EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 17, No. 3, 2024, 1618-1636 ISSN 1307-5543 – ejpam.com Published by New York Business Global Closed Geodetic Hop Domination in Graphs Niña Jeane A. Adolfo1,2,∗, Imelda S. Aniversario1,2, Ferdinand P. Jamil1,2 1 Department of Mathematics and Statistics, College of Science and Mathematics 2 Center for Mathematical and Theoretical Physical Sciences, Premier Research Institute of Science and Mathematics MSU-Iligan Institute of Technology, 9200 Iligan City, Philippines Abstract. Let G be a simple, undirected and connected graph. A subset S ⊆ V (G) is a geodetic cover of G if IG[S] = V (G), where IG[S] is the set of all vertices of G lying on any geodesic between two vertices in S. A geodetic cover S of G is a closed geodetic cover if the vertices in S are sequentially selected as follows: Select a vertex v1 and let S1 = {v1}. If G is nontrivial, select a vertex v2 ̸= v1 and let S2 = {v1, v2}. Where possible, for i ≥ 3, successively select vertex vi /∈ IG[Si−1] and let Si = {v1, v2, ..., vi}. Then there exists a positive integer k such that Sk = S. A geodetic cover S of G is a geodetic hop dominating set if every vertex in V (G) \S is of distance 2 from a vertex in S. A geodetic hop dominating set S is a closed geodetic hop dominating set if S is a closed geodetic cover of G. The minimum cardinality of a (closed) geodetic hop dominating set of G is the (closed) geodetic hop domination number of G. This study initiates the study of the closed geodetic hop domination. First, it characterizes all graphs G of order n whose closed geodetic hop domination numbers are 2 or n, and determines the closed geodetic hop domination number of paths, cycles and multigraphs. Next, it shows that any positive integers a and b with 2 ≤ a ≤ b are realizable as the closed geodetic number and closed geodetic hop domination number of a connected graph. Also, every positive integer n,m and k with 4 ≤ m ≤ k and 2k−m+2 ≤ n are realizable as the order, geodetic hop domination number and closed geodetic hop domination number, respectively of a connected graph. Furthermore, the study characterizes the closed geodetic hop dominating sets of graphs resulting from the join, corona and edge corona of graphs. 2020 Mathematics Subject Classifications: 05C69 Key Words and Phrases: Closed geodetic cover, hop dominating set, geodetic hop dominating set, closed geodetic hop dominating set, closed geodetic hop domination number, join, corona, edge corona ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v17i3.5241 Email addresses: ninajeane.adolfo@g.msuiit.edu.ph (N. Adolfo), imelda.aniversario@g.msuiit.edu.ph (I. Aniversario), ferdinand.jamil@g.msuiit.edu.ph (F. Jamil) https://www.ejpam.com 1618 © 2024 EJPAM All rights reserved. A. Adolfo, I. Aniversario, F. Jamil / Eur. J. Pure Appl. Math, 17 (3) (2024), 1618-1636 1619 1. Introduction F. Harary in [9] introduced two categories of graphical games called the achievement and avoidance games from which the concept of closed geodetic number evolved. The closed geodetic sets and closed geodetic numbers of connected graphs, which can find applications in location theory and convexity theory, have been extensively studied in [1, 2, 6, 7]. The hop domination in graphs is introduced in [14] by S.K. Ayyaswamy and C.Natarajan. Accordingly, this graph theoretic concept originated from the second electron affinity in Inorganic Chemistry. It has attracted relatively much attention and several further studies including investigations on some of its variations can be found in the existing literature (see [4, 5, 12–17]). In this present paper, inspired by the above-mentioned concepts, we introduce and initiate the study of closed geodetic hop domination in graphs. All graphs considered in this study are simple, undirected and connected. All graph terminologies which are not defined but are used here are adopted from [6]. As usual, we write G = (V (G), E(G)) for a graph G where V (G) and E(G) are the vertex set and edge set, respectively, of G. For S ⊆ V (G), |S| is the cardinality of S. In particular, |V (G)| is the order of G. Let G and H be two graphs with disjoint vertex sets. 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 ◦H of G and 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. The edge corona G ⋄ H of G and H is the graph obtained by taking one copy of G and |E(G)| copies of H and joining each of the end vertices u and v of each edge uv of G to every vertex of the copy Huv of H. For vertices u and v in G, the distance dG(u, v) between u and v is the length of a shortest path in G joining u and v. Any path joining u and v of length dG(u, v) is called a u-v geodesic. The diameter diam(G) of a graph G is the length of any longest geodesic of G. For every two vertices u and v of a graph G, the interval IG[u, v] refers the set of all vertices lying in some u-v geodesic. A vertex is called an end-vertex or a leaf if its degree is 1. The set of all end-vertices of G is denoted by L(G). A vertex v in a connected graph G is an support vertex if v is adjacent to a leaf vertex of G. A vertex v in a connected graph G is an extreme vertex if for every pair of distinct vertices u and w with {uv,wv} ⊆ E(G), uw ∈ E(G). The set of all extreme vertices in G is denoted by Ext(G). A vertex v in a connected graph G is a dominating vertex if uv ∈ E(G) for all u ∈ V (G) \ {v}. Dom(G) is the set of all dominating vertices in G. For S ⊆ V (G), the 2-path closure P2[S]G of S is the set P2[S]G = S∪{w ∈ V (G) : w ∈ IG[u, v] for some u, v ∈ S with dG(u, v) = 2}. A set S is called 2-path closure absorbing if P2[S]G = V (G) [7]. We denote by ρ2(G) the minimum cardinality of a 2-path closure A. Adolfo, I. Aniversario, F. Jamil / Eur. J. Pure Appl. Math, 17 (3) (2024), 1618-1636 1620 absorbing set of G. A set S ⊆ V (G) is a pointwise non-dominating set of G if for each v ∈ V (G) \ S, there exists u ∈ S such that uv /∈ E(G). A pointwise non-dominating set S ⊆ V (G) of a graph G is 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) A