The Zero Forcing Span of a Graph

In zero forcing, the focus is typically on finding the minimum cardinality of any zero forcing set in the graph; however, the number of cardinalities between 0 and the number of vertices in the graph for which there are both zero forcing sets and sets that fail to be zero forcing sets is not well known. In this paper, we introduce the zero forcing span of a graph, which is the number of distinct cardinalities for which there are sets that are zero forcing sets and sets that are not. We introduce the span within the context of standard zero forcing and skew zero forcing as well as for standard zero forcing on directed graphs. We characterize graphs with high span and low span of each type, and also investigate graphs with special zero forcing polynomials.