EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 16, No. 2, 2023, 1212-1227 ISSN 1307-5543 – ejpam.com Published by New York Business Global Connected Grundy Hop Dominating Sequences in Graphs 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 introduce and do an initial investigation of a variant of Grundy hop domination in a graph called the connected Grundy hop domination. We show that the connected Grundy hop domination number lies between the connected hop domination and Grundy hop domination number of a graph. In particular, we give realization results involving connected hop domination, connected Grundy hop domination, and Grundy hop domination numbers. Moreover, we determine the connected Grundy hop domination numbers of some graphs. 2020 Mathematics Subject Classifications: 05C69 Key Words and Phrases: Connected hop domination, closed hop neighborhood sequence, con- nected Grundy hop dominating sequence, connected Grundy hop domination number 1. Introduction In 2014, Bresar et al. [6] introduced Grundy domination in a graph and made an initial study of the concept. Subsequent studies on this newly defined parameter can be found in [3], [4], [5], [7], and [12]. The study in [5] specifically gave exact formulas for the Grundy domination numbers of Sierpinski graphs where the authors provided a linear algorithm for determining these numbers in arbitrary interval graphs. It is without a doubt that the concept of hop domination, just like domination, has ably attracted a lot of researchers to study it. Some of these researchers have actually introduced some variants of the concept by imposing additional properties on the standard definition (see [1], [2], [9], [10], [11], [13]). ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v16i2.4687 Email addresses: javierhassan@msutawi-tawi.edu.ph (J. Hassan), sergio.canoy@g.msuiit.edu.ph (S. Canoy) https://www.ejpam.com 1212 © 2023 EJPAM All rights reserved. J. Hassan, S. Canoy Jr. / Eur. J. Pure Appl. Math, 16 (2) (2023), 1212-1227 1213 Following the definition of Grundy domination (as a variant of the standard domination concept), Hassan et al. in [8] introduced and studied Grundy hop domination as a variation of hop domination. It was shown that difference of the Grundy hop domination number and the hop domination number can be made arbitrarily large. Values of the Grundy hop domination numbers had also been determined for some graphs under some binary operations. In this study, the concept of connected Grundy hop domination in a graph will be introduced. Realization results involving connected hop domination, Grundy hop domi- nation, and connected Grundy hop domination numbers are given. Moreover, graphs that attain some specific values for the parameter are characterized. 2. Terminology and Notation Let G be a simple undirected graph. Two vertices a and b of G are adjacent, or neighbors, if ab is an edge 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. A set D ⊆ V (G) is a dominating 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). 