EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 16, No. 2, 2023, 1154-1166 ISSN 1307-5543 – ejpam.com Published by New York Business Global Grundy Dominating and Grundy Hop Dominating Sequences in Graphs: Relationships and Some Structural Properties Javier A. Hassan1,∗, Sergio R. Canoy, Jr.2,3 1Mathematics and Sciences Department, College of Arts and Sciences, MSU Tawi-Tawi College of Technology and Oceanography, Bongao, Tawi-Tawi, Philippines 2Department of Mathematics and Statistics, College of Science and Mathematics 3Center for Mathematical and Theoretical Physical Sciences, Premier Research Institute of Science and Mathematics, MSU-Iligan Institute of Technology, 9200 Iligan City, Philippines Abstract. In this paper, we revisit the concepts of Grundy domination and Grundy hop dom- ination in graphs and give some realization results involving these parameters. We show that the Grundy domination number and Grundy hop domination number of a graph G are generally incomparable (one is not always less than or equal the other). It is shown that their absolute dif- ference can be made arbitrarily large. Moreover, the Grundy domination numbers of some graphs are determined. 2020 Mathematics Subject Classifications: 05C69 Key Words and Phrases: Grundy dominating sequence, Grundy hop dominating sequence, Grundy domination number, Grundy hop domination number 1. Introduction The concept of Grundy domination in a graph was introduced and initially studied by Bresar et al. [6]. This concept was also considered in other previous studies (see [3], [4], [5], [7]). In [5], exact formulas for Grundy domination numbers of Sierpinski graphs were generated and a linear algorithm for determining these numbers in arbitrary interval graphs was given. Grundy domination number was studied for Kneser graphs in [7] and graph products in [3] and [14]. Recently, Hassan and Canoy [8] introduced and investigated the concept of Grundy hop domination in a graph. They showed that the parameter is at least equal to the hop domination number of a graph G. The authors also characterized the Grundy hop ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v16i2.4690 Email addresses: javierhassan@msutawi-tawi.edu.ph (J. Hassan), sergio.canoy@g.msuiit.edu.ph (S. Canoy) https://www.ejpam.com 1154 © 2023 EJPAM All rights reserved. J. Hassan, S. Canoy Jr. / Eur. J. Pure Appl. Math, 16 (2) (2023), 1154-1166 1155 dominating sequences in graphs under some binary operations. Previous studies on hop domination can be found in [1], [2], [9], [10], [11], [12], [13], [15], and [16]. This paper revisits the concepts of Grundy domination and Grundy hop domination in a graph and shows, as a particular case of a realization result, that the absolute difference of the Grundy domination and Grundy hop domination numbers can be made arbitrarily large. Moreover, the Grundy domination numbers of some graphs are given. 2. Terminology and Notation Let G be a simple undirected 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. Any dominating set with cardinality equal to γ(G) is called a γ-set. Let S = (v1, v2, · · · , vk) be a sequence of distinct vertices of a graph G and let Ŝ = {v1, v2, · · · , vk} be its corresponding set. 