Related papers: Minimal Zero Forcing Sets
The zero forcing number Z(G), which is the minimum number of vertices in a zero forcing set of a graph G, is used to study the maximum nullity / minimum rank of the family of symmetric matrices described by G. It is shown that for a…
A set $Z$ of vertices of a graph $G$ is a zero forcing set of $G$ if initially labeling all vertices in $Z$ with $1$ and all remaining vertices of $G$ with $0$, and then, iteratively and as long as possible, changing the label of some…
Given a graph $G=(V,E)$ and a set of vertices marked as filled, we consider a color-change rule known as zero forcing. A set $S$ is a zero forcing set if filling $S$ and applying all possible instances of the color change rule causes all…
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 forcing rule results in all vertices being in $S$. The forcing rule is: if a vertex $v$…
For any simple graph $G$ on $n$ vertices, the (positive semi-definite) minimum rank of $G$ is defined to be the smallest possible rank among all (positive semi-definite) real symmetric $n\times n$ matrices whose entry in position $(i,j)$,…
Zero forcing in graphs is a coloring process where a colored vertex can force its unique uncolored neighbor to be colored. A zero forcing set is a set of initially colored vertices capable of eventually coloring all vertices of the graph.…
A forcing set for a perfect matching of a graph is defined as a subset of the edges of that perfect matching such that there exists a unique perfect matching containing it. A complete forcing set for a graph is a subset of its edges, such…
A subset $S$ of initially infected vertices of a graph $G$ is called forcing if we can infect the entire graph by iteratively applying the following process. At each step, any infected vertex which has a unique uninfected neighbour, infects…
Let $G$ be a graph, and $Z$ a subset of its vertices, which we color black, while the remaining are colored white. We define the skew color change rule as follows: if $u$ is a vertex of $G$, and exactly one of its neighbors $v$, is white,…
For a graph $G$ in which vertices are either black or white, a zero forcing process is an iterative vertex color changing process such that the only white neighbor of a black vertex becomes black in the next time step. A zero forcing set is…
Let $G$ be a simple and finite graph without isolated vertices. In this paper we study forcing sets (zero forcing sets) which induce a subgraph of $G$ without isolated vertices. Such a set is called a total forcing set, introduced and first…
Let each vertex of a graph G = (V(G), E(G)) be given one of two colors, say, "black" and "white". Let Z denote the (initial) set of black vertices of G. The color-change rule converts the color of a vertex from white to black if the white…
In zero forcing, the focus is typically on finding the minimum cardinality of any zero forcing set in the graph; however, the number of cardinalities between $0$ and the number of vertices in the graph for which there are both zero forcing…
This paper begins the study of reconfiguration of zero forcing sets, and more specifically, the zero forcing graph. Given a base graph $G$, its zero forcing graph, $\mathscr{Z}(G)$, is the graph whose vertices are the minimum zero forcing…
The forcing number of a perfect matching $M$ in a graph $G$ is the smallest number of edges inside $M$ that can not be contained in other perfect matchings. The anti-forcing number of $M$ is the smallest number of edges outside $M$ whose…
The zero forcing process is an iterative graph colouring process in which at each time step a coloured vertex with a single uncoloured neighbour can force this neighbour to become coloured. A zero forcing set of a graph is an initial set of…
Given a graph $G$ and a real number $0\le p\le 1$, we define the random set $B_p(G)\subset V(G)$ by including each vertex independently and with probability $p$. We investigate the probability that the random set $B_p(G)$ is a zero forcing…
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…
Zero forcing is an iterative graph coloring process, where given a set of initially colored vertices, a colored vertex with a single uncolored neighbor causes that neighbor to become colored. A zero forcing set is a set of initially colored…
In this paper, we study (zero) forcing sets which induce connected subgraphs of a graph. The minimum cardinality of such a set is called the connected forcing number of the graph. We provide sharp upper and lower bounds on the connected…