{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2023:2YRKWGU6HH3IEQODTA4VVJMS5F","short_pith_number":"pith:2YRKWGU6","canonical_record":{"source":{"id":"2303.13655","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2023-03-23T20:24:27Z","cross_cats_sorted":["cs.DM"],"title_canon_sha256":"622681a6be26663fd5f8b4f8ad3984fa4142155e1b578dfce5e16101ff7148c2","abstract_canon_sha256":"0d3010692e66607049e679acef3bdc32dfade8263d9ce2bdb51e6fe654bbd42c"},"schema_version":"1.0"},"canonical_sha256":"d622ab1a9e39f68241c398395aa592e94a68f2fab37a01eb4022ead40ce9da56","source":{"kind":"arxiv","id":"2303.13655","version":4},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2303.13655","created_at":"2026-05-21T01:04:11Z"},{"alias_kind":"arxiv_version","alias_value":"2303.13655v4","created_at":"2026-05-21T01:04:11Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2303.13655","created_at":"2026-05-21T01:04:11Z"},{"alias_kind":"pith_short_12","alias_value":"2YRKWGU6HH3I","created_at":"2026-05-21T01:04:11Z"},{"alias_kind":"pith_short_16","alias_value":"2YRKWGU6HH3IEQOD","created_at":"2026-05-21T01:04:11Z"},{"alias_kind":"pith_short_8","alias_value":"2YRKWGU6","created_at":"2026-05-21T01:04:11Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2023:2YRKWGU6HH3IEQODTA4VVJMS5F","target":"record","payload":{"canonical_record":{"source":{"id":"2303.13655","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2023-03-23T20:24:27Z","cross_cats_sorted":["cs.DM"],"title_canon_sha256":"622681a6be26663fd5f8b4f8ad3984fa4142155e1b578dfce5e16101ff7148c2","abstract_canon_sha256":"0d3010692e66607049e679acef3bdc32dfade8263d9ce2bdb51e6fe654bbd42c"},"schema_version":"1.0"},"canonical_sha256":"d622ab1a9e39f68241c398395aa592e94a68f2fab37a01eb4022ead40ce9da56","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-21T01:04:11.064830Z","signature_b64":"9rbfXVGRLP+IHOpfH8JIIzSjGD7T3oS/upximCI4xLrtWBs3h1ycfd6WqICr/2f/yLL+C9WXsFQaBxyBoEtgBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d622ab1a9e39f68241c398395aa592e94a68f2fab37a01eb4022ead40ce9da56","last_reissued_at":"2026-05-21T01:04:11.064033Z","signature_status":"signed_v1","first_computed_at":"2026-05-21T01:04:11.064033Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2303.13655","source_version":4,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-21T01:04:11Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"sjZ25GvSVCcAxjTJy6RgfWp6qyWiBhB7DwcJDSAjKV42pSpR2v75QMhfDBnFodMcr3+jdI6yNiJNYZjguPMzDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T00:03:47.685087Z"},"content_sha256":"8b7be00b50cb616327d8dceb4eb6861b5df938e3b9ce5fce6e627c3143091e3b","schema_version":"1.0","event_id":"sha256:8b7be00b50cb616327d8dceb4eb6861b5df938e3b9ce5fce6e627c3143091e3b"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2023:2YRKWGU6HH3IEQODTA4VVJMS5F","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Clustered independence and bounded treewidth","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM"],"primary_cat":"math.CO","authors_text":"Kolja Knauer, Torsten Ueckerdt","submitted_at":"2023-03-23T20:24:27Z","abstract_excerpt":"A set $S\\subseteq V$ of vertices of a graph $G$ is a $c$-clustered set if it induces a subgraph with components of order at most $c$ each, and $\\alpha_c(G)$ denotes the size of a largest $c$-clustered set. For any graph $G$ on $n$ vertices and treewidth $k$, we show that $\\alpha_c(G) \\geq \\frac{c}{c+k+1}n$, which improves a result of Dvo\\v{r}{\\'a}k and Wood [Innov.\\ Graph Theory, 2025], while we construct $n$-vertex graphs $G$ of treewidth $k$ with $\\alpha_c(G)\\leq \\frac{c}{c+k}n$. In the case $c\\leq 2$ or $k=1$ we prove the better lower bound $\\alpha_c(G) \\geq \\frac{c}{c+k}n$, which settles a"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2303.13655","kind":"arxiv","version":4},"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/2303.13655/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":"2026-05-21T01:04:11Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"U0VKFhoc0nKi3l0DEsQqRhQ1imqeRyjvxz80uOLPmi1b9lvX2kyPGCYs0Vw0GSB5azqjLKBa2YZXzdEuoHY0Bg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T00:03:47.685803Z"},"content_sha256":"1162cb4fa0049438e6170ed674abcec103214eb183b872ef764780a6321d0177","schema_version":"1.0","event_id":"sha256:1162cb4fa0049438e6170ed674abcec103214eb183b872ef764780a6321d0177"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/2YRKWGU6HH3IEQODTA4VVJMS5F/bundle.json","state_url":"https://pith.science