A splitter theorem for 3-connected 2-polymatroids

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 $M^{\mathrm{\prime }}$ that has an s-minor isomorphic to $N$ and has $\mathrm{\mid }E(M)\mathrm{\mid }-1$ elements unless $M$ is a whirl or the cycle matroid of a wheel. In the exceptional case, such an $M^{\mathrm{\prime }}$ can be found with $\mathrm{\mid }E(M)\mathrm{\mid }-2$ elements.