Abstract
Let M be a 3-connected matroid, and let N be a 3-connected minor of M. 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 second in a series of three papers where we describe the structures that arise when it is not possible to find an N-detachable pair in M. In the first paper in the series, we showed that, under mild assumptions, either M has an N-detachable pair, M has one of three particular 3-separators that can appear in a matroid with no N-detachable pairs, or there is a 3-separating set X with certain strong structural properties. In this paper, we analyse matroids with such a structured set X, and prove that they have either an N-detachable pair, or one of five particular 3-separators that can appear in a matroid with no N-detachable pairs.
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