The excluded minors for three classes of 2-polymatroids having special types of natural matroids

TL;DR

This paper determines the excluded minors for certain classes of 2-polymatroids with special natural matroids, expanding understanding of minor-closed classes and their properties in matroid theory.

Contribution

It identifies the excluded minors for classes of 2-polymatroids derived from specific minor-closed matroid classes, introducing the concept of $k$-natural matroids and their properties.

Findings

Excluded minors for classes of 2-polymatroids with binary and series-parallel natural matroids.

Finiteness of excluded minors only in certain classes.

Classes of 2-polymatroids are closed under 2-duality.

Abstract

If $\mathcal{C}$ is a minor-closed class of matroids, the class $\mathcal{C}'$ of integer polymatroids whose natural matroids are in $\mathcal{C}$ is also minor closed, as is the class $\mathcal{C}'_k$ of $k$-polymatroids in $\mathcal{C}'$. We find the excluded minors for $\mathcal{C}'_2$ when $\mathcal{C}$ is (i) the class of binary matroids, (ii) the class of matroids with no $M(K_4)$-minor, and, combining those, (iii) the class of matroids whose connected components are cycle matroids of series-parallel networks. In each case the class $\mathcal{C}$ has finitely many excluded minors, but that is true of $\mathcal{C}'_2$ only in case (ii). We also introduce the $k$-natural matroid, a variant of the natural matroid for a $k$-polymatroid, and use it to prove that these classes of 2-polymatroids are closed under 2-duality.