The zero forcing span of a graph

TL;DR

This paper introduces the concept of zero forcing span, measuring the diversity of set sizes that can or cannot force a graph to be colored, and characterizes graphs with extreme span values across different zero forcing variants.

Contribution

It defines and analyzes the zero forcing span, a new graph invariant, for standard, skew, and directed zero forcing, providing characterizations of graphs with high and low span.

Findings

Graphs with high span have diverse zero forcing set sizes.

Graphs with low span have uniform zero forcing set sizes.

The paper characterizes graphs with special zero forcing polynomials.

Abstract

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.