{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2023:7CJUST5LYIT2UVOKQS6XHMIT5V","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":"7c0aa9ad5a43f2beec801d6b478f350df5e3fd6098c2cf9496a4d5ae7da3e3a0","cross_cats_sorted":["cs.DM"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2023-11-30T06:44:11Z","title_canon_sha256":"6d2dc9c863e56922bd7b11583eaceb0ee92c5dfee5d9c6838fa0eaf9b4fdb254"},"schema_version":"1.0","source":{"id":"2311.18284","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2311.18284","created_at":"2026-07-05T07:18:40Z"},{"alias_kind":"arxiv_version","alias_value":"2311.18284v1","created_at":"2026-07-05T07:18:40Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2311.18284","created_at":"2026-07-05T07:18:40Z"},{"alias_kind":"pith_short_12","alias_value":"7CJUST5LYIT2","created_at":"2026-07-05T07:18:40Z"},{"alias_kind":"pith_short_16","alias_value":"7CJUST5LYIT2UVOK","created_at":"2026-07-05T07:18:40Z"},{"alias_kind":"pith_short_8","alias_value":"7CJUST5L","created_at":"2026-07-05T07:18:40Z"}],"graph_snapshots":[{"event_id":"sha256:0a552ffdc24a08aaedf70a1264242057e5e84e2e00fa512a14b227ae9dec33da","target":"graph","created_at":"2026-07-05T07:18:40Z","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/2311.18284/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"The Djokovi\\'{c}-Winkler relation $\\Theta$ is a binary relation defined on the edge set of a given graph that is based on the distances of certain vertices and which plays a prominent role in graph theory. In this paper, we explore the relatively uncharted ``reflexive complement'' $\\overline\\Theta$ of $\\Theta$, where $(e,f)\\in \\overline\\Theta$ if and only if $e=f$ or $(e,f)\\notin \\Theta$ for edges $e$ and $f$. We establish the relationship between $\\overline\\Theta$ and the set $\\Delta_{ef}$, comprising the distances between the vertices of $e$ and $f$ and shed some light on the intricacies of ","authors_text":"Bruno J. Schmidt, Guillaume E. Scholz, Marc Hellmuth, Sandhya Thekkumpadan Puthiyaveedu","cross_cats":["cs.DM"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2023-11-30T06:44:11Z","title":"The Complement of the Djokovic-Winkler Relation"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2311.18284","kind":"arxiv","version":1},"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:d28f67d7478f17abd3b7e2db44828ee03908a53cc0b19335fa1a55a02ab14e3b","target":"record","created_at":"2026-07-05T07:18:40Z","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":"7c0aa9ad5a43f2beec801d6b478f350df5e3fd6098c2cf9496a4d5ae7da3e3a0","cross_cats_sorted":["cs.DM"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2023-11-30T06:44:11Z","title_canon_sha256":"6d2dc9c863e56922bd7b11583eaceb0ee92c5dfee5d9c6838fa0eaf9b4fdb254"},"schema_version":"1.0","source":{"id":"2311.18284","kind":"arxiv","version":1}},"canonical_sha256":"f893494fabc227aa55ca84bd73b113ed798f5bd78c6c57f2b234685e42c71f0f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f893494fabc227aa55ca84bd73b113ed798f5bd78c6c57f2b234685e42c71f0f","first_computed_at":"2026-07-05T07:18:40.090141Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T07:18:40.090141Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"t2tlQW+tnE3Ne2VzfyzyVwz667Wt3VRobUT9QvrA3buhCam76QhYnZukjM3j7HWJHdWqE+x2aq8i7vavLAhiBg==","signature_status":"signed_v1","signed_at":"2026-07-05T07:18:40.090619Z","signed_message":"canonical_sha256_bytes"},"source_id":"2311.18284","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:d28f67d7478f17abd3b7e2db44828ee03908a53cc0b19335fa1a55a02ab14e3b","sha256:0a552ffdc24a08aaedf70a1264242057e5e84e2e00fa512a14b227ae9dec33da"],"state_sha256":"54378a7739287d0b6807cda6d67d2fa8dbaf509c14dadec5b979e614f3ca5438"}