The failed zero forcing number of a graph

Abstract

Given a graph $G$, the zero forcing number of $G$, $Z\left(G\right)$, is the smallest cardinality of any set $S$ of vertices on which repeated applications of the color change rule results in all vertices joining $S$. The color change rule is: if a vertex $v$ is in $S$, and exactly one neighbor $u$ of $v$ is not in $S$, then $u$ joins $S$ in the next iteration.

In this paper, we introduce a new graph parameter, the failed zero forcing number of a graph. The failed zero forcing number of $G$, $F\left(G\right)$, is the maximum cardinality of any set of vertices on which repeated applications of the color change rule will never result in all vertices joining the set.

We establish bounds on the failed zero forcing number of a graph, both in general and for connected graphs. We also classify connected graphs that achieve the upper bound, graphs whose failed zero forcing numbers are zero or one, and unusual graphs with smaller failed zero forcing number than zero forcing number. We determine formulas for the failed zero forcing numbers of several families of graphs and provide a lower bound on the failed zero forcing number of the Cartesian product of two graphs.

We conclude by presenting open questions about the failed zero forcing number and zero forcing in general.