On recognising frame and lifted-graphic matroids

TL;DR

This paper proves that there is no polynomial-time algorithm to determine whether a given matroid is a lifted-graphic or frame matroid, resolving two conjectures in the field.

Contribution

It establishes the computational hardness of recognizing lifted-graphic and frame matroids, settling two open conjectures.

Findings

No polynomial p exists for recognition algorithms

Recognition problem is computationally hard

Resolves conjectures by Geelen, Gerards, and Whittle

Abstract

We prove that there is no polynomial $p(\cdot)$ with the property that a matroid $M$ can be determined to be either a lifted-graphic or frame matroid using at most $p(|M|)$ rank evaluations. This resolves two conjectures of Geelen, Gerards and Whittle (Quasi-graphic matroids, arXiv:1512.03005v1).