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Equivalently, every edge belongs to a non-contractible curve of length k and there are no shorter non-contractible curves. We prove that a k-irreducible triangulation of an orientable surface of genus g has $O(k^2g)$ triangles, which is optimal. This is an improvement over the previous best bound $k^{O(k)} g^2$ of Gao, Richter and Seymour [Journal of Combinatorial Theory, Series B, 1996]."},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We prove that a k-irreducible triangulation of an orientable surface of genus g has O(k²g) triangles, which is optimal.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The two definitions of k-irreducible are equivalent and the triangulation remains a valid triangulation after any edge contraction that would violate the girth property; this equivalence and the topological properties of orientable surfaces are invoked to derive the size bound.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"k-irreducible triangulations of orientable genus-g surfaces have O(k²g) triangles.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"A k-irreducible triangulation of a genus g surface has O(k² g) triangles.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"db5591db7006a584902804f9c1a99aca7f392b461e6786f7b83408d98260799d"},"source":{"id":"2603.20030","kind":"arxiv","version":2},"verdict":{"id":"6b0db6f2-0891-4b76-bc5e-ab71c88b7a7f","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-19T18:17:27.716801Z","strongest_claim":"We prove that a k-irreducible triangulation of an orientable surface of genus g has O(k²g) triangles, which is optimal.","one_line_summary":"k-irreducible triangulations of orientable genus-g surfaces have O(k²g) triangles.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The two definitions of k-irreducible are equivalent and the triangulation remains a valid triangulation after any edge contraction that would violate the girth property; this equivalence and the topological properties of orientable surfaces are invoked to derive the size bound.","pith_extraction_headline":"A k-irreducible triangulation of a genus g surface has O(k² g) triangles."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2603.20030/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":44,"sample":[{"doi":"10.1137/1.9781611977073.11","year":2022,"title":"Untangling planar graphs and curves by staying positive","work_id":"051086c1-4a70-44a1-9151-d223d254db8b","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"10.1007/bf02764905","year":1989,"title":"D. 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