The family of bicircular matroids closed under duality

TL;DR

This paper characterizes 3-connected bicircular and cobicircular matroids, showing that, except for some small exceptional cases, they are mainly free swirls and their minors, revealing structural insights into this class.

Contribution

It provides a complete characterization of 3-connected members of the intersection of bicircular and cobicircular matroids, identifying the main classes involved.

Findings

Most 3-connected bicircular and cobicircular matroids are free swirls or their minors.

Exceptional cases with small rank and corank are explicitly described.

The class is closed under duality, with specific structural properties identified.

Abstract

We characterize the 3-connected members of the intersection of the class of bicircular and cobicircular matroids. Aside from some exceptional matroids with rank and corank at most 5, this class consists of just the free swirls and their minors.