Deletion–contraction to form a polymatroid

Let MM be a matroid with rank function rr, and let e∈E(M)e∈E(M). The deletion–contraction polymatroid with rank function f=rM∖e+rM/ef=rM∖e+rM/e will be denoted Pe(M)Pe(M). Notice that Pe(M)Pe(M) is uniquely determined by MM and ee. Similarly, a deletion–contraction polymatroid determines MM, unless ee 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)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)GF(q)-representable matroids would follow.