On the cop number of generalized Petersen graphs

We prove that the cop number of every generalized Petersen graph is at most 4. The strategy is to play a modified game of cops and robbers on an infinite cyclic covering space where the objective is to capture the robber or prevent the robber from visiting any vertex infinitely often. We also prove that finite isometric subtrees of any graph are 1-guardable, and we apply this to determine the exact cop number of some families of generalized Petersen graphs. Additionally, we extend these ideas to prove that the cop number of any connected I-graph is at most 5.