EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 16, No. 1, 2023, 319-335 ISSN 1307-5543 – ejpam.com Published by New York Business Global Convex Hop Domination in Graphs Javier A. Hassan1,∗, Sergio R. Canoy, Jr.2, Chrisley Jade Saromines2 1 Mathematics and Sciences Department, College of Arts and Sciences, MSU Tawi-Tawi College of Technology and Oceanography, Bongao, Tawi-Tawi, Philippines 2 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 connected graph with vertex and edge sets V (G) and E(G), respectively. A set C ⊆ V (G) is called convex hop dominating if for every two vertices x, y ∈ C, the vertex set of every x-y geodesic is contained in C and for every v ∈ V (G) \ C, there exists w ∈ C such that dG(v, w) = 2. The minimum cardinality of convex hop dominating set of G, denoted by γconh(G), is called the convex hop domination number of G. In this paper, we show that every two positive integers a and b, where 2 ≤ a ≤ b, are realizable as the connected hop domination number and convex hop domination number, respectively, of a connected graph. We also characterize the convex hop dominating sets in some graphs and determine their convex hop domination numbers. 2020 Mathematics Subject Classifications: 05C69 Key Words and Phrases: Hop domination, hop domination number, convex set, convex hop dominating set, convex hop domination number 1. Introduction Hop domination, a concept introduced and initially studied by Natarajan et al. in [18], has become one of the topics of investigation recently. So far, there is a significant number of variants of hop domination that have been defined and investigated. Some studies on hop domination, its variants, and related concepts can be found in [1], [2], [5], [8], [7], [9], [13], [14], [15], [19], [20], and [21]. Another interesting topic that had caught the attention of several researchers is con- vexity. Convexity is a concept that appears in many areas of mathematics (e.g. real analysis, topology, geometry, functional analysis). In Graph Theory, the concept can eas- ily find a graph-theoretic formulation. Convexity in graphs is discussed in the book by ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v16i1.4656 Email addresses: javier.hassan@g.msuiit.edu.ph (J. Hassan), sergio.canoy@g.msuiit.edu.ph (S. Canoy), chrisley.saromines@g.msuiit.edu.ph (C. Saromines) https://www.ejpam.com 319 © 2023 EJPAM All rights reserved. J. Hassan, S. Canoy Jr., C. Saromines / Eur. J. Pure Appl. Math, 16 (1) (2023), 319-335 320 Buckley and Harary [3]. The concept and other types of convexity are studied in [4], [6], and [11]. The concept is also combined with many other parameters. One well-known formed combination is convex domination. This variation of domination is studied in [4], [10], [16], and [17]. In this paper, we introduce and study convex hop domination. This study is motivated by the introduction of hop domination and convex domination. Just like convex domination, we believe that this new parameter will yield significant results in the topic of domination and can lead to other interesting research directions in the future. 2. Terminology and Notation Let G = V (G), E(G)) be an undirected graph. For any 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}. Vertices u and v are adjacent (or neighbors) if uv ∈ E(G). The set of neighbors of a vertex u in G, denoted by NG(u), is called the open neighborhood of u. The closed neighborhood of u is the set NG[u] = NG(u) ∪ {u}. If X ⊆ V (G), the open neighborhood of X is the set NG(X) = ⋃ u∈X NG(u). The closed neighborhood of X is the set NG[X] = NG(X) ∪X. A set D ⊆ V (G) is a dominating set (resp. total dominating set) of G if for every v ∈ V (G) \ D (resp. v ∈ V (G)), there exists u ∈ D such that uv ∈ E(G), that is, NG[D] = V (G) (resp. NG(D) = V (G)). The domination number (resp. total domination number) of G, denoted by γ(G) (resp. γt(G)), is the minimum cardinality of a dominating (resp. total dominating) set in G. Any dominating (resp. total dominating) set in G with cardinality γ(G) (resp. γt(G)), is called a γ-set (resp. γt-set) in G. If γ(G) = 1 and {v} is a dominating set in G, then we call v a dominating vertex in 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 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 is the set N2 G[X] = N2 G(X) ∪X. A set S ⊆ V (G) is a hop dominating set 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. Any hop dominating set with cardinality equal to γh(G) is called a γh-set. A hop dominating set S is connected hop dominating if ⟨S⟩ is connected. The minimum cardinality among all connected hop dominating sets of G, denoted by γch(G), is called the connected hop domination number of G. Any connected hop dominating set with cardinality equal to γch(G) is called a γch-set. A set C ⊆ V (G) is convex set if for every two vertices x, y ∈ C, the vertex set of every x-y geodesic is contained in C, that is, IG[x, y] ⊆ C. The largest cardinality of a proper convex set in G, denoted by con(G), is called the convexity number of G. A set C ⊆ V (G) J. Hassan, S. Canoy Jr., C. Saromines / Eur. J. Pure Appl. Math, 16 (1) (2023), 319-335 321 is called a convex dominating set (resp. convex hop dominating set) if C is both convex and dominating (resp. convex and hop dominating). The minimum cardinality among all convex dominating (resp. convex hop dominating) sets in G, denoted by γcon(G) (resp. γconh(G)), is called the convex domination number (resp. convex hop domination number) of G. Any convex dominating (resp. convex hop dominating set) with cardinality equal to γcon(G) (resp. γconh(G)) is called a γcon-set (resp. γconh-set). A nonempty set S ⊆ V (G) is non-connecting if for each pair of vertices v, w ∈ V (G)\S with dG(v, w) = 2, it holds that NG(v) ∩NG(w) ∩ S = ∅. A set S ⊆ V (G) is a clique if the subgraph ⟨S⟩ induced by S is a complete graph. The maximum cardinality of a clique in G, denoted by ω(G), is called the clique number of G. A clique S which is also hop dominating in G is called clique hop dominating. Whenever G admits a clique hop dominating set, we call the smallest cardinality of a clique hop dominating set in G, denoted by γclh(G), the clique hop domination number of G. A set C ⊆ V (G) is a pointwise non-dominating set if for every v ∈ V (G)\C, there exists u ∈ C such that v /∈ NG(u). The minimum cardinality of a pointwise non-dominating set in G, denoted by pnd(G), is called a pointwise non-domination number of G. A set S ⊆ V (G) is a clique pointwise non-dominating set if S is both a clique and a pointwise non-dominating set in G. The smallest cardinality of a clique pointwise non- dominating set in G, denoted by cpnd(G), is called the clique pointwise non-domination number of G. Any clique pointwise non-dominating set in G with cardinality cpnd(G) is called a cpnd-set in G. The shadow graph S(G) of graph G is constructed by taking two copies of G, say G1 and G2, and then joining each vertex u ∈ V (G1) to the neighbors of its corresponding vertex u′ ∈ V (G2). For a graph G, the complementary prism, denoted by GG, is formed from the disjoint union of G and its complement G by adding a perfect matching between corresponding vertices of G and G. For each v ∈ V (G), let v denote the vertex in G corresponding to v. In simple terms, the graph GG is form from G∪G by adding the edge vv for every vertex v ∈ V (G). 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. 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). Any non-empty set C ⊆ V (G) × V (H) can be expressed as C = ⋃ x∈S [{x} × Tx], where S ⊆ V (G) and Tx ⊆ V (H) for each x ∈ S. Specifically, Tx = {a ∈ V (H) : (x, a) ∈ C} for each x ∈ S. J. Hassan, S. Canoy Jr., C. Saromines / Eur. J. Pure Appl. Math, 16 (1) (2023), 319-335 322 3. Results Since every convex set in a connected graph induces a connected graph, every convex hop dominating set is connected hop dominating. We formally state a consequence of this fact here. Remark 1. Let G be any connected graph on n vertices. Then γch(G) ≤ γconh(G). Remark 2. The bound given in Remark 1 is tight. Moreover, strict inequality can also be attained. For tightness, consider G = K1,5. Then γch(G) = γconh(G) = 2. Next, consider the graph G in Figure 1. Let C = {c, d, f} and C ′ = {c, d, e, f}. Then C and C ′ are γch-set and γconh-set in G, respectively. Hence, γch(G) = 3 < 4 = γconh(G). .................................... .................................... .................................... .................................... .................................... .................................... .................................... .................................... ........... .......... .......... .......... .......... .......... .......... .......... .......... .......... .......... .......... .......... .......... .......... .......... .......... .......... .......... .......... .......... .......... .......... ...... ...................................................................................................................................................................................................................................................... ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ...... .................................... ......................................................................................................................................................................... ......................................................................................................................................................................... ......................................................................................................................................................................... ......................................................................................................................................................................... ......... ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ...... .................................... .................................... G : • • • • c d f e Figure 1: A graph G with γch(G) < γconh(G). Theorem 1. Let G be any connected graph on n ≥ 2 vertices. Then 2 ≤ γconh(G) ≤ n. Moreover, γconh(G) = 2 if and only if γch(G) = 2. Proof. Clearly, 2 ≤ γconh(G) ≤ n. Suppose γconh(G) = 2. By Remark 1, γch(G) ≤ γconh(G) = 2. Since γch(G) ≥ 2 for any connected graph of order n ≥ 2, it follows that γch(G) = 2. Conversely, suppose γch(G) = 2, say, S = {x, y} is a γch-set of G. Since the graph induced by S is K2, S is convex. Thus, S is a convex hop dominating set in G and γconh(G) ≤ 2. By Remark 1, γconh(G) = 2. Theorem 2. Let a and b be positive integers such that 3 ≤ a ≤ b. Then there exists a connected graph G such that γch(G) = a and γconh(G) = b. Proof. For a = b, consider G = Ka. Then γch(G) = a = γconh(G). Suppose a < b. Consider the following two cases: J. Hassan, S. Canoy Jr., C. Saromines / Eur. J. Pure Appl. Math, 16 (1) (2023), 319-335 323 Case 1: a = 3. Let m = b − a and consider the graph G in Figure 2. Let C = {x1, x2, x3} and C ′ = {x1, x2, x3, y1, y2, . . . , ym}. Then C and C ′ are γch-set and γconh-set in G, respectively. Thus, γch(G) = a and γconh(G) = a+m = b. .................................... .................................... .................................... .................................... .................................... .................................... .................................... .................................... .................................... ............................................................................ ............................................................................ ............................................................................ ............................................................................ ............................................................................ ............................................................................ .................................... .................................... .................................... ......... ........ ........ ........ ........ ........ ........ ........ ........ .................................................................................................................. ...................................................................................................................................................................................... ............................................................................................................................................................................................................................................................................................................................. ......... ........ ........ ........ ........ ........ ........ ........ ........ ... ............ ........... ........... ........... ........... ........... ........... ........... ........... ........... ................... .................. .................. .................. .................. .................. .................. .................. .................. .................. . ............................... .............................. .............................. .............................. .............................. .............................. .............................. .............................. .............................. .............................. ................ x1 x2 x3 y1 y2 . . . ym G : • • • • • • Figure 2: A graph G with γch(G) < γconh(G). Case 2: a ≥ 4. Let m = b − a and consider the graph G′ in Figure 3. Let D = {x1, x2, . . . , xa} and D′ = {x1, x2, . . . , xa, y1, y2, . . . , ym}. Then D and D′ are γch-set and γconh-set in G′, respectively. Thus, γch(G ′) = a and γconh(G ′) = a+m = b. .................................... .................................... .................................... .................................... .................................... .................................... .................................... .................................... .................................... ......... ........ ........ ........ ........ ........ ........ ........ ........ ... ......... ........ ........ ........ ........ ........ ........ ........ ........ ... ......... ........ ........ ........ ........ ........ ........ ........ ........ ... ......... ........ ........ ........ ........ ........ ........ ........ ........ ... ......... ........ ........ ........ ........ ........ ........ ........ ........ ... ......... ........ ........ ........ ........ ........ ........ ........ ........ ... ............................................................................ ............................................................................ . . . . . . . . . .................................... .................................... .................................... .................................... .................................... .................................... ......... ........ ........ ........ ........ ........ ........ ........ ........ ... ......... ........ ........ ........ ........ ........ ........ ........ ........ ... ......... ........ ........ ........ ........ ........ ........ ........ ........ ... ......... ........ ........ ........ ........ ........ ........ ........ ........ ... ............................................................................ .................................... .................................... .................................... .............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ..... ................... .................. .................. .................. .................. .................. .................. .................. .................. .................. . ................................. ................................ ................................ ................................ ................................ ...... .................................... .................................... ........................................................................................................................................................................... ..................................................................................................................................................................................................... .................................... ...................................................................................................................................................................................................................... ...................................................................................................................................................................................... ....................................................................................................................................................................... .......................... ......................... ......................... ......................... ......................... ......................... .................... ............... .............. .............. .............. .............. .............. .............. .............. .............. .............. .............. .............. .............. .............. .................................... ................................................................................................................ ............................................................................ ......... ........ ........ ........ ........ ...... .................................... ......... ........ ........ ........ .................................... .................................... ......... ........ ........ ........ ........ ...... ... G ′ : x1 x2 x3 xa−3 xa−2 xa−1 y1 y2 y3 ym xa • • • • • • • • • • • Figure 3: A graph G′ with γch(G ′) < γconh(G ′). This proves the assertion. Corollary 1. Let n be a positive integer. Then there exists a connected graph G such that γconh(G)− γch(G) = n. In other words, γconh − γch can be made arbitrarily large. Proposition 1. Let n be any positive integer. Then each of the following holds. (i) γconh(Pn) = { 2 if n = 2, 3, 4, 5 n− 4 if n ≥ 6. J. Hassan, S. Canoy Jr., C. Saromines / Eur. J. Pure Appl. Math, 16 (1) (2023), 319-335 324 (ii) γconh(Cn) =  2 if n = 4, 5 3 if n = 3 n− 4 if n ≥ 6 n− 4if6 ≤ n ≤ 9 nifn ≥ 10. (iii) γconh(Kn) = n for all n ≥ 1. Proof. (i) Clearly, γconh(Pn) = 2 for n ∈ {2, 3, 4, 5}. Suppose n ≥ 6. Let Pn = [v1, v2, . . . , vn] and consider C = {v3, v4, . . . , vn−3, vn−2}. Then C is a convex hop domi- nating set in Pn. Since every convex hop dominating set in Pn contains C, it follows that C is a γconh-set of Pn. Thus, γconh(Pn) = n− 4 for all n ≥ 6. (ii) Clearly, γconh(Cn) = 2 for n ∈ {4, 5} and γconh(Cn) = 3 for n = 3. Suppose 6 ≤ n ≤ 9.. Let Cn = [v1, v2, . . . , vn, v1] and let C ′ be a γconh-set of Cn. We may assume that v1 ∈ C ′ and vn /∈ C. Then C ′ = {v1, v2, . . . , vn−5, vn−4}. It follows that γconh(Cn) = n− 4 for all 6 ≤ n ≤ 9. Next, suppose that n ≥ 10.IfS’isaγconh-set of Cn, then |S′| ≥ n − 4 since S′ is a connected hop dominating set. We may assume that v1, v2, ..., vn−5, vn−4 ∈ S′. Then dCn(v1, vn−4) ≤ n− 5. It follows that vn−3, vn−2, vn−1, vnlieinthev1- vn−4 geodesic. Since S’ is convex, S′ = V (Cn) and γconh(Cn) = n. (iii) Since γch(Kn) = n for all n ≥ 1, it follows from Remark 1 that γconh(Kn) = n for all n ≥ 1. Theorem 3. Let G be a connected graph of order n. Then γconh(GG) = 2. In particular, {u, u} is a γconh-set of GG for any u ∈ V (G). Proof. Clearly, γconh(GG) = 2 if n = 1. Suppose n ≥ 2. Let S = {u, u} where u ∈ V (G) and u ∈ V (G). Clearly, S is a convex set. Let w ∈ V (GG) \ S and consider the following two cases: Case 1: w ∈ V (G). If uw ∈ E(G), then dGG(u,w) = 2. Suppose that uw /∈ E(G), then u w ∈ E(G). This implies that dGG(u,w) = 2. Case 2: w ∈ V (G). Let w = z, where z ∈ V (G). If u z ∈ E(G), then dGG(u,w) = 2. If u z /∈ E(G), then uz ∈ E(G). This means that dGG(u,w) = 2. Therefore, S is a convex hop dominating set in GG. Since GG is non-trivial, it follows that γconh(GG) = 2. If G1 and G2 are the copies of graph G in the definition of the shadow graph S(G) and if SG1 ⊆ V (G1) and SG2 ⊆ V (G2), then the sets S′ G1 and S′ G2 are the sets given by S′ G1 = {a′ ∈ V (G2) : a ∈ SG1} and S′ G2 = {a ∈ V (G1) : a ′ ∈ SG2}. J. Hassan, S. Canoy Jr., C. Saromines / Eur. J. Pure Appl. Math, 16 (1) (2023), 319-335 325 Theorem 4. Let G be a non-trivial connected