A matroid characterization of the combinatorially symmetric maximum nullity of a graph

Abstract

The graph parameter ${\mathrm N}(G)$ is the maximum nullity among all combinatorially symmetric matrices associated to a graph $G$. The zero forcing number ${\mathrm Z}(G)$ is a well-known upper bound for ${\mathrm N}(G)$, and a zero forcing set of $G$ is exactly a set that intersects each fort of the graph. So the zero forcing number can be defined as the size of a minimum set that intersects every fort of $G$. This paper provides a similar characterization of ${\mathrm N}(G)$ that is purely combinatorial, using a connection to matroid theory and collections of forts that are compatible in a certain sense. We provide several examples highlighting subtle differences among compatible collections of forts and how they relate to full spark matrices and matroids associated to $G$.