Cops, robbers, and infinite graphs

TL;DR

This paper introduces the concept of weakly cop-win graphs to extend the classical cops and robbers game results from finite to infinite graphs, establishing key properties and strategies.

Contribution

It defines weakly cop-win graphs, proves that all constructible graphs are weakly cop-win, and characterizes constructibility via protective strategies in infinite graphs.

Findings

Every constructible graph is weakly cop-win.

Weakly cop-win concept does not extend to all infinite graphs.

Constructibility is characterized by protective strategies in locally finite graphs.

Abstract

Cops and robbers is a game between two players, where one tries to catch the other by moving along the edges of a graph. It is well known that on a finite graph the cop has a winning strategy if and only if the graph is constructible and that finiteness is necessary for this result. We propose the notion of weakly cop-win graphs, a winning criterion for infinite graphs which could lead to a generalisation. In fact, we generalise one half of the result, that is, we prove that every constructible graph is weakly cop-win. We also show that a similar notion studied by Chastand et al. (which they also dubbed weakly cop-win) is not sufficient to generalise the above result to infinite graphs. In the locally finite case we characterise the constructible graphs as the graphs for which the cop has a so-called protective strategy and prove that the existence of such a strategy implies constructibility even for non-locally finite graphs.