{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2017:OPTZHDSSC6GNNVZANEJYDJXGCT","short_pith_number":"pith:OPTZHDSS","canonical_record":{"source":{"id":"1701.02651","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-01-10T15:54:40Z","cross_cats_sorted":[],"title_canon_sha256":"0f73e20965eae550bfce6f5bd11852467e137d21f400beb10df40e035967543d","abstract_canon_sha256":"7c7e80e3572aa28bead54bbaa9bd98774c438a1ba60b14c8f76eb37938ff1819"},"schema_version":"1.0"},"canonical_sha256":"73e7938e52178cd6d720691381a6e614ff1c3df92843d84212b08bdd7173ddbd","source":{"kind":"arxiv","id":"1701.02651","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1701.02651","created_at":"2026-07-05T00:35:08Z"},{"alias_kind":"arxiv_version","alias_value":"1701.02651v3","created_at":"2026-07-05T00:35:08Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1701.02651","created_at":"2026-07-05T00:35:08Z"},{"alias_kind":"pith_short_12","alias_value":"OPTZHDSSC6GN","created_at":"2026-07-05T00:35:08Z"},{"alias_kind":"pith_short_16","alias_value":"OPTZHDSSC6GNNVZA","created_at":"2026-07-05T00:35:08Z"},{"alias_kind":"pith_short_8","alias_value":"OPTZHDSS","created_at":"2026-07-05T00:35:08Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2017:OPTZHDSSC6GNNVZANEJYDJXGCT","target":"record","payload":{"canonical_record":{"source":{"id":"1701.02651","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-01-10T15:54:40Z","cross_cats_sorted":[],"title_canon_sha256":"0f73e20965eae550bfce6f5bd11852467e137d21f400beb10df40e035967543d","abstract_canon_sha256":"7c7e80e3572aa28bead54bbaa9bd98774c438a1ba60b14c8f76eb37938ff1819"},"schema_version":"1.0"},"canonical_sha256":"73e7938e52178cd6d720691381a6e614ff1c3df92843d84212b08bdd7173ddbd","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T00:35:08.858114Z","signature_b64":"BURakMcqhWBrjXAiZmwjxImTBFi/BjG0LWKX2TtGY1IA0LVv4UWwXMIYEOEyMzycoadLEzVy9+rqJ1+C72SlAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"73e7938e52178cd6d720691381a6e614ff1c3df92843d84212b08bdd7173ddbd","last_reissued_at":"2026-07-05T00:35:08.857673Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T00:35:08.857673Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1701.02651","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T00:35:08Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"XQWb049iJi68ClcV/PDhX2ZMXl0LEZrwQ5ku0Qa/F//5lpp+B7vvaDhATz4u+BpgJMykX+U7KM5R9PSLpF57AQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-13T08:40:52.707158Z"},"content_sha256":"9458a202b852f0477915e5c5dc398b16f3a9b45f86d36379413db64bb0a59e3b","schema_version":"1.0","event_id":"sha256:9458a202b852f0477915e5c5dc398b16f3a9b45f86d36379413db64bb0a59e3b"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2017:OPTZHDSSC6GNNVZANEJYDJXGCT","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Tangle-tree duality: in graphs, matroids and beyond","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Reinhard Diestel, Sang-il Oum","submitted_at":"2017-01-10T15:54:40Z","abstract_excerpt":"We apply a recent duality theorem for tangles in abstract separation systems to derive tangle-type duality theorems for width-parameters in graphs and matroids. We further derive a duality theorem for the existence of clusters in large data sets.\n  Our applications to graphs include new, tangle-type, duality theorems for tree-width, path-width, and tree-decompositions of small adhesion. Conversely, we show that carving width is dual to edge-tangles. For matroids we obtain a duality theorem for tree-width.\n  Our results can be used to derive short proofs of all the classical duality theorems fo"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1701.02651","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/1701.02651/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2026-07-05T00:35:08Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"aLVXTI3VHwWkY9QLJz2JBX/rCN+AMAhZXB3dEAbPJ8usE/nzBZctXe0XStxgsmCNW1NULbKlR3CYcmAn8EUQAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-13T08:40:52.707702Z"},"content_sha256":"9782601111334469fabdb5bf9ebe11d2c755f3389a41ca6ced39630d292250ed","schema_version":"1.0","event_id":"sha256:9782601111334469fabdb5bf9ebe11d2c755f3389a41ca6ced39630d292250ed"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/OPTZHDSSC6GNNVZANEJYDJXGCT/bundle.json","state_url":"https://pith.science/pith/OPTZHDSSC6GNNVZANEJYDJXGCT/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/OPTZHDSSC6GNNVZANEJYDJXGCT/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":"2026-08-13T08:40:52Z","links":{"resolver":"https://pith.science/pith/OPTZHDSSC6GNNVZANEJYDJXGCT","bundle":"https://pith.science/pith/OPTZHDSSC6GNNVZANEJYDJXGCT/bundle.json","state":"https://pith.science/pith/OPTZHDSSC6GNNVZANEJYDJXGCT/state.json","well_known_bundle":"https://pith.science/.well-known/pith/OPTZHDSSC6GNNVZANEJYDJXGCT/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:OPTZHDSSC6GNNVZANEJYDJXGCT","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":"7c7e80e3572aa28bead54bbaa9bd98774c438a1ba60b14c8f76eb37938ff1819","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-01-10T15:54:40Z","title_canon_sha256":"0f73e20965eae550bfce6f5bd11852467e137d21f400beb10df40e035967543d"},"schema_version":"1.0","source":{"id":"1701.02651","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1701.02651","created_at":"2026-07-05T00:35:08Z"},{"alias_kind":"arxiv_version","alias_value":"1701.02651v3","created_at":"2026-07-05T00:35:08Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1701.02651","created_at":"2026-07-05T00:35:08Z"},{"alias_kind":"pith_short_12","alias_value":"OPTZHDSSC6GN","created_at":"2026-07-05T00:35:08Z"},{"alias_kind":"pith_short_16","alias_value":"OPTZHDSSC6GNNVZA","created_at":"2026-07-05T00:35:08Z"},{"alias_kind":"pith_short_8","alias_value":"OPTZHDSS","created_at":"2026-07-05T00:35:08Z"}],"graph_snapshots":[{"event_id":"sha256:9782601111334469fabdb5bf9ebe11d2c755f3389a41ca6ced39630d292250ed","target":"graph","created_at":"2026-07-05T00:35:08Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/1701.02651/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We apply a recent duality theorem for tangles in abstract separation systems to derive tangle-type duality theorems for width-parameters in graphs and matroids. We further derive a duality theorem for the existence of clusters in large data sets.\n  Our applications to graphs include new, tangle-type, duality theorems for tree-width, path-width, and tree-decompositions of small adhesion. Conversely, we show that carving width is dual to edge-tangles. For matroids we obtain a duality theorem for tree-width.\n  Our results can be used to derive short proofs of all the classical duality theorems fo","authors_text":"Reinhard Diestel, Sang-il Oum","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-01-10T15:54:40Z","title":"Tangle-tree duality: in graphs, matroids and beyond"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1701.02651","kind":"arxiv","version":3},"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":"sha256:9458a202b852f0477915e5c5dc398b16f3a9b45f86d36379413db64bb0a59e3b","target":"record","created_at":"2026-07-05T00:35:08Z","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":"7c7e80e3572aa28bead54bbaa9bd98774c438a1ba60b14c8f76eb37938ff1819","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-01-10T15:54:40Z","title_canon_sha256":"0f73e20965eae550bfce6f5bd11852467e137d21f400beb10df40e035967543d"},"schema_version":"1.0","source":{"id":"1701.02651","kind":"arxiv","version":3}},"canonical_sha256":"73e7938e52178cd6d720691381a6e614ff1c3df92843d84212b08bdd7173ddbd","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"73e7938e52178cd6d720691381a6e614ff1c3df92843d84212b08bdd7173ddbd","first_computed_at":"2026-07-05T00:35:08.857673Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T00:35:08.857673Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"BURakMcqhWBrjXAiZmwjxImTBFi/BjG0LWKX2TtGY1IA0LVv4UWwXMIYEOEyMzycoadLEzVy9+rqJ1+C72SlAQ==","signature_status":"signed_v1","signed_at":"2026-07-05T00:35:08.858114Z","signed_message":"canonical_sha256_bytes"},"source_id":"1701.02651","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:9458a202b852f0477915e5c5dc398b16f3a9b45f86d36379413db64bb0a59e3b","sha256:9782601111334469fabdb5bf9ebe11d2c755f3389a41ca6ced39630d292250ed"],"state_sha256":"12401c4d291a39c45e206988d5695c0001cb675f2053eea2311eadd75dab37ab"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ZBwM7aLzC2774SrcasOwHvrDnzZ1l0rbMiKJLIUYraMEJtuYi4VqBAMnDLC/Kkf4v779rTV2xZCRNz+UJaqGCA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-13T08:40:52.712974Z","bundle_sha256":"4f5e50d827539068d6c2d3609bdf225e74a260203df5eb298df59795f40d2622"}}