ 3, γhgr(Cn) =  3 if n = 3 2 if n = 4 n− 4 if n ≥ 6 and even n− 2 if n ≥ 5 and odd. Proof. Let G = Cn = [v1, v2, · · · , vn, v1]. Clearly, γhgr(C3) = 3 and γhgr(C4) = 2. Let n ≥ 6. Let S0 = (v1, v2, · · · , vn−4). Then N2 G[v2] \ N2 G[v1] = {v2, v4, vn} ̸= ∅ and vi+2 ∈ N2 G[vi] \∪ i−1 j=1N 2 G[vj ] for all i ∈ {3, 4, · · · , n− 4}. It follows that S0 is a Grundy hop dominating sequence and γhgr(Cn) ≥ |Ŝ0| = n − 4. Suppose that n is even and suppose that S is a Grundy hop dominating sequence of Cn with γhgr(Cn) = |Ŝ|. Observe that if i is even and j is odd, then N2 G[vi]∩N2 G[vj ] = ∅. Hence, we may express S as concatenation S1⊕S2 where the subscripts of the terms of S1 and S2 are even and odd, respectively. Now, since N2 G[v1] = {v1, v3, vn−1}, N2 G[v2] = {v2, v4, vn}, N2 G[vn] = {v2, vn−2, vn}, N2 G[vn−1] = {v1, vn−3, vn−1}, and N2 G[vi] = {vi−2, vi, vi+2} for i ∈ {3, 4, · · · , n− 4}, it follows that each of S1 and S2 can only have at most n 2 − 2 terms. Thus, γhgr(Cn) = n− 4. Next, suppose that n ≥ 5 and is odd. Clearly, γhgr(Cn) = 3 if n = 5. For n ≥ 7, the sequence S = (v1, v3, · · · , vn, v2, v4, · · · , vn−5) = (v1, v3, · · · , vn)⊕(v2, v4, · · · , vn−5) can be verified to be a Grundy hop dominating sequence of Cn. This and Lemma 1 would imply that γhgr(G) = n− 2. Lemma 2. Let G be a graph. A sequence S is a co-legal closed neighborhood sequence in G if and only if S is a legal closed neighborhood sequence in G. Moreover, S is a co-Grundy dominating sequence in G if and only if it is a Grundy dominating sequence in G. In particular, γcogr(G) = γgr(G). Proof. Let S = (v1, · · · , vk) be a sequence in G. Since V (G) \ NG(vi) = NG[vi] for each i ∈ [k], it follows that [V (G) \NG(vi)] \ ∪i−1 j=1[V (G) \NG(vj)] = NG[vi] \ ∪ i−1 j=1NG[vj ]. Hence, S is a co-legal closed neighborhood sequence in G if and only if it is a legal closed neighborhood sequence in G. Clearly, a co-legal closed neighborhood sequence in G is a J. Hassan, S. Canoy / Eur. J. Pure Appl. Math, 15 (4) (2022), 1623-1636 1631 co-Grundy dominating sequence if and only if it is a Grundy dominating sequence in G. Hence, γcogr(G) = γgr(G). Theorem 5. Let G and H be any two graphs. A sequence S of distinct vertices of G+H is a legal closed hop neighborhood sequence if and only if one of the following holds: (i) S is a co-legal closed neighborhood sequence in G (legal closed neighborhood sequence in G). (ii) S is a co-legal closed neighborhood sequence in H (legal closed neighborhood sequence in H). (iii) S is a concatenation SG ⊕ SH , where SG and SH are co-legal closed neighborhood sequences in G and H, respectively. Proof. Suppose that S = (w1, · · · , wk) is a legal closed hop neighborhood sequence in G+H and let Ŝ = {w1, . . . , wk}. Suppose Ŝ ⊆ V (G). By the legality condition in S, we have N2 G+H [wi] \ ∪i−1 j=1N 2 G+H [wj ] ̸= ∅ for all i ∈ {2, 3, . . . , k}. Since N2 G+H [wi] = V (G) \NG(wi) for each i ∈ [k], it follows that [V (G) \NG(wi)] \ ∪i−1 j=1[V (G) \NG(wj)] ̸= ∅ for all i ∈ {2, 3, . . . , k}. Therefore, S is a co-legal closed neighborhood sequence in G, showing that (i) holds. Similarly, (ii) holds if Ŝ ⊆ V (H). Next, suppose that ŜG = Ŝ ∩ V (G) ̸= ∅ and ŜH = Ŝ ∩ V (H) ̸= ∅. Since N2 G+H [wj ] ⊆ V (G) for all wj ∈ ŜG and N2 G+H [ws] ⊆ V (H) for all ws ∈ ŜH , we may assume that ŜG = {w1, w2, . . . , wm} and ŜH = {wm+1, wm+1, . . . , wk}. Then S = SG ⊕ SH . Since S is a legal closed hop neighborhood sequence, [V (G) \NG(wi)] \ ∪i−1 j=1[V (G) \NG(wj)] = N2 G+H [wi] \ ∪i−1 j=1N 2 G+H [wj ] ̸= ∅ for all i ∈ {2, 3, . . . ,m}, showing that SG is a co-legal closed neighborhood sequence in G. Similarly, SH is a co-legal closed neighborhood sequence in H. This shows that (iii) holds. The converse is clear. Corollary 3. Let G and H be any two graphs. A sequence S of distinct vertices of G+H is a Grundy hop dominating sequence in G+H if and only if S = SG⊕SH , where SH and SH are co-Grundy dominating sequences in G and H, respectively (Grundy dominating sequences in G and H, respectively). Moreover, γhgr(G+H) = γcogr(G) + γcogr(H) = γgr(G) + γgr(H). In particular, each of the following holds. J. Hassan, S. Canoy / Eur. J. Pure Appl. Math, 15 (4) (2022), 1623-1636 1632 (i) γhgr(K1 +G) = 1 + γcogr(G) = 1 + γgr(G). (ii) γhgr(Km,n) = 2 for m,n ≥ 1. (iii) γhgr(Wn) = 1 + γcogr(Cn) = 1 + γgr(Cn) for all n ≥ 3. (iv) γhgr(Fn) = 1 + γcogr(Pn) = 1 + γgr(Pn) for all n ≥ 1. Theorem 6. Let G be a non-trivial connected graph on m vertices and let H be any graph. Then γhgr(G ◦H) ≥ m · γcogr(H) = m · γgr(H). Proof. Let V (G) = {v1, v2, . . . , vm} and let Svi = (w1 vi , w 2 vi , · · · , w k vi) be a co-Grundy dominating sequence in Hvi for each i ∈ [m] = {1, 2, . . . ,m}, where k = γcogr(H). Let S = Sv1 ⊕ Sv2 ⊕ · · · ⊕ Svm . Let x ∈ V (G ◦ H) \ Ŝ and let vt ∈ V (G) such that x ∈ V (vt +Hvt). Suppose first that x = vt. Let vs ∈ NG(vt) and pick any wj vs ∈ Ŝvs . Then wj vs ∈ Ŝ ∩ N2 G◦H(vt). Suppose x ̸= vt. Then x ∈ V (Hvt) \ Ŝvt . Since Ŝvt is a co-Grundy dominating set inHvt , it follows that there exists wl vt ∈ Ŝvt ⊆ Ŝ such that dHvt (x,wl vt) ̸= 1. It follows that dG◦H(x,wl vt) = 2. Thus, Ŝ is a hop dominating set in G ◦H. We relabel the terms in S, say S = (x1, x2, · · · , xk, · · · , xmk). Next, let i ∈ [mk] \ {1} and let xi = wt vr , where r ∈ [m] and t ∈ [k]. Then N2 G◦H [xi] \ ∪i−1 j=1N 2 G◦H [xj ] = N2 G◦H [wt vr ] \ [(∪ t−1 s=1N 2 G◦H [ws vr ]) ∪ (∪{N2 G◦H [wp vq ] : p ∈ [k] and 1 ≤ q ≤ r − 1})]. If t = 1, then N2 G◦H [wt vr ] \ (∪ t−1 s=1N 2 G◦H [ws vr ]) = N2 G◦H [wt vr ]. Clearly, wt vr ∈ N2 G◦H [wt vr ] \ [∪{N 2 G◦H [wp vq ] : p ∈ [k] and 1 ≤ q ≤ r − 1}]. Suppose t ̸= 1. Then N2 G◦H [wt vr ] \ (∪ t−1 s=1N 2 G◦H [ws vr ]) = [V (Hvr) \NHvr (wt vr)] \ [∪t−1 s=1(V (Hvr) \NHvr (ws vr))] ̸= ∅ because Svr is a co-legal neighborhood sequence in Hvr . Since N2 G◦H [wt vr ] \ (∪ t−1 s=1N 2 G◦H [ws vr ]) ∩ [∪{N2 G◦H [wp vq ] : p ∈ [k] and 1 ≤ q ≤ r − 1}] = ∅, N2 G◦H [xi] \ ∪i−1 j=1N 2 G◦H [xj ] ̸= ∅. Therefore, S is a Grundy hop dominating sequence in G ◦H. Accordingly, γhgr(G ◦H) ≥ |Ŝ| = m∑ i=1 |Ŝvi | = m · γcogr(H) = m · γgr(H). This proves the assertion. J. Hassan, S. Canoy / Eur. J. Pure Appl. Math, 15 (4) (2022), 1623-1636 1633 Remark 3. The bound given in Theorem 6 is tight. Moreover, strict inequality can also be attained. To see this, consider G = K3 and H = P3. Then γcogr(H) = 2 and γhgr(G ◦ H) = 3γcogr(H) = 6. For the strict inequality, consider the graphs G and G ◦ K1 in Figure 3. Then γcogr(K1) = 1 and S = (7, 5, 1, 4, 3) is a Grundy hop dominating sequence in G ◦ K1 with γhgr(G ◦K1) = |S| = 5 > 4 = 4γcogr(K1). G : G ◦K1 : 1 2 34 5 6 7 8 Figure 3: The corona G ◦K1 with γh gr(G ◦K1) = 5 Theorem 7. Let G and H be non-trivial connected graphs. Let SG = (v1, v2, · · · , vk) be a legal closed hop independent neighborhood sequence of G and let SH = (a1, a2, · · · , at) be a co-legal neighborhood sequence of H. Then S = ((v1, a1), (v1, a2), · · · , (v1, at), · · · , (vk, a1), (vk, a2), · · · , (vk, at)) is a legal closed hop neighborhood sequence of G[H]. Proof. Suppose SG is a legal closed hop independent neighborhood sequence of G and let SH be a co-legal neighborhood sequence of H. Let i ∈ [k] and j ∈ [t]. Then N2 G[H][(vi, aj)]\∪ j−1 l=1N 2 G[H][(vi, al)] = ({vi}×V (H)\NH(aj))\({vi}×∪j−1 l=1 [V (H)\NH(al)]). Now, since SH is a co-legal sequence in H, V (H) \NH(aj)) \ ∪j−1 l=1 [V (H) \NH(al)] ̸= ∅. Equality implies that N2 G[H][(vi, aj)] \ ∪ j−1 l=1N 2 G[H][(vi, al)] ̸= ∅. The assumption that SG is a legal closed hop independent neighborhood sequence would imply that ({vi}× V (H) \NH(aj)) \ ({vi}×∪j−1 l=1 [V (H) \NH(al)]) and ∪{N2 G[H][(vr, al)] : J. Hassan, S. Canoy / Eur. J. Pure Appl. Math, 15 (4) (2022), 1623-1636 1634 r ∈ [i− 1] and l ∈ [t]} are disjoint. Thus, N2 G[H][(vi, aj)] \ ∪{N 2 G[H][(vr, al)] : r ∈ [i], l ∈ [t] with (vi, aj) ̸= (vr, al)} ≠ ∅. Therefore, S is a legal closed hop neighborhood sequence in G[H]. Theorem 8. Let G and H be non-trivial connected graphs. If SG = (v1, v2, · · · , vk) is a Grundy hop independent hop dominating sequence in G and SH = (a1, a2, · · · , at) is a co-Grundy dominating sequence in H. Then S = ((v1, a1), (v1, a2), · · · , (v1, at), · · · , (vk, a1), (vk, a2), · · · , (vk, at)) is a Grundy hop dominating sequence of G[H]. In particular, γhgr(G[H]) ≥ γhihgr (G)γcogr(H). Proof. By Theorem 7, S is a legal closed hop neighborhood sequence in G[H]. Let (v, a) ∈ V (G[H])\Ŝ. Suppose that v ∈ V (G)\SG. Since SG is a hop dominating set, there exists vj ∈ SG such that v ∈ N2 G(vj). It follows that (vj , a1) ∈ Ŝ and (v, a) ∈ N2 G[H](vj , a1). Suppose v = vi for some i ∈ [k]. Then a ∈ V (H) \ SH . Since SH is a co-Grundy dominating sequence, there exists as ∈ SH such that a /∈ NH(as). Clearly, (vi, as) ∈ Ŝ and (v, a) ∈ N2 G[H](vi, as). Thus, Ŝ is a hop dominating set, showing that S is a Grundy hop dominating sequence in G[H]. Therefore, γhgr(G[H]) ≥ γhihgr (G)γcogr(H). Remark 4. The bound given in Theorem 8 is tight. To see this, consider G = H = P3. Then γhihgr (G) = 2 and γcogr(H) = 2. It can easily be verified that γhgr(G[H]) = γhihgr (G)γcogr(H) = 4. 