{"paper":{"title":"Paths and Intersections: Minimum Realization of Okamura-Seymour Instances","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["cs.CG","math.CO"],"primary_cat":"cs.DS","authors_text":"Pavlo Pylyavskyy, Yu Chen, Zihan Tan","submitted_at":"2026-07-03T02:29:36Z","abstract_excerpt":"We study the inverse problem for shortest-path metrics of Okamura-Seymour (OS) instances. Given an OS metric $D$ on a cyclically ordered terminal set $T$, the goal is to find minimum realizations of $D$, where minimum means having the fewest edges among all disk-embedded realizations with the prescribed terminal order. We show that $D$ determines a canonical medial graph template and every minimum realization is the primal graph of an arrangement of this template. Consequently, the underlying embedded graphs of minimum realizations of $D$ can be recovered, and for each such graph one can effic"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.02883","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/2607.02883/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"}