Comparing the power of cops to zombies in pursuit-evasion games

Abstract: We compare two kinds of pursuit-evasion games played on graphs. In Cops and Robbers, the cops can move strategically to adjacent vertices as they please, while in a new variant, called deterministic Zombies and Survivors, the zombies (the counterpart of the cops) are required to always move towards the survivor (the counterpart of the robber). The cop number of a graph is the minimum number of cops required to catch the robber on that graph; the zombie number of a graph is the minimum number of zombies required to catch the survivor on that graph. We answer two questions from the 2016 paper of Fitzpatrick, Howell, Messinger, and Pike. We show that for any $m \ge k \ge 1$, there is a graph with zombie number $m$ and cop number $k$. We also show that the zombie number of the $n$-dimensional hypercube is $\lceil 2n/3\rceil$.