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

TL;DR

This paper investigates a pursuit-evasion game variant on graphs involving zombies and a survivor, analyzing the minimum number of zombies needed for capture under various graph classes and the power of lazy zombies.

Contribution

It introduces and analyzes the zombie and lazy zombie pursuit game, establishing bounds on the number of zombies needed and comparing their power to cops across different graph families.

Findings

Existence of graphs requiring linear zombies for capture.

Lazy zombies are more powerful than normal zombies but less than cops.

Bounds on lazy zombies needed for various graph classes.

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.