{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2025:DCCK4TCAAAIEWTANYOHCY4DU5C","short_pith_number":"pith:DCCK4TCA","canonical_record":{"source":{"id":"2507.12697","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2025-07-17T00:10:48Z","cross_cats_sorted":[],"title_canon_sha256":"113c968bbacf785e023592a1fed702ef727820acd086f209d037a5ef21a562c2","abstract_canon_sha256":"614e8522876e4507ceb31058d3800529debf5b9513f91efa2e6a3b3a88cb8ffe"},"schema_version":"1.0"},"canonical_sha256":"1884ae4c4000104b4c0dc38e2c7074e88a5710167e23c9fa8a1f776d62278850","source":{"kind":"arxiv","id":"2507.12697","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2507.12697","created_at":"2026-07-05T11:38:37Z"},{"alias_kind":"arxiv_version","alias_value":"2507.12697v1","created_at":"2026-07-05T11:38:37Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2507.12697","created_at":"2026-07-05T11:38:37Z"},{"alias_kind":"pith_short_12","alias_value":"DCCK4TCAAAIE","created_at":"2026-07-05T11:38:37Z"},{"alias_kind":"pith_short_16","alias_value":"DCCK4TCAAAIEWTAN","created_at":"2026-07-05T11:38:37Z"},{"alias_kind":"pith_short_8","alias_value":"DCCK4TCA","created_at":"2026-07-05T11:38:37Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2025:DCCK4TCAAAIEWTANYOHCY4DU5C","target":"record","payload":{"canonical_record":{"source":{"id":"2507.12697","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2025-07-17T00:10:48Z","cross_cats_sorted":[],"title_canon_sha256":"113c968bbacf785e023592a1fed702ef727820acd086f209d037a5ef21a562c2","abstract_canon_sha256":"614e8522876e4507ceb31058d3800529debf5b9513f91efa2e6a3b3a88cb8ffe"},"schema_version":"1.0"},"canonical_sha256":"1884ae4c4000104b4c0dc38e2c7074e88a5710167e23c9fa8a1f776d62278850","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:38:37.517332Z","signature_b64":"u39NQr/cIx4fdIfViHlrEBNyRY0zPIXNELc1+zPT6/cxBCURttYk279K3JiLpoHxR3TDt5Hxj97IisVYXwXrBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1884ae4c4000104b4c0dc38e2c7074e88a5710167e23c9fa8a1f776d62278850","last_reissued_at":"2026-07-05T11:38:37.516802Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:38:37.516802Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2507.12697","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":"2026-07-05T11:38:37Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ElcubD8jBHcYcciRT868e0l3gzCufXLsa91O6cyKPwvPO1K+KDz4TGpbMKfKRnUjT0TyRVuJ0ETlWFLbisrgDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-11T18:58:46.847848Z"},"content_sha256":"7fc5a0eaf9c86c98221ece0d3eeb2ced602b02e12af0eb69eb4fbcdd8b63e397","schema_version":"1.0","event_id":"sha256:7fc5a0eaf9c86c98221ece0d3eeb2ced602b02e12af0eb69eb4fbcdd8b63e397"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2025:DCCK4TCAAAIEWTANYOHCY4DU5C","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Unavoidable pivot-minors in graphs of large rank-depth","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Jungho Ahn, Kevin Hendrey, O-joung Kwon, Sang-il Oum","submitted_at":"2025-07-17T00:10:48Z","abstract_excerpt":"Shrub-depth and rank-depth are related graph parameters that are dense analogs of tree-depth. We prove that for every positive integer $t$, every graph of sufficiently large rank-depth contains a pivot-minor isomorphic to a path on $t$ vertices or a graph consisting of two disjoint cliques of size $t$ joined by a half graph. This answers an open problem raised by Kwon, McCarty, Oum, and Wollan in 2021."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2507.12697","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2507.12697/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":"2026-07-05T11:38:37Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"0ZyuTL8rfQR/iLnXiisMkcucFEZZqB02bM2cLDEWuTUMhSIhnjMOxk4uglXtMsID0wTWTlZiqwE9wNAu2kfFAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-11T18:58:46.848373Z"},"content_sha256":"76e10586995dcdea4e21ca96bcc0ff9b3adf1bbfe02ce46b83e5f80fd3310a24","schema_version":"1.0","event_id":"sha256:76e10586995dcdea4e21ca96bcc0ff9b3adf1bbfe02ce46b83e5f80fd3310a24"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/DCCK4TCAAAIEWTANYOHCY4DU5C/bundle.json","state_url":"https://pith.science/pith/DCCK4TCAAAIEWTANYOHCY4DU5C/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/DCCK4TCAAAIEWTANYOHCY4DU5C/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":"2026-08-11T18:58:46Z","links":{"resolver":"https://pith.science/pith/DCCK4TCAAAIEWTANYOHCY4DU5C","bundle":"https://pith.science/pith/DCCK4TCAAAIEWTANYOHCY4DU5C/bundle.json","state":"https://pith.science/pith/DCCK4TCAAAIEWTANYOHCY4DU5C/state.json","well_known_bundle":"https://pith.science/.well-known/pith/DCCK4TCAAAIEWTANYOHCY4DU5C/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2025:DCCK4TCAAAIEWTANYOHCY4DU5C","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":"614e8522876e4507ceb31058d3800529debf5b9513f91efa2e6a3b3a88cb8ffe","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2025-07-17T00:10:48Z","title_canon_sha256":"113c968bbacf785e023592a1fed702ef727820acd086f209d037a5ef21a562c2"},"schema_version":"1.0","source":{"id":"2507.12697","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2507.12697","created_at":"2026-07-05T11:38:37Z"},{"alias_kind":"arxiv_version","alias_value":"2507.12697v1","created_at":"2026-07-05T11:38:37Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2507.12697","created_at":"2026-07-05T11:38:37Z"},{"alias_kind":"pith_short_12","alias_value":"DCCK4TCAAAIE","created_at":"2026-07-05T11:38:37Z"},{"alias_kind":"pith_short_16","alias_value":"DCCK4TCAAAIEWTAN","created_at":"2026-07-05T11:38:37Z"},{"alias_kind":"pith_short_8","alias_value":"DCCK4TCA","created_at":"2026-07-05T11:38:37Z"}],"graph_snapshots":[{"event_id":"sha256:76e10586995dcdea4e21ca96bcc0ff9b3adf1bbfe02ce46b83e5f80fd3310a24","target":"graph","created_at":"2026-07-05T11:38:37Z","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/2507.12697/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Shrub-depth and rank-depth are related graph parameters that are dense analogs of tree-depth. We prove that for every positive integer $t$, every graph of sufficiently large rank-depth contains a pivot-minor isomorphic to a path on $t$ vertices or a graph consisting of two disjoint cliques of size $t$ joined by a half graph. This answers an open problem raised by Kwon, McCarty, Oum, and Wollan in 2021.","authors_text":"Jungho Ahn, Kevin Hendrey, O-joung Kwon, Sang-il Oum","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2025-07-17T00:10:48Z","title":"Unavoidable pivot-minors in graphs of large rank-depth"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2507.12697","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":"sha256:7fc5a0eaf9c86c98221ece0d3eeb2ced602b02e12af0eb69eb4fbcdd8b63e397","target":"record","created_at":"2026-07-05T11:38:37Z","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":"614e8522876e4507ceb31058d3800529debf5b9513f91efa2e6a3b3a88cb8ffe","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2025-07-17T00:10:48Z","title_canon_sha256":"113c968bbacf785e023592a1fed702ef727820acd086f209d037a5ef21a562c2"},"schema_version":"1.0","source":{"id":"2507.12697","kind":"arxiv","version":1}},"canonical_sha256":"1884ae4c4000104b4c0dc38e2c7074e88a5710167e23c9fa8a1f776d62278850","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"1884ae4c4000104b4c0dc38e2c7074e88a5710167e23c9fa8a1f776d62278850","first_computed_at":"2026-07-05T11:38:37.516802Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T11:38:37.516802Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"u39NQr/cIx4fdIfViHlrEBNyRY0zPIXNELc1+zPT6/cxBCURttYk279K3JiLpoHxR3TDt5Hxj97IisVYXwXrBw==","signature_status":"signed_v1","signed_at":"2026-07-05T11:38:37.517332Z","signed_message":"canonical_sha256_bytes"},"source_id":"2507.12697","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:7fc5a0eaf9c86c98221ece0d3eeb2ced602b02e12af0eb69eb4fbcdd8b63e397","sha256:76e10586995dcdea4e21ca96bcc0ff9b3adf1bbfe02ce46b83e5f80fd3310a24"],"state_sha256":"bf826b0e3ea201d4cb3251e986839d14c2448fb3bda2e4c5fd8c77a933a6c05f"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"6UXwUi/0sGrClhLP/T/scbNxCdol6uh7IDT/NKwWoRwFgkuMXyelatGgIBw1n+BnQOj6BNv+6ktz8wFstfpUDQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-11T18:58:46.856239Z","bundle_sha256":"791fe409061e842c16c83cdb87d0e2607a11967df97782b75eed1d6c83751891"}}