Title: On the Total Forcing Number of a Graph,
Abstract: 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 $K_{\Delta + 1}$.