NP-Hardness of Non-Crossing Hamiltonian Path and Cycle in Non-Planar Graphs

2026-08-06

AI Summary

This study investigates the computational complexity of finding non-crossing Hamiltonian paths and cycles in non-planar graphs, aiming to determine whether the NP-hardness stems from the non-crossing constraint itself rather than the planarity of the graph. By directly constructing non-planar graph instances from 3-SAT and establishing a polynomial-time reduction, the work provides the first proof of NP-completeness for this problem in non-planar graphs without relying on traditional reductions through planar graphs. This approach cleanly disentangles the individual contributions of non-crossing constraints and graph planarity to computational hardness, demonstrating that the non-crossing requirement alone is sufficient to induce NP-hardness. The result establishes a simpler and more extensible theoretical framework for analyzing related complexity questions.

Abstract

We seek to disentangle the hardness of finding a Hamiltonian path or cycle from the hardness of finding a non-crossing path or cycle by giving a direct reduction from 3-SAT to the non-crossing Hamiltonian path and cycle problems on non-planar graphs. Prior hardness proofs proceed by reduction to planar graphs, where every path is automatically non-crossing; this conflates the two sources of difficulty and leaves unclear why forbidding crossings on the path alone makes the problem hard. Our reduction places the difficulty squarely in the non-crossing constraint, avoids planar gadget constructions, and yields a more transparent proof that may be easier to extend to related problems.

Problem

• NP-Hardness
• Non-Crossing
• Hamiltonian Path
• Hamiltonian Cycle
• Non-Planar Graphs

Innovation

• non-crossing Hamiltonian path
• NP-hardness
• direct reduction
• non-planar graphs
• 3-SAT