A PSPACE-complete Graph Nim

TL;DR

This paper introduces Neighboring Nim, a graph-based variation of Nim, and proves its PSPACE-hardness through reductions from Geography, demonstrating the complexity of certain combinatorial game variants.

Contribution

It constructs PSPACE-complete versions of Nim using graph structures and reductions from Geography, extending the understanding of game complexity.

Findings

Neighboring Nim is PSPACE-hard.

Reductions from Geography establish complexity.

Graph-based Nim variants can be PSPACE-complete.

Abstract

We build off the game, NimG to create a version named Neighboring Nim. By reducing from Geography, we show that this game is PSPACE-hard. The games created by the reduction share strong similarities with Undirected (Vertex) Geography and regular Nim, both of which are in P. We show how to construct PSPACE-complete versions with nim heaps *1 and *2. This application of graphs can be used as a form of game sum with any games, not only Nim.