EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 17, No. 4, 2024, 3772-3780 ISSN 1307-5543 – ejpam.com Published by New York Business Global Zero Forcing Domination in Some Graphs: Characterizations and Derived Formulas Javier A. Hassan1,2,∗, Maria Andrea O. Bonsocan3, Mercedita A. Langamin1, Vergel T. Bilar3, Sharifa Dianne A. Aming4, Bayah J. Amiruddin-Rajik1 1Mathematics and Statistics Department, College of Arts and Sciences, MSU-Tawi-Tawi College of Technology and Oceanography, Bongao, Tawi-Tawi, Philippines 2Department of Mathematics, College of Science, Korea University, Seoul 02841, South Korea 3Department of Mathematics, Ateneo de Davao University, E. Jacinto Street, 8016 Davao City, Philippines 4Office of the Chancellor, MSU-Tawi-Tawi College of Technology and Oceanography, Bongao, Tawi-Tawi, Philippines Abstract. In this paper, we initiate the study on zero forcing domination in a graph. We char- acterize zero forcing dominating sets in some special graphs and the join of two graphs, and we derive some formulas of the zero forcing domination using characterization results. Moreover, we present some relationships of this parameter with other known parameters in graph theory. 2020 Mathematics Subject Classifications: 05C69 Key Words and Phrases: Zero forcing set, zero forcing dominating set, zero forcing domination number 1. Introduction The concept of zero forcing set was initially introduced in [4] as a bound for the minimum rank problem. Given a starting set of blue vertices, with all other vertices white, and a color-change rule, zero forcing is a graph propagation mechanism that increases the number of blue vertices. According to the zero forcing color-change rule, a blue vertex next to a single white neighbor can force that neighbor to also be blue. The notation u → w means that if u is a blue vertex and w is the solitary white vertex in N(u), then u ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v17i4.5154 Email address: javierhassan@msutawi-tawi.edu.ph (J. A. Hassan), mariaandrea.bonsocan@g.msuiit.edu.ph (M. A. Bonsocan) merceditalangamin@msutawi-tawi.edu.ph (M. Langamin) vtbilar@addu.edu.ph (V. Bilar) sharifadianneaming@msutawi-tawi.edu.ph (S. D. Aming) bayahamiruddin@msutawi-tawi.edu.ph (B. Amiruddin) https://www.ejpam.com 3772 Copyright: © 2024 The Author(s). (CC BY-NC 4.0) J. A. Hassan et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3772-3780 3773 compels w to be blue. With graph G as input, a zero forcing set of G is a subset of vertices from V(G) such that the color-change rule given B is applied iteratively if B is originally colored blue and the remaining vertices in G are white. Apart from its use in the minimal rank problem, zero forcing sets have also proved interesting on their own. Numerous aspects and extensions of zero forcing have been examined. Among these are positive semi-definite zero forcing, a variant that bounds the minimum rank problem when the minimum is taken only over positive semi-definite matrices with graph G [3]. On the other hand, in the late 1950’s and 1960’s, the study on domination in graphs was developed. Beginning with C. Berge [1] in 1958, he referred to the domination number as the “coefficient of external stability” [13]. In 1962, O. Ore introduced the terms “dom- inating set” and “domination number”. Domination in a graph has been on the topics studied by researchers recently. Different variants of this parameter have been established and some of these studies can be found in [2, 5–12, 14, 15]. In this study, we initiate the study of zero forcing domination in graphs. We believe this parameter and its results would serve as reference for future researchers who will study on concept related to zero forcing domination in a graph. 