graph. Then a proper subset S of V (S(G)) is convex in S(G) if and only if one of the following conditions holds: (i) S is clique in G1. (ii) S is clique in G2. (iii) S = SG1 ∪ SG2 and satisfies the following conditions: (a) SG1 ∩ S′ G2 = ∅ and S′ G1 ∩ SG2 = ∅. (b) SG1 and SG2 are cliques in G1 and G2, respectively. (c) SG1 ∪ S′ G2 and S′ G1 ∪ SG2 are cliques in G1 and G2, respectively. Proof. Suppose S is convex in S(G). If SG2 = ∅, then S = SG1 . Suppose S is not a clique in G1. Then there exist a, b ∈ S such that dG1(a, b) = 2 = dS(G)(a, b). It follows that x ∈ S for all x ∈ NG1(a) ∩NG1(b). Hence, x′ ∈ S for all x ∈ NG1(a) ∩NG1(b). This contradicts the assumption that SG2 = ∅. Therefore, S is a clique in G1. Similarly, if CG1 = ∅, then S is a clique in G2. Hence, (i) and (ii) hold. Next, suppose SG1 and SG2 are both non-empty. Then S = SG1 ∪ SG2 . Suppose SG1 ∩ S′ G2 ̸= ∅, say v ∈ SG1 ∩ S′ G2 . Then v, v′ ∈ S. By convexity of S, x, x′ ∈ S for all x ∈ NG(v). This implies that S = V (S(G)), a contradiction. Therefore, SG1 ∩ S′ G2 = ∅. Similarly, S′ G1 ∩ SG2 = ∅, showing that (a) holds. Now, suppose SG1 is not clique. Then there exist a, b ∈ SG1 such that dG1(a, b) = 2 = dS(G)(a, b). Again, by convexity of S, it follows that x, x′ ∈ S for all x ∈ NG1(a) ∩ NG1(b). This implies that S = V (S(G)), a contradiction. Therefore, SG1 is a clique in G1. Similarly, SG2 is a clique in G2, showing that (b) holds. Suppose SG1 ∪S′ G2 is not a clique in G1. Then there exist x, y ∈ SG1 ∪S′ G2 such that dG1(x, y) = 2. Since SG1 and SG2 are cliques, we may assume that x ∈ SG1 and y ∈ S′ G2 . Then y′ ∈ SG2 . Let z ∈ NG(x) ∩ NG(y). Then z, z′ ∈ NS(G)(x) ∩ NS(G)(y ′). Since S is convex, z, z′ ∈ S. Since yz, yz′ ∈ E(S(G)), y ∈ S by convexity of S. This would imply that S = V (S(G)), a contradiction. Therefore, SG1 ∪ S′ G2 is a clique in G1. Similarly, S′ G1 ∪ SG2 is a clique in G2. Thus, (c) holds. The converse is clear. Corollary 2. Let G be a non-trivial connected graph. Then con(S(G)) = ω(G). Theorem 5. Let G be a non-trivial connected graph. Then S is a hop dominating set in S(G) if and only if one of the following conditions holds: (i) S is a hop dominating set in G1. (ii) S is a hop dominating set in G2. (iii) S = SG1 ∪ SG2 such that SG1 ∪ S′ G2 and S′ G1 ∪ SG2 are hop dominating sets in G1 and G2. J. Hassan, S. Canoy Jr., C. Saromines / Eur. J. Pure Appl. Math, 16 (1) (2023), 319-335 326 Proof. Let S be a hop dominating set in S(G). Set SG1 = S ∩ V (G1) and SG2 = S ∩ V (G2). If SG2 = ∅, then S = SG1 is a hop dominating set in G1. If SG1 = ∅, then S = SG2 is a hop dominating set in G2. Hence, (i) or (ii) holds. Next, suppose SG1 ̸= ∅ and SG2 ̸= ∅. Let x ∈ V (G1) \ SG1 ∪ S′ G2 . Then x ∈ V (S(G)) \ S. Since S is a hop dominating set in S(G), there exists y ∈ S such that dS(G)(x, y) = 2. If y ∈ SG1 , then we are done. Suppose y ∈ SG2 , say y = z′, where z ∈ V (G1). Then z ∈ S′ G2 and dS(G)(x, z) = dG1(x, z) = 2. Therefore, SG1 ∪S′ G2 is a hop dominating set in G1. Similarly, S′ G1 ∪ SG2 is a hop dominating set in G2. Hence, (iii) holds. For the converse, suppose (i) holds. Let a ∈ V (S(G)) \ S. If a ∈ V (G1) \ S, then there exists b ∈ S such that dG1(a, b) = dS(G)(a, b) = 2. Suppose a ∈ V (G2), say a = v′, where v ∈ V (G1). If v ∈ S, then dG1(a, v) = dS(G)(a, v) = 2. If v /∈ S, then there exists w ∈ S such that dG1(v, w) = 2. It follows that dS(G)(a,w) = dS(G)(v ′, w) = 2. Therefore, S is a hop dominating set in S(G). Similarly, if (ii) holds, then S is a hop dominating set in S(G). Now, suppose (iii) holds. Let y ∈ V (S(G)) \ S. Then y /∈ SG1 ∪ SG2 . Suppose y ∈ V (G2) \ SG2 , say y = z′, where z ∈ V (G1). Then z /∈ S′ G2 . If z ∈ SG1 , then dS(G)(y, z) = dS(G)(z ′, z) = 2. Suppose z /∈ SG1 . Since SG1∪S′ G2 is a hop dominating set in G1, there exists p ∈ SG1∪S′ G2 such that dG1(p, z) = 2 = dS(G)(p, z). If p ∈ SG1 , then p ∈ S and dS(G)(p, z ′) = 2. If p ∈ S′ G2 , then p′ ∈ SG2 ⊆ S and dG2(p ′, z′) = dS(G)(p ′, z′) = 2. Therefore, S is a hop dominating set in S(G). Corollary 3. Let G be a non-trivial connected graph. Then γh(S(G)) = γh(G). Theorem 6. Let G be a non-trivial connected graph. Then S is a convex hop dominating set in S(G) if and only if one of the following conditions holds: (i) S is clique hop dominating set in G1. (ii) S is clique hop dominating set in G2. (iii) S = SG1 ∪ SG2 where (a) SG1 ∩ S′ G2 = ∅ and S′ G1 ∩ SG2 = ∅. (b) SG1 and SG2 are cliques in SG1 and SG2, respectively. (c) SG1 ∪ S′ G2 and S′ G1 ∪ SG2 are clique hop dominating sets in SG1 and SG2, respectively. Proof. Follows from Theorem 4 and Theorem 5. Consider the following family of graphs: B = {G : G admits a clique hop domination}. Then the following result follows from Theorem 6. Corollary 4. Let G be a non-trivial connected graph. Then γconh(S(G)) = { γclh(G) if G ∈ B |V (S(G))| if G /∈ B. J. Hassan, S. Canoy Jr., C. Saromines / Eur. J. Pure Appl. Math, 16 (1) (2023), 319-335 327 Theorem 7. [7] Let G be a graph of order n. Then 1 ≤ cpnd(G) ≤ n. Moreover, (i) cpnd(G) = 1 if and only if G has an isolated vertex. (ii) cpnd(G) = n if and only if G is a complete graph. Corollary 5. [7] Let n be any positive integer. Then (i) cpnd(Pn) = 2 for any