EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 17, No. 1, 2024, 324-337 ISSN 1307-5543 – ejpam.com Published by New York Business Global Some Properties of Zero Forcing Hop Dominating Sets in a Graph Jahiri U. Manditong1,∗, Aziz B. Tapeing1, Javier A. Hassan1, Alcyn R. Bakkang2, Nurijam Hanna M. Mohommad1, Sisteta U. Kamdon1 1Mathematics and Sciences Department, College of Arts and Sciences, MSU Tawi-Tawi College of Technology and Oceanography, Bongao, Tawi-Tawi, Philippines 2 Secondary Education Department, College of Education, MSU Tawi-Tawi College of Technology and Oceanography, Bongao, Tawi-Tawi, Philippines Abstract. In this paper, we initiate the study of a zero forcing hop domination in a graph. We establish some properties of this parameter and we determine its connections with other known parameters in graph theory. Moreover, we obtain some exact values or bounds of the param- eter on the generalized graph, some families of graphs, and graphs under some operations via characterizations. 2020 Mathematics Subject Classifications: 05C69 Key Words and Phrases: Zero forcing hop dominating set, zero forcing hop domination number, zero forcing pointwise non-domination 1. Introduction Hop domination was introduced by Natarajan et al. in [12]. This parameter is incom- parable with the standard domination and just like domination, hop domination has many applications in different fields and in networks. A subset S of a vertex set V (G) is called a hop dominating in G if N2 G[S] = V (G), where N2 G[S] is the closed hop neighborhood of S in G. The minimum cardinality among all hop dominating sets in G, denoted by γh(G), is called the hop domination number of G. This concept had been studied on different types of graphs and graph theorists found some interesting results (see [1, 2, 10]). Since then, several researchers had studied this concepts and they had extended this parameter by ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v17i1.4920 Email addresses: jahirimanditong@msutawi-tawi.edu.ph (J. U. Manditong) aziztapeing@msutawi-tawi.edu.ph (A. Tapeing), javierhassan@msutawi-tawi.edu.ph (J. Hassan), hannamohammadg@msutawi-tawi.edu.ph (N. H. Mohammad), sistetakamdon@msutawi-tawi.edu.ph (S. Kamdon), alcynbakkang@msutawi-tawi.edu.ph (A. Bakkang) https://www.ejpam.com 324 © 2024 EJPAM All rights reserved. J. U. Manditong et al. / Eur. J. Pure Appl. Math, 17 (1) (2024), 324-337 325 introducing variants, that is, imposing additional properties or conditions on the standard hop domination (see [3–9, 11]). In this paper, we introduce and investigate zero forcing hop domination in a graph. Let G be a graph. A subset Z of a vertex-set V (G) of G is said to be a zero forcing hop dominating if Z is both zero forcing and hop dominating in G. The minimum cardinality among all zero forcing hop dominating sets in G, denoted by γzh(G), is called the zero forcing hop domination number of G. We study this parameter on some classes of graphs and graphs under some operations. We determine its connections with other known pa- rameters in graph theory such as zero forcing and hop domination. We believe that this study and its results would contribute a lot to the rapidly increasing number of studies in domination theory. 