Abstract. The natural matroid of an integer polymatroid was introduced to show that a simple construction of integer polymatroids from matroids yields all integer polymatroids. As we illustrate, the natural matroid can shed much more light on integer polymatroids. We focus on characterizations of integer polymatroids using their bases, their circuits, and their cyclic flats along with the rank of each cyclic flat and each element; we offer some new characterizations and insights into known characterizations.

A sufficiently large connected matroid $M$ contains a big circuit or a big cocircuit. Wu showed that we can ensure that $M$ has a big circuit or a big cocircuit containing any chosen element of $M$. In this paper, we prove that, for a fixed connected matroid $N$, if $M$ is a sufficiently large connected matroid having $N$ as a minor, then, up to duality, either $M$ has a big connected minor in which $N$ is a spanning restriction and the deletion of $E(N)$ is a large connected uniform matroid, or $M$ has, as a minor, the $2$-sum of a big circuit and a connected single-element extension or coextension of $N$. In addition, we find a set of unavoidable minors for the class of graphs that have a cycle and a bond with a big intersection.