{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:IOJ2WMOCOLNE6KXXBUCCIX27DI","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":"8ae4f38683bfb0d9dae2eddfa33752b598d8c01d326c93216129e53a3d28aa69","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2010-06-29T00:59:29Z","title_canon_sha256":"8220ef90c2d5e45bb91cabc76c3fbb95c6d332c4f770b2678411d13062e1170c"},"schema_version":"1.0","source":{"id":"1006.5485","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1006.5485","created_at":"2026-05-18T04:18:04Z"},{"alias_kind":"arxiv_version","alias_value":"1006.5485v2","created_at":"2026-05-18T04:18:04Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1006.5485","created_at":"2026-05-18T04:18:04Z"},{"alias_kind":"pith_short_12","alias_value":"IOJ2WMOCOLNE","created_at":"2026-05-18T12:26:09Z"},{"alias_kind":"pith_short_16","alias_value":"IOJ2WMOCOLNE6KXX","created_at":"2026-05-18T12:26:09Z"},{"alias_kind":"pith_short_8","alias_value":"IOJ2WMOC","created_at":"2026-05-18T12:26:09Z"}],"graph_snapshots":[{"event_id":"sha256:baa6249286b3fe4e6013844970241cd1c6a87ad5a25710b44853374644cbadc5","target":"graph","created_at":"2026-05-18T04:18:04Z","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"},"paper":{"abstract_excerpt":"A linkage of order k of a graph G is a subgraph with k components, each of which is a path. A linkage is vital if it spans all vertices, and no other linkage connects the same pairs of end vertices. We give a characterization of the graphs with a vital linkage of order 2: they are certain minors of a family of highly structured graphs.","authors_text":"Dillon Mayhew, Geoff Whittle, Stefan H. M. van Zwam","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2010-06-29T00:59:29Z","title":"The structure of graphs with a vital linkage of order 2"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1006.5485","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":"sha256:229156502e86ceb8cacd14b06e073338ecf8102e87a46781a96ae5860d0c29cc","target":"record","created_at":"2026-05-18T04:18:04Z","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":"8ae4f38683bfb0d9dae2eddfa33752b598d8c01d326c93216129e53a3d28aa69","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2010-06-29T00:59:29Z","title_canon_sha256":"8220ef90c2d5e45bb91cabc76c3fbb95c6d332c4f770b2678411d13062e1170c"},"schema_version":"1.0","source":{"id":"1006.5485","kind":"arxiv","version":2}},"canonical_sha256":"4393ab31c272da4f2af70d04245f5f1a1bbcc7890e7946ed5bc7083df4e3333f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"4393ab31c272da4f2af70d04245f5f1a1bbcc7890e7946ed5bc7083df4e3333f","first_computed_at":"2026-05-18T04:18:04.634915Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:18:04.634915Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"mPAkQqs7754Gn5TfINtinLPuxpgXrZDiM5BZhasLxpJMUH/NmWrnP587fJ6WOj1bHcEPNfFKIEa29MSvCsE1Cw==","signature_status":"signed_v1","signed_at":"2026-05-18T04:18:04.635570Z","signed_message":"canonical_sha256_bytes"},"source_id":"1006.5485","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:229156502e86ceb8cacd14b06e073338ecf8102e87a46781a96ae5860d0c29cc","sha256:baa6249286b3fe4e6013844970241cd1c6a87ad5a25710b44853374644cbadc5"],"state_sha256":"71c5853ac0fd23119ddc0eada503cced57d77e09d268d9f2a50a4993a54904e9"}