On the Total Forcing Number of a Graph

Let $G$ be a simple and finite graph without isolated vertices. In this paper we study forcing sets (zero forcing sets) which induce a subgraph of $G$ without isolated vertices. Such a set is called a total forcing set, introduced and first studied by Davila \cite{Davila}. The minimum cardinality of a total forcing set in $G$ is the total forcing number of $G$, denoted $F_t(G)$. We study basic properties of $F_t(G)$, relate $F_t(G)$ to various domination parameters, and establish $NP$-completeness of the associated decision problem for $F_t(G)$. We also prove that if $G$ is a connected graph of order $n \ge 3$ and maximum degree $\Delta$, then $F_t(G) \le ( \frac{\Delta}{\Delta +1} ) n$, with equality if and only if $G$ is a complete graph $K_{\Delta + 1}$.