Bounding the total forcing number of graphs

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∖ S no color, if one vertex u∈ S has a unique neighbor v in V∖ 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 δ (G[S])≥ 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 ϕ (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)≤ p(G)+q(G)+2ϕ (G) . Furthermore, all graphs achieving the equality are determined.