Brushing Directed Graphs

TL;DR

This paper introduces a new model for brushing directed graphs where brushes follow arc directions, providing bounds and exact values for various graph classes, advancing understanding of graph searching in directed networks.

Contribution

It proposes a novel brushing model for directed graphs and determines the brushing number for several important classes, including transitive tournaments and DAGs.

Findings

Brushing number for transitive tournaments is determined.

Upper bounds for brushing numbers of directed acyclic graphs are established.

Exact brushing numbers for complete directed graphs, rooted trees, and rotational tournaments are provided.

Abstract

Brushing of graphs is a graph searching process in which the searching agents are called brushes. We focus on brushing directed graphs based on a new model in which the brushes can only travel in the same direction as the orientation of the arcs that they traverse. We discuss strategies to brush directed graphs as well as values and bounds for the brushing number of directed graphs. We determine the brushing number for any transitive tournament, which we use to give an upper bound for the brushing number of directed acyclic graphs in general. We also establish exact values for the brushing numbers of complete directed graphs, rooted trees, and rotational tournaments.