Bounding the total forcing number of graphs

Abstract

In recent years, a dynamic coloring, named as zero forcing, of the vertices in a graph have attracted many researchers. For a given G and a vertex subset S, assigning each vertex of S black and each vertex of $V\setminus S$ no color, if one vertex $u\in S$ has a unique neighbor v in $V\setminus S$, then u forces v to color black. S is called a zero forcing set if S can be expanded to the entire vertex set V by repeating the above forcing process. S is regarded as a total forcing set if the subgraph G[S] satisfies $\delta (G[S])\ge 1$. The minimum cardinality of a total forcing set in G, denoted by $F_t(G)$, is named the total forcing number of G. For a graph G, p(G), q(G) and $\phi (G)$ denote the number of pendant vertices, the number of vertices with degree at least 3 meanwhile having one pendant path and the cyclomatic number of G, respectively. In the paper, by means of the total forcing set of a spanning tree regarding a graph G, we verify that $F_t(G)\le p(G)+q(G)+2\phi (G)$. Furthermore, all graphs achieving the equality are determined.