%0 Journal Article
%T GRAPHS WITH TOTAL FORCING NUMBER TWO, REVISITED
%J Journal of Algebraic Systems
%I Shahrood University of Technology
%Z 2345-5128
%A Alishahi, M.
%A Rezaei-Sani, E.
%D 2021
%\ 09/01/2021
%V 9
%N 1
%P 53-60
%! GRAPHS WITH TOTAL FORCING NUMBER TWO, REVISITED
%K Zero forcing set
%K Total forcing number
%K Connected forcing number
%R 10.22044/jas.2020.9229.1451
%X 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 $F_c(G)=2$.
%U https://jas.shahroodut.ac.ir/article_2054_f054a10538ec58a6a4782c9d7458d295.pdf