{"paper":{"title":"The Complement of the Djokovic-Winkler Relation","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["cs.DM"],"primary_cat":"math.CO","authors_text":"Bruno J. Schmidt, Guillaume E. Scholz, Marc Hellmuth, Sandhya Thekkumpadan Puthiyaveedu","submitted_at":"2023-11-30T06:44:11Z","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 "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2311.18284","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2311.18284/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"}