On the Zombie Number of Various Graph Classes

Abstract

The game of zombies and survivor on a graph is a pursuit-evasion game that is a variant of the game of cops and robber. The game proceeds in rounds where first the zombies move then the survivor moves. Zombies must move to an adjacent vertex that is on a shortest path to the survivor’s location. The survivor can move to any vertex in the closed neighborhood of its current location. The zombie number of a graph G is the smallest number of zombies required to catch the survivor on G. The graph G is said to be k-zombie-win if its zombie number is $k\ge 1$.

We first examine bounds on the zombie number of the Cartesian and strong products of various graphs. We also introduce graph classes which we call capped products and provide some bounds on zombie number of these as well. Fitzpatrick et al. (Discrete Applied Math., 2016) provided a sufficient condition for a graph to be 1-zombie-win. Using a capped product, we show that their condition is not necessary. We also provide bounds for two variants called lazy zombies and tepid zombies on some graph products. A lazy zombie is a zombie that does not need to move from its current location on its turn. A tepid zombie is a lazy zombie that can move to a vertex whose distance to the survivor does not increase. Finally, we design an algorithm (polynomial in n for constant k) that can decide, given a graph G, whether or not it is k-zombie-win for all the above variants of zombies.

Supported by the Natural Sciences and Engineering Research Council of Canada (NSERC).