clique in G is a complete subgraph of G. A maximal clique is a clique which is not a proper subgraph of a larger clique. The lower clique number ωL(G) is the minimum size of all maximal cliques of G. For S ⊆ V (G), the geodetic closure IG[S] is the union of intervals between all pairs of vertices from S, that is, IG[S] = ⋃ {IG[u, v] : u, v ∈ S}. S is a geodetic set provided IG[S] = V (G). The minimum cardinality gn(G) of a geodetic set is the geodetic number of G. A geodetic set of cardinality gn(G) is a geodetic basis. The introduction and further studies on geodetic sets and geodetic numbers can be found in [6–10]. A geodetic set S of G is a closed geodetic cover of G if S is obtained as follows: Choose v1 ∈ V (G) and put S1 = {v1}. Where possible, choose v2 ∈ V (G) \ {v1} and put S2 = {v1, v2}. For i ≥ 3, choose vi ∈ V (G) \ IG[Si−1], where Sk = {v1, v2, . . . , vi}, and there exists a positive integer k for which Sk = S. The closed geodetic number of G, denoted cgn(G), is the smallest positive integer k for which IG[Sk] = V (G), where Sk is obtained as illustrated above. If C∗(G) is the collection of all closed geodetic covers of G, then cgn(G) = min{|S| : S ∈ C∗(G)}. Any set S ∈ C∗(G) with |S| = cgn(G) is a closed geodetic basis 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. Let G be a connected graph. A set S ⊆ V (G) is a hop dominating set of G if for every v ∈ V (G) \ S, there exists u ∈ S such that dG(u, v) = 2. In particular, 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 G, denoted by γh(G), is called the hop domination number of G. Any hop dominating set with cardinality equals to γh(G) is called a γh-set of G. A subset S of vertex set 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 of G. The geodetic hop dominating set was first introduced by Anusha et al. in [3]. It is further investigated by Saromines et al. in [18, 19]. The following results concerning geodetic hop domination on paths and cycles are found in [19]. Proposition 1. [19] Let n be a positive integer. A. Adolfo, I. Aniversario, F. Jamil / Eur. J. Pure Appl. Math, 17 (3) (2024), 1618-1636 1621 (i) For a path Pn on n vertices, γhg(Pn) =  n if n = 1, 2, n+6 3 if n ≡ 0(mod 3) , n+2 3 if n ≡ 1(mod 3) , n+4 3 if n ≡ 2(mod 3) (ii) For a cycle Cn on n vertices, γhg(Cn) =  3 if n = 3, 4, 5, n 3 if n ≡ 0(mod 3) , n+2 3 if n ≡ 1(mod 3) , n+4 3 if n ≡ 2(mod 3) . 2. Results In this section, we introduce and initiate the study of closed geodetic hop domination in graphs. 2.1. Closed Geodetic Hop Domination A subset S of vertices of G is a closed geodetic hop dominating set if it is both a geodetic hop dominating set and a closed geodetic cover of G. The minimum cardinality among all closed geodetic hop dominating sets in G, denoted by γhcg(G) is called the closed geodetic hop domination number of G. A closed geodetic hop dominating set S of G with |S| = γhcg(G) is called a γhcg-set of G. We remark that not every graph admits a closed geodetic hop dominating set. Con- sider, for example, the graph G = C12. It is easy to verify that every closed geodetic set of G is not a hop dominating set. Observation 1. Let G be a connected graph. If V (G) is a closed geodetic set, then G admits a closed geodetic hop dominating set. The following also provides some other conditions under which a graph admits a closed geodetic hop dominating set. Proposition 2. If G is a connected graph with diam(G) ≤ 2, then G admits a closed geodetic hop dominating set. Proof. If diam(G) = 1, then G is complete so that V (G) is a closed geodetic set. As observed above, G admits a closed geodetic hop dominating set. Suppose diam(G) = 2. Then G is not complete. Let G1 be a maximal clique of G. Let V (G1) = Sr = A. Adolfo, I. Aniversario, F. Jamil / Eur. J. Pure Appl. Math, 17 (3) (2024), 1618-1636 1622 {u1, u2, ..., ur}. Since G is not complete, r < |V (G)|. For i ≥ 1, choose ur+i ∈ V (G) such that ur+i /∈ IG[Sr+i−1] where Sr+i−1 = {u1, u2, ..., ur, ur+1, ..., ur+i−1}. Since V (G) is finite, there exists a smallest positive integer n > r for which IG[Sn] = V (G). If V (G) \ Sn = ∅ then Sn = V (G). Suppose V (G) \ Sn ̸= ∅, and let a ∈ V (G) \ Sn. By maximality of G1, there exists b in Sr such that dG(a, b) = 2. Hence Sn is a hop dominating set of G. Therefore, Sn is a closed geodetic hop dominating set of G. We denote by C ∗ h the family of all connected graphs that admit a closed geodetic hop dominating set. Since closed geodetic hop dominating sets are themselves geodetic hop dominating sets, 2 ≤ γhg(G) ≤ γhcg(G) ≤ n (1) for all graphs G ∈ C ∗ h . Theorem 2. Let G ∈ C ∗ h . Then γhcg(G) = 2 if and only if either G = K2 or G has a geodetic set S = {u, v} such that dG(u, v) = 3. Proof. Suppose that γhcg(G) = 2. If G = K2, then we are done. Suppose that G ̸= K2. Let S = {u, v} be a closed geodetic hop dominating set of G. Since G ̸= K2, V (G) \ S ̸= ∅ and w ∈ IG(u, v) for every w ∈ V (G) \ S. Let [u = x1, x2, x3, . . . , xk = v] be a u-v geodesic in G. Then k ≥ 3. Since S is a hop dominating set, in particular, dG(x2, v) = 2. Necessarily, k = 4 and dG(u, v) = 3. Clearly, if G = K2, then γhcg(G) = 2. Suppose that G has a geodesic set S = {u, v} with dG(u, v) = 3. Then S is a closed geodetic set of G. Let w ∈ V (G) \ S. Being a geodetic set, there exists a u-v geodesic [u, x, y, v] on which w lies. If w = x, then dG(w, v) = 2. If w = y, then dG(u,w) = 2. Accordingly, S is a hop dominating set of G. Thus, γhcg(G) ≤ |S| = 2. Equation 1 completes the desired equality. Lemma 1. Let G ∈ C ∗ h of order n. If γhcg(G) = n, then G has a dominating vertex. Proof. This is clear for n = 1, 2. Let n ≥ 3. Assume that γhcg(G) = n. Suppose that G does not contain a dominating vertex. The assumption implies that there exists a sequence of sets of vertices of G, say Sk = {v1, v2, . . . , vk}, k = 1, 2, . . . , n, such that v1 ̸= v2 and vk /∈ IG[Sk−1] for all k ≥ 3. Now, since G is not complete and n ≥ 3, G has vertices u and v such that dG(u, v) = 2. Let [u,w, v] be a u-v geodesic in G. For some distinct