{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2016:L276AZ6Q4H3BEXGYOB4KKPTALG","short_pith_number":"pith:L276AZ6Q","schema_version":"1.0","canonical_sha256":"5ebfe067d0e1f6125cd87078a53e6059bcd9a0da02961f8b4e9d2d5494250973","source":{"kind":"arxiv","id":"1608.08623","version":2},"attestation_state":"computed","paper":{"title":"On the purity of minor-closed classes of graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Colin McDiarmid, Micha{\\l} Przykucki","submitted_at":"2016-08-30T19:59:58Z","abstract_excerpt":"Given a graph $H$ with at least one edge, let $\\operatorname{gap}_{H}(n)$ denote the maximum difference between the numbers of edges in two $n$-vertex edge-maximal graphs with no minor $H$. We show that for exactly four connected graphs $H$ (with at least two vertices), the class of graphs with no minor $H$ is pure, that is, $\\operatorname{gap}_{H}(n) = 0$ for all $n \\geq 1$; and for each connected graph $H$ (with at least two vertices) we have the dichotomy that either $\\operatorname{gap}_{H}(n) = O(1)$ or $\\operatorname{gap}_{H}(n) = \\Theta(n)$. Further, if $H$ is 2-connected and does not yi"},"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":"1608.08623","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-08-30T19:59:58Z","cross_cats_sorted":[],"title_canon_sha256":"d39dabe7dceb926fa7203adf8243072208fe6b15387766a5afd552b9da6613e8","abstract_canon_sha256":"8b2eafbc1c98ac7e9622f284a9cf7eb2ef24de4f3c72257198efbe64fbc36a51"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:06:43.847617Z","signature_b64":"BHAlwoEmbwcm7k9ubgf/0aqUtLI+HUKUCDnfyl29jmIcHD7SU0CXDoN5b88lvWxLPKMDy0tenpbTp3NGefsoDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5ebfe067d0e1f6125cd87078a53e6059bcd9a0da02961f8b4e9d2d5494250973","last_reissued_at":"2026-05-18T00:06:43.847183Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:06:43.847183Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the purity of minor-closed classes of graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Colin McDiarmid, Micha{\\l} Przykucki","submitted_at":"2016-08-30T19:59:58Z","abstract_excerpt":"Given a graph $H$ with at least one edge, let $\\operatorname{gap}_{H}(n)$ denote the maximum difference between the numbers of edges in two $n$-vertex edge-maximal graphs with no minor $H$. We show that for exactly four connected graphs $H$ (with at least two vertices), the class of graphs with no minor $H$ is pure, that is, $\\operatorname{gap}_{H}(n) = 0$ for all $n \\geq 1$; and for each connected graph $H$ (with at least two vertices) we have the dichotomy that either $\\operatorname{gap}_{H}(n) = O(1)$ or $\\operatorname{gap}_{H}(n) = \\Theta(n)$. Further, if $H$ is 2-connected and does not yi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1608.08623","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1608.08623","created_at":"2026-05-18T00:06:43.847245+00:00"},{"alias_kind":"arxiv_version","alias_value":"1608.08623v2","created_at":"2026-05-18T00:06:43.847245+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1608.08623","created_at":"2026-05-18T00:06:43.847245+00:00"},{"alias_kind":"pith_short_12","alias_value":"L276AZ6Q4H3B","created_at":"2026-05-18T12:30:29.479603+00:00"},{"alias_kind":"pith_short_16","alias_value":"L276AZ6Q4H3BEXGY","created_at":"2026-05-18T12:30:29.479603+00:00"},{"alias_kind":"pith_short_8","alias_value":"L276AZ6Q","created_at":"2026-05-18T12:30:29.479603+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/L276AZ6Q4H3BEXGYOB4KKPTALG","json":"https://pith.science/pith/L276AZ6Q4H3BEXGYOB4KKPTALG.json","graph_json":"https://pith.science/api/pith-number/L276AZ6Q4H3BEXGYOB4KKPTALG/graph.json","events_json":"https://pith.science/api/pith-number/L276AZ6Q4H3BEXGYOB4KKPTALG/events.json","paper":"https://pith.science/paper/L276AZ6Q"},"agent_actions":{"view_html":"https://pith.science/pith/L276AZ6Q4H3BEXGYOB4KKPTALG","download_json":"https://pith.science/pith/L276AZ6Q4H3BEXGYOB4KKPTALG.json","view_paper":"https://pith.science/paper/L276AZ6Q","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1608.08623&json=true","fetch_graph":"https://pith.science/api/pith-number/L276AZ6Q4H3BEXGYOB4KKPTALG/graph.json","fetch_events":"https://pith.science/api/pith-number/L276AZ6Q4H3BEXGYOB4KKPTALG/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/L276AZ6Q4H3BEXGYOB4KKPTALG/action/timestamp_anchor","attest_storage":"https://pith.science/pith/L276AZ6Q4H3BEXGYOB4KKPTALG/action/storage_attestation","attest_author":"https://pith.science/pith/L276AZ6Q4H3BEXGYOB4KKPTALG/action/author_attestation","sign_citation":"https://pith.science/pith/L276AZ6Q4H3BEXGYOB4KKPTALG/action/citation_signature","submit_replication":"https://pith.science/pith/L276AZ6Q4H3BEXGYOB4KKPTALG/action/replication_record"}},"created_at":"2026-05-18T00:06:43.847245+00:00","updated_at":"2026-05-18T00:06:43.847245+00:00"}