{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:SGHA7VTLXIXPSQ62Y7LKZTIPKG","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":"df6c2090cb753dcce9305153c69de1a649f9a4a87bc49fee9dc0708482528746","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2012-04-13T08:28:11Z","title_canon_sha256":"7c1a000638e3796803ce885b097b66f4ada1ca5ef64800080ddeb1b9b2a6fb92"},"schema_version":"1.0","source":{"id":"1204.2915","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1204.2915","created_at":"2026-05-18T03:58:02Z"},{"alias_kind":"arxiv_version","alias_value":"1204.2915v1","created_at":"2026-05-18T03:58:02Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1204.2915","created_at":"2026-05-18T03:58:02Z"},{"alias_kind":"pith_short_12","alias_value":"SGHA7VTLXIXP","created_at":"2026-05-18T12:27:20Z"},{"alias_kind":"pith_short_16","alias_value":"SGHA7VTLXIXPSQ62","created_at":"2026-05-18T12:27:20Z"},{"alias_kind":"pith_short_8","alias_value":"SGHA7VTL","created_at":"2026-05-18T12:27:20Z"}],"graph_snapshots":[{"event_id":"sha256:f6be1248f3f12fa909e3e1549ffd39a30fb848fc93dc66b1692083697610fa4b","target":"graph","created_at":"2026-05-18T03:58:02Z","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 partially embedded graph (or PEG) is a triple (G,H,\\H), where G is a graph, H is a subgraph of G, and \\H is a planar embedding of H. We say that a PEG (G,H,\\H) is planar if the graph G has a planar embedding that extends the embedding \\H.\n  We introduce a containment relation of PEGs analogous to graph minor containment, and characterize the minimal non-planar PEGs with respect to this relation. We show that all the minimal non-planar PEGs except for finitely many belong to a single easily recognizable and explicitly described infinite family. We also describe a more complicated containment ","authors_text":"Ignaz Rutter, Jan Kratochv\\'il, V\\'it Jel\\'inek","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2012-04-13T08:28:11Z","title":"A Kuratowski-Type Theorem for Planarity of Partially Embedded Graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1204.2915","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":"sha256:2a7e89becca203467b930a0eee1b31e4d17d9ca0f97d21edd4f373a80dde200e","target":"record","created_at":"2026-05-18T03:58:02Z","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":"df6c2090cb753dcce9305153c69de1a649f9a4a87bc49fee9dc0708482528746","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2012-04-13T08:28:11Z","title_canon_sha256":"7c1a000638e3796803ce885b097b66f4ada1ca5ef64800080ddeb1b9b2a6fb92"},"schema_version":"1.0","source":{"id":"1204.2915","kind":"arxiv","version":1}},"canonical_sha256":"918e0fd66bba2ef943dac7d6accd0f5184cd96804d992bd9e81d66f079de988d","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"918e0fd66bba2ef943dac7d6accd0f5184cd96804d992bd9e81d66f079de988d","first_computed_at":"2026-05-18T03:58:02.539301Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:58:02.539301Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"3olQnq59gc6DF9XmBsyQbe1ZkCdGfSszAZqRfht3K2Gg6mnlTNbw0FRuKMm3bKsLCibqe4R6zOg6E73wncZCCA==","signature_status":"signed_v1","signed_at":"2026-05-18T03:58:02.539926Z","signed_message":"canonical_sha256_bytes"},"source_id":"1204.2915","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:2a7e89becca203467b930a0eee1b31e4d17d9ca0f97d21edd4f373a80dde200e","sha256:f6be1248f3f12fa909e3e1549ffd39a30fb848fc93dc66b1692083697610fa4b"],"state_sha256":"f628fc68ff8bca4a48ebb7380609dd2f083418b1073026d0fd96b7f7d03f6ced"}