There are only a finite number of excluded minors for the class of bicircular matroids

TL;DR

This paper proves that the class of bicircular matroids has only finitely many excluded minors, using biased graph representations and quasi-graphic matroids, and establishes bounds on their size based on rank.

Contribution

It demonstrates the finiteness of excluded minors for bicircular matroids and introduces bounds on their size using advanced matroid representation techniques.

Findings

Finite number of excluded minors for bicircular matroids.

Excluded minors of rank at least ten are quasi-graphic.

An upper bound on the size of excluded minors in terms of rank.

Abstract

We show that the class of bicircular matroids has only a finite number of excluded minors. Key tools used in our proof include representations of matroids by biased graphs and the recently introduced class of quasi-graphic matroids. We show that if $N$ is an excluded minor of rank at least ten, then $N$ is quasi-graphic. Several small excluded minors are quasi-graphic. Using biased-graphic representations, we find that $N$ already contains one of these. We also provide an upper bound, in terms of rank, on the number of elements in an excluded minor, so the result follows.