The multi-robber damage number of a graph

TL;DR

This paper introduces and analyzes the s-robber damage number in a multi-robber, single-cop game variant on graphs, providing bounds, exact values for certain graph families, and characterizations of extremal cases.

Contribution

It defines the s-robber damage number for multiple robbers against one cop and derives bounds, exact values, and characterizations for various graph classes.

Findings

Determined the s-robber damage number for several graph families.

Established bounds on the s-robber damage number.

Characterized graphs with extreme 2-robber damage numbers.

Abstract

In many variants of the game of Cops and Robbers on graphs, multiple cops play against a single robber. In 2019, Cox and Sanaei introduced a variant of the game that gives the robber a more active role than simply evading the cop. In their version, the robber tries to damage as many vertices as possible and the cop attempts to minimize this damage. While the damage variant was originally studied with one cop and one robber, it was later extended to play with multiple cops by Carlson et. al in 2021. We take a different approach by studying the damage variant with multiple robbers against one cop. Specifically, we introduce the $s$-robber damage number of a graph and obtain a variety of bounds on this parameter. Applying these bounds, we determine the $s$-robber damage number for a variety of graph families and characterize graphs with extreme $2$-robber damage number.