EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 16, No. 4, 2023, 2597-2612 ISSN 1307-5543 – ejpam.com Published by New York Business Global Grundy Total 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 MSU-Iligan Institute of Technology, 9200 Iligan City, Philippines 3 Center for Mathematical and Theoretical Physical Sciences, PRISM MSU-Iligan Institute of Technology, 9200 Iligan City, Philippines Abstract. Let G = (V (G), E(G)) be an undirected graph with γ(C) ̸= 1 for each component C of G. Let S = (v1, v2, · · · , vk) be a sequence of distint vertices of a graph G, and let Ŝ = {v1, v2, . . . , vk}. Then S is a legal open hop neighborhood sequence if N2 G(vi) \ ⋃i−1 j=1 N 2 G(vj) ̸= ∅ for every i ∈ {2, . . . , k}. If, in addition, Ŝ is a total hop dominating set of G, then S is a Grundy total hop dominating sequence. The maximum length of a Grundy total hop dominating sequence in a graph G, denoted by γth gr(G), is the Grundy total hop domination number of G. In this paper, we show that the Grundy total hop domination number of a graph G is between the total hop domination number and twice the Grundy hop domination number of G. Moreover, determine values or bounds of the Grundy total hop domination number of some graphs. 2020 Mathematics Subject Classifications: 05C69 Key Words and Phrases: total hop domination, total hop domination number, open hop neigh- borhood sequence, Grundy total hop dominating sequence, Grundy total hop domination number 1. Introduction Domination has attracted many researchers because of its nice applications in various fields and in networks. A number of variations of the domination concept (see for example, [7, 18, 19, 21]) have been introduced and studied. Recently, hop domination was defined and studied by Natarajan and Ayyaswamy in [17]. From then on a lot of investigations of the concept and some of its variants have been done (see [1, 2, 8–15, 20]). In 2014, Bresar et al. [4] introduced another concept called Grundy dominating se- quence in a graph. The newly defined concept has subsequently attracted other researchers in the area to study and generate more interesting results (see [6, 16]) on it. ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v16i4.4877 Email addresses: javierhassan@msutawi-tawi.edu.ph (J. Hassan), sergio.canoy@g.msuiit.edu.ph (S. Canoy) https://www.ejpam.com 2597 © 2023 EJPAM All rights reserved. J.A. Hassan, S. Canoy / Eur. J. Pure Appl. Math, 16 (4) (2023), 2597-2612 2598 In 2016, the concept of Grundy total domination in graphs was investigated by Bresar et al. [5]. Bresar [3] studied further the concept on the product of graphs. In this study, the concept of Grundy total hop domination number of a graph will be introduced and investigated. Its relationship with total hop domination, and Grundy hop domination numbers of a graph will be given. Bounds for the parameter will be determined for the shadow graph as well as the join and the corona of two 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 total dominating set of G if for every v ∈ V (G), there exists u ∈ D such that uv ∈ E(G), that is, NG(D) = V (G). The total domination number of G, denoted by γt(G), is the minimum cardinality of a total dominating set of G. Any total dominating set with cardinality equal to γt(G) is called a γt-set. Let S = (v1, v2, · · · , vk) be a sequence of distint vertices of a graph G, and let Ŝ = {v1, v2, . . . , vk}. Then S is a legal open neighborhood sequence if NG(vi)\ ⋃i−1 j=1NG(vj) ̸= ∅ for every i ∈ {2, . . . , k}. If, in addition, Ŝ is a total dominating set of G, then S is called a Grundy total dominating sequence. The maximum length of a Grundy total dominating sequence in a graph G is called the Grundy total domination number of G, and is denoted by γtgr(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. Any Grundy