Describing Quasi-Graphic Matroids

This is a revised version of our original paper (arXiv:1808.00489v2) incorporating the corrections published in a corrigendum (arXiv:1808.00489v3). Our main theorem as originally stated was missing the required assumption that matroids should be connected. Those unfamiliar with the original paper will find in this version a complete, correct description of quasi-graphic matroids, sparing them the inconvenience of having to read both the original paper and a separate corrigendum. We also present here some new results that do not appear in our original paper nor its corrigendum. These appear in Section 6. Of particular interest to readers familiar with the original paper and its corrigendum may be the following result. Given a matroid and a graph, of the four axioms for quasi-graphic matroids, three may be checked in time polynomial in the size of the ground set, but the fourth axiom in general cannot. It is desirable to have such a check that could be carried out in polynomial time. We provide such an alternative (Theorem 6.20).