# Quasi‐graphic matroids *

@article{Geelen2018QuasigraphicM, title={Quasi‐graphic matroids *}, author={James F. Geelen and Bert Gerards and Geoff Whittle}, journal={Journal of Graph Theory}, year={2018}, volume={87}, pages={253 - 264} }

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

## 12 Citations

### The 9-Connected Excluded Minors for the Class of Quasi-graphic Matroids

- MathematicsSIAM J. Discret. Math.
- 2022

The class of quasi-graphic matroids, recently introduced by Geelen, Gerards, and Whittle, is minor closed and contains both the class of lifted-graphic matroids and the class of frame matroids, each…

### On Recognizing Frame and Lifted‐Graphic Matroids

- MathematicsJ. Graph Theory
- 2018

We prove that there is no polynomial p(·) with the property that a matroid M can be determined to be either a lifted‐graphic or frame matroid using at most p(|M|) rank evaluations. This resolves two…

### Infinitely many excluded minors for frame matroids and for lifted-graphic matroids

- MathematicsJ. Comb. Theory, Ser. B
- 2018

### Defining Bicircular Matroids in Monadic Logic

- Mathematics
- 2020

We conjecture that the class of frame matroids can be characterised by a sentence in the monadic second-order logic of matroids, and we prove that there is such a characterisation for the class of…

### Obstructions for Bounded Branch-depth in Matroids

- Mathematics
- 2020

DeVos, Kwon, and Oum introduced the concept of branch-depth of matroids as a natural analogue of tree-depth of graphs. They conjectured that a matroid of sufficiently large branch-depth contains the…

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

- Mathematics
- 2021

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…

### Projective planarity of matroids of 3-nets and biased graphs

- MathematicsAustralas. J Comb.
- 2020

Criteria for embeddability of biased-graphic matroids in Desarguesian projective spaces is established, that is, embeddable in an arbitrary projective plane that is not necessarily Desargue'sian.

### A Whitney type theorem for surfaces: characterising graphs with locally planar embeddings

- Mathematics
- 2020

We prove that a graph G embeds r-locally planarly in a pseudo-surface if and only if a certain matroid associated to the graph G is co-graphic. This extends Whitney's abstract planar duality theorem…

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We prove that there is no polynomial p(·) with the property that a matroid M can be determined to be either a lifted‐graphic or frame matroid using at most p(|M|) rank evaluations. This resolves two…

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