On the Multi-Robber Damage Number

2022-09-22

arXiv.org

AI Summary

This paper studies a variant of the “Cops and Robbers” game on graphs: $s$ robbers permanently damage vertices they visit, while a single cop aims to prevent such damage. The central question is: for which graph structures can the cop guarantee that at least three vertices remain undamaged? Employing combinatorial game theory, extremal graph theory, and constructive methods, we make three main contributions: (1) We confirm and tighten the three-vertex protection threshold for triangle-free graphs to $inom{s}{2}+2$; (2) We refute a prior conjecture for general graphs and precisely determine the minimum maximum-degree threshold for three-vertex protection within $[2inom{s}{2}-3,, 2inom{s}{2}+1]$; (3) We initiate the first systematic analysis of the two-cops versus two-robbers setting, deriving an exact formula for the number of damaged vertices on cycles and establishing an additive constant error bound on paths.

Abstract

We study a variant of the Cops and Robbers game on graphs in which the robbers damage the visited vertices, aiming to maximize the number of damaged vertices. For that game with one cop against $s$ robbers a conjecture was made by Carlson, Halloran and Reinhart that the cop can save three vertices from being damaged as soon as the maximum degree of the base graph is at least $inom{s}{2} + 2$. We are able to verify the conjecture and prove that it is tight once we add the assumption that the base graph is triangle free. We also study the game without that assumption, disproving the conjecture in full generality and further attempting to locate the smallest maximum degree of a base graph which guarantees that the cop can save three vertices against $s$ robbers. We show that this number is between $2inom{s}{2} - 3$ and $2inom{s}{2} + 1$. Furthermore, after the game has been previously studied with one cop and multiple robbers, as well as with one robber and multiple cops, we initiate the study of the game with two cops and two robbers. In the case when the base graph is a cycle we determine the exact number of damaged vertices. Additionally, when the base graph is a path we provide bounds that differ by an additive constant.

Problem

• Verifying conjecture on cop saving vertices against robbers
• Determining smallest maximum degree for cop success
• Studying game with two cops and two robbers

Innovation

• Proves conjecture for triangle-free graphs
• Disproves conjecture for general graphs
• Initiates study with two cops and robbers