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.