Failed zero forcing and critical sets on directed graphs

TL;DR

This paper investigates failed zero forcing sets in directed graphs, introduces the failed zero forcing number, and characterizes digraphs based on this parameter, including specific classes like acyclic graphs and cycles.

Contribution

It introduces the concept of failed zero forcing number for digraphs and characterizes digraphs with specific failed zero forcing numbers, expanding understanding of zero forcing dynamics in directed graphs.

Findings

Characterized digraphs with F(D)<Z(D)

Determined F(D) for classes like acyclic graphs and cycles

Constructed weak cycles with any failed zero forcing number

Abstract

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 $F(D)=k$.