The highly connected even-cycle and even-cut matroids
  The classes of even-cycle matroids, even-cycle matroids with a blocking pair, and even-cut matroids each have hundreds of excluded minors. We show that the number of excluded minors for these classes can be drastically reduced if we consider in each class only the highly connected matroids of sufficient size. 
Kevin Grace and Stefan H. M. van Zwam
math.CO