相关论文: Matching, Path Covers, and Total Forcing Sets

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 is an iterative graph coloring process studied for its wide array of applications. In this process, the vertices of the graph are initially designated as blue or white, and a zero forcing set is a set of initially blue vertices…

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…

Zero forcing (also called graph infection) on a simple, undirected graph $G$ is based on the color-change rule: If each vertex of $G$ is colored either white or black, and vertex $v$ is a black vertex with only one white neighbor $w$, then…

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…

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…

A well-studied concept is that of the total chromatic number. A proper total colouring of a graph is a colouring of both vertices and edges so that every pair of adjacent vertices receive different colours, every pair of adjacent edges…

Let $G$ be a simple, finite graph with vertex set $V(G)$ and edge set $E(G)$, where each vertex is either colored blue or white. Define the standard zero forcing process on $G$ with the following color-change rule: let $S$ be the set of all…

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…

\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…

Let $G=(V,E)$ be a finite connected graph along with a coloring of the vertices of $G$ using the colors in a given set $X$. In this paper, we introduce multi-color forcing, a generalization of zero-forcing on graphs, and give conditions in…

Let $S$ be a set of vertices of a graph $G$. Let $cl(S)$ be the set of vertices built from $S$, by iteratively applying the following propagation rule: if a vertex and all but exactly one of its neighbors are in $cl(S)$, then the remaining…

Zero forcing is an iterative graph coloring process where at each discrete time step, a colored vertex with a single uncolored neighbor forces that neighbor to become colored. The zero forcing number of a graph is the cardinality of the…

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…

A graph is said to be {\it total-colored} if all the edges and the vertices of the graph is colored. A path in a total-colored graph is a {\it total proper path} if $(i)$ any two adjacent edges on the path differ in color, $(ii)$ any two…

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 $V(G)-S$ are colored white) such that $V(G)$ is turned black after finitely many applications of "the…

We consider vertex colorings of graphs in which adjacent vertices have distinct colors. A graph is $s$-chromatic if it is colorable in $s$ colors and any coloring of it uses at least $s$ colors. The forcing chromatic number $F(G)$ of an…

In this paper, we introduce a new concept in graph coloring, namely the \textit{packing total coloring}, which extends the idea of packing coloring to both the vertices and the edges of a given graph. More precisely, for a graph $G$, a…

A power dominating set of a graph $G=(V,E)$ is a set $S\subset V$ that colors every vertex of $G$ according to the following rules: in the first timestep, every vertex in $N[S]$ becomes colored; in each subsequent timestep, every vertex…

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…