The $9$-connected Excluded Minors for the Class of Quasi-graphic Matroids

TL;DR

This paper characterizes the 9-connected excluded minors for quasi-graphic matroids, showing that only two specific matroids, U_{3,7} and U_{4,7}, serve as such minors, thereby advancing understanding of this matroid class.

Contribution

It identifies the only 9-connected excluded minors for quasi-graphic matroids, providing a complete classification within this connectivity level.

Findings

U_{3,7} and U_{4,7} are the only 9-connected excluded minors.

The class of quasi-graphic matroids is minor closed.

This result narrows the structure of excluded minors in this class.

Abstract

The class of quasi-graphic matroids, recently introduced by Geelen, Gerards, and Whittle, is minor closed and contains both the class of lifted-graphic matroids and the class of frame matroids, each of which generalises the class of graphic matroids. In this paper, we prove that the matroids $U_{3,7}$ and $U_{4,7}$ are the only $9$-connected excluded minors for the class of quasi-graphic matroids.