{"id":"https://openalex.org/W2019830026","doi":"https://doi.org/10.1145/1998196.1998250","title":"Contact representations of planar graphs with cubes","display_name":"Contact representations of planar graphs with cubes","publication_year":2011,"publication_date":"2011-06-13","ids":{"openalex":"https://openalex.org/W2019830026","doi":"https://doi.org/10.1145/1998196.1998250","mag":"2019830026"},"language":"en","primary_location":{"id":"doi:10.1145/1998196.1998250","is_oa":false,"landing_page_url":"https://doi.org/10.1145/1998196.1998250","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Proceedings of the twenty-seventh annual symposium on Computational geometry","raw_type":"proceedings-article"},"type":"conference-paper","indexed_in":["crossref"],"open_access":{"is_oa":false,"oa_status":"closed","oa_url":null,"any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5005468020","display_name":"Stefan Felsner","orcid":"https://orcid.org/0000-0002-6150-1998"},"institutions":[{"id":"https://openalex.org/I4577782","display_name":"Technische Universit\u00e4t Berlin","ror":"https://ror.org/03v4gjf40","country_code":"DE","type":"education","lineage":["https://openalex.org/I4577782"]}],"countries":["DE"],"is_corresponding":false,"raw_author_name":"Stefan Felsner","raw_affiliation_strings":["Technische Universit\u00e4t Berlin, Berlin, Germany"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Technische Universit\u00e4t Berlin, Berlin, Germany","institution_ids":["https://openalex.org/I4577782"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5053182876","display_name":"Mathew C. Francis","orcid":"https://orcid.org/0000-0002-0498-7856"},"institutions":[{"id":"https://openalex.org/I21250087","display_name":"Charles University","ror":"https://ror.org/024d6js02","country_code":"CZ","type":"education","lineage":["https://openalex.org/I21250087"]}],"countries":["CZ"],"is_corresponding":false,"raw_author_name":"Mathew C. Francis","raw_affiliation_strings":["Charles University, Prague, Czech Rep"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Charles University, Prague, Czech Rep","institution_ids":["https://openalex.org/I21250087"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":2,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":false,"cited_by_count":39,"citation_normalized_percentile":null,"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"first_page":"315","last_page":"320"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10996","display_name":"Computational Geometry and Mesh Generation","score":0.9998999834060669,"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":0.9998999834060669,"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":0.9993000030517578,"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/T12288","display_name":"Optimization and Search Problems","score":0.9768999814987183,"subfield":{"id":"https://openalex.org/subfields/1705","display_name":"Computer Networks and Communications"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/planar-graph","display_name":"Planar graph","score":0.827660083770752},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.7122954726219177},{"id":"https://openalex.org/keywords/planar","display_name":"Planar","score":0.6723840236663818},{"id":"https://openalex.org/keywords/vertex","display_name":"Vertex (graph theory)","score":0.6672365665435791},{"id":"https://openalex.org/keywords/planar-straight-line-graph","display_name":"Planar straight-line graph","score":0.5921263694763184},{"id":"https://openalex.org/keywords/outerplanar-graph","display_name":"Outerplanar graph","score":0.5673030018806458},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.5464311242103577},{"id":"https://openalex.org/keywords/cube","display_name":"Cube (algebra)","score":0.5263831615447998},{"id":"https://openalex.org/keywords/intersection","display_name":"Intersection (aeronautics)","score":0.5252801179885864},{"id":"https://openalex.org/keywords/intersection-graph","display_name":"Intersection graph","score":0.522439181804657},{"id":"https://openalex.org/keywords/book-embedding","display_name":"Book embedding","score":0.5183092951774597},{"id":"https://openalex.org/keywords/graph","display_name":"Graph","score":0.5162156224250793},{"id":"https://openalex.org/keywords/representation","display_name":"Representation (politics)","score":0.4557672142982483},{"id":"https://openalex.org/keywords/polyhedral-graph","display_name":"Polyhedral graph","score":0.44254592061042786},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.32119297981262207},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.2999211549758911},{"id":"https://openalex.org/keywords/chordal-graph","display_name":"Chordal graph","score":0.25118446350097656},{"id":"https://openalex.org/keywords/1-planar-graph","display_name":"1-planar graph","score":0.18883371353149414},{"id":"https://openalex.org/keywords/pathwidth","display_name":"Pathwidth","score":0.149898499250412},{"id":"https://openalex.org/keywords/line-graph","display_name":"Line graph","score":0.12374326586723328},{"id":"https://openalex.org/keywords/computer-graphics","display_name":"Computer graphics (images)","score":0.11863437294960022},{"id":"https://openalex.org/keywords/engineering","display_name":"Engineering","score":0.04348766803741455}],"concepts":[{"id":"https://openalex.org/C101837359","wikidata":"https://www.wikidata.org/wiki/Q547823","display_name":"Planar graph","level":3,"score":0.827660083770752},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.7122954726219177},{"id":"https://openalex.org/C134786449","wikidata":"https://www.wikidata.org/wiki/Q3391255","display_name":"Planar","level":2,"score":0.6723840236663818},{"id":"https://openalex.org/C80899671","wikidata":"https://www.wikidata.org/wiki/Q1304193","display_name":"Vertex (graph theory)","level":3,"score":0.6672365665435791},{"id":"https://openalex.org/C131930043","wikidata":"https://www.wikidata.org/wiki/Q7200964","display_name":"Planar straight-line graph","level":5,"score":0.5921263694763184},{"id":"https://openalex.org/C125792925","wikidata":"https://www.wikidata.org/wiki/Q3115604","display_name":"Outerplanar graph","level":5,"score":0.5673030018806458},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5464311242103577},{"id":"https://openalex.org/C53051483","wikidata":"https://www.wikidata.org/wiki/Q861555","display_name":"Cube (algebra)","level":2,"score":0.5263831615447998},{"id":"https://openalex.org/C64543145","wikidata":"https://www.wikidata.org/wiki/Q162942","display_name":"Intersection (aeronautics)","level":2,"score":0.5252801179885864},{"id":"https://openalex.org/C54540088","wikidata":"https://www.wikidata.org/wiki/Q3498041","display_name":"Intersection graph","level":4,"score":0.522439181804657},{"id":"https://openalex.org/C180222743","wikidata":"https://www.wikidata.org/wiki/Q4942980","display_name":"Book embedding","level":5,"score":0.5183092951774597},{"id":"https://openalex.org/C132525143","wikidata":"https://www.wikidata.org/wiki/Q141488","display_name":"Graph","level":2,"score":0.5162156224250793},{"id":"https://openalex.org/C2776359362","wikidata":"https://www.wikidata.org/wiki/Q2145286","display_name":"Representation (politics)","level":3,"score":0.4557672142982483},{"id":"https://openalex.org/C91233654","wikidata":"https://www.wikidata.org/wiki/Q4555371","display_name":"Polyhedral graph","level":5,"score":0.44254592061042786},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.32119297981262207},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.2999211549758911},{"id":"https://openalex.org/C160446614","wikidata":"https://www.wikidata.org/wiki/Q1322892","display_name":"Chordal graph","level":3,"score":0.25118446350097656},{"id":"https://openalex.org/C102192266","wikidata":"https://www.wikidata.org/wiki/Q4545823","display_name":"1-planar graph","level":4,"score":0.18883371353149414},{"id":"https://openalex.org/C43517604","wikidata":"https://www.wikidata.org/wiki/Q7144893","display_name":"Pathwidth","level":4,"score":0.149898499250412},{"id":"https://openalex.org/C203776342","wikidata":"https://www.wikidata.org/wiki/Q1378376","display_name":"Line graph","level":3,"score":0.12374326586723328},{"id":"https://openalex.org/C121684516","wikidata":"https://www.wikidata.org/wiki/Q7600677","display_name":"Computer graphics (images)","level":1,"score":0.11863437294960022},{"id":"https://openalex.org/C127413603","wikidata":"https://www.wikidata.org/wiki/Q11023","display_name":"Engineering","level":0,"score":0.04348766803741455},{"id":"https://openalex.org/C199539241","wikidata":"https://www.wikidata.org/wiki/Q7748","display_name":"Law","level":1,"score":0.0},{"id":"https://openalex.org/C94625758","wikidata":"https://www.wikidata.org/wiki/Q7163","display_name":"Politics","level":2,"score":0.0},{"id":"https://openalex.org/C17744445","wikidata":"https://www.wikidata.org/wiki/Q36442","display_name":"Political science","level":0,"score":0.0},{"id":"https://openalex.org/C146978453","wikidata":"https://www.wikidata.org/wiki/Q3798668","display_name":"Aerospace engineering","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1145/1998196.1998250","is_oa":false,"landing_page_url":"https://doi.org/10.1145/1998196.1998250","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Proceedings of the twenty-seventh annual symposium on Computational geometry","raw_type":"proceedings-article"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":13,"referenced_works":["https://openalex.org/W1490017425","https://openalex.org/W1971244706","https://openalex.org/W1973094571","https://openalex.org/W1973624728","https://openalex.org/W1983580040","https://openalex.org/W2003154502","https://openalex.org/W2021586503","https://openalex.org/W2028582766","https://openalex.org/W2104901438","https://openalex.org/W2130702552","https://openalex.org/W2134272545","https://openalex.org/W2166735327","https://openalex.org/W3031005930"],"related_works":["https://openalex.org/W1759199114","https://openalex.org/W4390306783","https://openalex.org/W2995965358","https://openalex.org/W1491152805","https://openalex.org/W2942112873","https://openalex.org/W2009558719","https://openalex.org/W3216500139","https://openalex.org/W4287758186","https://openalex.org/W2039281885","https://openalex.org/W3034858646"],"abstract_inverted_index":{"We":[0],"prove":[1],"that":[2,19,73],"every":[3,74],"planar":[4,30,75],"graph":[5,31,76],"has":[6,77],"a":[7,17,22,36,53,64,67,79],"representation":[8,80],"using":[9,81],"axis-parallel":[10,82],"cubes":[11,34,51],"in":[12,15],"three":[13],"dimensions":[14],"such":[16,78],"way":[18],"there":[20],"is":[21,63],"cube":[23],"corresponding":[24,44],"to":[25],"each":[26,59],"vertex":[27],"of":[28,66],"the":[29],"and":[32,40],"two":[33,50],"have":[35,52],"non-empty":[37,54],"intersection":[38],"if":[39,42],"only":[41],"their":[43],"vertices":[45],"are":[46],"adjacent.":[47],"Moreover,":[48],"when":[49],"intersection,":[55],"they":[56],"just":[57],"touch":[58],"other.":[60],"This":[61],"result":[62,68],"strengthening":[65],"by":[69],"Thomassen":[70],"which":[71],"states":[72],"boxes.":[83]},"counts_by_year":[{"year":2025,"cited_by_count":1},{"year":2023,"cited_by_count":3},{"year":2022,"cited_by_count":1},{"year":2021,"cited_by_count":1},{"year":2019,"cited_by_count":4},{"year":2017,"cited_by_count":3},{"year":2016,"cited_by_count":1},{"year":2015,"cited_by_count":7},{"year":2014,"cited_by_count":4},{"year":2013,"cited_by_count":10},{"year":2012,"cited_by_count":2}],"updated_date":"2026-07-14T23:27:15.235271","created_date":"2025-10-10T00:00:00"}