{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2023:JNBSGNGAKXXE2ZY37QPP7PPQKJ","short_pith_number":"pith:JNBSGNGA","canonical_record":{"source":{"id":"2303.08726","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2023-03-15T16:11:55Z","cross_cats_sorted":["cs.CG"],"title_canon_sha256":"f205d298f92af9e7f9672d5d9385307d67ae3a028e863530890c070d79a66de2","abstract_canon_sha256":"f1c8b05ba1c90fba32ab78a9c06af57a575d3ac062eae06ac4df2f1cfd1ad452"},"schema_version":"1.0"},"canonical_sha256":"4b432334c055ee4d671bfc1effbdf05270c910030ba9f5a17e57c8893b274190","source":{"kind":"arxiv","id":"2303.08726","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2303.08726","created_at":"2026-07-05T05:51:36Z"},{"alias_kind":"arxiv_version","alias_value":"2303.08726v1","created_at":"2026-07-05T05:51:36Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2303.08726","created_at":"2026-07-05T05:51:36Z"},{"alias_kind":"pith_short_12","alias_value":"JNBSGNGAKXXE","created_at":"2026-07-05T05:51:36Z"},{"alias_kind":"pith_short_16","alias_value":"JNBSGNGAKXXE2ZY3","created_at":"2026-07-05T05:51:36Z"},{"alias_kind":"pith_short_8","alias_value":"JNBSGNGA","created_at":"2026-07-05T05:51:36Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2023:JNBSGNGAKXXE2ZY37QPP7PPQKJ","target":"record","payload":{"canonical_record":{"source":{"id":"2303.08726","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2023-03-15T16:11:55Z","cross_cats_sorted":["cs.CG"],"title_canon_sha256":"f205d298f92af9e7f9672d5d9385307d67ae3a028e863530890c070d79a66de2","abstract_canon_sha256":"f1c8b05ba1c90fba32ab78a9c06af57a575d3ac062eae06ac4df2f1cfd1ad452"},"schema_version":"1.0"},"canonical_sha256":"4b432334c055ee4d671bfc1effbdf05270c910030ba9f5a17e57c8893b274190","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T05:51:36.740671Z","signature_b64":"sbDhyZfhBZ/CYV27kHQcGUEA8it9NT6mU20jPO/ORyrFo7hOftpEIxN3AwCR7skAIzOBZJlmUwp/tNXpDHpGBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4b432334c055ee4d671bfc1effbdf05270c910030ba9f5a17e57c8893b274190","last_reissued_at":"2026-07-05T05:51:36.740226Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T05:51:36.740226Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2303.08726","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":"2026-07-05T05:51:36Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"bhlVUki0FC90BLz+Ior87yrQLyS1qgwznxPTAC2TMEve8lHXTk+c/flaZOH5gL/w1wtpZNMeGal7ffR0/6IbDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-20T03:32:49.890482Z"},"content_sha256":"a826662d0e6342187fc0f40d5d8e28e1522bdb97e7827814afa12f5302e3031b","schema_version":"1.0","event_id":"sha256:a826662d0e6342187fc0f40d5d8e28e1522bdb97e7827814afa12f5302e3031b"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2023:JNBSGNGAKXXE2ZY37QPP7PPQKJ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The Number of Edges in Maximal 2-planar Graphs","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["cs.CG"],"primary_cat":"math.CO","authors_text":"Meghana M. Reddy, Michael Hoffmann","submitted_at":"2023-03-15T16:11:55Z","abstract_excerpt":"A graph is $2$-planar if it has local crossing number two, that is, it can be drawn in the plane such that every edge has at most two crossings. A graph is maximal $2$-planar if no edge can be added such that the resulting graph remains $2$-planar. A $2$-planar graph on $n$ vertices has at most $5n-10$ edges, and some (maximal) $2$-planar graphs -- referred to as optimal $2$-planar -- achieve this bound. However, in strong contrast to maximal planar graphs, a maximal $2$-planar graph may have fewer than the maximum possible number of edges. In this paper, we determine the minimum edge density "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2303.08726","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2303.08726/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2026-07-05T05:51:36Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"apPWZS9LLabmF9IOSJBIJPn333oOwEkB2PKtvX0ZYM1k4w9xkMbYxjuS/3S6KWsZVDNlMH5Hp+f6O41z3ixaDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-20T03:32:49.890974Z"},"content_sha256":"89a6d1ae7772c02e564f2bba4b65cbc56a63e9f5fd350c556b4cd3e0c6675897","schema_version":"1.0","event_id":"sha256:89a6d1ae7772c02e564f2bba4b65cbc56a63e9f5fd350c556b4cd3e0c6675897"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/JNBSGNGAKXXE2ZY37QPP7PPQKJ/bundle.json","state_url":"https://pith.science