Recognizing graphic matroids

There is no polynomially bounded algorithm to test if a matroid (presented by an “independence oracle”) is binary. However, there is one to test graphicness. Finding this extends work of previous authors, who have given algorithms to test binary matroids for graphicness. Our main tool is a new result that ifM′ is the polygon matroid of a graphG, andM is a different matroid onE(G) with the same rank, then there is a vertex ofG whose star is not a cocircuit ofM.