/pith/2YRKWGU6HH3IEQODTA4VVJMS5F/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/2YRKWGU6HH3IEQODTA4VVJMS5F/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":"2026-05-26T00:03:47Z","links":{"resolver":"https://pith.science/pith/2YRKWGU6HH3IEQODTA4VVJMS5F","bundle":"https://pith.science/pith/2YRKWGU6HH3IEQODTA4VVJMS5F/bundle.json","state":"https://pith.science/pith/2YRKWGU6HH3IEQODTA4VVJMS5F/state.json","well_known_bundle":"https://pith.science/.well-known/pith/2YRKWGU6HH3IEQODTA4VVJMS5F/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2023:2YRKWGU6HH3IEQODTA4VVJMS5F","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":"0d3010692e66607049e679acef3bdc32dfade8263d9ce2bdb51e6fe654bbd42c","cross_cats_sorted":["cs.DM"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2023-03-23T20:24:27Z","title_canon_sha256":"622681a6be26663fd5f8b4f8ad3984fa4142155e1b578dfce5e16101ff7148c2"},"schema_version":"1.0","source":{"id":"2303.13655","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2303.13655","created_at":"2026-05-21T01:04:11Z"},{"alias_kind":"arxiv_version","alias_value":"2303.13655v4","created_at":"2026-05-21T01:04:11Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2303.13655","created_at":"2026-05-21T01:04:11Z"},{"alias_kind":"pith_short_12","alias_value":"2YRKWGU6HH3I","created_at":"2026-05-21T01:04:11Z"},{"alias_kind":"pith_short_16","alias_value":"2YRKWGU6HH3IEQOD","created_at":"2026-05-21T01:04:11Z"},{"alias_kind":"pith_short_8","alias_value":"2YRKWGU6","created_at":"2026-05-21T01:04:11Z"}],"graph_snapshots":[{"event_id":"sha256:1162cb4fa0049438e6170ed674abcec103214eb183b872ef764780a6321d0177","target":"graph","created_at":"2026-05-21T01:04:11Z","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/2303.13655/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"A set $S\\subseteq V$ of vertices of a graph $G$ is a $c$-clustered set if it induces a subgraph with components of order at most $c$ each, and $\\alpha_c(G)$ denotes the size of a largest $c$-clustered set. For any graph $G$ on $n$ vertices and treewidth $k$, we show that $\\alpha_c(G) \\geq \\frac{c}{c+k+1}n$, which improves a result of Dvo\\v{r}{\\'a}k and Wood [Innov.\\ Graph Theory, 2025], while we construct $n$-vertex graphs $G$ of treewidth $k$ with $\\alpha_c(G)\\leq \\frac{c}{c+k}n$. In the case $c\\leq 2$ or $k=1$ we prove the better lower bound $\\alpha_c(G) \\geq \\frac{c}{c+k}n$, which settles a","authors_text":"Kolja Knauer, Torsten Ueckerdt","cross_cats":["cs.DM"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2023-03-23T20:24:27Z","title":"Clustered independence and bounded treewidth"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2303.13655","kind":"arxiv","version":4},"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":"sha256:8b7be00b50cb616327d8dceb4eb6861b5df938e3b9ce5fce6e627c3143091e3b","target":"record","created_at":"2026-05-21T01:04:11Z","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":"0d3010692e66607049e679acef3bdc32dfade8263d9ce2bdb51e6fe654bbd42c","cross_cats_sorted":["cs.DM"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2023-03-23T20:24:27Z","title_canon_sha256":"622681a6be26663fd5f8b4f8ad3984fa4142155e1b578dfce5e16101ff7148c2"},"schema_version":"1.0","source":{"id":"2303.13655","kind":"arxiv","version":4}},"canonical_sha256":"d622ab1a9e39f68241c398395aa592e94a68f2fab37a01eb4022ead40ce9da56","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"d622ab1a9e39f68241c398395aa592e94a68f2fab37a01eb4022ead40ce9da56","first_computed_at":"2026-05-21T01:04:11.064033Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-21T01:04:11.064033Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"9rbfXVGRLP+IHOpfH8JIIzSjGD7T3oS/upximCI4xLrtWBs3h1ycfd6WqICr/2f/yLL+C9WXsFQaBxyBoEtgBQ==","signature_status":"signed_v1","signed_at":"2026-05-21T01:04:11.064830Z","signed_message":"canonical_sha256_bytes"},"source_id":"2303.13655","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:8b7be00b50cb616327d8dceb4eb6861b5df938e3b9ce5fce6e627c3143091e3b","sha256:1162cb4fa0049438e6170ed674abcec103214eb183b872ef764780a6321d0177"],"state_sha256":"e39acb45e1e14d05c7796fc4dbe03e4d45975944c917b16994af5d210ce0fda1"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"s4RhTbWcTMoaydiHI+aFOlNnZFbBjDKpf5SDps+8qEYEovgqmilc66x0HkcGXxFyRldsDeLyE6lSw1eMmPeLCQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-26T00:03:47.689274Z","bundle_sha256":"b6b9b3518b1240f70a7970908a00680b90666b7c792dec1a63646695b79f7c67"}}