Shahrood University of TechnologyJournal of Algebraic Systems2345-51289120210901GRAPHS WITH TOTAL FORCING NUMBER TWO, REVISITED5360205410.22044/jas.2020.9229.1451ENM.AlishahiFaculty of Mathematical Sciences, Shahrood University of Technology, P.O. Box:
316-3619995161, Shahrood, Iran.0000-0001-6588-8520E.Rezaei-SaniFaculty of Mathematical Sciences, Shahrood University of Technology, P.O. Box:
316-3619995161, Shahrood, Iran.Journal Article20191228A 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 $F_c(G)=2$.https://jas.shahroodut.ac.ir/article_2054_f054a10538ec58a6a4782c9d7458d295.pdf