{"paper":{"title":"Excluded minors are almost fragile II: essential elements","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Charles Semple, Geoff Whittle, James Oxley, Nick Brettell","submitted_at":"2022-06-27T03:58:48Z","abstract_excerpt":"Let $M$ be an excluded minor for the class of $\\mathbb{P}$-representable matroids for some partial field $\\mathbb{P}$, let $N$ be a $3$-connected strong $\\mathbb{P}$-stabilizer that is non-binary, and suppose $M$ has a pair of elements $\\{a,b\\}$ such that $M\\backslash a,b$ is $3$-connected with an $N$-minor. Suppose also that $|E(M)| \\geq |E(N)|+11$ and $M \\backslash a,b$ is not $N$-fragile. In the prequel to this paper, we proved that $M \\backslash a,b$ is at most five elements away from an $N$-fragile minor. An element $e$ in a matroid $M'$ is $N$-essential if neither $M'/e$ nor $M' \\backsla"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2206.13036","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/2206.13036/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"}