EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 16, No. 1, 2023, 454-464 ISSN 1307-5543 – ejpam.com Published by New York Business Global Weakly Connected Hop Domination in Graphs Resulting from Some Binary Operations Jamil J. Hamja1,∗, Imelda S. Aniversario2, Catherine I. Merca2 1 Office of the Vice Chancellor for Academic Affairs, MSU-Tawi-Tawi College of Technology and Oceanography, 7500 Tawi-Tawi, Philippines 2 Department of Mathematics and Statistics, College of Science in Mathematics, MSU-Iligan Institute of Technology, 9200 Iligan City, Philippines Abstract. Let G = (V (G), E(G)) be a simple connected graph. A set S ⊆ V (G) is a weakly connected hop dominating set of G if for every q ∈ V \S, there exists r ∈ S such that dG(q, r) = 2, the subgraph weakly induced by S, denoted by ⟨S⟩w = ⟨NG[S], Ew⟩ where Ew = {qr ∈ E(G) : q ∈ S or r ∈ S} is connected and S is a dominating set of G. The minimum cardinality of a weakly connected hop dominating set of G is called weakly connected hop domination number and is denoted by γwh(G). In this paper, the authors show and explore the concept of weakly connected hop dominating set. The weakly connected hop dominating set of some special graphs, shadow of graphs, join, corona and Lexicographic product of two graphs are characterized. Also, the weakly connected domination number of the aforementioned graphs are determined. 2020 Mathematics Subject Classifications: 05C69 and 0576 Key Words and Phrases: Weakly connected set, hop dominating set, hop domination number, weakly connected hop dominating set, and weakly connected hop domination number 1. Introduction The weakly connected domination in graph was studied by Dunbar, et.al. in [3]. They considered the weakly connected domination number γw(G) of a graph G and some re- lated domination parameters. They have shown that the problem of computing γw(G) is NP-hard in general but linear for trees. In addition, several sharp upper and lower bounds for γw(G) are obtained, and it was further investigated by Domke, et. al. in [2]. They provided a constructive characterization for trees T for which γ(T ) = γwc(T ). Moreover, some related variations and parameters are studied in many classes of graphs including those results under some binary operations (see [4], [9], [10] and [11]). ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v16i1.4587 Email addresses: jamilhamja@msutawi-tawi.edu.ph (J. Hamja), imelda.aniversario@g.msuiit.edu.ph (I. Aniversario), catherine.merca@g.msuiit.edu.ph (C. Merca) https://www.ejpam.com 454 © 2023 EJPAM All rights reserved. J. Hamja, I. Aniversario, C. Merca / Eur. J. Pure Appl. Math, 16 (1) (2023), 454-464 455 The concept of hop dominating set of a graph G was investigated and introduced by Ayyaswamy, et. al. in [1]. Later on, they presented the strong equality of hop domination and hop independent domination numbers for trees. Hop domination numbers of shadow graph and mycielskian graph are also discussed in [8]. Canoy, et.al. in [7], revisited the concept of hop domination, related it with other domination concepts, and investigated it in graphs resulting from some binary operations. In this paper, we introduce the concept of weakly connected hop dominating set of a graph G. We show and investigate a parameter that is, defined in a manner that a well- known weakly connected set and hop dominating set are put into one. Indeed, notwith- standing