i, j, k ∈ {1, 2, . . . , n}, we have u = vi, w = vj and vk = v. Without loss of generality, assume i < k. Since w ∈ IG[u, v] and vj /∈ IG[Sj−1], j < k. Define Tl = {x1, x2, . . . , xl} for l = 1, 2, . . . ,m with k ≤ m ≤ n − 1 such that • xl = vl for all l ∈ {1, 2, . . . , j − 1}; • xl = vl+1 for all l ∈ {j, j + 1, . . . ,m}. A. Adolfo, I. Aniversario, F. Jamil / Eur. J. Pure Appl. Math, 17 (3) (2024), 1618-1636 1623 Then • IG[Tl] = IG[Sl] = Sl = Tl for all l ∈ {1, 2, . . . , k − 2}; • IG[Tk−1] = Tk−1 ∪ {w} = Sk; and • IG[Tl] = Sl+1 for all l ∈ {k, k + 1, . . . ,m}. This means that Tm = V (G) \ {w} is a closed geodetic set of G. Finally, since w is not a dominating vertex of G, there exists z ∈ V (G) \ {w} = Tm such that dG(w, z) = 2. Thus, Tm is a hop dominating set of G. Hence, γhcg(G) ≤ |Tm| = m < n, a contradiction. Therefore, G contains a dominating vertex. If G ∈ C ∗ h , then a closed geodetic hop dominating set of G contains the extreme vertices. It also contains all dominating vertices. Theorem 3. Let G ∈ C ∗ h of order n. Then γhcg(G) = n if and only if either (i) G = Kn; or (ii) G ̸= Kn such that the set S of dominating vertices is nonempty and each of the components of ⟨V (G) \ S⟩ is complete. Proof. If G = Kn, then V (G) is the unique closed geodetic hop dominating set of G. Thus, γhcg(G) = n. Suppose that G ̸= Kn. First, assume γhcg(G) = n. By Lemma 1, the set S of dominating vertices of G is nonempty. Let C be a component of ⟨V (G) \ S⟩. We claim that C is complete. Let x ∈ V (C) and let u, v ∈ NC(x). Suppose, to the contrary, that uv /∈ E(C). Following a similar proof to that of Lemma 1, T = V (G)\{x} is a closed geodetic hop dominating set of G, a contradiction. Thus, uv ∈ E(G), showing that C is complete. Conversely, suppose that G is as described in condition (ii). Let T ⊆ V (G) be a closed geodetic hop dominating set of G. By the preceding remark, S ⊆ T . Let C be a component of G∗ = ⟨V (G) \ S⟩. Let x ∈ V (C) and u, v ∈ NG(x). If u, v ∈ V (C), then uv ∈ E(G) since C is complete. Suppose that u /∈ V (C). Then u ∈ S, i.e., u is a dominating vertex in G. Thus, uv ∈ E(G). This shows that x ∈ Ext(G) ⊆ T . Thus, V (C) ⊆ T . Since C is arbitrary, V (G) = S ∪ (∪C component of G∗ V (C)) = T. Since T is arbitrary, γhcg(G) = |V (G)| = n. The star graph K1,n is an example of the infinite family of graphs described in Theorem 3(ii). A. Adolfo, I. Aniversario, F. Jamil / Eur. J. Pure Appl. Math, 17 (3) (2024), 1618-1636 1624 2.2. For Paths Pn, Cycles Cn and Multipartite Graphs Since every geodetic hop dominating set of Pn is a closed geodetic hop dominating set of Pn, we have the following: Proposition 3. For a path Pn on n vertices, γhcg(Pn) =  n if n = 1, 2, n+6 3 if n ≡ 0(mod 3) , n+2 3 if n ≡ 1(mod 3) , n+4 3 if n ≡ 2(mod 3) Proposition 4. A cycle graph Cn of order n admits a closed geodetic hop dominating set if and only if n < 12. Moreover precisely, γhcg(Cn) =  3 if n = 3, 4, 5 n 3 if n = 6, 9 n+2 3 if n = 7, 10 n+4 3 if n = 8, 11 (2) Proof. The case where 3 ≤ n ≤ 11 can be readily verified. Suppose that n ≥ 12. Let P = [x1, x2, ..., xk], k = ⌈n2 ⌉, be a path in Cn and v ∈ V (Cn) \ V (P ). Then M = {x1, x2, ..., xk, v} is a closed geodetic cover of Cn with |M | = ⌈n2 ⌉ + 1. Since n ≥ 12, |V (Cn)\V (P )| ≥ 6. Thus, Cn has at least 2 adjacent vertices which are not hop dominated by V (P ). Consequently, Cn has at least one vertex which is not hop dominated by M . This means that M is not a hop dominating set of Cn (see, for example, Figure 1). We claim that every closed geodetic cover S of Cn is contained in a closed geodetic cover M of Cn as constructed above with |M | = ⌈n2 ⌉+ 1. Let S = Sk = {v1, v2, ..., vk} be a closed geodetic cover of Cn. Note that here, IG[Sj ] ̸= V (Cn) for all j ∈ {1, ..., k − 1} and IG[Sk] = V (Cn). If k = ⌈n2 ⌉+1, then by relabelling of vertices where necessary, Sk is the desired M . If k = 2, then dCn(v1, vk) = ⌈n2 ⌉. Take M = {x1, x2, ..., xj}, j = ⌈n2 ⌉ + 1 where P = [x1, x2, ..., xj ] is a v1-vk geodesic in Cn. Then M is a closed geodetic cover of Cn with |M | = ⌈n2 ⌉+1 and S ⊆ M . Now assume 2 < k < ⌈n2 ⌉+1. Choose v ∈ V (Cn) and a v-vk−1 geodesic P = [x1, x2, ..., xm], where m = ⌈n2 ⌉ such that dCn(v, vk−1) = ⌈n2 ⌉ − 1 and vk /∈ V (P ). Define M = {x1, x2, ..., xm, vk}. Consequently, M is a closed geodetic cover of Cn with |M | = ⌈n2 ⌉ + 1 as described above. Now, let j ∈ {1, 2, ..., k − 1}. Then dCn(vj , vk−1) ≤ ⌈n2 ⌉ − 1. By the choice of P , vj ∈ V (P ). Thus, S ⊆ M . Therefore, being a subset of a non-hop dominating set, any closed geodetic set S is not a hop dominating set of Cn. Thus, Cn does not admit a closed geodetic hop dominating set. A. Adolfo, I. Aniversario, F. Jamil / Eur. J. Pure Appl. Math, 17 (3) (2024), 1618-1636 1625 .................................... ........................................................................ .................................... .................................... .................. ................. . ...... .............................. .................................... .................................... ........... .......... .......... ..... .................................... .................................... .................................... ......... ........ ........ ........ ... .................................... .......... ......... ......... ........ .................................... .................................... .................................... ............. ............ ........... .................................... .................................... ...................................................................... .. ........................................................................ ........................................................................•• • • • • • • .................................... ........................................................................ .................................... .................................... .................. ................. . ...... .............................. .................................... .................................... ........... .......... .......... ..... .................................... .................................... .................................... ......... ........ ........ ........ ... .................................... .......... ......... ......... ........ .................................... .................................... .................................... ............. ............ ........... .................................... .................................... ...................................................................... .. ........................................................................ ........................................................................•• • • • • • • .................................... ........................................................................ .................................... .................................... .................. ................. . ...... .............................. .................................... .................................... ........... .......... .......... ..... .................................... .................................... .................................... ......... ........ ........ ........ ... .................................... .......... ......... ......... ........ .................................... .................................... .................................... ............. ............ ........... .................................... .................................... ...................................................................... .. ........................................................................ ........................................................................•• • • • • • • .................................... ........................................................................ .................................... .................................... .................. ................. . ...... .............................. .................................... .................................... ........... .......... .......... ..... .................................... .................................... .................................... ......... ........ ........ ........ ... .................................... .......... ......... ......... ........ .................................... .................................... .................................... ............. ............ ........... .................................... .................................... ...................................................................... .. ........................................................................ ........................................................................•• • • • • • • .................................... ........................................................................ .................................... .................................... .................. ................. . ...... .............................. .................................... .................................... ........... .......... .......... ..... .................................... .................................... .................................... ......... ........ ........ ........ ... .................................... .......... ......... ......... ........ .................................... .................................... .................................... ............. ............ ........... .................................... .................................... ...................................................................... .. ........................................................................ ........................................................................•• • • • • • • Figure 1: Cycle graph C13 illustrating the first part of proof of Proposition 4 Proposition 5. Let p ≥ 2, 2 ≤ n1 ≤ n2 ≤ ... ≤ np and G = Kn1,n2,...,np with partite sets Uni, i = 1, 2, . . . , p. Then S ⊆ V (G) is a closed geodetic hop dominating set of G if and only if for some i, S = Uni ∪ ( ∪p k=1;k ̸=i{xnk } ) , (3) where xnk ∈ Unk . Consequently, γhcg(Kn1,n2,...,np) = n1+p−1. In particular, γhcg(Km,n) = 1 +min{m,n} for m,n ≥ 2. Proof. Clearly, if S ⊆ V (G) satisfies Equation 3, then S is a closed geodetic hop dominating set of G. Conversely, let S be a closed geodetic hop dominating set of G. Since S is a hop dominating set, S ∩ Unj ̸= ∅ for all j = 1, 2, ..., p. Since S is a closed geodetic set, Uni ⊆ S for some i and |S ∩ Unk | = 1 for all k ̸= i. The remaining statements follow immediately. 2.3. Realization Problems Theorem 4. Let a and b be positive integers such that 2 ≤ a ≤ b. Then there exists a connected graph G such that cgn(G) = a and γhcg(G) = b. Proof. Let m = b − a + 1. Consider the tree G in Figure 2 below obtained from the P3m = [y1, y2, . . . , y3m] on 3m vertices by adding (a − 1) pendant edges xky1, k = 1, 2, . . . , a− 1. x1 x2 x3 xa−1 y1 y2 y3 y4 y5 y6 y3(m−1) y3m−2 y3m−1 y3m G : Figure 2: Graph G complying with the specifications of Theorem 4 A. Adolfo, I. Aniversario, F. Jamil / Eur. J. Pure Appl. Math, 17 (3) (2024), 1618-1636 1626 Then set Ext(G) = {x1, x2, ..., xa−1, y3m} is a closed geodetic basis of G. Hence cgn(G) = a − 1 + 1 = a. On the other hand, the set {x1, x2, ..., xa−1, y3, y6, ..., y3m} is a γhcg-set of G. Hence γhcg(G) = a− 1 +m = b. Theorem 5. If n, m, and k are integers with 4 ≤ m ≤ k and 2k−m+ 2 ≤ n, then there exists a connected graph G such that |V (G)| = n, γhg(G) = m and γhcg(G) = k. Proof. Let r = k − m + 3 and s = n − k + 1. Let U = {u1, u2, . . . , ur} and W = {v1, v2, . . . , vs} be the partite sets ofKr,s. ObtainG as in Figure 3 by adding toKr,s (m−4) new pendant edges wjv1, j = 1, 2, . . . ,m − 4. Then |V (G)| = r + s + (m − 4) = n. The v1 v2 v3 v4 vs−2 vs−1 vs