Related papers: A Comparison between the Zero Forcing Number and t…

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…

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…

\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…

In this note, we study a dynamic vertex coloring for a graph $G$. In particular, one starts with a certain set of vertices black, and all other vertices white. Then, at each time step, a black vertex with exactly one white neighbor forces…

The metric dimension dim(G) of a graph $G$ is the minimum cardinality of a subset $S$ of vertices of $G$ such that each vertex of $G$ is uniquely determined by its distances to $S$. It is well-known that the metric dimension of a graph can…

The zero forcing number of a graph $G$, denoted by $Z(G)$, is the minimum cardinality of a set $S$ of black vertices (where vertices in $V(G)\setminus S$ are colored white) such that $V(G)$ is turned black after finitely many applications…

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 $v$…

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 $v$…

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 simple, finite graph with vertex set $V(G)$, we define a zero forcing set of $G$ as follows. Choose $S\subseteq V(G)$ and color all vertices of $S$ blue and all vertices in $V(G) - S$ white. The color change rule is if $w$ is the…

In this paper, we study a dynamic coloring of the vertices of a graph $G$ that starts with an initial subset $S$ of colored vertices, with all remaining vertices being non-colored. At each discrete time interval, a colored vertex with…

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 $V(G)\setminusS$ are colored white) such that $V(G)$ is turned black after finitely many applications of…

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 $(i,j)$,…

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,…

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…

The zero forcing number of a simple graph, written $Z(G)$, is a NP-hard graph invariant which is the result of the zero forcing color change rule. This graph invariant has been heavily studied by linear algebraists, physicists, and graph…

A dynamic coloring of the vertices of a graph $G$ starts with an initial subset $F$ of colored vertices, with all remaining vertices being non-colored. At each time step, a colored vertex with exactly one non-colored neighbor forces this…

Let $G$ be a simple, finite, and undirected graph with vertices each given an initial coloring of either blue or white. Zero forcing on graph $G$ is an iterative process of forcing its white vertices to become blue after a finite…

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…

Zero forcing number has recently become an interesting graph parameter studied in its own right since its introduction by the "AIM Minimum Rank -- Special Graphs Work Group", whereas metric dimension is a well-known graph parameter. We…