EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 16, No. 2, 2023, 1196-1211 ISSN 1307-5543 – ejpam.com Published by New York Business Global Weakly Convex Hop Dominating Sets in Graphs Sergio R. Canoy, Jr.1,2, Javier A. Hassan3,∗ 1 Department of Mathematics and Statistics, College of Science and Mathematics, Center for Graph Theory, Premier Research Institute of Science and Mathematics, MSU-Iligan Institute of Technology, 9200 Iligan City, Philippines 2 Center for Mathematical and Theoretical Physical Sciences, Premier Research Institute of Science and Mathematics, MSU-Iligan Institute of Technology, 9200 Iligan City, Philippines 3 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 an undirected connected graph with vertex and edge sets V (G) and E(G), respectively. A set C ⊆ V (G) is called weakly convex hop dominating if for every two vertices x, y ∈ C, there exists an x-y geodesic P (x, y) such that V (P (x, y)) ⊆ C and for every v ∈ V (G)\C, there exists w ∈ C such that dG(v, w) = 2. The minimum cardinality of a weakly convex hop dominating set of G, denoted by γwconh(G), is called the weakly convex hop domination number of G. In this paper, we introduce and initially investigate the concept of weakly convex hop domination. We show that every two positive integers a and b with 3 ≤ a ≤ b are realizable as the weakly convex hop domination number and convex hop domination number of some connected graph. Furthermore, we characterize the weakly convex hop dominating sets in some graphs under some binary operations. 2020 Mathematics Subject Classifications: 05C69 Key Words and Phrases: Weakly convex set, weakly convex hop dominating set, weakly convex hop domination number 1. Introduction Domination has been one of the interesting and widely studied topics in graph theory. A number of concepts had already been used to introduce variations of the standard concept of domination. In particular, concepts such as convex and weakly convex had been considered to define convex domination and weakly convex domination, respectively (see [4], [12], [14], [15], [16], [17], [22], [23]). Some variations of domination can be found in [5], [6], [10], and [20]. ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v16i2.4761 Email addresses: sergio.canoy@g.msuiit.edu.ph (S. Canoy), javierhassan@msutawi-tawi.edu.ph (J. Hassan) https://www.ejpam.com 1196 © 2023 EJPAM All rights reserved. S. Canoy Jr., J. Hassan / Eur. J. Pure Appl. Math, 16 (2) (2023), 1196-1211 1197 Recently, Natarajan et al. in [19] introduced and studied the concept of hop domina- tion. Follow-up studies are done in [2], [3], [11], and [13]. Since its introduction, a number of variations of the concept have already been defined and studied (see [7], [8], [18], [21], and [24]). Previously, the authors (see [9]) introduced and made an initial investigation of convex hop domination. Motivated by the studies on weakly convex domination and hop domination, we introduce and study a new variant of hop domination that incorpo- rates the concept of weakly convex. This new parameter lies between the connected hop domination number and convex hop domination number of a graph. 2. Terminology and Notation Let G = (V (G), E(G)) be a simple undirected graph and let u and v be vertices of G. The distance dG(u, v) of u and v is the length of a shortest path joining them. 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 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}. The open neighborhood of X ⊆ V (G) is the set NG(X) = ⋃ w∈X NG(w). The closed neighborhood of X is the set NG[X] = NG(X) ∪X. A set D ⊆ V (G) is dominating (total dominating) in 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)). 