{"paper":{"title":"On Two Unsolved Problems Concerning Matching Covered Graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Cl\\'audio L. Lucchesi, Marcelo H. de Carvalho, Nishad Kothari, U. S. R. Murty","submitted_at":"2017-05-26T04:22:52Z","abstract_excerpt":"A cut $C:=\\partial(X)$ of a matching covered graph $G$ is a separating cut if both its $C$-contractions $G/X$ and $G/\\overline{X}$ are also matching covered. A brick is solid if it is free of nontrivial separating cuts. In 2004, we (Carvalho, Lucchesi and Murty) showed that the perfect matching polytope of a brick may be described without recourse to odd set constraints if and only if it is solid. In 2006, we proved that the only simple planar solid bricks are the odd wheels. The problem of characterizing nonplanar solid bricks remains unsolved.\n  A bi-subdivision of a graph $J$ is a graph obt"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1705.09428","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":""},"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"}