Checking the admissibility of odd-vertex pairings is hard

TL;DR

This paper proves that determining the admissibility of odd-vertex pairings and certain orientation properties in graphs is computationally hard, specifically co-NP-complete and NP-complete, resolving open questions in graph theory.

Contribution

It establishes the computational complexity of checking admissibility of odd-vertex pairings and local arc-connectivity orientations, answering longstanding open problems.

Findings

Deciding admissibility of odd-vertex pairings is co-NP-complete.

Deciding the existence of orientations satisfying local arc-connectivity is NP-complete.

Resolves a question posed by Frank regarding these complexities.

Abstract

Nash-Williams proved that every graph has a well-balanced orientation. A key ingredient in his proof is admissible odd-vertex pairings. We show that for two slightly different definitions of admissible odd-vertex pairings, deciding whether a given odd-vertex pairing is admissible is co-NP-complete. This resolves a question of Frank. We also show that deciding whether a given graph has an orientation that satisfies arbitrary local arc-connectivity requirements is NP-complete.