A wheels-and-whirls theorem for 3-connected 2-polymatroids

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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.

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SIAM Journal on Discrete Mathematics

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