hop dominating sequence S with |Ŝ| = γhgr(G) is called a maximum J.A. Hassan, S. Canoy / Eur. J. Pure Appl. Math, 16 (4) (2023), 2597-2612 2599 Grundy hop dominating sequence or a γhgr-sequence of G. In this case, we call Ŝ a γhgr-set of G. A subset S of V (G) is a total hop dominating set of G if for every v ∈ V (G), there exists u ∈ S such that dG(u, v) = 2. The smallest cardinality of a total hop dominating set of G denoted by γth(G), is called the total hop domination number of G. Any hop dominating set with cardinality equal to γth(G) is called a γth-set. Let G be any graph with γ(C) ̸= 1 for each component C of G. Let S = (v1, v2, · · · , vk) be a sequence of distint vertices of a graph G, and let Ŝ = {v1, v2, . . . , vk}. Then S is a legal open hop neighborhood sequence if N2 G(vi)\ ⋃i−1 j=1N 2 G(vj) ̸= ∅ for every i ∈ {2, . . . , k}. If, in addition, Ŝ is a total hop dominating set of G, then S is called a Grundy total hop dominat- ing sequence. The maximum length of a Grundy total hop dominating sequence in a graph G is called the Grundy total hop domination number of G, and is denoted by γthgr(G). Any Grundy total hop dominating sequence S with |Ŝ| = γthgr(G) is called a maximum Grundy total hop dominating sequence or a γthgr-sequence of G. In this case, we call Ŝ a γthgr-set of G. A legal open hop neighborhood sequence S = (v1, v2, · · · , vk) with maximum length, i.e., k = max{p ∈ N : ∃ a legal open hop neighborhood sequence (x1, · · · , xp) of G}, will be referred to as a maximum legal open hop neighborhood sequence. We say that vertex vi hop-footprints the vertices from N2 G[vi] \ ∪i j=1N 2 G[vj ] ( resp. N2 G(vi) \ ∪i j=1N 2 G(vj) ) , and that vi is their hop-footprinter. Two sequences S and S′ in G are loh-identical if they are legal open hop neighborhood sequences (or Grundy total hop dominating sequences) and Ŝ = Ŝ′ (i.e., one is a reaarrangement of the terms of the other). A sequence S = (v1, v2, · · · , vk) of distinct vertices of a graph G is a co-legal open 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 open neighborhood sequence S = (v1, v2, . . . , vk) is a co-Grundy total dominating sequence if V (G) = ∪k i=1[V (G) \NG[vi]]. The maximum length of a co- Grundy total dominating sequence in a graph G is called the co-Grundy total domination number of G, and is denoted by γtcogr(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). The shadow graph S(G) of a graph G is constructed by taking two copies of G, say G1 and G2 and joining each vertex u ∈ G1 to the neighbors of the corresponding vertex u′ ∈ G2. 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. J.A. Hassan, S. Canoy / Eur. J. Pure Appl. Math, 16 (4) (2023), 2597-2612 2600 3. Results Theorem 1. Let G be a graph of order n with γ(C) ̸= 1 for each component C of G. Then the following statements hold. (i) If γth(G) = t and D = {u1, u2, . . . , ut} is a minimum total hop dominating set of G, then S = (u1, u2, · · · , ut) is a Grundy total hop dominating sequence. In particular, γth(G) ≤ γthgr(G). (ii) If S = (u1, u2, · · · , us) is a minimum Grundy total hop dominating sequence, then γth(G) = |Ŝ|. Proof. (i) Suppose that there exists i ∈ {2, 3, . . . , t} such that N2 G(ui)\∪ i−1 j=1N 2 G(uj) = ∅. Then N2 G(ui) ⊆ ∪i−1 j=1N 2 G(uj). This means that D\{ui} is a total hop dominating set of G, which is a contradiction to the minimality of D. Hence, N2 G(ui) \ ∪ i−1 j=1N 2 G(uj) ̸= ∅ for each i ∈ {2, 3, . . . , t}, and so S is Grundy total hop dominating sequence. Consequently, γth(G) ≤ γthgr(G). (ii) From (i), every γth-set of G forms a Grundy total hop dominating sequence. Since S is a minimum Grundy total hop dominating sequence, it follows that |Ŝ| ≤ γth(G). On the other hand, since every Grundy total hop dominating sequence forms a total hop dominating set by definition, it follows that γth(G) ≤ |Ŝ|. Consequently, γth(G) = |Ŝ|. Theorem 2. Let G be a graph of order n with γ(C) ̸= 1 for each component C of G. Then S = (u1, u2, · · · , ul) is a maximum legal open hop neighborhood sequence of G if and only if S is a Grundy total hop dominating sequence of G and γthgr(G) = l. Proof. Let S = (u1, · · · , ul) be a maximum legal open hop neighborhood sequence of G. Suppose on the contrary that Ŝ is not a total hop dominating set of G. Then there exists u ∈ V (G) such that u /∈ N2 G(Ŝ). This means that u /∈ N2 G(v) for every v ∈ Ŝ. Since u is not hop dominated by any v ∈ Ŝ, u ∈ N2 G(t) for some t ∈ V (G) \ Ŝ. This means that N2 G(t) \ ⋃k i=1N 2 G(ui) ̸= ∅. Thus, S′ = (u1, · · · , ul, t) is a legal open hop neighborhood sequence of G, a contradiction to the maximality of S. Therefore, Ŝ is a total hop dominating set of G. Consequently, S is a Grundy total hop dominating sequence of G and γthgr(G) = l. The converse is clear. The next result follows from Theorem 2. Corollary 1. Let G be a graph of order n with γ(C) ̸= 1 for each component C of G and let T = (x1, · · · , xj) be a legal open hop neighborhood sequence of G. Then |T̂ | = j ≤ γthgr(G). Theorem 3. Let G be a graph of order n with γ(C) ̸= 1 for each component C of G. Then 4 ≤ γthgr(G) ≤ n and these bounds cannot be improved. J.A. Hassan, S. Canoy / Eur. J. Pure Appl. Math, 16 (4) (2023), 2597-2612 2601 Proof. Clearly γthgr(G) = 1 is not possible. Suppose γthgr(G) = 2, say, S = (v1, v2) is a Grundy total hop dominating sequence. Since Ŝ is a total hop dominating set, v1 ∈ N2 G(v2). Let v ∈ NG(v1) ∩ NG(v2). Then v /∈ N2 G(v1) ∪ N2 G(v2), a contradiction. Next, suppose γthgr(G) = 3, say S = (v1, v2, v3) is a Grundy total hop dominating sequence. Since Ŝ is a total hop dominating set, v1 is hop dominated by v2 or v3. Suppose dG(v1, v2) = 2 and let p ∈ NG(v1) ∩ NG(v2). Then p ∈ N2 G(v3) and v3 ∈ N2 G(v1) ∪ N2 G(v2). Let S∗ = (p, v1, v2, v3). Then Ŝ∗ is a total hop dominating set. Moreover, observe that v3 ∈ N2 G(p), v2 ∈ N2 G(v1) \ N2 G(p), v1 ∈ [N2 G(v2) \ (N2 G(v1) ∪ N2 G(p))] and p ∈ [N2 G(v3) \ (N2 G(v2) ∪ N2 G(v1) ∪ N2 G(p))]. Hence, S∗ is a legal open hop neighborhood sequence, and so S∗ is a Grundy total hop dominating sequence of G, contrary to our assumption that γthgr(G) = 3. Therefore, γthgr(G) ≥ 4. For tightness of the bounds, consider G = C4 and H = P8. Then γthgr(G) = 4 and γthgr(H) = 8. Theorem 4. Let G be a connected graph of order n such that γ(G) ̸= 1. If n = 2m,m ≥ 2 and the vertices of G can be labeled as u1, . . . , um, v1, . . . , vm in such a way that (i) dG(ui, vi) = 2 for each i, (ii) {u1, . . . , um} is a hop independent set of G, and (iii) dG(ui, vj) = 2 implies that i ≥ j, then γthgr(G) = n. Proof. Suppose the vertices of G can be labeled as described. Clearly, Ŝ = {u1, · · · , um, vm, · · · , v1} is a total hop dominating set of G. Observe that vi ∈ N2 G(ui) \ ⋃i−1 j=1N 2 G(uj) for each i ∈ {2, . . . ,m} by (i) and (iii) and um ∈ N2 G(vm) \⋃m j=1N 2 G(uj) by (i) and (ii). Now, for any k < m, uk ∈ N2 G(vk) \  m⋃ j=1 N2 G(uj)  ∪ ( k+1⋃ i=m N2 G(vi) ) by (i), (ii), and (iii). Therefore, S is a Grundy total hop dominating sequence, showing that γthgr(G) = n. Proposition 1. Let n and m be any positive integers such that n ≥ 4. Then γthgr(Pn) =  n− 2 if n = 4m+ 2 n− 1 if n ≥ 5 and odd n if n = 4m Proof. Let Pn = [v1, v2, . . . , vn]. Clearly, γthgr(P6) = 4. For n = 4m + 2 ≥ 10, let S = (v1, v2, . . . , vn−2). Clearly, S is a Grundy total hop dominating sequence of Pn. Thus, J.A. Hassan, S. Canoy / Eur. J. Pure Appl. Math, 16 (4) (2023), 2597-2612 2602 γthgr(Pn) ≥ n − 2. On the other hand, let S′ = (w1, w2, . . . , wk) be a Grundy total hop dominating sequence of Pn. Notice that one of the vertices v1, v5, v9, . . . , vn−5, vn−1 is not in Ŝ′. Suppose all the vertices v1, v5, v9, . . . , vn−5, vn−1 are in Ŝ. If v5 