# Optimizing the trade-off between number of cops and capture time in Cops and Robbers

## TL;DR

This paper investigates the cop throttling number in the game of Cops and Robbers, providing bounds, constructing specific graph families, and introducing a new parameter to optimize resource use during capture.

## Contribution

It establishes new bounds on the cop throttling number, constructs graphs with specific throttling properties, and introduces the product cop throttling number for resource optimization.

## Key findings

- Constructed graphs with $th_c(G)= oldsymbol{ ext{Omega}(n^{2/3})}$
- Established a sublinear upper bound on $th_c(G)$
- Characterized $th_c^{ imes}(G)$ for chordal graphs as $1+rad(G)$

## Abstract

The cop throttling number $th_c(G)$ of a graph $G$ for the game of Cops and Robbers is the minimum of $k + capt_k(G)$, where $k$ is the number of cops and $capt_k(G)$ is the minimum number of rounds needed for $k$ cops to capture the robber on $G$ over all possible games in which both players play optimally. In this paper, we construct a family of graphs having $th_c(G)= \Omega(n^{2/3})$, establish a sublinear upper bound on the cop throttling number, and show that the cop throttling number of chordal graphs is $O(\sqrt{n})$. We also introduce the product cop throttling number $th_c^{\times}(G)$ as a parameter that minimizes the person-hours used by the cops. This parameter extends the notion of speed-up that has been studied in the context of parallel processing and network decontamination. We establish bounds on the product cop throttling number in terms of the cop throttling number, characterize graphs with low product cop throttling number, and show that for a chordal graph $G$, $th_c^{\times}=1+rad(G)$.