4. Conclusion This study did introduce the concept of Grundy hop domination and make an initial investigation of the concept. It was pointed out and proved that every graph admits a Grundy hop dominating sequence. Extremal values of the Grundy hop domination number were given. Moreover, exact value or tight lower bound for each of the Grundy hop domination numbers of the join, corona, and lexicographic product of two graphs was determined. Bounds for this newly defined parameter in terms of other parameters (e.g. minimum degree, maximum degree, diameter, etc.) may be obtained. The parameter can be investigated further for trees and graphs under other binary operations. Acknowledgements The authors would like to thank the referees for the invaluable assistance they gave us through their comments and suggestions which led to the improvement of the paper. Moreover, the authors are extremely grateful to the Department of Science and Technology - Accelerated Science and Technology Human Resource Development Program (DOST- ASTHRDP)-Philippines and MSU-Iligan Institute of Technology for funding this research. REFERENCES 1635 References [1] S. Ayyaswamy, B. Krishnakumari, B. Natarjan, and Y. Venkatakrishnan. Bounds on the hop domination number of a tree. Proceedings-Mathematical Sciences., 125(4):449–455, 2015. [2] S. Ayyaswamy, C. Natarajan, and G. Sathiamoorphy. A note on hop domination number of some special families of graphs. International Journal of Pure and Applied Mathematics., 119(12):11465–14171, 2018. [3] B. Bresar, Cs. Bujtas, T. Gologranc, S. Klavzar, G. Kosmrlj, B. Patkos, Zs. Tuza, and M. Vizer. Dominating sequence in grid-like and toroidal graphs. Electron. J. Combin., (23):1–17, 2016. [4] B. Bresar, Cs. Bujtas, T. Gologranc, S. Klavzar, G. Kosmrlj, B. Patkos, Zs. Tuza, and M. Vizer. Grundy dominating sequence and zero forcing sets. Discrete Optim., (26):66–77, 2017. [5] B. Bresar, T. Gologranc, and T. Kos. Dominating sequences under atomic changes with applications in sierpinski and interval graphs. Appl. Anal. Discrete Math., (10):518–531, 2016. [6] B. Bresar, T. Gologranc, M. Milanic, D. Rall, and R. Rizzi. Dominating sequence in graphs. Discrete Math., (336):22–36, 2014. [7] B. Bresar, T. Kos, and P. Torres. Grundy domination and zero forcing in kneser graphs. Ars Math. Contemp., (17):419–430, 2019. [8] J. Hassan, S. Canoy Jr., and A. Aradais. Hop independent sets in graphs. Eur. J. Pure Appl. Math., 15(2):467–477, 2022. [9] M. Henning and N. Rad. On 2-step and hop dominating sets in graphs. Graphs and Combinatorics., 33(4):913–927, 2017. [10] S. Canoy Jr. 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. [11] S. Canoy Jr. and G. Malacas. Differentiating-dominating sets in graphs under binary operations. Tamkang J. Math., 46(1):51–60, 2015. [12] S. Canoy Jr., R. Mollejon, and J. G. Canoy. Hop dominating sets in graphs under binary operations. Eur. J. Pure Appl. Math., 12(4):1455–1463, 2019. [13] S. Canoy Jr. and G. Salasalan. Locating-hop domination in graphs. Kyungpook Mathematical Journal., 62:193–204, 2022. [14] G. Nasini and P. Torres. Grundy dominating sequences on x-join product. Discrete Applied Mathematics., (284):138–149, 2020. REFERENCES 1636 [15] C. Natarajan and S. Ayyaswamy. Hop domination in graphs ii. Versita, 23(2):187– 199, 2015. [16] G. Salasalan and S. Canoy Jr. Global hop domination numbers of graphs. Eur. J. Pure Appl. Math., 14(1):112–125, 2021.