{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:W4L3OZJLARNZIEOM5KVHR2OUY6","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":"e2cb0845b2d7d0de9e2647e39f0bdfde4c4dd603354a16390e72c7fefced34e0","cross_cats_sorted":["cs.DM"],"license":"http://creativecommons.org/licenses/by-nc-sa/3.0/","primary_cat":"math.CO","submitted_at":"2012-07-23T09:05:57Z","title_canon_sha256":"734245a483203d374aaf4b3e3aff80212718189a26beddc927bcb3bbe41e7085"},"schema_version":"1.0","source":{"id":"1207.5329","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1207.5329","created_at":"2026-05-18T03:50:23Z"},{"alias_kind":"arxiv_version","alias_value":"1207.5329v1","created_at":"2026-05-18T03:50:23Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1207.5329","created_at":"2026-05-18T03:50:23Z"},{"alias_kind":"pith_short_12","alias_value":"W4L3OZJLARNZ","created_at":"2026-05-18T12:27:25Z"},{"alias_kind":"pith_short_16","alias_value":"W4L3OZJLARNZIEOM","created_at":"2026-05-18T12:27:25Z"},{"alias_kind":"pith_short_8","alias_value":"W4L3OZJL","created_at":"2026-05-18T12:27:25Z"}],"graph_snapshots":[{"event_id":"sha256:4870fc57f4362b1fe5a727243ce3c777e2f450c5e87d09eb1b356cc68352af9c","target":"graph","created_at":"2026-05-18T03:50:23Z","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":"The immersion relation is a partial ordering relation on graphs that is weaker than the topological minor relation in the sense that if a graph $G$ contains a graph $H$ as a topological minor, then it also contains it as an immersion but not vice versa. Kuratowski graphs, namely $K_{5}$ and $K_{3,3}$, give a precise characterization of planar graphs when excluded as topological minors. In this note we give a structural characterization of the graphs that exclude Kuratowski graphs as immersions. We prove that they can be constructed by applying consecutive $i$-edge-sums, for $i\\leq 3$, starting","authors_text":"Archontia C. Giannopoulou, Dimitrios M. Thilikos, Marcin Kaminski","cross_cats":["cs.DM"],"headline":"","license":"http://creativecommons.org/licenses/by-nc-sa/3.0/","primary_cat":"math.CO","submitted_at":"2012-07-23T09:05:57Z","title":"Forbidding Kuratowski Graphs as Immersions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1207.5329","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":"sha256:18ed57dc57b7031bbcaee6a29f5b52b8a4bb4e1f33bb03b75952693ad6ccc0dc","target":"record","created_at":"2026-05-18T03:50:23Z","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":"e2cb0845b2d7d0de9e2647e39f0bdfde4c4dd603354a16390e72c7fefced34e0","cross_cats_sorted":["cs.DM"],"license":"http://creativecommons.org/licenses/by-nc-sa/3.0/","primary_cat":"math.CO","submitted_at":"2012-07-23T09:05:57Z","title_canon_sha256":"734245a483203d374aaf4b3e3aff80212718189a26beddc927bcb3bbe41e7085"},"schema_version":"1.0","source":{"id":"1207.5329","kind":"arxiv","version":1}},"canonical_sha256":"b717b7652b045b9411cceaaa78e9d4c7aa57a7d8d9401eabadd676bd570c4de9","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"b717b7652b045b9411cceaaa78e9d4c7aa57a7d8d9401eabadd676bd570c4de9","first_computed_at":"2026-05-18T03:50:23.859725Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:50:23.859725Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"s8gfdTl68lLj5meBo/3V4W6K0HyucBTHc17+HZZjffGM79rDgUUGOyzKh5EGQJzD1nzdeCfe44NpcPaAZIg3BQ==","signature_status":"signed_v1","signed_at":"2026-05-18T03:50:23.860380Z","signed_message":"canonical_sha256_bytes"},"source_id":"1207.5329","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:18ed57dc57b7031bbcaee6a29f5b52b8a4bb4e1f33bb03b75952693ad6ccc0dc","sha256:4870fc57f4362b1fe5a727243ce3c777e2f450c5e87d09eb1b356cc68352af9c"],"state_sha256":"0b61865f124a81ea53d2f4e0d5ddb1b966d2b7ebf3224209cc749958580d07d9"}