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.

近年来，图中顶点的动态着色（称为迫零）吸引了许多研究人员。对于给定的G和顶点子集S，如果一个顶点\ (u\in S \ )在\ ( V\setminus S\)，然后u强制v变为黑色。如果S可以通过重复上述强制过程扩展到整个顶点集V ，则S称为迫零集。如果子图G [ S ] 满足\(\delta (G[S])\ge 1\) ，则S被视为总强迫集。G中总强迫集的最小基数用F_t(G)表示，称为G的总强迫数。对于图G，p ( G ) 、q ( G ) 和\(\phi (G)\)表示下垂顶点的数量、度数至少为 3 同时具有一条下垂路径的顶点数量以及分别为G。在本文中，通过图G的生成树的总强迫集，我们验证了\(F_t(G)\le p(G)+q(G)+2\phi (G)\)。此外，确定了所有实现相等的图。