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 hop dominating (total hop dominating) in G if N2 G[S] = V (G) (resp. N2 G(S) = V (G)), that is, for every v ∈ V (G)\S (resp. v ∈ V (G)), there exists u ∈ S such that dG(u, v) = 2. The minimum cardinality among all hop dominating (resp. total hop dominating) sets in G, denoted by γh(G) (resp. γth(G)), is called the hop domination number (resp. total hop domination number) of G. Any hop dominating (resp. total hop dominating) set with cardinality equal to γh(G) (resp. γth(G)) is called a γh-set (resp. γth-set). A hop dominating set S is connected hop dominating if ⟨S⟩ is connected. The minimum cardinality among all connected hop dominating sets in 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 if for every two vertices x, y ∈ C, 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) is convex dominating (resp. convex hop dominating) if C is both convex and dominating (resp. convex and hop dominating). The minimum cardinality S. Canoy Jr., J. Hassan / Eur. J. Pure Appl. Math, 16 (2) (2023), 1196-1211 1198 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 set W ⊆ V (G) is weakly convex if for every two vertices x, y ∈ W , there exists an x-y geodesic P (x, y) such that V (P (x, y)) ⊆ W . The largest cardinality of a proper weakly convex set in G, denoted by wcon(G), is called the weakly convexity number of G. A set W ⊆ V (G) is weakly convex dominating (resp. weakly convex hop dominating, weakly convex total hop dominating) if C is both weakly convex and dominating (resp. weakly convex and hop dominating, weakly convex and total hop dominating). The minimum cardinality among all weakly convex dominating (resp. weakly convex hop dominating, weakly convex total hop dominating) sets in G, denoted by γwcon(G) (resp. γwconh(G), γwconth(G)), is called the weakly convex domination number (resp. weakly convex hop domination number, weakly convex total hop domination number) ofG. Any weakly convex dominating (resp. weakly convex hop dominating, weakly convex total hop dominating) set with cardinality equal to γwcon(G) (resp. γwconh(G), γwconth(G)) is called a γwcon-set (resp. γwconh-set, γwconth-set). A set C ⊆ V (G) is pointwise non-dominating 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 the pointwise non-domination number of 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 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 formed 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. S. Canoy Jr., J. Hassan / Eur. J. Pure Appl. Math, 16 (2) (2023), 1196-1211 1199 3. Results Proposition 1. Let G be any connected graph G on n ≥ 2 vertices. Then the following hold: (i) If S is a weakly convex hop dominating set in G, then the induced graph ⟨S⟩ is connected. (ii) γch(G) ≤ γwconh(G) ≤ γconh(G) and every equality and strict inequality can be at- tained. Proof. (i) Let S be a weakly convex hop dominating set in G and let x, y ∈ S with x ̸= y. Since S weakly convex, there exists an x-y geodesic P (x, y) such that V (P (x, y) ⊆ S. This implies that the induced graph ⟨S⟩ is connected. (ii) Since every weakly convex hop dominating set in G is connected hop dominating, γch(G) ≤ γwconh(G). Also, since every convex hop dominating set in G is weakly convex hop dominating, γwconh(G) ≤ γconh(G). For equality, consider the graph G in Figure 1. Let W = {v1, v2, v3, v4}. Then W is both a γch-set, a γwconh-set, and a γconh-set of G. Thus, γch(G) = γwconh(G) = γconh(G) = 4. G : v3v4 v1 v5 v2 Figure 1: A graph G with γch(G) = γwconh(G) = γconh(G) = 4 For strict inequalities, consider first C14. Then it can be verified that γch(C14) = 10 and γwconh(C14) = 14. Thus, γch(C14) < γwconh(C14). Next, consider the graph G′ in Figure 2. Let W ′ = {u1, u2, u3} and W ′′ = {u1, u2, u3, u4, u5}. Then W ′ and W ′′ are γwconh-set and γconh-set of G ′, respectively. Accordingly, γwconh(G ′) = 3 < 5 = γconh(G ′). S. Canoy Jr., J. Hassan / Eur. J. Pure Appl. Math, 16 (2) (2023), 1196-1211 1200 G′ : u5 u3u1 u2 u4 Figure 2: A graph G′ with γwconh(G ′) = 3 < 5 = γconh(G ′) This completes the proof of the assertion. Theorem 1. Let G be any connected graph on n ≥ 2 vertices. Then 2 ≤ γwconh(G) ≤ n. Moreover, each of the following holds: (i) γwconh(G) = 2 if and only if there exist two adjacent vertices x and y of G such that NG(x) ∩ NG(y) = ∅ and for each z ∈ V (G) \ NG({x, y}), there exists w ∈ [NG({x, y}) \ {x, y}] ∩NG(z). (ii) γconh(G) = n if and only if for every connected hop dominating set S ̸= V (G), there exists a pair of distinct vertices x and y in S such that dG(x, y) < d⟨S⟩(x, y). Proof. Since G is a non-trivial graph, every hop dominating has at least two elements. Hence, 2 ≤ γwconh(G) ≤ n. (i) Suppose that γwconh(G) = 2, say S = {x, y} is a γwconh-set in G. By Proposition 1(i), xy ∈ E(G). Since S is a hop dominating set in G, NG(x) ∩ NG(y) = ∅. Let z ∈ V (G) \ NG({x, y}). Since S is a hop dominating set, it follows that z ∈ N2 G({x, y}). We may assume that z ∈ N2 G(x). Let [z, w, x] be a z-x geodesic. Since z /∈ NG({x, y}), w ̸= y. Thus, w ∈ [NG({x, y}) \ {x, y}] ∩NG(z). For the converse, suppose there exist adjacent vertices x and y of G such that NG(x) ∩ NG(y) = ∅ and for each z ∈ V (G) \ NG({x, y}), there exists w ∈ [NG({x, y})\{x, y}]∩NG(z). Let S = {x, y} and let v ∈ V (G)\S. If v ∈ NG({x, y}), then by assumption dG(x, v) = 2 or dG(y, v) = 2. If v ∈ V (G) \ NG({x, y}), then there exists w ∈ [NG({x, y}) \ {x, y}] ∩NG(v) by assumption. With the assumption that NG(x)∩NG(y) = ∅, v ∈ N2 G({x, y}). Therefore, S is a weakly convex hop dominating set in G and γwconh(G) = |S| = 2. S. Canoy Jr., J. Hassan / Eur. J. Pure Appl. Math, 16 (2) (2023), 1196-1211 1201 (ii) Suppose that γwconh(G) = n. Suppose further that there exists a connected hop dominating set S ̸= V (G) such that dG(a, b) = d⟨S⟩(a, b) for each pair of distinct vertices a and b in S. Since ⟨S⟩ is connected, there exists an a-b geodesic in ⟨S⟩ for each pair of distinct vertices a and b in S. Hence, S is a weakly convex hop dominating set in G, contrary to the assumption that γwconh(G) = n. Therefore, for each connected hop dominating set S in G, there exist distinct vertices x and y in S such that dG(x, y) < d⟨S⟩(x, y). Conversely, suppose that for each connected hop dominating set S in G, there exist distinct vertices x and y in S such that dG(x, y) < d⟨S⟩(x, y). Let S0 be a γwconh-set in G. Suppose S0 ̸= V (G). Then, by Proposition 1(i), S0 is a connected hop dominating set. Hence, by assumption, there exists a pair of distinct vertices p, q ∈ S0 such that dG(p, q) < d⟨S0⟩(p, q). This implies that every p-q path in ⟨S0⟩ is not a geodesic in G. Hence, S0 is not a weakly convex set, a contradiction. Therefore, S0 = V (G), showing that γwconh(G) = n. Since γh(Kn) = n, it follows that γwconh(G) = n. The same conclusion can be deduced from Theorem 1 because V (Kn) is the only (connected) hop dominating set of Kn. Corollary 1. Let n be a positive integer. Then γwconh(Kn) = n. Observe that for a non-trivial connected graph G, γch(G) = 2 (or γconh(G) = 2) is equivalent to the condition given in Theorem 1(i). The next result states this formally. Corollary 2. Let G be any connected graph on n ≥ 2 vertices. Then the following statements are equivalent: (i) γch(G) = 2. (ii) γwconh(G) = 2. (iii) 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 γwconh(G) = a and γconh(G) = b. Proof. For a = b, consider the complete graph Ka = G. Then γwconh(G) = a = γconh(G). Suppose a < b. Consider the following two cases: Case 1: a = 3. Let m = b − a and consider the graph G in Figure 3. Let W = {w1, w2, w3} and W ′ = {w1, w2, w3, v1, v2, . . . , vm}. Then W and W ′ are γwconh-set and γconh-set in G, respectively. Therefore, γwconh(G) = a and γconh(G) = a+m = b. Case 2: a ≥ 4. Let m = b − a and consider the graph G′ in Figure 4. Let W1 = {x1, x2, . . . , xa} and W2 = {x1, x2, . . . , xa, y1, y2, . . . , ym}. Then W1 and W2 are γwconh-set and γconh-set in G′, respectively. Hence, γwconh(G ′) = a and γconh(G ′) = a+m = b. S. Canoy Jr., J. Hassan / Eur. J. Pure Appl. Math, 16 (2) (2023), 1196-1211 1202 v1 v2 v3 vm G : . . . w1 w2 w3 Figure 3: A graph G with γwconh(G) < γconh(G). v1 v2 v3 vn u1 u2 ua−3u3 . . .G′ : . . . ua−2 ua−1 ua Figure 4: A graph G′ with γwconh(G ′) < γconh(G ′). Therefore, the assertion holds. Corollary 3. Let n be a positive integer. Then there exists a connected graph G such that γconh(G)− γwconh(G) = n. In other words, γconh − γwconh can be made arbitrarily large. Proposition 2. Let n be any positive integer. Then each of the following holds. (i) γwconh(Pn) = { 2 if n = 2, 3, 4, 5, 6 n− 4 if n ≥ 7. (ii) γwconh(Cn) =  2 if n = 4, 5 3 if n = 3 n− 4 if 6 ≤ n ≤ 10 n if n ≥ 11. S. Canoy Jr., J. Hassan / Eur. J. Pure Appl. Math, 16 (2) (2023), 1196-1211 1203 Proof. (i) Clearly, γwconh(Pn) = 2 for n ∈ {2, 3, 4, 5, 6}. Suppose n ≥ 7. Let Pn = [v1, v2, . . . , vn] and consider W = {v3, v4, . . . , vn−3, vn−2}. Clearly, W is weakly convex set in Pn. Now, observe that N 2 G[W ] = V (Pn) and so W is a hop dominating set in Pn. Therefore, W is a weakly convex hop dominating set in Pn. Notice that every weakly convex hop dominating set in Pn contains W . It follows that W is a γwconh-set of Pn. Hence, γwconh(Pn) = n− 4 for all n ≥ 7. (ii) Clearly, γwconh(Cn) = 2 for n ∈ {4, 5} and γwconh(C3) = 3. Suppose 6 ≤ n ≤ 10. Let Cn = [v1, v2, . . . , vn, v1] and let W ′ = {v1, v2, . . . , vn−4}. Then W ′ is a minimum weakly convex hop dominating set in Cn. Thus, γwconh(Cn) = n − 4 for all n, where 6 ≤ n ≤ 10. Next, suppose that n ≥ 11. Clearly, γch(Cn) = n− 4 and if S is a connected hop dominating set in Cn with S ̸= V (Cn), then S satisfies the property given in Theorem 1(ii). Hence, γwconh(Cn) = n. Theorem 3. [9] Let G be a connected graph of order n. Then γconh(GG) = 2. In particular, {u, u} is a γconh-set of GG for each u ∈ V (G). The next result is immediate from Corollary 2 and Theorem 3. Theorem 4. Let G be any connected graph of order n. Then γwconh(GG) = 2. In particular, W = {a, a} is a γwconh-set of GG for any a ∈ V (G). Theorem 5. [9] 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, respectively. Theorem 6. Let G be a non-trivial connected graph. Then W is a weakly convex hop dominating set in S(G) if and only if one of the following conditions holds: (i) W is weakly convex hop dominating set in G1. (ii) W is weakly convex hop dominating set in G2. (iii) W = WG1 ∪ WG2 such that WG1 ∪ W ′ G2 and W ′ G1 ∪ WG2 are weakly convex hop dominating sets in G1 and G2, respectively, where W ′ G2 = {a ∈ V (G1) : a ′ ∈ WG2} and W ′ G1 = {a ∈ V (G2) : a ′ ∈ WG1} Proof. Let W be a weakly convex hop dominating set in S(G). Let WG1 = W ∩V (G1) and WG2 = W ∩ V (G2). If WG2 = ∅, then W = WG1 is a weakly convex hop dominating set in G1, showing that (i) holds. Similarly, ifWG1 = ∅, thenW = WG2 is a weakly convex S. Canoy Jr., J. Hassan / Eur. J. Pure Appl. Math, 16 (2) (2023), 1196-1211 1204 hop dominating set in G2. Hence, (ii) holds. Next, suppose WG1 ̸= ∅ and WG2 ̸= ∅. Then WG1 ∪ W ′ G2 is a hop dominating set in G1 by Theorem 5. Let x, y ∈ WG1 ∪ W ′ G2 where x ̸= y. Suppose x, y ∈ WG1 . Since W is weakly convex, there exists an x-y geodesic P (x, y) in S(G) such that V (P (x, y)) ⊆ W . Let P (x, y) = [x1, x2, · · · , xk], where x1 = x and xk = y. Let j ∈ {2, · · · , k − 1} such that xj ∈ WG2 , say xj = y′j . Then yj ∈ W ′ G2 . Replacing each xj ∈ WG2 by yj , we obtain an x-y geodesic P ′(x, y) in G1 such that V (P ′(x, y)) ⊆ WG1 ∪W ′ G2 . Suppose now that x, y ∈ W ′ G2 . Then x′, y′ ∈ WG2 ⊂ W . By assumption, we