EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 16, No. 1, 2023, 5-17 ISSN 1307-5543 – ejpam.com Published by New York Business Global Geodetic Hop Dominating Sets in a Graph Chrisley Jade C. Saromines 1,∗, Sergio R. Canoy, Jr.1 1 Department of Mathematics and Statistics, College of Science and Mathematics, Center for Graph Theory, Algebra and Analysis-PRISM, 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. A subset S of vertices of G is a geodetic hop dominating set if it is both a geodetic and a hop dominating set. The geodetic hop domination number of G, γhg(G), is the minimum cardinality among all geodetic hop dominating sets in G. Geodetic hop dominating sets in a graph resulting from some binary operations have been characterized. These characterizations have been used to determine some tight bounds for the geodetic hop domination number of each of the graphs considered. 2020 Mathematics Subject Classifications: 05C69 Key Words and Phrases: Geodetic domination, hop domination, corona, lexicographic 1. Introduction Frank Harary et al. in [10] introduced a graph theoretical parameter called geodetic number of a graph. Geodetic sets and geodetic numbers are studied further in Chartrand [7]. In 2011, H. Escuadro et al. (see [8]) introduced the concept of geodetic domination in graphs. After their introduction, more studies have been done on the concepts. Some of the studies dealing with geodetic sets, geodetic number, and geodetic dominating sets can be found in [4], [5], [6], [7], [8], [9], [10], [14], and [24]. The concept of hop domination in graphs was introduced and initially investigated by Natarajan and S. K. Ayyaswamy [19]. The study was then followed by numerous studies on the topic. In particular, a lot of variations of the concept have been introduced and studied (see [2], [3], [11], [12], [13], [15], [16], [18], [20], [21], [17], [22], and [23]). Recently, Anusha and Robin [1] introduced the concept of geodetic hop domination and studied it for join and corona of graphs. In this present paper, we revisit the concept of geodetic hop domination and give further results of the new parameter. ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v16i1.4646 Email addresses: chrisleyjade.saromines@g.msuiit.edu.ph (C.J. Saromines), sergio.canoy@g.msuiit.edu.ph (S. Canoy, Jr.) https://www.ejpam.com 5 © 2023 EJPAM All rights reserved. C.J. Saromines, S. Canoy, Jr., / Eur. J. Pure Appl. Math, 16 (1) (2023), 5-17 6 2. Terminology and Notation For any two vertices u and v in an undirected connected graph G, the distance dG(u, v) is the length of a shortest path joining u and v. Any u-v path of length dG(u, v) is called a u-v geodesic. The interval IG [u, v] consists u, v and all vertices lying on a u-v geodesic. The interval IG(u, v) = IG [u, v] \ {u, v}. The open neighborhood of a vertex u is the set NG(u) consisting of all vertices v which are adjacent to u. The closed neighborhood of u is NG[u] = NG(u) ∪ {u}. For any A ⊆ V (G), NG(A) = ⋃ v∈A NG(v) is called the open neighborhood of A andNG[A] = NG(A)∪A is called the closed neighborhood of A. The open hop neighborhood of a vertex u is the set N2 G(u) = {v ∈ V (G) : dG(v, u) = 2}. The closed hop neighborhood of u is N2 G[u] = N2 G(u) ∪ {u}. For any A ⊆ V (G), N2 G(A) = ⋃ v∈A N2 G(v) is called the open hop neighborhood of A and N2 G[A] = N2 G(A)∪A is called the closed hop neighborhood of A. A set S ⊆ V (G) is a dominating set in G if NG[S] = V (G). The smallest cardinality of a dominating set in G, denoted by γ(G) is called the domination number of G. The geodetic closure of a set S ⊆ V (G), denoted by IG [S], is the union of the intervals IG[u, v], where u, v ∈ S. Set S is geodetic set in G if IG[S] = V (G). The smallest cardinality among all geodetic sets in G, denoted by g(G), is called the geodetic