Playing Snake on a Graph

TL;DR

This paper extends the classic Snake game to arbitrary graphs, analyzing the complexity of winning strategies, characterizing certain graph classes, and establishing NP-hardness results for the problem.

Contribution

It introduces the concept of snake-winnability on graphs, proves NP-hardness of determining winnability, and characterizes specific classes of graphs where Snake can always win.

Findings

Determining snake-winnability is NP-hard, even on grid graphs.

Hamiltonian graphs are always snake-winnable.

Non-Hamiltonian snake-winnable graphs have girth at most 6.

Abstract

Snake is a classic computer game, which has been around for decades. Based on this game, we study the game of Snake on arbitrary undirected graphs. A snake forms a simple path that has to move to an apple while avoiding colliding with itself. When the snake reaches the apple, it grows longer, and a new apple appears. A graph on which the snake has a strategy to keep eating apples until it covers all the vertices of the graph is called snake-winnable. We prove that determining whether a graph is snake-winnable is NP-hard, even when restricted to grid graphs. We fully characterize snake-winnable graphs for odd-sized bipartite graphs and graphs with vertex-connectivity 1. While Hamiltonian graphs are always snake-winnable, we show that non-Hamiltonian snake-winnable graphs have a girth of at most 6 and that this bound is tight.