{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:LRZ2L7TZ5QZFOE6QGP7I6HGQWB","short_pith_number":"pith:LRZ2L7TZ","canonical_record":{"source":{"id":"1510.05891","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CG","submitted_at":"2015-10-20T13:40:33Z","cross_cats_sorted":[],"title_canon_sha256":"2c2cbb6a1701ae416cba9e2c9d1890824354cbd26cf9e546e5e6304c747e7d47","abstract_canon_sha256":"bbd2c68ad08a0825bb3ae6dbe0cac3df3791b233c5ec7d4624c81fe331365867"},"schema_version":"1.0"},"canonical_sha256":"5c73a5fe79ec325713d033fe8f1cd0b05e87567c343cbd72e4e6e715fc1a5daa","source":{"kind":"arxiv","id":"1510.05891","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1510.05891","created_at":"2026-05-18T01:29:39Z"},{"alias_kind":"arxiv_version","alias_value":"1510.05891v1","created_at":"2026-05-18T01:29:39Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1510.05891","created_at":"2026-05-18T01:29:39Z"},{"alias_kind":"pith_short_12","alias_value":"LRZ2L7TZ5QZF","created_at":"2026-05-18T12:29:29Z"},{"alias_kind":"pith_short_16","alias_value":"LRZ2L7TZ5QZFOE6Q","created_at":"2026-05-18T12:29:29Z"},{"alias_kind":"pith_short_8","alias_value":"LRZ2L7TZ","created_at":"2026-05-18T12:29:29Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:LRZ2L7TZ5QZFOE6QGP7I6HGQWB","target":"record","payload":{"canonical_record":{"source":{"id":"1510.05891","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CG","submitted_at":"2015-10-20T13:40:33Z","cross_cats_sorted":[],"title_canon_sha256":"2c2cbb6a1701ae416cba9e2c9d1890824354cbd26cf9e546e5e6304c747e7d47","abstract_canon_sha256":"bbd2c68ad08a0825bb3ae6dbe0cac3df3791b233c5ec7d4624c81fe331365867"},"schema_version":"1.0"},"canonical_sha256":"5c73a5fe79ec325713d033fe8f1cd0b05e87567c343cbd72e4e6e715fc1a5daa","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:29:39.412428Z","signature_b64":"3DqiaiZ95ZT0i8skxrnpT84UIJHDpBpU6TNdQ6KtMGVcoRLa5GP6XpQauL+ZtqDsLLik7jHIIVrS6jfvHDBTAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5c73a5fe79ec325713d033fe8f1cd0b05e87567c343cbd72e4e6e715fc1a5daa","last_reissued_at":"2026-05-18T01:29:39.411775Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:29:39.411775Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1510.05891","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":"2026-05-18T01:29:39Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"d7tY6F+1cn5XxVthyuH198GjQFM4P5NKBVFF8zRi6YG+ngdrlTVfPWhZwDQsFazSILufGnCi7ZzRny+exQI/Cg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T13:16:45.821571Z"},"content_sha256":"546841e505a12ef4451b53f1b0f8a5bfddb4952a2238253b72d62a3a2b8c6e84","schema_version":"1.0","event_id":"sha256:546841e505a12ef4451b53f1b0f8a5bfddb4952a2238253b72d62a3a2b8c6e84"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:LRZ2L7TZ5QZFOE6QGP7I6HGQWB","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On the Book Thickness of 1-Planar Graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.CG","authors_text":"Franz J. Brandenburg, Md. Jawaherul Alam, Stephen G. Kobourov","submitted_at":"2015-10-20T13:40:33Z","abstract_excerpt":"In a book embedding of a graph G, the vertices of G are placed in order along a straight-line called spine of the book, and the edges of G are drawn on a set of half-planes, called the pages of the book, such that two edges drawn on a page do not cross each other. The minimum number of pages in which a graph can be embedded is called the book-thickness or the page-number of the graph. It is known that every planar graph has a book embedding on at most four pages. Here we investigate the book-embeddings of 1-planar graphs. A graph is 1-planar if it can be drawn in the plane such that each edge "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1510.05891","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":"2026-05-18T01:29:39Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"H24BegHdNGFa/DOHjbQ2DZ9CPLZxfkSf6urOlkNErowDi1Wdd/0meGgKhjpGXcqKrGROnNccTD7KK8aRq999BA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T13:16:45.821915Z"},"content_sha256":"f66c3c93e193d0d854b824c64f2c1e37a05d4da36e30cb0ab78d2f4956d1e2ea","schema_version":"1.0","event_id":"sha256:f66c3c93e193d0d854b824c64f2c1e37a05d4da36e30cb0ab78d2f4956d1e2ea"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/LRZ2L7TZ5QZFOE6QGP7I6HGQWB/bundle.json","state_url":"https://pith.science/pith/LRZ2L7TZ5QZFOE6QGP7I6HGQWB/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/LRZ2L7TZ5QZFOE6QGP7I6HGQWB/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":"2026-05-28T13:16:45Z","links":{"resolver":"https://pith.science/pith/LRZ2L7TZ5QZFOE6QGP7I6HGQWB","bundle":"https://pith.science/pith/LRZ2L7TZ5QZFOE6QGP7I6HGQWB/bundle.json","state":"https://pith.science/pith/LRZ2L7TZ5QZFOE6QGP7I6HGQWB/state.json","well_known_bundle":"https://pith.science/.well-known/pith/LRZ2L7TZ5QZFOE6QGP7I6HGQWB/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:LRZ2L7TZ5QZFOE6QGP7I6HGQWB","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":"bbd2c68ad08a0825bb3ae6dbe0cac3df3791b233c5ec7d4624c81fe331365867","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CG","submitted_at":"2015-10-20T13:40:33Z","title_canon_sha256":"2c2cbb6a1701ae416cba9e2c9d1890824354cbd26cf9e546e5e6304c747e7d47"},"schema_version":"1.0","source":{"id":"1510.05891","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1510.05891","created_at":"2026-05-18T01:29:39Z"},{"alias_kind":"arxiv_version","alias_value":"1510.05891v1","created_at":"2026-05-18T01:29:39Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1510.05891","created_at":"2026-05-18T01:29:39Z"},{"alias_kind":"pith_short_12","alias_value":"LRZ2L7TZ5QZF","created_at":"2026-05-18T12:29:29Z"},{"alias_kind":"pith_short_16","alias_value":"LRZ2L7TZ5QZFOE6Q","created_at":"2026-05-18T12:29:29Z"},{"alias_kind":"pith_short_8","alias_value":"LRZ2L7TZ","created_at":"2026-05-18T12:29:29Z"}],"graph_snapshots":[{"event_id":"sha256:f66c3c93e193d0d854b824c64f2c1e37a05d4da36e30cb0ab78d2f4956d1e2ea","target":"graph","created_at":"2026-05-18T01:29:39Z","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":"In a book embedding of a graph G, the vertices of G are placed in order along a straight-line called spine of the book, and the edges of G are drawn on a set of half-planes, called the pages of the book, such that two edges drawn on a page do not cross each other. The minimum number of pages in which a graph can be embedded is called the book-thickness or the page-number of the graph. It is known that every planar graph has a book embedding on at most four pages. Here we investigate the book-embeddings of 1-planar graphs. A graph is 1-planar if it can be drawn in the plane such that each edge ","authors_text":"Franz J. Brandenburg, Md. Jawaherul Alam, Stephen G. Kobourov","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CG","submitted_at":"2015-10-20T13:40:33Z","title":"On the Book Thickness of 1-Planar Graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1510.05891","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":"sha256:546841e505a12ef4451b53f1b0f8a5bfddb4952a2238253b72d62a3a2b8c6e84","target":"record","created_at":"2026-05-18T01:29:39Z","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":"bbd2c68ad08a0825bb3ae6dbe0cac3df3791b233c5ec7d4624c81fe331365867","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CG","submitted_at":"2015-10-20T13:40:33Z","title_canon_sha256":"2c2cbb6a1701ae416cba9e2c9d1890824354cbd26cf9e546e5e6304c747e7d47"},"schema_version":"1.0","source":{"id":"1510.05891","kind":"arxiv","version":1}},"canonical_sha256":"5c73a5fe79ec325713d033fe8f1cd0b05e87567c343cbd72e4e6e715fc1a5daa","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"5c73a5fe79ec325713d033fe8f1cd0b05e87567c343cbd72e4e6e715fc1a5daa","first_computed_at":"2026-05-18T01:29:39.411775Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:29:39.411775Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"3DqiaiZ95ZT0i8skxrnpT84UIJHDpBpU6TNdQ6KtMGVcoRLa5GP6XpQauL+ZtqDsLLik7jHIIVrS6jfvHDBTAg==","signature_status":"signed_v1","signed_at":"2026-05-18T01:29:39.412428Z","signed_message":"canonical_sha256_bytes"},"source_id":"1510.05891","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:546841e505a12ef4451b53f1b0f8a5bfddb4952a2238253b72d62a3a2b8c6e84","sha256:f66c3c93e193d0d854b824c64f2c1e37a05d4da36e30cb0ab78d2f4956d1e2ea"],"state_sha256":"4cb805b607c7a78a50d080e235a707019ccc724aadcbbda40302aa75facfed81"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"RFPKlAf3CWuAMd6uUf95JZyLMV38Ui1LmrGC2zsGRso+ROEgW/5FqBW5YXQWYocxQvPRbcNYubnClx9F6NvZBQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-28T13:16:45.823871Z","bundle_sha256":"7b6fff919622e602333dd45a715995db922c104778a90a1e376a00899432f9b5"}}