number of G. A geodetic set of cardinality g(G) is called a g-set of G. A set S ⊆ V (G) is a geodetic dominating set in G if it is both a dominating and a geodetic set. A set S ⊆ V (G) is a hop dominating set if N2 G[S] = V (G). The minimum cardinality of a hop dominating set of a graph G, denoted by γh(G), is called the hop domination number 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 total hop dominating set of G with cardinality γth(G) is called a γth-set. A subset S of vertices of G is a geodetic hop dominating set if it is both a geodetic and a hop dominating set. The geodetic hop domination number γhg(G) of G is the minimum cardinality among all geodetic hop dominating sets in G. Any geodetic hop dominating set of G with cardinality γhg(G) is called a γhg-set. A set S ⊆ V (G) of a graph G is called a 2-path closure absorbing if for each x ∈ V (G)\S there exist u, v ∈ S such that dG(u, v) = 2 and x ∈ IG(u, v). The minimum cardinality of a 2-path closure absorbing set in G is denoted by ρ2(G). Any 2-path closure absorbing set of G with cardinality ρ2(G) is called a ρ2-set. A set D ⊆ V (G) is a pointwise non-dominating set of G if for each v ∈ V (G) \ S, there exists u ∈ S such that v /∈ NG(u). The smallest cardinality of a pointwise non-dominating set of G, denoted by pnd(G), is called the pointwise non-domination number of G. A pointwise non-dominating set S ⊆ V (G) of a graph G is called a 2-path closure absorbing pointwise non-dominating set if it is a 2-path closure absorbing set. The minimum cardinality of a 2-path closure absorbing pointwise non-dominating set in G is denoted by ρ2pnd(G). Any 2-path closure absorbing pointwise non-dominating set of G C.J. Saromines, S. Canoy, Jr., / Eur. J. Pure Appl. Math, 16 (1) (2023), 5-17 7 with cardinality ρ2pnd(G) is called a ρ2pnd-set. Let Kn be the complete graph of order n ≥ 3 and Ω a family of complete proper subgraphs of Kn. We say that Ω is an independent set if no two distinct subgraphs in Ω have common vertex. The graph G of order n obtained from Kn by deleting the edges in Ω is denoted Kn \ E(Ω). Hence, xy ∈ E(G) if and only if xy is not an edge in any subgraph in Ω. Let G and H be two graphs. The corona G ◦H is 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 ∈ V (G) and write v + Hv for ⟨v⟩ + Hv. The lexicographic product 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). 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. 3. Results Remark 1. Let G be a connected graph of order n. If S is a geodetic hop dominating set then S is a hop dominating set. In particular, γh(G) ≤ γhg(G). Theorem 1. Let G be a connected graph of order n ≥ 2. Then γhg(G) = n if and only if one of the following holds: (i) G = Kn (ii) G ̸= Kn and there exist dominating vertices v1, v2, ..., vk such that H1 = G \ v1, H2 = H1 \ v2, . . . , Hk−1 = Hk−2 \ vk−1 are connected graphs and Hk = Hk−1 \ vk is the union of atleast 2 complete components. Proof. Suppose γhg(G) = n and suppose that G ̸= Kn. Then there exist x, y ∈ V (G) such that dG(x, y) = 2. Let v1 ∈ NG(x) ∩ NG(y). Suppose there exists z ∈ V (G) \ {v1} such that v1z /∈ E(G). We may pick z so that dG(v1, z) = 2. Then V (G) \ {v1} is a geodetic hop dominating set of G, contrary to our assumption that γhg(G) = n. Thus, NG [v1] = V (G). Next, let H1 = G \ v1. Suppose that H1 is disconnected and suppose that H1 has a component H ′ 1 that is not complete. Then there exists s, t ∈ V (H ′ 1) such that d H ′ 1 (s, t) = dH1(s, t) = dG(s, t) = 2. Let r ∈ NG(s) ∩ NG(t). Then V (G) \ {r} is a geodetic hop dominating set of G, a contradiction. Therefore, all components of H1 are complete. Suppose H1 is connected. Suppose further that dH1(x, y) ≥ 3, say [x1, x2, ..., xk], where x1 = x and xk = y, be an x-y geodesic inH. Then V (G)\{x2} is a geodetic hop dominating set, a