# On excluded minors for classes of graphical matroids

## TL;DR

This paper proves that for the classes of frame, lifted-graphic, and quasi-graphic matroids, there are only finitely many excluded minors of any fixed rank, advancing the understanding of their structural properties.

## Contribution

It establishes the finiteness of excluded minors of fixed rank for three important classes of matroids that generalize graphic matroids.

## Key findings

- Finite number of excluded minors of fixed rank for each class
- Extension of known results to quasi-graphic matroids
- Improved understanding of minor-closed classes of matroids

## Abstract

Frame matroids and lifted-graphic matroids are two distinct minor-closed classes of matroids, each of which generalises the class of graphic matroids. The class of quasi-graphic matroids, recently introduced by Geelen, Gerards, and Whittle, simultaneously generalises both the classes of frame and lifted-graphic matroids. Let $\mathcal{M}$ be one of these three classes, and let $r$ be a positive integer. We show that $\mathcal{M}$ has only a finite number of excluded minors of rank $r$.