{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2022:3O5F7ATMMFO2JNU4YM7A2N2TK2","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":"007bafcb2300dfa994a896283d0754df4e9d13ec908d9f308f1fa4d524a08062","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2022-10-19T14:31:59Z","title_canon_sha256":"d5354d57dcd61d6e0ed1ef4c463d353f84876474957b2195a181ef9564992eec"},"schema_version":"1.0","source":{"id":"2210.10590","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2210.10590","created_at":"2026-07-05T05:08:21Z"},{"alias_kind":"arxiv_version","alias_value":"2210.10590v1","created_at":"2026-07-05T05:08:21Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2210.10590","created_at":"2026-07-05T05:08:21Z"},{"alias_kind":"pith_short_12","alias_value":"3O5F7ATMMFO2","created_at":"2026-07-05T05:08:21Z"},{"alias_kind":"pith_short_16","alias_value":"3O5F7ATMMFO2JNU4","created_at":"2026-07-05T05:08:21Z"},{"alias_kind":"pith_short_8","alias_value":"3O5F7ATM","created_at":"2026-07-05T05:08:21Z"}],"graph_snapshots":[{"event_id":"sha256:3be4b443a9b53e6984c02b876bc3a4f2a8dc3b8d4b532d60aa95af6c60305a30","target":"graph","created_at":"2026-07-05T05:08:21Z","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/2210.10590/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"A simple graph $G=(V,E)$ on $n$ vertices is said to be recursively partitionable (RP) if $G \\simeq K_1$, or if $G$ is connected and satisfies the following recursive property: for every integer partition $a_1, a_2, \\dots, a_k$ of $n$, there is a partition $\\{A_1, A_2, \\dots, A_k\\}$ of $V$ such that each $|A_i|=a_i$, and each induced subgraph $G[A_i]$ is RP ($1\\leq i \\leq k$). We show that if $S$ is a vertex cut of an RP graph $G$ with $|S|\\geq 2$, then $G-S$ has at most $3|S|-1$ components. Moreover, this bound is sharp for $|S|=3$. We present two methods for constructing new RP graphs from ol","authors_text":"Brandon Du Preez, Calum Buchanan, K. E. Perry, Puck Rombach","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2022-10-19T14:31:59Z","title":"Toughness of recursively partitionable graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2210.10590","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":"sha256:5ec13ab31f81577e7ef32ffd49ace08d573a7b3b5d7795b6cd77f9d1f377738c","target":"record","created_at":"2026-07-05T05:08:21Z","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":"007bafcb2300dfa994a896283d0754df4e9d13ec908d9f308f1fa4d524a08062","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2022-10-19T14:31:59Z","title_canon_sha256":"d5354d57dcd61d6e0ed1ef4c463d353f84876474957b2195a181ef9564992eec"},"schema_version":"1.0","source":{"id":"2210.10590","kind":"arxiv","version":1}},"canonical_sha256":"dbba5f826c615da4b69cc33e0d375356b6af676f17e0c15a9c1a0c3717b6624f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"dbba5f826c615da4b69cc33e0d375356b6af676f17e0c15a9c1a0c3717b6624f","first_computed_at":"2026-07-05T05:08:21.293474Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T05:08:21.293474Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"073ly4wKXCY87TQZ8QRuHHp5efdWmUDRboVuFJF4lPcO/QITetimFt4ogHSFxhnUEHPo9fZqsX9tMKRqCs6IDw==","signature_status":"signed_v1","signed_at":"2026-07-05T05:08:21.293884Z","signed_message":"canonical_sha256_bytes"},"source_id":"2210.10590","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:5ec13ab31f81577e7ef32ffd49ace08d573a7b3b5d7795b6cd77f9d1f377738c","sha256:3be4b443a9b53e6984c02b876bc3a4f2a8dc3b8d4b532d60aa95af6c60305a30"],"state_sha256":"351b0ac9de1ecd8363b6c86b959c2a47f9b08a0166715cfd72bf2c7caf8ffaf3"}