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.