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 GF(q)-representable deletion-contraction polymatroids is well-quasi-ordered. From this attractive conjecture, both Rota's Conjecture and the WQO Conjecture for GF(q)-representable matroids would follow.