Frame matroids and lifted-graphic matroids are two interesting generalizations of graphic matroids. Here we introduce a new generalization, quasi-graphic matroids, that uni fies 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 rame matroid.
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Networks and Optimization

Geelen, J., Gerards, B., & Whittle, G. (2015). Quasi-graphic matroids.