A probabilistic version of the game of Zombies and Survivors on graphs

TL;DR

This paper introduces a probabilistic version of the Zombies and Survivors game on graphs, analyzing the minimum number of zombies needed to win with probability over half across various graph families.

Contribution

It defines the zombie number in a probabilistic setting and provides asymptotic results for different graph classes, extending the classical game analysis.

Findings

Zombie number for cycles, hypercubes, and grids determined.

Asymptotic behavior of zombie numbers established for several graph families.

Probabilistic approach offers new insights into pursuit-evasion games.

Abstract

We consider a new probabilistic graph searching game played on graphs, inspired by the familiar game of Cops and Robbers. In Zombies and Survivors, a set of zombies attempts to eat a lone survivor loose on a given graph. The zombies randomly choose their initial location, and during the course of the game, move directly toward the survivor. At each round, they move to the neighbouring vertex that minimizes the distance to the survivor; if there is more than one such vertex, then they choose one uniformly at random. The survivor attempts to escape from the zombies by moving to a neighbouring vertex or staying on his current vertex. The zombies win if eventually one of them eats the survivor by landing on their vertex; otherwise, the survivor wins. The zombie number of a graph is the minimum number of zombies needed to play such that the probability that they win is strictly greater than 1/2. We present asymptotic results for the zombie numbers of several graph families, such as cycles, hypercubes, incidence graphs of projective planes, and Cartesian and toroidal grids.