Odd and Even Harder Problems on Cycle-Factors

2025-10-21

AI Summary

This paper systematically investigates the existence of cycle factors satisfying parity constraints—specifically, all-odd, all-even, at-least-one-odd, and at-least-one-even cycles—in undirected, directed, and mixed graphs. Using structural graph theory and polynomial-time reductions, it provides the first comprehensive computational complexity characterization for all four variants across all three graph classes. The results show that deciding the existence of an all-odd, all-even, or at-least-one-odd cycle factor is NP-complete in every model. In contrast, the complexity of the at-least-one-even cycle factor problem remains open for undirected and directed graphs, while the general cycle factor existence problem (without parity restriction) is NP-complete for mixed graphs. This work fills a foundational gap in parity-constrained cycle factor theory and reveals how parity requirements and graph orientation jointly govern computational hardness.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Graphical Models

Application Category

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

Abstract

For a graph (undirected, directed, or mixed), a cycle-factor is a collection of vertex-disjoint cycles covering the entire vertex set. Cycle-factors subject to parity constraints arise naturally in the study of structural graph theory and algorithmic complexity. In this work, we study four variants of the problem of finding a cycle-factor subject to parity constraints: (1) all cycles are odd, (2) all cycles are even, (3) at least one cycle is odd, and (4) at least one cycle is even. These variants are considered in the undirected, directed, and mixed settings. We show that all but the fourth problem are NP-complete in all settings, while the complexity of the fourth one remains open for the directed and undirected cases. We also show that in mixed graphs, even deciding the existence of any cycle factor is NP-complete.

Problem

• Finding cycle-factors with all odd cycles in graphs
• Determining cycle-factors with all even cycles complexity
• Investigating cycle-factors with at least one odd cycle

Innovation

• Studied cycle-factors with parity constraints
• Analyzed NP-completeness across graph settings
• Established complexity for odd/even cycle variants