\(N\)-detachable pairs in 3-connected matroids. III: The theorem. (English) Zbl 1481.05023
Summary: Let \(M\) be a 3-connected matroid, and let \(N\) be a 3-connected minor of \(M\). A pair \(\{ x_1, x_2 \} \subseteq E(M)\) is \(N\)-detachable if one of the matroids \(M / x_1 / x_2\) or \(M / x_1 / x_2\) is 3-connected and has an \(N\)-minor. This is the third and final paper in a series where we prove that if \(| E(M) | - | E(N) | \geq 10\), then either \(M\) has an \(N\)-detachable pair after possibly performing a single \(\Delta-Y\) or \(Y-\Delta\) exchange, or \(M\) is essentially \(N\) with a spike attached. Moreover, we describe the additional structures that arise if we require only that \(| E(M) | - | E(N) | \geq 5\).
For Part I and II see [the authors, ibid. 141, 295–342 (2020; Zbl 1430.05011); ibid. 149, 222–271 (2021; Zbl 1466.05025)].
For Part I and II see [the authors, ibid. 141, 295–342 (2020; Zbl 1430.05011); ibid. 149, 222–271 (2021; Zbl 1466.05025)].
MSC:
| 05B35 | Combinatorial aspects of matroids and geometric lattices |
| 05C40 | Connectivity |
| 52B40 | Matroids in convex geometry (realizations in the context of convex polytopes, convexity in combinatorial structures, etc.) |
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