EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 16, No. 4, 2023, 2035-2048 ISSN 1307-5543 – ejpam.com Published by New York Business Global Outer-Convex Hop Domination in Graphs Under Some Binary Operations Al-Amin Y. Isahac1, Javier A. Hassan1,∗, Ladznar S. Laja1, Hounam B. Copel1 1 Mathematics and Sciences Department, College of Arts and Sciences, MSU Tawi-Tawi College of Technology and Oceanography, Bongao, Tawi-Tawi, Philippines Abstract. Let G be a graph with vertex and edge sets V (G) and E(G), respectively. A set C ⊆ V (G) is called an outer-convex hop dominating if for every two vertices x, y ∈ V (G) \ C, the vertex set of every x− y geodesic is contained in V (G) \C and for every a ∈ V (G) \C, there exists b ∈ C such that dG(a, b) = 2. The minimum cardinality of an outer-convex hop dominating set of G, denoted by γ̃conh(G), is called the outer-convex hop domination number of G. In this paper, we generate some formulas for the parameters of some special graphs and graphs under some binary operations by characterizing first the outer-convex hop dominating sets of each of these graphs. Moreover, we establish realization result that identifies and determines the connection of this parameter with the standard hop domination parameter. It shows that given any graph, this new parameter is always greater than or equal to the standard hop domination parameter. 2020 Mathematics Subject Classifications: 05C69 Key Words and Phrases: Outer-convex set, outer-convex hop dominating set, outer-convex hop domination number 1. Introduction A subset S of a vertex set of a simple graph G is called a hop dominating in 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 in G, denoted by γh(G), is called the hop domination number of G. This concept was introduced and investigated by Natarajan et al. in [11]. They have studied this concept on some types of graphs and generated some interesting results. Some extreme values and properties of the said parameter can be found in [1, 2, 11]. Recently, Canoy et al. [9] investigated hop domination parameter on graphs under some binary operations. ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v16i4.4862 Email addresses: al-aminisahac@msutawi-tawi.edu.ph (A. Isahac) javierhassan@msutawi-tawi.edu.ph (J. Hassan), ladznarlaja@msutawi-tawi.edu.ph (L. Laja) hounamcopel@msutawi-tawi.edu.ph (H. Copel) https://www.ejpam.com 2035 © 2023 EJPAM All rights reserved. J. A. Hassan et al. / Eur. J. Pure Appl. Math, 16 (4) (2023), 2035-2048 2036 Because of its nice application in networks and different fields, researchers have intro- duced variants of hop domination and they studied these variants for some special graphs and graphs under some operations. (see [3–7, 10, 12]). In this paper, we will introduce new variant of hop domination called outer-convex hop domination. This study is motivated by the introduction of convex hop domination in [5]. We will investigate this concept on some classes of graphs, and join and corona of two graphs. We believe that this new parameter and its results can lead to other interesting research directions in the future. 