A Note on Broom Number and Maximum Edge-Cut of Graphs

Abstract

A model for cleaning a graph with brushes was first introduced by Messinger, Nowakowski, and Pralat in 2008. Later, they focused on the problem of determining the maximum number of brushes needed to clean a graph. This maximum number of brushes needed to clean a graph in the model is called the broom number of the graph. In this paper, we show that the broom number of a graph is equal to the size of a maximum edge-cut of the graph, and prove the \(\mathcal{NP}\)-completeness of the problem of determining the broom number of a graph. As an application, we determine the broom number exactly for the Cartesian product of two graphs.