{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2010:ZLA7HDXRFCTCIVFTXL7E2KVWXZ","short_pith_number":"pith:ZLA7HDXR","canonical_record":{"source":{"id":"1011.2136","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DS","submitted_at":"2010-11-09T16:29:47Z","cross_cats_sorted":["cs.DM"],"title_canon_sha256":"637a50531a202442d9d85fa34d2bc37744d9dac8bcee81b5550ae20e9cecd705","abstract_canon_sha256":"4ea210afc81eac85b967ed40bc8a9b73af05619957c40f48740049bfc31049f4"},"schema_version":"1.0"},"canonical_sha256":"cac1f38ef128a62454b3bafe4d2ab6be53fb38a716e4ddf4a3458e461b203fb3","source":{"kind":"arxiv","id":"1011.2136","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1011.2136","created_at":"2026-05-18T04:36:42Z"},{"alias_kind":"arxiv_version","alias_value":"1011.2136v1","created_at":"2026-05-18T04:36:42Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1011.2136","created_at":"2026-05-18T04:36:42Z"},{"alias_kind":"pith_short_12","alias_value":"ZLA7HDXRFCTC","created_at":"2026-05-18T12:26:18Z"},{"alias_kind":"pith_short_16","alias_value":"ZLA7HDXRFCTCIVFT","created_at":"2026-05-18T12:26:18Z"},{"alias_kind":"pith_short_8","alias_value":"ZLA7HDXR","created_at":"2026-05-18T12:26:18Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2010:ZLA7HDXRFCTCIVFTXL7E2KVWXZ","target":"record","payload":{"canonical_record":{"source":{"id":"1011.2136","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DS","submitted_at":"2010-11-09T16:29:47Z","cross_cats_sorted":["cs.DM"],"title_canon_sha256":"637a50531a202442d9d85fa34d2bc37744d9dac8bcee81b5550ae20e9cecd705","abstract_canon_sha256":"4ea210afc81eac85b967ed40bc8a9b73af05619957c40f48740049bfc31049f4"},"schema_version":"1.0"},"canonical_sha256":"cac1f38ef128a62454b3bafe4d2ab6be53fb38a716e4ddf4a3458e461b203fb3","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:36:42.494689Z","signature_b64":"bm4liuiDAxiNrLEDScOcwbxju3i4fV3Qc2OZa3XAcM6fb4bFJ1/YYv9wnpvzN8cJsfiyaioVE0G3VrnD9B4vAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"cac1f38ef128a62454b3bafe4d2ab6be53fb38a716e4ddf4a3458e461b203fb3","last_reissued_at":"2026-05-18T04:36:42.494137Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:36:42.494137Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1011.2136","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:36:42Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"wX1tJzKNRz+8B3nvMR1sA5ZUv0j8EvRh9ZNI8OAFVpm8PFdU4bYdCzj3LL2v27jTsQ1GJoBcjoj3Sun9/+ypCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T12:19:16.691725Z"},"content_sha256":"f9232c580ada211bb6981bbc0eed598300097fb45a15c594d2b9b0f3acf6a8e0","schema_version":"1.0","event_id":"sha256:f9232c580ada211bb6981bbc0eed598300097fb45a15c594d2b9b0f3acf6a8e0"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2010:ZLA7HDXRFCTCIVFTXL7E2KVWXZ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A lower bound for the tree-width of planar graphs with vital linkages","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM"],"primary_cat":"cs.DS","authors_text":"Isolde Adler, Philipp Klaus Krause","submitted_at":"2010-11-09T16:29:47Z","abstract_excerpt":"The disjoint paths problem asks, given an graph G and k + 1 pairs of terminals (s_0,t_0), ...,(s_k,t_k), whether there are k+1 pairwise disjoint paths P_0, ...,P_k, such that P_i connects s_i to t_i. Robertson and Seymour have proven that the problem can be solved in polynomial time if k is fixed. Nevertheless, the constants involved are huge, and the algorithm is far from implementable. The algorithm uses a bound on the tree-width of graphs with vital linkages, and deletion of irrelevant vertices. We give single exponential lower bounds both for the tree-width of planar graphs with vital link"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1011.2136","kind":"arxiv","version":1},"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":"2026-05-18T04:36:42Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"18E5n+J0H1ZwTbMYBbZTY2Bam7Fqq2x9rBMNU00K4oQLVKbJKwnW3SCpnfVGU+rOcmMXL/S2L4pDZI8dMPTCDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T12:19:16.692063Z"},"content_sha256":"4d548cf1c2de4a099a9b859dc236802fcf2a6b5908d03bd0a2c29575d04ca252","schema_version":"1.0","event_id":"sha256:4d548cf1c2de4a099a9b859dc236802fcf2a6b5908d03bd0a2c29575d04ca252"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/ZLA7HDXRFCTCIVFTXL7E2KVWXZ/bundle.json","state_url":"https://pith.science/pith/ZLA7HDXRFCTCIVFTXL7E2KVWXZ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/ZLA7HDXRFCTCIVFTXL7E2KVWXZ/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":"2026-06-02T12:19:16Z","links":{"resolver":"https://pith.science/pith/ZLA7HDXRFCTCIVFTXL7E2KVWXZ","bundle":"https://pith.science/pith/ZLA7HDXRFCTCIVFTXL7E2KVWXZ/bundle.json","state":"https://pith.science/pith/ZLA7HDXRFCTCIVFTXL7E2KVWXZ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/ZLA7HDXRFCTCIVFTXL7E2KVWXZ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:ZLA7HDXRFCTCIVFTXL7E2KVWXZ","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":"4ea210afc81eac85b967ed40bc8a9b73af05619957c40f48740049bfc31049f4","cross_cats_sorted":["cs.DM"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DS","submitted_at":"2010-11-09T16:29:47Z","title_canon_sha256":"637a50531a202442d9d85fa34d2bc37744d9dac8bcee81b5550ae20e9cecd705"},"schema_version":"1.0","source":{"id":"1011.2136","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1011.2136","created_at":"2026-05-18T04:36:42Z"},{"alias_kind":"arxiv_version","alias_value":"1011.2136v1","created_at":"2026-05-18T04:36:42Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1011.2136","created_at":"2026-05-18T04:36:42Z"},{"alias_kind":"pith_short_12","alias_value":"ZLA7HDXRFCTC","created_at":"2026-05-18T12:26:18Z"},{"alias_kind":"pith_short_16","alias_value":"ZLA7HDXRFCTCIVFT","created_at":"2026-05-18T12:26:18Z"},{"alias_kind":"pith_short_8","alias_value":"ZLA7HDXR","created_at":"2026-05-18T12:26:18Z"}],"graph_snapshots":[{"event_id":"sha256:4d548cf1c2de4a099a9b859dc236802fcf2a6b5908d03bd0a2c29575d04ca252","target":"graph","created_at":"2026-05-18T04:36:42Z","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":"The disjoint paths problem asks, given an graph G and k + 1 pairs of terminals (s_0,t_0), ...,(s_k,t_k), whether there are k+1 pairwise disjoint paths P_0, ...,P_k, such that P_i connects s_i to t_i. Robertson and Seymour have proven that the problem can be solved in polynomial time if k is fixed. Nevertheless, the constants involved are huge, and the algorithm is far from implementable. The algorithm uses a bound on the tree-width of graphs with vital linkages, and deletion of irrelevant vertices. We give single exponential lower bounds both for the tree-width of planar graphs with vital link","authors_text":"Isolde Adler, Philipp Klaus Krause","cross_cats":["cs.DM"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DS","submitted_at":"2010-11-09T16:29:47Z","title":"A lower bound for the tree-width of planar graphs with vital linkages"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1011.2136","kind":"arxiv","version":1},"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:f9232c580ada211bb6981bbc0eed598300097fb45a15c594d2b9b0f3acf6a8e0","target":"record","created_at":"2026-05-18T04:36:42Z","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":"4ea210afc81eac85b967ed40bc8a9b73af05619957c40f48740049bfc31049f4","cross_cats_sorted":["cs.DM"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DS","submitted_at":"2010-11-09T16:29:47Z","title_canon_sha256":"637a50531a202442d9d85fa34d2bc37744d9dac8bcee81b5550ae20e9cecd705"},"schema_version":"1.0","source":{"id":"1011.2136","kind":"arxiv","version":1}},"canonical_sha256":"cac1f38ef128a62454b3bafe4d2ab6be53fb38a716e4ddf4a3458e461b203fb3","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"cac1f38ef128a62454b3bafe4d2ab6be53fb38a716e4ddf4a3458e461b203fb3","first_computed_at":"2026-05-18T04:36:42.494137Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:36:42.494137Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"bm4liuiDAxiNrLEDScOcwbxju3i4fV3Qc2OZa3XAcM6fb4bFJ1/YYv9wnpvzN8cJsfiyaioVE0G3VrnD9B4vAQ==","signature_status":"signed_v1","signed_at":"2026-05-18T04:36:42.494689Z","signed_message":"canonical_sha256_bytes"},"source_id":"1011.2136","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:f9232c580ada211bb6981bbc0eed598300097fb45a15c594d2b9b0f3acf6a8e0","sha256:4d548cf1c2de4a099a9b859dc236802fcf2a6b5908d03bd0a2c29575d04ca252"],"state_sha256":"4a8c54a76d07a905b334abbab82ded81dec846496b6ffcc6837b49b1d758721d"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"WwUttYOW9BL4/SFfEpMcEbnNAg3VfibJH9ZJPQa9LBlJDBOV9gV25q8BZf9Dq28EnB54R+mslp4FmuwIw03BBw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-02T12:19:16.693906Z","bundle_sha256":"24130bbf5c5aa3245d4d911f5161992fa7f6bb464c793bd75f7a406e0c82ca2d"}}