GRAPHS WITH TOTAL FORCING NUMBER TWO, REVISITED

A subset of the vertex set of a graph $G$ is called a zero forcing set if by considering them colored and, as far as possible, a colored vertex with exactly one non-colored neighbor forces its non-colored neighbor to get colored, then the whole vertices of $G$ become colored. The total forcing number of a graph $G$, denoted by $F_t(G)$, is the cardinality of a smallest zero forcing set of $G$ which induces a subgraph with no isolated vertex. The connected forcing number, denoted by $F_c(G)$, is the cardinality of a smallest zero forcing set of $G$ which induces a connected subgraph. In this paper, we first characterize the graphs with $F_t(G)=2$ and, as a corollary, we characterize the graphs with $F_c(G)=2$.