The structure of matroids with a spanning clique or projective geometry

TL;DR

This paper provides a structural characterization of certain matroids with spanning clique or projective geometry, showing they are close to well-understood classes under minor exclusions.

Contribution

It offers a new structural description of matroids spanned by frame or projective geometry minors, extending understanding of their proximity to representable or frame matroids.

Findings

Matroids spanned by a complete graph frame are close to frame matroids.

Matroids with a spanning projective geometry are close to GF(q)-representable matroids.

Excluded minors constrain the structure, leading to near-classification.

Abstract

Let $s,n \ge 2$ be integers. We give a qualitative structural description of every matroid $M$ that is spanned by a frame matroid of a complete graph and has no $U_{s,2s}$-minor and no rank-$n$ projective geometry minor, showing that every such matroid is `close' to a frame matroid. We also give a similar description of every matroid $M$ with a spanning projective geometry over a field GF$(q)$ as a restriction and with no $U_{s,2s}$-minor and no PG$(n,q')$-minor for any $q' > q$, showing that such an $M$ is `close' to a GF$(q)$-representable matroid.