may let P (x′, y′) = [v1, v2, · · · , vt], where x′ = v1 and y′ = vt, be an x′-y′ geodesic in S(G) such that V (P (x′, y′)) ⊆ W . Let r ∈ {1, 2, · · · , t} such that vr ∈ WG2 , say vr = z′r. Then zr ∈ W ′ G2 . Note that z1 = x and zt = y. Replacing each vr ∈ WG2 by zj , we obtain an x-y geodesic P ∗(x, y) in G1 such that V (P ∗(x, y)) ⊆ WG1 ∪W ′ G2 . Finally, suppose that x ∈ WG1 and y ∈ W ′ G2 . Then y′ ∈ WG2 . Let P (x, y′) = [q1, q2, · · · , qm], where x = q1 and y′ = qm, be an x-y′ geodesic in S(G) such that V (P (x, y′)) ⊆ W . Replacing each qi = p′i ∈ WG2 by pi ∈ W ′ G2 (pm = y), we obtain an x-y geodesic P ∗∗(x, y) such that V (P ∗∗(x, y)) ⊆ WG1 ∪ W ′ G2 . Therefore, WG1 ∪ W ′ G2 is a weakly convex set in G1. Similarly, W ′ G1 ∪WG2 is weakly convex in G2. This shows that (iii) holds. For the converse, suppose first that (i) or (ii) holds. Then W is hop dominating by Theorem 5. Since weakly convex sets in G1 and G2 are weakly convex sets in S(G), W is weakly convex. Suppose (iii) holds. Again, by Theorem 5, W is a hop dominating set in S(G). By the additional assumption, W is weakly convex in S(G). Therefore, W is a weakly convex hop dominating set in S(G). The next result follows from Theorem 6. Corollary 4. Let G be a non-trivial connected graph. Then γwconh(S(G)) = γwconh(G). Theorem 7. [1] Let G be a graph of order n. Then 1 ≤ pnd(G) ≤ n. Moreover, (i) pnd(G) = 1 if and only if G has an isolated vertex. (ii) pnd(G) = n if and only if G = Kn. Corollary 5. [1] Let n be any positive integer. Then (i) pnd(Pn) = 2 for any n ≥ 2. (ii) pnd(Cn) = 2 for any n ≥ 4. Theorem 8. [13] 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. Corollary 6. [13] Let G and H be any two graphs. Then γh(G+H) = pnd(G) + pnd(H). Theorem 9. Let G and H be any two graphs. Then W ⊆ V (G + H) is weakly convex hop dominating in G+H if and only it is hop dominating in G+H. S. Canoy Jr., J. Hassan / Eur. J. Pure Appl. Math, 16 (2) (2023), 1196-1211 1205 Proof. SupposeW is a weakly convex hop dominating set inG+H. Then, by definition, W is hop dominating in G+H. For the converse, suppose that W is a hop dominating set in G + H. By Theorem 8, W = WG + WH where WG and WH are pointwise non-dominating sets in G and H, respectively. Let v, w ∈ W and v ̸= w. If dG+H(v, w) = 1, then IG+H [v, w] = {v, w} ⊆ W . Suppose that dG+H(v, w) = 2. Then v, w ∈ WG or v, w ∈ WH . We may assume that v, w ∈ WG. Pick any z ∈ WH . Then P (v, w) = [v, z, w] is a v-w geodesic in G +H and V (P (v, w)) = {v, z, w} ⊆ W . Thus, W is weakly convex hop dominating in G+H. The next result follows from Theorem 9 and Corollary 6. Corollary 7. Let G and H be two graphs. Then γwconh(G+H) = pnd(G) + pnd(H). In particular, we have (i) γwconh(Pn + Pm) = 4 for all n,m ≥ 2; (ii) γwconh(Cn + Cm) = 4 for all n,m ≥ 4; (iii) γwconh(Fn) = 3 for all n ≥ 2; (iv) γwconh(Wn) = 3 for all n ≥ 4; and (v) γwconh(K1,n) = 2 for all n ≥ 1. The result that follows is a restatement of a result in [13]. Theorem 10. 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 11. Let G be a non-trivial connected graph and let H be any graph. Then W is a weakly convex hop dominating set in G ◦H if and only if W = B ∪ (∪v∈V (G)Wv), where B ⊆ V (G), Wv ⊆ V (Hv) for each v ∈ V (G), and satisfies the following conditions: (i) B is a weakly convex dominating set in G. (ii) Wv = ∅ for each v ∈ V (G) \B. (iii) For each a ∈ V (G) \ B, there exists b ∈ B with dG(a, b) = 2 or there exists y ∈ B ∩NG(a) such that Wy ̸= ∅. S. Canoy Jr., J. Hassan / Eur. J. Pure Appl. Math, 16 (2) (2023), 1196-1211 1206 (iv) Wx is a pointwise non-dominating set in Hx for each x ∈ B \NG(B). Proof. Suppose W is a weakly convex hop dominating set in G ◦H. Then W is hop dominating and W = B ∪ (∪v∈V (G)Wv) where the sets B and Wvs satisfy (i) and (ii) of Theorem 10. Suppose B = ∅. Then