Color-change Rule Research Articles

Neutrosophic 1-Failed SuperHyperForcing in the SuperHyperFunction to Use Neutrosophic SuperHyperGraphs on Cancer’s Neutrosophic Recognition and Beyond

Neutrosophic 1-Failed SuperHyperForcing in the SuperHyperFunction to Use Neutrosophic SuperHyperGraphs on Cancer’s Neutrosophic Recognition and Beyond

Open global shadow graph and it’s zero forcing number

Zero forcing number of a graph is the minimum cardinality of the zero forcing set. A zero forcing set is a set of black vertices of minimum cardinality that can colour the entire graph black using the color change rule: each vertex of G is coloured either white or black, and vertex v is a black vertex and can force a white neighbour only if it has one white neighbour. In this paper we identify a class of graph where the zero forcing number is equal to the minimum rank of the graph and call it as a new class of graph that is open global shadow graph”. Some of the basic properties of open global shadow graph are studied. The zero forcing number of open global shadow graph of a graph with upper and lower bound is obtained. Hence giving the upper and lower bound for the minimum rank of the graph.

K-Forcing number for Cartesian product of some graphs

$k$-Forcing is an iterative graph coloring process based on a color change rule that describes how to color the vertices. $k$-Forcing is a generalization of zero forcing that is useful in multiple scientific branches, such as quantum control. In this paper, we investigate the $k$-forcing number of the Cartesian product of some graphs. The main contribution of this paper is to determine the $k$-forcing number of the Cartesian product of two complete bipartite graphs using a new representation of this graph.

Various characterizations of throttling numbers

Zero forcing can be described as a graph process that uses a color change rule in which vertices change white vertices to blue. The throttling number of a graph minimizes the sum of the number of vertices initially colored blue and the number of time steps required to color the entire graph. Positive semidefinite (PSD) zero forcing is a commonly studied variant of standard zero forcing that alters the color change rule. This paper introduces a method for extending a graph using a PSD zero forcing process. Using this extension method, graphs with PSD throttling number at most t are characterized as specific minors of the Cartesian product of complete graphs and trees. A similar characterization is obtained for the minor monotone floor of PSD zero forcing. Finally, the set of connected graphs on n vertices with throttling number at least n−k is characterized by forbidding a finite family of induced subgraphs. These forbidden subgraphs are constructed for standard throttling.

Improved Computational Approaches and Heuristics for Zero Forcing

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.

Using Markov Chains to Determine Expected Propagation Time for Probabilistic Zero Forcing

Zero forcing is a coloring game played on a graph where each vertex is initially colored blue or white and the goal is to color all the vertices blue by repeated use of a (deterministic) color change rule starting with as few blue vertices as possible. Probabilistic zero forcing yields a discrete dynamical system governed by a Markov chain. Since in a connected graph any one vertex can eventually color the entire graph blue using probabilistic zero forcing, the expected time to do this is studied. Given a Markov transition matrix for a probabilistic zero forcing process, an exact formula is established for expected propagation time. Markov chains are applied to determine bounds on expected propagation time for various families of graphs.

Connected zero forcing sets and connected propagation time of graphs

The zero forcing number $Z(G)$ of a graph $G$ is the minimum cardinality of a set $S$ with colored (black) vertices which forces the set $V(G)$ to be colored (black) after some times. ``color change rule'': a white vertex is changed to a black vertex when it is the only white neighbor of a black vertex. In this case, we say that the black vertex forces the white vertex. We investigate here the concept of connected zero forcing set and connected zero forcing number. We discusses this subject for special graphs and some products of graphs. Also we introduce the connected propagation time. Graphs with extreme minimum connected propagation times and maximum propagation times $|G|-1$ and $|G|-2$ are characterized.

Computing the zero forcing number for generalized Petersen graphs

Let $G$ be a simple undirected graph with each vertex colored
 either white or black, $ u $ be a black vertex of $ G, $ and
 exactly one neighbor $ v $ of $ u $ be white. Then change the
 color of $ v $ to black. When this rule is applied, we say $ u $
 forces $ v, $ and write $ u \rightarrow v $. A $zero\ forcing\ set$ of a graph $ G$ is a
 subset $Z$ of vertices such that if initially the vertices in $ Z $ are colored
 black and remaining vertices
 are colored white, the entire graph $ G $ may be colored black
 by repeatedly applying the
 color-change rule.
 The zero forcing number of $ G$, denoted $Z(G), $ is the minimum size of a
 zero forcing set.\\
 In this paper, we investigate the zero forcing number for the generalized Petersen graphs (It is denoted by $P(n,k)$). We obtain upper and lower bounds for the zero forcing number for $P(n,k)$. We show that $Z(P(n,2))=6$ for $n\geq 10$, $Z(P(n,3))=8$ for $n\geq 12$ and $Z(P(2k+1,k))=6$ for $k\geq 5$.

