Matroids and graphs with few non-essential edges

Finally, Theorem 1.2 is quite straightforward to deduce from Corollary 5.4. Alternatively , it is not diicult to prove this theorem directly using Theorem 2.1. We conclude by presenting the rst of these proofs. Proof of Theorem 1.2. From the list given in Corollary 5.4 of binary 3{connected matroids whose set of non-essential elements has rank two, we eliminate the ma-troids under (i) and (iii) as having more than two non-essential elements. If m 3, then an m{dimensional wheel has K 3;3 as a minor and so the cocycle matroid of this graph is not graphic.) have k as a minor and so have 3 as a minor. But 3 has the dual of the Fano matroid as a restriction and so is not graphic. The only matroids that remain in the list in Corollary 5.4 are the cycle matroids of twisted wheels and m{dimensional wheels for m 3, so the theorem is proved.