Wv is pointwise non-dominating in Hv for each v ∈ V (G) by Theorem 10(ii). Let a, b ∈ V (G) (a ̸= b) and choose any p ∈ Wa and q ∈ Wb. Since every p-q geodesic in G ◦ H contains a and b, W is not weakly convex in G ◦ H, a contradiction. Hence, B ̸= ∅. Let w ∈ V (G) \ B and suppose that w /∈ NG(B). By Theorem 10(ii), Ww is a pointwise non-dominating set in Hw. Pick any p ∈ Ww and z ∈ B (z exists because B ̸= ∅). Then there exists no p-z geodesic P (p, z) in G ◦ H with V (P (p, z)) ⊆ W , contrary to the assumption that W is weakly convex. Thus, B is a dominating set in G. Let x and y be distinct vertices in B. Then x, y ∈ W . Since W is weakly convex in G ◦ H, there exists an x-y geodesic P (x, y) in G ◦ H such that V (P (x, y) ⊆ W . Clearly, P (x, y) is an x-y geodesic in G. Hence, V (P (x, y)) ⊆ B. Therefore, B is weakly convex in G, showing that (i) holds. Next, let v ∈ V (G) \ B. Suppose Wv ̸= ∅, say p ∈ Wv. Choose any z ∈ B. Since every p-z geodesic in G ◦ H contains v as a vertex, it follows that W is not weakly convex in G ◦H, a contradiction. Therefore, Wv = ∅, showing that (ii) holds. This and (i) in Theorem 10 imply that (iii) holds. Moreover, because B is a dominating set, and (ii) in Theorem 10 holds, condition (iv) also holds. For the converse, suppose that W = B ∪ (∪v∈V (G)Wv) and satisfies (i), (ii), (iii), and (iv). Then (i) and (ii) in Theorem 10 hold, that is, W is a hop dominating set in G ◦H. Next, let x, y ∈ W and let v, w ∈ V (G) such that x ∈ V (v + Hv) and y ∈ V (w + Hw). Consider the following cases: Case 1. x = v and y = w. Then x, y ∈ B. Since B is weakly convex in G, there exists an x-y geodesic P (x, y) in G (also an x-y geodesic in G ◦H) such that V (P (x, y)) ⊆ B ⊆ W . Case 2. x = v and y ∈ Ww (or x ∈ Wv and y = w). Then w ∈ B by (iv). By (ii), we may let P ′(x,w) be an x-w geodesic in G such that V (P (x,w)) ⊆ B ⊆ W . Let P ′(x,w) = [x1, x2, · · · , xk], where x = x1 and w = xk. Then P ∗(x, y) = [x1, x2, · · · , xk, y] is an x-y geodesic in G ◦H and V (P ∗(x, y)) ⊆ W . Case 3. x ∈ V (Hv) and y ∈ V (Hw). Then v, w ∈ B by (iv). Property (ii) will imply that there exists a v-w geodesic P (v, w) in G◦H such that V (P (v, w)) ⊆ W . Let P (v, w) = [v1, v2, · · · , vk], where v = v1 and w = vk. Then P (x, y) = [x, x1, x2, · · · , xk, y] is an x-y geodesic in G ◦H and V (P (x, y)) ⊆ W . Therefore, W is weakly convex in G ◦H. Accordingly, W is weakly convex hop dominating in G ◦H. Corollary 8. Let G be a non-trivial connected graph and let H be any graph. Then γwcon(G) ≤ γwconh(G ◦H) ≤ γhtwcon(G), S. Canoy Jr., J. Hassan / Eur. J. Pure Appl. Math, 16 (2) (2023), 1196-1211 1207 where γhtwcon(G) = min{|S| : S is hop dominating and weakly convex total dominating in G}. Note that the bounds in Corollary 8 are tight. In fact, γwconh(C4 ◦H) = 2 = γwcon(C4) for any graph H. Consider the graph G in Figure 5 and the sets W1 = {a, b, c} and W2 = W1 ∪ {v}. It can be verified that W1 is weakly convex dominating, W2 is weakly convex total dominating and hop dominating, and γwcon(G) = |W1| and γhtwcon(G) = |W2|. For any graph H, we find that γwcon(G) < γhtwcon(G) = γwconh(G ◦H) = 4. G : a b c v Figure 5: A graph G with γwcon(G) = 3 < 4 = γh twcon(G) = γwconh(G ◦H) The next result is found in [13]. Theorem 12. 