{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2025:RC2DYNKZTMRWZOCPAEST3JKBNM","short_pith_number":"pith:RC2DYNKZ","canonical_record":{"source":{"id":"2504.08266","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2025-04-11T05:43:21Z","cross_cats_sorted":["cs.DM"],"title_canon_sha256":"4b2a60d249b2e5f982558c34ca6f7bc858f1a119d3a21867c47aecfa05dd4bab","abstract_canon_sha256":"c8d29f51352dfa12e252f4531846bb010f0ed5850a89b0cc09f48c2f47f1a874"},"schema_version":"1.0"},"canonical_sha256":"88b43c35599b236cb84f01253da5416b22bc42bdd17be67135b4d7595cb4d4e7","source":{"kind":"arxiv","id":"2504.08266","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2504.08266","created_at":"2026-07-05T11:21:45Z"},{"alias_kind":"arxiv_version","alias_value":"2504.08266v2","created_at":"2026-07-05T11:21:45Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2504.08266","created_at":"2026-07-05T11:21:45Z"},{"alias_kind":"pith_short_12","alias_value":"RC2DYNKZTMRW","created_at":"2026-07-05T11:21:45Z"},{"alias_kind":"pith_short_16","alias_value":"RC2DYNKZTMRWZOCP","created_at":"2026-07-05T11:21:45Z"},{"alias_kind":"pith_short_8","alias_value":"RC2DYNKZ","created_at":"2026-07-05T11:21:45Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2025:RC2DYNKZTMRWZOCPAEST3JKBNM","target":"record","payload":{"canonical_record":{"source":{"id":"2504.08266","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2025-04-11T05:43:21Z","cross_cats_sorted":["cs.DM"],"title_canon_sha256":"4b2a60d249b2e5f982558c34ca6f7bc858f1a119d3a21867c47aecfa05dd4bab","abstract_canon_sha256":"c8d29f51352dfa12e252f4531846bb010f0ed5850a89b0cc09f48c2f47f1a874"},"schema_version":"1.0"},"canonical_sha256":"88b43c35599b236cb84f01253da5416b22bc42bdd17be67135b4d7595cb4d4e7","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:21:45.408957Z","signature_b64":"NOmp+V7Fq2YiqWLcSaec/dfAa8PI1U4+1QqwLnniBc2B+099vIJMg8Yb3GisGEYRmZ1BG16pFUVKc+y6whaXDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"88b43c35599b236cb84f01253da5416b22bc42bdd17be67135b4d7595cb4d4e7","last_reissued_at":"2026-07-05T11:21:45.408284Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:21:45.408284Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2504.08266","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":"2026-07-05T11:21:45Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"q82D4r9bii/FooZS2HB+961rL+gr671LketQXAu2FUhO+3+uzIdOqO8jPFKdAljWJYWiXozPU2epJBitepDCAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-11T14:12:25.902288Z"},"content_sha256":"67f61955d812565b5989ae8d5c68fe15d856e005e401d372598ebe859a5413a8","schema_version":"1.0","event_id":"sha256:67f61955d812565b5989ae8d5c68fe15d856e005e401d372598ebe859a5413a8"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2025:RC2DYNKZTMRWZOCPAEST3JKBNM","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"$\\chi$-Boundedness and Neighbourhood Complexity of Bounded Merge-Width Graphs","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["cs.DM"],"primary_cat":"math.CO","authors_text":"Colin Geniet, Marthe Bonamy","submitted_at":"2025-04-11T05:43:21Z","abstract_excerpt":"Merge-width, recently introduced by Dreier and Toru\\'nczyk, is a common generalisation of bounded expansion classes and twin-width for which the first-order model checking problem remains tractable. We prove that a number of basic properties shared by bounded expansion and bounded twin-width graphs also hold for bounded merge-width graphs: they are $\\chi$-bounded, they satisfy the strong Erd\\H{o}s-Hajnal property, and their neighbourhood complexity is linear."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2504.08266","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2504.08266/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":"2026-07-05T11:21:45Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"gnSH4Ts2x0RLc38cCovTMF/nTPcfQ7GSDyb98zY+SBG1QEEC4hwC7g/CD3kWoN4NWBiM7zfOJm/BPPAXBATVAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-11T14:12:25.903228Z"},"content_sha256":"e31e4df6d4edcb2833537f461d4648081084999666aea8e3176bcc665055bbb7","schema_version":"1.0","event_id":"sha256:e31e4df6d4edcb2833537f461d4648081084999666aea8e3176bcc665055bbb7"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/RC2DYNKZTMRWZOCPAEST3JKBNM/bundle.json","state_url":"https://pith.science/pith/RC2DYNKZTMRWZOCPAEST3JKBNM/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/RC2DYNKZTMRWZOCPAEST3JKBNM/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":"2026-08-11T14:12:25Z","links":{"resolver":"https://pith.science/pith/RC2DYNKZTMRWZOCPAEST3JKBNM","bundle":"https://pith.science/pith/RC2DYNKZTMRWZOCPAEST3JKBNM/bundle.json","state":"https://pith.science/pith/RC2DYNKZTMRWZOCPAEST3JKBNM/state.json","well_known_bundle":"https://pith.science/.well-known/pith/RC2DYNKZTMRWZOCPAEST3JKBNM/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2025:RC2DYNKZTMRWZOCPAEST3JKBNM","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":"c8d29f51352dfa12e252f4531846bb010f0ed5850a89b0cc09f48c2f47f1a874","cross_cats_sorted":["cs.DM"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2025-04-11T05:43:21Z","title_canon_sha256":"4b2a60d249b2e5f982558c34ca6f7bc858f1a119d3a21867c47aecfa05dd4bab"},"schema_version":"1.0","source":{"id":"2504.08266","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2504.08266","created_at":"2026-07-05T11:21:45Z"},{"alias_kind":"arxiv_version","alias_value":"2504.08266v2","created_at":"2026-07-05T11:21:45Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2504.08266","created_at":"2026-07-05T11:21:45Z"},{"alias_kind":"pith_short_12","alias_value":"RC2DYNKZTMRW","created_at":"2026-07-05T11:21:45Z"},{"alias_kind":"pith_short_16","alias_value":"RC2DYNKZTMRWZOCP","created_at":"2026-07-05T11:21:45Z"},{"alias_kind":"pith_short_8","alias_value":"RC2DYNKZ","created_at":"2026-07-05T11:21:45Z"}],"graph_snapshots":[{"event_id":"sha256:e31e4df6d4edcb2833537f461d4648081084999666aea8e3176bcc665055bbb7","target":"graph","created_at":"2026-07-05T11:21:45Z","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/2504.08266/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Merge-width, recently introduced by Dreier and Toru\\'nczyk, is a common generalisation of bounded expansion classes and twin-width for which the first-order model checking problem remains tractable. We prove that a number of basic properties shared by bounded expansion and bounded twin-width graphs also hold for bounded merge-width graphs: they are $\\chi$-bounded, they satisfy the strong Erd\\H{o}s-Hajnal property, and their neighbourhood complexity is linear.","authors_text":"Colin Geniet, Marthe Bonamy","cross_cats":["cs.DM"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2025-04-11T05:43:21Z","title":"$\\chi$-Boundedness and Neighbourhood Complexity of Bounded Merge-Width Graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2504.08266","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":"sha256:67f61955d812565b5989ae8d5c68fe15d856e005e401d372598ebe859a5413a8","target":"record","created_at":"2026-07-05T11:21:45Z","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":"c8d29f51352dfa12e252f4531846bb010f0ed5850a89b0cc09f48c2f47f1a874","cross_cats_sorted":["cs.DM"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2025-04-11T05:43:21Z","title_canon_sha256":"4b2a60d249b2e5f982558c34ca6f7bc858f1a119d3a21867c47aecfa05dd4bab"},"schema_version":"1.0","source":{"id":"2504.08266","kind":"arxiv","version":2}},"canonical_sha256":"88b43c35599b236cb84f01253da5416b22bc42bdd17be67135b4d7595cb4d4e7","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"88b43c35599b236cb84f01253da5416b22bc42bdd17be67135b4d7595cb4d4e7","first_computed_at":"2026-07-05T11:21:45.408284Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T11:21:45.408284Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"NOmp+V7Fq2YiqWLcSaec/dfAa8PI1U4+1QqwLnniBc2B+099vIJMg8Yb3GisGEYRmZ1BG16pFUVKc+y6whaXDg==","signature_status":"signed_v1","signed_at":"2026-07-05T11:21:45.408957Z","signed_message":"canonical_sha256_bytes"},"source_id":"2504.08266","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:67f61955d812565b5989ae8d5c68fe15d856e005e401d372598ebe859a5413a8","sha256:e31e4df6d4edcb2833537f461d4648081084999666aea8e3176bcc665055bbb7"],"state_sha256":"304c5bf5b8b4138efdf92789ff79fd8181081ebbf27d13b1968cb1df9102c563"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"gEFLhDhqitgj+V8XzfSXHJOWmW6YbuPvyMBr2FHEETzqsSNK5eP1HC87lMlXhoHyNwpreS6x0klGXB4tO8bgAQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-11T14:12:25.915133Z","bundle_sha256":"3fb1c64dfbe401d7422fdf1c2f15481c3dd5239ecc261c147f2d82ed9500f2f0"}}