A Wheels-and-Whirls Theorem for $3$-Connected $2$-Polymatroids

Abstract

Tutte's wheels-and-whirls theorem is a basic inductive tool for dealing with $3$-connected matroids. This paper proves a generalization of that theorem for the class of $2$-polymatroids. Such structures include matroids, and they model both sets of points and lines in a projective space and sets of edges in a graph. The main result proves that, in a $3$-connected $2$-polymatroid that is not a whirl or the cycle matroid of a wheel, one can obtain another $3$-connected $2$-polymatroid by deleting or contracting some element, or by performing a new operation that generalizes series contraction in a graph. Moreover, we show that unless one uses some reduction operation in addition to deletion and contraction, the set of minimal $2$-polymatroids that are not representable over a fixed field ${\mathbb F}$ is infinite, irrespective of whether ${\mathbb F}$ is finite or infinite.