u1 u2 u3 ur−1 ur w1 w2 w3 wm−4 Figure 3: Graph G complying with the specifications of Theorem 5 vertices w1, w2, ..., wm−4 are extreme vertices, thus are in any geodetic cover of G. Since the set {w1, w2, ..., wm−4, u1, ur, v1, vs} is a γhg-set of G, it follows that γhg(G) = m. Since the set {v1, w1, w2, ..., wm−4, u1, ..., ur} is a γhcg-set of G, we have γhcg(G) = 1+m−4+r = 1 +m− 4 + k −m+ 3 = k. 2.4. In the Join of Graphs Since diam(G+H) ≤ 2, G+H ∈ C ∗ h for any graphs G and H. A set S ⊆ V (G) is a closed 2-path closure absorbing set of G if P2[S] = V (G) and S = Sk = {v1, v2, ..., vk} where v1 ̸= v2 and vi /∈ P2[Si−1] for 3 ≤ i ≤ k. The minimum cardinality of a closed 2-path closure absorbing set in G is denoted by ρc2(G). A 2-path closure absorbing set of G with cardinality ρc2(G) is called ρc2-set. A set S ⊆ V (G) is a closed 2-path closure absorbing pointwise non-dominating set of G provided S is a closed 2-path closure absorbing set and at the same time pointwise non-dominating set of G. The minimum cardinality of a closed 2-path closure absorbing pointwise non- dominating set of G is denoted by ρc2pnd(G). A closed 2-path closure absorbing pointwise non-dominating set of G with cardinality ρc2pnd(G) is called ρc2pnd-set. A. Adolfo, I. Aniversario, F. Jamil / Eur. J. Pure Appl. Math, 17 (3) (2024), 1618-1636 1627 Since any closed 2-path closure absorbing pointwise non-dominating set is a 2-path closure absorbing pointwise non-dominating set and a closed 2-path closure absorbing set, ρ2pnd(G) ≤ ρc2pnd(G) and ρc2(G) ≤ ρc2pnd(G)for all connected graphs. Example 1. Consider the graph K5,6 in Figure 4, the sets {v1, v2, v3, v4, v5}, {v1, v3, u1, u2}, and {u1, v1, v2, v3, v4, v5} are ρ2c-set, ρ2pnd-set and ρc2pnd-set of K5,6, respectively. There- fore, ρc2(K5,6) = 5 ρ2pnd(K5,6) = 4 and ρc2pnd(K5,6) = 6. K5,6 : v1 v2 v3 v4 v5 u1 u2 u3 u4 u5 u6 Figure 4: The bipartite graph K5,6 Observation 6. Let n be a positive integer. Then (i) ρc2pnd(Kn) = n and ρc2(Kn) = n; (ii) ρc2pnd(Pn) = { n if n = 1, 2, 3, ⌈n+1 2 ⌉ if n ≥ 4, and ρc2(Pn) = { 2 if n = 3, ⌈n+1 2 ⌉ if n ≥ 4; (iii) ρc2pnd(Cn) = { 3 if n = 3, 4, ⌈n2 ⌉ if n ≥ 5 and ρc2(Cn) = { 3 if n = 3, ⌈n2 ⌉ if n ≥ 4; (iv) ρc2pnd(Km,n) = { m+ n if m = 1 or n = 1 min{m,n}+ 1 if m,n ≥ 2 Lemma 2. [11] Let G be a connected noncomplete graph, and let S ⊆ V (G). If S is a 2-path closure absorbing set of G, then ⟨S⟩ is not complete. Theorem 7. Let G be a noncomplete connected graph and n ≥ 1. Then S ⊆ V (G+Kn) is a closed geodetic hop dominating set of G+Kn if and only if S = V (Kn) ∪ C, where C ⊆ V (G) and is a closed 2-path closure absorbing pointwise non-dominating set in G. A. Adolfo, I. Aniversario, F. Jamil / Eur. J. Pure Appl. Math, 17 (3) (2024), 1618-1636 1628 Proof. Let S ⊆ V (G + Kn). Suppose that S is a closed geodetic hop dominat- ing set of G + Kn, say S = Sk = {x1, x2, . . . , xk} with x1 ̸= x2 and for k ≥ 3, xk /∈ IG+Kn [Sk−1]. Since V (Kn) ⊆ Dom(G + Kn), V (Kn) ⊆ S. Let j = |S ∩ V (H)|. Write Ai = {xn1 , xn2 , . . . , xni} for i = 1, 2, . . . , j such that n1 < n2 < · · · < nj . First, we claim that C = Aj is a closed 2-path closure absorbing set of G. Suppose that for some 3 ≤ l ≤ j, xnl ∈ P2[Al−1]. This means that there exist r < s < l such that [xnr , xnl , xns ] is a geodesic in G. Since diam(G+Kn) = 2, [xnr , xnl , xns ] is also a geodesic in G+Kn. Thus, xnl ∈ IG+Kn [Snl−1], a contradiction to the definition of S = Sk. Hence, xnl /∈ P2[Al−1] for each 3 ≤ l ≤ j. Let x ∈ V (G) \ Aj . There exist a, b ∈ {1, 2, . . . , k} such that x ∈ IG+Kn(xa, xb). Necessarily, xa, xb ∈ V (G)∩S = Aj and each xa-xb geodesic containing x lies entirely in G. Since diam(G + Kn) = 2, dG(xa, xb) = 2. Therefore, P2[Aj ] = V (G), and the first claim is done. We next claim that C is a pointwise non- dominating set of G. Let x ∈ V (G) \C. Since S is a hop dominating set of G+Kn, there exists v ∈ S such that dG+Kn(x, v) = 2. Clearly, v ∈ V (G) ∩ S = C and dG(x, v) = 2. This shows that the second claim holds. Conversely, suppose that S = V (Kn) ∪ C, where C ⊆ V (G) and is a closed 2-path closure absorbing pointwise non-dominating set in G. Let k = |C|. Being a closed 2-path closure absorbing set, there is a sequence of sets Aj = {v1, v2, . . . , vj} (j = 1, 2, . . . , k) such that v1 ̸= v2, vj /∈ P2[Aj−1] for 2 ≤ j ≤ k and P2[Ak] = V (G). For i = 1, 2, . . . , n+k, write Si = {x1, x2, . . . , xi}, where V (Kn) = {x1, x2, . . . , xn} and xn+j = vj for all j = 1, 2, . . . , k. Observe that • IG+Kn [Si] = Si for all i = 1, 2, . . . , n; • xn+1 /∈ IG+Kn [Sn] and xn+2 /∈ IG+Kn [Sn−1]; • xn+i /∈ IG+Kn [Sn+i−1] = V (Kn) ∪ P2[Ai−1] for all i = 1, 2, . . . , k; and • IG+Kn [S] = V (G+Kn). This means that S is a closed geodetic set of G + Kn. Finally, let x ∈ V (G + Kn) \ S. Then x /∈ C. Since C is a pointwise non-dominating set, there exists y ∈ C ⊆ S such that dG+Kn(x, y) = dG(x, y) = 2. Therefore, S is a closed geodetic hop dominating set of G+Kn. Corollary 1. Let G be a noncomplete connected graph and n ≥ 1. Then γhcg(G+Kn) = n+ ρc2pnd(G). Example 2. (i) γhcg(Pn +Kp) = { p+ 3 if n = 3, p+ ⌈n+1 2 ⌉ if n ≥ 4, (ii) γhcg(Cn +Kp) = { p+ 3 if n = 3, 4, p+ ⌈n2 ⌉ if n ≥ 5. A. Adolfo, I. Aniversario, F. Jamil / Eur. J. Pure Appl. Math, 17 (3) (2024), 1618-1636 1629 Theorem 8. [18, Theorem 4] Let G and H be any graphs. A set S ⊆ V (G+H) is geodetic hop dominating set of G+H if and only if S = SG ∪SH , where SG and SH are pointwise non-dominating sets of G and H, respectively, such that (i) SG is a 2-path closure absorbing set in G whenever ⟨SH⟩ is a complete subgraph of H and (ii) SH is a 2-path closure absorbing set in H whenever ⟨SG⟩ is a complete subgraph of G. Part of Theorem 9 asserts that under the same condition, S is a hop dominating set of G+H if and only if SG and SH are pointwise non-dominating sets of G andH, respectively. Theorem 9. Let G and H be connected noncomplete graphs. Then S is a closed geodetic hop dominating set of G if and only if S = SG ∪ SH where SG and SH are pointwise non-dominating sets of G and H, respectively, such