Cops and an Insightful Robber

TL;DR

This paper studies a variant of the Cops and Robbers game where the robot has observational advantages, analyzing how this affects the number of cops needed and providing strategies for specific graph classes.

Contribution

It introduces a new 'Cheating Robot' variant, establishing bounds and strategies for capturing a robot with observational capabilities on various graphs.

Findings

More cops are needed to catch a robot than a robber.

On trees, one cop can guarantee capture of a robot.

Exact cop numbers are determined for hypercubes and k-dimensional grids.

Abstract

The 'Cheating Robot' version of Cops and Robbers is played on a finite, simple, connected graph. The players move in the same time period. However, before moving, the robot observes to which vertices the cops are moving and it is fast enough to complete its move in the time period. The cops also know that the robot will use this information. More cops are required to capture a robot than to capture a robber. Indeed, the minimum degree is a lower bound on the number of cops required to capture a robot. Only on a tree is one cop guaranteed to capture a robot, although two cops are sufficient to capture both a robber and a robot on outerplanar graphs. In graphs where retracts are involved, we show how cop strategies against a robber can be modified to capture a robot. This approach gives exact numbers for hypercubes, and $k$-dimensional grids in general.