Brettel Whittle Williams 2018 N-detachable pairs in 3-connected matroids I - Unveiling X.pdf (584.46 kB)
N-detachable pairs in 3-connected matroids I: Unveiling X
journal contribution
posted on 2020-02-26, 20:49 authored by N Brettell, G Whittle, A Williams© 2019 Elsevier Inc. Let M be a 3-connected matroid, and let N be a 3-connected minor of M. We say that a pair {x1,x2}⊆E(M) is N-detachable if one of the matroids M/x1/x2 or M\x1\x2 is both 3-connected and has an N-minor. This is the first in a series of three papers where we describe the structures that arise when M has no N-detachable pairs. In this paper, we prove that if no N-detachable pair can be found in M, then either M has a 3-separating set, which we call X, with certain strong structural properties, or M has one of three particular 3-separators that can appear in a matroid with no N-detachable pairs.