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 Ft(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 Fc(G), is the cardinality of a smallest zero forcing set of G which induces a connected subgraph. In this paper, we first recharacterize the graphs with Ft(G) = 2 and, as a corollary, we recharacterize the graphs with Fc(G) = 2.