Brushing Number and Zero-Forcing Number of Graphs and their Line Graphs

TL;DR

This paper establishes tight bounds relating the brushing number and zero-forcing number of graphs and their line graphs, providing new insights into graph parameters and their interrelations.

Contribution

It proves that the zero-forcing number of a line graph bounds the brushing number, and confirms conjectures about the relationships between these parameters for graphs and their line graphs.

Findings

Zero-forcing number of line graph bounds brushing number

Zero-forcing number of graph is no more than that of its line graph

All bounds are shown to be tight

Abstract

In this paper we compare the brushing number of a graph with the zero-forcing number of its line graph. We prove that the zero-forcing number of the line graph is an upper bound for the brushing number by constructing a brush configuration based on a zero-forcing set for the line graph. Using a similar construction, we also prove the conjecture that the zero-forcing number of a graph is no more than the zero-forcing number of its line graph; moreover we prove that the brushing number of a graph is no more than the brushing number of its line graph. All three bounds are shown to be tight.