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“警察与强盗”游戏的一种变体，该变体由[D. Cox 和 A. Sanaei 在《图的损伤数》，[Aust. J. of Comb. 75(1) (2019) 1-16]中引入，其中强盗访问过的顶点被视为受损，而一名警察的目标是尽量减少强盗造成的不同受损顶点数量。受输入图与其笛卡尔积之间常出现的有趣关系的启发，我们研究了图笛卡尔积的损伤数。我们给出了一个一般上界，并考虑了两个树或环的乘积的损伤数。我们还研究了具有小损伤数的图。