Variations on Cops and Robbers

TL;DR

This paper explores various extensions of the Cops and Robbers game, providing new bounds on cop numbers and capture times for different variants, including multiple moves per turn and directed graphs.

Contribution

It generalizes known bounds for the classical game to cases where the robber moves multiple edges, and introduces bounds for directed graphs, using expansion techniques.

Findings

Upper bound for robber with R moves: N / α^{(1-o(1))√log_α N}

Maximum cop number for finite R: N^{1 - 1/(R-2)}

Cop number for strongly connected digraphs: O(N(log log N)^2 / log N)

Abstract

We consider several variants of the classical Cops and Robbers game. We treat the version where the robber can move R > 1 edges at a time, establishing a general upper bound of N / \alpha ^{(1-o(1))\sqrt{log_\alpha N}}, where \alpha = 1 + 1/R, thus generalizing the best known upper bound for the classical case R = 1 due to Lu and Peng. We also show that in this case, the cop number of an N-vertex graph can be as large as N^{1 - 1/(R-2)} for finite R, but linear in N if R is infinite. For R = 1, we study the directed graph version of the problem, and show that the cop number of any strongly connected digraph on N vertices is at most O(N(log log N)^2/log N). Our approach is based on expansion.