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, denoted by γgr(G), is called the Grundy domination number of G. We say that vertex vi footprints the vertices from NG[vi] \ ∪i j=1NG[vj ], and that vi is their footprinter. Any Grundy dominating sequence S with |Ŝ| = γgr(G) is called a maximum Grundy dominating sequence or a γgr-sequence of G. In this case, we call Ŝ a γgr-set of G. A legal closed neighborhood sequence S = (v1, v2, · · · , vk) of G is a maximum legal closed neighborhood sequence if for any legal closed neighborhood sequence (w1, w2, · · · , wt) of G, we have t ≤ k. A legal closed neighborhood sequence S is non-dominating if Ŝ is a non-dominating set of 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} be the set induced by S. 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. Any Grundy hop dominating J. Hassan, S. Canoy Jr. / Eur. J. Pure Appl. Math, 16 (2) (2023), 1154-1166 1156 sequence S with |Ŝ| = γhgr(G) is called a maximum Grundy hop dominating sequence or a γhgr-sequence of G. In this case, we call Ŝ a γhgr-set of G. A set S ⊆ V (G) is an independent set of G if for any two distinct vertices v and w of S, dG(v, w) ̸= 1. The maximum cardinality of an independent set of G, denoted by α(G), is called the independence number of G. Any independent set with cardinality α(G) is referred to as a maximum independent set or α-set of G. 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). 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)}. 3. Main Results Theorem 1. Let a and b be positive integers such that 3 ≤ a ≤ b. Then (i) there exists a connected graph G such that γgr(G) = a and γhgr(G) = b, and (ii) there exists a connected graph G′ such that γhgr(G ′) = a and γgr(G ′) = b. Proof. For a = b, consider the graphG given in Figure 1. Let S = (u1, u2, . . . , ua−1, ua). Then S is both a γgr-sequence and a γhgr-sequence of G. Hence, γgr(G) = a = γhgr(G). u2 ua−1u1 G : . . . ua Figure 1: A graph G with γgr(G) = γh gr(G) Next, suppose a < b and let m = b− a. For (i), consider the graph G given in Figure 2, where the graph G[{xa, z1, z2, . . . , zm}] induced by {xa, z1, z2, . . . , zm} is complete. It can easily be verified that S1 = (x1, x2, · · · , xa) and S2 = (u, v, x1, . . . , xa−2, z1, · · · , zm) are γgr-sequence and γhgr-sequence of G, respectively. Therefore, γgr(G) = a and γhgr(G) = m+ a = b. J. Hassan, S. Canoy Jr. / Eur. J. Pure Appl. Math, 16 (2) (2023), 1154-1166 1157 u v x1 x2 G : x3 . . . xa−1 xa z1 z2 zm . . . Figure 2: A graph G with γgr(G) = a < γh gr(G) = b For (ii), consider the graph G′ given in Figure 3. Let S1 = (s1, s2, · · · , sa) and S2 = (s1, s2, . . . , sa, u1, u2, · · · , um). Then S1 and S2 are γhgr-sequence and γgr-sequence of G′, respectively. Thus, γhgr(G ′) = a and γgr(G ′) = m+ a = b. G′ : . . . s1 s2 um sa u1 u2. . . Figure 3: A graph G′ with γh gr(G ′) = a < γgr(G′) = b This proves the assertion. Corollary 1. Let n be a positive integer. Then each of the following statements holds. (i) There exists a connected graph G such that γhgr(G)− γgr(G) = n. (ii) There exists a connected graph H such that γgr(H)− γhgr(H) = n. In other words, the absolute difference |γhgr(G)− γgr(G)| can be made arbitrarily