2. Terminology and Notation Let G be a graph. The distance dG(u, v) of two vertices u, v in G is the length of a shortest u-v path in G. The greatest distance between any two vertices in G, denoted by diam(G), is called the diameter of G. Two distinct vertices v, w of G are said to be neighbors, if dG(v, w) = 1. The open neighborhood (resp. closed neighborhood) of v in G is the set defined by NG(v) = {w ∈ V (G) : dG(v, w) = 1} (resp. NG[v] = NG(v) ∪ {v}). If X ⊆ V (G), then the open neighborhood (resp. closed neighborhood) of X in G is the set defined by NG(X) =⋃ x∈X NG(x) (resp. NG[X] = NG(X) ∪X). The color change rule is: If u is a blue vertex and exactly one neighbor w of u is white, then change the color of w to blue. We say u forces w and denote this by u → w. A zero forcing set for G is a subset of vertices B such that when the vertices in Z are colored blue and the remaining vertices are colored white initially, repeated application of the color change rule can color all vertices of G blue. The zero forcing number of G, denoted by Z(G), is the minimum cardinality among all zero forcing sets 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} (resp. N2 G[u] = N2 G(u) ∪ {u}) is called the open hop neighborhood (resp. closed hop neighborhood) of u. Let A be a subset of V (G). Then the open hop neighborhood (resp. closed hop neighborhood) of A is the set defined by N2 G(A) = ⋃ u∈A N2 G(u) (resp. N 2 G[A] = N2 G(A) ∪A). A subset S of V (G) is called a hop dominating ofG if for every v ∈ V (G)\S, there exists u ∈ S such that dG(u, v) = 2. The minimum cardinality among all hop dominating sets of G, denoted by γh(G), is called the hop domination number of G. Any hop dominating set with cardinality equal to γh(G) is called a γh-set of G. A subset C of V (G) is called a pointwise non-dominating (PND) if for every v ∈ V (G) \ C, there exists u ∈ C such that v /∈ NG(u). The minimum cardinality of a pointwise non-dominating (PND) set of G, denoted by pnd(G), is called the pointwise non-domination number of G. Any PND set of G with cardinality pnd(G) is called a J. U. Manditong et al. / Eur. J. Pure Appl. Math, 17 (1) (2024), 324-337 326 minimum PND set or a pnd-set of G. Let G and H be two graphs. The join G+H of G and H is the graph with vertex set V (G+H) = V (G) ∪ V (H) and edge set E(G+H) = E(G) ∪ E(H) ∪ {ab : a ∈ V (G), b ∈ V (H)} . The corona G ◦ H of G and H is the graph obtained by taking one copy of G and |V (G)| copies of H, and then joining the ith vertex of G to every vertex of the ith copy of H. We denote by Ha the copy of H in G ◦H corresponding to the vertex a ∈ V (G). 3. Results We begin this section by defining the concept of zero forcing hop domination in a graph as follows: Definition 1. Let G be a graph. A subset Z of V (G) is said to be a zero forcing hop dominating if Z is both a zero forcing and a hop dominating in G. The minimum cardinality among all zero forcing hop dominating sets in G, denoted by γzh(G), is called the zero forcing hop domination number of G. A zero forcing hop dominating set Z with |Z| = γzh(G), is called the minimum zero forcing hop dominating set of G or a γzh-set of G. Example 1. Consider the graph G below. G2 : h g f ed c b a Figure 1: Graph G with γZh(G) = 5 Let Z = {a, b, e, f, g}. Then Z is a zero forcing set in G. Observe that N2 G[a] = {a, b, c, e} = N2 G[b] = N2 G[e] and N2 G[f ] = {d, f, g, h} = N2 G[g]. Thus, N2 G[Z] = {a, b, c, d, e, f, g, h} = V (G), showing that Z is a hop dominating set in G. Hence, Z is a zero forcing hop dominating set of G. Moreover, since Z is a minimum zero forcing set of G, it follows that Z is a minimum zero forcing hop dominating set of G, and so γzh(G) = 5. J. U. Manditong et al. / Eur. J. Pure Appl. Math, 17 (1) (2024), 324-337 327 Proposition 1. Let G be a graph. Then (i) a zero forcing set may not be a hop dominating; and (ii) a hop dominating set may not be a zero forcing. Proof. (i) Consider the graph G below. G : g h fec b d a Let Z = {a, b, e, h}. Then, Z is a zero forcing set in G. However, c, f /∈ N2 G[Z]. Thus, N2 G[Z] ̸= V (G), showing that Z is not a hop dominating set of G. Hence, the result follows. (ii) Consider again the graph G in (i) and let S = {c, d, e}. Then, N2 g [S] = V (G), and so S is a hop dominating set in G. However, S is not a zero forcing set in G since it cannot forces vertices a, b, g and h in G. Thus, the assertion follows. Remark 1. The Proposition 1 says that a zero forcing (resp. hop dominating) set may not be a zero forcing hop dominating set. Theorem 1. Let G be any graph. Then (i) Z(G) ≤ γzh(G); (ii) γh(G) ≤ γzh(G); (iii) 1 ≤ γzh(G) ≤ |V (G)|; and (iv) γzh(G) = |V (G)| if and only if γh(G) = |V (G)|. Proof. (i) Let G be a graph and let Z be a γzh-set of G. Then Z is a zero forcing in G and |Z| = γzh(G). Since Z(G) is the minimum cardinality among all zero forcing sets in G, we have Z(G) ≤ |Z| = γzh(G). J. U. Manditong et al. / Eur. J. Pure Appl. Math, 17 (1) (2024), 324-337 328 (ii) Let S be a γzh-set of G. Then S is a hop dominating set in G and |S| = γzh(G). Since γh(G) is the minimum cardinality among all hop dominating sets in G, hence γh(G) ≤| S |= γzh(G). (iii) Since γh(G) ≥ 1 for any graph G, it follows that γzh(G) ≥ 1 by (ii). Since any zero forcing hop dominating set S′ is always a subset of V (G), we have γzh(G) ≤| V (G) |. Consequently, 1 ≤ γzh(G) ≤ |V (G)|. (iv) Suppose that γzh(G) = |V (G)|. Then V (G) is the minimum zero forcing hop dominating set in G. Assume that G is connected. Suppose further that G is non- complete. Then dG(v, w) = 2 for some v, w ∈ V (G). Hence, Z ′ = V (G) \ {w} is a zero forcing hop dominating set of G, showing that γzh(G) ≤ |V (G)| − 1, which is a contradiction. Therefore, G is complete, and so γh(G) = |V (G)|. Now, let G1, . . . , Gk, k ≥ 2 be components of G. Suppose that Gi is non-complete for some i ∈ {1, . . . , k}. Then dGi(s, t) = 2 = dG(s, t) for some s, t ∈ V (Gi). Thus, Z ′′ = V (G) \ {t} is a zero forcing hop dominating set of G, and so γzh(G) ≤ |V (G)| − 1, a contradiction. Hence, every component of G is complete. Therefore, γh(Gi) = |V (Gi)| for each i ∈ {1 . . . , k}. Consequently, γh(G) = γh(G1) + · · ·+ γh(Gk) = |V (G1)|+ · · ·+ |V (Gk)| = |V (G)|. Conversely, suppose that γh(G) = |V (G)|. Then by (ii) and (iii), γzh(G) = |V (G)|. The following result follows immediately from Theorem 1(iv). Corollary 1. γzh(Kr) = r = γzh(Kr) for all positive integer r ≥ 1. Proposition 2. Let G be any graph with |V (G)| ≥ 2. If γzh(G) = 2, then γh(G) = 