{"id":"https://openalex.org/W2978125219","doi":"https://doi.org/10.1137/18m1221606","title":"Representation Complexities of SemiAlgebraic Graphs","display_name":"Representation Complexities of SemiAlgebraic Graphs","publication_year":2019,"publication_date":"2019-01-01","ids":{"openalex":"https://openalex.org/W2978125219","doi":"https://doi.org/10.1137/18m1221606","mag":"2978125219"},"language":"en","primary_location":{"id":"doi:10.1137/18m1221606","is_oa":false,"landing_page_url":"https://doi.org/10.1137/18m1221606","pdf_url":null,"source":{"id":"https://openalex.org/S117679221","display_name":"SIAM Journal on Discrete Mathematics","issn_l":"0895-4801","issn":["0895-4801","1095-7146"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Discrete Mathematics","raw_type":"journal-article"},"type":"article","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/A5103077611","display_name":"Thao Do","orcid":"https://orcid.org/0000-0002-9813-081X"},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Thao Do","raw_affiliation_strings":[],"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5103077611"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":false,"cited_by_count":2,"citation_normalized_percentile":{"value":0.15624399,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":95},"biblio":{"volume":"33","issue":"4","first_page":"1864","last_page":"1877"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10374","display_name":"Advanced Graph Theory Research","score":0.9998999834060669,"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"}},"topics":[{"id":"https://openalex.org/T10374","display_name":"Advanced Graph Theory Research","score":0.9998999834060669,"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/T10720","display_name":"Complexity and Algorithms in Graphs","score":0.9995999932289124,"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/T11329","display_name":"Limits and Structures in Graph Theory","score":0.9994000196456909,"subfield":{"id":"https://openalex.org/subfields/2607","display_name":"Discrete Mathematics and Combinatorics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.8679485321044922},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.8131332397460938},{"id":"https://openalex.org/keywords/bipartite-graph","display_name":"Bipartite graph","score":0.8011845350265503},{"id":"https://openalex.org/keywords/disjoint-sets","display_name":"Disjoint sets","score":0.6115627288818359},{"id":"https://openalex.org/keywords/graph","display_name":"Graph","score":0.4969671070575714},{"id":"https://openalex.org/keywords/upper-and-lower-bounds","display_name":"Upper and lower bounds","score":0.49135103821754456},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.4518900513648987},{"id":"https://openalex.org/keywords/integer","display_name":"Integer (computer science)","score":0.43828433752059937}],"concepts":[{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.8679485321044922},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8131332397460938},{"id":"https://openalex.org/C197657726","wikidata":"https://www.wikidata.org/wiki/Q174733","display_name":"Bipartite graph","level":3,"score":0.8011845350265503},{"id":"https://openalex.org/C45340560","wikidata":"https://www.wikidata.org/wiki/Q215382","display_name":"Disjoint sets","level":2,"score":0.6115627288818359},{"id":"https://openalex.org/C132525143","wikidata":"https://www.wikidata.org/wiki/Q141488","display_name":"Graph","level":2,"score":0.4969671070575714},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.49135103821754456},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.4518900513648987},{"id":"https://openalex.org/C97137487","wikidata":"https://www.wikidata.org/wiki/Q729138","display_name":"Integer (computer science)","level":2,"score":0.43828433752059937},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.0},{"id":"https://openalex.org/C199360897","wikidata":"https://www.wikidata.org/wiki/Q9143","display_name":"Programming