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). Theorem 13. Let G and H be connected non-trivial graphs. Then C = ⋃ x∈S [{x}×Tx] is a weakly convex hop dominating set in G[H] if and only if the following conditions hold. (i) S is a weakly convex hop dominating set in G. (ii) Tx is a pointwise non-dominating set in H for each x ∈ S \N2 G(S). Proof. Suppose C is weakly convex hop dominating in G[H]. Then S is a hop dominating set in G and property (ii) holds by Theorem 12(ii). Let v, w ∈ S with v ̸= w. Choose any p ∈ Tv and q ∈ Tw. By assumption, there exists a (v, p)-(w, q) geodesic P ((v, p), (w, q)) = [(v1, p1), (v2, p2), · · · , (vk, pk)] in G[H], where (v, p) = (v1, p1) and (w, q) = (vk, pk), such that V (P ((v, p), (w, q))) ⊆ C. This implies that P (v, w) = [v1, v2, · · · , vk] is a v-w geodesic in G and V (P (v, w)) ⊆ S. This shows that S is weakly convex, showing that (i) holds. S. Canoy Jr., J. Hassan / Eur. J. Pure Appl. Math, 16 (2) (2023), 1196-1211 1208 For the converse, suppose that C satisfies conditions (i) and (ii). Then C is a hop dominating set G[H] by Theorem 12. Let (x, a), (y, b) ∈ C with (x, a) ̸= (y, b). Consider the following cases: Case 1. x ̸= y. Since S is weakly convex in G, there exists an x-y geodesic P (x, y) = [x1, x2, · · · , xt], where x = x1 and y = xt, such that V (P (x, y)) ⊆ S. Let a1 = a and at = b. For each i ∈ {2, . . . , t− 1}, choose any ai ∈ Txi . Then P ((x, a), (y, b)) = [(x1, a1), (x2, a2), · · · , (xt, at)] is an (x, a)-(y, b) geodesic in G[H] and V (P ((x, a), (y, b))) ⊆ C. Case 2. x = y. Then a ̸= b and dG[H]((x, a), (y, b)) = 2. Since ⟨S⟩ is non-trivial and connected, choose any z ∈ S∩NG(x) and let c ∈ Tz. Then P ((x, a), (y, b)) = [(x, a), (z, c), (y, b)] is an (x, a)-(y, b) geodesic in G[H] and V (P ((x, a), (y, b))) ⊆ C. Therefore, C is weakly convex in G[H]. The next result follows from Theorem 13. Corollary 9. Let G and H be non-trivial connected graphs. Then γwconh(G[H]) = min{|S∩N2 G(S)|+pnd(H)|S\N2 G(S)| : S weakly convex hop dominating in G}. Moreover, each of the following holds: (i) If γ(G) ̸= 1, then γwconh(G[H]) ≤ γwconth(G). (ii) If G = Kn, then γwconh(G[H]) = n · pnd(H). Proof. Let α = min{|S ∩ N2 G(S)| + pnd(H)|S \ N2 G(S)| : S is a weakly convex hop dominating set in G}. Let S0 be a weakly convex hop dominating set in G such that α = |S ∩ N2 G(S)| + pnd(H)|S \ N2 G(S)|. Choose any a ∈ V (H) and let D be a pnd- set in H. Set Tx = {a} if x ∈ S0 ∩ N2 G(S0) and Tx = D if x ∈ S0 \ N2 G(S0). Then C = ⋃ x∈S0 [{x} × Tx] is a weakly convex hop dominating set in G[H] by Theorem 13. Hence, γwconh(G[H]) ≤ |C| = |S ∩NG(S)|+ pnd(H)|S \N2 G(S)| = α. Next, suppose that C0 = ⋃ x∈A[{x} × Rx] is a γwconh-set in G[H]. By Theoren 13, A is weakly convex hop dominating in G and Rx is pointwise non-dominating in H for each x ∈ A \ N2 G(A). It follows that γwconh(G[H]) = |C0| ≥ |A ∩N2 G(A)| + pnd(H)|A \N2 G(A)| ≥ α, showing the desired equality. For (i), let S be a weakly convex total hop dominating set in G with |S| = γwconth(G) (S exists because γ(G) ̸= 1). Then S ∩ N2 G(S) = S and S \ N2 G(S) = ∅. Set Qx = {p}, where p ∈ V (H), for each x ∈ S. Then C = ⋃ x∈S [{x} × Qx] is a weakly convex hop dominating set in G[H] by Theorem 13. It follows that γwconh(G[H]) ≤ |C| = γwconth(G). For (ii), suppose that G = Kn. Let C = ⋃ x∈S [{x} × Tx] be a γwconh-set in G[H]. By Theorem 13, S is a weakly convex hop dominating set in G and Tx is a pointwise non-dominating set in H for each x ∈ S \N2 G(S). Since V (G) is the only hop dominating set in G, it follows that S = V (G). Also, since C is a γwconh-set, each Tx is a pnd-set in H. Hence, γwconh(G[H]) = |C| = n · pnd(H). REFERENCES 1209 Note that the bound in Corollary 9(i) is tight. To see this, consider G = P7. For any connected graph H, γwconh(P7[H]) = 4 = γwconth(P7). 4. Conclusion Weakly convex hop domination was introduced and initially investigated in this study. Characterizations of weakly convex hop dominating sets in the shadow graph, and in the join, corona, and lexicographic product of two graphs were formulated. These characteri- zations were used to determine the weakly convex hop domination number of each of these graphs. Moreover, it was shown that any two positive integers a and b with 3 ≤ a ≤ b are realizable as weakly connected hop domination number and convex hop domination number, respectively, of some connected graph. The parameter may be studied further for trees and other graphs. Also, bounds involving other known parameters may be obtained. Moreover, interested readers may find the complexity aspect of the weakly convex hop dominating set problem worth considering. 