hop dominating set of a graph G requires the subgraph weakly induced by S, denoted by ⟨S⟩w = ⟨NG[S], Ew⟩ where Ew = {qr ∈ E(G) : q ∈ S or r ∈ S} is connected. The motivation of introducing the concept is to give further investigation on weakly connected, hop domination and some of its variations. In fact, it can be shown that every weakly connected hop dominating set is a hop dominating set of a graph G. Therefore, the hop domination number a graph G is at most equal to the weakly connected hop domination number of a graph G. 2. Terminology and Notation A simple connected graph G = (V (G), E(G)), where V (G) is a vertex-set of G and E(G) is an edge-set of G. The elements of V (G) are called vertices and the cardinality |V (G)| of V (G) is the order of G. The elements of E(G) are called edges and the cardinality |E(G)| of E(G) is the size of G. Two vertices u, v of a graph G are adjacent, or neighbors, if uv is an edge of G. The set of neighbors of a vertex u in G is called the open neighborhood of u in G and is denoted by NG(u). The closed neighborhood of u in G is the set NG[u] = NG(u)∪{u}, defined by Harary in [5]. A set S ⊆ V (G) is a dominating set of G if NG[S] = V (G). The domination num- ber of G, denoted by γ(G), is the minimum cardinality among the dominating sets of G. A dominating set S with |S| = γ(G) is said to be γ-set of G, defined by J. Tarr, et.al in [12]. Let S ⊆ V (G). The subgraph weakly induced by S is the graph ⟨S⟩w = ⟨NG[S], Ew⟩, where Ew = {qr ∈ E(G) : q ∈ S or r ∈ S}, Patangan, et.al [9]. A dominating set S ⊆ V (G) is a weakly connected dominating set in G if the subgraph ⟨S⟩w weakly in- duced by S is connected. The weakly connected domination number γw(G) of G is the minimum cardinality among all weakly connected dominating sets of G. A weakly con- nected dominating set S with |S| = γw(G) is said to be γw-set of G, Sandueta, et.al in [10]. A set S ⊆ V (G) is a hop dominating set of G if for every q ∈ V (G) \ S, there exists r ∈ S such that dG(q, r) = 2. The hop domination number of G, denoted by γh(G), is the minimum cardinality among the hop dominating sets of G. Any hop dominating set S of G with |S|= γh(G) is refered to as γh − set of G, Ayyaswamy, et.al. in [1]. J. Hamja, I. Aniversario, C. Merca / Eur. J. Pure Appl. Math, 16 (1) (2023), 454-464 456 3. Results Definition 1. A set S ⊆ V (G) is a weakly connecetd hop dominating set of G if S is both a weakly connected set and hop dominating set of G. The minimum cardinality of a weakly connected hop dominating set of G is called weakly connected hop domination number and is denoted by γwh(G). A weakly connected hop dominating set S with |S| = γwh(G) is said to be γwh-set of G. Example 1. Let G be the graph in Figure 1 and S = {b, d, h} ⊆ V (G). Observe that for all vertices a, c, e, f, g ∈ V (G) \ S, there are b, d, h ∈ S such that dG(b, g) = 2, dG(d, f) = 2, dG(h, a) = 2, dG(h, e) = 2, and dG(h, c) = 2, and NG[S] = V (G). Thus, S is a hop dominating set. Moreover, ⟨S⟩w = ⟨NG(S), {ba, bc, bh, dc, de, dh, hg, hf}⟩ is connected. Therefore, S is a weakly connected hop dominating set of G. a b d e h g f c G : a b d e h g f c ⟨S⟩w : Figure 1: Graph G with γwh(G) = 3 and its ⟨S⟩w Remark 1. Every weakly connected hop dominating set of a connected graph G is a weakly connected dominating set of G. Proposition 1. Every weakly connected hop dominating set of a connected