  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.
