Even Circuits in Oriented Matroids

TL;DR

This paper extends the concept of even directed cycles from digraphs to regular matroids, providing characterizations and complexity results for detecting even circuits in this broader context.

Contribution

It introduces non-even oriented matroids, establishes polynomial equivalence between detecting even circuits and recognizing non-even matroids, and characterizes non-even oriented bond matroids via forbidden minors.

Findings

Detection of even directed circuits is polynomially equivalent to recognizing non-even oriented matroids.

Characterization of non-even oriented bond matroids through forbidden minors.

Extension of the even cycle problem to the setting of regular matroids.

Abstract

In this paper we generalise the even directed cycle problem, which asks whether a given digraph contains a directed cycle of even length, to orientations of regular matroids. We define non-even oriented matroids generalising non-even digraphs, which played a central role in resolving the computational complexity of the even dicycle problem. Then we show that the problem of detecting an even directed circuit in a regular matroid is polynomially equivalent to the recognition of non-even oriented matroids. Our main result is a precise characterisation of the class of non-even oriented bond matroids in terms of forbidden minors, which complements an existing characterisation of non-even oriented graphic matroids by Seymour and Thomassen.