Odd and Even Harder Problems on Cycle-Factors

TL;DR

This paper investigates the computational complexity of finding cycle-factors with parity constraints in various types of graphs, revealing NP-completeness in most cases and highlighting open problems.

Contribution

It classifies the complexity of four parity-constrained cycle-factor problems across different graph settings, identifying NP-completeness and open questions.

Findings

All but one problem are NP-complete in all settings.

Deciding the existence of any cycle factor in mixed graphs is NP-complete.

Complexity remains open for some parity-constrained problems in undirected and directed graphs.

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.