The Splitter Theorem

Abstract

This chapter states and proves Seymour's Splitter Theorem. This result, which is a generalization of Tutte's Wheels-and-Whirls Theorem (8.8.4), is a very powerful general tool for deriving matroid structure results. The Wheels-and-Whirls Theorem determines when we can find some element in a 3-connected matroid M to delete or contract in order to preserve 3-connectedness. The Splitter Theorem considers when such an element removal is possible that will not only preserve 3-connectedness but will also maintain the presence of an isomorphic copy of some specified minor of M. The chapter illustrates the power of the Splitter Theorem by noting a variety of consequences of it. It also discusses some extensions and generalizations of the theorem.