n ≥ 2. (ii) cpnd(Cn) = 2 for any n ≥ 4. The next result is found in [13]. Theorem 8. Let G and H be any two graphs. A set S ⊆ V (G +H) is hop dominating set in G+H if and only if S = SG ∪SH , where SG and SH are pointwise non-dominating sets in G and H, respectively. The following two results are obtained in [12]. Theorem 9. Let G be a connected graph and Kn the complete graph of order n. Then a proper subset C = S1 ∪ S2 of V (G + Kn), where S1 ⊆ V (G) and S2 ⊆ V (Kn), is a convex set in G+H if and only if S1 induces a complete subgraph of G or V (G) \ S1 is a non-connecting set and S2 = V (Kn). Theorem 10. Let G and H be two non-complete 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 11. Let G and H be two non-complete connected graphs. A set S ⊆ V (G+H) is a convex hop dominating set in G +H if and only if S = SG ∪ SH , where SG and SH are clique pointwise non-dominating sets in G and H, respectively. Proof. Suppose S is a convex hop dominating set in G+H. Then SG and SH are both non-empty. Since S is a hop dominating set, SG and SH are pointwise non-dominating sets in G and H, respectively by Theorem 8. Since S is a convex set, SG and SH are cliques in G and H, respectively, by Theorem 10. Therefore, SG and SH are clique pointwise non-dominating sets in G and H, respectively. Conversely, suppose that S = SG ∪ SH , where SG and SH are clique pointwise non- dominating sets in G and H, respectively. Since SG and SH are pointwise non-dominating sets, S = SG ∪ SH is a hop dominating set in G+H by Theorem 8. Since SG and SH are cliques, it follows that S = SG∪SH is a convex set in G+H by Theorem 10. Consequently, S = SG ∪ SH is a convex hop dominating set in G+H. The next result follows from Theorem 7, Corollary 5 and Theorem 11. Corollary 6. Let G and H be two non-complete connected graphs. Then γconh(G+H) = cpnd(G) + cpnd(H). In particular, we have J. Hassan, S. Canoy Jr., C. Saromines / Eur. J. Pure Appl. Math, 16 (1) (2023), 319-335 328 (i) γconh(Pn + Pm) = 4 for all n,m ≥ 3, and (ii) γconh(Cn + Cm) = 4 for all n,m ≥ 4. Theorem 12. Let G be a connected graph and Kn the complete graph of order n. A set S ⊆ V (G+Kn) is a convex hop dominating set in G+Kn if and only if S = V (Kn)∪SG where V (G) \ SG is a non-connecting set and SG is a pointwise non-dominating set in G. Proof. Suppose S = SKn ∪SG is a convex hop dominating set of G+Kn. By Theorem 8, SKn and SG are pointwise non-dominating sets of Kn and G, respectively. Hence, SKn = V (Kn). Moreover, by Theorem 9, V (G) \ SG is a non-connecting set in G. Conversely, suppose that S = V (Kn) ∪ SG such that V (G) \ SG is a non-connecting set and SG is a pointwise non-dominating set in G. Then, by Theorem 8 and Theorem 9, S is a convex hop dominating set of G+Kn. The next result follows from Theorem 12. Corollary 7. Let G a connected graph and Kn the complete graph of order n. Then γconh(G+Kn) = n+ rG, where rG = min{|S| : V (G)\S is non-connecting and S is a pointwise non-dominating set in G}. In particular, the following hold: (i) γconh(Kn + Cn) = { n+ 3 if n = 3 n+ 2 if n ≥ 4. (ii) γconh(Kn + Pn) = n+ 2 for all n ≥ 2. The result that follows is a restatement of a result in [13]. Theorem 13. Let G and H be any two graphs. A set C ⊆ V (G) is a hop dominating set in G ◦H if and only if C = A∪ (∪v∈V (G)Cv), where A ⊆ V (G) and Cv ⊆ V (Hv) for each v ∈ V (G), and satisfies the following conditions: (i) For each w ∈ V (G) \ A, there exists x ∈ A with dG(w, x) = 2 or there exists y ∈ NG(w) with Cy ̸= ∅. (ii) Cw is a pointwise non-dominating set in Hw for each w ∈ V (G) \NG(A). Theorem 14. Let G be a non-trivial connected graph and let H be any graph. Then C is a convex hop dominating set in G◦H if and only if C = A∪ (∪v∈V (G)Cv), where A ⊆ V (G), Cv ⊆ V (Hv) for each v ∈ V (G), and satisfies the following conditions: (i) For each a ∈ V (G) \ A, there exists b ∈ A with dG(a, b) = 2 or there exists y ∈ A ∩NG(a) with Cy ̸= ∅. J. Hassan, S. Canoy Jr., C. Saromines / Eur. J. Pure Appl. Math, 16 (1) (2023), 319-335 329 (ii) A is a convex dominating set in G. (iii) Cv = ∅ for each v ∈ V (G) \A. (iv) V (Hv) \ Cv is a non-connecting in Hv for each v ∈ A ∩NG(A). (v) V (Hv) \Cv is a non-connecting set and Cv is a pointwise non-dominating set in Hv if A = {v} (that is, if v ∈ A \NG(A)). Proof. Suppose C is a convex hop dominating set in G ◦ H. By Theorem 13(ii), statement (i) holds. Let x, y ∈ A with x ̸= y. Then x and y are in C. Since C is convex and IG◦H [x, y] = IG[x, y], it follows that IG[x, y] ⊆ A. Hence, A is convex. Suppose A is not a dominating set in G. Then there exists v ∈ V (G) \NG[A]. By Theorem 13(ii), Cv is a pointwise non-dominating set in Hv. Also, by Theorem 13(i), there exists x ∈ A with dG(v, x) = 2 or there exists y ∈ NG(v) with Cy ̸= ∅. Pick any p ∈ Cv and let q ∈ C such that dG◦H(v, q) = 2 (q = x or q ∈ Cy). Then v ∈ IG◦H(p, q). By convexity of C, it follows that v ∈ C, a contradiction. Thus, A is a dominating set in G. This shows that (ii) holds. Next, let y ∈ V (G) \A. Since A is a dominating set in G, y ∈ NG(A). By convexity of C, Cy = ∅. Hence, (iii) holds. Let v ∈ A. Suppose V (Hv)\Cv is not a non-connecting set in Hv. Then there exist p, q ∈ Cv