A Cop-Win Graph with Maximum Capture Time $\omega$

TL;DR

This paper constructs an infinite cop-win graph with unbounded finite capture times, demonstrating that the maximum capture time can be infinite, which disproves a previous conjecture in the study of cops and robbers on graphs.

Contribution

It provides a counterexample of an infinite cop-win graph with unbounded capture times, challenging existing beliefs about maximum capture time in such graphs.

Findings

Constructed an infinite cop-win graph with maximum capture time 

Showed that finite capture times can be unbounded in cop-win graphs

Disproved the conjecture that maximum capture time cannot be 

Abstract

The game of cops and robbers is played on a fixed (finite or infinite) graph $G$. The cop chooses his starting position, then the robber chooses his. After that, they take turns and move to adjacent vertices, or stay at their current vertex, with the cop moving first. The game finishes if the cop lands on the robber's vertex. In that case we say that the cop wins, while if the robber is never caught then we say that the robber wins. The graph $G$ is called cop-win if the cop has a winning strategy. In this paper we construct an infinite cop-win graph in which, for any two given starting positions of the cop and the robber, we can name in advance a finite time in which the cop can capture the robber, but these finite times are not bounded above. This shows that this graph has maximum capture time (CR-ordinal) $\omega$, disproving a conjecture of Bonato, Gordinowicz and Hahn that no such graph should exist.