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.
Publication Source (Journal or Book title)
SIAM Journal on Discrete Mathematics
Oxley, J., Semple, C., & Whittle, G. (2016). A wheels-and-whirls theorem for 3-connected 2-polymatroids. SIAM Journal on Discrete Mathematics, 30 (1), 493-524. https://doi.org/10.1137/140996549