Sprague-Grundy Function of Matroids and Related Hypergraphs

TL;DR

This paper extends the classical game of NIM to hypergraph NIM, providing explicit formulas for the Sprague-Grundy function for certain hypergraph classes, including all 2-uniform hypergraphs and matroids, especially self-dual ones.

Contribution

It introduces explicit formulas for the Sprague-Grundy function in hypergraph NIM, characterizing all applicable 2-uniform hypergraphs and matroids, notably including self-dual matroids.

Findings

Explicit formulas for hypergraph NIM's Sprague-Grundy function.

Characterization of 2-uniform hypergraphs and matroids where formulas apply.

Inclusion of all self-dual matroids in the characterized class.

Abstract

We consider a generalization of the classical game of $NIM$ called hypergraph $NIM$. Given a hypergraph $\cH$ on the ground set $V = \{1, \ldots, n\}$ of $n$ piles of stones, two players alternate in choosing a hyperedge $H \in \cH$ and strictly decreasing all piles $i\in H$. The player who makes the last move is the winner. In this paper we give an explicit formula that describes the Sprague-Grundy function of hypergraph $NIM$ for several classes of hypergraphs. In particular we characterize all $2$-uniform hypergraphs (that is graphs) and all matroids for which the formula works. We show that all self-dual matroids are included in this class.