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For a graph $G$ and $d \\in \\{2,3\\}$, let $\\rho^1_d(G)$ denote the smallest number of lines in $\\mathbb{R}^d$ whose union contains a crossing-free straight-line drawing of $G$. For $d=2$, $G$ must be planar. Similarly, let $\\rho^2_3(G)$ denote the smallest number of planes in $\\mathbb{R}^3$ whose union contains a crossing-free straight-line drawing of $G$.\n  We investigate the complexity of computing these three parameters and obtain th"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1607.06444","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CC","submitted_at":"2016-07-21T19:50:36Z","cross_cats_sorted":["cs.CG"],"title_canon_sha256":"7e13685e7a4850bdc62ca906480f7ffdaa17dd5cf16bef5a1a12c542684f47c3","abstract_canon_sha256":"78df397b0c94c292fe456bb7f617f3568fa69242a323c9503acb0d9c3ff2b828"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T07:50:46.751773Z","signature_b64":"h7bxAJbEHAzvgCQ4FSvsQOEiKhTp4sOFeGKg8v3ObomLmYmihZcg3kRuICr2G6Z2vOcTWfMSOTa12zkIZoLhDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ed75b2e72929978010f4b983836df88876ea1a76495e0bd5fb6c49576e042ceb","last_reissued_at":"2026-07-05T07:50:46.751299Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T07:50:46.751299Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The Complexity of Drawing Graphs on Few Lines and Few Planes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CG"],"primary_cat":"cs.CC","authors_text":"Alexander Ravsky, Alexander Wolff, Fabian Lipp, Krzysztof Fleszar, Oleg Verbitsky, Steven Chaplick","submitted_at":"2016-07-21T19:50:36Z","abstract_excerpt":"It is well known that any graph admits a crossing-free straight-line drawing in $\\mathbb{R}^3$ and that any planar graph admits the same even in $\\mathbb{R}^2$. For a graph $G$ and $d \\in \\{2,3\\}$, let $\\rho^1_d(G)$ denote the smallest number of lines in $\\mathbb{R}^d$ whose union contains a crossing-free straight-line drawing of $G$. For $d=2$, $G$ must be planar. Similarly, let $\\rho^2_3(G)$ denote the smallest number of planes in $\\mathbb{R}^3$ whose union contains a crossing-free straight-line drawing of $G$.\n  We investigate the complexity of computing these three parameters and obtain th"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1607.06444","kind":"arxiv","version":4},"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/1607.06444/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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1607.06444","created_at":"2026-07-05T07:50:46.751367+00:00"},{"alias_kind":"arxiv_version","alias_value":"1607.06444v4","created_at":"2026-07-05T07:50:46.751367+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1607.06444","created_at":"2026-07-05T07:50:46.751367+00:00"},{"alias_kind":"pith_short_12","alias_value":"5V23FZZJFGLY","created_at":"2026-07-05T07:50:46.751367+00:00"},{"alias_kind":"pith_short_16","alias_value":"5V23FZZJFGLYAEHU","created_at":"2026-07-05T07:50:46.751367+00:00"},{"alias_kind":"pith_short_8","alias_value":"5V23FZZJ","created_at":"2026-07-05T07:50:46.751367+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"1908.09400","citing_title":"Optimal Curve Straightening is $\\exists\\mathbb{R}$-Complete","ref_index":20,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/5V23FZZJFGLYAEHUXGBYG3PYRB","json":"https://pith.science/pith/5V23FZZJFGLYAEHUXGBYG3PYRB.json","graph_json":"https://pith.science/api/pith-number/5V23FZZJFGLYAEHUXGBYG3PYRB/graph.json","events_json":"https://pith.science/api/pith-number/5V23FZZJFGLYAEHUXGBYG3PYRB/events.json","paper":"https://pith.science/paper/5V23FZZJ"},"agent_actions":{"view_html":"https://pith.science/pith/5V23FZZJFGLYAEHUXGBYG3PYRB","download_json":"https://pith.science/pith/5V23FZZJFGLYAEHUXGBYG3PYRB.json","view_paper":"https://pith.science/paper/5V23FZZJ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1607.06444&json=true","fetch_graph":"https://pith.science/api/pith-number/5V23FZZJFGLYAEHUXGBYG3PYRB/graph.json","fetch_events":"https://pith.science/api/pith-number/5V23FZZJFGLYAEHUXGBYG3PYRB/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/5V23FZZJFGLYAEHUXGBYG3PYRB/action/timestamp_anchor","attest_storage":"https://pith.science/pith/5V23FZZJFGLYAEHUXGBYG3PYRB/action/storage_attestation","attest_author":"https://pith.science/pith/5V23FZZJFGLYAEHUXGBYG3PYRB/action/author_attestation","sign_citation":"https://pith.science/pith/5V23FZZJFGLYAEHUXGBYG3PYRB/action/citation_signature","submit_replication":"https://pith.science/pith/5V23FZZJFGLYAEHUXGBYG3PYRB/action/replication_record"}},"created_at":"2026-07-05T07:50:46.751367+00:00","updated_at":"2026-07-05T07:50:46.751367+00:00"}