scholarly journals Zero forcing number of degree splitting graphs and complete degree splitting graphs

2019 ◽  
Vol 11 (1) ◽  
pp. 40-53
Author(s):  
Charles Dominic
Keyword(s):  
Color Change ◽  
Simple Graph ◽  
White Vertex ◽  
Zero Forcing ◽  
Zero Forcing Number ◽  
Forcing Number ◽  
Minimum Number ◽  
Splitting Graph ◽  
Forcing Set ◽  
Zero Forcing Set

Abstract A subset ℤ ⊆ V(G) of initially colored black vertices of a graph G is known as a zero forcing set if we can alter the color of all vertices in G as black by iteratively applying the subsequent color change condition. At each step, any black colored vertex has exactly one white neighbor, then change the color of this white vertex as black. The zero forcing number ℤ (G), is the minimum number of vertices in a zero forcing set ℤ of G (see [11]). In this paper, we compute the zero forcing number of the degree splitting graph (𝒟𝒮-Graph) and the complete degree splitting graph (𝒞𝒟𝒮-Graph) of a graph. We prove that for any simple graph, ℤ [𝒟𝒮(G)] k + t, where ℤ (G) = k and t is the number of newly introduced vertices in 𝒟𝒮(G) to construct it.

scholarly journals Families of graphs with maximum nullity equal to zero forcing number

2018 ◽  
Vol 6 (1) ◽  
pp. 56-67
Author(s):  
Joseph S. Alameda ◽  
Emelie Curl ◽  
Armando Grez ◽  
Leslie Hogben ◽  
O’Neill Kingston ◽  
...  
Keyword(s):  
Color Change ◽  
Simple Graph ◽  
Zero Forcing ◽  
Generalized Petersen Graphs ◽  
Zero Forcing Number ◽  
Regularity Properties ◽  
Maximum Nullity ◽  
Forcing Number ◽  
Minimum Number ◽  
Petersen Graphs

Abstract The maximum nullity of a simple graph G, denoted M(G), is the largest possible nullity over all symmetric real matrices whose ijth entry is nonzero exactly when fi, jg is an edge in G for i =6 j, and the iith entry is any real number. The zero forcing number of a simple graph G, denoted Z(G), is the minimum number of blue vertices needed to force all vertices of the graph blue by applying the color change rule. This research is motivated by the longstanding question of characterizing graphs G for which M(G) = Z(G). The following conjecture was proposed at the 2017 AIM workshop Zero forcing and its applications: If G is a bipartite 3- semiregular graph, then M(G) = Z(G). A counterexample was found by J. C.-H. Lin but questions remained as to which bipartite 3-semiregular graphs have M(G) = Z(G). We use various tools to find bipartite families of graphs with regularity properties for which the maximum nullity is equal to the zero forcing number; most are bipartite 3-semiregular. In particular, we use the techniques of twinning and vertex sums to form new families of graphs for which M(G) = Z(G) and we additionally establish M(G) = Z(G) for certain Generalized Petersen graphs.

Zero forcing number of fuzzy graphs with application

2020 ◽  
Vol 39 (3) ◽  
pp. 3873-3882
Author(s):  
Asefeh Karbasioun ◽  
R. Ameri
Keyword(s):  
Upper Bounds ◽  
Zero Forcing ◽  
Medical Treatments ◽  
Zero Forcing Number ◽  
Fuzzy Graphs ◽  
Forcing Number ◽  
Zero Forcing Numbers ◽  
Forcing Set ◽  
Zero Forcing Set

We introduce and study forcing number for fuzzy graphs. Also, we compute zero forcing numbers for some classes of graphs and extend this concept to fuzzy graphs. In this regard we obtain upper bounds for zero forcing of some classes of fuzzy graphs. We will proceed to obtain a new algorithm to computing zero forcing set and finding a formula for zero forcing number, and by some examples we illustrate these notions. Finally, we introduce some applications of fuzzy zero forcing in medical treatments.

scholarly journals The Bipartite Zero Forcing Set for a Full Sign Pattern Matrix

Mathematics ◽  
2020 ◽  
Vol 8 (3) ◽  
pp. 354
Author(s):  
Gu-Fang Mou ◽  
Tian-Fei Wang ◽  
Zhong-Shan Li
Keyword(s):  
Sign Pattern ◽  
Zero Forcing ◽  
Minimum Rank ◽  
Zero Forcing Number ◽  
Maximum Nullity ◽  
New Variant ◽  
Forcing Number ◽  
Pattern Matrix ◽  
Forcing Set ◽  
Zero Forcing Set

For an m × n sign pattern P, we define a signed bipartite graph B ( U , V ) with one set of vertices U = { 1 , 2 , … , m } based on rows of P and the other set of vertices V = { 1 ′ , 2 ′ , … , n ′ } based on columns of P. The zero forcing number is an important graph parameter that has been used to study the minimum rank problem of a matrix. In this paper, we introduce a new variant of zero forcing set−bipartite zero forcing set and provide an algorithm for computing the bipartite zero forcing number. The bipartite zero forcing number provides an upper bound for the maximum nullity of a square full sign pattern P. One advantage of the bipartite zero forcing is that it can be applied to study the minimum rank problem for a non-square full sign pattern.

