Extremal values and bounds for the zero forcing number

Abstract

A set $Z$ of vertices of a graph $G$ is a zero forcing set of $G$ if iteratively adding to $Z$ vertices from $V\left(G\right)\setminus Z$ that are the unique neighbor in $V\left(G\right)\setminus Z$ of some vertex in $Z$, results in the entire vertex set $V\left(G\right)$ of $G$. The zero forcing number $Z\left(G\right)$ of $G$ is the minimum cardinality of a zero forcing set of $G$.

Amos et al. (2015) proved $Z\left(G\right)\le ((\Delta -2)n+2)/(\Delta -1)$ for a connected graph $G$ of order $n$ and maximum degree $\Delta \ge 2$. Verifying their conjecture, we show that $C_n$, $K_n$, and $K_{\Delta ,\Delta }$ are the only extremal graphs for this inequality. Confirming a conjecture of Davila and Kenter [5], we show that $Z\left(G\right)\ge 2\delta -2$ for every triangle-free graph $G$ of minimum degree $\delta \ge 2$. It is known that $Z\left(G\right)\ge P\left(G\right)$ for every graph $G$ where $P\left(G\right)$ is the minimum number of induced paths in $G$ whose vertex sets partition $V\left(G\right)$. We study the class of graphs $G$ for which every induced subgraph $H$ of $G$ satisfies $Z\left(H\right)=P\left(H\right)$.