There are Only a Finite Number of Excluded Minors for the Class of Bicircular Matroids

We show that the class of bicircular matroids has only a finite number of excluded minors. Key tools used in our proof include representations of matroids by biased graphs and the recently introduced class of quasi-graphic matroids. We show that if N is an excluded minor of rank at least ten, then N is quasi-graphic. Several small excluded minors are quasi-graphic. Using biased-graphic representations, we find that N already contains one of these. We also provide an upper bound, in terms of rank, on the number of elements in an excluded minor, so the result follows.