2. However, the converse is not true. Proof. Let G be a graph with |V (G)| ≥ 2. Then γh(G) ≥ 2. Since γzh(G) = 2, γh(G) ≤ 2 by Theorem 1(ii). Hence, γh(G) = 2. To see that the converse is not true, consider the graph G below. G : c d e b a J. U. Manditong et al. / Eur. J. Pure Appl. Math, 17 (1) (2024), 324-337 329 Let Z1 = {a, e}. Then N2 G[Z1] = V (G), showing that Z1 is a hop dominating set of G. Thus, γh(G) ≤ 2. Since, |V (G)| ≥ 2, it follows that γh(G) ≥ 2. Hence, γh(G) = 2. Now, let Z2 = {a, b, c, e}. Then, Z2 is minimum zero forcing hop dominating set in G. Thus, fzG(G) = 4 Theorem 2. Let r, q ∈ N with 2 ≤ r ≤ q. Then there exists a connected graph K such that γh(K) = r and γzh(K) = q. Proof. Suppose that r < q. Let l = q − r and consider the graph K below. K : x1 x2 x3 xr−1 xr . .. . . . y1 y2 yl−1 yl Figure 2: Graph K with γh(K) < γzh(K) Let A = {x1, x2, . . . , xr} and B = {x1, x2, . . . , xr, y1, y2, . . . , yl}. Then A is a minimum hop dominating set of K. Thus, γh(K) = r. Observe that B is a minimum zero forcing set of K. Since A ⊆ B, it follows that B is also a hop dominating set of K. Hence, B is a minimum zero forcing hop dominating set of K. Consequently, γh(K) = r < q = l + r = γzh(K). For r = q, consider a complete graph G with order r. Then the sharpness of γzh(G) and γh(G) follows. J. U. Manditong et al. / Eur. J. Pure Appl. Math, 17 (1) (2024), 324-337 330 The next definition will be used to calculate the exact value of parameter of the join of two graphs. Definition 2. Let J be any graph. Then F ⊆ V (J) is called a zero forcing point- wise non-dominating (ZFPND) in J if F is both a zero forcing and a pointwise non- dominating (PND) in J . The minimum cardinality among all zero forcing pointwise non- dominating (ZFPND) sets in J , denoted by zfpnd(J), is called the zero forcing pointwise non-domination number of J . Any ZFPND set F with |F | = zfpnd(J), is called the minimum ZFPND set or a zfpnd-set of J . Example 2. Consider the graph G below. a b c d e f g hG : i Figure 3: A graph G with zfpnd(G) = 5 Let F = {a, b, e, g, h}. Notice that c, f, i /∈ NG(a) and d /∈ NG(g). It follows that F is a PND set of G. Since F is a minimum zero forcing set of G, F is a minimum ZFPND set of G. Thus, zfpnd(G) = 5. Consequently, F is a zfpnd-set of G. Proposition 3. Let G be any graph. Then every ZFPND set F ⊆ V (G) is a PND. But the converse is not true. Proof. Let F be a ZFPND. Then F is a PND set (by definition). To see that the converse is not true, consider the graph G below. G : f e dc b a J. U. Manditong et al. / Eur. J. Pure Appl. Math, 17 (1) (2024), 324-337 331 Let N = {a, b, c}. Observe that d, e, f /∈ NG(a). It follows that N is a PND set in G. However, N is not a zero forcing set in G since it cannot forces vertices e and f . Hence, N is not a ZFPND set of G, and so the assertion follows. Theorem 3. Let G be any graph. Then (i) pnd(G) ≤ zfpnd(G); (ii) 1 ≤ zfpnd(G) ≤ |V (G)|; and (iii) zfpnd(G) = 1 if and only if G = K1. Proof. (i) Let G be any graph and let F be a minimum ZFPND set of G. Then zfpnd(G) = |F | and F is a PND set of G. Since pnd(G) is the minimum cardinality among all PND sets in G, it follows that pnd(G) ≤ |F | = zfpnd(G). (ii) Since pnd(G) ≥ 1 for any graph G, we have zfpnd(G) ≥ 1 