Failed skew zero forcing on a graph

Given a graph G and an initial designation of each vertex in V(G) as “filled” or “empty,” we apply the skew color change rule, which states that a vertex v becomes filled if and only if it is the unique empty neighbor of some other vertex in the graph. If repeated application of the color change rule results in all vertices in the graph eventually being filled, then the initial set of filled vertices is a skew zero forcing set. The minimum cardinality of such a set is the skew zero forcing number of G, Z−(G). In this paper, we introduce and investigate a new parameter, the failed skew zero forcing number F−(G), which is the maximum cardinality of a starting set that will not result in all vertices in the graph eventually being filled.