Cops and Robbers: A $\times$-homotopy Invariant Variant

TL;DR

This paper introduces a new variant of the Cops and Robbers game on graphs, establishing its invariance under $ imes$-homotopy and analyzing cop numbers across various graph families and products.

Contribution

It defines the Sneaky-Active Cops and Robbers variant, proves its equivalence to classical game on reflexive graphs, and explores cop number invariance under $ imes$-homotopy.

Findings

Cop number is invariant under $ imes$-homotopy for reflexive graphs.

Computed cop numbers for multiple graph families.

Analyzed behavior of categorical and box products of graphs.

Abstract

Cops and Robbers is a pursuit-evasion game played on graphs, of which many variants have been developed and studied. We introduce a variant of this game, "Sneaky-Active Cops and Robbers", where all cops and robber must move on their turn, and where the robber is allowed to move onto a cop position without being captured. We show that for reflexive graphs, this game is equivalent to the classical cops and robbers and that the cop number for a graph is invariant under $\times$-homotopy equivalence. We then develop further properties of this game, computing cop numbers for a number of graph families and developing results about the behavior of categorical and box products of graphs.