language","level":1,"score":0.0},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1137/18m1221606","is_oa":false,"landing_page_url":"https://doi.org/10.1137/18m1221606","pdf_url":null,"source":{"id":"https://openalex.org/S117679221","display_name":"SIAM Journal on Discrete Mathematics","issn_l":"0895-4801","issn":["0895-4801","1095-7146"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Discrete Mathematics","raw_type":"journal-article"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":20,"referenced_works":["https://openalex.org/W1444168583","https://openalex.org/W1972978715","https://openalex.org/W1986701615","https://openalex.org/W1991581432","https://openalex.org/W2014178973","https://openalex.org/W2023839655","https://openalex.org/W2083179104","https://openalex.org/W2091376083","https://openalex.org/W2100503751","https://openalex.org/W2101788557","https://openalex.org/W2130760369","https://openalex.org/W2144337080","https://openalex.org/W2144898438","https://openalex.org/W2166961358","https://openalex.org/W2170386298","https://openalex.org/W2235559321","https://openalex.org/W2343027471","https://openalex.org/W2962915177","https://openalex.org/W2963063257","https://openalex.org/W2963208722"],"related_works":["https://openalex.org/W2371352078","https://openalex.org/W2953461625","https://openalex.org/W4256429076","https://openalex.org/W1971174658","https://openalex.org/W2077383796","https://openalex.org/W2099195351","https://openalex.org/W2080136900","https://openalex.org/W2372768926","https://openalex.org/W2999799752","https://openalex.org/W2072510267"],"abstract_inverted_index":{"The":[0],"representation":[1,82],"complexity":[2,83],"of":[3,27,58,67,84,116,132,168],"a":[4,22,56,65,103,130,156,162],"bipartite":[5,29,159],"graph":[6,164],"$G=(P,Q)$":[7],"is":[8,50,55,64,86,107,118,127,151,170],"the":[9,73,81,140,166],"minimum":[10],"size":[11],"$\\sum_{i=1}^s":[12],"(|A_i|+|B_i|)$":[13],"over":[14],"all":[15],"possible":[16],"ways":[17],"to":[18],"write":[19],"$G$":[20,49,85,106],"as":[21],"(not":[23],"necessarily":[24],"disjoint)":[25],"union":[26],"complete":[28,158],"subgraphs":[30],"$G=\\cup_{i=1}^s":[31],"A_i\\times":[32],"B_i$":[33],"where":[34],"$A_i\\subset":[35],"P,":[36],"B_i\\subset":[37],"Q$":[38],"for":[39,90,109,175],"$i=1,\\dots,":[40],"s$.":[41],"In":[42],"this":[43],"paper":[44],"we":[45,153],"prove":[46],"that":[47,152],"if":[48],"semialgebraic,":[51],"i.e.,":[52],"when":[53,105,139,165],"$P$":[54],"set":[57,66],"$m$":[59],"points":[60,69],"in":[61,70,161],"$\\mathbb{R}^{d_1}$,":[62],"$Q$":[63],"$n$":[68],"$\\mathbb{R}^{d_2}$,":[71],"and":[72,100,137,144],"edges":[74,117,169],"are":[75],"defined":[76],"by":[77,98],"some":[78,110],"semialgebraic":[79,163,176],"relations,":[80],"$O(":[87],"m^{\\frac{d_1d_2-d_2}{d_1d_2-1}+\\varepsilon}":[88,120],"n^{\\frac{d_1d_2-d_1}{d_1d_2-1}+\\varepsilon}+m^{1+\\varepsilon}+n^{1+\\varepsilon})$":[89],"arbitrarily":[91],"small":[92],"positive":[93,111],"$\\varepsilon$.":[94],"This":[95,125],"generalizes":[96],"results":[97,173],"Apfelbaum--Sharir":[99],"Solomon--Sharir.":[101],"As":[102],"consequence,":[104],"$K_{u,u}$-free":[108],"integer":[112],"$u$,":[113],"its":[114],"number":[115,167],"$O(u":[119],"n^{\\frac{d_1d_2-d_1}{d_1d_2-1}+\\varepsilon}+":[121],"u":[122],"m^{1+\\varepsilon}+u":[123],"n^{1+\\varepsilon})$.":[124],"bound":[126],"stronger":[128],"than":[129],"result":[131],"Fox,":[133],"Pach,":[134],"Sheffer,":[135],"Suk,":[136],"Zahl":[138],"first":[141],"term":[142],"dominates":[143],"$u$":[145],"grows":[146],"with":[147],"$m,n$.":[148],"Another":[149],"consequence":[150],"can":[154],"find":[155],"large":[157],"subgraph":[160],"large.":[171],"Similar":[172],"hold":[174],"hypergraphs.":[177]},"counts_by_year":[{"year":2025,"cited_by_count":1},{"year":2024,"cited_by_count":1}],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2025-10-10T00:00:00"}