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 $n=|V(D)|$, and each arc is an ordered pair of distinct vertices. If $(v,u) \in A(D)$, then $u$ is considered an \emph{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. The process continues until the CCR does not allow any empty vertex to become filled. If all vertices in $V(D)$ are eventually filled, then the initial set is called a \emph{zero forcing set} (ZFS); if not, it is a \emph{failed zero forcing set} (FZFS). We introduce the \emph{failed zero forcing number} $F(D)$ on a digraph, which is the maximum cardinality of any FZFS. The \emph{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 directed acyclic graphs, weak paths and cycles, and weakly connected line digraphs such as 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 $n \geq 3$ and any non-negative integer $k$ with $k <n$, there exists a weak cycle $D$ with $F(D)=k$.