Cop number of graphs without long holes

TL;DR

This paper establishes a new upper bound on the number of cops needed to catch a robber in graphs without long holes, improving previous results and linking cop number to the Dilworth number.

Contribution

It provides a simplified winning strategy for t-3 cops in graphs without long holes, strengthening prior bounds and connecting cop number to graph parameters.

Findings

t-3 cops suffice in graphs without holes of length at least t

Improved bound over previous t-2 cops result

Derived inequality relating cop number and Dilworth number

Abstract

A hole in a graph is an induced cycle of length at least 4. We give a simple winning strategy for t-3 cops to capture a robber in the game of cops and robbers played in a graph that does not contain a hole of length at least t. This strengthens a theorem of Joret-Kaminski-Theis, who proved that t-2 cops have a winning strategy in such graphs. As a consequence of our bound, we also give an inequality relating the cop number and the Dilworth number of a graph.