Representations of bicircular lift matroids

TL;DR

This paper characterizes when two graphs produce identical bicircular lift matroids, showing they are mostly equivalent if the graphs are 2-isomorphic, thus advancing understanding of graph-matroid relationships.

Contribution

It provides a characterization of graph equivalence in bicircular lift matroids, answering a question by Irene Pivotto and linking graph isomorphism to matroid theory.

Findings

Two graphs have the same bicircular lift matroid if and only if they are 2-isomorphic, except for some small graphs.

The result clarifies the relationship between graph structure and matroid representation.

This characterization helps in understanding the structure of bicircular lift matroids and their graph origins.

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 are the edge sets of those subgraphs of $G$ that contain at least two cycles, and are minimal with respect to this property. The main result of this paper is a characterization of when two graphs give rise to the same bicircular lift matoid, which answers a question proposed by Irene Pivotto. In particular, aside from some appropriately defined "small" graphs, two graphs have the same bicircular lift matroid if and only if they are $2$-isomorphic in the sense of Whitney.