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 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 D ⊆ V (G) is a hop dominating set of G if N2 G[D] = V (G), that is, for every v ∈ V (G)\D, there exists u ∈ D 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. A hop dominating set D is called a connected hop dominating set if ⟨D⟩ is connected. The minimum cardinality among all connected hop dominating sets of G, denoted by J. Hassan, S. Canoy Jr. / Eur. J. Pure Appl. Math, 16 (2) (2023), 1212-1227 1214 γch(G), is called the connected hop domination number of G. Any connected hop domi- nating set with cardinality equal to γch(G) is called a γch-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. Any Grundy hop dominating 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 Grundy hop dominating sequence S is called a connected Grundy hop dominating sequence if ⟨Ŝ⟩ is connected. The maximum length of a connected Grundy hop dominating sequence in a graph G, denoted by γchgr (G), is called the connected 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 connected Grundy hop dominating sequence S with |Ŝ| = γchgr (G) is called a maximum connected Grundy hop dominating sequence or a γchgr -sequence of G. In this case, we call Ŝ a γchgr -set of G. 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}, i.e., S is legal closed neighborhood sequence in G. A co-legal closed neighborhood sequence S = (v1, v2, . . . , vk) is a co-Grundy dominating sequence if V (G) = ∪k i=1[V (G) \NG(vi)], i.e., S is Grundy dominating sequence in G. 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). Clearly, γcogr(G) = γgr(G). The shadow graph S(G) of a graph G is constructed by taking two copies of G, say G1 and G2, and then joining each vertex u ∈ G1 to the neighbors of its corresponding vertex u′ ∈ G2. Let G and H be 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. Results Remark 1. Let G be a connected graph. Then each of the following is true. (i) The vertex set V (G) may not form a connected Grundy hop dominating sequence. (ii) A proper connected hop dominating set may not form a connected Grundy hop dom- inating sequence. (iii) A Grundy hop dominating sequence need not be a connected Grundy hop dominating sequence. J. Hassan, S. Canoy Jr. / Eur. J. Pure Appl. Math, 16 (2) (2023), 1212-1227 1215 To see (i), consider the graph G in Figure 1. Let S = (v1, v2, v3, v4, v5, v6). Then Ŝ is a connected hop dominating set of G. Observe that N2 G[v6] = {v3, v6} ⊆ NG[v3]. It follows that N2 G[v6] \ ⋃5 j=1N 2 G[vj ] = ∅. Hence, S is not a legal closed hop neighborhood sequence of G. Consequently, S is not connected Grundy hop dominating sequence of G. In fact (as to be shown later), γchgr (G) ̸= 6. G : v1 v2 v3 v4 v5 v6 Figure 1: A graph G where vertex set does not form a connected Grundy hop dominating sequence For (ii), consider the graph G = C5 = [x, z, y, w, v, x]. Clearly, Ŝ = {x, y, z, w} is a connected hop dominating set of G. Observe that N2 G[w] ⊆ N2 G[x] ∪ N2 G[y] ∪ N2 G[z]. It follows that N2 G[w] \ [N2 G[x] ∪ N2 G[y] ∪ N2 G[z]] = ∅. Thus, S is not a legal closed hop neighborhood sequence of G. Therefore, S is not a connected Grundy hop dominating sequence of G. Finally, for (iii), consider G = P5 = [v1, v2, v3, v4, v5] and let S′ = (v1, v3, v4). Clearly, Ŝ′ is a hop dominating set of G. Observe that v5 ∈ N2 G[v3] \N2 G[v1] and v2, v4 ∈ NG[v4] \ (N2 G[v1] ∪ N2 G[v3]). It follows that S′ is a legal closed hop neighborhood sequence of G. Thus, S′ is a Grundy hop dominating sequence of G. However, ⟨Ŝ⟩ is not connected. Hence, S′ is not a connected Grundy hop dominating sequence of G. Remark 2. Let G be a connected graph. Then γch(G) ≤ γchgr (G) ≤ γhgr(G) and these bounds are tight. Moreover, both strict inequalities are attainable. Note that the first inequality follows from the fact that every connected Grundy hop dominating sequence induces a connected hop dominating set (by definition). Moreover, since