{"id":"https://openalex.org/W2489413438","doi":"https://doi.org/10.7155/jgaa.00630","title":"The Complexity of Drawing Graphs on Few Lines and Few Planes","display_name":"The Complexity of Drawing Graphs on Few Lines and Few Planes","publication_year":2023,"publication_date":"2023-07-01","ids":{"openalex":"https://openalex.org/W2489413438","doi":"https://doi.org/10.7155/jgaa.00630","mag":"2489413438"},"language":"en","primary_location":{"id":"doi:10.7155/jgaa.00630","is_oa":true,"landing_page_url":"https://doi.org/10.7155/jgaa.00630","pdf_url":null,"source":{"id":"https://openalex.org/S33451491","display_name":"Journal of Graph Algorithms and Applications","issn_l":"1526-1719","issn":["1526-1719"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Journal of Graph Algorithms and Applications","raw_type":"journal-article"},"type":"article","indexed_in":["arxiv","crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"diamond","oa_url":"https://doi.org/10.7155/jgaa.00630","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5018484255","display_name":"Steven Chaplick","orcid":"https://orcid.org/0000-0003-3501-4608"},"institutions":[{"id":"https://openalex.org/I34352273","display_name":"Maastricht University","ror":"https://ror.org/02jz4aj89","country_code":"NL","type":"education","lineage":["https://openalex.org/I34352273"]}],"countries":["NL"],"is_corresponding":true,"raw_author_name":"Steven Chaplick","raw_affiliation_strings":["Maastricht University, The Netherlands"],"affiliations":[{"raw_affiliation_string":"Maastricht University, The Netherlands","institution_ids":["https://openalex.org/I34352273"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5056277334","display_name":"Krzysztof Fleszar","orcid":"https://orcid.org/0000-0002-1129-3289"},"institutions":[{"id":"https://openalex.org/I4654613","display_name":"University of Warsaw","ror":"https://ror.org/039bjqg32","country_code":"PL","type":"education","lineage":["https://openalex.org/I4654613"]}],"countries":["PL"],"is_corresponding":false,"raw_author_name":"Krzysztof Fleszar","raw_affiliation_strings":["Institute of Informatics, University of Warsaw, Poland"],"affiliations":[{"raw_affiliation_string":"Institute of Informatics, University of Warsaw, Poland","institution_ids":["https://openalex.org/I4654613"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5068437293","display_name":"Fabian Lipp","orcid":"https://orcid.org/0000-0001-7833-0454"},"institutions":[{"id":"https://openalex.org/I25974101","display_name":"University of W\u00fcrzburg","ror":"https://ror.org/00fbnyb24","country_code":"DE","type":"education","lineage":["https://openalex.org/I25974101"]}],"countries":["DE"],"is_corresponding":false,"raw_author_name":"Fabian Lipp","raw_affiliation_strings":["Institut f\u00fcr Informatik, Universit\u00e4t W\u00fcrzburg, Germany"],"affiliations":[{"raw_affiliation_string":"Institut f\u00fcr Informatik, Universit\u00e4t W\u00fcrzburg, Germany","institution_ids":["https://openalex.org/I25974101"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5021041638","display_name":"Alexander Ravsky","orcid":null},"institutions":[{"id":"https://openalex.org/I4210145787","display_name":"Pidstryhach Institute for Applied Problems of Mechanics and Mathematics","ror":"https://ror.org/05dry3213","country_code":"UA","type":"facility","lineage":["https://openalex.org/I149851306","https://openalex.org/I4210145787"]},{"id":"https://openalex.org/I149851306","display_name":"National Academy of Sciences of Ukraine","ror":"https://ror.org/00je4t102","country_code":"UA","type":"funder","lineage":["https://openalex.org/I149851306"]}],"countries":["UA"],"is_corresponding":false,"raw_author_name":"Alexander Ravsky","raw_affiliation_strings":["Pidstryhach