A chain theorem for matroids

Tutteʹs Wheels-and-Whirls Theorem proves that if M is a 3-connected matroid other than a wheel or a whirl, then M has a 3-connected minor N such that | E ( M ) | − | E ( N ) | = 1 . Geelen and Whittle extended this theorem by showing that when M is sequentially 4-connected, the minor N can also be guaranteed to be sequentially 4-connected, that is, for every 3-separation ( X , Y ) of N, the set E ( N ) can be obtained from X or Y by successively applying the operations of closure and coclosure. Hall proved a chain theorem for a different class of 4-connected matroids, those for which every 3-separation has at most five elements on one side. This paper proves a chain theorem for those sequentially 4-connected matroids that also obey this size condition.