Deletion–contraction to form a polymatroid

Let M be a matroid with rank function r , and let e ∈ E ( M ) . The deletion–contraction polymatroid with rank function f = r M ∖ e + r M / e will be denoted P e ( M ) . Notice that P e ( M ) is uniquely determined by M and e . Similarly, a deletion–contraction polymatroid determines M , unless e is a loop or co-loop. This paper will characterize all polymatroids of this deletion–contraction form by giving the set of excluded minors. Vertigan conjectured that the class of G F ( q ) -representable deletion–contraction polymatroids is well-quasi-ordered. From this attractive conjecture, both Rota’s Conjecture and the WQO Conjecture for G F ( q ) -representable matroids would follow.