large. Proof. Let n be a positive integer and let a = n + 2 and b = 2n + 2. By Theorem 1(i), there exists a connected graph G with γgr(G) = a and γhgr(G) = b. Hence, γhgr(G)− γgr(G) = n. By Theorem 1(ii), there exists a connected graph H such that γhgr(H) = a and γgr(H) = b. Hence, γgr(H)− γhgr(H) = n. J. Hassan, S. Canoy Jr. / Eur. J. Pure Appl. Math, 16 (2) (2023), 1154-1166 1158 The following result shows that difference of the Grundy domination number (Grundy hop domination number) and domination number (resp. hop domination number) of a graph G can be made arbitrarily large. Proposition 1. Let n be a positive integer. Then each of the following statements holds. (i) There exists a connected graph G such that γgr(G)− γ(G) = n. (ii) There exists a connected graph G′ such that γhgr(G ′)− γh(G ′) = n. Proof. (i) Consider the graph G in Figure 4. Let D = {u} and S = (v1, v2, . . . , vn+1). Then D is a γ-set and S is a γgr-sequence of G. Hence, γ(G) = 1 and γgr(G) = n + 1, that is, γgr(G)− γ(G) = n. G : v1 v2 vn+1 u . . . Figure 4: A graph G with γgr(G)− γ(G) = n (ii) Consider the graph G′ given in Figure 5. Let S1 = {u,w} and S2 = (x1, x2, . . . , xn+1, w). Then S1 is a γh-set and S2 is a γhgr-sequence of G′. Therefore, γh(G ′) = 2 and γhgr(G ′) = n+ 2, that is, γhgr(G ′)− γh(G ′) = n. x2 xn+1x1 G′ : . . . w u Figure 5: A graph G′ with γh gr(G ′)− γh(G ′) = n This proves the assertion. J. Hassan, S. Canoy Jr. / Eur. J. Pure Appl. Math, 16 (2) (2023), 1154-1166 1159 Proposition 2. If D = {v1, . . . , vk} is a γ-set of G, then (v1, . . . , vk) is a Grundy domi- nating sequence. In particular, γ(G) ≤ γgr(G). Proof. Let D = {v1, . . . , vk} be a γ-set of G. Suppose S = (v1, v2, . . . , vk) is not a legal closed neighborhood sequence of G. Then there exists i ∈ {2, 3, . . . , k} such that NG[vi] \ ∪i−1 j=1NG[vj ] = ∅. This implies that NG[vi] ⊆ ∪i−1 j=1NG[vj ]. Thus, Ŝ \ {vi} = D \ {vi} is a dominating set of G, contradicting the the minimality of D. Therefore, NG[vi]\∪i−1 j=1NG[vj ] ̸= ∅ for each i ∈ {2, 3, . . . , k}, showing that S is a Grundy dominating sequence of G. Proposition 3. Let G be a graph on n vertices. If S = (u1, v2, · · · , uk) is a Grundy dominating sequence of smallest length k, then γ(G) = |Ŝ| = k. Proof. Since Ŝ is a dominating set of G, it follows that γ(G) ≤ |Ŝ|. On the other hand, by Proposition 2 and the assumption that S a Grundy dominating sequence of smallest length k, |Ŝ| ≤ γ(G). Consequently, γ(G) = |Ŝ| = k. Theorem 2. Let G be a graph. Then S = (s1, s2, · · · , sk) is a maximum legal closed neighborhood sequence of G if and only if S is a Grundy dominating sequence of G with γgr(G) = k. Proof. Let S = (s1, s2, · · · , sk) be a maximum legal closed neighborhood sequence of G. Suppose Ŝ is not a dominating set of G. Then there exists v ∈ V (G)\NG[Ŝ]. This implies that v /∈ NG[u] for every u ∈ Ŝ. Let S∗ = (s1, s2, · · · , sk, v). Then NG[v]\∪k j=1NG[si] ̸= ∅. It follows that S∗ is a legal closed neighborhood sequence of G, a contradiction to the maximality of S. Thus, Ŝ is a dominating set of G. Therefore, by assumption, S is a Grundy dominating sequence of G and γgr(G) = k. The converse is clear. The next result follows from Theorem 2 Corollary 2. Let