2. Terminology and Notation Let G = (V (G), E(G)) be a simple and undirected graph. The distance dG(u, v) in G of two vertices u, v is the length of a shortest u-v path in G. The greatest distance between any two vertices in G, denoted by diam(G), is called the diameter of G. Two vertices x, y of G are adjacent, or neighbors, if xy is an edge of G. The open neighborhood of x in G is the set NG(x) = {y ∈ V (G) : xy ∈ E(G)}. The closed neighborhood of x inG is the setNG[x] = NG(x)∪{x}. IfX ⊆ V (G), the open neighborhood of X in G is the set NG(X) = ⋃ x∈X NG(x). The closed neighborhood of X in G is the set NG[X] = NG(X) ∪X. A subset S of V (G) is a dominating of G if for every a ∈ V (G)\S, there exists b ∈ S such that dG(a, b) = 1. The minimum cardinality among all dominating sets of G, denoted by γ(G), is called the domination number of G. Any dominating set with cardinality equal to γh(G) is called a γ-set of G. The color change rule states that a blue vertex adjacent to a single white neighbor can force its neighbor to blue. A zero forcing set for a graph G is a subset Z of V(G) such that if initially the vertices in Z are colored blue and the remaining vertices are colored white, the entire graph G may be colored blue by repeatedly applying the color-change rule. Furthermore, the zero forcing number of G, denoted by Z(G), is the minimum cardinality of zero forcing set of G. 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 J. A. Hassan et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3772-3780 3774 forcing hop dominating set of G or a γzh-set of G. Let G and H be any two graphs. The join of 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)}. 3. Results We begin this section by introducing the concept of zero forcing domination in a graph. Definition 1. Let G be a graph. If B is zero forcing set and for every u ∈ V (G) \ B, there exists v ∈ B such that uv ∈ E (G), then B is called a zero forcing dominating set of G. The minimum cardinality of a zero forcing dominating set of G, denoted by γz(G), is called zero forcing domination number of G. Example 1. Consider C5 below: a b cd e C5 : Let B = {a, c, d}. Then NC5 [B] = V (C5) . Thus, B is a dominating set of C5. Observe that vertices b and e can be forced by c and d, respectively. It follows that B is a zero forcing set of C5. Therefore, B is a zero forcing dominating set of C5. Moreover, it can be verified that γz (C5) = 3. Remark 1. Let G be a graph. Then the zero forcing domination and zero forcing hop domination parameter of G are incomparable. To see this, consider the graph G below: a b c d e fG : Let Q = {b, c, e}. Then NG[Q] = V (G). Thus, Q is a dominating set of G. Moreover, observe that vertices a, d and f are forced by vertices b, c and e. It follows that Q is a zero J. A. Hassan et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3772-3780 3775 forcing set of G. Since {c, d} is not a zero forcing set of G, it follows that Q = {b, c, e} is a minimum zero forcing dominating set of G. Hence, γz(G) = 3. Now, let S = {a, c, d, e}. Then S is a minimum zero forcing hop dominating set of G. Therefore, γzh(G) = 4. Next, consider the graph H below: a b c H : f e d i h g Let P = {c, d, g}. Then N2 G[P ] = V (H), and so P is a hop dominating set of H. Observe that vertices b and a are forced by c and b, respectively, vertices e and f are forced by vertices d and e, respectively, and vertices h and i are forced by vertices g and h, respectively. Thus, P is a zero forcing hop dominating set of H. Since P is the minimum hop domnating set of H, it follows that γzh(H) = 3. Now, let D = {b, e, h, g}. Then D is a dominating set of H. Observe that vertices d and c are forced by