Characterization of Graphs With Failed Skew Zero Forcing Number of 1

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. Hence the failed zero forcing number of a graph was defined to be the size of the largest set of vertices which fails to force all vertices in the graph. A similar property called skew zero forcing was defined so that if there is exactly one neighbor $u$ of $v$ is not in $S$, then $u$ is added to $S$ in the next iteration. The difference is that vertices that are not in $S$ can force other vertices. This leads to the failed skew zero forcing number of a graph, which is denoted by $F^{-}(G)$. In this paper we provide a complete characterization of all graphs with $F^{-}(G)=1$. Fetcie, Jacob, and Saavedra showed that the only graphs with a failed zero forcing number of $1$ are either: the union of two isolated vertices; $P_3$; $K_3$; or $K_4$. In this paper we provide a surprising result: changing the forcing rule to a skew-forcing rule results in an infinite number of graphs with $F^{-}(G)=1$.