Pursuit-Evasion in Graphs: Zombies, Lazy Zombies and a Survivor

Abstract: We study zombies and survivor, a variant of the game of cops and robber on graphs. In this variant, the single survivor plays the role of the robber and attempts to escape from the zombies that play the role of the cops. The zombies are restricted, on their turn, to always follow an edge of a shortest path towards the survivor. Let $z(G)$ be the smallest number of zombies required to catch the survivor on a graph $G$ with $n$ vertices. We show that there exist outerplanar graphs and visibility graphs of simple polygons such that $z(G) = \Theta(n)$. We also show that there exist maximum-degree-$3$ outerplanar graphs such that $z(G) = \Omega\left(n/\log(n)\right)$. Let $z_L(G)$ be the smallest number of lazy zombies (zombies that can stay still on their turn) required to catch the survivor on a graph $G$. We establish that lazy zombies are more powerful than normal zombies but less powerful than cops. We prove that $z_L(G) = 2$ for connected outerplanar graphs. We show that $z_L(G)\leq k$ for connected graphs with treedepth $k$. This result implies that $z_L(G)$ is at most $(k+1)\log n$ for connected graphs with treewidth $k$, $O(\sqrt{n})$ for connected planar graphs, $O(\sqrt{gn})$ for connected graphs with genus $g$ and $O(h\sqrt{hn})$ for connected graphs with any excluded $h$-vertex minor. Our results on lazy zombies still hold when an adversary chooses the initial positions of the zombies.