# A splitter theorem for 3-connected 2-polymatroids

## TL;DR

This paper extends Seymour's Splitter Theorem to 3-connected 2-polymatroids, providing a structural result that identifies smaller 3-connected s-minors containing a given s-minor, with specific exceptions.

## Contribution

It generalizes the splitter theorem from matroids to 2-polymatroids, including new cases involving series compressions and dual-contractions.

## Key findings

- Identifies conditions for the existence of smaller 3-connected s-minors
- Establishes bounds on element reduction in such minors
- Specifies exceptions for whirl and wheel cycle matroids

## Abstract

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.