Pursuit-evasion in graphs: Zombies, lazy zombies and a survivor

We study zombies and survivor, a variant of the game of cops and robber on graphs where the single survivor plays the role of the robber and attempts to escape from the zombies that play the role of the cops. The difference is that zombies must follow an edge of a shortest path towards the survivor on their turn. Let $z\left(G\right)$ 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 $z\left(G\right)=\Theta \left(n\right)$. We also show that there exist maximum-degree-3 outerplanar graphs such that $z\left(G\right)=\Omega (n/\log (n\left)\right)$.

A zombie that can remain at its current vertex on its turn is called lazy. Let $z_L\left(G\right)$ be the smallest number of lazy zombies required to catch the survivor. The ability to remain at its current vertex on its turn makes lazy zombies more powerful than normal zombies but less powerful than cops. We prove that $z_L\left(G\right)\le 2$ for connected outerplanar graphs which is tight in the worst case. We also show that in this case, the survivor is caught after $O\left(n\right)$ rounds. We then show that $z_L\left(G\right)\le k$ for connected graphs with treedepth k and that $O\left(n^{2k}\right)$ rounds are sufficient to catch the survivor. The bound on treedepth implies that $z_L\left(G\right)$ is at most $(k+1)\log n$ for connected graphs with treewidth k, $O\left(\sqrt{n}\right)$ for connected planar graphs, $O\left(\sqrt{gn}\right)$ for connected graphs with genus g and $O\left(h\sqrt{hn}\right)$ 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.