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The tournament T is self dual if it is isomorphic to its dual. T is {-k}-self dual if for each set X of k vertices, T[V \\ X] is self dual. T is strongly self dual if each of its induced subtournaments is self dual. A subset I of V is an interval of T if for a, b \\in I and for x \\in V \\ I, (a,x) \\in A if and only if ("},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1204.2513","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/3.0/","primary_cat":"math.CO","submitted_at":"2012-04-11T18:26:56Z","cross_cats_sorted":[],"title_canon_sha256":"776f153fa2c930b8d075867951b4b49d55e1fbd30a3ee8b958c4ba9371afb6b0","abstract_canon_sha256":"36bddeb7fcd5debb000e5479b7c12e23e58fe2ab5b41044384e414ec7428dee3"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:58:04.881272Z","signature_b64":"TId11WREIcCrPTwCOOi33B0tkzrVxGdPKFvPlufGEl7JrdUoxVPvjmWB0D/XWmfX6ffo9YxJ6SD514QcAq+cCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"deaa06fef101a30f468611bd99efe9099bf4038860cb792de30e9c183f06bf92","last_reissued_at":"2026-05-18T03:58:04.880863Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:58:04.880863Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The {-3}-reconstruction and the {-3}-self duality of tournaments","license":"http://creativecommons.org/licenses/by/3.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Abderrahim Boussairi, Mouna Achour, Youssef Boudabbous","submitted_at":"2012-04-11T18:26:56Z","abstract_excerpt":"Let T = (V,A) be a (finite) tournament and k be a non negative integer. 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A subset I of V is an interval of T if for a, b \\in I and for x \\in V \\ I, (a,x) \\in A if and only if ("},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1204.2513","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1204.2513","created_at":"2026-05-18T03:58:04.880925+00:00"},{"alias_kind":"arxiv_version","alias_value":"1204.2513v1","created_at":"2026-05-18T03:58:04.880925+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1204.2513","created_at":"2026-05-18T03:58:04.880925+00:00"},{"alias_kind":"pith_short_12","alias_value":"32VAN7XRAGRQ","created_at":"2026-05-18T12:26:50.516681+00:00"},{"alias_kind":"pith_short_16","alias_value":"32VAN7XRAGRQ6RUG","created_at":"2026-05-18T12:26:50.516681+00:00"},{"alias_kind":"pith_short_8","alias_value":"32VAN7XR","created_at":"2026-05-18T12:26:50.516681+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/32VAN7XRAGRQ6RUGCG6ZT37JBG","json":"https://pith.science/pith/32VAN7XRAGRQ6RUGCG6ZT37JBG.json","graph_json":"https://pith.science/api/pith-number/32VAN7XRAGRQ6RUGCG6ZT37JBG/graph.json","events_json":"https://pith.science/api/pith-number/32VAN7XRAGRQ6RUGCG6ZT37JBG/events.json","paper":"https://pith.science/paper/32VAN7XR"},"agent_actions":{"view_html":"https://pith.science/pith/32VAN7XRAGRQ6RUGCG6ZT37JBG","download_json":"https://pith.science/pith/32VAN7XRAGRQ6RUGCG6ZT37JBG.json","view_paper":"https://pith.science/paper/32VAN7XR","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1204.2513&json=true","fetch_graph":"https://pith.science/api/pith-number/32VAN7XRAGRQ6RUGCG6ZT37JBG/graph.json","fetch_events":"https://pith.science/api/pith-number/32VAN7XRAGRQ6RUGCG6ZT37JBG/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/32VAN7XRAGRQ6RUGCG6ZT37JBG/action/timestamp_anchor","attest_storage":"https://pith.science/pith/32VAN7XRAGRQ6RUGCG6ZT37JBG/action/storage_attestation","attest_author":"https://pith.science/pith/32VAN7XRAGRQ6RUGCG6ZT37JBG/action/author_attestation","sign_citation":"https://pith.science/pith/32VAN7XRAGRQ6RUGCG6ZT37JBG/action/citation_signature","submit_replication":"https://pith.science/pith/32VAN7XRAGRQ6RUGCG6ZT37JBG/action/replication_record"}},"created_at":"2026-05-18T03:58:04.880925+00:00","updated_at":"2026-05-18T03:58:04.880925+00:00"}