scholarly journals On zero forcing number of graphs and their complements

2015 ◽  
Vol 07 (01) ◽  
pp. 1550002 ◽  
Author(s):  
Linda Eroh ◽  
Cong X. Kang ◽  
Eunjeong Yi
Keyword(s):  
Color Change ◽  
Minimum Degree ◽  
Work Group ◽  
Black Vertex ◽  
White Vertex ◽  
Zero Forcing ◽  
Minimum Rank ◽  
Minimum Cardinality ◽  
Zero Forcing Number ◽  
Forcing Number

The zero forcing number, Z(G), of a graph G is the minimum cardinality of a set S of black vertices (whereas vertices in V(G)\S are colored white) such that V(G) is turned black after finitely many applications of "the color-change rule": a white vertex is converted to a black vertex if it is the only white neighbor of a black vertex. Zero forcing number was introduced and used to bound the minimum rank of graphs by the "AIM Minimum Rank-Special Graphs Work Group". It is known that Z(G) ≥ δ(G), where δ(G) is the minimum degree of G. We show that Z(G) ≤ n - 3 if a connected graph G of order n has a connected complement graph [Formula: see text]. Further, we characterize a tree or a unicyclic graph G which satisfies either [Formula: see text] or [Formula: see text].

scholarly journals On the Relationships between Zero Forcing Numbers and Certain Graph Coverings

2014 ◽  
Vol 2 (1) ◽  
Author(s):  
Fatemeh Alinaghipour Taklimi ◽  
Shaun Fallat ◽  
Karen Meagher
Keyword(s):  
Upper Bound ◽  
Tree Cover ◽  
Positive Zero ◽  
Zero Forcing ◽  
Outerplanar Graphs ◽  
Zero Forcing Number ◽  
Forcing Number ◽  
Minimum Number ◽  
Graph Parameters ◽  
Graph Coverings

AbstractThe zero forcing number and the positive zero forcing number of a graph are two graph parameters that arise from two types of graph colourings. The zero forcing number is an upper bound on the minimum number of induced paths in the graph that cover all the vertices of the graph, while the positive zero forcing number is an upper bound on the minimum number of induced trees in the graph needed to cover all the vertices in the graph. We show that for a block-cycle graph the zero forcing number equals the path cover number.We also give a purely graph theoretical proof that the positive zero forcing number of any outerplanar graphs equals the tree cover number of the graph. These ideas are then extended to the setting of k-trees, where the relationship between the positive zero forcing number and the tree cover number becomes more complex.

scholarly journals Signed zero forcing number and controllability for a networks system with a directed hypercube

2022 ◽  
Vol 355 ◽  
pp. 01012
Author(s):  
Gufang Mou ◽  
Qiuyan Zhang
Keyword(s):  
Effective Control ◽  
Signed Graph ◽  
Directed Network ◽  
Network System ◽  
Zero Forcing ◽  
Global Networks ◽  
Zero Forcing Number ◽  
Forcing Number ◽  
Minimum Number ◽  
The Relationship

The controllability for complex network system is to find the minimum number of leaders for the network system to achieve effective control of the global networks. In this paper, the problem of controllability of the directed network for a family of matrices carrying the structure under directed hypercube is considered. The relationship between the minimum number of leaders for the directed network system and the number of the signed zero forcing set is established. The minimum number of leaders of the directed networks system under a directed hypercube is obtained by computing the zero forcing number of a signed graph.

scholarly journals Zero Forcing Number of Some Families of Graphs

2021 ◽  
Vol 3 ◽  
pp. 48-52
Author(s):  
Victoria Petruk
Keyword(s):  
Minimum Degree ◽  
Colour Change ◽  
Discrete Mathematics ◽  
White Vertex ◽  
Zero Forcing ◽  
Minimum Cardinality ◽  
Zero Forcing Number ◽  
Ladder Graph ◽  
Forcing Number ◽  
Wheel Graph

The work is devoted to the study of the zero forcing number of some families of graphs. The concept of zero forcing is a relatively new research topic in discrete mathematics, which already has some practical applications, in particular, is used in studies of the minimum rank of the matrices of adjacent graphs. The zero forcing process is an example of the spreading process on graphs. Such processes are interesting not only in terms of mathematical and computer research, but also interesting and are used to model technical or social processes in other areas: statistical mechanics, physics, analysis of social networks, and so on. Let the vertices of the graph G be considered white, except for a certain set of S black vertices. We will repaint the vertices of the graph from white to black, using a certain rule.Colour change rule: A white vertex turns black if it is the only white vertex adjacent to the black vertex.[5] The zero forcing number Z(G) of the graph G is the minimum cardinality of the set of black vertices S required to convert all vertices of the graph G to black in a finite number of steps using the ”colour change rule”.It is known [10] that for any graph G, its zero forcing number cannot be less than the minimum degree of its vertices. Such and other already known facts became the basis for finding the zero forcing number for two given below families of graphs:A gear graph, denoted W2,n is a graph obtained by inserting an extra vertex between each pair of adjacent vertices on the perimeter of a wheel graph Wn. Thus, W2,n has 2n + 1 vertices and 3n edges.A prism graph, denoted Yn, or in general case Ym,n, and sometimes also called a circular ladder graph, is a graph corresponding to the skeleton of an n-prism.A wheel graph, denoted Wn is a graph formed by connecting a single universal vertex to all vertices of a cycle of length n.In this article some known results are reviewed, there is also a definition, proof and some examples of the zero forcing number and the zero forcing process of gear graphs and prism graphs.

