{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:T4ETNTRJNJ2TZN6CGRRVDI4Q5N","short_pith_number":"pith:T4ETNTRJ","canonical_record":{"source":{"id":"1108.0064","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-07-30T12:27:50Z","cross_cats_sorted":["cs.DM"],"title_canon_sha256":"77296a26968b5ce50a63ae9a8e87d76c9c986bfdae26d58e15f6fd82c31c3f2e","abstract_canon_sha256":"8a244839bca7f426b36feca82a003e3a1d438bad870e83b884b20a1327bba17c"},"schema_version":"1.0"},"canonical_sha256":"9f0936ce296a753cb7c2346351a390eb676eb9b07bbacc5618c22f1aee1f406f","source":{"kind":"arxiv","id":"1108.0064","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1108.0064","created_at":"2026-05-18T04:15:07Z"},{"alias_kind":"arxiv_version","alias_value":"1108.0064v1","created_at":"2026-05-18T04:15:07Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1108.0064","created_at":"2026-05-18T04:15:07Z"},{"alias_kind":"pith_short_12","alias_value":"T4ETNTRJNJ2T","created_at":"2026-05-18T12:26:41Z"},{"alias_kind":"pith_short_16","alias_value":"T4ETNTRJNJ2TZN6C","created_at":"2026-05-18T12:26:41Z"},{"alias_kind":"pith_short_8","alias_value":"T4ETNTRJ","created_at":"2026-05-18T12:26:41Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:T4ETNTRJNJ2TZN6CGRRVDI4Q5N","target":"record","payload":{"canonical_record":{"source":{"id":"1108.0064","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-07-30T12:27:50Z","cross_cats_sorted":["cs.DM"],"title_canon_sha256":"77296a26968b5ce50a63ae9a8e87d76c9c986bfdae26d58e15f6fd82c31c3f2e","abstract_canon_sha256":"8a244839bca7f426b36feca82a003e3a1d438bad870e83b884b20a1327bba17c"},"schema_version":"1.0"},"canonical_sha256":"9f0936ce296a753cb7c2346351a390eb676eb9b07bbacc5618c22f1aee1f406f","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:15:07.029952Z","signature_b64":"MllQsg+BNH6WzWSILtNeoyh9k8HrJFBT8DOM4fE/BxnUVA67oZTwDV6H85rkjCiJQ55P/u8m8WGetCgBkFM+Aw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9f0936ce296a753cb7c2346351a390eb676eb9b07bbacc5618c22f1aee1f406f","last_reissued_at":"2026-05-18T04:15:07.029288Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:15:07.029288Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1108.0064","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":"2026-05-18T04:15:07Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"TpZydxmCD15x0A2LL+P73RepbGZgoqARzaqhCYY8k8xCr6w5aJl326/q4PPDxu7dD7QtTcaS5AQ5yvgdWJuSCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-08T01:09:07.714996Z"},"content_sha256":"1330f78fc5cac823fa9d1c03c40843b7a8a11ded7c92309eaedc8565e4b23c9e","schema_version":"1.0","event_id":"sha256:1330f78fc5cac823fa9d1c03c40843b7a8a11ded7c92309eaedc8565e4b23c9e"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:T4ETNTRJNJ2TZN6CGRRVDI4Q5N","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On the Complexity of Planar Covering of Small Graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM"],"primary_cat":"math.CO","authors_text":"Jan Volec, Jozef Jir\\'asek, Martin Tancer, Ond\\v{r}ej B\\'ilka, Pavel Klav\\'ik","submitted_at":"2011-07-30T12:27:50Z","abstract_excerpt":"The problem Cover(H) asks whether an input graph G covers a fixed graph H (i.e., whether there exists a homomorphism G to H which locally preserves the structure of the graphs). Complexity of this problem has been intensively studied. In this paper, we consider the problem PlanarCover(H) which restricts the input graph G to be planar.\n  PlanarCover(H) is polynomially solvable if Cover(H) belongs to P, and it is even trivially solvable if H has no planar cover. Thus the interesting cases are when H admits a planar cover, but Cover(H) is NP-complete. This also relates the problem to the long-sta"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1108.0064","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":""},"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":"2026-05-18T04:15:07Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"rklBGfBDHoFrt+2MEch1ows/nu9oTgP68CVb6mDaw35ktwtM9/rpUv9Lwptrh4kfL7qe2Rvb0DL3RbYdgdp5Cw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-08T01:09:07.715734Z"},"content_sha256":"cab880b043d13ae0991a15ced2f9f620fe7f9bc6793d057485afb3739ba716e9","schema_version":"1.0","event_id":"sha256:cab880b043d13ae0991a15ced2f9f620fe7f9bc6793d057485afb3739ba716e9"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/T4ETNTRJNJ2TZN6CGRRVDI4Q5N/bundle.json","state_url":"https