The essential bound of a polymatroid and its applications to excluded minor problems

Abstract

The singleton and doubleton minors of a polymatroid $\rho$ encode a surprising amount of information about the structural complexity of $\rho$. Given any polymatroid $\rho$, we can subtract from it a maximally-separated polymatroid, resulting in a $k$-polymatroid. We introduce a notion of boundedness for $\rho$ that corresponds to $k$. Our results provide an organized framework for thinking about polymatroid excluded minor problems. In particular, let $\mathcal{C}$ denote the minor-closed class of matroids characterized by excluding the uniform matroid $U_{2,b}$ and its dual $U_{b-2,b}$. We show that the list of excluded minors for the class of $k$-polymatroids whose $k$-natural matroids are in $\mathcal{C}$ is finite. We also investigate the more general case of excluding $U_{a,b}$ and its dual $U_{b-a,b}$.