{"paper":{"title":"Problems, proofs, and disproofs on the inversion number","license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","headline":"","cross_cats":["cs.DM"],"primary_cat":"math.CO","authors_text":"Cl\\'ement Rambaud, Felix Klingelhoefer, Florian H\\\"orsch, Fr\\'ed\\'eric Havet, Guillaume Aubian, Nicolas Nisse, Quentin Vermande","submitted_at":"2022-12-18T23:02:24Z","abstract_excerpt":"The {\\it inversion} of a set $X$ of vertices in a digraph $D$ consists in reversing the direction of all arcs of $D\\langle X\\rangle$. The {\\it inversion number} of an oriented graph $D$, denoted by ${\\rm inv}(D)$, is the minimum number of inversions needed to transform $D$ into an acyclic oriented graph. In this paper, we study a number of problems involving the inversion number of oriented graphs. Firstly, we give bounds on ${\\rm inv}(n)$, the maximum of the inversion numbers of the oriented graphs of order $n$. We show $n - \\mathcal{O}(\\sqrt{n\\log n}) \\ \\leq \\ {\\rm inv}(n) \\ \\leq \\ n - \\lcei"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2212.09188","kind":"arxiv","version":3},"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/2212.09188/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"}