A chain theorem for matroids
Tutte's 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. © 2007 Elsevier Inc. All rights reserved.
Publication Source (Journal or Book title)
Journal of Combinatorial Theory. Series B
Oxley, J., Semple, C., & Whittle, G. (2008). A chain theorem for matroids. Journal of Combinatorial Theory. Series B, 98 (3), 447-483. https://doi.org/10.1016/j.jctb.2007.08.005