Throttling for the game of Cops and Robbers on graphs

TL;DR

This paper studies the cop-throttling number in the game of Cops and Robbers on graphs, providing bounds, characterizations, and analyzing various graph families, with all bounds being on the order of the square root of the number of vertices.

Contribution

It introduces tools for bounding the cop-throttling number, relates it to PSD throttling, characterizes graphs with low cop-throttling, and analyzes multiple graph families with bounds of O(√n).

Findings

PSD throttling bounds cop-throttling from above.

Graphs with low cop-throttling are characterized.

All studied graph families have cop-throttling bounds of O(√n).

Abstract

We consider the cop-throttling number of a graph $G$ for the game of Cops and Robbers, which is defined to be the minimum of $(k + \text{capt}_k(G))$, where $k$ is the number of cops and $\text{capt}_k(G)$ is the minimum number of rounds needed for $k$ cops to capture the robber on $G$ over all possible games. We provide some tools for bounding the cop-throttling number, including showing that the positive semidefinite (PSD) throttling number, a variant of zero forcing throttling, is an upper bound for the cop-throttling number. We also characterize graphs having low cop-throttling number and investigate how large the cop-throttling number can be for a given graph. We consider trees, unicyclic graphs, incidence graphs of finite projective planes (a Meyniel extremal family of graphs), a family of cop-win graphs with maximum capture time, grids, and hypercubes. All the upper bounds on the cop-throttling number we obtain for families of graphs are $ O(\sqrt n)$.