2. Terminology and Notation Let G = V (G), E(G)) be an undirected graph. Given two vertices u and v of 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 of u, v, and all vertices lying on a u-v geodesic. The interval IG(u, v) = IG [u, v] \ {u, v}. Let G be a graph. A set C ⊆ V (G) is called a convex if for every two vertices x, y ∈ C, the vertex set of every x− y geodesic is contained in C. A set C ′ ⊆ V (G) is a clique if the subgraph ⟨C ′⟩ induced by C ′ is complete. The maximum cardinality among all clique sets of G, denoted by ω(G), is called the clique number of G. Two vertices x, y of G are adjacent, or neighbors, if xy is an edge of G. The open neighborhood of x in G is the set NG(x) = {y ∈ V (G) : xy ∈ E(G)}. The closed neighborhood of x inG is the setNG[x] = NG(x)∪{x}. IfX ⊆ V (G), the open neighborhood of X in G is the set NG(X) = ⋃ x∈X NG(x). The closed neighborhood of X in G is the set NG[X] = NG(X) ∪X. A set P ⊆ V (G) is a pointwise non-dominating set if for every v ∈ V (G)\P , there exists u ∈ P such that v /∈ NG(u). The minimum cardinality of a pointwise non-dominating set of G, denoted by pnd(G), is called a pointwise non-domination number of G. A path graph is a non-empty graph with vertex-set {x1, x2, ..., xn} and edge-set {x1x2, x2x3, ..., xn−1xn}, where the x ′ is are all distinct. The path of order n is denoted by Pn. If G is a graph and u and v are vertices of G, then a path from vertex u to vertex v is sometimes called a u-v path. The cycle graph Cn = [x1, x2, . . . , xn, x1] is the graph of order n ≥ 3 with vertex-set {x1, x2, ..., xn} and edge-set {x1x2, x2x3, ..., xn−1xn, xnx1}. Let G and H be any two graphs. The join 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 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 ∈ G and write v + Hv for ⟨{v}⟩+Hv. J. A. Hassan et al. / Eur. J. Pure Appl. Math, 16 (4) (2023), 2035-2048 2037 The distance dG(u, v) in G of two vertices u, v is the length of a shortest u-v path in G. The greatest distance between any two vertices in G, denoted by diam(G), is called the diameter 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 neigh- borhood 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 of G. 3. Results We begin this section by defining the new concept called outer-convex hop domination in a graph. Definition 1. Let G be a simple graph with vertex and edge-sets V (G) and E(G), re- spectively. Then C ⊆ V (G) is called an outer-convex hop dominating set if C is hop dom- inating and V (G) \ C is convex in G. The minimum cardinality among all outer-convex hop dominating sets of G, denoted by γ̃conh(G), is called the outer-convex hop domination number of G. Any outer-convex hop dominating set C satisfying |C| = γ̃conh(G), is called a γ̃conh-set of G. Example 1. Consider the graph G given in Figure 1. Let C = {a5, a6, . . . , a13}. Then N2 G[C] = V (G) and so C is a hop dominating set of G. Observe that V (G) \ C = {a1, a2, a3, a4} is convex in G. Hence, C is an outer-convex hop dominat- ing set of G. Next, consider C ′ = {a5, a6, a7, a11, a12}. Then C ′ is a hop dominating set in G. However, C ′ is not an outer-convex hop dominating set in G since V (G) \C ′ is not convex in G. Moreover, it can be verified that γ̃conh(G) = 9. J. A. Hassan et al. / Eur. J. Pure Appl. Math, 16 (4) (2023), 2035-2048 2038 a1 a2 a3 a4 G : a5 a6 a7 a8 a9 a10 a11 a12 a13 Figure 1: Graph G with γ̃conh(G)=9 Remark 1. (i) Any graph G admits an outer-convex hop domination. (ii) If S is an outer-convex hop dominating set of G, then S and V (G) \ C are not necessarily convex and hop dominating sets in G, respectively. Remark 2. Let G be any graph. Then every outer-convex hop dominating set C in G is hop dominating but the converse is not always true. The converse part can be seen by considering C ′ in the previous example. Proposition 1. Let G be any graph. Then γh(G) ≤ γ̃conh(G). Proof. Let G be any graph and let S be a minimum outer-convex hop dominating set of G. Then γ̃conh(G) = |S|. By Remark 2, S is a hop dominating set of G. It follows that γh(G) ≤ |S| = γ̃conh(G). Remark 3. The bound given in Proposition 1 is sharp. Moreover, strict inequality can also be attained. To see this, consider the graph G1 in Figure 2. Let C = {f, g}. Then C is both a γh-set and a γ̃conh-set of G1. Hence, γh(G1) = 2 = γ̃conh(G1). J. A. Hassan et al. / Eur. J. Pure Appl. Math, 16 (4) (2023), 2035-2048 2039 G1 : c d b e a f g Figure 2: Graph G1 with γh(G1) = γ̃conh(G1) For strict inequality, consider the graph G2 in Figure 3. Let C ′ = {e, f} and C ′′ = {e, f, g, h, i}. Then C ′ and C ′′ are γh-set and γ̃conh-set of G2, respectively. Thus, γh(G2) = 2 < 5 = γ̃conh(G2). G2 : a b c d e f g h i Figure 3: Graph G2 with γh(G2) < γ̃conh(G2) Theorem 1. Let G be any graph. Then 1 ≤ γ̃conh(G) ≤ |V (G)|. Moreover, (i) γ̃conh(G) = 1 if and only if G is trivial. (ii) γ̃conh(G) = 2 if and only if G has γh-set C = {x, y} such that V (G) \ C is convex set of G. (iii) γ̃conh(G) = |V (G)| if and only if every component of G is complete. Proof. Clearly, 1 ≤ γ̃conh(G) ≤ |V (G)|. (i) Suppose that γ̃conh(G) = 1. Then γh(G) = 1 by Proposition 1. It follows that G = K1 which is a trivial graph. The converse is clear. J. A. Hassan et al. / Eur. J. Pure Appl. Math, 16 (4) (2023), 2035-2048 2040 (ii) Suppose that γ̃conh(G) = 2, say C = {x, y} is a γ̃conh-set of G. Then G is non-trivial by (i) and so γh(G) ≥ 2. By assumption and Proposition 1, γh(G) ≤ 2. Thus, γh(G) = 2. In particular, C is a γh-set of G. Moreover, V (G) \C is a convex set of G by assumption. Conversely, suppose that G has γh-set C = {x, y} of G such that V (G) \ C is convex set of G. Then C is an outer-convex hop dominating set of G. Thus, γ̃conh(G) ≤ 2. Since γh(G) = 2, it follows that γ̃conh(G) = 2 by Proposition 1. (iii) Assume that γ̃conh(G) = |V (G)|. Suppose that there is a component C of G which is non-complete. Then there exist u, v ∈ V (C) ⊆ V (G) such that dC(u, v) = 2 = dG(u, v). Let S∗ = V (G) \ {u}. Then S∗ is an outer-convex hop dominating set in G. Hence, γ̃conh(G) ≤ |V (G)| − 1, a contradiction. Therefore, every component of G is complete. Conversely, suppose every component of G is complete. Then V (G) is the minimum outer-convex hop dominating set of G. Therefore, γ̃conh(G) = |V (G)|. The next result is a direct consequence of Theorem 1. Corollary 1. Let G be any graph of order k ≥ 1. Then each of the following statements holds. (i) γ̃conh(G) = k if G is complete. (ii) γ̃conh(G) ≤ k − 1 if G is non-complete. The following result is a realization result involving outer-convex hop domination and hop domination. Theorem 2. Let a and b be positive integers such that 2 ≤ a ≤ b. Then there exists a connected graph G such that γh(G) = a and γ̃conh(G) = b. Proof. For a = b, consider the graph G in Figure 4. Let C = {v1, v2, . . . , va}. Then C is both γh-set and γ̃conh-set of G, respectively. Hence, γh(G) = a = γ̃conh(G). . . . v1 v2 va G : v3 va−1va−2 Figure 4: Graph G with γh(G) = γ̃conh(G) Suppose that a < b. For a = 2, let s = b − 2 and consider the graph H given in Figure 5. Let C = {v1, v2} and C ′ = {v1, v2, y1, y2, . . . , ys}. Then C and C ′ are γh-set and γ̃conh-set of H, respectively. Therefore, γh(H) = 2 and γ̃conh(H) = s+ 2 = b. J. A. Hassan et al. / Eur. J. Pure