{"paper":{"title":"Joins and ear decompositions beyond graphic matroids","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["cs.DM"],"primary_cat":"math.CO","authors_text":"Chaitanya Nalam, Krist\\'of B\\'erczi, Yuhang Bai","submitted_at":"2026-08-02T07:51:24Z","abstract_excerpt":"For a matroid $M$, a join is a set $J\\subseteq E(M)$ that meets every circuit $C$ in at most $|C|/2$ elements. Let $\\mu(M)$ denote the maximum size of a join. Motivated by Frank's min--max theorem for graphic matroids, we compare $\\mu(M)$ with an ear-decomposition parameter $\\eta(M)=(r(M)+\\varphi(M))/2$, where $\\varphi(M)$ is the minimum number of even lobes in an ear decomposition of $M$. Frank's theorem implies $\\mu(M)=\\eta(M)$ for connected graphic matroids.\n  Here we study how far this equality extends beyond graphic matroids. We show that the exact equality does not hold in general: it al"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2608.01059","kind":"arxiv","version":1},"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/2608.01059/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"}