Maximal failed zero forcing sets for products of two graphs

Abstract

Let $G$ be a simple, finite graph with vertex set $V(G)$ and edge set $E(G)$, where each vertex is either colored blue or white. Define the standard zero forcing process on $G$ with the following color-change rule: let $S$ be the set of all initially blue vertices of $G$ and let $u \in S$. If $v$ is the unique white vertex adjacent to $u$ in $G$, color $v$ blue and update $S$ by adding $v$ to $S$. If $S = V(G)$ after a finite number of iterations of the color-change rule, we say that $S$ is a zero forcing set for $G$. Otherwise, we say that $S$ is a failed zero forcing set. In this paper, we construct maximal failed zero forcing sets for graph products such as Cartesian products, strong products, lexicographic products, and coronas. In particular, we consider products of two paths, two cycles, and two complete graphs.