comes before v1, then N2 Pn (v1) ⊆ N2 Pn (v5), a contradiction. So, v5 comes after v1. Next, suppose v9 comes before v5, then N2 Pn (v5) ⊆ N2 Pn (v1) ∪ N2 Pn (v9), a contradiction. Thus, v9 comes after v5. Continuing in this manner, we find that the following order of appearance (not necessarily consecutive) of the given vertices in the Grundy total hop dominating sequence S′: v1, v5, v9, . . . , vn−5, vn−1. However, N2 Pn (vn−1) ⊆ NPn(vn−5). Hence, vn−1 /∈ Ŝ′, a contradiction. Similarly, one of the vertices v2, v6, v10, . . . , vn−4, vn is not in Ŝ′. Therefore, γthgr(Pn) = k ≤ n− 2. Consequently, γthgr(Pn) = n− 2 for all n = 4m+ 2. Next, let n ≥ 5 and odd. Clearly, γthgr(P5) = 4. Suppose n ≥ 7 and odd. For n ∈ {7, 11, 15, . . .}, let S1 = (v1, v2, v5, v6, · · · , vn−6, vn−5, vn−2, vn, vn−3, vn−4, vn−7, vn−8, · · · , v4, v3). Then S1 is a Grundy total hop dominating sequence of Pn. Hence, γthgr(Pn) ≥ n− 1. Next, for n ∈ {9, 13, 17, . . .}, let S2 = (v1, v2, v5, v6, · · · , vn−4, vn−3, vn−2, vn−1, vn−6, vn−5, · · · , v3, v4). Observe that S2 is a Grundy total hop dominating sequence of Pn. Hence, γthgr(Pn) ≥ n− 1. Suppose γthgr(Pn) = n, say S0 = (w1, w2, · · · , wn) is a Grundy total hop dominating se- quence of Pn. Observe that for n ∈ {7, 11, 15, . . .}, one of the vertices v2, v6, v10, . . . , vn−4, vn is not in Ŝ0. Suppose all vertices v2, v6, v10, . . . , vn−4, vn are in Ŝ0. If v6 comes before v2, then N2 Pn (v2) ⊆ N2 Cn (v6), a contradiction. So, v6 comes after v2. Next, suppose v10 comes before v6, then N2 Pn (v6) ⊆ N2 Pn (v2) ∪N2 Pn (v10), a contradiction. Thus, v10 comes after v6. Continuing in this manner, we find that the following order of appearance (not necessar- ily consecutive) of the given vertices in the Grundy total hop dominating sequence S0: v2, v6, v10, . . . , vn−4, vn. However, N 2 Pn (vn) ⊆ NPn(vn−4). Hence, vn /∈ Ŝ0, a contradiction. Similarly, for n ∈ {9, 13, 17, . . .}, one of the vertices v1, v5, v9, . . . , vn−5, vn−1 is not in the Grundy total hop dominating sequence, say S′′. Thus, γthgr(Pn) ≤ n − 1. Consequently, γthgr(Pn) = n− 1 for all n ≥ 5 and odd. Lastly, assume that n = 4m. Clearly, γthgr(P4) = 4. For n = 4m ≥ 8, let C = (v1, v2, v5, v6, · · · , vn−3, vn−2, vn−1, vn, vn−5, vn−4, · · · , v3, v4). Then C is a Grundy total hop dominating sequence of Pn. Thus, γthgr(Pn) = n for all n = 4m. J.A. Hassan, S. Canoy / Eur. J. Pure Appl. Math, 16 (4) (2023), 2597-2612 2603 Proposition 2. Let G be a graph of order n with γ(C) ̸= 1 for each component C of G. If |N2 G(v)| ≥ m for every v ∈ V (G), then γthgr(G) ≤ n− (m− 1). Proof. Suppose γthgr(G) = k, say S = (w1, w2, · · · , wk) is a Grundy total hop dominating sequence of G. Assume w1 = vi for some i ∈ {1, . . . , n}. Then |N2 G(w1)| = |N2 G(vi)| ≥ m for some i ∈ {1, . . . , n}. 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, γthgr(G) = k ≤ n−m+ |{vi}| = n−m+ 1 = n− (m− 1). The next result follows from Proposition 2. Corollary 2. Let G be a graph of order n with γ(C) ̸= 1 for each component C of G. If |N2 G(u)| ≥ 2 for every u ∈ V (G), then γthgr(G) ≤ n− 1. Proposition 3. Let n and m be any positive integers such that n ≥ 4. Then γthgr(Cn) =  4 if n = 4, 5, 6 6 if n = 8 n− 4 if n = 4m ≥ 12 n− 2 if n = 4m+ 2 ≥ 10 n− 1 if n ≥ 7 and odd Proof. Clearly for n = 4, 5, 6 and n = 8, γthgr(Cn) = 4 and γthgr(Cn) = 6, respectively. For n = 4m ≥ 12, let V (Cn) = {v1, v2, . . . , vn}. Observe that S = (v1, v2, · · · , vn−4) is a Grundy total hop dominating sequence of Cn. Thus, γthgr(Cn) ≥ n − 4. On the other hand, let S′ = (w1, w2, . . . , wk) be a Grundy total hop dominating sequence of Cn. Notice that one of the v1, v5, v9, . . . , vn−7, vn−3 is not in Ŝ′. Suppose all the vertices v1, v5, v9, . . . , vn−7, vn−3 are in Ŝ′. WLOG, assume that w1 = v1. If v9 comes before v5, then N2 Cn (v5) ⊆ N2 Cn (v1)∪N2 Cn (v9), a