vertices g and d, respectively, and vetices a, f and i are forced by vertices b, e and h, respectively. Thus, D is a zero forcing dominating set of H. Since {b, e, h} is not a zero forcing set, it follows that D is a minimum zero forcing dominating set of H. Therefore, γzh(H) = 4. Proposition 1. Let G be a graph. Then (i) a dominating set D of G may not be a zero forcing set of G; and (ii) a zero forcing set T of G may not be a dominating set of G; Proof. (i) Consider the G below: a1 a4a2 a5a3 G : a6 a7 a8 LetD = {a4, a6}. ThenNG [a4] = {a1, a2, a3, a4, a5} andNG [a6] = {a5, a6, a7, a8}. Thus, NG [D] = V (G), showing that D is a dominating set of G. However, D is not a J. A. Hassan et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3772-3780 3776 zero forcing set of G since vertices a4 and a6 cannot force any other vertices in V (G) \D. (ii) Consider T = {a1, a2,, a3, a7, a8}. Then vertices a4, a5 and a6 are forced by vertices a1, a4 and a5, respectively. Thus, T is a zero forcing set ofG. However, T is not dominating set of G since a5 cannot be dominated by any vertex in T . Proposition 2. Let G be a graph. Then (i) Z (G) ≤ γz(G); (ii) γ (G) ≤ γz(G); and (iii) 1 ≤ γz (G) ≤ |V (G) |. Proof. (i) Let G be a graph and let Q be a minimum zero forcing dominating set of G. Then Q is a zero forcing set of G. Since Z(G) is the minimum cardinality of a zero forcing set of G, it follows that γz (G) = |Q| ≥ Z(G). (ii) Let Q1 be minimum zero forcing domination of G. Then Q1 is a dominating set of G. Since γ (G) is the minimum cardinality of a dominating set of G, we have γz (G) = |Q1| ≥ γ (G) . (iii) Since γ (G) ≥ 1 for any graph G, by (ii), γz (G) ≥ 1. Sine every zero forcing dominating set P is always a subset of a vertex-set V (G), we have γz (G) ≤ |V (G) |. Therefore, 1 ≤ γz (G) ≤ |V (G) |. Lemma 1. Let G be graph. Then Z (G) = 1 if and only if G = Pn for all n ≥ 1. Theorem 1. Let G be a graph. Then γz (G) = Z (G) = 1 if and only if G = P1 or G = P 2. Proof. Suppose that γz (G) = Z (G) = 1. Then by Lemma 1, G = Pn ∀ n ≥ 1. If n ≥ 4, then γ (Pn) ≥ 2. Since γz (G) ≥ γ (G)for any graph G, γz (G) ≥ 2, a contradic- tion. Assume that n = 3. Let V (P3 ) = { a1, a2, a3}. Note that {a2} is a dominating but not a zero forcing set in P3 and {a1} or {a3} is a zero forcing but not a dominating set in P3. Let N = {a1, a2}. Then N is a minimum zero forcing dominating set P3. Thus, γz (P3) = 2, which is a contradiction. Therefore, either n = 1 or n = 2, that is G = P1 or G = P2. Conversely, suppose that G = P1. Then γz (G) = 1 = Z(P1). Similarly, when G = P2, then γz (G) = 1 = Z(P2). Theorem 2. Let G be a graph. Then γz (G) = |V (G)| = Z (G) if and only if G = Kn. J. A. Hassan et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3772-3780 3777 Proof. Suppose that γz (G) = |V (G)| = Z (G) . Then V (G) is the minimum zero forcing set of G. Suppose there is a component H of G which is non-trivial. Let V (H) = {x1, . . . , xk}, where k ≥ 2. Then V (G) \ {x1} is a zero forcing set of G. Thus, Z (G) ≤ |V (G)| − 1, a contradiction. Therefore, every component of G is trivial, that is, G = Kn. Conversely, suppose that G = Kn. Then V (G) is the minimum dominating set of G, and so γ (G) = |V (G) |. Since γz (G) ≥ γ (G), it follows that γz (G) = |V (G) |. Clearly, V (G) is the only zero forcing set of G. Consequently, Z (G) = |V (G)| = γz (G) . Corollary 1. Let G be a graph. Then γZ (G) ≤ |V (G)|−1 If and only if G has non-trivial component. Theorem 3. Let n ≥ 2 be a positive integer and let S be a zero forcing dominating set of Kn. Then S is a minimum zero forcing dominating set of Kn if and only if |S| = n− 1. Proof. Let S be a minimum zero forcing dominating set of Kn and let V (Kn) = {v1, v2, . . . , vn}. Suppose