by (i). Moreover, since any ZFPND set F is always a subset of V (G), it follows that zfpnd(G) ≤ |V (G)|. Therefore, 1 ≤ zfpnd(G) ≤ |V (G)|. (iii) Suppose that zfpnd(G) = 1. Assume that G ̸= K1. If G is connected, then pnd(G) ≥ 2, a contradiction. Assume that G is disconnected. Let G1, . . . , Gk, k ≥ 2 be components of G. Then Z(G) ≥ 2. Since every ZFPND set is a zero forcing, we have zfpnd(G) ≥ Z(G). Thus, zfpnd(G) ≥ 2, a contradiction. Therefore, G = K1. The converse is clear. Theorem 4. Let G be non-trivial graph. Then zfpnd(G) = |V (G)| if and only if every component of G is complete. Proof. Suppose that zfpnd(G) = |V (G)|. Then V (G) is the minimum ZFPND set in G. Assume that G is connected. Suppose further that G is non-complete. Then dG(v, w) = 2 for some v, w ∈ V (G). Hence, Z ′ = V (G) \ {w} is a ZFPND set of G, showing that zfpnd(G) ≤ |V (G)| − 1, a contradiction. Therefore, G is complete. Now, let Q1, . . . , Qk, k ≥ 2 be components of G. Suppose that Qi is non-complete for some i ∈ {1, . . . , k}. Then dQi(s, t) = 2 = dG(s, t) for some s, t ∈ V (Qi). Thus, V (G) \ {t} is a ZFPND set of G, and so zfpnd(G) ≤ |V (G)|−1, a contradiction. Hence, every component of G is complete. Conversely, let G1, . . . , Gk, k ≥ 2 be complete components of G. If Gi is non-trivial for each i ∈ {1, . . . , k}, then pnd(G) = |V (G)| = k. Thus, zfpnd(G) = k by Theorem 3(i). Assume that Gi is trivial for some i ∈ {1, . . . , k}. Since every ZFPND set F is a zero J. U. Manditong et al. / Eur. J. Pure Appl. Math, 17 (1) (2024), 324-337 332 forcing, V (Gi) ⊆ F . Since vertices of every non-trivial complete component of G are also in any ZFPND set of G, it follows that V (G) is the minimum ZFPND set of G. Thus, zfpnd(G) = |V (G)|. The following result follows from Theorem 3(iii) and Theorem 4. Corollary 2. zfpnd(Kq) = q = zfpnd(Kq) for all positive integer q ≥ 1. Proposition 4. Let n be any positive integer. Then each of the following holds. (i) zfpnd(Pn) = { n if n = 1, 2 2 if n ≥ 3. (ii) zfpnd(Cn) = { 3 if n = 3 2 if n ≥ 4. Proof. (i) Clearly, zfpnd(Pn) = n for n = 1, 2. Suppose that n ≥ 3. Let V (Pn) = {a1, a2, . . . , an} and consider F = {a1, a2}. Clearly, F is a zero forcing set of Pn. Observe that for every w ∈ V (Pn) \ F , w /∈ NPn(a1). Thus, F is a PND set of Pn, showing that F is a ZFPND set of Pn. Since {x} is not a ZFPND set in Pn ∀ x ∈ V (Pn), it follows that F is a minimum ZFPND set of Pn. Hence, zfpnd(Pn) = 2 for all n ≥ 3. (ii) Since pnd(C3) = 3, it follows that zfpnd(C3) = 3 by Theorem 3(i)(ii). Suppose that n ≥ 4. Let V (Cn) = {x1, x2, . . . , xn} and consider F ′ = {x1, x2}. Notice that for all j ∈ {3, 4, ...n} xj /∈ NCn(x1) and xn /∈ NCn(x2). Thus, F ′ is a PND set of Cn, and so F ′ is a ZFPND set of Cn. Since {xi} is not a ZFPND set of Cn for each i ∈ {1, 2, . . . , n}, it follows that F ′ is a minimum ZFPND set of Cn. Consequently, zfpnd(Cn) = 2 for all n ≥ 4. Theorem 5. [10] Let G and H be two graphs. A set S ⊆ V (G +H) is hop dominating set of G +H if and only if S = SG ∪ SH , where SG and SH are PND sets of G and H, respectively. Theorem 6. Let S and T be two non-complete graphs. A subset Z of V (S + T ) is a zero forcing hop dominating set in S + T if and only if Z = ZS ∪ ZT and satisfies one of the