{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:L276AZ6Q4H3BEXGYOB4KKPTALG","short_pith_number":"pith:L276AZ6Q","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"},"canonical_sha256":"5ebfe067d0e1f6125cd87078a53e6059bcd9a0da02961f8b4e9d2d5494250973","source":{"kind":"arxiv","id":"1608.08623","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1608.08623","created_at":"2026-05-18T00:06:43Z"},{"alias_kind":"arxiv_version","alias_value":"1608.08623v2","created_at":"2026-05-18T00:06:43Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1608.08623","created_at":"2026-05-18T00:06:43Z"},{"alias_kind":"pith_short_12","alias_value":"L276AZ6Q4H3B","created_at":"2026-05-18T12:30:29Z"},{"alias_kind":"pith_short_16","alias_value":"L276AZ6Q4H3BEXGY","created_at":"2026-05-18T12:30:29Z"},{"alias_kind":"pith_short_8","alias_value":"L276AZ6Q","created_at":"2026-05-18T12:30:29Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:L276AZ6Q4H3BEXGYOB4KKPTALG","target":"record","payload":{"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"},"canonical_sha256":"5ebfe067d0e1f6125cd87078a53e6059bcd9a0da02961f8b4e9d2d5494250973","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"},"source_kind":"arxiv","source_id":"1608.08623","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:06:43Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"9vv66EZbRxx5zBkgAjvyIiSUylpLROtvB+cqNAZIcyWelh8Rgwads5MmJtyPaAaSbRbAZ1PpTEM9tlRwHQ+UCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T03:54:54.004557Z"},"content_sha256":"ae82b4a819fc2bc80addabdb9edfefb37313542ad2c251ac9836bf7b62008dfe","schema_version":"1.0","event_id":"sha256:ae82b4a819fc2bc80addabdb9edfefb37313542ad2c251ac9836bf7b62008dfe"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:L276AZ6Q4H3BEXGYOB4KKPTALG","target":"graph","payload":{"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:06:43Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"XIjp0Fd2zKWTUcXOqbRNrfRs9DoljJ8yCL0sH+2DCZERPkRAnPFDmQ6/iJfjRTWPjWDm+iOJC62DGzYIf4lYDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T03:54:54.004914Z"},"content_sha256":"1552a6732dec301adc4f3e7254dbb3130e1b75f5adc4e7f6eefba53c44b8dc5a","schema_version":"1.0","event_id":"sha256:1552a6732dec301adc4f3e7254dbb3130e1b75f5adc4e7f6eefba53c44b8dc5a"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/L276AZ6Q4H3BEXGYOB4KKPTALG/bundle.json","state_url":"https://pith.science/pith/L276AZ6Q4H3BEXGYOB4KKPTALG/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/L276AZ6Q4H3BEXGYOB4KKPTALG/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-05-31T03:54:54Z","links":{"resolver":"https://pith.science/pith/L276AZ6Q4H3BEXGYOB4KKPTALG","bundle":"https://pith.science/pith/L276AZ6Q4H3BEXGYOB4KKPTALG/bundle.json","state":"https://pith.science/pith/L276AZ6Q4H3BEXGYOB4KKPTALG/state.json","well_known_bundle":"https://pith.science/.well-known/pith/L276AZ6Q4H3BEXGYOB4KKPTALG/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:L276AZ6Q4H3BEXGYOB4KKPTALG","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":"8b2eafbc1c98ac7e9622f284a9cf7eb2ef24de4f3c72257198efbe64fbc36a51","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-08-30T19:59:58Z","title_canon_sha256":"d39dabe7dceb926fa7203adf8243072208fe6b15387766a5afd552b9da6613e8"},"schema_version":"1.0","source":{"id":"1608.08623","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1608.08623","created_at":"2026-05-18T00:06:43Z"},{"alias_kind":"arxiv_version","alias_value":"1608.08623v2","created_at":"2026-05-18T00:06:43Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1608.08623","created_at":"2026-05-18T00:06:43Z"},{"alias_kind":"pith_short_12","alias_value":"L276AZ6Q4H3B","created_at":"2026-05-18T12:30:29Z"},{"alias_kind":"pith_short_16","alias_value":"L276AZ6Q4H3BEXGY","created_at":"2026-05-18T12:30:29Z"},{"alias_kind":"pith_short_8","alias_value":"L276AZ6Q","created_at":"2026-05-18T12:30:29Z"}],"graph_snapshots":[{"event_id":"sha256:1552a6732dec301adc4f3e7254dbb3130e1b75f5adc4e7f6eefba53c44b8dc5a","target":"graph","created_at":"2026-05-18T00:06:43Z","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":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"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","authors_text":"Colin McDiarmid, Micha{\\l} Przykucki","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-08-30T19:59:58Z","title":"On the purity of minor-closed classes of graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1608.08623","kind":"arxiv","version":2},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:ae82b4a819fc2bc80addabdb9edfefb37313542ad2c251ac9836bf7b62008dfe","target":"record","created_at":"2026-05-18T00:06:43Z","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":"8b2eafbc1c98ac7e9622f284a9cf7eb2ef24de4f3c72257198efbe64fbc36a51","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-08-30T19:59:58Z","title_canon_sha256":"d39dabe7dceb926fa7203adf8243072208fe6b15387766a5afd552b9da6613e8"},"schema_version":"1.0","source":{"id":"1608.08623","kind":"arxiv","version":2}},"canonical_sha256":"5ebfe067d0e1f6125cd87078a53e6059bcd9a0da02961f8b4e9d2d5494250973","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"5ebfe067d0e1f6125cd87078a53e6059bcd9a0da02961f8b4e9d2d5494250973","first_computed_at":"2026-05-18T00:06:43.847183Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:06:43.847183Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"BHAlwoEmbwcm7k9ubgf/0aqUtLI+HUKUCDnfyl29jmIcHD7SU0CXDoN5b88lvWxLPKMDy0tenpbTp3NGefsoDA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:06:43.847617Z","signed_message":"canonical_sha256_bytes"},"source_id":"1608.08623","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:ae82b4a819fc2bc80addabdb9edfefb37313542ad2c251ac9836bf7b62008dfe","sha256:1552a6732dec301adc4f3e7254dbb3130e1b75f5adc4e7f6eefba53c44b8dc5a"],"state_sha256":"408dbbe70c4191cea8fa808ace550860bc0dc1e2e2ca24266a1247d44da04f6c"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"S7GAyjhpapR7ruWJsnCvwA7O1JuVYJXcJld2ZzuwxXNTJS1/L6+J590V3uAdK1k4MnXNxO5pScQIezwFOO2gDA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-31T03:54:54.006973Z","bundle_sha256":"b3368d8aa910db967ad73fad3cdac29c63bbae22336e1468bc072ec6b7929146"}}