Failed zero forcing and critical sets on directed graphs

Let D be a simple digraph (directed graph) with vertex set V(D) and arc set A(D) where n = vertical bar V(D)vertical bar, and each arc is an ordered pair of distinct vertices. If (v, u) is an element of A(D), then u is considered an out-neighbor of v in D. Initially, we designate each vertex to be either filled or empty. Then, the following color change rule (CCR) is applied: if a filled vertex v has exactly one empty out-neighbor u, then u will be filled. If all vertices in V(D) are eventually filled under repeated applications of the CCR, then the initial set is called a zero forcing set (ZFS); if not, it is a failed zero forcing set (FZFS). We introduce the failed zero forcing number F(D) on a digraph, which is the maximum cardinality of any FZFS. The zero forcing number, Z(D), is the minimum cardinality of any ZFS. We characterize digraphs that have F(D) < Z(D) and determine F(D) for several classes of digraphs including de Bruijn and Kautz digraphs. We also characterize digraphs with F(D) = n - 1, F(D) = n - 2, and F(D) = 0, which leads to a characterization of digraphs in which any vertex is a ZFS. Finally, we show that for any integer n >= 3 and any non-negative integer k with k < n, there exists a weak cycle D with F(D) = k.