Playing Snake on a Graph
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.
Cite
@article{arxiv.2506.21281,
title = {Playing Snake on a Graph},
author = {Denise Graafsma and Bodo Manthey and Alexander Skopalik},
journal= {arXiv preprint arXiv:2506.21281},
year = {2025}
}