All Graphs with a Failed Zero Forcing Number of Two

Given a graph $G$, the zero-forcing number of $G$, $Z(G)$, is the smallest cardinality of any set $S$ of vertices on which repeated applications of the forcing rule results in all vertices being in $S$. The forcing rule is: if a vertex $v$ is in $S$, and exactly one neighbor $u$ of $v$ is not in $S$, then $u$ is added to $S$ in the next iteration. Zero-forcing numbers have attracted great interest over the past 15 years and have been well studied. In this paper we investigate the largest size of a set $S$ that does not force all of the vertices in a graph to be in $S$. This quantity is known as the failed zero-forcing number of a graphs and will be denoted by $F(G)$, and has received attention in recent years. We present new results involving this parameter. In particular, we completely characterize all graphs $G$ where $F(G)=2$, solving a problem posed in 2015 by Fetcie, Jacob, and Saavedra.