Visibility Graphs, Dismantlability, and the Cops and Robbers Game

TL;DR

This paper proves that in pursuit-evasion games on visibility graphs of polygons and certain planar regions, the cop can always win, highlighting the significance of dismantlability in these graph classes.

Contribution

It establishes that visibility graphs are dismantlable, ensuring the cop's victory in pursuit-evasion games within polygons and planar regions, extending known results to infinite graphs.

Findings

Cop always wins on visibility graphs of polygons.

Visibility graphs are shown to be dismantlable.

Results extend to polygons with curved sides and planar regions.

Abstract

We study versions of cop and robber pursuit-evasion games on the visibility graphs of polygons, and inside polygons with straight and curved sides. Each player has full information about the other player's location, players take turns, and the robber is captured when the cop arrives at the same point as the robber. In visibility graphs we show the cop can always win because visibility graphs are dismantlable, which is interesting as one of the few results relating visibility graphs to other known graph classes. We extend this to show that the cop wins games in which players move along straight line segments inside any polygon and, more generally, inside any simply connected planar region with a reasonable boundary. Essentially, our problem is a type of pursuit-evasion using the link metric rather than the Euclidean metric, and our result provides an interesting class of infinite cop-win graphs.