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.

翻译：我们考虑一种“警察与强盗”游戏的变体，该变体由[D. Cox 和 A. Sanaei, The damage number of a graph, Aust. J. of Comb. 75(1) (2019) 1-16]引入，其中被强盗访问过的顶点被视为受损，而单个警察的目标是尽量减少被强盗损坏的不同顶点的数量。受输入图与其笛卡尔积之间经常出现的有趣关系的启发，我们研究了图的笛卡尔积的损伤数。我们给出了一个通用的上界，并考虑了两棵树或两个圈的乘积的损伤数。我们还研究了具有较小损伤数的图。