following conditions: (i) ZS = V (S) and ZT is a ZFPND set in T . (ii) ZT = V (T ) and ZS is a ZFPND set in S. (iii) ZS = V (S) \ {a} and ZT = V (T ) \ {b} are ZFPND sets in S and T , respectively, for some a ∈ V (S), b ∈ V (T ). J. U. Manditong et al. / Eur. J. Pure Appl. Math, 17 (1) (2024), 324-337 333 Proof. Let Z = ZS ∪ ZT be a zero forcing hop dominating set in S + T . Then Z is a zero forcing in S+T . Suppose that ZS = V (S). If ZT = V (T ), then we are done. Assume that ZT ̸= V (T ). Suppose ZT is not a zero forcing set in T . Then there exists w ∈ ZT such that w cannot be forced by any element in ZT . Thus, w cannot be forced by any element of Z, which is a contradiction. Hence, ZT is a zero forcing set in T . Since Z is a hop dominating, ZT is a PND set in T by Theorem 5. Consequently, ZT is a ZFPND set in T , and so (i) holds. The (ii) can be proved in similar manner. Next, suppose that ZS ̸= V (S) and ZT ̸= V (T ) then there exists u ∈ V (S) \ ZS and v ∈ V (T ) \ ZT . If |ZS | ≤ |V (G)|−2, then there exist at least two vertices s, t ∈ V (S)\ZS . However, any ele- ment of ZS and ZT cannot forces vertices s and t, a contradiction. Thus, |ZS | = |V (S)|−1. Similarly, |ZT | = |V (T )|−1. Let V (S) = {v1, v2, . . . , vm} and V (T ) = {u1, u2, . . . , un} and let ZS = V (S) \ {vi} and ZT = V (T ) \ {uj} for some i ∈ {1, 2, . . . ,m} , j ∈ {1, 2, . . . , n}. Clearly, ZS and ZT are zero forcing sets in S and T , respectively. Since Z = ZS ∪ZT is a hop dominating set in S + T , it follows that ZS and ZT are PND sets in S and T , respec- tively, by Theorem 5. Therefore, ZS and ZT are ZFPND sets in S and T , respectively, showing that (iii) holds. Conversely, suppose that (i) holds. Since ZT is a PND set in T , it follows that Z = V (S) ∪ ZT is a hop dominating set S + T by Theorem 5. Since ZT is also a zero forcing set in T , Z = V (S) ∪ ZT is a zero forcing set in S + T . Hence, Z = V (S) ∪ ZT is a zero forcing hop dominating set in S + T . Similarly, if (ii) holds, then the assertion follows. Now, suppose that (iii) holds. Then Z is a hop dominating set in S + T by Theorem 5. Since S is non-complete, there exist x, y ∈ V (S) such that dS(x, y) = 2. Since ZS = V (S) \ {a} for some a ∈ V (S), we let y = a and so x ∈ ZS . Then x forces all the vertices in V (S + T ) \ Z, that is, Z is a zero forcing set in S + T . Therefore, Z is a zero forcing hop dominating set in S + T . The following result follows from Theorem 6. Corollary 3. Let S and T be two non-complete graphs. Then γzh(S + T ) = min{|V (S)|+ |V (T )| − 2, |V (S)|+ zfpnd(T ), |V (T )|+ zfpnd(S)}. Theorem 7. Let J and K be complete and non-complete graphs, respectively. A subset Z of V (J +K) is a zero forcing hop dominating set in J +K if and only if Z = V (J)∪ZK , where ZK is a ZFPND set in K. Proof. Let Z be a zero forcing hop dominating set in J + K. Since J is complete, Z = V (J) ∪ ZK , ZK ̸= ∅. Thus, by Theorem 6(i), ZK is a ZFPND set in K. Conversely, suppose that Z = V (J) ∪ ZK , where ZK is a ZFPND set in K. Since ZK is a zero forcing in K, Z = V (J) ∪ ZK is a zero forcing in J + K. Moreover, since ZK is PND set in K, it follows that Z = V (J) ∪ ZK is a hop dominating set in J + K by Theorem 5. Therefore, Z is a zero forcing hop dominating set