Quasi-graphic matroids

TL;DR

This paper introduces quasi-graphic matroids, a new class that unifies frame and lifted-graphic matroids, and shows that 3-connected representable quasi-graphic matroids are either of these two types.

Contribution

It defines quasi-graphic matroids, providing a unifying framework, and proves a characterization for 3-connected representable cases.

Findings

Quasi-graphic matroids unify frame and lifted-graphic classes.

Certification of quasi-graphic matroids is straightforward.

3-connected representable quasi-graphic matroids are either lifted-graphic or frame.

Abstract

Frame matroids and lifted-graphic matroids are two interesting generalizations of graphic matroids. Here we introduce a new generalization, {\em quasi-graphic matroids}, that unifies these two existing classes. Unlike frame matroids and lifted-graphic matroids, it is easy to certify that a matroid is quasi-graphic. The main result of the paper is that every $3$-connected representable quasi-graphic matroid is either a lifted-graphic matroid or a frame matroid.