{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2022:DVF7BUG56KGCI7DXDVWUOVVU33","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"8e1e57172c37e6b3fb14605913102c13d03726d31a1fc1a3f89ae80229505325","cross_cats_sorted":["cs.DM"],"license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"math.CO","submitted_at":"2022-12-18T23:02:24Z","title_canon_sha256":"a62335025e146ebe0657fcc30fd3b59b8e91cd71fc287d4aff3235413c65787d"},"schema_version":"1.0","source":{"id":"2212.09188","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2212.09188","created_at":"2026-07-05T05:27:49Z"},{"alias_kind":"arxiv_version","alias_value":"2212.09188v3","created_at":"2026-07-05T05:27:49Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2212.09188","created_at":"2026-07-05T05:27:49Z"},{"alias_kind":"pith_short_12","alias_value":"DVF7BUG56KGC","created_at":"2026-07-05T05:27:49Z"},{"alias_kind":"pith_short_16","alias_value":"DVF7BUG56KGCI7DX","created_at":"2026-07-05T05:27:49Z"},{"alias_kind":"pith_short_8","alias_value":"DVF7BUG5","created_at":"2026-07-05T05:27:49Z"}],"graph_snapshots":[{"event_id":"sha256:ad9834e52ec9cb0b6f641ab2d84912a5c7be58c93140b37309207b6382d9be99","target":"graph","created_at":"2026-07-05T05:27:49Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2212.09188/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"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","authors_text":"Cl\\'ement Rambaud, Felix Klingelhoefer, Florian H\\\"orsch, Fr\\'ed\\'eric Havet, Guillaume Aubian, Nicolas Nisse, Quentin Vermande","cross_cats":["cs.DM"],"headline":"","license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"math.CO","submitted_at":"2022-12-18T23:02:24Z","title":"Problems, proofs, and disproofs on the inversion number"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2212.09188","kind":"arxiv","version":3},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:7d95d4c5431f0dcbb419e97c9268d055c3c7c46b07e31eb4396a832a4f50c328","target":"record","created_at":"2026-07-05T05:27:49Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"8e1e57172c37e6b3fb14605913102c13d03726d31a1fc1a3f89ae80229505325","cross_cats_sorted":["cs.DM"],"license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"math.CO","submitted_at":"2022-12-18T23:02:24Z","title_canon_sha256":"a62335025e146ebe0657fcc30fd3b59b8e91cd71fc287d4aff3235413c65787d"},"schema_version":"1.0","source":{"id":"2212.09188","kind":"arxiv","version":3}},"canonical_sha256":"1d4bf0d0ddf28c247c771d6d4756b4dec52d1a9f14b7cbb830ddd2f5a312f253","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"1d4bf0d0ddf28c247c771d6d4756b4dec52d1a9f14b7cbb830ddd2f5a312f253","first_computed_at":"2026-07-05T05:27:49.945058Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T05:27:49.945058Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"XJtmJ4dBX3inWCNjqj841JyhJyRD/w2KxeO7S6Qk5bPcAFPiZ1aUzVjIF6B79JynjxlsUxWAomx0zQWcT9vsDA==","signature_status":"signed_v1","signed_at":"2026-07-05T05:27:49.947598Z","signed_message":"canonical_sha256_bytes"},"source_id":"2212.09188","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:7d95d4c5431f0dcbb419e97c9268d055c3c7c46b07e31eb4396a832a4f50c328","sha256:ad9834e52ec9cb0b6f641ab2d84912a5c7be58c93140b37309207b6382d9be99"],"state_sha256":"e2bd82157ee9bb4bc0d88b7e4ab8ef0750d57e31cfe32b3f42538f95706c800e"}