Even flying cops should think ahead

TL;DR

This paper investigates the entanglement game on sparse graphs, revealing that even with low maximum degree, a linear number of cops may be necessary to catch a robber, challenging previous bounds based on minimum degree.

Contribution

It demonstrates that the number of cops needed can be significantly larger than the minimum degree, including for 3-regular graphs, providing new insights into the complexity of the game.

Findings

Minimum degree is not a tight lower bound for cops needed.

Existence of 3-regular graphs requiring a linear number of cops.

Cops and robbers game complexity increases on sparse graphs.

Abstract

We study the entanglement game, which is a version of cops and robbers, on sparse graphs. While the minimum degree of a graph G is a lower bound for the number of cops needed to catch a robber in G, we show that the required number of cops can be much larger, even for graphs with small maximum degree. In particular, we show that there are 3-regular graphs where a linear number of cops are needed.