/pith/JNBSGNGAKXXE2ZY37QPP7PPQKJ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/JNBSGNGAKXXE2ZY37QPP7PPQKJ/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":"2026-08-20T03:32:49Z","links":{"resolver":"https://pith.science/pith/JNBSGNGAKXXE2ZY37QPP7PPQKJ","bundle":"https://pith.science/pith/JNBSGNGAKXXE2ZY37QPP7PPQKJ/bundle.json","state":"https://pith.science/pith/JNBSGNGAKXXE2ZY37QPP7PPQKJ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/JNBSGNGAKXXE2ZY37QPP7PPQKJ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2023:JNBSGNGAKXXE2ZY37QPP7PPQKJ","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":"f1c8b05ba1c90fba32ab78a9c06af57a575d3ac062eae06ac4df2f1cfd1ad452","cross_cats_sorted":["cs.CG"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2023-03-15T16:11:55Z","title_canon_sha256":"f205d298f92af9e7f9672d5d9385307d67ae3a028e863530890c070d79a66de2"},"schema_version":"1.0","source":{"id":"2303.08726","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2303.08726","created_at":"2026-07-05T05:51:36Z"},{"alias_kind":"arxiv_version","alias_value":"2303.08726v1","created_at":"2026-07-05T05:51:36Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2303.08726","created_at":"2026-07-05T05:51:36Z"},{"alias_kind":"pith_short_12","alias_value":"JNBSGNGAKXXE","created_at":"2026-07-05T05:51:36Z"},{"alias_kind":"pith_short_16","alias_value":"JNBSGNGAKXXE2ZY3","created_at":"2026-07-05T05:51:36Z"},{"alias_kind":"pith_short_8","alias_value":"JNBSGNGA","created_at":"2026-07-05T05:51:36Z"}],"graph_snapshots":[{"event_id":"sha256:89a6d1ae7772c02e564f2bba4b65cbc56a63e9f5fd350c556b4cd3e0c6675897","target":"graph","created_at":"2026-07-05T05:51:36Z","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/2303.08726/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"A graph is $2$-planar if it has local crossing number two, that is, it can be drawn in the plane such that every edge has at most two crossings. A graph is maximal $2$-planar if no edge can be added such that the resulting graph remains $2$-planar. A $2$-planar graph on $n$ vertices has at most $5n-10$ edges, and some (maximal) $2$-planar graphs -- referred to as optimal $2$-planar -- achieve this bound. However, in strong contrast to maximal planar graphs, a maximal $2$-planar graph may have fewer than the maximum possible number of edges. In this paper, we determine the minimum edge density ","authors_text":"Meghana M. Reddy, Michael Hoffmann","cross_cats":["cs.CG"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2023-03-15T16:11:55Z","title":"The Number of Edges in Maximal 2-planar Graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2303.08726","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:a826662d0e6342187fc0f40d5d8e28e1522bdb97e7827814afa12f5302e3031b","target":"record","created_at":"2026-07-05T05:51:36Z","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":"f1c8b05ba1c90fba32ab78a9c06af57a575d3ac062eae06ac4df2f1cfd1ad452","cross_cats_sorted":["cs.CG"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2023-03-15T16:11:55Z","title_canon_sha256":"f205d298f92af9e7f9672d5d9385307d67ae3a028e863530890c070d79a66de2"},"schema_version":"1.0","source":{"id":"2303.08726","kind":"arxiv","version":1}},"canonical_sha256":"4b432334c055ee4d671bfc1effbdf05270c910030ba9f5a17e57c8893b274190","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"4b432334c055ee4d671bfc1effbdf05270c910030ba9f5a17e57c8893b274190","first_computed_at":"2026-07-05T05:51:36.740226Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T05:51:36.740226Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"sbDhyZfhBZ/CYV27kHQcGUEA8it9NT6mU20jPO/ORyrFo7hOftpEIxN3AwCR7skAIzOBZJlmUwp/tNXpDHpGBA==","signature_status":"signed_v1","signed_at":"2026-07-05T05:51:36.740671Z","signed_message":"canonical_sha256_bytes"},"source_id":"2303.08726","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:a826662d0e6342187fc0f40d5d8e28e1522bdb97e7827814afa12f5302e3031b","sha256:89a6d1ae7772c02e564f2bba4b65cbc56a63e9f5fd350c556b4cd3e0c6675897"],"state_sha256":"e4e4bef8405ff44c6ab2d1fe93106418bf32fcda5b072f632144776352b66f0d"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"7MxBUTUzZarbIKFbszaX4rVd7YSsg7pS0aK3N7oyRJCtA5h6BuwQwKY4+0YrdQ3C5nGeagHvi1uT6H2H94SQCg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-20T03:32:49.894571Z","bundle_sha256":"d8df2224a4a50189f0a39375a3b2b6aa7ec2c7b8f4e8a81016a85818640d9c89"}}