Forcing Brushes

TL;DR

This paper provides concise proofs of inequalities relating the brushing number, zero forcing number, and line graph of a graph, simplifying previous complex proofs in graph theory.

Contribution

It introduces short and simple proofs for established inequalities involving brushing and zero forcing numbers in graphs and their line graphs.

Findings

Proved that B(G) ≤ Z(L(G)) for graphs without isolated vertices.

Proved that Z(G) ≤ Z(L(G)) for graphs without isolated vertices.

Simplified the understanding of relationships between graph invariants.

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 $G$.