10.26686/wgtn.11904942.v1 N Brettell N Brettell G Whittle G Whittle A Williams A Williams N-detachable pairs in 3-connected matroids I: Unveiling X Open Access Te Herenga Waka-Victoria University of Wellington 2020 math.CO 05B35 Computation Theory & Mathematics Pure Mathematics Pure Mathematics not elsewhere classified Combinatorics and Discrete Mathematics (excl. Physical Combinatorics) 2020-02-26 20:49:02 Journal contribution https://openaccess.wgtn.ac.nz/articles/journal_contribution/N-detachable_pairs_in_3-connected_matroids_I_Unveiling_X/11904942 © 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.