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In 2002, Chudnovsky, Robertson, Seymour and Thomas proved a decomposition theorem for Berge graphs saying that every Berge graph either is in a well understood basic class, or has some kind of decomposition. Then, Chudnovsky proved stronger theorems. One of them restricts the allowed decompositions to 2-joins and balanced skew partitions.\n  We prove that the problem of deciding whether a graph has a balanced skew partition is NP-hard. We give an $O(n^9)"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1309.0680","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-09-03T13:54:25Z","cross_cats_sorted":[],"title_canon_sha256":"035f03a6dad926a0dd004126c4e0b5c8a51f5c78fbf2c46135cebcc07df822e3","abstract_canon_sha256":"879579735afd7169ef4fabd3a92a52d44fa6f909ff042769ca6a64cc1f7068a5"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:14:18.378067Z","signature_b64":"+UfPOHecyMWBYg82NAo34CE/SuNFbAH72PieQ06cXExMi73SVKElo7WBE381hnEp0cupxUaNZwOLpP9M86WZDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"04a3c3c5beea9cc4f23773da0f56fd0cf4d4f27cc72509f1408d39d866946784","last_reissued_at":"2026-05-18T03:14:18.377432Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:14:18.377432Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Decomposing Berge graphs and detecting balanced skew partitions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Nicolas Trotignon","submitted_at":"2013-09-03T13:54:25Z","abstract_excerpt":"A hole in a graph is an induced cycle on at least four vertices. 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