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, MSU-Iligan Institute of Technology, and MSU Tawi-Tawi College of Technol- ogy and Oceanography for funding this research. References [1] S. Arriola and S. Canoy Jr. (1,2) domination in graphs. Advances and Applications in Discrete Mathematics., 18(2):179–190, 2017. [2] S. Ayyaswamy, B. Krishnakumari, B. Natarjan, and Y. Venkatakrishnan. Bounds on the hop domination number of a tree. Proceedings-Mathematical Sciences., 125(4):449–455, 2015. [3] S. Ayyaswamy, C. Natarajan, and G. Sathiamoorphy. A note on hop domination number of some special families of graphs. International Journal of Pure and Applied Mathematics., 119(12):11465–14171, 2018. [4] B. Bresar, T. Gologranc, and T. Kos. Convex and isometric domination of (weak) domination pair graphs. Theoretical Computer Science., 730:32–43, 2018. [5] E. Cockayne, R. Dawes, and S. Hedetnniemi. Total domination in graphs. Networks, 10:211–219, 1980. [6] T. Daniel and S. Canoy Jr. Clique domination in graphs. Applied Mathematical Sciences, 9(116):5449–5455, 2015. REFERENCES 1210 [7] J. Hassan and S. Canoy Jr. Grundy hop domination in graphs. Eur. J. Pure Appl. Math., 15(4):1623–1636, 2022. [8] J. Hassan and S. Canoy Jr. Hop independent hop domination in graphs. Eur. J. Pure Appl. Math., 15(4):1783–1796, 2022. [9] J. Hassan and S. Canoy Jr. Convex hop domination in graphs. Eur. J. Pure Appl. Math., 16(1):319–335, 2023. [10] S. Hedetniemi and R. Lascar. Connected domination in graphs. Chapter 18 in Graph Theory and Combinatorics, London Academic Press., pages 209–217, 1984. [11] M. Henning and N. Rad. On 2-step and hop dominating sets in graphs. Graphs and Combinatorics., 33(4):913–927, 2017. [12] S. Canoy Jr. A short note on convexity and convex domination in g[km]. Applied Mathematical Sciences, 8(115):5737–5741, 2014. [13] S. Canoy Jr., R. Mollejon, and J. G. Canoy. Hop dominating sets in graphs under binary operations. Eur. J. Pure Appl. Math., 12(4):1455–1463, 2019. [14] M. Labendia and S. Canoy Jr. Convex domination in the composition and cartesian product of graphs. Czechoslovak Mathematical Journal, 62(4):1003–1009, 2012. [15] M. Lemanska. Weakly convex and convex domination numbers. Opusc. Math., 24:181– 188, 2004. [16] R. Leonida and S. Canoy Jr. Weakly convex and weakly connected independent domination in the corona of graphs. International Mathematical Forum., 31(8):1515– 1522, 2013. [17] R. Leonida and S. Canoy Jr. Weakly convexity and weakly convex domination in graphs. Applied Mathematical Sciences, 3(9):137–141, 2015. [18] J. Mohamad and H. Rara. Strong resolving hop domination in graphs. European Journal of Pure and Applied Mathematics., 16(1):131–143, 2023. [19] C. Natarajan and S. Ayyaswamy. Hop domination in graphs ii. Versita, 23(2):187– 199, 2015. [20] B. Omamalin, S. Canoy Jr., and H. Rara. Locating total dominating sets in the join, corona and composition of graphs. Applied Mathematical Sciences., 8(48):2363–2374, 2014. [21] Y. Pabilona and H. Rara. Connected hop domination in graphs under some binary operations. Asian-Eur. J. Math., 11(5):1850075–1–1850075–11, 2018. [22] J. Raczed and M. Lemanska. A note on weakly convex and convex domination num- bers of a torus. Discrete Appl. Math., (158):1708–1713, 2010. REFERENCES 1211 [23] M. Roscika. Convex and weak convex domination in prism graphs. Discussiones Mathematicae Graph Theory., 39:741–755, 2019. [24] G. Salasalan and S. Canoy Jr. Global hop domination numbers of graphs. Eur. J. Pure Appl. Math., 14(1):112–125, 2021.