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

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.

Forcing brushes

“…Erzurumluoglu, Meagher, and Pike [4] recently discovered an inequality between the brushing number of a graph and the zero forcing number of its line graph; thereby linking two graph parameters whose connections have not been studied so far. Our goal in this note is to provide short and simple proofs for two of their main results.…”