graph G is a hop dominating set of G. Thus, γh(G) ≤ γwh(G). Proof. Let S ⊆ V (G) be a weakly connected hop dominating set of G. Then S is a weakly connected and a hop dominating set of G. Thus, the inequality holds. Proposition 2. Let G be any connected graph. Then S ⊆ V (G) is weakly connected hop dominating set of G if and only if S is hop dominating set of G. Proof. By Proposition 1, we only need to show that a hop dominating set S of G is a weakly connected hop dominating set of G. Suppose there exists a set S ⊆ V (G) such that S is not a weakly connected hop dominating set. Since by assumption, S is hop dominating set of G. It follows that for all q ∈ V (G) \ S, there exists r ∈ S such that dG(q, r) = 2. Thus, rq ∈ E(⟨S⟩w) where q ∈ V (G) \ S and r ∈ S. Hence, ⟨S⟩w = ⟨NG[S], Ew⟩ with J. Hamja, I. Aniversario, C. Merca / Eur. J. Pure Appl. Math, 16 (1) (2023), 454-464 457 Ew = {rq ∈ E(G) : r ∈ S or q ∈ S}. Finally, since for all q ∈ V (G) \ S, there exists r ∈ S such that dG(q, r) = 2 which implies that there exists w ∈ V (G) with qw, wr ∈ E(⟨S⟩w). Thus, ⟨S⟩w is connected. Consequently, S is a weakly connected hop dominating set of G. Remark 2. For any connected graph G of order n ≥ 3, then 2 ≤ γh(G) ≤ γwh(G) ≤ n. Theorem 1. Let G be any connected graph of order n ≥ 3. Then γwh(G) = 2 if and only if γh(G) = 2. Proof. If γh(G) = 2. By Remark 2, γwh(G) = 2. Let S = {v1, v2} be a weakly connected hop dominating set of G. Then by Remark 1, S is a hop dominating set of G. Therefore, γh(G) = 2. Theorem 2. Let G be any connected graph of order n ≥ 3. Then, γwh(G) = n if and only if G = Kn. Proof. Let G be any connected graph of order n ≥ 3. If G = Kn, then γwh(G) = n. Suppose γwh(G) = n and G ̸= Kn. Let S ⊆ V (G) be a weakly connected hop dominating set of G. Then for all q ∈ V (G) \ S there exists r ∈ S such that dG(q, r) = 2. Thus, S = V (G) \ {q}. Consequently, γwh(G) ≤ |S| = n− 1, this contradicts to the assumption. Theorem 3. For a complete bipartite Km,n of order m,n ≥ 2, γwh(Km,n) = 2. Proof. Write Km,n = Km + Kn. Let X and Y be the partite sets of V (Km,n) where V (Km) = X and V (Kn) = Y and let S = {x1, y1} where x1 ∈ X and y1 ∈ Y be a weakly connected and dominating set of Km,n. Since for every vertex x ∈ X \ S there exists yi ∈ S, 1 ≤ i ≤ m, such that dG(x, yi) = 2, and for every vertex y ∈ Y \ S there exists xj ∈ S, 1 < j < n such that dG(xj , y) = 2. It follows that, S is a hop dominating set of Km,n. Thus, S is a weakly connected hop dominating set of Km,n. Consquently, γwh(Km,n) =|S| = 2. Corollary 1. Let G beany connected grap with γ(G) = 1. Then γwh(G) = 2 if and only if G = Km,n for m = 1 and n ≥ 2. Corollary 2. For a star Sn of order n ≥ 2, γwh(Sn) = 2. Theorem 4. Let G be a connected graph of order n = 4. Then γwh(G) = 2 if and only if G is either P4, C4, or S4. Proof. If G = P4, G = C4 or G = S4, then γwh(G) = 2. Let S = {v1, v2} be a weakly connected hop dominating set of G. Since S is a hop dominating set there exist q, r ∈ V (G) \ S such that dG(q, v1) = 2 and dG(r, v2) = 2. Now, observe that J. Hamja, I. Aniversario, C. Merca / Eur. J. Pure Appl. Math, 16 (1) (2023), 454-464 458 NG[v1] = {v1, q, v2} and NG(v2) = {v2, v1, r}. Thus, NG[S] = {q, r, v1, v2} = V (G) and E(⟨S⟩w) = {qv1, v1v2, v1r} = Ew which implies that ⟨Sw⟩ is connected for E(G) = {qv1, v1v2, v2r} or E(G) = {qv1, v1v2, v2r, rq}. Therefore, G is either path P4, cycle C4 