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

TL;DR

This paper compares traditional cops and robbers pursuit games with a new zombie variant on graphs, analyzing their effectiveness and providing bounds for specific graph classes like hypercubes.

Contribution

It introduces the zombie pursuit game, compares it to the cop game, and establishes bounds for zombie and cop numbers on various graphs.

Findings

For any m ≥ k ≥ 1, there exists a graph with zombie number m and cop number k.

The zombie number of the n-dimensional hypercube is approximately two-thirds of n.

The paper answers two open questions from prior research.

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 $\lceil 2n/3\rceil$.