Cops and robbers on planar directed graphs

TL;DR

This paper extends the study of the Cops and Robbers pursuit game to directed planar graphs, demonstrating that three cops may be insufficient for capture due to advanced robber strategies, unlike the undirected case.

Contribution

It introduces a geometric construction showing that directed planar graphs can require more than three cops, highlighting a separation from the undirected case.

Findings

Three cops are not always sufficient in directed planar graphs.

A specific strongly connected planar directed graph allows indefinite evasion.

The study reveals fundamental differences between directed and undirected graph pursuit games.

Abstract

Aigner and Fromme initiated the systematic study of the cop number of a graph by proving the elegant and sharp result that in every connected planar graph, three cops are sufficient to win a natural pursuit game against a single robber. This game, introduced by Nowakowski and Winkler, is commonly known as Cops and Robbers in the combinatorial literature. We extend this study to directed planar graphs, and establish separation from the undirected setting. We exhibit a geometric construction which shows that a more sophisticated robber strategy can indefinitely evade three cops on a particular strongly connected planar directed graph.