that |S| ≤ n − 2. Then there exist ui, uj ∈ V (Kn) such that ui, uj /∈ S for some i, j ∈ {1, 2, . . . , n} . Now, let ut ∈ S, where t ̸= i, j. Since the graph is complete, ut is adjacent to both ui and uj . Thus, ut cannot force either ui or uj . Since ut is arbitrary, it follows that S is not a zero forcing set of Kn, which is a contradiction. Therefore, |S| ≥ n−1. Since S = V (Kn)\{vk} is a zero forcing dominating set of Kn for some k ∈ {1, 2, . . . , n}, the assertion follows. Conversely, suppose that |S| = n−1, say S = V (Kn)\{vi} for some i ∈ {1, 2, . . . , n}. Then S is a zero forcing dominating set of Kn. If S is not the minimum, then there exists T ⊂ S ⊂ V (Kn) such that T is a zero forcing dominating set of Kn. Let vs, vt ∈ V (Kn) such that vs, vt /∈ T . Then any element of T cannot force either vs or vt, a contradiction. Hence, S must be a minimum zero forcing dominating set of Kn. Corollary 2. Let m be a positive integer. Then γZ (Km) = { 1 , if m = 1 m− 1 , if m ≥ 2. Theorem 4. Let G and H be two non-complete graphs. Then Q ⊆ V (G + H) is a zero forcing dominating set in G +H if and only if Q = QG ∪ QH and satisfies one of the following: (i) QG = V (G) and QH is a zero forcing set of H. (ii) QH = V (H) and QG is a zero forcing set of G. (iii) QG = V (G)\{x} and QH is a zero forcing set of H with NH [y]∩ (V (H) \QH) = ∅ for some y ∈ QH and x ∈ V (G) . J. A. Hassan et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3772-3780 3778 (iv) QH = V (H)\{u} and QG is a zero forcing set of G with NG [v]∩ (V (G) \QG) = ∅ for some v ∈ QG and u ∈ V (G) . Proof. Suppose that Q = QG ∪ QH is a zero forcing dominating set of G+H. Assume that QG = V (G). If QH = V (H), then (i) and (ii) follows. Suppose that QH ̸= V (H). If QH is not a zero forcing set, then there exists two vertices x, y ∈ V (H) \QH such that x and y cannot be forced by any element in QH . Therefore, Q = QG ∪ QH is not a zero forcing set, a contradiction. Hence, QH must be a zero forcing set of H, and so (i) holds. Similarly, (ii) holds. Now, suppose that QG ̸= V (G) and QH ̸= V (H). Since Q is zero forcing, either |QG| = |V (G)| − 1 or |QH | = |V (H)| − 1. Assume that |QG| = |V (G)| − 1. Since Q is a zero forcing set of G + H, QH must be a zero forcing in H. Now, suppose that for every element w ∈ QH , NH [w] ∩ (V (G) \QH) ̸= ∅. Since |QG| = |V (G)| − 1, and H is non-complete graph, it follows that any element w ∈ QH cannot force any element in V (G) \ {x} and in V (H) \QH , respectively, a contradiction. Hence (iii) holds. Similarly, (iv) holds. Conversely, suppose that (i) holds. Since QH is a zero forcing set in H,Q = QG ∪ QH is a zero forcing set in G+H. Thus, Q is a zero forcing dominating set of G+H. Similarly, the assertion follows when (ii) holds. Next, suppose that (iii) holds. Then Q is a dominating set of G+H. Let y ∈ Q such that NH [y] ∩ (V (H) \QH) = ∅. Then y forces vertex x in G. Since QH is a zero forcing set in H, it follows that Q = QG ∪ QH is a zero forcing set of G +H. Therefore, Q is zero forcing dominating set of G+H. Similarly, the assertion holds, when (iv) is true. Corollary 3. Let G and H be two non-complete graphs. Then γz (G+H) = min{|V (G)|+ Z (H) , |V (H)|+ Z (G} . 4. Conclusion The concept of zero forcing domination has been introduced and initially investigated in this study. Its bounds concerning other known parameters in graph theory have been established. Moreover, the zero forcing domination number of some graphs has been deter- mined. In addition, characterizations of zero forcing dominating sets in some graphs were formulated and were used to solve the exact value of the parameter of these graphs. 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