contradiction. So, v9 comes after v5. Next, suppose v13 comes before v9, then N2 Cn (v9) ⊆ N2 Cn (v1)∪N2 Cn (v5)∪N2 Cn (v13), a contradiction. Thus, v13 comes after v9. Continuing in this manner, we find that the following order of appear- ance (not necessarily consecutive) of the given vertices in the Grundy total hop dominat- ing sequence S′: v1, v5, v9, . . . , vn−7, vn−3. However, N2 Cn (vn−3) ⊆ N2 Cn (v1) ∪ NCn(vn−7). Hence, vn−3 /∈ Ŝ′, a contradiction. Similarly, one of the vertices v2, v6, v10, . . . , vn−6, vn−2, v3, v7, v11, . . . , vn−5, vn−1, and v4, v8, v12, . . . , vn−4, vn, respectively, is not in Ŝ′. Therefore, γthgr(Pn) = k ≤ n− 4. Consequently, γthgr(Pn) = n− 4 for all n = 4m. Next, for n = 4m+ 2 ≥ 10, let S1 = (v1, v5, . . . , vn−5, vn−1, v3, v7, . . . , vn−7, v2, v6, . . . , vn−4, vn, v4, v8, . . . , vn−6). Then S1 is a Grundy total hop dominating sequence of Cn. Hence, γthgr(Cn) ≥ n− 2. On the other hand, let S = (w1, w2, . . . , wk) be a Grundy total hop dominating sequence of Cn. Then applying the same arguments with the first part, one can show that one of the J.A. Hassan, S. Canoy / Eur. J. Pure Appl. Math, 16 (4) (2023), 2597-2612 2604 v1, v5, v9, . . . , vn−5, vn−1, v3, v7, . . . , vn−7, vn−3 and v2, v6, v10, . . . , vn−4, vn, v4, v8, . . . , vn−6, vn−2 is not in Ŝ1, respectively. Hence, γ th gr(Cn) = k ≤ n−2. Consequently, γthgr(Cn) = n−2. Let n ≥ 7 and odd. Clearly, γthgr(C7) = 6. Suppose n ≥ 9 and odd. For n ∈ {9, 13, 17, . . .}, let S2 = (v1, v5, . . . , vn, v4, . . . , vn−1, v3, . . . , vn−2, v2, . . . , vn−7). Then S2 is a Grundy total hop dominating sequence of Cn. Hence, γ th gr(Cn) ≥ n−1. Next, for n ∈ {11, 15, 19, . . .}, let S3 = (v1, v5, . . . , vn−2, v2, . . . , vn−1, v3, . . . , vn, v4, . . . , vn−7). Then S3 is a Grundy total hop dominating sequence of Cn. Hence, γthgr(Cn) ≥ n− 1. On the other hand, since |NCn(v)| = 2 for every v ∈ V (Cn), it follows that γthgr(Cn) ≤ n − 1 by Corollary 2. Therefore, γthgr(Cn) = n− 1. Theorem 5. Let G be a graph of order n with γ(C) ̸= 1 for each component C of G. Then γthgr(G) ≤ 2γhgr(G). Proof. Let S = (v1, v2, · · · , vk) be a Grundy total hop dominating sequence of G, where k = γthgr(G). We will prove that at most k/2 vertices can be removed from S in such a way the resulting sequence S′ forms a legal closed hop neighborhood sequence of G. Notice that a vertex vi ∈ Ŝ prevents S from being a legal closed hop neighborhood sequence only if N2 G[vi] \ ⋃i−1 j=1N 2 G[vj ] = ∅ for each i ∈ {1, . . . , k}. Since S is a Grundy total hop dominating sequence, vi hop footprinted only vertices from S that precedes vi. That is, h−1 S (vi) ⊆ {v1, . . . , vi−1}, where hS : V (G) → Ŝ is a hop footprinter function, mapping each vertex to its hop footprinter. Set T = {vi ∈ Ŝ : h−1 S (vi) ⊆ {v1, . . . , vi−1}}. Since v1 /∈ T , T ̸= Ŝ. Suppose that h−1 S (vj)∩T ̸= ∅ for some vj ∈ T . Let vi ∈ h−1 S (vj)∩T . Since vj ∈ T , the vertex vi that is hop footprinted by vj satisties i < j. Since vi ∈ T , vi hop footprints some vertex vt, where t < i. This means that hS(vt) = vi, where 1 ≤ t ≤ i− 1. It follows that vi /∈ N2 G(vj) \∪ j−1 k=1N 2 G(vk), that is, hS(vi) ̸= vj , contrary to the assumption that vi ∈ h−1 S (vj). Therefore, h −1 S (vj)∩T = ∅ for every vertex vj ∈ T . Now, suppose that vi, vj ∈ T , where i < j. By definition, h−1 S (vi) ⊆ {v1, . . . , vi−1} and h−1 S (vj) ⊆ {v1, . . . , vj−1}. Since every vertex is hop footprinted by a unique vertex in Ŝ, it follows that h−1 S (vi) ∩ h−1 S (vj) = ∅. Since h−1 S (vj) ∩ T = ∅ for every vertex vj ∈ T , {h−1 S (vi) : vi ∈ T} forms a partition of a subset of Ŝ \ T . Note that for each vi ∈ T , |h−1 S (vi)| ≥ 1, and so |T | ≤ | ⋃ vi∈T h−1 S (vi)| ≤ |Ŝ| − |T | implying that 2|T | ≤ |Ŝ| = k. Hence, |T | ≤ k 2 . Let S′ be a sequence obtained from S by deleting vertices from T . Then S′ is a