Loop zero forcing and grundy domination in planar graphs and claw-free cubic graphs

Abstract

Given a simple, finite graph with vertex set $V(G)$, we define a zero forcing set of $G$ as follows. Choose $S\subseteq V(G)$ and color all vertices of $S$ blue and all vertices in $V(G) - S$ white. The color change rule is if $w$ is the only white neighbor of blue vertex $v$, then we change the color of $w$ from white to blue. If after applying the color change rule as many times as possible eventually every vertex of $G$ is blue, we call $S$ a zero forcing set of $G$. $Z(G)$ denotes the minimum cardinality of a zero forcing set. Davila and Henning proved in \cite{zerocubic} that for any claw-free cubic graph $G$, $Z(G) \le \frac{1}{3}|V(G)| + 1$. We show that if $G$ is $2$-edge-connected, claw-free, and cubic, then $Z(G) \le \left\lceil\frac{5n(G)}{18}\right\rceil+1$. We also study a similar graph invariant known as the loop zero forcing number of a graph $G$ which happens to be the dual invariant to the Grundy domination number of $G$. Specifically, we study the loop zero forcing number in two particular types of planar graphs.