{"paper":{"title":"$N$-detachable pairs in 3-connected matroids III: the theorem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Alan Williams, Geoff Whittle, Nick Brettell","submitted_at":"2018-04-18T07:39:40Z","abstract_excerpt":"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 $M \\backslash x_1 \\backslash x_2$ is both 3-connected and has an $N$-minor. This is the third and final paper in a series where we prove that if $|E(M)|-|E(N)| \\ge 10$, 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)"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1804.06588","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/1804.06588/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}