every connected Grundy hop dominating sequence is a Grundy hop dominating sequence, the second inequality follows. To see that the bounds are tight, consider G = K4. Then γch(G) = γchgr (G) = γhgr(G) = 4. For strict inequalities, consider first the graph G in Figure 2. Let S1 = {s3, s4, s5, s6} and S2 = (s1, s2, s3, s4, s5, s6). Then it can be verified that S1 and S2 are γch- and γchgr - sequences of G, respectively. Hence, γch(G) = 4 < 6 = γchgr (G) = 6. J. Hassan, S. Canoy Jr. / Eur. J. Pure Appl. Math, 16 (2) (2023), 1212-1227 1216 s2 s5 s1 G : s4s3 s6 Figure 2: A graph G with γch(G) < γch gr (G) Lastly, consider the graphG′ in Figure 3. Let S′ = (u3, u4, u5) and S′′ = (u1, u2, u6, u7). Then S′ and S′′ are γchgr -sequence and γhgr-sequence of G′, respectively. Hence, γchgr (G ′) = 3 and γhgr(G ′) = 4, that is, γchgr (G ′) = 3 < 4 = γhgr(G ′). u1 u2 u6 u7 u3 u5 u4 G′ : Figure 3: A graph G′ with γch gr (G ′) < γh gr(G ′) Proposition 1. Let G be a connected graph. Then S = (s1, s2, · · · , sk) is a legal closed hop neighborhood sequence of G with maximum length and ⟨Ŝ⟩ connected if and only if S is a connected Grundy hop dominating sequence of G with γchgr (G) = k. Proof. Let S = (s1, s2, · · · , sk) be a legal closed hop neighborhood sequence of G with maximum length k and ⟨Ŝ⟩ connected. Suppose Ŝ is not a connected 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. Pick u0 = st ∈ Ŝ such that dG(v, u0) ≤ dG(v, sj) for all j ∈ {1, 2, . . . , k}. Let [q1, q2, . . . , qm], where q1 = u0 and qm = v, be a u0-v geodesic. Thenm ≥ 4 and q4 /∈ N2 G[u] for every u ∈ Ŝ. Let S∗ = (s1, s2, · · · , sk, q2). Then ⟨Ŝ∗⟩ is connected and q4 ∈ N2 G[q2] \ ∪k j=1N 2 G[si] ̸= ∅. It follows that S∗ is a legal closed hop neighborhood sequence of G, a contradiction to the maximality of S. Thus, Ŝ is a connected dominating set of G. Therefore, by assumption, S is a connected Grundy hop dominating sequence of G and γchgr (G) = k. The converse is clear. The next result follows from Proposition 1 J. Hassan, S. Canoy Jr. / Eur. J. Pure Appl. Math, 16 (2) (2023), 1212-1227 1217 Corollary 1. Let G be a connected graph and let S = (s1, s2, . . . , sm) be a legal closed hop neighborhood sequence of G such that ⟨Ŝ⟩ is connected. Then |Ŝ| = m ≤ γchgr (G). Theorem 1. [8] Let G be any graph on n (n ≥ 2) vertices. Then γhgr(G) = n if and only if every component C of G is complete. Theorem 2. Let G be a connected graph on n vertices. Then 1 ≤ γchgr (G) ≤ n. Moreover, each of the following statements holds. (i) γchgr (G) = 1 if and only if G is trivial. (ii) γchgr (G) = 2 if and only if G a non-trivial graph, has no induced cycles C3 and C5, and {a, b} is a (connected) hop dominating set for each pair of adjacent vertices a, b ∈ V (G). (iii) γchgr (G) = n if and only if G is complete. Proof. Clearly, 1 ≤ γchgr (G) ≤ n. (i) Assume that γhgr(G) = 1. Suppose on the contrary that G is non-trivial. Then γch(G) ≥ 2. By Proposition 2, γchgr (G) ≥ 2, a contradiction. Therefore, G is trivial. The converse is clear. (ii) Suppose γchgr (G) = 2. Then G is non-trivial by (i). Suppose G has a triangle, say C3 = [x, y, z, x]. Then ⟨{x, y, z}⟩ is connected and (x, y, z) is a legal closed hop neighborhood sequence of G. By Corollary 1, γchgr (G) ≥ 3, a contradiction. Hence, G is triangle-free. Next, suppose that G has an induced cycle C5 = [p1, p2, . . . , p5, p1]. Then (p1, p3, p2) is a legal closed hop neighborhood sequence and ⟨{p1, p3, p2}⟩ is connected. Again, by Corollary 1, this implies that