Min-max theorem for the game of Cops and Robber on geodesic spaces

TL;DR

This paper extends the Cops and Robber pursuit-evasion game to geodesic spaces, establishing a min-max theorem by approximating the game with finite discrete versions, thus bridging discrete and continuous settings.

Contribution

It introduces a new formulation of the Cops and Robber game on geodesic spaces and proves a min-max theorem through finite game approximations.

Findings

Game can be approximated by finite discrete games

Min-max theorem established for geodesic spaces

Framework bridges discrete graphs and continuous spaces

Abstract

The game of Cops and Robber is traditionally played on a finite graph. The purpose of this note is to introduce and analyze the game that is played on an arbitrary geodesic space. The game is defined in such a way that it preserves the beauty and power of discrete games played on graphs and also keeps the specialties of the pursuit-evasion games played on polyhedral complexes. It is shown that the game can be approximated by finite games of discrete type and as a consequence a min-max theorem is obtained.