scholarly journals Edge Forcing in Butterfly Networks

2021 ◽  
Vol 182 (3) ◽  
pp. 285-299
Author(s):  
G. Jessy Sujana ◽  
T.M. Rajalaxmi ◽  
Indra Rajasingh ◽  
R. Sundara Rajan
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Higher Dimensions ◽  
Zero Forcing ◽  
Minimum Cardinality ◽  
Forcing Number ◽  
Np Complete ◽  
Forcing Set ◽  
Zero Forcing Set

A zero forcing set is a set S of vertices of a graph G, called forced vertices of G, which are able to force the entire graph by applying the following process iteratively: At any particular instance of time, if any forced vertex has a unique unforced neighbor, it forces that neighbor. In this paper, we introduce a variant of zero forcing set that induces independent edges and name it as edge-forcing set. The minimum cardinality of an edge-forcing set is called the edge-forcing number. We prove that the edge-forcing problem of determining the edge-forcing number is NP-complete. Further, we study the edge-forcing number of butterfly networks. We obtain a lower bound on the edge-forcing number of butterfly networks and prove that this bound is tight for butterfly networks of dimensions 2, 3, 4 and 5 and obtain an upper bound for the higher dimensions.

Improved Computational Approaches and Heuristics for Zero Forcing

2021 ◽  
Author(s):  
Boris Brimkov ◽  
Derek Mikesell ◽  
Illya V. Hicks
Keyword(s):  
Graph Coloring ◽  
Color Change ◽  
State Of The Art ◽  
Exact Algorithms ◽  
Boolean Satisfiability ◽  
The State ◽  
Zero Forcing ◽  
Computational Approaches ◽  
Forcing Set ◽  
Zero Forcing Set

Zero forcing is a graph coloring process based on the following color change rule: all vertices of a graph [Formula: see text] are initially colored either blue or white; in each timestep, a white vertex turns blue if it is the only white neighbor of some blue vertex. A zero forcing set of [Formula: see text] is a set of blue vertices such that all vertices eventually become blue after iteratively applying the color change rule. The zero forcing number [Formula: see text] is the cardinality of a minimum zero forcing set. In this paper, we propose novel exact algorithms for computing [Formula: see text] based on formulating the zero forcing problem as a two-stage Boolean satisfiability problem. We also propose several heuristics for zero forcing based on iteratively adding blue vertices which color a large part of the remaining white vertices. These heuristics are used to speed up the exact algorithms and can also be of independent interest in approximating [Formula: see text]. Computational results on various types of graphs show that, in many cases, our algorithms offer a significant improvement on the state-of-the-art algorithms for zero forcing. Summary of Contribution: This paper proposes novel algorithms and heuristics for an NP-hard graph coloring problem that has numerous applications. Our exact methods combine Boolean satisfiability modeling with a constraint generation framework commonly used in operations research. The paper also includes an analysis of the facets of the polytope associated with this problem and decomposition techniques which can reduce the size of the problem. Our computational approaches are implemented and tested on a wide variety of graphs and are compared with the state-of-the-art algorithms from the literature. We show that our proposed algorithms based on Boolean satisfiability, in conjunction with the heuristics and order-reduction techniques, yield a significant speedup in some cases.

Odd Cycle Zero Forcing Parameters and the Minimum Rank of Graph Blowups

2016 ◽  
Vol 31 ◽  
pp. 42-59 ◽  
Author(s):  
Jephian Chin-Hung Lin
Keyword(s):  
Simple Graph ◽  
Symmetric Matrices ◽  
Zero Forcing ◽  
Minimum Rank ◽  
Simple Graphs ◽  
Zero Forcing Number ◽  
Forcing Number ◽  
Odd Cycle ◽  
One Step

The minimum rank problem for a simple graph G and a given field F is to determine the smallest possible rank among symmetric matrices over F whose i, j-entry, i ≠ j, is nonzero whenever i is adjacent to j, and zero otherwise; the diagonal entries can be any element in F. In contrast, loop graphs \mathscr{G} go one step further to restrict the diagonal i, i-entries as nonzero whenever i has a loop, and zero otherwise. When char F ≠ 2, the odd cycle zero forcing number and the enhanced odd cycle zero forcing number are introduced as bounds for loop graphs and simple graphs, respectively. A relation between loop graphs and simple graphs through graph blowups is developed, so that the minimum rank problem of some families of simple graphs can be reduced to that of much smaller loop graphs.

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