://pith.science/pith/T4ETNTRJNJ2TZN6CGRRVDI4Q5N/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/T4ETNTRJNJ2TZN6CGRRVDI4Q5N/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":"2026-06-08T01:09:07Z","links":{"resolver":"https://pith.science/pith/T4ETNTRJNJ2TZN6CGRRVDI4Q5N","bundle":"https://pith.science/pith/T4ETNTRJNJ2TZN6CGRRVDI4Q5N/bundle.json","state":"https://pith.science/pith/T4ETNTRJNJ2TZN6CGRRVDI4Q5N/state.json","well_known_bundle":"https://pith.science/.well-known/pith/T4ETNTRJNJ2TZN6CGRRVDI4Q5N/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:T4ETNTRJNJ2TZN6CGRRVDI4Q5N","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":"8a244839bca7f426b36feca82a003e3a1d438bad870e83b884b20a1327bba17c","cross_cats_sorted":["cs.DM"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-07-30T12:27:50Z","title_canon_sha256":"77296a26968b5ce50a63ae9a8e87d76c9c986bfdae26d58e15f6fd82c31c3f2e"},"schema_version":"1.0","source":{"id":"1108.0064","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1108.0064","created_at":"2026-05-18T04:15:07Z"},{"alias_kind":"arxiv_version","alias_value":"1108.0064v1","created_at":"2026-05-18T04:15:07Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1108.0064","created_at":"2026-05-18T04:15:07Z"},{"alias_kind":"pith_short_12","alias_value":"T4ETNTRJNJ2T","created_at":"2026-05-18T12:26:41Z"},{"alias_kind":"pith_short_16","alias_value":"T4ETNTRJNJ2TZN6C","created_at":"2026-05-18T12:26:41Z"},{"alias_kind":"pith_short_8","alias_value":"T4ETNTRJ","created_at":"2026-05-18T12:26:41Z"}],"graph_snapshots":[{"event_id":"sha256:cab880b043d13ae0991a15ced2f9f620fe7f9bc6793d057485afb3739ba716e9","target":"graph","created_at":"2026-05-18T04:15:07Z","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 problem Cover(H) asks whether an input graph G covers a fixed graph H (i.e., whether there exists a homomorphism G to H which locally preserves the structure of the graphs). Complexity of this problem has been intensively studied. In this paper, we consider the problem PlanarCover(H) which restricts the input graph G to be planar.\n  PlanarCover(H) is polynomially solvable if Cover(H) belongs to P, and it is even trivially solvable if H has no planar cover. Thus the interesting cases are when H admits a planar cover, but Cover(H) is NP-complete. This also relates the problem to the long-sta","authors_text":"Jan Volec, Jozef Jir\\'asek, Martin Tancer, Ond\\v{r}ej B\\'ilka, Pavel Klav\\'ik","cross_cats":["cs.DM"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-07-30T12:27:50Z","title":"On the Complexity of Planar Covering of Small Graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1108.0064","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":"sha256:1330f78fc5cac823fa9d1c03c40843b7a8a11ded7c92309eaedc8565e4b23c9e","target":"record","created_at":"2026-05-18T04:15:07Z","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":"8a244839bca7f426b36feca82a003e3a1d438bad870e83b884b20a1327bba17c","cross_cats_sorted":["cs.DM"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-07-30T12:27:50Z","title_canon_sha256":"77296a26968b5ce50a63ae9a8e87d76c9c986bfdae26d58e15f6fd82c31c3f2e"},"schema_version":"1.0","source":{"id":"1108.0064","kind":"arxiv","version":1}},"canonical_sha256":"9f0936ce296a753cb7c2346351a390eb676eb9b07bbacc5618c22f1aee1f406f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"9f0936ce296a753cb7c2346351a390eb676eb9b07bbacc5618c22f1aee1f406f","first_computed_at":"2026-05-18T04:15:07.029288Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:15:07.029288Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"MllQsg+BNH6WzWSILtNeoyh9k8HrJFBT8DOM4fE/BxnUVA67oZTwDV6H85rkjCiJQ55P/u8m8WGetCgBkFM+Aw==","signature_status":"signed_v1","signed_at":"2026-05-18T04:15:07.029952Z","signed_message":"canonical_sha256_bytes"},"source_id":"1108.0064","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:1330f78fc5cac823fa9d1c03c40843b7a8a11ded7c92309eaedc8565e4b23c9e","sha256:cab880b043d13ae0991a15ced2f9f620fe7f9bc6793d057485afb3739ba716e9"],"state_sha256":"c493637dea4653422083dfa03a6409ca40594685936efcaecdd383d138aef40f"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"3xheFSLQzBWTeW6mLYkID9x6TOBlxR7sHP/TBcfNyqcaMyIFzJ/UEW6C4jIqJLN+IWvhbMskUGobZdko4ziICQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-08T01:09:07.720269Z","bundle_sha256":"b3ad36e2291a82be59da1980657e383afc4d820de57a89a0ae3a77d20f7c7dc7"}}