Forcing Brushes

Abstract

We give short and simple proofs of the inequalities $B(G)\leq Z(L(G))$ and $Z(G)\leq Z(L(G))$ first established by Erzurumluo\u{g}lu, Meagher, and Pike, where $G$ is a graph without isolated vertices, $B(G)$ is the brushing number of $G$, $Z(G)$ is the zero forcing number of $G$, and $L(G)$ is the line graph of $G$.