Appl. Math, 16 (4) (2023), 2035-2048 2041 v1 v2 y1 y2 ys . . . u H : Figure 5: Graph G with γh(G) < γ̃conh(G) For a ≥ 3, let s = b−a and consider the graphG′ in Figure 6. Let C ′ = {v1, v2, . . . , va−2, u, va} and C ′′ = {v1, v2, . . . , va, y1, y2, . . . , ys}. Then C ′ and C ′′ are γh-set and γ̃conh-set of G′, respectively. Therefore, γh(G ′) = a and γ̃conh(G ′) = s+ a = b. . . . v1 v2 v3 va−2 va−1 va y1 y2 ys . . . u G′ : Figure 6: Graph G′ with γh(G ′) < γ̃conh(G ′) Corollary 2. Let n be a positive integer. Then there exists a connected graph G such that γ̃conh(G)− γh(G) = n. In other words, γ̃conh(G)− γh(G) can be made arbitrarily large. Proposition 2. Let n be any positive integer. Then each of the following holds. (i) γ̃conh(Pn) = { 2 if n = 2, 3 n− 2 if n ≥ 4. J. A. Hassan et al. / Eur. J. Pure Appl. Math, 16 (4) (2023), 2035-2048 2042 (ii) γ̃conh(Cn) =  2 if n = 4, 5 3 if n = 3, 6 n− 4 if n ≥ 7. Proof. (i) Clearly, γ̃conh(Pn) = 2 for n = 2, 3. Suppose that n ≥ 4. Let Pn = [v1, v2, . . . , vn] and consider S = {v1, v2, . . . , vn−3, vn−2}. Clearly, S is a hop domi- nating set of Pn. Observe that V (Pn) \ S = {vn−1, vn} is convex set of Pn. It follows that S is an outer-convex hop dominating set of Pn. Since the induced subgraph of a convex set is always connected, it follows that S is a minimum outer-convex hop dominating set of Pn. Thus, γ̃conh(Pn) = n− 2 for all n ≥ 4. (ii) Clearly, γ̃conh(C4) = 2 = γ̃conh(C5) and γ̃conh(C3) = 3 = γ̃conh(C6). Suppose that n ≥ 7. Let Cn = [v1, v2, . . . , vn, v1] and consider S∗ = {v1, v2, . . . , vn−5, vn−4}. Clearly, S∗ is hop dominating set of Cn. Notice that V (Cn) \ S∗ = {vn−3, vn−2, vn−1, vn} is a convex set of Cn. It follows that S∗ is an outer-convex hop dominating set of Cn. Since the induced subgraph of any convex set is connected, it follows that S∗ is a minimum outer-convex hop dominating set of Cn. Therefore, γ̃conh(Cn) = n− 4 for all n ≥ 7. Proposition 3. Let G be any graph on n ≥ 2 vertices. If γ̃conh(G) = 2, then γh(G) = 2. However, the converse is not always true. Proof. Suppose γ̃conh(G) = 2. Then γh(G) ≤ 2 by Remark 1. Since γh(G) ≥ 2 for any graph of order n ≥ 2, it follows that γh(G) = 2. To see the converse is not necessarily true, consider P5 = [v1, v2, v3, v4, v5]. Let C = {v2, v3}. Then C is a γh-set of P5. Hence, γh(P5) = 2. However, γ̃conh(P5) = 3 by Proposition 2. The following concept can be found in [6] and it will be used to characterize outer- convex hop dominating sets in the join of two graphs. Definition 2. Let G be a non-complete graph. Then D ⊆ V (G) is called an outer-clique pointwise non-dominating set in G if D is pointwise non-dominating set and V (G) \D is clique set in G. The smallest cardinality of an outer-clique pointwise non-dominating set of G, denoted by ocpnd(G), is called the outer-clique pointwise non-domination number of G. Any outer-clique pointwise non-dominating set D of G with |D| = ocpnd(G), is called an ocpnd-set of G. The following results will be used to calculate the exact values of the parameter on the join of two graphs. Theorem 3. Let G be a non-complete graph of order n. Then 1 ≤ ocpnd(G) ≤ n − 1. Moreover, (i) pnd(G) ≤ ocpnd(G). J. A. Hassan et al. / Eur. J. Pure Appl. Math, 16 (4) (2023), 2035-2048 2043 (ii) ocpnd(G) = 1 if and only if G has an isolated vertex v such that ⟨V (G) \ {v}⟩ is complete. Proof. Since ∅ is not an outer-clique pointwise non-dominating set of G, it follows that ocpnd(G) ≥ 