  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.