Institute for Applied Problems of Mechanics and Mathematics, National Academy of Sciences of Ukraine, Lviv, Ukraine"],"affiliations":[{"raw_affiliation_string":"Pidstryhach Institute for Applied Problems of Mechanics and Mathematics, National Academy of Sciences of Ukraine, Lviv, Ukraine","institution_ids":["https://openalex.org/I4210145787","https://openalex.org/I149851306"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5064842516","display_name":"Oleg Verbitsky","orcid":"https://orcid.org/0000-0002-9524-1901"},"institutions":[{"id":"https://openalex.org/I39343248","display_name":"Humboldt-Universit\u00e4t zu Berlin","ror":"https://ror.org/01hcx6992","country_code":"DE","type":"education","lineage":["https://openalex.org/I39343248"]}],"countries":["DE"],"is_corresponding":false,"raw_author_name":"Oleg Verbitsky","raw_affiliation_strings":["Institut f\u00fcr Informatik, Humboldt Universit\u00e4t, Germany"],"affiliations":[{"raw_affiliation_string":"Institut f\u00fcr Informatik, Humboldt Universit\u00e4t, Germany","institution_ids":["https://openalex.org/I39343248"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5014138072","display_name":"Alexander Wolff","orcid":"https://orcid.org/0000-0001-5872-718X"},"institutions":[{"id":"https://openalex.org/I25974101","display_name":"University of W\u00fcrzburg","ror":"https://ror.org/00fbnyb24","country_code":"DE","type":"education","lineage":["https://openalex.org/I25974101"]}],"countries":["DE"],"is_corresponding":false,"raw_author_name":"Alexander Wolff","raw_affiliation_strings":["Institut f\u00fcr Informatik, Universit\u00e4t W\u00fcrzburg, Germany"],"affiliations":[{"raw_affiliation_string":"Institut f\u00fcr Informatik, Universit\u00e4t W\u00fcrzburg, Germany","institution_ids":["https://openalex.org/I25974101"]}]}],"institutions":[],"countries_distinct_count":4,"institutions_distinct_count":6,"corresponding_author_ids":["https://openalex.org/A5018484255"],"corresponding_institution_ids":["https://openalex.org/I34352273"],"apc_list":null,"apc_paid":null,"fwci":1.0061,"has_fulltext":false,"cited_by_count":3,"citation_normalized_percentile":{"value":0.80237328,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":95},"biblio":{"volume":"27","issue":"6","first_page":"459","last_page":"488"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10996","display_name":"Computational Geometry and Mesh Generation","score":0.9995999932289124,"subfield":{"id":"https://openalex.org/subfields/1704","display_name":"Computer Graphics and Computer-Aided Design"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T10996","display_name":"Computational Geometry and Mesh Generation","score":0.9995999932289124,"subfield":{"id":"https://openalex.org/subfields/1704","display_name":"Computer Graphics and Computer-Aided Design"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10374","display_name":"Advanced Graph Theory Research","score":0.9970999956130981,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11522","display_name":"VLSI and FPGA Design Techniques","score":0.9879000186920166,"subfield":{"id":"https://openalex.org/subfields/2208","display_name":"Electrical and Electronic Engineering"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.4768139719963074},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.4404427111148834},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.3639541268348694},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.3517456650733948}],"concepts":[{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.4768139719963074},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.4404427111148834},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.3639541268348694},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.3517456650733948}],"mesh":[],"locations_count":5,"locations":[{"id":"doi:10.7155/jgaa.00630","is_oa":true,"landing_page_url":"https://doi.org/10.7155/jgaa.00630","pdf_url":null,"source":{"id":"https://openalex.org/S33451491","display_name":"Journal