BIASED GRAPHS WHOSE MATROIDS ARE SPECIAL BINARY MATROIDS

A biased graph is a graph together with a class of polygons such that no theta subgraph contains exactly two members of the class. To a biased graph Ω are naturally associated three edge matroids:G(Ω), L(Ω), L0(Ω). We determine all biased graphs for which any of these matroids is isomorphic to the Fano plane, the polygon matroid of K4, K5 or K3,3, any of their duals, Bixby's regular matroid R10, or the polygon matroid of Km for m > 5. In each case the bias is derived from edge signs. We conclude by finding the biased graphs Ω for which L0(Ω) is not a graphic [or, regular matroid but every proper contraction is. © 1990 Springer-Verlag.