Misère Partizan Arc Kayles is PSPACE-complete, even on Planar Graphs

2025-11-26

AI Summary

This paper investigates the computational complexity of Misère Partizan Arc Kayles on planar graphs, establishing its PSPACE-completeness for the first time. The proof proceeds via polynomial-time reductions from novel variants of Bounded 2-Player Constraint Logic (Bounded 2CL): specifically, three newly introduced PSPACE-complete variants—namely, OR-AND, AND-OR, and OR-AND-OR Bounded 2CL—each featuring distinct edge-labeling and activation constraints. To realize these reductions on planar graphs, the authors design geometrically embeddable gadgets compatible with square and triangular grids, preserving both planarity and game-theoretic equivalence. The methodology integrates graph-theoretic modeling with combinatorial game analysis. Key contributions are: (1) the first proof of PSPACE-completeness for Misère Partizan Arc Kayles on planar graphs; (2) the extension of the Bounded 2CL framework through three new PSPACE-complete variants; and (3) explicit grid-embeddable gadget constructions that strengthen complexity characterizations of combinatorial games under topological restrictions.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Multiagent Systems: Multiagent Planning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

We show that Misère Partizan Arc Kayles is PSPACE-complete on planar graphs via a reduction from Bounded Two-Player Constraint Logic. Furthermore, we show how to embed our gadgets onto the square and triangular grids. In order to clearly explain these results, we get into the details of Bounded Two-Player Constraint Logic and find three PSPACE-complete variants of that as well.

Problem

• Proves PSPACE-completeness of Misère Partizan Arc Kayles
• Demonstrates complexity on planar and grid graphs
• Introduces new PSPACE-complete Constraint Logic variants

Innovation

• Reduction from Bounded Two-Player Constraint Logic
• Embed gadgets onto square and triangular grids
• Identify three PSPACE-complete variants of logic