or star S4. Proposition 3. Let n be a positive integer, then (i) For a path Pn, γwh(Pn) =  ⌊n+1 2 ⌋, if n = 3, 4, 5 ; 3r + 1, if n = 6r ; 3r + ⌊ s+1 2 ⌋, if n = 6r + s ; 1 ≤ s ≤ 5. (ii) For a cycle Cn, γwh(Cn) =  ⌊n+1 2 ⌋, if n = 4, 5 ; 3r + 1, if n = 6r ; 3r + ⌊ s+1 2 ⌋, if n = 6r + s ; 1 ≤ s ≤ 5. (iii) For a wheel Wn of order n with n− 1 spokes, γwh(Wn) = 3. (iv) For a fan Fn of order n, γwh(Fn) = 3. (v) For a Petersen graph P , γwh(P ) = 4. The shadow graph Sh(G) of a graph G is constructed by taking two copies of G, say G1 and G2. Join each vertex u ∈ V (G1) to the neighbors of the corresponding vertex u ′ ∈ V (G2), defined by Jayagopal, et.al. [6]. Theorem 5. (Natarajan, et. al., in [8]) For any connected graph G, γh(Sh(G)) = γh(G). Theorem 6. Let G be any graph of order n ≥ 3, then γwh(Sh(G)) = γwh(G). Proof. Let S be a weakly connected hop dominating set of G. By Remark 1, S is a hop dominating set of G and by Theorem 5, γh(Sh(G)) = γh(G). So we are left to show that ⟨S⟩w = ⟨NG[S], Ew⟩ is connected, where Ew = {uv′ ∈ E(Sh(G)) : u ∈ S or v ′ ∈ S}. Let u ′ ∈ V (Sh(G)) be a twin vertex of u ∈ V (G). For every vertex u ′ ∈ V (Sh(G)), there exists v ∈ S \ {u} such that v ∈ NSh(G)[v] ∪NSh(G)[u]. Now, u′ ∈ NSh(G)(v) implies that u ′ v ∈ E(Sh(G)). Hence, ⟨S⟩w is connected. Since dSh(G)(u ′ , v) = 1. This implies that dSh(G)(u ′ , u) = 2. Thus, S is a weakly connected hop dominating set of Sh(G). This concludes that γwh(Sh(G)) = γwh(G). The next results show the equality of γwh(Sh(G)) = γwh(G). Corollary 3. For a shadow graph of Pn, Cn, Kn, and Km,n with m,n vertices. (i) For a path Pn of order n, γwh(Sh(Pn)) =  ⌊n+1 2 ⌋, if n = 3, 4, 5 ; 3r + 1, if n = 6r ; 3r + ⌊ s+1 2 ⌋, if n = 6r + s ; 1 ≤ s ≤ 5. = γwh(Pn) J. Hamja, I. Aniversario, C. Merca / Eur. J. Pure Appl. Math, 16 (1) (2023), 454-464 459 (ii) For a cycle Cn of order n, γwh(Sh(Cn)) =  ⌊n+1 2 ⌋, if n = 4, 5 ; 3r + 1, if n = 6r ; 3r + ⌊ s+1 2 ⌋, if n = 6r + s ; 1 ≤ s ≤ 5. = γwh(Cn) (iii) For a complete graph Kn of order n, γwh(Sh(Kn)) = n = γwh(Kn). (iv) For a complete bipartite Km,n of order m,n ≥ 2, γwh(Sh(Km,n)) = 2 = γwh(Km,n). The join of two graphs G and H, denoted by G + H, is the graph with vertex-set V (G+H) = V (G) • ∪ V (H) and edge-set E(G+H) = E(G) • ∪E(H) • ∪ {uv : u ∈ V (G), v ∈ V (H)}, Harary [5]. Lemma 1. Let G+H be the join of G and H. Then S ⊆ V (G+H) is a weakly connected hop dominating set of G+H if and only if either S is a hop dominating set of G or S is a hop dominating set of H. Proof. Suppose S is a weakly connected hop dominating set of G + H. Suppose S ∩ V (G) ̸= ∅ and S ∩ V (H) ̸= ∅. Let u, v ∈ S and let u, v ∈ V (G) or u, v ∈ V (H). Pick u ∈ S ∩ V (G) and v ∈ S ∩ V (H). Then for every x ∈ V (G) \ S there exists u ∈ S ∩ V (G) such that dG(x, u) = 2 or for every x ∈ V (H) \ S, there exists v ∈ S ∩ V (H) such that dG(x, v) = 2. Thus, S is a hop dominating set of G or S is a hop dominating set of H. Conversely, suppose without loss of generality, S is a hop dominating set of G or S is a hop dominating set of H. It follows that S is a hop dominating set of G+H. Consider the following cases: Case i. Let u, v ∈ S. Pick x ∈ V (G). Then there exists xu, xv ∈ E(G + H) such that u, v ∈ S. Hence, x ∈ NG+H(S). It follows that, NG+H [S] = V (G). Hence, ⟨S⟩w is connected. Case ii. Let u, v ∈ S. Pick x ∈ V (H). Similar process to Case i, ⟨S⟩w is connected. Therefore, we conclude the