γchgr (G) ≥ 3, a contradiction. Thus, G does not have an induced cycle of order 5. Now let u and v be adjacent vertices. Then (u, v) is a legal closed hop neighborhood sequence of G. By assumption and Proposition 1, {a, b} is a hop dominating set of G. For the converse, suppose that G satisfies the given conditions. Let (v1, v2, . . . , vk) be a γchgr -sequence of G. Suppose further that k ≥ 2. Let i ∈ {1, 2, . . . , k − 1}. Since ⟨Ŝ⟩ is connected, there exists 1 ≤ j ≤ k, where j ̸= i, such that vivj ∈ E(G). If j < k, then N2 G[vk] ⊆ N2 G[vi] ∪ N2 G[vj ] because {vi, vj} is a hop dominating set of G by assumption. Thus, N2 G[vk]\∪ k−1 r=1N 2 G[vr] = ∅, a contradiction. Therefore, j = k and vivk ∈ E(G) for all i ∈ {1, 2, . . . , k− 1}. Moreover, since G is triangle-free, ⟨Ŝ⟩ is a star. Now let w ∈ N2 G[v2]. Suppose w /∈ N2 G[v1]. Let [v2, z, w] be a v2-w geodesic. Since {v1, vk} is a hop dominating set of G and w ∈ N2 G[v1], it follows that w /∈ N2 G[vk]. Let [vk, y, w] be a vk-w geodesic. Since v2vk ∈ E(G) and G is tringle-free, zvk /∈ E(G); hence, y ̸= z. This implies that [vk, v2, z, w, y, vk] is an induced cycle of G of order 5, a contradiction to an assumption. Thus, w ∈ N2 G[v1], implying that N2 G[v2] ⊆ N2 G[v1]. This contradicts the legality property of S. Therefore, k = 2, i.e., γchgr (G) = 2. (iii) Suppose γchgr (G) = n. Then by Remark 2, γhgr(G) = n. Since G is connected, it follows that G is complete by Theorem 1. J. Hassan, S. Canoy Jr. / Eur. J. Pure Appl. Math, 16 (2) (2023), 1212-1227 1218 Conversely, suppose that G is complete. Then N2 G[u] = {u} for each u ∈ V (G). Let V (G) = {a1, a2, . . . , an}. Then N2 G[ai] \ ∪i−1 j=1N 2 G[aj ] = {ai} \ {aj : j ̸= i} = {ai} ≠ ∅ for each i ∈ {2, 3, . . . , n}. It follows that (a1, a2, · · · , an) is a Grundy hop dominating sequence of G. Since G is connected, it follows that γchgr (G) = n. The next results are immediate from Theorem 2. Corollary 2. Let T be a non-trivial tree. Then γchgr (G) = 2 if and only if {a, b} is a (connected) hop dominating set for each pair of adjacent vertices a, b ∈ V (T ). Corollary 3. Let G be a connected graph on n vertices. Then γchgr (G) ≤ n− 1 if and only if G is non-complete. Proposition 2. Let n be any positive integer. Then each of the following holds. (i) There exists a connected graph G such that γchgr (G)− γch(G) = n. (ii) There exists a connected graph H such that γhgr(H)− γchgr (H) = n. Proof. For (i), consider the graph G given in Figure 4. Let S1 = {u, vn+2} and S2 = (v1, v2, · · · , vn+2). Then S1 and S2 are γch-set and γchgr -sequence of G, respectively. Hence, γch(G) = 2 and γchgr (G) = n+ 2. Consequently, γchgr (G)− γch(G) = n+ 2− 2 = n. v1 v3 v2 vn+2G : u . . . Figure 4: A graph G with γch gr (G)− γch(G) = n For (ii), consider the graph H given in Figure 5. Let S′ = (v1, u, w) and S′′ = (v1, v2, . . . , vn+2, u). Then S′ and S′′ are γchgr - and γhgr-sequences of H, respectively. Therefore, γchgr (H) = 3 and γhgr(H) = n+3. Consequently, γhgr(H)−γchgr (H) = n+3−3 = n. J. Hassan, S. Canoy Jr. / Eur. J. Pure Appl. Math, 16 (2) (2023), 1212-1227 1219 H : . . . v1 v2 v3 vn+2 vn+1 u w Figure 5: A graph H with γh gr(H)− γch gr (H) = n This proves the assertion. Corollary 4. Let G be a connected graph. Then γchgr (G) − γch(G) and γhgr(G) − γchgr (G) can be made arbitrarily large. Next, we give a more general result (than Proposition 2) involving the connected hop domination, connected Grundy hop domination and Grundy hop domination parameters. Theorem 3. Let