that either (i) ⟨SG⟩ is complete and SH is a closed 2-path closure absorbing set of H; or (ii) ⟨SH⟩ is complete and SG is a closed 2-path closure absorbing set of G. Proof. Suppose that S is a closed geodetic hop dominating set of G + H. Then SG and SH are pointwise non-dominating sets of G and H, respectively. Since S is a closed geodetic set of G+H, there is a positive integer k and sequence of sets Sj = {x1, x2, . . . , xj} (j = 1, 2, . . . , k) such that x1 ̸= x2, IG+H [Sk] = IG+H [S] = V (G) and xj /∈ IG+H [Sj−1] for 3 ≤ j ≤ k. First, we claim that ⟨SG⟩ is a complete subgraph of G or ⟨SH⟩ is a complete subgraph of H. Suppose this claim is false. If ⟨SG⟩ and ⟨SH⟩ are noncomplete, then there exist distinct integers i, j, l, r such that xi, xj ∈ SG with dG(xi, xj) = 2 and xl, xr ∈ SH with dG(xl, xr) = 2. Without loss of generality, assume that l = max{i, j, l, r}. Since xl ∈ IG+H(xi, xj), xl ∈ IG+H [Sl−1], a contradiction. The claim, therefore, is true. Next, suppose ⟨SG⟩ is a complete subgraph of G. Write SH = {xn1 , xn2 , . . . , xnl } ⊆ Sk with n1 < n2 < · · · < nl, and let Aj = {xn1 , xn2 , . . . , xnj} for each j = 1, 2, . . . , l. As shown in the proof of Theorem 7, xnj /∈ P2[Aj−1] for 3 ≤ j ≤ l, and P2[Al] = V (H). Therefore, SH is a closed 2-path closure absorbing set of H. Similarly, if ⟨SH⟩ is complete, then SG is a closed 2-path closure absorbing set of G. In view of Lemma 2, conditions (i) and (ii) cannot hold at the same time. Conversely, suppose that SG and SH are pointwise non-dominating sets of G and H, respectively. Then S = SG ∪ SH is a hop dominating set of G+H. Suppose further that condition (i) holds, i.e., ⟨SG⟩ is complete and SH is a closed 2-path closure absorbing set of H. Let k = |SG| = k and j = |SH |. There is a sequence of sets Ci = {v1, v2, . . . , vi} (i = 1, 2, . . . , j) such that v1 ̸= v2, vi /∈ P2[Ci−1] for 2 ≤ i ≤ j and P2[Cj ] = V (H). For i = 1, 2, . . . , k + j, write Si = {x1, x2, . . . , xi}, where SG = {x1, x2, . . . , xk} and xk+i = vi for all i = 1, 2, . . . , j. As observed in the proof of Theorem 7, S is a closed geodetic set of G+H. Similarly, if condition (ii) holds, then S is a closed geodetic set of G+H. A. Adolfo, I. Aniversario, F. Jamil / Eur. J. Pure Appl. Math, 17 (3) (2024), 1618-1636 1630 Lemma 3. Let G be a connected noncomplete graph and S ⊆ V (G) such that ⟨S⟩ is complete. Then ⟨S⟩ is a maximal clique if and only if S is a pointwise non-dominating set of G. Proof. Assume that ⟨S⟩ is a maximal clique of G. Let v ∈ V (G) \ S. Suppose that uv ∈ E(G) for all u ∈ S. Then ⟨S ∪ {u}⟩ is a complete subgraph of G, contradicting the maximality of ⟨S⟩. Thus, there exists u ∈ S for which dG(u, v) ≥ 2. Since v is arbitrary, S is pointwise non-dominating. Conversely, suppose that S is pointwise non-dominating set of G. Let C ⊆ V (G) for which ⟨C⟩ is a complete subgraph of G and S ⊆ C. Suppose that C \S ̸= ∅, say x ∈ C \S. Since S is pointwise non-dominating, there exists y ∈ S such that xy /∈ E(G). However, y ∈ C since S ⊆ C. This is a contradiction since ⟨C⟩ is complete. In view of Lemma 3, Theorem 9 can be rephrased as follows: Theorem 10. Let G and H be connected noncomplete graphs. Then S is a closed geodetic hop dominating set of G if and only if S = SG ∪ SH where SG ⊆ V (G) and SH ⊆ V (H) such that either (i) ⟨SG⟩ is a maximal clique of G and SH is a closed 2-path closure absorbing pointwise non-dominating set of H; or (ii) ⟨SH⟩ is maximal clique of H and SG is a closed 2-path closure absorbing pointwoise non-dominating set of G. Corollary 2. Let G and H be connected noncomplete graphs. Then γhcg(G+H) = min{ρc2pnd(G) + ωL(H), ρc2pnd(H) + ωL(G)}. (4) Example 3. (i) γhcg(Pr+Km,n) =  5 if r = 3 and m,n ≥ 2 min{5,m+ n+ 2} if r = 3 and m = 1 or n = 1 min{⌈ r+1 2 ⌉+ 2,m+ n+ 2} if r ≥ 4 and m = 1 or n = 1 min{⌈ r+1 2 ⌉+ 2,min{m,n}+ 3} if r ≥ 4 and m,n ≥ 2, (ii) γhcg(Cr +Km,n) =  5 if r = 4 and m,n ≥ 2 min{5,m+ n+ 2} if r = 4 and m = 1 or n = 1 min{⌈ r2⌉+ 2,m+ n+ 2} if r ≥ 5 and m = 1 or n = 1 min{⌈ r2⌉+ 2,min{m,n}+ 3} if r ≥ 5 and m,n ≥ 2 . A. Adolfo, I. Aniversario, F. Jamil / Eur. J. Pure Appl. Math, 17 (3) (2024), 1618-1636 1631 2.5. In the Corona and Edge Corona of Graphs For the purpose of this section, a sequence of subsets Sk = {v1, v2, . . . , xk} (k = 1, 2, . . . , n) of V (G) is said to be a closed geodetic sequence of sets if v1 ̸= v2 and vk /∈ IG[Sk−1] for 3 ≤ k ≤ n. A closed geodetic sequence of sets Sk = {v1, v2, . . . , xk} (k = 1, 2, . . . , n) is a maximal closed geodetic sequence if IG[Sn] = V (G). More precisely, S ⊆ V (G) is a closed geodetic set of G if and only if there exists a positive integer n and a maximal closed geodetic sequence of sets Sk = {v1, v2, . . . , xk} (k = 1, 2, . . . , n) such that S = Sn. Parallel definitions are adopted for a closed 2-path closure absorbing sequence of sets and maximal closed 2-path closure absorbing sequence of sets. Theorem 11. Let G and H be connected graphs where G is nontrivial, and let S ⊆ V (G ◦H). Then S is a closed geodetic hop dominating set of G ◦H if and only if S = A ∪ ( ∪v∈V (G)Sv ) , (5) where A ⊆ V (G) and Sv ⊆ V (Hv) satisfying the following conditions: (i) Sv is a pointwise non-dominating set of Hv for each v ∈ V (G) \NG(A); (ii) Sv is a closed 2-path closure absorbing set of Hv; and (iii) The vertices in A constitute a closed geodetic sequence of sets of G Proof. Assume S is a closed geodetic hop dominating set of G ◦H. Let A = S ∩ V (G) and Sv = S∩V (Hv) for each v ∈ V (G). Then S = A∪ ( ∪v∈V (G)Sv ) . Let v ∈ V (G)\NG(A), and let u ∈ V (Hv) \ Sv. Since S is a hop dominating set of G ◦ H, there exists w ∈ S such that dG◦H(u,w) = 2. If w ∈ V (G), then w ∈ A and wv ∈ E(G), which is impossible. Thus, w /∈ V (G) so that w ∈ Sv. In this case, dG◦H(u,w) = dHv(u,w) = 2. This means that Sv is pointwise non-dominating in Hv, showing (i). To show, (ii), let v ∈ V (G). Let n = |S|. There exists a closed geodetic sequence of sets Sk = {x1, x2, . . . , xk}, 3 ≤ k ≤ n, such that IG◦H [Sn] = V (G ◦ H). Write Sv = {xn1 , xn2 , . . . , xnj} with