Combinatorial Analysis of a Subtraction Game on Graphs

TL;DR

This paper introduces and analyzes a graph-based subtraction game, exploring strategies, automorphisms, computational methods, and random graph behavior, with a specific focus on paths and connections to open problems in combinatorial game theory.

Contribution

It provides a comprehensive analysis of a new graph subtraction game, including strategies, automorphism properties, computational approaches, and insights into random graph play, linking to open questions in the field.

Findings

Strategies for specific graph families identified

Automorphism group properties analyzed

Connection to open problems in Octal games established

Abstract

We define a two-player combinatorial game in which players take alternate turns; each turn consists on deleting a vertex of a graph, together with all the edges containing such vertex. If any vertex became isolated by a player's move then it would also be deleted. A player wins the game when the other player has no moves available. We study this game under various viewpoints: by finding specific strategies for certain families of graphs, through using properties of a graph's automorphism group, by writing a program to look at Sprague-Grundy numbers, and by studying the game when played on random graphs. When analyzing Grim played on paths, using the Sprague-Grundy function, we find a connection to a standing open question about Octal games.