a and b be positive integers such that 3 ≤ a ≤ b. Then each of the following holds. (i) There exists a connected graph G such that γch(G) = a and γchgr (G) = b. (ii) There exists a connected graph G′ such that γchgr (G ′) = a and γhgr(G ′) = b. Proof. For a = b, consider G = Ka. Then γch(G) = a = γchgr (G) = γhgr(G). Suppose now that a < b. For (i), let m = b− a and consider the following cases: Case 1: a is odd. Consider the graph G given in Figure 6, where ⟨{y1, y2, · · · , ym−1, w}⟩ is complete. One can verify that S1 = (v1, v2, . . . , va} and S2 = (y1, y2, · · · , ym−1, w, va, va−1 · · · , v1) are γch- and γchgr -sequences of G, respectively. Hence, γch(G) = a and γchgr (G) = m+a = b. J. Hassan, S. Canoy Jr. / Eur. J. Pure Appl. Math, 16 (2) (2023), 1212-1227 1220 . . .v1 v2 va−1 va y1 w ym−1 ...G : y2 v3 Figure 6: A graph G with γch(G) < γch gr (G) Case 2: a is even. Consider the graph G′ given in Figure 7, where ⟨{y1, y2, · · · , ym−1, w}⟩ is complete. One can verify that S′ = {v1, v2, . . . , va} and S′′ = (y1, y2, . . . , ym−1, w, va, va−1 . . . , v1) are γch- and γchgr -sequences of G ′, respectively. Thus, γch(G ′) = a and γchgr (G ′) = m+ a = b. . . .v1 v2 va−1 va y1 w ym−1 ...G′ : y2 v3 va−2 Figure 7: A graph G′ with γch(G ′) < γch gr (G ′) For (ii), let m = b − a + 1. Consider the graph H given in Figure 8. One can verify that C ′ = {x1, x2, · · · , xa) and C ′′ = (x1, x2, · · · , xa−1, y1, y2, · · · , ym) are γchgr -sequence and γhgr-sequence of H, respectively. Therefore, γchgr (H) = a and γhgr(H) = m+ a− 1 = b. J. Hassan, S. Canoy Jr. / Eur. J. Pure Appl. Math, 16 (2) (2023), 1212-1227 1221 H : . . . y1 y2 ym x1 x2 x3 xa−1 xa. . . Figure 8: A graph H with γch gr (H) < γh gr(H) This proves the assertion. Proposition 3. [8] For any positive integer n ≥ 2, γhgr(Pn) = { 2 if n = 2, 3 n− 2 if n ≥ 4. Proposition 4. For any positive integer n ≥ 2, γchgr (Pn) = { 2 if n = 2, 3 n− 2 if n ≥ 4. Proof. Let Pn = [v1, v2, · · · , vn]. Clearly, γchgr (Pn) = 2 for n = 2, 3. Next, suppose that n ≥ 4. Let S′ = (v1, v2 · · · , vn−2). Clearly, S′ is a connected Grundy hop dominating sequence of Pn. Thus, γchgr (Pn) ≥ n − 2. Since γhgr(Pn) = n − 2 for all n ≥ 4, it follows that γchgr (Pn) ≤ n − 2 for all n ≥ 4 by Remark 2. Consequently, γchgr (Pn) = n − 2 for all n ≥ 4. Proposition 5. Let G be a connected graph of order n. If |N2 G[v]| ≥ m for every v ∈ V (G), then γchgr (G) ≤ n− (m− 1). Proof. Let V (G) = {v1, v2, . . . , vn}. Suppose γchgr (G) = k, say S = (s1, s2, · · · , sk) is a connected Grundy hop dominating sequence of G. Assume s1 = vi for some i ∈ {1, . . . , n}. Then |N2 G[s1]| = |N2 G[vi]| ≥ m by assumption. It follows that there are at most n − m remaining vertices of G that could be hop footprinted by the next terms of S. Therefore, γchgr (G) = k ≤ n−m+ |{vi}| = n−m+ 1 = n− (m− 1). The next result follows immediately form Proposition 5. Corollary 5. Let G be a connected graph on n vertices. If |N2 G[u]| = 3 for every u ∈ V (G), then γchgr (G) ≤ n− 2. J. Hassan, S. Canoy Jr. / Eur. J. Pure Appl. Math, 16 (2) (2023), 1212-1227 1222 Proposition 6. [8] 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. Proposition 7. For any positive integer n ≥ 3, γchgr (Cn) =  2 if n = 4 3 if n = 3, 5 n− 4 if n ≥ 6 and even n− 3 if n ≥ 7 and odd Proof. Let G = Cn = [v1, v2, · · · , vn, v1]. Clearly, γchgr (C3) = 3 = γchgr (C5) and γchgr (C4) = 2. Suppose that n ≥ 6 and is even. Let S = (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 fol- lows that S is