{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:LAI7FJVDMC3ACT5YD3PGXNBR46","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":"e46554f1ba2e1854cf3965b31635a5f0af562932cdc814fdf0fa25f93f66b2e4","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.CO","submitted_at":"2026-05-17T08:44:20Z","title_canon_sha256":"8fda682d1cf35835b3d96cc13cffb4f4c34a0368f65e97692992941bf4c6753b"},"schema_version":"1.0","source":{"id":"2605.17330","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2605.17330","created_at":"2026-05-20T00:03:52Z"},{"alias_kind":"arxiv_version","alias_value":"2605.17330v1","created_at":"2026-05-20T00:03:52Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.17330","created_at":"2026-05-20T00:03:52Z"},{"alias_kind":"pith_short_12","alias_value":"LAI7FJVDMC3A","created_at":"2026-05-20T00:03:52Z"},{"alias_kind":"pith_short_16","alias_value":"LAI7FJVDMC3ACT5Y","created_at":"2026-05-20T00:03:52Z"},{"alias_kind":"pith_short_8","alias_value":"LAI7FJVD","created_at":"2026-05-20T00:03:52Z"}],"graph_snapshots":[{"event_id":"sha256:01eb1b6a94e6515626e3cc0456ea69b3ffdcaefaa7772d84a70d3d5e54124780","target":"graph","created_at":"2026-05-20T00:03:52Z","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":4,"items":[{"attestation":"unclaimed","claim_id":"C1","kind":"strongest_claim","source":"verdict.strongest_claim","status":"machine_extracted","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."},{"attestation":"unclaimed","claim_id":"C2","kind":"weakest_assumption","source":"verdict.weakest_assumption","status":"machine_extracted","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."},{"attestation":"unclaimed","claim_id":"C3","kind":"one_line_summary","source":"verdict.one_line_summary","status":"machine_extracted","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."},{"attestation":"unclaimed","claim_id":"C4","kind":"headline","source":"verdict.pith_extraction.headline","status":"machine_extracted","text":"The outerplanar Turán number for double stars S_{p,q} is determined exactly except for the case p=2 and q=3."}],"snapshot_sha256":"474fa213beb4f01da4869512ac8623cc9afa27583c41f817effd18e4a0a249f3"},"formal_canon":{"evidence_count":2,"snapshot_sha256":"f307a30af02418876b0b48affdcad03d3930fcf85847cc4063f44f4b3f1efff9"},"integrity":{"available":true,"clean":true,"detectors_run":[{"findings_count":0,"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"}],"endpoint":"/pith/2605.17330/integrity.json","findings":[],"snapshot_sha256":"d745a889fb4e594d9bebf8955ca80dab87bd28d99e425fd6155eb8f7ec45484c","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"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","authors_text":"Changqing Xu, Chaofan Zhang, Yongxin Lan","cross_cats":[],"headline":"The outerplanar Turán number for double stars S_{p,q} is determined exactly except for the case p=2 and q=3.","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.CO","submitted_at":"2026-05-17T08:44:20Z","title":"The Outerplanar Tur\\'{a}n Number of Double Stars"},"references":{"count":20,"internal_anchors":1,"resolved_work":20,"sample":[{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":1,"title":"Graph Theory","work_id":"4753d058-8ba5-4775-8c2f-c20b29acb15d","year":2008},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":2,"title":"On some problems in graph theory, combinatorial analysis and combinatorial num- ber theory","work_id":"017486a0-6916-484c-b20d-dd0e66292973","year":1984},{"cited_arxiv_id":"1201.3587","doi":"","is_internal_anchor":true,"ref_index":3,"title":"Tur\\'an densities of hypercubes","work_id":"ae0e4b6f-6070-4a89-bac4-b5dc2be56f88","year":2012},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":4,"title":"Subgraphs of a hypercube containing no small even cycles","work_id":"77ae397f-3ed3-4334-bf6d-3b7ff0fe1e9e","year":1992},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":5,"title":"Hexagon-free subgraphs of hypercubes","work_id":"0062ae83-8534-4401-8695-1b03a608d514","year":1993}],"snapshot_sha256":"0b708c06b7e738d7396c09749eca1906bf99d47f9d3fe86d6edf41aa5687b4a2"},"source":{"id":"2605.17330","kind":"arxiv","version":1},"verdict":{"created_at":"2026-05-19T23:09:22.166832Z","id":"89f5ffc7-a754-46b2-ba9b-419989b93688","model_set":{"reader":"grok-4.3"},"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","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.","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.","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."}},"verdict_id":"89f5ffc7-a754-46b2-ba9b-419989b93688"}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:85283fb577ae57cf851b2ceed96ec90891d92293769cc0d63ceacc8f4441df32","target":"record","created_at":"2026-05-20T00:03:52Z","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":"e46554f1ba2e1854cf3965b31635a5f0af562932cdc814fdf0fa25f93f66b2e4","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.CO","submitted_at":"2026-05-17T08:44:20Z","title_canon_sha256":"8fda682d1cf35835b3d96cc13cffb4f4c34a0368f65e97692992941bf4c6753b"},"schema_version":"1.0","source":{"id":"2605.17330","kind":"arxiv","version":1}},"canonical_sha256":"5811f2a6a360b6014fb81ede6bb431e781e32b251c51823ba113e429be38b119","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"5811f2a6a360b6014fb81ede6bb431e781e32b251c51823ba113e429be38b119","first_computed_at":"2026-05-20T00:03:52.449088Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-20T00:03:52.449088Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"lLnLVf6j4pxRZNOooqAWPdNRhd5SDCncU5pcXQX23MP/bN+ocFIes00GId/JFxpVAS/MUoONBeyd8u7UyBBUBg==","signature_status":"signed_v1","signed_at":"2026-05-20T00:03:52.449893Z","signed_message":"canonical_sha256_bytes"},"source_id":"2605.17330","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:85283fb577ae57cf851b2ceed96ec90891d92293769cc0d63ceacc8f4441df32","sha256:01eb1b6a94e6515626e3cc0456ea69b3ffdcaefaa7772d84a70d3d5e54124780"],"state_sha256":"f32af4f4190bfc2274b91d06aa2036bd6fe60b68d2da282a31614615faec84a2"}