fact that S is a weakly connected hop dominating set of G+H. Proposition 4. Let G + H be the join of G and H, A set S ⊆ V (G + H) is a weakly connected hop dominating set if and only if one of the following is satisfied: (i) S ⊆ V (G) and S is a weakly connected hop dominating set in G. (ii) S ⊆ V (H) and S is a weakly connected hop dominating set in H. (iii) S ∩ V (G) ̸= ∅ and S ∩ V (H) ̸= ∅. Theorem 7. Let G+H be the join of G of order m and H of order n with γh(G) = 2 or γh(H) = 2. Then γwh(G+H) = 4. J. Hamja, I. Aniversario, C. Merca / Eur. J. Pure Appl. Math, 16 (1) (2023), 454-464 460 Proof. Let S = {u1, u2, v1, v2} where u1, u2 ∈ V (G) and v1, v2 ∈ V (H). Observe that u1, u2, v1, v2 ∈ V (G+H) will compromise a set that will induce a complete graph of order 4 in G+H. By Theorem 2, γwh(K4) = 4. Also, for every vertex q ∈ V (G+H) \ S, there exists an element in S say u2 or v2 such that dG(q, u2) = 2 or dG(q, v2) = 2. Moreover, ⟨S⟩w is connected since for every vertex q in V (G+H) \S is adjacent to u1, u2 ∈ V (G) or v1, v2 ∈ V (H) and S is dominating. Hence, S is a weakly connected hop dominating set in G+H. Thus γwh(G+H) =|S|= 4. Corollary 4. The weakly connected hop domination number of the join of path and cycle graphs of order m,n ≥ 2 are given as follows: (i) γwh(Pm + Pn) = 4. (ii) γwh(Cm + Cn) = 4. (iii) γwh(Pm + Cn) = 4. Proposition 5. Let G +H be the join of G of order m ≥ 2 and H of order n ≥ 2, then γwh(G+H) = 4. The corona G ◦H of graphs G and H, is the graph obtained by taking one copy of G and |V (G)| copies of H, and then joining the ith vertex of G to every vertex of the ith copy of H. For every v ∈ V (G), denote by Hv the copy of H whose vertices are attached one by one to the vertex v. Subsequently, denote by v+Hv the subgraph of the corona G ◦H corresponding to the join ⟨{v}⟩+Hv, v ∈ V (G), Harary [5]. Theorem 8. Let G and H be connected graph of orders m and n, respectively. Let S ⊆ V (G ◦ H) be a weakly connected hop dominating set of G ◦ H and H1, H2, ...,Hn be the copies of H. Then (i) S ∩ V (H i) = ∅ for all 1 ≤ i ≤ n. (ii) S ∩ V (G) ̸= ∅. Proof. Let S ⊆ V (G ◦H) be γwh-set of G ◦H. (i) Suppose that S ∩ V (H i) = ∅ for some i, 1 ≤ i ≤ n. Since every vertex of H i for all i, 1 ≤ i ≤ n is adjacent to exactly one vertex of G. It follows that S is a γwh-set of G ◦H. (ii) Suppose that S ∩ V (G) ̸= ∅. Let u be a vertex of S ∩ V (G). Hence u ∈ S and u ∈ V (G). Since |S| = |V (G)|, dG(u, v) = 2 for all v ∈ V (H i) for all i, 1 ≤ i ≤ n. Therefore, S is a γwh-set of G ◦H. J. Hamja, I. Aniversario, C. Merca / Eur. J. Pure Appl. Math, 16 (1) (2023), 454-464 461 Theorem 9. (Sandueta, et. al., in [10]) Let G be a connected graph of order m ≥ 2 and H a graph of order n. Then S ⊆ V (G ◦H) is a weakly connected dominating set of G ◦H if and only if V (v+Hv)∩C is a dominating set of v+Hv for all v ∈ V (G) and V (G)∩C is a weakly connected dominating set of G. Theorem 10. Let G be a connected graph of order m ≥ 2 and H a graph of order n. Then S ⊆ V (G ◦ H) is a weakly connected hop dominating set of G ◦ H if and only if S ∩ V (v +Hv) is a dominating set of v +Hv for all v ∈ V (G) and S ∩ V (G) is a weakly connected hop dominating set of G. Proof. Suppose S is a weakly conncted hop dominating set of G◦H and u ∈ V (G). By Remark 1, S is a hop dominating set of G ◦H. By Theorem 9, if S is a weakly connected dominating set of G◦H and u ∈ V (G), then S∩V (v+Hv) is a dominating set of (v+Hv). Since S is a weakly connected hop dominating set of G ◦ H, S ∩ V (G) ̸= ∅ (unless, ⟨S⟩w is disconneted). By Theorem 9, S ∩ V (G) is a weakly connected dominating set of G. So we are left to show that S ∩ V (G) is hop dominating set of G. Let u ∈ S ∩ V (G) for all u ∈ S and u ∈ V (G). Then |S|= |V (G)|. Since every vertex of Hv is adjacent to exactly one vertex of V (G), say x ∈ V (v +Hv), this implies that for each x ∈ V (v +Hv), there exists u ∈ S ∩ V (G) such that dG(u, x) = 2. Therefore, the conclusion follows. Conversely, assume that S ∩ V (v+Hv) is a dominating set of v+Hv for all v ∈ V (G) and S ∩ V (G) is a weakly connected hop dominating set of G. Suppose S ⊆ V (G ◦H) is not a dominating set of G ◦H. Let x ∈ V (G) \ S or x ∈ V (v+Hv) \ S. Then there exists u ∈ V (G ◦H) \ S such that ux ∈ E(G ◦H), which is a contradiction in both cases. Suppose ⟨S⟩w is disconnected. Then ⟨S ∩ V (G)⟩w must be disconnected. This implies that S ∩ V (G) is not weakly connected hop dominating set of G, which is a contradiction to our assumption. Therefore, S ⊆ V (G ◦H) is a weakly connected hop dominating set of G ◦H. Theorem 11. Let G be any connected graph of order m ≥ 2 and H a graph of order n. A set S ⊆ V (G ◦H) is a weakly connected hop dominating set if |S|= |V (G)|. Proof. Let S1 = V (G). Then by Theorem 10, S1 is a weakly connected hop domi- nating set of G ◦ H. Thus, γwh(G ◦ H) ≤ |S1| = |V (G)|. Next, suppose S2 is a γwh-set of G ◦ H. Then |S2 ∩ V (v + Hv)| = 1 for all v ∈ V (G) since every vertex of Hv is ad- jacent to exactly one vertex of V (G). Thus, γwh(G ◦ H) = |S2| = |V (G)|. Therefore, γwh(G ◦H) = |V (G)|. Proposition 6. Let G be a complete graph of order m ≥ 2 and H be any noncomplete graph of order n. Then γwh(G ◦H) = |V (G)|. Proposition 7. Let G be a path of order m ≥ 2 and H be any noncomplete graph of order n. Then γwh(G ◦H) = |V (G)|. Proposition 8. For any positive integers m,n ≥ 2, we have (i) γwh(Pm ◦Kn) = |V (Pm)|. J. Hamja, I. Aniversario, C. Merca / Eur. J. Pure Appl. Math, 16 (1) (2023), 454-464 462 (ii) γwh(Cm ◦Kn) = |V (Cm)|. The lexicographic product of graphs G and H, denoted by G[H], is the graph with vertex - set V (G[H]) = V (G)×V (H) such that (v, a)(u, b) ∈ E(G[H]) if and only if either uv ∈ E(G) or u = v and ab ∈ E(H). Observe that every non-empty set F ⊆ V (G)×V (H) can be expressed as F = ⋃ x∈s [{x}× Tx], where S ⊆ V (G), Tx ⊆ V (H) for each x ∈ S and Tx = {y ∈ V (H)|(x, y) ∈ C}, Harary [5]. Lemma 2. For any non-trivial connected graphs G and H, a set F = ⋃ x∈s [{x} × Tx] ⊆ V (G[H]) where S ⊆ V (G) and Tx ⊆ V (H) for all x ∈ S, is a weakly connected hop dominating set of G[H] if and only if S is a weakly connected hop dominating set of G. Proof. Suppose F is a weakly connected hop dominating set of G[H]. Let S ⊆ V (G) and y /∈ V (G) \ S. Choose k ∈ V (H). Then (y, k) /∈ V (G[F ]). Observe that for (y, k) /∈ V (G[H]) \ F , there exists (x, l) ∈ F such that dG[H]((y, k), (x, l)) = 2. for all x ∈ S and l ∈ Tx. It follows that dG(x, l) = 2. Thus, S is a hop dominating set of G. Now, let w, x ∈ S, with x ̸= y and xy ∈ E(G). If m ∈ Tw and l ∈ Tx. Then (m,w), (l, x) ∈ F and dG[H]((m,w), (x, l)) = 1. This implies that (m,w)(x, l) ∈ E(G[H]). Since dG[H]((y, k), (x, l)) = 2 for all (y, k) ∈ V (G[H]) \ F , (y, k)(m,w), (y, k)(x, l) ∈ E(G[H]). It follows that xy, kl ∈ E(G[H]) for all x, y ∈ S. Hence, NG(S) = V (G). So, ⟨S⟩w is connected. Therefore, S is a weakly connected hop dominating set of G. Conversely, suppose that ⟨S⟩w is connected and S is a weakly connected hop dominating set of G. Observe that for every w ∈ S, there exists y ∈ S such that dG(w, y) = 2. Choose m ∈ Tx. Then (m,x) ∈ F and dG[H]((y, k), (x,m)) = 2. Hence, (y, k)(x, l) ∈ E(G[H]) for all (y, k) ∈ V (G(H)) \ F . Thus, NG[H](F ) = V (G[H]). Therefore, ⟨F ⟩w is connected. Consequently, F is a weakly connected hop dominating set of G[H]. Theorem 12. For any non-trivial connected graphs G and H. a set F = ⋃ x∈s [{x} × Tx] ⊆ V (G[H]) where S ⊆ V (G) and Tx ⊆ V (H) for all x ∈ S, is a weakly connected hop dominating set of G[H] if and only if S is a weakly connected hop dominating set of G and Tx is a dominating set in H for all x ∈ S. Proof. Suppose F is a weakly connected hop dominating set of G[H]. Let y ∈ V (G)\S and choose k ∈ V (H). Then (y, k) ∈ V [G]. Since F is a dominating set of G[H], there exists (x, l) ∈ F such that (y, k)(x, l) ∈ E(G[H]). It follows that x ∈ S and y ∈ NG(x). This shows that S is a dominating set of G. Furthermore, by Lemma 2, ⟨S⟩w is connected. Now, let v ∈ V (G) \ S and choose k ∈ V (H) for all k ∈ Tx. Then (x,m) ∈ V (G[H]) \ F which implies that km ∈ E(H) for all k ∈ Tx. Therefore, NH [Tx] = V (H). That is Tx is a dominating set in H. Conversely, suppose that S is weakly connected hop dominating set of G. Let y ∈ V (G)\S and choose n ∈ V (H). Then (y, n) ∈ V (G[H]) \ F and ny ∈ E(H) for all n ∈ Tx. Hence, REFERENCES 463 NH [Tx] = V (H). This shows that Tx is a dominating set of G. Since (y, n) ∈ V (G[H])\S, there exists (x, l), (x,m) ∈ F such that dG[H]((x, l), (y, n)) = 2 for all x ∈ S and l,m ∈ V (H). This implies that (x,m)(y, n) ∈ E(G[H]). Hence, NG[H][F ] = V (G[H]) and ⟨F ⟩w is connected. Consequently, F is a weakly connected hop dominating set of G[H]. Proposition 9. The weakly connected hop domination numbers of Pm[Pn], Pm[Cn], and Pm[Kn] for positive integers m,n are given as follows: (i) γwh(Pm[Pn]) = γwh(Pm) · γwh(Pn), m,n ≥ 2 (ii) γwh(Pm[Cn]) = γwh(Pm) · γwh(Cn), m ≥ 2 and n ≥ 3 (ii) γwh(Pm[Kn]) = γwh(Pm) · 2, m,n ≥ 2 4. Conclusion This paper was able to introduced the concept of weakly connected hop dominating sets of some graphs and discussed its characterizations in the shadow graph, join, corona and lexicographic product of two graphs. The weakly connected hop domination number of these graphs are determined. For further study, the authors recommend to establish other variants related to the concept of the weakly connected hop dominating sets. Also, the characterization of the weakly connected hop dominating sets under some binary operations that are not discussed in the study and determine the exact values of the parameters of the said graphs are also encouraged. Acknowledgements The authors would like to thank the referees for their invaluable comments and sug- gestions which led to the enhancement of the paper. This study has been supported by the Department of Science and Technology-Accelerated Science and Technology Human Resource Development Program (DOST-ASTHRDP), Mindanao State University - Tawi- Tawi College of Technology and Oceanography and Mindanao State University - Iligan Institute of Technology. References [1] S.K. Ayyaswamy and C. Natarajan. Hop domination in graphs. 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