1. Also, since V (G) \ {v} is an outer-clique pointwise non-dominating set of G for every v ∈ V (G), by definition we have ocpnd(G) ≤ n − 1. Consequently, 1 ≤ ocpnd(G) ≤ n− 1. (i) Since every outer-clique pointwise non-dominating set is a pointwise non-dominating set, it follows that pnd(G) ≤ ocpnd(G). (ii) Suppose that ocpnd(G) = 1, say {v} is an ocpnd-set of G. Then pnd(G) = 1 by (i). It follows that G is either a trivial or G has an isolated vertex v. Since outer-clique pointwise non-domination is not defined on any complete graph, it follows that G has an isolated vertex v. Moreover, ⟨V (G) \ {v}⟩ is complete by assumption. Conversely, supposeG has an isolated vertex v such that ⟨V (G)\{v}⟩ is complete. Then {v} is an outer-clique pointwise non-dominating set ofG. It follows that ocpnd(G) = 1. Proposition 4. Let n ≥ 2 be any positive integer. Then each of the following holds: (i) ocpnd(Pn) = { 2 if n = 3 n− 2 if n ≥ 4. (ii) ocpnd(Cn) = n− 2 for all n ≥ 4. (iii) ocpnd(Kn) = n− 1 for all n ≥ 2. Proof. (i) Clearly, ocpnd(P3) = 2. Suppose that n ≥ 4. Let Pn = [v1, v2, . . . , vn] and consider C = {v1, v2, . . . , vn−3, vn−2}. Observe that C is a pointwise non-dominating set of Pn. Since ⟨V (Pn) \ C⟩ ∼= K2, it follows that V (Pn) \ C is clique in Pn. Hence, C is an outer-clique pointwise non-dominating set in Pn. Since K1 and K2 are the only complete subgraphs of Pn for all n ≥ 4, it follows that C is a minimum outer-clique pointwise non- dominating set of Pn. Therefore, ocpnd(G) = n− 2 for all n ≥ 4. (ii) Suppose that n ≥ 4. Then applying the same argument as in the proof of (i), we have ocpnd(Cn) = n− 2 for all n ≥ 4. (iii) Let V (Kn) = {v1, v2, . . . , vn}, where n ≥ 2 and consider C∗ = {v1, v2, . . . , vn−1}. Then C∗ is a minimum outer-clique pointwise non-dominating set of Kn for all n ≥ 2. Hence, ocpnd(Kn) = n− 1 for all n ≥ 2. Theorem 4. [9] Let G and H be any two graphs. A set S ⊆ V (G+H) is 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. J. A. Hassan et al. / Eur. J. Pure Appl. Math, 16 (4) (2023), 2035-2048 2044 Theorem 5. [8] Let G and H be two connected graphs. Then a proper subset C = S1∪S2 of V (G+H), where S1 ⊆ V (G) and S2 ⊆ V (H), is a convex set in G+H if and only if S1 and S2 induce complete subgraphs of G and H, respectively, where it may occur that S1 = ∅ or S2 = ∅. Theorem 6. Let G and H be two non-complete graphs. A set C ⊆ V (G + H) is an outer-convex hop dominating set of G+H if and only if C = CG∪CH , where CG and CH are outer-clique pointwise non-dominating sets of G and H, respectively. Proof. Suppose that C is an outer-convex hop dominating set ofG+H. If CG = ∅, then V (G) ⊈ N2 G[C]. However, this is a contradiction to the fact that C is a hop dominating set of G + H. Thus, CG ̸= ∅. Similarly, SH ̸= ∅. Now, since C is a hop dominating set of G + H, it follows that CG and CH are pointwise non-dominating sets of G and H, respectively by Theorem 4. Moreover, since V (G + H) \ C is convex set in G + H, V (G) \CG and V (H) \CH are cliques in G and H, respectively by Theorem 5. Therefore, CG and CH are outer-clique pointwise non-dominating sets of G and H, respectively. Conversely, suppose that C = CG ∪ CH , where CG and CH are outer-clique pointwise non-dominating sets of G and H, respectively. Then ⟨V (G) \ CG⟩ and ⟨V (H) \ CH⟩ are complete in G and H, respectively. Hence, by Theorem 5, V (G + H) \ C is convex set in G +H. Since CG and CH are pointwise non-dominating sets, C = CG ∪ CH is a hop dominating set of G+H by Theorem 4. Therefore, C = CG ∪CH is an outer-convex hop dominating set of G+H. The next result follows from Proposition 4 and Theorem 6. Corollary 3. Let G and H be two non-complete graphs. Then γ̃conh(G+H) = ocpnd(G) + ocpnd(H). In particular, given positive integers n and m, we have (i) γ̃conh(Pn + Pm) =  4 if n,m = 3 m if n = 3,m ≥ 4 n if n ≥ 4,m = 3 n+m− 4 if n,m ≥ 4; (ii) γ̃conh(Cn + Cm) = n+m− 4 for all n,m ≥ 4; and (iii) γ̃conh(Pn + Cm) = { m if n = 3,m ≥ 4 n+m− 4 if n,m ≥ 4. Theorem 7. Let G be any non-complete graph and H be any complete graph. Then C ⊆ V (G + H) is an outer-convex hop dominating set of G + H if and only if C = CG ∪ V (H), where CG is an outer-clique pointwise non-dominating set of G. J. A. Hassan et al. / Eur. J. Pure Appl. Math, 16 (4) (2023), 2035-2048 2045 Proof. Suppose that C is an outer-convex hop dominating set in G + H. Since H is complete, it follows that C = CG ∪ V (H), where CG ̸= ∅. By Theorem 6, CG is an outer-clique pointwise non-dominating set of G. Conversely, assume that C = CG ∪ V (H), where CG is an outer-clique pointwise non- dominating set of G. Then N2 G+H [C] = V (G+H), that is, C is a hop dominating set in G + H. Since CG is an outer-clique set in G, it follows that V (G + H) \ C is clique in G +H, that is, V (G +H) \ C is convex in G +H. Therefore, C is an outer-convex hop dominating set of G+H. The next result follows from Corollary 1, Theorem 3, Proposition 4, and Theorem 7. Corollary 4. Let G be any non-complete graph and H be any complete graph. Then γ̃conh(G+H) = ocpnd(G) + |V (H)|. In particular, given positive integers n and m, we have (i) γ̃conh(Pn +Km) = { m+ 2 if n = 3,m ≥ 1 n+m− 2 if n ≥ 4, m ≥ 1; (ii) γ̃conh(Cn +Km) = n+m− 2 for all n ≥ 4,m ≥ 1. Theorem 8. Let G be a connected non-trivial graph and H be any graph. If C = ⋃ x∈V (G) V (Hx), then C is an outer-convex hop dominating set of G ◦ H. More- over, γ̃conh(G ◦H) ≤ |V (G)||V (H)|. Proof. Suppose that C = ⋃ x∈V (G) V (Hx). Let a ∈ V (G ◦ H) \ C. Then a ∈ V (G). Since G is connected non-trivial graph, there exists b ∈ V (Hx) for some x ∈ V (G) such that dG◦H(a, b) = 2. Thus, C is a hop dominating set in G ◦H. Clearly, V (G ◦H) \ C is convex in G◦H. Therefore, C is an outer-convex hop dominating set in G◦H. Moreover, since |C| = |V (G)||V (H)|, it follows that γ̃conh(G ◦H) ≤ |V (G)||V (H)|. Remark 4. The sharpness of Theorem 8 is attainable. Moreover, strict inequality can be attained. For the sharpness, consider the graph P4 ◦ P2 in Figure 7. Let C = {u1, u2, . . . , u8}. Then C is the minimum outer-convex hop dominating set of P4 ◦ P2. Thus, γ̃conh(P4 ◦ P2) = 4(2) = |V (G)||V (H)|. J. A. Hassan et al. / Eur. J. Pure Appl. Math, 16 (4) (2023), 2035-2048 2046 P4 ◦ P2 : u3 u5u1 u2 u4 u6 u7 u8 u9 u10 u11 u12 Figure 7: Graph P4 ◦ P2 with γ̃conh(P4 ◦ P2) = |V (P4)||V (P2)| For strict inequality, consider the graph C4 ◦K3 given in Figure 8. Let C ′ = {a, b, c, d, e, f, g, h}. Then C ′ is a minimum outer-convex hop dominating of C4 ◦K3. Hence, γ̃conh(C4 ◦K3) = 8 < 12 = |V (C4)||V (K3)|. C4 ◦K3 : a b c d e f g h i j k l m n o p Figure 8: Graph C4 ◦K3 with γ̃conh(C4 ◦K3) < |V (C4)||V (K3)| 4. Conclusion The concept of an outer-convex hop domination has been introduced and investigated in this study. Necessary and sufficient conditions for sets in some special graphs and the join of two graphs have been formulated. 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