On the Computational Complexities of Various Geography Variants

TL;DR

This paper investigates the computational complexity of various rule variants of the Generalized Geography game, proving that certain variants like Undirected Partizan Geography remain PSPACE-complete even under specific restrictions.

Contribution

It introduces multiple new variants of Generalized Geography and establishes their computational complexities, including PSPACE-completeness results for Undirected Partizan Geography on bipartite graphs.

Findings

Undirected Partizan Geography is PSPACE-complete on bipartite graphs.

Multiple rule variants of Generalized Geography are analyzed for complexity.

Complexity results extend understanding of combinatorial game computational hardness.

Abstract

Generalized Geography is a combinatorial game played on a directed graph. Players take turns moving a token from vertex to vertex, deleting a vertex after moving the token away from it. A player unable to move loses. It is well known that the computational complexity of determining which player should win from a given position of Generalized Geography is PSPACE-complete. We introduce several rule variants to Generalized Geography, and we explore the computational complexity of determining the winner of positions of many resulting games. Among our results is a proof that determining the winner of a game known in the literature as Undirected Partizan Geography is PSPACE-complete, even when restricted to being played on a bipartite graph.