a connected Grundy hop dominating sequence. Hence, γchgr (Cn) ≥ |Ŝ| = n−4. Now, since γhgr(Cn) = n − 4 for even integers n ≥ 6, it follows that γchgr (Cn) ≤ n − 4 by Proposition 2. Consequently, γchgr (Cn) = n− 4 for all even integers n ≥ 6. Next, suppose that n ≥ 7 and is odd. Let S = (v1, v3, · · · , vn−4, vn−3, vn−5, · · · , v2). Then S is a maximum connected Grundy hop dominating sequence of Cn. Hence, γchgr (G) = n− 3 for all n ≥ 7 and odd. Lemma 1. [11] Let G be a non-trivial connected graph and let G1 and G2 be two copies of G in the graph S(G). If w ∈ V (G1) and w′ ∈ V (G2) is the corresponding vertex of w, then N2 S(G)[w] = N2 G1 [w] ∪N2 G2 [w′] = N2 S(G)[w ′]. In what follows, if G1 and G2 are copies of G in the shadow graph S(G), and D ⊆ V (G1), and Q ⊆ V (G2), then the sets D′ and Q′ are given by D′ = {v′ ∈ V (G2) : v ∈ D} and Q′ = {w ∈ V (G1) : w ∈ Q}. Theorem 4. Let G be a non-trivial connected graph and let G1 and G2 be copies of G in the shadow graph S(G). Then C is a connected hop dominating set of S(G) if and only if one of the following conditions holds: (i) C is a connected hop dominating set in G1. (ii) C is a connected hop dominating set in G2. (iii) C = CG1 ∪ CG2 such that CG1 ∪ C ′ G2 and C ′ G1 ∪ CG2 are connected hop dominating sets of G1 and G2, respectively. J. Hassan, S. Canoy Jr. / Eur. J. Pure Appl. Math, 16 (2) (2023), 1212-1227 1223 Proof. Let CG1 = C ∩ V (G1) and CG2 = C ∩ V (G2). If CG2 = ∅, then C = CG1 is a connected hop dominating set of G1. If CG1 = ∅, then C = CG2 is a connected hop dominating set of G2. Hence, (i) or (ii) holds. Next, suppose CG1 ̸= ∅ and CG2 ̸= ∅. Let x ∈ V (G1) \ (CG1 ∪ C ′ G2 ). Then x ∈ V (S(G)) \ C. Since C is a hop dominating of S(G), there exists y ∈ C such that dS(G)(x, y) = 2. If y ∈ CG1 , then we are done. Suppose y ∈ CG2 , say y = z′, where z ∈ V (G1). Then z ∈ C ′ G2 and dS(G)(x, y) = dG1(x, z) = 2 by Lemma 1. Therefore, CG1 ∪ C ′ G2 is a hop dominating set of G1. Clearly, ⟨CG1 ∪ C ′ G2 ⟩ is connected. Consequently, CG1 ∪ C ′ G2 is a connected hop dominating set of G1. Similarly, C ′ G1 ∪ CG2 is a connected hop dominating set of G2. Hence, (iii) holds. Conversely, suppose (i) holds. Let a ∈ V (S(G)) \ C. If a ∈ V (G1), then there exists b ∈ C such that dG1(a, b) = dS(G)(a, b) = 2. Suppose a ∈ V (G2), say a = u′, where u ∈ V (G1). If u ∈ C, then dG1(a, u) = dS(G)(a, u) = 2. If u /∈ C, then there exists v ∈ C such that dG1(u, v) = 2. It follows that dS(G)(a, u) = dS(G)(u ′, v) = 2. Therefore, C is a hop dominating set of S(G). Clearly, ⟨C⟩ is connected. Consequently, C is a connected hop dominating set of S(G). Similarly, if (ii) holds, then C is a connected hop dominating set of S(G). Next, suppose that (iii) holds. Let x ∈ V (S(G)) \ C. Then x /∈ CG1 ∪ CG2 . Suppose x ∈ V (G2) \ CG2 , say x = y′, where y ∈ V (G1). Then y /∈ C ′ G2 . If y ∈ CG1 , then dS(G)(x, y) = dS(G)(y ′, y) = 2. Suppose y /∈ CG1 . Since CG1 ∪ C ′ G2 is a hop dominating set of G1, there exists w ∈ CG1 ∪ C ′ G2 such that dG1(w, y) = 2 = dS(G)(w, y). If w ∈ CG1 , then w ∈ C and dS(G)(w, y ′) = 2 by Lemma 1. If w ∈ C ′ G2 , then w′ ∈ CG2 ⊆ C and dG2(w ′, y′) = dS(G)(w ′, y′) = 2 by Lemma 1. Therefore, C is a hop dominating set of S(G). Clearly, ⟨C⟩ is connected. Consequently, C is a connected hop dominating set of S(G). The next result follows from Theorem 4. Corollary 6. Let G be a non-trivial connected graph and let G1 and G2 be copies of G in the shadow graph S(G). Then γch(S(G)) = γch(G). Theorem 5. Let G be a