A Unifying Framework for Strong Structural Controllability

This article deals with strong structural controllability of linear systems. In contrast to the existing work, the structured systems studied in this article have a so-called zero/nonzero/arbitrary structure, which means that some of the entries are equal to zero, some of the entries are arbitrary but nonzero, and the remaining entries are arbitrary (zero or nonzero). We formalize this in terms of pattern matrices, whose entries are either fixed zero, arbitrary nonzero, or arbitrary. We establish necessary and sufficient algebraic conditions for strong structural controllability in terms of full rank tests of certain pattern matrices. We also give a necessary and sufficient graph-theoretic condition for the full rank property of a given pattern matrix. This graph-theoretic condition makes use of a new color change rule that is introduced in this article. Based on these two results, we then establish a necessary and sufficient graph-theoretic condition for strong structural controllability. Moreover, we relate our results to those that exist in the literature and explain how our results generalize previous work.

Two distance forcing number of a graph

Motivated from the graph parameters namely zero forcing number, k-forcing number and the connected k-forcing number, in this article, we introduce a new parameter known as the 2-distance forcing number. Assume that each vertex of a graph G = (V(G),E(G)) is colored as either white or black. Consider the set Z2d of black colored vertices of the graph G. The color change rule changes the color of a white vertex v to black if the white vertex v is the only 2-distance white neighbor of a black vertex u. The set Z2d is called a two distance forcing set of G if all vertices of the graph G will be turned black after limited applications of the color change rule. The 2-distance forcing number of G, denoted by Z2d(G), is the minimum of |Z2d| over all 2- distance forcing sets Z2d ⊆ V(G). This manuscript is intended to study the 2-distance forcing number of some graphs. We find the exact value of the 2-distance forcing number of graphs such as the pineapple graph, gear graph, jelly fish graph, helm graph, sunflower graph, comet graph and the n-pan graph.

Throttling positive semidefinite zero forcing propagation time on graphs

Zero forcing is a process on a graph that colors vertices blue by starting with some of the vertices blue and applying a color change rule. Throttling minimizes the sum of the size of the initial blue vertex set and the number of the time steps needed to color the graph. We study throttling for positive semidefinite zero forcing. We establish a tight lower bound on the positive semidefinite throttling number as a function of the order, maximum degree, and positive semidefinite zero forcing number of the graph, and determine the positive semidefinite throttling numbers of paths, cycles, and full binary trees. We characterize the graphs that have extreme positive semidefinite throttling numbers.

Effects of vertex degrees on the zero-forcing number and propagation time of a graph

In this paper, we present new bounds on the zero-forcing propagation time and zero-forcing number. Zero-forcing is based on iterating the following color change rule. If a vertex is colored and has only one neighbor that is not colored, then that uncolored neighbor becomes colored in the next iteration. The zero-forcing number is the size of a minimum set of vertices such that repeated iterations of the color change rule will color the entire graph. The zero-forcing propagation time is the minimum number of iterations required to color the entire graph from a minimum zero-forcing set. We present new bounds on the zero-forcing propagation time for cubic graphs. These bounds reduce the known bounds that are based on the minimum degree of the graph. We introduce zero-forcing forts, a special type of subgraph that must contain a vertex in any minimum zero-forcing set, and use these forts to bound the zero-forcing number of a graph. We also show that the zero-forcing number of a graph is bounded from below by its branchwidth.

On the error of a priori sampling: Zero forcing sets and propagation time

Zero forcing is an iterative process on a graph used to bound the maximum nullity. The process begins with select vertices as colored, and the remaining vertices can become colored under a specific color change rule. The goal is to find a minimum set of vertices such that after iteratively applying the rule, all of the vertices become colored (i.e., a minimum zero forcing set). Of particular interest is the propagation time of a chosen set which is the number of steps the rule must be applied in order to color all the vertices of a graph.We give a purely linear algebraic view of zero forcing: Find a set of vertices S such that for any weighted adjacency matrix A, whenever Ax=0, the entirety of x can be recovered using only xS, the entries corresponding to S. The key here is that S must be chosen before A. In this light, we are able to give a linear algebraic interpretation of the propagation time: Any error in xS effects the error of x exponentially in the propagation time. This error can be quantitatively measured using newly defined zero forcing-related parameters, the error polynomial vector and the variance polynomial vector. In this sense, the quality of two zero forcing sets can objectively be compared even if the sets are the same size and their propagation times are the same. Examples and constructions are given.

Families of graphs with maximum nullity equal to zero forcing number

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 propagation time on oriented graphs

Zero forcing is an iterative coloring procedure on a graph that starts by initially coloring vertices white and blue and then repeatedly applies the following rule: if any blue vertex has a unique (out-)neighbor that is colored white, then that neighbor is forced to change color from white to blue. An initial set of blue vertices that can force the entire graph to blue is called a zero forcing set. In this paper we consider the minimum number of iterations needed for this color change rule to color all of the vertices blue, also known as the propagation time, for oriented graphs. We produce oriented graphs with both high and low propagation times, consider the possible propagation times for the orientations of a fixed graph, and look at balancing the size of a zero forcing set and the propagation time.

