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

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 F is infinite, irrespective of whether F is finite or infinite.