non-trivial connected graph and let G1 and G2 be copies of G in the shadow graph S(G). If S is a connected Grundy hop dominating sequence of G1 or G2, then S is a connected Grundy hop dominating sequence of S(G). In particular, γchgr (G) ≤ γchgr (S(G)). Proof. Suppose S = (v1, v2, . . . , vk) is a connected Grundy hop dominating sequence of G1. Then Ŝ is a connected hop dominating set of G1. Hence, Ŝ is a connected hop domi- nating set of S(G) by Theorem 4. Let i ∈ {2, 3, . . . , k}. Since N2 G1 [vi]∩(∪i−1 j=1N 2 G2 [v′j ]) = ∅ and N2 G2 [v′i] ∩ (∪i−1 j=1N 2 G1 [vj ]) = ∅, Lemma 1 implies that N2 S(G)[vi] \ ∪ i−1 j=1N 2 S(G)[vj ] = (N2 G1 [vi] \ (∪i−1 j=1N 2 G1 [vj ]) ∪ (N2 G2 [v′i] \ (∪i−1 j=1N 2 G2 [v′j ]). By the legality property of S, ∅ ̸= N2 G1 [vi] \ ∪i−1 j=1N 2 G1 [vj ] ⊆ N2 S(G)[vi] \ ∪ i−1 j=1N 2 S(G)[vj ]. J. Hassan, S. Canoy Jr. / Eur. J. Pure Appl. Math, 16 (2) (2023), 1212-1227 1224 Thus, S is a legal closed hop neighborhood sequence in S(G), showing that S is a con- nected Grundy hop dominating sequence in S(G). Similarly, S is a connected Grundy hop dominating sequence in S(G) if S is a connected Grundy hop dominating sequence in G2. Therefore, γchgr (G) ≤ γchgr (S(G)). Lemma 2. Let G be a connected graph of order n. If |NG[v]| ≥ k for every v ∈ V (G), then γgr(G) ≤ n− (k − 1). Proof. Let V (G) = {v1, v2, . . . , vn}. Suppose γgr(G) = t, say S = (s1, s2, · · · , st) is a Grundy dominating sequence of G. Then |NG[s1]| ≥ k by assumption. It follows that there are at most n − k remaining vertices of G that could be footprinted by the next terms of S. Therefore, γcgr(G) = t ≤ n− k + 1. Proposition 8. Let n be a positive integer. Then γgr(Pn) =  1, n = 1, 2, n = 2, 3 3, n ≥ 4. Proof. Clearly, for n = 1 and n = 2, 3, γgr(Pn) = 1 and γgr(Pn) = 2, respectively. Let {v1, v2, · · · , vn} be a vertex set of G = Pn. Let = {v2, v4, v3}. Notice that S is a Grundy dominating sequence of G. Hence, γgr(G) ≥ 3. Now, notice that |NG[vi]| = n − 2 for every i, where i ̸= 1, n. Fix i. If we let vi to be the first element of a Grundy dominating sequence S′, then there are only two remaining vertices which are not in NG[vi]. Thus, γgr(G) ≤ 3. Therefore, γhgr(G) = 3. Proposition 9. Let n be a positve integer. Then γgr(Cn) =  3, n = 3, 2, n = 4 3, n ≥ 5. Proof. Clearly, for n = 3 and n = 4, γgr(Cn) = 3 and γgr(Cn) = 2, respectively. Let {v1, v2, · · · , vn} be a vertex set of G = Cn. Let = {v1, v3, v2}. Notice that S is a Grundy dominating sequence of G. Hence, γgr(G) ≥ 3. Now, notice that |NG[vi]| = n−2 for every i ∈ {1, · · · , n}. Fix i. If we let vi to be the first element of a Grundy dominating sequence S′, then there are only two remaining vertices which are not in NG[vi]. Thus, γgr(G) ≤ 3. Therefore, γgr(G) = 3. The next result is the correction of a result found in an earlier paper of the authors in [8]. Theorem 6. Let G and H be two graphs and let C be a sequence of distinct vertices of G +H. Then C is a legal closed hop neighborhood sequence in G +H if and only if one of the following holds: (i) C is a co-legal closed neighborhood sequence in G. J. Hassan, S. Canoy Jr. / Eur. J. Pure Appl. Math, 16 (2) (2023), 1212-1227 1225 (ii) C is a co-legal closed neighborhood sequence in H. (iii) The subsequences CG and CH of C, where Ĉ = ĈG ∪ ĈH , are co-legal closed neigh- borhood sequences of G and H, respectively. Proof. Assume that C = (a1, a2, · · · , am) is a legal closed hop neighborhood sequence of G+H. If Ĉ ⊆ V (G), then N2 G[ai] = N2 G+H [ai] = V (G)\NG(ai) for