It turns out that F−(G) is undefined for certain graphs: we classify such graphs. We provide an algorithm that determines if F−(G) is undefined, and also finds the unique perfect matching guaranteed in the case F−(G) is undefined. We provide classifications of graphs with high and low values of F−(G), compare the parameter to other zero forcing parameters, and finally, establish F−(G) for multiple graph families.
The skew zero forcing number was originally introduced for its application to the problem of finding the minimum rank among all skew-symmetric matrices associated with a graph, and it also applies to the problem among all symmetric matrices with zero diagonal. We establish connections between F−(G) and these minimum rank problems. We also classify all graphs such that any set of vertices of order Z−(G) is a skew zero forcing set.