The Cop Number of the One-Cop-Moves Game on Planar Graphs

TL;DR

This paper investigates a variant of the cops-and-robbers game on planar graphs, demonstrating that three cops are insufficient to guarantee capture when only one cop can move per turn, contrary to the classical case.

Contribution

It constructs a planar graph where three cops cannot guarantee capture in the one-cop-moves game, disproving a previous generalization of the classical result.

Findings

Three cops cannot always capture a robber in the one-cop-moves game on certain planar graphs.

The classical bound of three cops is not valid for the one-cop-moves variant.

A specific planar graph is constructed as a counterexample.

Abstract

Cops and robbers is a vertex-pursuit game played on graphs. In the classical cops-and-robbers game, a set of cops and a robber occupy the vertices of the graph and move alternately along the graph's edges with perfect information about each other's positions. If a cop eventually occupies the same vertex as the robber, then the cops win; the robber wins if she can indefinitely evade capture. Aigner and Frommer established that in every connected planar graph, three cops are sufficient to capture a single robber. In this paper, we consider a recently studied variant of the cops-and-robbers game, alternately called the one-active-cop game, one-cop-moves game or the lazy-cops-and-robbers game, where at most one cop can move during any round. We show that Aigner and Frommer's result does not generalise to this game variant by constructing a connected planar graph on which a robber can indefinitely evade three cops in the one-cop-moves game. This answers a question recently raised by Sullivan, Townsend and Werzanski.