Odd and Even Harder Problems on Cycle-Factors

TL;DR

This work studies parity-constrained cycle-factors across undirected, directed, and mixed graphs, focusing on four variants: all cycles odd, all cycles even, at least one odd, and at least one even. The authors establish NP-completeness for three variants in the directed setting, while the complexity of the Exists Even variant remains open for both directed and undirected graphs; they also show NP-completeness for Cycle-Factors in mixed graphs even without parity constraints. Key methodological contributions include reductions from classical NP-hard problems (e.g., Hamiltonian path, 3-edge-coloring, 2VDP, 3DM) and the introduction of intermediate problems PRCF and SMCF to transfer hardness to the mixed-graph setting. A parity-preserving reduction from directed to undirected graphs extends the NP-hardness results, painting a comprehensive hardness landscape and highlighting open questions for the Exists Even case and potential tractable subclasses. Overall, the paper clarifies the complexity of parity-constrained cycle-factors and suggests algebraic techniques as a promising avenue for resolving the remaining open cases.

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.

Figures (7)

• Figure 1: The gadget for an arc $a=uv \in A(H)$ in the reduction of Theorem \ref{['thm:all-odd-2-factor_NPhard']}.
• Figure 2: The gadget for an edge $uv \in A(H)$ in the reduction of Theorem \ref{['thm:all-even-2-factor_NPhard']}.
• Figure 3: Illustration of the reduction of Theorem \ref{['thm:exists-odd-2-factor_NPhard']}.
• Figure 4: The gadget for a vertex $v \in V(H)$ in the reduction from Even Dicycle.
• Figure 5: Illustration for Lemma \ref{['lem:connection_digraph_undirected']}.

Theorems & Definitions (28)

• Theorem 3.1
• proof
• Claim 3.2
• proof
• Claim 3.3
• proof
• Theorem 3.4
• proof
• Theorem 3.5
• proof