The excluded minors for the class of matroids that are graphic or bicircular lift

TL;DR

This paper characterizes the finite set of minimal matroids that are not graphic or bicircular lift, establishing a key structural property of these matroid classes.

Contribution

It proves that the class of matroids that are either graphic or bicircular lift has a finite list of excluded minors, a significant structural insight.

Findings

Finite list of excluded minors for the class

Characterization of bicircular lift matroids

Connection between graphic and bicircular lift matroids

Abstract

Bicircular lift matroids are a class of matroids defined on the edge set of a graph. For a given graph $G$, the circuits of its bicircular lift matroid $L(G)$ are the edge sets of those subgraphs of $G$ that contain at least two cycles, and are minimal with respect to this property. For each cycle $C$ of $G$, since $L(G)/C$ is graphic and most graphic matroids are not bicircular lift, the class of bicircular lift matroids is not minor-closed. In this paper, we prove that the class of matroids that are graphic or bicircular lift has a finite list of excluded minors.