G be a graph and let S = (s1, s2, . . . , sm) be a legal closed neighborhood sequence of G. Then |Ŝ| = m ≤ γgr(G). Theorem 3. Let G be a graph on n vertices. Then γgr(G) = α(G) if and only if every α-set is induced by a maximum legal closed neighborhood sequence of G. Proof. Suppose γgr(G) = α(G). Let D = {v1, . . . , vk} be a maximum independent set of G and let S = (v1, . . . , vk). Then Ŝ = D is a dominating set of G. Moreover, since Ŝ is an independent set, vi ∈ NG[vi] \ ⋃i−1 j=1NG[vj ] for each i ∈ {2, . . . , k}. Hence, S is a legal closed neighborhood sequence of G. Since γgr(G) = α(G), S is a maximum legal closed neighborhood sequence of G by Theorem 2. Therefore, α-set is induced by a maximum legal closed neighborhood sequence of G. For the converse, suppose that every α-set is induced by a maximum legal closed neighborhood sequence of G. By Theorem 2, γgr(G) = α(G). J. Hassan, S. Canoy Jr. / Eur. J. Pure Appl. Math, 16 (2) (2023), 1154-1166 1160 Corollary 3. γgr(K1,n) = α(K1,n) = n for every positive integer n. Proposition 4. Let G be a graph with components G1, G2, . . . , Gk, k ≥ 2. Then γgr(G) = k∑ i=1 γgr(Gi). Proof. For each i ∈ {1, 2, . . . , k}, let Si be a γgr-sequence of Gi. Then S = S1 ⊕ S2 ⊕ · · · ⊕ Sk is a Grundy dominating sequence of G. Thus, γgr(G) ≥ |Ŝ| = k∑ i=1 |Ŝi| = k∑ i=1 γgr(Gi). Next, suppose that S′ = (w1, w2, . . . , wm) is a γgr-sequence of G. For each i ∈ {1, 2, . . . , k}, let S′ i = (wi,1, wi,2, . . . , wi,mi) be a subsequence of S′ such that Ŝ′ i = Ŝ′ ∩ V (Gi). Since S′ is a Grundy dominating sequence of G, S′ i is a Grundy dominating sequence of Gi for each i ∈ {1, 2, . . . , k}. Hence, S∗ = S′ 1 ⊕ S′ 2 ⊕ · · · ⊕ S′ k is a Grundy dominating sequence of G. Therefore, γgr(G) = |Ŝ′| = |Ŝ∗| = k∑ i=1 |Ŝ′ i| ≤ k∑ i=1 γgr(Gi). Consequently, γgr(G) = k∑ i=1 γgr(Gi). Lemma 1. Let G be a graph on n vertices and k be any positive integer. If |NG[a]| ≥ k for every a ∈ V (G), then γgr(G) ≤ n− k + 1. Proof. Suppose S = (s1, s2, . . . , st) is γgr-sequence of G. Then |NG[s1]| ≥ k by assumption. It follows that there are only at most n − k remaining vertices that can be footprinted by the remaining terms of S. Thus, γgr(G) = t ≤ n− k + 1. Theorem 4. Let G be any non-trivial graph on n ≥ 1 vertices. Then 1 ≤ γgr(G) ≤ n. Moreover, each of the following statements holds. (i) γgr(G) = 1 if and only if G is a complete graph. (ii) γgr(G) = 2 if and only if G is non-complete and {a, b} is a dominating set of G for each pair of distinct vertices a, b ∈ V (G) with NG[a] ̸= NG[b]. (iii) γgr(G) = n if and only if G = Kn. J. Hassan, S. Canoy Jr. / Eur. J. Pure Appl. Math, 16 (2) (2023), 1154-1166 1161 Proof. Clearly, 1 ≤ γgr(G) ≤ n. (i) Assume that γgr(G) = 1. Suppose G is not a complete graph. Then there exists a, b ∈ V (G) such that dG(a, b) = 2. This implies that b ∈ NG[b] \NG[a]. Hence, (a, b) is a legal closed neighborhood sequence of G. Therefore, γgr(G) ≥ 2, a contradiction. Conversely, suppose G is a complete graph. Then γgr(G) = 1. (ii) Suppose that γgr(G) = 2. Then G is non-complete by (i). Let a, b be two distinct vertices of G such that NG[a] ̸= NG[b]. Then NG[a]\NG[b] ̸= ∅ or NG[b]\NG[a] ̸= ∅. We may ssume that NG[b] \NG[a] ̸= ∅. This implies that (a, b) is a legal closed neighborhood sequence of G. Suppose there exists c ∈ V (G)\(NG[a]∪NG[b]). Since c ∈ NG[c], it follows that NG[c] \ (NG[a] ∪NG[b]) ̸= ∅. Hence, (a, b, c) is a legal closed neighborhood sequence of G. Thus, γgr(G) ≥ 3, a contradiction to the assumption that γgr(G) = 2. Therefore, {a, b} is a dominating set of G. Conversely, suppose that G is non-complete and {a, b} is a dominating set of G for each pair of distinct vertices a, b ∈ V (G) with NG[a] ̸= NG[b]. Then γgr(G) ≥ 2. Let (s1, s2, · · · , st) be a γgr-sequence of G. ThenNG[s2]\NG[s1] ̸= ∅, that is, NG[s2] ̸= NG[s1]. By assumption, {s1, s2} is a dominating set of G, that is, V (G) = N2 G[s1]∪N2 G[s2]. There- fore, γgr(G) = t = 2. (iii) Suppose γgr(G) = n. Suppose there exists component C of G which is non-trivial. Then |NC [v]| ≥ 2 for every v ∈ V (C). By Lemma 1, γgr(C) ≤ |V (C)| − 1. By Proposition 4, it follows that γgr(G) ≤ n− 1, a contradiction. Therefore, every component C of G is trivial, i.e., G = Kn. Conversely, suppose that every component C of G is trivial. Then by (i) and Propo- sition 4, γgr(G) = n. The next result follows from Theorem 4 and Proposition 4. Corollary 4. Let G be a graph on n vertices. Then each of the following statements holds. (i) γgr(G) ≥ 2 if and only if G is non-complete graph. (ii) If G has k components and every component is complete, then γgr(G) = k. (iii) If G is complete, then γgr(G) + γgr(G) = n+ 1. (iv) If G is non-complete, then (a) 3 ≤ γgr(G) + γgr(G) ≤ 2n− 1, and (b) 2 ≤ γgr(G) · γgr(G) ≤ n2 − n. In particular, equality in (a) and (b) holds if and only if G = K2. Proposition 5. For any positive integer n, each of the following holds. (i) γgr(Cn) = { 1 if n = 3 n− 2 if n ≥ 4. J. Hassan, S. Canoy Jr. / Eur. J. Pure Appl. Math, 16 (2) (2023), 1154-1166 1162 (ii) γgr(Pn) = { 1 if n = 1, 2 n− 1 if n ≥ 3. (iii) γgr(Pn) = { 2, n = 2, 3 3, n ≥ 4. (iv) γgr(Cn) = { 2, ifn = 4 3, if n = 3 or n ≥ 5. Proof. (i) Clearly, γgr(Cn) = 1 for n = 3. Suppose n ≥ 4. Let Cn = [v1, v2, . . . , vn, v1] and S = (v1, v2, · · · , vn−2). Clearly, Ŝ is a dominating set of Cn. Observe that vi+1 ∈ NG[vi]\ ⋃i−1 j=1NG[vj ] for each i ∈ {2, . . . , n−2}. Thus, S is a Grundy dominating sequence of Cn and γgr(Cn) ≥ n − 2. On the other hand, let S0 = (a1, a2, . . . , ak) be a Grundy dominating sequence of Cn. Since |NG[vi]| = 3 for each i ∈ {1, . . . , n}, it follows that γgr(Cn) = k ≤ n− 2 by Lemma 1. Consequently, γgr(G) = n− 2 for all n ≥ 4. (ii) Clearly, γgr(Pn) = 1 for n = 1, 2. Suppose n ≥ 3. Let Pn = [v1, v2, . . . , vn] and let D = (v1, v2, · · · , vn−1). Clearly, D̂ is a dominating set of Pn. Observe that vi+1 ∈ NG[vi] \ ⋃i−1 j=1NG[vj ] for each i ∈ {2, . . . , n− 2}. Thus, D is a Grundy dominating sequence of Pn showing that γgr(Pn) ≥ n − 1. Since |NPn [a]| ≥ 2 for every a ∈ V (Pn), γgr(Pn) ≤ n− 1 by Lemma 1. Therefore, γgr(Pn) = n− 1 for all n ≥ 3. (iii) Clearly, γgr(Pn) = 2 for n = 2, 3. Suppose n ≥ 4. Let {v1, v2, · · · , vn} be a ver- tex set of Pn and consider S = (v2, v4, v3). Then S is a Grundy dominating sequence of Pn. Hence, γgr(Pn) ≥ 3. On the other hand, let S′ = (w1, · · · , wk) be a Grundy dominating sequence of Pn. Notice that |NPn [vi]| ≥ n− 2 for every i ∈ {1, . . . , n}. Thus, γgr(Pn) = k ≤ 3 by Lemma 1. Consequently, γgr(Pn) = 3 for all n ≥ 4. (iv) Clearly, γgr(C3) = 3 and γgr(C4) = 2. Suppose n ≥ 5. Let {v1, v2, · · · , vn} be a vertex set of Cn and consider S′ = {v1, v3, v2}. Then S′ is a Grundy dominating sequence of Cn. Hence, γgr(Cn) ≥ 3. On the other hand, let S′′ = (u1, · · · , ul) be a Grundy dominating sequence of Cn. Observe that |NCn [vi]| = n − 2 for every i ∈ {1, · · · , n}. Thus, γgr(Cn) = l ≤ 3 by Lemma 1. Therefore, γgr(Cn) = 3 for all n ≥ 5. Theorem 5. Let G and H be two non-complete graphs. A sequence D of distinct vertices of G +H is a Grundy dominating sequence in G +H if and only if one of the following condition holds: (i) D is a Grundy dominating sequence of G. (ii) D is a Grundy dominating sequence of H. (iii) D = DG ⊕ (w) for some non-dominating legal closed neighborhood sequence DG of G and w ∈ V (H). J. Hassan, S. Canoy Jr. / Eur. J. Pure Appl. Math, 16 (2) (2023), 1154-1166 1163 (iv) D = DH ⊕ (v) for some non-dominating legal closed neighborhood sequence DH of H and v ∈ V (G). Proof. Let DG and DH be subsequences D such that D̂G = D̂ ∩ V (G) and D̂H = D̂ ∩ V (H). If D̂H = ∅, then D = DG is a Grundy dominating sequence of G. If D̂G = ∅, then D = DH is a Grundy dominating sequence of H. Hence, (i) or (ii) holds. Suppose now that D̂G and D̂H are both non-empty. Since D is a legal closed neighborhood sequence of G+H, DG and DH are legal closed neighborhood sequences of G and H, respectively. If |D̂G| = |D̂H | = 1, then both (iii) and (iv) hold. Next, suppose that |D̂G| ≥ 2. Suppose further that D̂G is dominating. Let D = (x1, x2, . . . , xm). Since D̂G is dominating, none of the terms in DH comes after (succeeds) all the terms of DG in D by the legality property of D. Hence, if xj ∈ D̂H and xk ∈ D̂G, then k > j. However, if xj ∈ D̂H for some j, then V (G) ⊆ NG+H [xj ]. This would imply that |D̂G| = 1, a contradiction. Thus, D̂H = ∅, a contradiction. Thus, DG is a non-dominating legal closed neighborhood sequence of G. Let DG = (v1, v2, . . . , vr) and DH = (w1, w2, . . . , wt). If in D the term w1 does not precede vr, where r ≥ 2, then vr does not satisfy the property in the legality condition, a contradiction. Hence, in D, w1 comes after the term vr in DG. Since V (H) ⊆ ⋃r j=1NG+H [vj ] and V (G) \ ( ⋃r j=1NG[vj ]) ⊆ NG+H [w1], DH cannot have other terms, i.e., t = 1. Thus, DH = (w), where w = w1, and D = DG ⊕ (w). This shows that (iii) holds. Similarly, (iv) holds if |D̂H | ≥ 2. For the converse, suppose first that (i) or (ii) holds. Then clearly, D is a Grundy dominating sequence of G +H. Next, suppose that (iii) holds. Then D̂ is a dominating set of G+H. Let DG = (v1, v2, . . . , vr). Since DG is a legal closed neighborhood sequence of G, it is legal closed neighborhood sequence of G + H. Moreover, since D̂G is non- dominating in G, NG+H [w] \ ( ⋃r j=1NG+H [vj ]) ̸= ∅. Thus, D = DG ⊕ (w) is a legal closed neighborhood sequence of G+H. Similarly, D is a Grundy dominating sequence of G+H if (iv) holds. The next result follows from Proposition 5 and Theorem 5. Corollary 5. Let G and H be two non-complete graphs. Then γgr(G+H) = max{γgr(G), γgr(H)} In particular, each of the following holds: (i) γgr(Pn + Pm) = { n− 1 if n ≥ m ≥ 3 m− 1 if m ≥ n ≥ 3. (ii) γgr(Pn + Cm) = { n− 1 if n ≥ m = 4 m− 2 if m ≥ n+ 1 = 4. (iii) γgr(Cn + Cm) = { n− 2 if n ≥ m ≥ 4 m− 2 if m ≥ n ≥ 4. J. Hassan, S. Canoy Jr. / Eur. J. Pure Appl. Math, 16 (2) (2023), 1154-1166 1164 (iv) γgr(Km,n) = max {m,n} for all m,n ≥ 2. Then next result can be proved easily. Theorem 6. Let G be a complete graph and let H be a non-complete graph. A sequence D of distinct vertices of G+H is a Grundy dominating sequence in G+H if and only if one of the following condition holds: (i) D = (v) for some v ∈ V (G). (ii) D is a Grundy dominating sequence of H. (iii) D = DH ⊕ (v) for some non-dominating legal closed neighborhood sequence DH of H and v ∈ V (G). Since γgr(H) ≥ 2 for any non-complete graph H, the following result follows from Theorem 6. Corollary 6. Let G be a complete graph and let H be any non-complete graph. Then γgr(G+H) = γgr(H). In particular, each of the following holds: (i) γgr(K1,n) = n for all n ≥ 1. (ii) γgr(Wn) = γgr(K1 + Cn) = γgr(Cn) = n− 2 for all n ≥ 4. (iii) γgr(Fn) = γgr(K1 + Pn) = γgr(Pn) = n− 1 for all n ≥ 3. 4. Conclusion The study revisited the concepts of Grundy domination and Grundy hop domination in graphs which have been considered previously by various authors. In general, the Grundy domination and Grundy hop domination numbers do not satisfy a consistent relationship as any one of them can be larger than the other. In fact, it was shown that the absolute difference of these two parameters can be made arbitrarily large. The Grundy domination numbers of some graphs were determined. For the join of two graphs, the Grundy domi- nation number was obtained by first characterizing all the Grundy dominating sequences in the graph. These two parameters can still be studied for other graphs; in particular, for graphs under some binary operations not yet considered in previous studies. Moreover, it still remains a conjecture that the Grundy hop 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, MSU-Iligan Institute of Technology and MSU Tawi-Tawi College of Technol- ogy and Oceanography for funding this research. REFERENCES 1165 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 and S. Canoy Jr. Grundy hop domination in graphs. Eur. J. Pure Appl. Math., 15(4):1623–1636, 2022. [9] J. Hassan and S. Canoy Jr. Hop independent hop domination in graphs. Eur. J. Pure Appl. Math., 15(4):1783–1796, 2022. [10] J. Hassan, S. Canoy Jr., and A. Aradais. Hop independent sets in graphs. Eur. J. Pure Appl. Math., 15(2):467–477, 2022. [11] M. Henning and N. Rad. On 2-step and hop dominating sets in graphs. Graphs and Combinatorics., 33(4):913–927, 2017. [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. Revisiting domination, hop domination, and global hop domination in graphs. Eur. J. Pure Appl. Math., 14:1415–1428, 2021. [14] G. Nasini and P. Torres. Grundy dominating sequences on x-join product. Discrete Applied Mathematics., (284):138–149, 2020. REFERENCES 1166 [15] C. Natarajan and S. Ayyaswamy. Hop domination in graphs ii. Versita, 23(2):187– 199, 2015. [16] Y. Pabilona and H. Rara. Connected hop domination in graphs under some binary operations. Asian-Eur. J. Math., 11(5):1850075–1–1850075–11, 2018.