of J +K. J. U. Manditong et al. / Eur. J. Pure Appl. Math, 17 (1) (2024), 324-337 334 Corollary 4. Let J and K be complete and non-complete graphs, respectively. Then γzh(J +K) = |V (J)|+ zfpnd(K). In particular, for any positive integers m,n ≥ 1, we have (i) γzh(Km + Pn) = { m+ n if n = 1, 2 m+ 2 if n ≥ 3, and (ii) γzh(Km + Cn) = { m+ 3 if n = 3 m+ 2 if n ≥ 4 Proof. Let Z be a minimum zero forcing hop dominating set in J + K. Then by Theorem 7, Z = V (J) ∪ ZK , where ZK is a ZFPND set in K. Hence, γzh(J +K) = |Z| = |V (J)|+ |ZK | ≥ |V (J)|+ zfpnd(K). Conversely, suppose that Z = V (J) ∪ ZK , where ZK is a minimum ZFPND set in K. Then Z is a zero forcing hop dominating set of J +K by Theorem 7. Thus, |V (J)|+ zfpnd(K) = |Z| ≥ γzh(J +K). Consequently, γzh(J +K) = |V (J)|+ zfpnd(K). The particular case, follows from Proposition 4. Theorem 8. Let J and K be any non-trivial connected and any graph, respectively. Then, M = V (J)∪ ( ⋃ v∈V (K) Mv) is a zero forcing hop dominating set in J ◦K if Mv is a ZFPND set in Kv for each v ∈ V (J). Moreover, γzh(J ◦K) ≤ |V (J)| · zfpnd(K) + |V (J)| . Proof. Let M = V (J) ∪ ( ⋃ v∈V (K) Mv), where Mv is a ZFPND set in Kv for each v ∈ V (J). Let u ∈ V (J ◦ K)\M . Then u ∈ Kw for some w ∈ V (G). Since Mw is PND set in Kw, there exists y ∈ Mw such that dJ◦K(u, y) = 2. Hence, M is a hop dominating set of J ◦K. Now, since Mv is a zero forcing set in Kv for each v ∈ V (J), it follows that M = V (J)∪ ( ⋃ v∈V (G) Mv) is a zero forcing set in J ◦K. Therefore, M is a zero forcing hop dominating set in J ◦K. Since γzh(J ◦K) is the minimum cardinality among all zero forcing hop dominating sets in J ◦K, we have γzh(J ◦K) ≤ M = |V (J)| · zfpnd(K) + |V (J)|. J. U. Manditong et al. / Eur. J. Pure Appl. Math, 17 (1) (2024), 324-337 335 Remark 2. The sharpness and strict inequality given in Theorem 8 are attainable. For the sharpness, consider the graph P3 ◦ C4 below. u v w u1 u2 u3 u4 v1 v2 v3 v4 w1 w2 w3 w4 P3 C4 : Figure 4: Graph P3 ◦ C4 with γzh(P3 ◦ C4) = |V (P3)| · zfpnd(C4) + |V (P3)| . Let M = {u1, u2, u, v1, v2, v, w1, w2, w}. Then, N2 P3◦C3 [M ] = V (P3 ◦ C4). Thus, M is a hop dominating set of P3◦C4. Observe that {u1, u2}, {v1, v2} and {w1, w2} are zero forcing sets in Cu 4 , C v 4 and Cw 4 , respectively. Hence, M is a zero forcing set in P3 ◦ C4, and so M is a zero forcing hop dominating set in P3 ◦ C4. Moreover, it can be verified that γzh(P3 ◦ C4) = |V (P3)| · zfpnd(C4) + |V (P3)| = 3 · 2 + 3 = 9. For strict inequality, consider the graph C3 ◦K4 below. a4 a2 a3 a1 a b4 b2b1 b b3c3 c1 c2 c c4 C3 K4 : Figure 5: Graph C3 ◦K4 with γzh(C3 ◦K4) < |V (C3)| · zfpnd(K4) + |V (C3)| . REFERENCES 336 Let S = {a1, a2, a3, a4, b, c, c1, c2, c3, b1, b2, b3}. Clearly, S is a zero forcing set in C3◦K4. Notice that N2 C3◦K4 [S] = V (C3 ◦ K4). Thus, S is a zero forcing hop dominating set in C3 ◦ K4, showing that γzh(C3 ◦ K4) ≤ |S| = 12. Now, since, zfpnd(K4) = 4, it follows that |V (C3)| · zfpnd(K4) + |V (C3)| = 3 · 4 + 3 = 15. Consequently, γzh(C3 ◦K4) ≤ 12 < 15 = |V (C3)| · zfpnd(K3) + |V (K4)| . Acknowledgements The authors would like to thank Mindanao State University - Tawi-Tawi College of Technology and Oceanography for funding this research. 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