Cops and Robbers ordinals of cop-win trees

TL;DR

This paper investigates the ordinal values associated with cop-win graphs, especially trees, extending the understanding from finite to infinite graphs and highlighting new classifications and open problems.

Contribution

It classifies CR-ordinals of cop-win trees and provides examples of infinite graphs with non-standard CR-ordinals, advancing the theoretical understanding of the game.

Findings

CR-ordinals of cop-win trees are either finite or of the form α + ω

Infinite cop-win graphs can have CR-ordinals not of the classified form

Open problems are posed for characterizing CR-ordinals in general graphs

Abstract

A relational characterization of cop-win graphs was provided by Nowakowski and Winkler in their seminal paper on the game of Cops and Robbers. As a by-product of that characterization, each cop-win graph is assigned a unique ordinal, which we refer to as a CR-ordinal. For finite graphs, CR-ordinals correspond to the length of the game assuming optimal play, with the cop beginning the game in a least favourable initial position. For infinite graphs, however, the possible values of CR-ordinals have not been considered in the literature until the present work. We classify the CR-ordinals of cop-win trees as either a finite ordinal, or those of the form $\alpha + \omega$, where $\alpha$ is a limit ordinal. For general infinite cop-win graphs, we provide an example whose CR-ordinal is not of this form. We finish with some problems on characterizing the CR-ordinals in the general case of cop-win graphs.