A splitter theorem for connected 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.