legal closed hop neighborhood sequence of G. Thus, γhgr(G) ≥ |S′| = k − |T | ≥ k − k 2 = k 2 = 1 2γ th gr(G). Consequently, γthgr(G) ≤ 2γhgr(G). J.A. Hassan, S. Canoy / Eur. J. Pure Appl. Math, 16 (4) (2023), 2597-2612 2605 Remark 1. The bound given in Theorem 5 is tight. To see this, consider C4 in Fig. 1. Let S = (u1, u2, u3, u4). Then S is a γthgr-sequence of C4. Thus, γthgr(C4) = 4. Next, let S∗ = (u1, u2). Then S∗ is a γhgr-sequence of C4. Hence, γhgr(C4) = 2. Consequently, γthgr(C4) = 4 = 2γhgr(C4). C4 : u2 u3 u4 u1 Figure 1: A graph C4 with γth gr(C4) = 4 = 2γh gr(C4) Remark 2. Let G be a graph of order n with γ(C) ̸= 1 for each component C of G. Then γthgr(G) ≥ γhgr(G) does not hold in general. To see this, consider K5 ◦ K2 in Fig. 2. Let S = (v1, v2, . . . , v10). Then S is a γhgr- sequence of K5 ◦K2, that is, γ h gr(K5 ◦K2) = 10. Next, let S′ = (v1, v12, v3, v13). Then S′ is a γthgr-sequence of K5 ◦K2. Thus, γ th gr(K5 ◦K2) = 4. K5 ◦K2 : v2 v3 v1 v4 v5 v7 v8 v6 v10 v9 v11 v12 v13 v14v15 Figure 2: A graph K5 ◦K2 with γth gr(K5 ◦K2) = 4 < 10 = γh gr(K5 ◦K2) Lemma 1. Let G be a non-trivial connected graph and let G1 and G2 be two copies of G in the graph S(G). If v ∈ V (G1) and v′ ∈ V (G2) is the corresponding vertex of v, then J.A. Hassan, S. Canoy / Eur. J. Pure Appl. Math, 16 (4) (2023), 2597-2612 2606 (i) N2 S(G)(v) = N2 G1 (v) ∪N2 G2 [v′] and (ii) N2 S(G)(v ′) = N2 G1 [v] ∪N2 G2 (v′). Proof. (i) Let a ∈ N2 S(G)(v). Then dS(G)(a, v) = 2. If a ∈ V (G1), then a ∈ N2 G1 (v). Suppose a ∈ V (G2). By assumption, it follows that dG2(a, v ′) = 2. Thus, a ∈ N2 G2 (v′). Hence, N2 S(G)(v) ⊆ N2 G1 (v)∪N2 G2 [v′]. Clearly, N2 G1 (v)∪N2 G2 [v′] ⊆ N2 S(G)(v). Consequently, N2 S(G)(v) = N2 G1 (v) ∪N2 G2 [v′]. (ii) can be proved similarly. Theorem 6. Let G be a graph of order n with γ(C) ̸= 1 for each component C of G. If S is a Grundy total hop dominating sequence of G1 or G2, then S is a Grundy total hop dominating sequence of S(G). Moreover, γthgr(G) ≤ γthgr(S(G)). Proof. Let G1 and G2 be two copies of G. Let S = (v1, v2, . . . , vk) be a Grundy total hop dominating sequence in G1 and let v′ ∈ V (G2). Then N2 G1 (vi) \ i−1⋃ j=1 N2 G1 (vj) ̸= ∅ for each i ∈ {2, 3, . . . , k}. Thus, by Lemma 1 N2 S(G)(vi) \ i−1⋃ j=1 N2 S(G)(vj) = [ N2 G1 (vi) ∪N2 G2 [v′i] ] \ i−1⋃ j=1 [ N2 G1 (vj) ∪N2 G2 [v′j ] ] = (N2 G1 (vi) ∪N2 G2 [v′i] ) \ i−1⋃ j=1 N2 G1 (vj)  ∪ (N2 G1 (vi) ∪N2 G2 [v′i] ) \ i−1⋃ j=1 N2 G2 [v′j ]  ̸= ∅ for each i ∈ {2, . . . , k}. Since Ŝ is a total hop dominating set of G1, there exists w ∈ Ŝ ∩N2 G1 (v). By Lemma 1, w ∈ Ŝ ∩ N2 S(G)(v ′), i.e., w ∈ Ŝ and dS(G)(v ′, w) = 2. Consequently, Ŝ is a Grundy total hop dominating sequence of S(G). Remark 3. The bound given in Theorem 6 is tight. Moreover, strict inequality can also be attained. For equality, consider C4. Then γthgr(C4) = 4. Now, consider the shadow graph of C4 given in Fig. 3. Let S = (a, a′, b, b′). Observe that S is a Grundy total hop dominating sequence of S(C4). Moreover, it can be verified that γthgr(S(C4)) = 4. Consequently, γthgr(S(C4)) = 4 = γthgr(S(C4)). J.A. Hassan, S. Canoy / Eur. J. Pure Appl. Math, 16 (4) (2023), 2597-2612 2607 a b c d a′ b′ c′ d′ S(C4) : Figure 3: A graph C4 with γth gr(C4) = γth gr(S(C4)) For strict inequality, consider P5. Then γthgr(P5) = 4. Now, consider the shadow graph of P5 given in Fig. 4. Let S = (a, a′, e, e′, d, d′). Observe that S is a Grundy total hop dominating sequence of S(P5). Moreover, it can be verified that γthgr(S(P5)) = 6. Hence, γthgr(S(P5)) = 4 < 6 = γthgr(S(P5)). a b S(P5) : a′ b′ c c′ d d′ e e′ Figure 4: A graph G with γth gr(G) < γth gr(S(G)) Lemma 2. Let G be a graph of order n with no isolated vertices. If |NG(v)| ≥ l for every v ∈ V (G), then γtgr(G) ≤ n− (l − 1). Proof. Suppose γtgr(G) = k, say S = (w1, w2, · · · , wk) is a Grundy total dominating sequence of G. Assume w1 = vi for some i ∈ {1, . . . , n}. Then |NG(w1)| = |NG(vi)| ≥ l for some i ∈ {1, . . . , n}. It follows that there are at most n − l remaining vertices of G that could be footprinted by the next terms of S. Therefore, γtgr(G) = k ≤ n− l + |{vi}| = n− l + 1 = n− (l − 1). Proposition 4. Let n ≥ 4 be any positive integer. Then γtgr(Pn) = 4 = γtgr(Cn). J.A. Hassan, S. Canoy / Eur. J. Pure Appl. Math, 16 (4) (2023), 2597-2612 2608 Proof. Clearly, γtgr(P 4) = 4. For n ≥ 5, let Pn = [u1, u2, · · · , un] and S = (u3, u2, u1, un). Notice that Ŝ is a total dominating set of Pn. Now, observe that v4 ∈ NPn (v2) \NPn (v3), v3 ∈ NPn (v1) \ (NPn (v2)∪NPn (v3)), and v2 ∈ NPn (vn) \ (NPn (v1)∪NPn (v2)∪NPn (v3)). It follow that S is a Grundy total dominating sequence of Pn. Hence, γtgr(Pn) ≥ 4. On the other hand, suppose γtgr(Pn) = k, say S = (w1, w2, · · · , wk) is a Grundy total domi- nating sequence of Pn. Since |NPn (vj)| ≥ n − 3 for every j ∈ {1, . . . , n}, it follows that γtgr(Pn) = k ≤ n− (n− 3− 1) = 4 by Lemma 2. Therefore, γtgr(Pn) = 4. Next, let Cn = [v1, v2, · · · , vn, v1] and S = (v1, v2, v3, v4). Clearly, S is a total dominat- ing set of Cn. Observe that vn ∈ NCn (v2) \NCn (v1), v1 ∈ NCn (v3) \ (NCn (v2)∪NCn (v1)), and v2 ∈ NCn (v4) \ (NCn (v3) ∪ NCn (v2) ∪ NCn (v1)). Thus, S is a Grundy total domi- nating sequence of Cn and γtgr(Cn) ≥ 4. On the other hand, suppose γtgr(Cn) = k, say S = (u1, u2, · · · , uk) is a Grundy total dominating sequence of Cn. We may assume that u1 = v1. Then |NCn (u1)| = |NCn (v1)| = n− 3. Thus, γtgr(Cn) = k ≤ n− (n− 3− 1) = 4 by Lemma 2. Therefore, γtgr(Cn) = 4. Throughout, [n] = {1, 2, . . . , n} for each positive integer n. Lemma 3. Let G be a graph such that γ(G) ̸= 1. A sequence S is a co-legal open neighborhood sequence in G if and only if S is a legal open neighborhood sequence in G. Moreover, S is a co-Grundy total dominating sequence in G if and only if it is a Grundy total dominating sequence in G. In particular, γtcogr(G) = γtgr(G). Proof. Let S = (u1, · · · , ut) be a sequence in G. Notice that V (G) \NG[ui] = NG(ui) for each i ∈ [t]. So, [V (G) \NG[ui]] \ ∪i−1 j=1[V (G) \NG[uj ]] = NG(ui) \ ∪ i−1 j=1NG(uj). Hence, S is a co-legal open neighborhood sequence in G if and only if it is a legal open neighborhood sequence in G. Clearly, a co-legal open neighborhood sequence in G is a co- Grundy total dominating sequence if and only if it is a Grundy total dominating sequence in G. Consequently, γtcogr(G) = γtgr(G). Theorem 7. Let G and H be any two graphs such that γ(G) ̸= 1 and γ(H) ̸= 1. A sequence S of distinct vertices of G+H is a legal open hop neighborhood sequence if and only if one of the following holds: (i) S is a co-legal open neighborhood sequence in G (legal open neighborhood sequence in G). (ii) S is a co-legal open neighborhood sequence in H (legal open neighborhood sequence in H). (iii) S is loh-identical to S′ = SG⊕SH , where SG and SH are co-legal open neighborhood sequences in G and H, respectively. J.A. Hassan, S. Canoy / Eur. J. Pure Appl. Math, 16 (4) (2023), 2597-2612 2609 Proof. Suppose that S = (u1, · · · , ut) is a legal open hop neighborhood sequence in G +H and let Ŝ be the corresponding set of S. Suppose further that Ŝ ⊆ V (G). Then by the legality condition in S, we have N2 G+H(ui) \ ∪i−1 j=1N 2 G+H(uj) ̸= ∅ for each i ∈ {2, 3, . . . , t}. Since N2 G+H(ui) = V (G) \NG[ui] for each i ∈ [t], it follows that [V (G) \NG[ui]] \ ∪i−1 j=1[V (G) \NG[uj ]] ̸= ∅ for each i ∈ {2, 3, . . . , t}. Hence, S is a co-legal open neighborhood sequence in G, and so (i) holds. Similarly, (ii) holds if Ŝ ⊆ V (H). Next, suppose that Ŝ ∩ V (G) ̸= ∅ and Ŝ ∩ V (H) ̸= ∅. Since N2 G+H(uj) ⊆ V (G) for all uj ∈ Ŝ ∩V (G) and N2 G+H(us) ⊆ V (H) for all us ∈ Ŝ ∩V (H), S is loh-identical to S′ = SG ⊕ SH , where ŜG = Ŝ ∩ V (G) = {u1, u2, . . . , um}, ŜH = Ŝ ∩ V (H) = {w1, w2, . . . , wt}, |Ŝ| = m+t, and the orders of appearances of the terms of both sequences in S are retained. Since S is a legal open hop neighborhood sequence, it follows that [V (G) \NG[ui]] \ ∪i−1 j=1[V (G) \NG[uj ]] = N2 G+H(ui) \ ∪i−1 j=1N 2 G+H(uj) ̸= ∅ for each i ∈ {2, 3, . . . ,m}. Thus, SG is a co-legal open neighborhood sequence in G. Similarly, SH is a co-legal open neighborhood sequence in H, and so (iii) holds. The converse is clear. The next result follows from Lemma 3 and Theorem 7. Corollary 3. Let G and H be any two graphs such that γ(G) ̸= 1 and γ(H) ̸= 1. A sequence S of distinct vertices of G + H is a Grundy total hop dominating sequence in G+H if and only if S is loh-identical to S′ = SG ⊕SH , where SG and SH are co-Grundy total dominating sequences in G and H, respectively (Grundy total dominating sequences in G and H, respectively). Moreover, γthgr(G+H) = γtcogr(G) + γtcogr(H) = γtgr(G) + γtgr(H). In particular, we have (i) γthgr(Km,n) = γthgr(Km +Kn) = γtgr(Km) + γtgr(Kn) = 4 for any m,n ≥ 2, (ii) γthgr(Kn + Pm) = γtgr(Kn) + γtgr(Pm) = 6 for any n ≥ 2 and m ≥ 4, (iii) γthgr(Kn + Cm) = γtgr(Kn) + γtgr(Cm) = 6 for any n ≥ 2 and m ≥ 4, (iv) γthgr(Pn + Pm) = γtgr(Pn) + γtgr(Pm) = 8 for any n,m ≥ 4, (v) γthgr(Pn + Cm) = γtgr(Pn) + γtgr(Cm) = 8 for any n,m ≥ 4, and (vi) γthgr(Cn + Cm) = γtgr(Cn) + γtgr(Cm) = 8 for any n,m ≥ 4. J.A. Hassan, S. Canoy / Eur. J. Pure Appl. Math, 16 (4) (2023), 2597-2612 2610 Theorem 8. Let G be a non-trivial connected graph on n vertices and let H be any graph such that γ(H) ̸= 1. Then γthgr(G ◦H) ≥ n · γtcogr(H) = n · γtgr(H). Proof. Let V (G) = {u1, u2, . . . , un} and let Sui = (w1 ui , w2 ui , · · · , wk ui ) be a co-Grundy total dominating sequence in Hui for each i ∈ [n], where k = γtcogr(H). Let S = Su1 ⊕ Su2 ⊕ · · · ⊕ Sun . Let v ∈ V (G ◦H) \ Ŝ and let ut ∈ V (G) such that v ∈ V (ut +Hut) for some t ∈ [n]. Suppose first that v = ut. Let us ∈ NG(ut) and pick any wj us ∈ Ŝus for some s ∈ [n]. Then wj us ∈ Ŝ ∩N2 G◦H(ut). Suppose v ̸= ut. Then v ∈ V (Hut) \ Ŝut . Since Ŝut is a co-Grundy total dominating sequence in Hut , it follows that there exists wl ut ∈ Ŝut ⊆ Ŝ such that dHut (v, wl ut ) ̸= 1. It follows that dG◦H(v, wl ut ) = 2. Therefore, Ŝ is a total hop dominating set in G◦H. Now, we relabel the terms in S, say S = (v1, v2, · · · , vk, · · · , vnk). Let i ∈ [nk] \ {1} and let vi = wt ur for some r ∈ [n] and t ∈ [k]. Then N2 G◦H(vi) \ ∪i−1 j=1N 2 G◦H(vj) = N2 G◦H(wt ur ) \ [(∪t−1 s=1N 2 G◦H(ws ur )) ∪ (∪{N2 G◦H(wp uq ) : p ∈ [k] and 1 ≤ q ≤ r − 1})]. If t = 1, then N2 G◦H(wt ur ) \ (∪t−1 s=1N 2 G◦H(ws ur )) = N2 G◦H(wt ur ). Clearly, wt ur ∈ N2 G◦H(wt ur ) \ [∪{N2 G◦H(wp uq ) : p ∈ [k] and 1 ≤ q ≤ r − 1}]. Suppose t ̸= 1. Since Sur is a co-legal open neighborhood sequence in Hur , N2 G◦H(wt ur ) \ (∪t−1 s=1N 2 G◦H(ws ur )) = [V (Hur) \NHur (wt ur )] \ [∪t−1 s=1(V (Hur) \NHur (ws ur ))] ̸= ∅. Observe that N2 G◦H(wt ur ) \ (∪t−1 s=1N 2 G◦H(ws ur )) ∩ [∪{N2 G◦H(wp uq ) : p ∈ [k] and 1 ≤ q ≤ r − 1}] = ∅. Hence, N2 G◦H(vi)\∪i−1 j=1N 2 G◦H(vj) ̸= ∅ for all i ∈ [nk]\{1} and so S is a Grundy total hop dominating sequence in G ◦H. Consequently, γthgr(G ◦H) ≥ |Ŝ| = n∑ i=1 |Ŝvi | = n · γtcogr(H) = n · γtgr(H). Remark 4. The bound given in Theorem 8 is tight. To see this, consider the graph K5 ◦ P4 in Fig. 5. Let S = (a1, a2, · · · , a20). Then S is a Grundy total hop dominating sequence of K5 ◦ P4. Moreover, it can be verified that γthgr(K5 ◦ P4) = 20. Since γtgr(P 4) = 4, the assertion follows. REFERENCES 2611 K5 ◦ P4 : a2a1 a3 a4 a5 a6 a7 a8 a9 a10 a11 a12 a13 a14 a15 a16 a17 a18 a19 a20 Figure 5: A graph K5 ◦ P4 with γth gr(K5 ◦ P4) = |K5|γt gr(P 4). Acknowledgements The authors would like to thank the referees for the comments and suggestions they made in the initial manuscript. 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