A comparison between the metric dimension and zero forcing number of trees and unicyclic graphs

The \emph{metric dimension} $\dim(G)$ of a graph $G$ is the minimum number of vertices such that every vertex of $G$ is uniquely determined by its vector of distances to the chosen vertices. The \emph{zero forcing number} $Z(G)$ of a graph $G$ is the minimum cardinality of a set $S$ of black vertices (whereas vertices in $V(G)\!\setminus\!S$ are colored white) such that $V(G)$ is turned black after finitely many applications of "the color-change rule": a white vertex is converted black if it is the only white neighbor of a black vertex. We show that $\dim(T) \leq Z(T)$ for a tree $T$, and that $\dim(G) \le Z(G)+1$ if $G$ is a unicyclic graph, along the way, we characterize trees $T$ attaining $\dim(T)=Z(T)$. For a general graph $G$, we introduce the "cycle rank conjecture". We conclude with a proof of $\dim(T)-2 \leq \dim(T+e) \le \dim(T)+1$ for $e \in E(\overline{T})$.

Failed skew zero forcing on a graph

Given a graph G and an initial designation of each vertex in V(G) as “filled” or “empty,” we apply the skew color change rule, which states that a vertex v becomes filled if and only if it is the unique empty neighbor of some other vertex in the graph. If repeated application of the color change rule results in all vertices in the graph eventually being filled, then the initial set of filled vertices is a skew zero forcing set. The minimum cardinality of such a set is the skew zero forcing number of G, Z−(G). In this paper, we introduce and investigate a new parameter, the failed skew zero forcing numberF−(G), which is the maximum cardinality of a starting set that will not result in all vertices in the graph eventually being filled.It turns out that F−(G) is undefined for certain graphs: we classify such graphs. We provide an algorithm that determines if F−(G) is undefined, and also finds the unique perfect matching guaranteed in the case F−(G) is undefined. We provide classifications of graphs with high and low values of F−(G), compare the parameter to other zero forcing parameters, and finally, establish F−(G) for multiple graph families.The skew zero forcing number was originally introduced for its application to the problem of finding the minimum rank among all skew-symmetric matrices associated with a graph, and it also applies to the problem among all symmetric matrices with zero diagonal. We establish connections between F−(G) and these minimum rank problems. We also classify all graphs such that any set of vertices of order Z−(G) is a skew zero forcing set.

Positive semidefinite propagation time

Let G be a simple, undirected graph. Positive semidefinite (PSD) zero forcing on G is based on the following color-change rule: Let W1,W2,…,Wk be the sets of vertices of the k connected components in G−B (where B is a set of blue vertices). If w∈Wi is the only white neighbor of some b∈B in the graph G[B∪Wi], then we change w to blue. A minimum positive semidefinite zero forcing set (PSDZFS) is a set of blue vertices that colors the entire graph blue and has minimum cardinality. The PSD propagation time of a PSDZFS B of graph G is the minimum number of iterations that it takes to color the entire graph blue, starting with B blue, such that at each iteration as many vertices are colored blue as allowed by the color-change rule. The minimum and maximum PSD propagation times are taken over all minimum PSD zero forcing sets of the graph. It is conjectured that every propagation time between the minimum and maximum propagation time is attainable by some minimum PSDZFS (this is not the case for the standard color-change rule). Tools are developed that aid in the computation of PSD propagation time. Several graph families and graphs with extreme PSD propagation times (|G|−2,|G|−1,1,0) are analyzed.

On zero forcing number of graphs and their complements

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].

The failed zero forcing number of a graph

Given a graph G, the zero forcing number of G, Z(G), is the smallest cardinality of any set S of vertices on which repeated applications of the color change rule results in all vertices joining S. The color change rule is: if a vertex v is in S, and exactly one neighbor u of v is not in S, then u joins S in the next iteration. In this paper, we introduce a new graph parameter, the failed zero forcing number of a graph. The failed zero forcing number of G, F(G), is the maximum cardinality of any set of vertices on which repeated applications of the color change rule will never result in all vertices joining the set. We establish bounds on the failed zero forcing number of a graph, both in general and for connected graphs. We also classify connected graphs that achieve the upper bound, graphs whose failed zero forcing numbers are zero or one, and unusual graphs with smaller failed zero forcing number than zero forcing number. We determine formulas for the failed zero forcing numbers of several families of graphs and provide a lower bound on the failed zero forcing number of the Cartesian product of two graphs. We conclude by presenting open questions about the failed zero forcing number and zero forcing in general.