such that p ̸= q and NHv(p)∩NHv(q)∩ [V (Hv) \Cv] ̸= ∅. This implies that C is not convex, a contradiction. Therefore, V (Hv) \ Cv is a non- connecting set in Hv, showing that (iv) holds. Suppose now that v ∈ A \NG(A). Then, by Theorem 13(ii), Cv is a pointwise non-dominating set in Hv. Hence, (v) also holds. Conversely, suppose that C has the given form and satisfies (i), (ii), (iii) (iv) and (v). Since (i) and (v) hold and A is a dominating set in G, the conditions (i) and (ii) of The- orem 13 hold. Thus, C is a hop dominating set in G ◦H. Next, let x, y ∈ C with x ̸= y. Let v, w ∈ V (G) such that x ∈ V (v+Hv) and y ∈ V (w+Hw). Consider the following cases: Case 1: v = w. If one of x and y is v, say x = v, then y ∈ Cv and IG◦H [x, y] = {x, y} ⊆ C. Suppose x, y ∈ Cv. Since Cv ̸= ∅, v ∈ A by (iii). By (iv), V (Hv) \ Cv is a non-connecting set in Hv. Hence, IG◦H [x, y] ⊆ C. Case 2: v ̸= w. Suppose x = v and y = w. Since A is convex, IG[x, y] ⊆ A. Since IG◦H [x, y] = IG[x, y], IG◦H [x, y] ⊆ C. Suppose x = v and y ∈ Cw. Then w ∈ A and, by convexity of A, IG[x,w] ⊆ A. Since IG◦H [x, y] = IG[x,w] ∪ IG◦H [w, y] = IG◦H [x,w] ∪ {y}, it follows that IG◦H [x, y] ⊆ C. The same conclusion holds when x ∈ Cv and y = w. Finally, let x ∈ Cv and y ∈ Cw. Then, by (iii), v, w ∈ A. Again, by convexity of A, IG◦H [v, w] = IG[v, w] is contained in A ⊆ C. This implies that IG◦H [x, y] = IG◦H [v, w] ∪ IG◦H [x, v] ∪ IG◦H [y, w] = IG◦H [v, w] ∪ {x, y} J. Hassan, S. Canoy Jr., C. Saromines / Eur. J. Pure Appl. Math, 16 (1) (2023), 319-335 330 is contained in C. Therefore, C is a convex set in G ◦H. Accordingly, C is a convex hop dominating set in G ◦H. Let G be a graph. We denote by DG, LG, and IH the sets containing the dominating vertices, leaves, and isolated vertices of G, respectively. Note that if γ(G) = 1, then |DG| ≥ 1. Corollary 8. Let G be a non-trivial connected graph with γ(G) = 1 and let H be any graph. Then γconh(G ◦H) = { 2, if |LG| ≥ 1 or |IH | ≥ 1 3, otherwise Proof. Let v ∈ DG. Suppose |LG| ≥ 1, say w ∈ LG. Set A1 = {v, w}. Then A1 is a convex dominating set in G. Let Cu = ∅ for each u ∈ V (G). Then C1 = A1 ∪ (∪u∈V (G)Cu) = A1 is a convex hop dominating set in G ◦H by Theorem 14. Thus, γconh(G ◦ H) = 2. Next, suppose that |IH | ≥ 1. Pick any p ∈ IHv . Then A2 = {v} is a convex dominating set in G. Set Cv = {p} and let Cu = ∅ for all u ∈ V (G) \ {v}. Then V (Hv)\Cv is a non-connecting set and Cv is a pointwise non-dominating set in Hv. Hence, C2 = A2 ∪ (∪z∈V (G)Cz) = A2 ∪ Cv is a convex hop dominating set in G ◦ H by Theorem 14. It follows that γconh(G ◦H) = 2. Suppose now that |LG| = 0 and |IH | = 0. Again, let v ∈ DG. Pick any z ∈ V (G) \ {v} and let A = {v, z}. Then A is a convex dominating set of G. Choose any q ∈ V (Hv) and let Cv = {q}. Put Cx = ∅ for all x ∈ V (G) \ {v}. Then Cz = ∅ and V (Hv) \ Cv and V (Hz) \ Cz are non-connecting sets in Hv and Hz, respectively. By Theorem 14, C = A ∪ (∪y∈V (G)Cy) = A ∪ Cv is a convex hop dominating set in G ◦ H. It follows that γconh(G ◦H) ≤ 3. Suppose now that C = A0 ∪ (∪u∈V (G)Su) is a γconh-set of G ◦H. Suppose first that |A0| = 1, say A0 = {z}. Then A0 is (convex) dominating set in G by Theorem 14(ii). Moreover, Su = ∅ for all u ∈ V (G) \ A0 by Theorem 14(iii). Since |IH | = 0, any pointwise non-dominating set in Hz contains at least two elements, that is, |Sz| ≥ 2. It follows that γconh(G ◦H) = |C0| ≥ 3. Suppose that |A0| = 2, say A0 = {x, y}. If x, y /∈ DG, then Cx ̸= ∅ or Cy ̸= ∅ (since x and y are not hop neighbors of a dominating vertex of G). Suppose one of x and y, say x, is a dominating vertex in G. Since y /∈ LG, there exists a vertex d ∈ NG(y) ∩NG(x). This implies that Cx ̸= ∅ or Cy ̸= ∅. In either case, γconh(G ◦H) = |C0| ≥ 3. Therefore, γconh(G ◦H) = 3. For a connected graph G, γhcon(G) = min{|S| : S is a convex dominating and hop dominating set in G}. Since V (G) is a convex dominating and hop dominating set, G admits a convex dominating and hop dominating set. Moreover, γcon(G) ≤ γhcon(G). Corollary 9. Let G be a non-trivial connected graph with γ(G) ̸= 1 and let H be any J. Hassan, S. Canoy Jr., C. Saromines / Eur. J. Pure Appl. Math, 16 (1) (2023), 319-335 331 graph. Then γconh(G ◦H) = { γcon(G), if γcon(G) = γhcon(G) γcon(G) + 1, otherwise. Proof. Suppose γcon(G) = γhcon(G). Let A be a γhcon-set ofG. Then |A| ≥ 2. Set Cv = ∅ for all v ∈ V (G). Then, by Theorem 14, C = A is a convex hop dominating set in G ◦H. Hence, γconh(G◦H) ≤ γcon(G). By Theorem 14(ii), it follows that γconh(G◦H) = γcon(G). Next, suppose that γcon(G) < γhcon(G). Let A′ be a γcon-set of G. Since γ(G) ̸= 1, |A′| ≥ 2. The assumption that γcon(G) < γhcon(G) implies that A′ is not a hop dominating set in G. Hence, there exists v /∈ N2 G[A ′]. Let x, y ∈ A′ with x ̸= y. Since A′ is a dominating set, there exists w ∈ A′ ∩NG(v). Because A′ is convex, ⟨A′⟩ is connected. Let [w1, w2, ..., wk], where w1 = w and wk = x, be a w-x geodesic in ⟨A′⟩. Since v /∈ N2 G[A ′], vwj ∈ E(G) for all j ∈ {1, 2, ..., k}. In particular, vx ∈ E(G). Let [x1, x2, ..., xt], where x1 = x and xt = y, be an x-y geodesic in ⟨A′⟩. Again, since v /∈ N2 G[A ′], vxi ∈ E(G) for all i ∈ {1, 2, ..., t}. Moreover, by convexity