Path cover number, maximum nullity, and zero forcing number of oriented graphs and other simple digraphs

Abstract

An oriented graph is a simple digraph obtained from a simple graph by choosing exactly one of the two arcs $\left(u,v\right)$ or $\left(v,u\right)$ to replace each edge $\left\{u,v\right\}$. A simple digraph describes the zero-nonzero pattern of off-diagonal entries of a family of (not necessarily symmetric) matrices. The minimum rank of a simple digraph is the minimum rank of this family of matrices; maximum nullity is defined analogously. The simple digraph zero forcing number and path cover number are related parameters. We establish bounds on the range of possible values of all these parameters for oriented graphs, establish connections between the values of these parameters for a simple graph $G$, for various orientations $\overrightarrow{G}$ and for the doubly directed digraph of $G$, and establish an upper bound on the number of arcs in a simple digraph in terms of the zero forcing number.