The 2-Cops-Move Cop Number of Graphs on the Torus, Klein Bottle, and Projective Plane

We study the $2$-cops-move variant of the game of cops and robber, in which at most two cops may move in each turn. Let $c_2(G)$ denote the corresponding cop number. We prove that every graph embeddable on the torus, Klein bottle, or projective plane satisfies $c_2(G)\le 3$, extending a recent result of Gonz'alez Hermosillo de la Maza and Mohar for planar graphs.