of A′, ⟨{x1, x2, ..., xt}⟩ is complete (otherwise, v ∈ A′, a contradiction). Hence, xy ∈ E(G). Thus, ⟨A′⟩ is complete. Pick any w ∈ A′ and p ∈ V (Hw). Set Cw = {p} and Cz = ∅ for all z ∈ V (G) \ {w}. Then C ′ = A′ ∪ Cw is a convex hop dominating set in G ◦H by Theorem 14. Hence, γconh(G ◦H) ≤ |C ′| = γcon(G)+1. Now let C∗ = A∗∪ (∪v∈V (G)Rv) be a γconh-set of G ◦H. Then A∗ is a convex dominating set in G by Theorem 14. If |A∗| > γcon(G), then |C∗| ≥ |A∗| ≥ γcon(G) + 1. Suppose |A∗| = γcon(G). Since γcon(G) < γhcon(G), A∗ is not a hop dominating set, say v /∈ N2 G[A ∗]. Hence, by Theorem 14(i), there exists y ∈ A∗ ∩ NG(v) with Ry ̸= ∅. It follows that γconh(G ◦H) = |C∗| ≥ |A∗|+ |Ry| ≥ γcon(G) + 1. This establishes the desired equality. The next result is found in [13]. Theorem 15. Let G and H be connected non-trivial graphs. Then C = ⋃ x∈S [{x} × Tx] is a hop dominating set in G[H] if and only if the following conditions hold. (i) S is a hop dominating set in G. (ii) Tx is a pointwise non-dominating set in H for each x ∈ S \N2 G(S). The next result is a restatement of the one obtained by Canoy and Garces in [12]. Theorem 16. Let G and H be connected non-complete graphs. Then C = ⋃ x∈S({x}×Tx) is convex in G[H] if and only if S is a clique in G and Tx is a clique in H for each x ∈ S. Theorem 17. Let G and H be connected non-complete graphs. Then C = ⋃ x∈A [{x}× Tx], where A ⊆ V (G) and Tx ⊆ V (H) for each x ∈ A, is a convex hop dominating set in G[H] if and only if C = V (G[H]) or C satisfies the following conditions: (i) A is a clique hop dominating set in G. (ii) Tx is a clique pointwise non-dominating set in H for each x ∈ A. J. Hassan, S. Canoy Jr., C. Saromines / Eur. J. Pure Appl. Math, 16 (1) (2023), 319-335 332 Proof. If C = V (G[H]), then we are done. Suppose C ̸= V (G[H]). Then A is a clique in G and Tx is a clique in H for each x ∈ A by Theorem 16. Since C is hop dominating set, A is a hop dominating set in G by Theorem 15. Since A is a clique, x /∈ N2 G(A) for all x ∈ A. Thus, Tx is a pointwise non-dominating set in H for every x ∈ A by Theorem 15(ii). Therefore, (i) and (ii) hold. For the converse, suppose that C = V (G[H]). Then C is convex hop dominating in G[H]. Next, suppose C satisfies i and (ii). Then by Theorem 15, C is a hop dominating set in G[H]. By (i), (ii) and Theorem 16, C is a convex set in G[H]. Hence, C is a convex hop dominating set in G[H]. In the next result, we shall consider the family C of graphs given by C = {G : G is a connected non-complete graph that admits a clique hop dominating set}. Corollary 10. Let G and H be connected non-complete graphs of orders m and n, re- spectively. Then γconh(G[H]) = { nm if G /∈ C γclh(G)cpnd(H) if G ∈ C. The next result is taken from [10]. Theorem 18. Let G be a connected graph and Km the complete graph of order m. A subset C = ⋃ x∈S({x} × Tx) of V (G[Km]) is convex in G[Km] if and only if S is convex in G and Tx = V (Km) for each x ∈ S ∩ IG(S). Theorem 19. Let G be a connected graph and Km the complete graph of order m. Then C = ⋃ x∈A [{x} × Tx], where A ⊆ V (G) and Tx ⊆ V (Km) for each x ∈ A, is a convex hop dominating set in G[Km] if and only if C = V (G[H]) or C satisfies the following conditions: (i) A is a convex hop dominating set in G. (ii) Tx = V (Km) for each x ∈ (A ∩ IG(A)) ∪ (A \N2 G(A)). Proof. Suppose C is a convex hop dominating set of G[Km]. By Theorem 15 and Theorem 18, A is a convex hop dominating set in G and Tx = V (Km) for each x ∈ (A ∩ IG(A)) ∪ (A \N2 G(A)). Hence, (i) and (ii) hold. Conversely, suppose that (i) and (ii) hold. Then, by Theorem 15 and Theorem 18, C is a convex hop dominating set in G[Km]. Corollary 11. Let G be a connected graph and Km the complete graph of order m. Then γconh(G[Km]) = min{|S|+(m−1)|S0∪(S\N2 G(S))| : S is a convex hop dominating set in G}, where S0 = S ∩ IG(S). REFERENCES 333 Proof. Let C = ⋃ x∈S [{x} × Tx] be a γconh-set of G[Km]. Then S is a convex hop dominating set and Tx = V (Km) for all x ∈ S0 ∪ (S \N2 G(S)) by Theorem 19. Since C is a γconh-set, |Tx| = 1 for all x ∈ S \ [S0 ∪ (S \N2 G(S))]. It follows that |C| = ∑ x∈[S0∪(S\N2 G(S))] |Tx|+ ∑ x∈S\[S0∪(S\N2 G(S))] |Tx| = m|S0 ∪ (S \N2 G(S))|+ |S| − |S0 ∪ (S \N2 G(S))| = |S|+ (m− 1)|S0 ∪ (S \N2 G(S))|. This proves the desired equality. It is worth mentioning that the value of the parameter given in Corollary 11 is not necessarily attained when S is a γconh-set in G. To see this, consider P5[K3]. It is easily verified that γconh(P5) = 2. If S is γconh-set in P5, then ⟨S⟩ = K2 and S0∪(S\N2 G(S)) = S. Hence, |S| + (3 − 1)|S0 ∪ (S \ N2 G(S))| = 6. However, by taking any three consecutive vertices of P5, one can see that γconh(P5[K3]) = 5. 4. Conclusion The concept of convex hop domination has been introduced and initially investigated in this study. Graphs which attained some specific convex hop domination number have been characterized. The convex hop domination number of the complementary prism has been obtained and necessary and sufficient conditions for a subset to be convex hop dominating in the shadow graph, join, corona, and lexicographic product of two graphs have been obtained. 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