# A splitter theorem for connected clutters

## TL;DR

This paper generalizes the concept of connectivity from matroids to clutters and proves a splitter theorem for connected clutters, extending known results in matroid theory.

## Contribution

It introduces a connectivity notion for clutters and establishes a splitter theorem that generalizes the matroid case.

## Key findings

- Established a connectivity concept for clutters
- Proved a splitter theorem for connected clutters
- Extended matroid splitter theorem to clutters

## Abstract

A clutter consists of a finite set and a collection of pairwise incomparable subsets. Clutters are natural generalisations of matroids, and they have similar operations of deletion and contraction. We introduce a notion of connectivity for clutters that generalises that of connectivity for matroids. We prove a splitter theorem for connected clutters that has the splitter theorem for connected matroids as a special case: if $M$ and $N$ are connected clutters, and $N$ is a proper minor of $M$, then there is an element in $E(M)$ that can be deleted or contracted to produce a connected clutter with $N$ as a minor.