A note on deterministic zombies

TL;DR

This paper investigates the deterministic zombie game variant, proving an upper bound on the zombie number for Cartesian product graphs and presenting a graph family with low cop number but high zombie number.

Contribution

It establishes that the zombie number of the Cartesian product of two graphs is at most the sum of their zombie numbers and introduces a graph family with a low cop number but arbitrarily large zombie number.

Findings

Zombie number of Cartesian product graphs is at most the sum of individual zombie numbers.

Existence of graphs with cop number 2 and arbitrarily large zombie number.

Abstract

"Zombies and Survivor" is a variant of the well-studied game of "Cops and Robber" where the zombies (cops) can only move closer to the survivor (robber). We consider the deterministic version of the game where a zombie can choose their path if multiple options are available. The zombie number, like the cop number, of a graph is the minimum number of zombies, or cops, required to capture the survivor. In this short note, we solve a question by Fitzpatrick et al., proving that the zombie number of the Cartesian product of two graphs is at most the sum of their zombie numbers. We also give a simple graph family with cop number $2$ and an arbitrarily large zombie number.