Some results on the total (zero) forcing number of a graph

Abstract

Let F(G) and \(F_t(G)\) be the zero forcing number and the total forcing number of a graph G, respectively. In this paper, we study the relationship between the total forcing number of a graph and its vertex covering number (or independence number), and prove that \(F_t(G) \le \Delta \alpha (G)\) and \(F_t(G) \le (\Delta - 1)\beta (G) + 1\) for any connected graph G with the maximum degree \(\Delta \), where \(\alpha (G)\) and \(\beta (G)\) are the independence number and the vertex covering number of G. In particular, we prove that \(F_t(T) \le F(T) + \beta (T)\) for any tree T and characterize all trees T with \(F_t(T) = F(T) + \beta (T)\). At the same time, all trees T with \(F_t(T) = (\Delta - 1)\beta (T) + 1\) are completely characterized. In addition, we explore trees, unicycle graphs and Halin graphs satisfying \(F(G) \le \alpha (G)+1\).