each i ∈ {1, 2, . . . ,m}. Thus, by the legality condition property of C, [V (G) \NG(ai)] \ ∪i−1 j=1[V (G) \NG(aj)] ̸= ∅ for each i ∈ {2, 3, . . . ,m}. It follows that C is a co-legal closed neighborhood sequence in G. Hence, (i) holds. Similarly, if Ĉ ⊆ V (H), then (ii) holds. Next, suppose that Ĉ∩V (G) ̸= ∅ and Ĉ∩V (H) ̸= ∅. Let CG and CH be subsequences of C such that ĈG = Ĉ ∩V (G) and ĈH = Ĉ ∩V (H). Let CG = (an1 , an2 , . . . , ant) and CH = (am1 , am2 , . . . , amr). Note that N 2 G+H [anj ] ⊆ V (G) for all j ∈ {1, 2, . . . , t} and N2 G+H [ams ] ⊆ V (H) for all s ∈ {1, 2, . . . , r}. Since C is a connected legal closed hop neighborhood sequence in G+H, it follows that [V (G) \NG(ani)] \ ∪i−1 j=1[V (G) \NG(anj )] = N2 G+H [ani ] \ ∪i−1 j=1N 2 G+H [anj ] ̸= ∅ for all i ∈ {2, 3, . . . , t}. Hence, CG is a co-legal closed neighborhood sequence in G. Simi- larly, CH is a co-legal closed neighborhood sequence in H. Therefore, (iii) holds. For the converse, suppose (i) or (ii) holds. Then C is a legal closed hop neigh- borhood sequence in G + H. Suppose (iii) holds. Let C = (a1, a2, · · · , am) and let CG = (an1 , an2 , . . . , ant) and CH = (am1 , am2 , . . . , amr). Let i ∈ {2, 3, . . . ,m}. Suppose ai ∈ ĈG. Then ai = ank for some k ∈ {1, 2, . . . , t}. Since CG is a co-legal closed neighbor- hood sequence in G, N2 G+H [ai] \ ∪i−1 j=1N 2 G+H [aj ] = N2 G+H [ank ] \ ∪k−1 p=1N 2 G+H [anp ] = [V (G) \NG(ank )] \ ∪k−1 p=1[V (G) \NG(anp)] ̸= ∅. If ai ∈ ĈH , then ai = amq for some q ∈ {1, 2, . . . ,m}. Since CH is a co-legal closed neighborhood sequence in H, N2 G+H [ai] \ ∪i−1 j=1N 2 G+H [aj ] = N2 G+H [amq ] \ ∪ q−1 r=1N 2 G+H [amr ] = [V (H) \NH(amq)] \ ∪ q−1 r=1[V (H) \NH(amr)] ̸= ∅. This shows that C is a legal closed hop neighborhood sequence in G+H. Theorem 7. Let G and H be two graphs and let C be a sequence of distinct vertices of G + H. Then C is a connected Grundy hop dominating sequence in G + H if and only if the subsequences CG and CH of C, where Ĉ = ĈG ∪ ĈH , are co-Grundy dominating sequences in G and H, respectively. Proof. Suppose C is a connected Grundy hop dominating sequence in G +H. Since C is a legal closed hop neighborhood sequence and Ĉ is a connected hop dominating set J. Hassan, S. Canoy Jr. / Eur. J. Pure Appl. Math, 16 (2) (2023), 1212-1227 1226 in G+H, it follows that C satisfies (iii) of Theorem 6. Hence, the subsequences CG and CH of C, where Ĉ = ĈG ∪ ĈH , are co-legal closed neighborhood sequences in G and H, respectively. Since Ĉ is a hop dominating set in G + H, V (G) = ∪v∈CG [V (G) \ NG(v)] and V (H) = ∪w∈CH [V (H) \ NH(w)]. Thus, CG and and CH are co-Grundy dominating sets in G and H, respectively. Conversely, suppose that the subsequences CG and CH of C, where Ĉ = ĈG ∪ ĈH , are co-Grundy dominating sequences in G and H, respectively. By Theorem 6, C is a legal closed hop neighborhood sequence in G + H. Since CG and CH co-Grundy dominating sequences in G and H, respectively, Ĉ is a hop dominating set. Moreover, because ⟨Ĉ⟩ is connected, C is a connected Grundy hop dominating sequence in G+H. The following result follows from Theorem 7, Proposition 8, and Proposition 9. Corollary 7. Let G and H be graphs. Then γchgr (G+H) = γcogr(G) + γcogr(H). In particular, each of the following holds. (i) γchgr (K1 +G) = 1 + γcogr(G). (ii) γchgr (Km,n) = 2 for m,n ≥ 1. (iii) γchgr (Wn) = 4 for all n ≥ 5. (iv) γchgr (Fn) = 4 for all n ≥ 4. (v) γchgr (Pn + Pm) = 6 for all n,m ≥ 4. 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