The damage number of the Cartesian product of graphs

TL;DR

This paper investigates the damage number in the Cartesian product of graphs, extending the concept from a variation of Cops and Robber, providing bounds and specific cases like trees and cycles.

Contribution

It introduces bounds for the damage number of Cartesian product graphs and analyzes specific cases such as trees and cycles, advancing understanding of this graph parameter.

Findings

Provided a general upper bound for the damage number of Cartesian product graphs.

Analyzed the damage number for products of two trees or cycles.

Explored graphs with small damage numbers.

Abstract

We consider a variation of Cops and Robber, introduced in [D. Cox and A. Sanaei, The damage number of a graph, [Aust. J. of Comb. 75(1) (2019) 1-16] where vertices visited by a robber are considered damaged and a single cop aims to minimize the number of distinct vertices damaged by a robber. Motivated by the interesting relationships that often emerge between input graphs and their Cartesian product, we study the damage number of the Cartesian product of graphs. We provide a general upper bound and consider the damage number of the product of two trees or cycles. We also consider graphs with small damage number.