contradiction. Thus, dH(x, y) = 2. Let v2 ∈ NH1(x)∩NH2(y). Suppose there exists p ∈ V (H1) such that dH1(v2, p) = 2. Then V (G) \ {v2} is geodetic hop dominating set of G, a contradiction. Thus, NH1(v2) = V (H1) and NG [v2] = V (G). C.J. Saromines, S. Canoy, Jr., / Eur. J. Pure Appl. Math, 16 (1) (2023), 5-17 8 Continuing in this manner, there exists a finite sequence of dominating vertices v1, v2, ..., vk such that H1 = G \ v1, H2 = H1 \ v2, . . . , Hk−1 = Hk−2 \ vk−1 are connected graphs and Hk = Hk−1 \ vk is the union of at least 2 complete components. For the converse, suppose that G = Kn. Then, clearly, γgh(G) = n. Let v1, v2, ..., vk be dominating vertices such that H1 = G \ v1, ...,Hk−1 = Hk−2 \ vk−1 are connected and Hk = Hk−1 \ vk is the union of at least two complete graphs. Let S be a γhg-set of G. Then v1, v2, ..., vk ∈ S. Let v ∈ V (G) \ {v1, v2, ..., vk}. Suppose v ∈ S. Since S is a hop dominating set, there exists w ∈ S ∩ N2 G(v). Also, since S is a geodetic set, there exists p, q ∈ S such that [p, v, q] is a p-q geodesic. Since p, v and q are not dominating vertices, p, v, q ∈ V (Hk \ vk). It follows that the component of Hk \ vk containing p, v and q is not complete, contrary to our assumption. Therefore, v ∈ S. Accordingly, S = V (G) and γhg(G) = n. Proposition 1. Let n be a positive integer. (i) For a path Pn on n vertices , γhg(Pn) =  n, if n = 1, 2. n+6 3 , if n ≡ 0(mod3), n+2 3 , if n ≡ 1(mod3), n+4 3 , if n ≡ 2(mod3), (ii) For a cycle Cn on n vertices, γhg(Cn) =  3, if n = 3, 4, 5 n 3 , if n ≡ 0(mod3), n+2 3 , if n ≡ 1(mod3), n+4 3 , if n ≡ 2(mod3), Proof. (i) Let Pn = [v1, v2, ..., vn] and S be γhg-set of Pn. Since S is a geodetic set, v1, vn ∈ S. Consider the following cases: Case 1. n ≡ 0(mod3) Let n = 3r, for some positive integer r. Then S1 = {v1, v4, ..., v3r−2, v3r−1, v3r} and S2 = {v3r, v3r−3, ..., v3, v2, v1} are the only γhg-sets of Pn. Hence, γhg(Pn) = |S1| = n+6 3 . Case 2. n ≡ 1(mod3) Let n = 3t+ 1, for some non-negative integer t. Then S3 = {v1, v4, ..., v3t+1} is the unique γhg-set of Pn. Hence, γhg(Pn) = |S3| = n+2 3 . Case 3. n ≡ 2(mod3) Let n = 3s+ 2, for some non-negative integer s. Then S4 = {v1, v4, ..., v3s+1, v3s+2} and S5 = {v3s+2, v3s−1, ..., v2, v1} are the only γhg-sets of Pn. Hence, γhg(Pn) = |S4| = n+4 3 . C.J. Saromines, S. Canoy, Jr., / Eur. J. Pure Appl. Math, 16 (1) (2023), 5-17 9 (ii) Let Cn = [v1, v2, ..., vn, v1] and let D be γhg-set of Cn. By circularity property of Cn, we may assume that v1 ∈ D. Consider the following cases: Case 1. n ≡ 0(mod3) Let n = 3r for some positive integer r. Then D = {v1, v4, ..., v3r−2}. Hence, γhg(Pn) = |D| = n 3 . Case 2. n ≡ 1(mod3) Let n = 3t+ 1 for some non-negative integer t. Then D = {v1, v4, ..., v3t+1}. Hence, γhg(Cn) = |D| = n+2 3 . Case 3. n ≡ 2(mod3) Let n = 3s+ 1 for some non-negative integer s. Then D = {v1, v4, ..., v3s+1, v3s+2}. Hence, γhg(Cn) = |D| = n+4 3 . Remark 2. If Pn = [v1, v2, ..., vn] and S is a geodetic set of Pn, then {v1, vn} ⊆ S. Theorem 2. Let Kn be the complete graph of order n ≥ 3 and Ω an independent family of complete proper subgraph of Kn, each of order at least 2. If G = Kn \ E(Ω), then γhg(G) =  n, if |Ω| = 1 |Ω| − 2 + n− ∑ Kq∈Ω q +min {4, p+ 1} , if |Ω| ≥ 2 where p = min {q : Kq ∈ Ω} . Proof. Suppose that S is a γhg-set of G and let Ω = {Kp}. If Kp = Kn, then we are done. If Kp ̸= Kn, then there exist dominating vertices v1, v2, ..., vk such that H1 = G\v1, H2 = H1 \ v2, . . . , Hk−1 = Hk−2 \ vk−1 are connected graphs and Hk = Hk−1 \ vk is the union of at least 2 complete components. By Theorem 2, γhg(G) = n. Suppose |Ω| ≥ 2. Let D1 = V (G) \ ⋃ Kq∈Ω V (Kq) and let D2 be a smallest subset of S such that V (G) \ S ⊆ IG(D2). Since G is non-complete, there exist u, v ∈ S such that dG(u, v) = 2. Since Ω is an independent set, u, v ∈ V (Kq) for a unique