The damage throttling number of a graph

TL;DR

This paper introduces the damage throttling number, a new graph parameter balancing the number of cops and damage caused by a robber, extending previous concepts and analyzing its properties and bounds.

Contribution

It formalizes the damage throttling number, compares it with cop throttling, and explores bounds and examples, advancing understanding of pursuit-evasion game parameters.

Findings

Damage throttling is bounded above by the cop throttling number minus one.

Infinite families of graphs demonstrate tightness and non-tightness of bounds.

Existence of graphs with damage throttling number growing as Omega(n^{2/3}).

Abstract

The cop throttling number of a graph, introduced in 2018 by Breen et al., optimizes the balance between the number of cops used and the number of rounds required to catch the robber in a game of Cops and Robbers. In 2019, Cox and Sanaei studied a variant of Cops and Robbers in which the robber tries to occupy (or damage) as many vertices as possible and the cop tries to minimize this damage. In their paper, they study the minimum number of vertices damaged by the robber over all games played on a given graph $G$, called the damage number of $G$. We introduce the natural parameter called the damage throttling number of a graph, denoted $\operatorname{th}_d(G)$, which optimizes the balance between the number of cops used and the number of vertices damaged in the graph. To this end, we formalize the definition of $k$-damage number, which extends the damage number to games played with $k$ cops. We show that damage throttling and cop throttling share many properties, yet they exhibit interesting differences. We prove that the damage throttling number is tightly bounded above by one less than the cop throttling number. Infinite families of examples and non-examples of tightness in this bound are given. We also find an infinite family of connected graphs $G$ of order $n$ for which $\operatorname{th}_d(G) = \Omega(n^{2/3})$.