n1 < n2 < · · · < nj . Define Ti = {xn1 , xn2 , . . . , xni} for i = 1, 2, . . . , j. Suppose that for a < b < c, [xna , xnc , xnb ] is a geodesic in Hv. Then [xna , xnc , xnb ] is a geodesic in G ◦H so that xnc ∈ IG◦H [Snb ], a contradiction. Therefore, xni /∈ P2[Ti−1] for 3 ≤ i ≤ j, and therefore, Ti = {xn1 , xn2 , . . . , xni}, i = 1, 2, . . . , j, is a closed 2-path closure absorbing sequence of sets in Hv. Let x ∈ V (Hv) \ Tj . Since IG◦H [Sn] = V (G ◦ H), there exist 1 ≤ a, b ≤ n such that x ∈ IG◦H(xa, xb). Because yv ∈ E(G ◦ H for all y ∈ V (Hv), any xa-xb geodesic lies completely in V (Hv). Thus, a, b ∈ {n1, n2, . . . , nj}. This means that P2[Tj ] = V (Hv) and Tj = Sv is a closed 2-path closure absorbing set of Hv. Statement (iii) is done similarly. The sequence Ai = {xk1 , xk2 , . . . , xki}, i = 1, 2, . . . , j, such that Aj = A is a closed geodetic sequence of sets of G. To prove the converse, assume that Equation 3 holds for S together with conditions (i), (ii) and (iii). Let n = |S|, j = |A|, and for each v ∈ V (G), let Sj v = {x1v, x2v, . . . , x j v} ⊆ Sv, A. Adolfo, I. Aniversario, F. Jamil / Eur. J. Pure Appl. Math, 17 (3) (2024), 1618-1636 1632 j = 1, 2, . . . , kv, be a maximal closed 2-path closure absorbing sequence of sets in Hv. Define for each k = 1, 2, . . . , n, Sk = {x1, x2, . . . , xk} ⊆ S such that • A = {x1, x2, . . . , xj}; • If for i < k, xi, xk ∈ Sv, say xi = xsv and xk = xrv, then s < r. Then Sk = {x1, x2, . . . , xk} (k = 1, 2, . . . , n) is a closed geodetic sequence of sets in G ◦H. Let w ∈ V (G ◦H) \ S, and let v ∈ V (G) for which w ∈ V (Hv + v). If w ∈ V (Hv), then w ∈ P2[Sv] = IG◦H [Sv]. Suppose that w = v. Let z ∈ V (G) ∩ NG(v). Pick u ∈ Sv and y ∈ Sz. Then w ∈ IG◦H [u, z] ⊆ IG◦H [S]. Hence, S is a closed geodetic set of G ◦H. Finally, we show S is a hop dominating set of G ◦H. Let w ∈ V (G ◦H) \ S, and let v ∈ V (G) for which w ∈ V (Hv + v). If w = v, then for any z ∈ NG(v), dG◦H(w, y) = 2 for all y ∈ Sz. Suppose that w ∈ V (Hv). If v ∈ NG(A), then dG◦H(w, y) = 2 for all y ∈ A ∩ NG(v). If v /∈ NG(A), then since Sv is pointwise non-dominating, there exists y ∈ Sv for which wy /∈ E(Hv). Then dG◦H(w, y) = 2. Corollary 3. Let G and H be connected graphs where G is nontrivial of order n. Then n · ρ2(H) ≤ γhcg(G ◦H) ≤ n · ρc2pnd(H), and these bounds are sharp. Proof. Let S ⊆ V (G◦H) be a γhcg-set of G◦H. By Theorem 11, S = A∪ ( ∪v∈V (G)Sv ) , where Sv is a closed 2-path closure absorbing set of Hv. Thus, n · ρ2(H) ≤ ∑ v∈V (G) |Sv| ≤ |S| = γhcg(G ◦H). To get the other inequality, for each v ∈ V (G), let Sv ⊆ V (Hv) be a closed 2-path closure absorbing pointwise non-dominating set of Hv. By Theorem 11, S = ∪v∈V (G)Sv is a closed geodetic hop dominating set of G ◦H. Hence, γhcg(G ◦H) ≤ |S| = n · ρc2pnd(H). For a graph G, let τ(G) be the set of all support vertices v of G for which NG(x) = {v} for all x ∈ NG(v)}. In particular, if G = K1,n with central vertex v, then τ(G) = {v}. Theorem 12. Let G be a nontrivial connected graph and n ≥ 1, and let S ⊆ V (G ⋄Kn). Then S is a closed geodetic hop dominating set of G ⋄Kn if and only if S = A ∪ ( ∪uv∈E(G)V (Huv) ) , (6) where A ⊆ V (G) such that L(G) ∪ τ(G) ⊆ A and the elements of A constitute a closed geodetic sequence of sets of G. A. Adolfo, I. Aniversario, F. Jamil / Eur. J. Pure Appl. Math, 17 (3) (2024), 1618-1636 1633 Proof. Put H = Kn. Suppose that S is a closed geodetic hop dominating set of G ⋄H. Since L(G) ∪ V (Huv) ⊆ Ext(G ⋄ H), L(G) ∪ V (Huv) ⊆ S for each uv ∈ E(G). Let A = S ∩ V (G). Since the vertices in S constitute a closed geodetic sequence of sets in G ⋄H, it follows that the vertices in A constitute a closed geodetic sequence of sets in G. Let w ∈ τ(G). Suppose that w /∈ A. Since S is a hop dominating set, there exists x ∈ S such that dG⋄H(w, x) = 2. If x ∈ V (G), then dG(x,w) = 2. Thus, G contains a geodesic [x, v, w]. This means that there exists v ∈ NG(w) with NG(v) ̸= {w}, a contradiction since w ∈ τ(G). Suppose there exist uv ∈ E(G) such that x ∈ Suv. Then either wu ∈ E(G) or wv ∈ E(G). Assume wv ∈ E(G). Then there exists v ∈ NG(w) with NG(v) ̸= {w}, a contradiction. Hence, τ(G) ⊆ A. Conversely, suppose that S is as described in Equation 3 together with the indicated properties. Let n = |S| and |A| = k. For each j = 1, 2, . . . , k, let Aj = {x1, x2, . . . , xj} ⊆ A be a closed geodetic sequence inG. Extend the sequence by defining for each i = 1, 2, . . . , n, Si = {x1, x2, . . . , xk, xk+1, . . . , xi} ⊆ S. This means that Sn \ Ak = ∪uv∈E(G)V (Huv). Thus, Si = {x1, x2, . . . , xk, xk+1, . . . , xi} (i = 1, 2, . . . , n) is a closed geodetic sequence in G ⋄ H. Let w ∈ V (G) \ A. Since w /∈ L(G), there exist distinct x, y ∈ V (G) ∩NG(w). Pick z ∈ V (Hxw) and t ∈ V (Hyw). Then z, t ∈ S and w ∈ IG⋄H(z, y). Since w is arbitrary, IG⋄H [S] = V (G ⋄H) and S is a closed geodetic set of G ⋄H. To show that S is a hop dominating set, let w ∈ V (G) \ A. Since w /∈ τ(G), there exists v ∈ NG(w) such that NG(v) \ {w} ≠ ∅, say u ∈ NG(v) \ {w}. Pick z ∈ Suv. Then z ∈ S and dG⋄H(w, z) = 2. Accordingly, S is a hop dominating set of G ⋄H. Corollary 4. Let G be a nontrivial connected graph of size n and p ≥ 1. Then γhcg(G ⋄Kp) = np+ |L(G)|+ |τ(G)|. (7) In particular, if L(G) = ∅ and τ(G) = ∅, then γhcg(G ⋄Kp) = np. (8) If G = K2, then G ⋄H = K2 +H. This case is taken in Theorem 7. In what follows, we consider G of order n ≥ 3. Theorem 13. Let G and H be connected graphs where |V (G)| ≥ 3 and H is not complete, and let S ⊆ V (G ⋄ H). Then S is a closed geodetic hop dominating set of G ⋄ H if and only if S = A ∪ ( ∪uv∈E(G)Suv ) , (9) where A ⊆ V (G) and Suv ⊆ V (Huv) satisfying the following: (i) The elements in A constitute a closed geodetic sequence of sets of G and τ(G) ⊆ A; (ii) Suv is a closed 2-path closure absorbing set of Huv for each uv ∈ E(G). A. Adolfo, I. Aniversario, F. Jamil / Eur. J. Pure Appl. Math, 17 (3) (2024), 1618-1636 1634 Proof. Assume S is a closed geodetic set of G ◦ H. Let A = S ∩ V (G), and Suv = S ∩ V (Huv) for each uv ∈ E(G). Then S = A ∪ ( ∪uv∈E(G)Suv ) . At this far, showing that the elements of A and Suv constitute a closed geodetic sequence and a closed 2-path closure absorbing sequence of sets in G and Huv, respectively, for each uv ∈ E(G), is already a routine. Since S is a hop dominating set, τ(G) ⊆ A. Thus, (i) holds. To completely show (ii), observe that for each z ∈ V (Huv), every x-y geodesic (with x ̸= z ̸= y) in G ⋄ H containing z lies entirely in Huv. Thus, since IG⋄H [S] = V (G ⋄ H), P2[Suv] = V (Huv). This makes Suv a closed 2-path absorbing set of Huv. Conversely, suppose that S is as given in Equation 4 and satisfies conditions (i) and (ii). Assume |S| = n. Obtain from S a closed geodetic sequence of sets in G⋄H as follows: Construct Sk = {x1, x2, . . . , xk} for k = 1, 2, . . . , n such that Sk for k ∈ {1, 2, . . . , |A|} is a closed geodetic sequence constituted by the vertices in A and Sn \A = ∪uv∈E(G)Suv. Then Sk, k = 1, 2, . . . , n, is a closed geodetic sequence of sets in G ⋄H. Let w ∈ V (G ⋄H) \ S and let uv ∈ E(G) such that w ∈ V (Huv + uv). Suppose that u = w. If Suv = V (Huv), then since Huv is not complete, there exist x, y ∈ Suv such that xy /∈ E(Huv). Then dG⋄H(x, y) = 2 and w ∈ IG⋄H(x, y). Suppose that Suv ̸= V (Huv). Since Suv is a 2-path closure absorbing set of Huv, for w ∈ V (Huv)\Suv, there exist x, y ∈ Suv such that [x,w, y] is a geodesic in Huv. This means that dG⋄H(x, y) = 2 and w ∈ IG⋄H(x, y). The case where w = v is handled similarly. Now, suppose that w ∈ V (Huv). Since Suv is 2-path closure absorbing, there exist x, y ∈ Suv such that dHuv(x, y) = 2 and w ∈ IHuv(x, y). This means that dG⋄H(x, y) = 2 and w ∈ IG⋄H(x, y). We have just shown that S is a closed geodetic set of G ⋄H. Finally, to show that S is a hop dominating set, let w ∈ V (G⋄H)\S. If w ∈ V (G), then since w /∈ τ(G), there exists v ∈ NG(w) such that NG(v) \ {w} ≠ ∅. Let u ∈ NG(v) \ {w}. Pick z ∈ Suv. Then dG⋄H(w, z) = 2. Suppose that w ∈ V (Huv) for some uv ∈ V (G). Then w ∈ V (Huv)\Suv. Since |V (G)| ≥ 3 and G is connected, there exists z ∈ V (G) such that uz or vz is an edge in G. Let uz ∈ E(G). Take x ∈ Suz. Then dG⋄H(x,w) = 2. Same goes for the case where vz ∈ E(G). Therefore, S is a hop dominating set of G ⋄H. Corollary 5. Let G and H be connected graphs where G is of order n ≥ 3 and H is not complete. Then γhcg(G ⋄H) = n · ρc2(H) + |τ(G)|. (10) 3. Conclusion The concept of closed geodetic hop domination in graphs has been introduced and initially investigated in this study. As shown, not all graphs admit this concept. Some conditions under which a graph admits a closed geodetic hop dominating set are provided. Realizations results involving closed geodetic number, geodetic hop domination number and closed geodetic hop domination number are also provided. The closed geodetic hop dominating sets of the join corona, and edge corona of two graphs have been obtained. These characterizations have been used to obtain bounds or exact values of the closed REFERENCES 1635 geodetic hop domination number of each of these graphs. Exploring necessary and suffi- cient conditions for a graph to admit closed geodetic hop dominating set may be interesting and worthwhile to possibly provide insightful results. Acknowledgements The authors would like to thank the Department of Science and Technology - Acceler- ated Science and Technology Human Resource Development Program (DOST-ASTHRDP)- Philippines, and MSU-Iligan Institute of Technology for funding this research. References [1] I. Aniversario, F. Jamil, and S. Canoy Jr. The closed geodetic numbers of graphs. Utilitas Mathematica, 74:3–18, 2007. [2] I. Aniversario, F. Jamil, and S. Canoy Jr. The closed geodetic numbers of the corona and composition of graphs. Utilitas Mathematica, 82, 2010. [3] D. Anusha and S. Joseph Robin. Geodetic hop domination in join and corona of graphs. Journal of Combinatorial Mathematics and Combinatorial Computing, 21:1117–1127, 2011. [4] S. Ayyaswamy, B. Krishnakumari, C. Natarajan, and Y. Venkatakrishman. Bounds on the hop domination number of a tree. Proc. Math. Sci., 125:449–455, 2015. [5] M.A. Bonsocan and F. Jamil. Transversal hop domination in graphs. European Journal of Pure and Applied Mathematics, 16(1):192–206, 2023. [6] F. Buckley and F. Harary. Distance in Graphs. Addison-Wesley Publishing Company, Inc., Redwood City, CA, 1990. [7] G. Cagaanan. On Geodesic Convexity in Graphs. PhD thesis, MSU-Iligan Institute of Technology, 2004. [8] G. Chartrand, F. Harary, and P. Zhang. Geodetic sets in graphs. Discussiones Mathematicae Graph Theory, 20:129–138, 2000. [9] F. Harary. Convexity in graphs: Achievement and avoidance games. Ann. Discrete Math, 20:323, 1983. [10] F. Harary, E. Loukakis, and C. Tsouros. The geodetic number of a graph. Mathl. Comput. Modelling, 17(11):89–95, 1993. [11] F. Jamil and I Aniversario S. Canoy Jr. On closed and upper closed geodetic numbers of graphs. ARS Combinatoria, 84:191–203, 2007. REFERENCES 1636 [12] S. Canoy Jr., R. Mollejon, and J.G. Canoya. Hop dominating sets in graphs under binary operations. European Journal of Pure and Applied Mathematics, 12(4):1455– 1463, 2019. [13] S. Canoy Jr. and G. Salasalan. Locating-hop domination in graphs. Kyungpook Mathematical Journal, 62:193–204, 2022. [14] C. Natarajan and S. Ayyaswamy. Hop domination in graphs ii. Versita, 23:187–199, 2015. [15] Y. Pabilona and H. Rara. Connected hop domination in graphs under some binary operations. Asian-European Journal of Mathematics, 11, 2018. [16] G. Salasalan and S. Canoy Jr. Some Related Concepts of Hop Domination in a Graph. PhD thesis, 2021. [17] C.J. Saromines and S. Canoy Jr. Outer-connected hop dominating sets in graphs. European Journal of Pure and Applied Mathematics, 15(4):1966–1981, 2022. [18] C.J. Saromines and S. Canoy Jr. Another look at geodetic hop domination in a graph. European Journal of Pure and Applied Mathematics, 16(3):1568–1579, 2023. [19] C.J. Saromines and S. Canoy Jr. Geodetic hop dominating sets in a graph. European Journal of Pure and Applied Mathematics, 16(1):5–17, 2023.