{"paper":{"title":"The Outerplanar Tur\\'{a}n Number of Double Stars","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"The outerplanar Turán number for double stars S_{p,q} is determined exactly except for the case p=2 and q=3.","cross_cats":[],"primary_cat":"math.CO","authors_text":"Changqing Xu, Chaofan Zhang, Yongxin Lan","submitted_at":"2026-05-17T08:44:20Z","abstract_excerpt":"Let $H$ be a nonempty graph. A graph is $H$-free if it does not contain any copy of $H$ as a subgraph. The outerplanar Tur\\'{a}n number of $H$, denoted by $ex_{_\\mathcal{OP}}(n,H)$, is the maximum number of edges among all $H$-free outerplanar graphs on $n$ vertices. A double star $S_{p,q}$ is the graph obtained from an edge by joining its two endpoints with $p$ and $q$ isolated vertices respectively, where $q \\ge p\\ge 1$. In this paper, we determine the exact values of $ex_{_\\mathcal{OP}}(n,S_{p,q})$ for all $q\\ge p\\ge 2$, with the sole exception of $p=2$ and $q=3$; for the latter, we establi"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We determine the exact values of ex_OP(n,S_{p,q}) for all q≥p≥2, with the sole exception of p=2 and q=3; for the latter, we establish a lower bound.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The determination rests on the premise that the extremal outerplanar graphs for these forbidden subgraphs admit a uniform structural description (e.g., a cycle plus pendant trees of bounded size) that can be verified by case analysis on p and q; this structural claim is not stated in the abstract but is required for the exact formulas to hold.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"The paper computes exact outerplanar Turán numbers ex_OP(n, S_p,q) for q ≥ p ≥ 2 except p=2 q=3, where only a lower bound is shown.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"The outerplanar Turán number for double stars S_{p,q} is determined exactly except for the case p=2 and q=3.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"474fa213beb4f01da4869512ac8623cc9afa27583c41f817effd18e4a0a249f3"},"source":{"id":"2605.17330","kind":"arxiv","version":1},"verdict":{"id":"89f5ffc7-a754-46b2-ba9b-419989b93688","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-19T23:09:22.166832Z","strongest_claim":"We determine the exact values of ex_OP(n,S_{p,q}) for all q≥p≥2, with the sole exception of p=2 and q=3; for the latter, we establish a lower bound.","one_line_summary":"The paper computes exact outerplanar Turán numbers ex_OP(n, S_p,q) for q ≥ p ≥ 2 except p=2 q=3, where only a lower bound is shown.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The determination rests on the premise that the extremal outerplanar graphs for these forbidden subgraphs admit a uniform structural description (e.g., a cycle plus pendant trees of bounded size) that can be verified by case analysis on p and q; this structural claim is not stated in the abstract but is required for the exact formulas to hold.","pith_extraction_headline":"The outerplanar Turán number for double stars S_{p,q} is determined exactly except for the case p=2 and q=3."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.17330/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"doi_title_agreement","ran_at":"2026-05-19T23:31:20.129826Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T23:21:25.343984Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"claim_evidence","ran_at":"2026-05-19T21:41:57.810215Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"ai_meta_artifact","ran_at":"2026-05-19T21:33:23.743902Z","status":"skipped","version":"1.0.0","findings_count":0}],"snapshot_sha256":"d745a889fb4e594d9bebf8955ca80dab87bd28d99e425fd6155eb8f7ec45484c"},"references":{"count":20,"sample":[{"doi":"","year":2008,"title":"Graph Theory","work_id":"4753d058-8ba5-4775-8c2f-c20b29acb15d","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1984,"title":"On some problems in graph theory, combinatorial analysis and combinatorial num- ber theory","work_id":"017486a0-6916-484c-b20d-dd0e66292973","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2012,"title":"Tur\\'an densities of hypercubes","work_id":"ae0e4b6f-6070-4a89-bac4-b5dc2be56f88","ref_index":3,"cited_arxiv_id":"1201.3587","is_internal_anchor":true},{"doi":"","year":1992,"title":"Subgraphs of a hypercube containing no small even cycles","work_id":"77ae397f-3ed3-4334-bf6d-3b7ff0fe1e9e","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1993,"title":"Hexagon-free subgraphs of hypercubes","work_id":"0062ae83-8534-4401-8695-1b03a608d514","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":20,"snapshot_sha256":"0b708c06b7e738d7396c09749eca1906bf99d47f9d3fe86d6edf41aa5687b4a2","internal_anchors":1},"formal_canon":{"evidence_count":2,"snapshot_sha256":"f307a30af02418876b0b48affdcad03d3930fcf85847cc4063f44f4b3f1efff9"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}