Seymour’s Splitter 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. A series compression in such a structure is an analogue of contracting an edge of a graph that is in a series pair. A 2-polymatroid N is an s-minor of a 2-polymatroid M if N can be obtained from M by a sequence of contractions, series compressions, and dual-contractions, where the last are modified deletions. The main result proves that if M and N are 3-connected 2-polymatroids such that N is an s-minor of M, then M has a 3-connected s-minor M′ that has an s-minor isomorphic to N and has |E(M)| − 1 elements unless M is a whirl or the cycle matroid of a wheel. In the exceptional case, such an M′ can be found with |E(M)| − 2 elements.
Publication Source (Journal or Book title)
Electronic Journal of Combinatorics
Oxley, J., Semple, C., & Whittle, G. (2019). A splitter theorem for 3-connected 2-polymatroids. Electronic Journal of Combinatorics, 26 (2), 1-95. https://doi.org/10.37236/7308