of Graph Algorithms and Applications","issn_l":"1526-1719","issn":["1526-1719"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Journal of Graph Algorithms and Applications","raw_type":"journal-article"},{"id":"pmh:oai:cris.maastrichtuniversity.nl:openaire/fa8c6cae-0054-4e08-a1d6-0e20c35798ca","is_oa":true,"landing_page_url":"https://cris.maastrichtuniversity.nl/en/publications/fa8c6cae-0054-4e08-a1d6-0e20c35798ca","pdf_url":null,"source":{"id":"https://openalex.org/S4306402616","display_name":"Research Publications (Maastricht University)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I34352273","host_organization_name":"Maastricht University","host_organization_lineage":["https://openalex.org/I34352273"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Chaplick, S, Fleszar, K, Lipp, F, Ravsky, A, Verbitsky, O & Wolff, A 2023, 'The Complexity of Drawing Graphs on Few Lines and Few Planes', Journal of Graph Algorithms and Applications, vol. 27, no. 6, pp. 459-488. https://doi.org/10.7155/jgaa.00630","raw_type":"info:eu-repo/semantics/publishedVersion"},{"id":"pmh:oai:arXiv.org:1607.06444","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1607.06444","pdf_url":"https://arxiv.org/pdf/1607.06444","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"pmh:oai:escidoc.org:escidoc:2554146","is_oa":false,"landing_page_url":"http://hdl.handle.net/21.11116/0000-0000-B552-7","pdf_url":null,"source":{"id":"https://openalex.org/S7407052962","display_name":"Max Planck Digital Library","issn_l":null,"issn":[],"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"info:eu-repo/semantics/conferenceObject"},{"id":"pmh:um:oai:cris.maastrichtuniversity.nl:publications/0799c180-8d13-400e-a992-0acdc19b40c1","is_oa":false,"landing_page_url":"https://cris.maastrichtuniversity.nl/en/publications/0799c180-8d13-400e-a992-0acdc19b40c1","pdf_url":null,"source":{"id":"https://openalex.org/S4306401843","display_name":"Data Archiving and Networked Services (DANS)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I1322597698","host_organization_name":"Royal Netherlands Academy of Arts and Sciences","host_organization_lineage":["https://openalex.org/I1322597698"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"VOLUME=abs/1607.06444;TITLE=None","raw_type":"info:eu-repo/semantics/other"}],"best_oa_location":{"id":"doi:10.7155/jgaa.00630","is_oa":true,"landing_page_url":"https://doi.org/10.7155/jgaa.00630","pdf_url":null,"source":{"id":"https://openalex.org/S33451491","display_name":"Journal of Graph Algorithms and Applications","issn_l":"1526-1719","issn":["1526-1719"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Journal of Graph Algorithms and Applications","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":49,"referenced_works":["https://openalex.org/W94985437","https://openalex.org/W176071698","https://openalex.org/W176606683","https://openalex.org/W574688124","https://openalex.org/W1527367709","https://openalex.org/W1549363509","https://openalex.org/W1967415230","https://openalex.org/W1981172165","https://openalex.org/W1987886060","https://openalex.org/W1991813962","https://openalex.org/W2001255903","https://openalex.org/W2006753964","https://openalex.org/W2024271250","https://openalex.org/W2056925460","https://openalex.org/W2067754486","https://openalex.org/W2081580682","https://openalex.org/W2082388593","https://openalex.org/W2094502103","https://openalex.org/W2102797028","https://openalex.org/W2115124828","https://openalex.org/W2116909526","https://openalex.org/W2117950870","https://openalex.org/W2126186592","https://openalex.org/W2131662609","https://openalex.org/W2139230981","https://openalex.org/W2150278829","https://openalex.org/W2167816765","https://openalex.org/W2284193075","https://openalex.org/W2296066415","https://openalex.org/W2301555989","https://openalex.org/W2394950630","https://openalex.org/W2404730450","https://openalex.org/W2468630637","https://openalex.org/W2569493303","https://openalex.org/W2573654584","https://openalex.org/W2610183793","https://openalex.org/W2785162637","https://openalex.org/W2950863166","https://openalex.org/W2964278127","https://openalex.org/W2990967247","https://openalex.org/W2991639878","https://openalex.org/W3098189637","https://openalex.org/W3103264504","https://openalex.org/W3105020981","https://openalex.org/W3105946615","https://openalex.org/W3108168007","https://openalex.org/W3156690625","https://openalex.org/W4287028296","https://openalex.org/W4300869738"],"related_works":["https://openalex.org/W2899084033","https://openalex.org/W2748952813","https://openalex.org/W4391375266","https://openalex.org/W1979597421","https://openalex.org/W2007980826","https://openalex.org/W2061531152","https://openalex.org/W3002753104","https://openalex.org/W2077600819","https://openalex.org/W2142036596","https://openalex.org/W2072657027"],"abstract_inverted_index":{"It":[0],"is":[1,114,123,136,184,195],"well":[2],"known":[3],"that":[4,15,101,133,162,164,179],"any":[5,16,187],"graph":[6,18,27,55,109],"admits":[7,19],"a":[8,26,46,74,107],"crossing-free":[9,47,75],"straight-line":[10,48,76],"drawing":[11,49,77],"in":[12,23,41,69,153],"$\\mathbb{R}^3$":[13,70],"and":[14,29,89,94,111,149],"planar":[17],"the":[20,36,54,64,82,91,128,134,157,193],"same":[21],"even":[22],"$\\mathbb{R}^2$.":[24],"For":[25,52,97],"$G$":[28,56,110],"$d":[30],"\\in":[31],"\\{2,3\\}$,":[32],"let":[33,61],"$\\rho^1_d(G)$":[34],"denote":[35,63],"smallest":[37,65],"number":[38,66],"of":[39,50,67,78,84],"lines":[40],"$\\mathbb{R}^d$":[42],"whose":[43,71],"union":[44,72],"contains":[45,73],"$G$.":[51,79],"$d=2$,":[53],"must":[57],"be":[58],"planar.":[59],"Similarly,":[60],"$\\rho^2_3(G)$":[62],"planes":[68],"We":[80,177],"investigate":[81],"complexity":[83],"computing":[85],"these":[86],"three":[87],"parameters":[88],"obtain":[90],"following":[92],"hardness":[93],"algorithmic":[95],"results.":[96],"$d\\in\\{2,3\\}$,":[98],"we":[99,131,160],"prove":[100,178],"deciding":[102,120,180],"whether":[103,181],"$\\rho^1_d(G)\\le":[104,121],"k$":[105,122,183],"for":[106,125,186],"given":[108],"integer":[112],"$k$":[113,202],"$\\exists\\mathbb{R}$-complete.":[115],"Since":[116,143],"$NP":[117],"\\subseteq":[118,145],"\\exists\\mathbb{R}$,":[119],"$NP$-hard":[124,185],"$d\\in\\{2,3\\}$.":[126],"On":[127,156],"positive":[129],"side,":[130,159],"show":[132,161],"problem":[135,194],"fixed-parameter":[137,197],"tractable":[138,198],"with":[139,167,199],"respect":[140,168,200],"to":[141,169,201],"$k$.":[142],"$\\exists\\mathbb{R}":[144],"PSPACE$,":[146],"both":[147],"$\\rho^1_2(G)$":[148],"$\\rho^1_3(G)$":[150],"are":[151,165],"computable":[152],"polynomial":[154],"space.":[155],"negative":[158],"drawings":[163],"optimal":[166],"$\\rho^1_2$":[170],"or":[171],"$\\rho^1_3$":[172],"sometimes":[173],"require":[174],"irrational":[175],"coordinates.":[176],"$\\rho^2_3(G)\\le":[182],"fixed":[188],"$k":[189],"\\ge":[190],"2$.":[191],"Hence,":[192],"not":[196],"unless":[203],"$P=NP$.":[204]},"counts_by_year":[{"year":2025,"cited_by_count":1},{"year":2023,"cited_by_count":1},{"year":2017,"cited_by_count":1}],"updated_date":"2026-04-05T17:49:38.594831","created_date":"2025-10-10T00:00:00"}