Kq ∈ Ω. We may assume that u, v ∈ D2. Consider the following cases: Case 1. p < 4. Then V (G) \ S ⊆ IG(V (Kp)). If there exists w ∈ V (Kp) \ S, then there exist x, y ∈ S such that [x,w, y] is an x-y geodesic. It follows that x, y ∈ V (Kr) for some Kr ∈ Ω\{Kp}. Since u, v, x, y ∈ D2, |D2| ≥ 4 > p, a contradiction. Thus, V (Kp) ⊆ S and D2 = V (Kp). Next, let Kq ∈ Ω \ {Kp}. Since S is a hop dominating set of G, S ∩ V (Kq) ̸= ∅. Moreover, |S ∩ V (Kq)| = 1 since S is a γhg-set of G. Let S ∩ V (Kq) = {xq} for each Kq ∈ Ω \ {Kp}. Note that if D1 ̸= ∅, then D1 contains all the dominating vertices of G. Hence D1 ⊆ S. Therefore, S = D1 ∪D2 ∪ ⋃ q ̸=p {xq}  C.J. Saromines, S. Canoy, Jr., / Eur. J. Pure Appl. Math, 16 (1) (2023), 5-17 10 and γhg(G) = |S| = n− ∑ Kq∈Ω q + p+ |Ω| − 1 = n− ∑ Kq∈Ω q + p+ 1 + |Ω| − 2. Case 2. p ≥ 4. Suppose D2 = V (Kp). Then |S| = n − ∑ Kq∈Ω q + p + |Ω| − 1. Let Kt ∈ Ω \ {Kp}. Pick a, b ∈ V (Kt) with a ̸= b. Let D ′ = {u, v, a, b}. Then V (G) \ S ⊆ I(D ′ ). For each Kq ∈ Ω \ {Kp,Kt}, pick yq ∈ V (Kq). Let S ′ = D1 ∪ D ′ ∪ (⋃ q ̸=p,t {yq} ) . Then S ′ is a geodetic hop dominating set of G and ∣∣∣S′ ∣∣∣ = n− ∑ Kq∈Ω q + 4 + |Ω| − 2 < n− ∑ Kq∈Ω q + p+ |Ω| − 1. This is a contradiction to the above assumption. Thus, D ̸= V (Kp). Using the preceding arguments, we have γhg(G) = |S| = n− ∑ Kq∈Ω q + 4 + |Ω| − 2. This proves the assertion. Corollary 1. Let Kn be the complete graph of order n ≥ 4 and Ω an independent family of complete proper subgraphs of Kn, each of order at least 2. If K2 ∈ Ω and G = Kn \E(Ω), then γhg(G) = { n, if |Ω| = 1 |Ω|+ 1 + n− ∑ Kq∈Ω, if |Ω| ≥ 2. Corollary 2. Let Kn be the complete graph of order n ≥ 4. If G is a graph of order n obtained from Kn by deleting an edge, then γhg(G) = n The next result is a restatement of the one obtained in [15]. C.J. Saromines, S. Canoy, Jr., / Eur. J. Pure Appl. Math, 16 (1) (2023), 5-17 11 Theorem 3. Let G and H be any two graphs. A set C ⊆ V (G ◦H) is a hop dominating set of G ◦H if and only if C = A ∪ ( ∪v∈V (G)Sv ) , where A ⊆ V (G) and Sv ⊆ V (Hv) for each v ∈ V (G), and satisfies the following condi- tions: (i) For each w ∈ V (G) \ A, there exists x ∈ A with dG(w, x) = 2 or there exists y ∈ V (G) ∩NG(w) with Sy ̸= ∅. (ii) Sv ⊆ V (Hv) is a pointwise non-dominating set of Hv for each v ∈ V (G) \NG(A). Theorem 4. Let G and H be any two graphs. A set C ⊆ V (G ◦ H) is a geodetic hop dominating set of G ◦H if and only if C = A ∪ ( ∪v∈V (G)Sv ) , where A ⊆ V (G) and Sv ⊆ V (Hv) for each v ∈ V (G), and satisfies the following condi- tions: (i) Sv ⊆ V (Hv) is a pointwise non-dominating set of Hv for each v ∈ V (G) \NG(A). (ii) For each w ∈ V (G) \A, one of the following condition holds: (1) ∃ a, b ∈ Sw with dHw(a, b) ̸= 1. (2) ∃ x, y ∈ V (G) with w ∈ IG(x, y). (3) ∃ s ∈ Sw and t ∈ A. (iii) Sv is a 2-path closure absorbing set in Hv ∀v ∈ V (G). Proof. Suppose C is a geodetic hop dominating set of G ◦H. Let A = C ∩ V (G) and Sv = C ∩ V (Hv) for each v ∈ V (G). Since C is a geodetic set, Sv ̸= ∅ for each v ∈ V (G). By Theorem 3, (i) holds. Let w ∈ V (G) \ A. Since C is a geodetic set of G ◦H, at least one of the three statements in (ii) holds. Let v ∈ V (G). Let p ∈ V (Hv) \ Sv. Since C is a geodetic set, there exist s, t ∈ C such that p ∈ IG◦H(s, t). It follows that s, t ∈ Sv and dHv(s, t) = 2 and [s, p, t] is an s-t geodesic in Hv. Thus, Sv is a 2-path closure absorbing set in Hv, showing that (iii) holds. Conversely, suppose C satisfies the given conditions. Let v ∈ V (G) \A and choose any y ∈ NG(v). By assumption, Sy ̸= ∅. Hence, by Theorem 3, C is hop dominating set of G ◦H. Let z ∈ V (G ◦H) \ C and let w ∈ V (G) such that z ∈ V (w +Hw). Consider the following cases: Case 1. z = w. Then w ∈ V (G) \ A. Suppose condition (1) of (ii) holds. Then a, b ∈ C and z ∈ IG◦H(a, b). Suppose (2) holds. Let p ∈ Sx and q ∈ Sy. Then p, q ∈ C and z ∈ IG◦H(p, q). Next, suppose that (3) holds. Then s, t ∈ C and z ∈ IG◦H(s, t). Case 2. z ̸= w. C.J. Saromines, S. Canoy, Jr., / Eur. J. Pure Appl. Math, 16 (1) (2023), 5-17 12 Then z ∈ V (Hw) \ Sw. By (iii), Sw is a 2-path closure absorbing set in Hw; hence, there exists a, b ∈ Sw such that [a, z, b] is an a-b geodesic in Hw. Therefore, a, b ∈ C and [a, z, b] is an a-b geodesic in G ◦H. Accordingly, C is a geodetic hop dominating set of G ◦H. Corollary 3. Let G be a connected non-trivial graph on n vertices and let H be any noncomplete graph. Then γhg(G ◦H) = min {nρ2pnd(H), γ(H) + nρ2(H)} . Proof. For each v ∈ V (G), let Sv be a ρ2pnd-set of Hv. Then C = ∪v∈V (G)Sv is a geodetic hop dominating set of G◦H by Theorem 4. Next, let A be a γ-set of G. For each v ∈ V (G), let Tv be a ρ2 set of Hv. By Theorem 4, C ′ = A ∪ ( ∪v∈V (G)Tv ) is a geodetic hop dominating set of G ◦H. Thus, γhg(G ◦H) ≤ min { |C|, |C ′ | } = min {nρ2pnd(H), γ(H) + nρ2(H)} . Let Rv be a ρ2pnd-set of Hv. Let A1 = V (G) \ NG(A0) and A2 = NG(A0). Then C0 = A0 ∪ ( ∪v∈V (G)Rv ) is a geodetic hop dominating set of G ◦H by Theorem 4. Thus |C0| = |A0|+ ∑ v∈A1 |Rv|+ ∑ v∈A2 |Rv| ≥ |A0|+ |A1| ρ2pnd(H) + |A2| ρ2(H). Suppose γ(G) + nρ2(H) ≤ nρ2pnd(H). Then ρ2(H) < ρ2pnd(H), that is, ρ2(H) + 1 ≤ ρ2pnd(H). It follows that |C0| ≥ |A0|+ |A1| (ρ2(H) + 1) + |A2| ρ2(H) = |A0|+ |A1|+ |A1|+ |A2| ρ2(H) = |A0|+ |A1|+ nρ2(H) ≥ γ(G) + nρ2(H) since A0 ∪A1 is a dominating set of G and |A0 +A1| ≤ |A0|+ |A1| . Suppose nρ2pnd(H) < γ(G) + nρ2(H). Then ρ2pnd(H) = ρ2(H). Thus, |C0| ≥ |A0|+ |A1| ρ2pnd(H) + |A2| ρ2(H) = |A0|+ (|A1|+ |A2|) ρ2pnd(H) = |A0|+ nρ2pnd(H) ≥ nρ2pnd(H). Therefore, γhg(G ◦H) = |C0| ≥ min {nρ2pnd(H), γ(G) + nρ2pnd(H)} C.J. Saromines, S. Canoy, Jr., / Eur. J. Pure Appl. Math, 16 (1) (2023), 5-17 13 Accordingly, γhg(G ◦H) = min {nρ2pnd(H), γ(G) + nρ2pnd(H)} . Canoy et al. in [15] obtained the next result. Theorem 5. Let G and H be connected non-trivial graphs. A subset C = ⋃ x∈S [x × Tx] of V (G[H]), where S ⊆ V (G) and Tx ⊆ V (H) for each x ∈ S, is a hop dominating set of G[H] if and only if the following conditions hold: (i) S is a hop dominating set of G; (ii) Tx is a pointwise non-dominating set of H for each x ∈ S \N2 G(S). Theorem 6. Let G and H be connected non-trivial graphs. A subset C = ⋃ x∈S [x×Tx] of V (G[H]), where S ⊆ V (G) and Tx ⊆ V (H) for each x ∈ S, is a geodetic hop dominating set of G[H] if and only if the following conditions hold: (i) S is a geodetic hop dominating set of G, (ii) Tx is a pointwise non-dominating set of H for each x ∈ S \N2 G(S). (iii) Tx is a 2-path closure absorbing set of H for each x ∈ S \ IG(S). Proof. Suppose C is a geodetic hop dominating set of G [H]. By Theorem 5, S is a hop dominating set of G and (ii) holds. Suppose v ∈ V (G) \ S. Let a ∈ V (H). Since C is a geodetic set, there exist (x, p), (y, q) ∈ C such that (v, a) ∈ IG[H]((x, p), (y, q)). Then x, y ∈ S and v ∈ IG(x, y). This shows that S is a geodetic set of G, showing that (i) holds. Next, let x ∈ S \ IG(S). If Tx = V (H), then it is a 2-path closure absorbing set of H. Suppose Tx ̸= V (H) and let b ∈ V (H) \ Tx. Since (x, b) /∈ C and C is a geodetic set, there exist (u, k), (w, t) ∈ C such that (x, b) ∈ IG[H]((u, k), (w, t)). Since x ∈ S \ IG(S), u = w = x and dG[H]((u, k)(w, t)) = 2. Because (x, b) ∈ IG[H]((u, k), (w, t)), this would imply that dH(k, t) = 2 and x ∈ IH(k, t). This shows that Tx is a 2-path closure absorbing set of H, that is, (iii) holds. Conversely, suppose C satisfies (i), (ii) and (iii). By Theorem 5, S is a hop dominating set of G. Let (v, p) ∈ V (G[H]) \ C. Consider the following cases: Case 1. v /∈ S. Since S is a geodetic set of G, there exist u,w ∈ S such that v ∈ IG(u,w). Let [v1, v2, ..., vk], where v1 = u and vk = w, a u-w geodesic in G. Let v = vj where 1 < j < k. Let s ∈ Tu and t ∈ Tw. Then [(v1, s), (v2, p), ..., (vj , p), ..., (vk−1, p), (vk, t)] is (u, s)-(w, t) geodesic in G[H] containing (v, p). Case 2. v ∈ S. Then p /∈ Tv. If v ∈ IG(S), then following the arguments of Case 1, there exist (x, a), (y, b) ∈ C such that (v, p) ∈ IG[H]((x, a), (y, b)). Suppose v /∈ IG(S). By (iii), C.J. Saromines, S. Canoy, Jr., / Eur. J. Pure Appl. Math, 16 (1) (2023), 5-17 14 Tv is a 2-path closure absorbing set of H. This implies that there exists c, d ∈ Tv such that dH(c, d) = 2 and p ∈ IH(c, d). Hence, (v, c), (v, d) ∈ C and [(v, c), (v, p), (v, d)] is a (v, c)-(v, d) geodesic in G[H]. Therefore, C is a geodetic hop dominating set of G[H]. Corollary 4. Let G and H be connected non-trivial graphs. Then γhg(G[H]) ≤ γhg(G) ρ2pnd(H) Proof. Let S be a γhg-set of G and let D be a ρ2pnd-set of H. For each x ∈ S, let Tx = D. Then C = ⋃ x∈S ({x} × Tx) = S ×D is a geodetic hop dominating set by Theorem 6. Therefore, γhg(G [H]) ≤ |C| = |S| |D| = γhg(G)ρ2pnd(H). Remark 3. Strict inequality in Corollary 4 can be attained. Example 1. Consider the graph P3 [P3]. It can be verified that γhg(P3 [P3]) = 7 < 9 = γhg(P3)ρ2pnd(P3) . .................................... .................................... .................................... .................................... .................................... .................................... .................................... .................................... .................................... ......... ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ....... ......... ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ....... ......... ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ....... ............ ........... ........... ........... ........... ........... ........... ........... ........... ........... ........... ........... ........... ........ ................... .................. .................. .................. .................. .................. .................. .................. .................. .................. .................. .................. .................. ........... ........................................................................................................ ........................................................................................................ ........................................................................................................ ........................................................................................................ ........................................................................................................ ................................................................................................................................................................................................................................................................ ............ ........... ........... ........... ........... ........... ........... ........... ........... ........... ........... ........... ........... .............................................................................................................................................................................................................................................................. ........................................................................................................................................................ ......... ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ....... ......... ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ....... ......... ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ....... ........................................................................................................................................................ ...................................................................................................................................................................................................................................................... ........................................................................................................ ................................................................................................................................................................................................................ ........................................................................................................ ........................................................................................................ .................................................................................................................... ........... ........... ........... ........... ........... ........... ........... ........... ........... ........... ........... ........... ........ ........................................................................................................................................................................... .................. .................. .................. .................. .................. .................. .................. .................. .................. .................. .................. .................. ........... ............ ........... ........... ........... ........... ........... ........... ........... ........... ........... ........... ........... ........... ........ Figure 1: The lexicographic product P3[P3] Corollary 5. Let n ≥ 2 be a positive integer and let H be any connected non-trivial graph. Then γhg (Kn [H]) = nρ2pnd(H) . Proof. Let C = ⋃ x∈S ({x} × Tx) be a γhg-set of Kn [H]. Then S = V (Kn) by Theorem 6(i). Also, by (ii) and (iii) of Theorem 6, Tx is a pointwise non-dominating and 2-path C.J. Saromines, S. Canoy, Jr., / Eur. J. Pure Appl. Math, 16 (1) (2023), 5-17 15 closure absorbing set of H for every x ∈ S. Thus, γhg (Kn [H]) = |C| = ∑ x∈S |Tx| ≥ |S| ρ2pnd(H) = nρ2pnd(H) By Corollary 4, γhg (Kn [H]) = nρ2pnd(H). Example 2. Consider the graph K3 [P4]. It can be verified that γhg(K3 [P4]) = 9 = γhg(K3)ρ2pnd(P4) . .................................... .................................... .................................... .................................... .................................... .................................... .................................... .................................... .................................... .................................... .................................... .................................... ......... ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ... ......... ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ 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Figure 2: The lexicographic product K3[P4] 4. Conclusion This paper investigated the concept of geodetic hop domination, a variant of hop domination, which was definedand studied previously by some authors. Some bounds of the parameter are determined and graphs attaining these bounds are also characterized. Characterizations of geodetic hop dominating sets in the corona and lexicographic product of two graphs are given. These characterizations were used to ob- tain exact or tight bounds for the geodetic hop domination number of the corresponding graphs. It is recommended that some other bounds for the geodetic hop domination be determined and that the parameter be studied for other interesting graphs. Acknowledgements The authors are very much grateful to the referees for the corrections and suggestions they made in the initial manuscript. 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