Take-Away Impartial Combinatorial Games on Hypergraphs and Other Related Geometric and Discrete Structures

TL;DR

This paper develops a new winning strategy for Take-Away Games played on a broader class of hypergraphs that are neither uniformly odd nor even, expanding understanding of combinatorial game strategies on complex structures.

Contribution

It introduces a novel winning strategy for Take-Away Games on hypergraphs that are neither uniformly odd nor even, under specific structural conditions.

Findings

New winning strategy for non-uniform hypergraphs

Extends game theory to broader hypergraph classes

Identifies structural conditions for strategy applicability

Abstract

In a Take-Away Game on hypergraphs, two players take turns to remove the vertices and the hyperedges of the hypergraphs. In each turn, a player must remove either a single vertex or a hyperedge. When a player chooses to remove one vertex, all of the hyperedges that contain the chosen vertex are also removed. When a player chooses to remove one hyperedge, only that chosen hyperedge is removed. Whoever removes the last vertex wins the game. Following from the winning strategy for the Take-Away Impartial Combinatorial Games on only Oddly Uniform or only Evenly Uniform Hypergraphs, this paper is about the new winning strategy for Take-Away Games on neither Oddly nor Evenly Uniform Hypergraphs. These neither Oddly nor Evenly Uniform Hypergraphs, however, have to satisfy the specific given requirements.