{"id":"https://openalex.org/W2100842600","doi":"https://doi.org/10.37236/656","title":"Extremal Problems for Independent Set Enumeration","display_name":"Extremal Problems for Independent Set Enumeration","publication_year":2011,"publication_date":"2011-08-26","ids":{"openalex":"https://openalex.org/W2100842600","doi":"https://doi.org/10.37236/656","mag":"2100842600"},"language":"en","primary_location":{"id":"doi:10.37236/656","is_oa":true,"landing_page_url":"https://doi.org/10.37236/656","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v18i1p169/pdf","source":{"id":"https://openalex.org/S38448739","display_name":"The Electronic Journal of Combinatorics","issn_l":"1077-8926","issn":["1077-8926","1097-1440"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310317970","host_organization_name":"Electronic Journal of Combinatorics","host_organization_lineage":["https://openalex.org/P4310317970"],"host_organization_lineage_names":["Electronic Journal of Combinatorics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"The Electronic Journal of Combinatorics","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"diamond","oa_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v18i1p169/pdf","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5010808215","display_name":"Jonathan Cutler","orcid":null},"institutions":[{"id":"https://openalex.org/I166088655","display_name":"Montclair State University","ror":"https://ror.org/01nxc2t48","country_code":"US","type":"education","lineage":["https://openalex.org/I166088655"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Jonathan Cutler","raw_affiliation_strings":["Montclair State University New Jersey, U.S.A","Montclair State University"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Montclair State University New Jersey, U.S.A","institution_ids":["https://openalex.org/I166088655"]},{"raw_affiliation_string":"Montclair State University","institution_ids":["https://openalex.org/I166088655"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5032520716","display_name":"A. J. Radcliffe","orcid":"https://orcid.org/0000-0002-8817-0383"},"institutions":[{"id":"https://openalex.org/I114395901","display_name":"University of Nebraska\u2013Lincoln","ror":"https://ror.org/043mer456","country_code":"US","type":"education","lineage":["https://openalex.org/I114395901"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"A. J. Radcliffe","raw_affiliation_strings":["University of Nebraska-Lincoln Nebraska, U.S.A","#N##TAB##TAB##TAB##TAB# University of Nebraska-Lincoln#N##TAB##TAB##TAB#"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"University of Nebraska-Lincoln Nebraska, U.S.A","institution_ids":["https://openalex.org/I114395901"]},{"raw_affiliation_string":"#N##TAB##TAB##TAB##TAB# University of Nebraska-Lincoln#N##TAB##TAB##TAB#","institution_ids":["https://openalex.org/I114395901"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":5.3752,"has_fulltext":true,"cited_by_count":34,"citation_normalized_percentile":{"value":0.96695761,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":90,"max":99},"biblio":{"volume":"18","issue":"1","first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11329","display_name":"Limits and Structures in Graph Theory","score":0.9983999729156494,"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"}},"topics":[{"id":"https://openalex.org/T11329","display_name":"Limits and Structures in Graph Theory","score":0.9983999729156494,"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"}},{"id":"https://openalex.org/T11476","display_name":"Graph theory and applications","score":0.9959999918937683,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10374","display_name":"Advanced Graph Theory Research","score":0.9896000027656555,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.8709596395492554},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.8208013772964478},{"id":"https://openalex.org/keywords/independent-set","display_name":"Independent set","score":0.6020305156707764},{"id":"https://openalex.org/keywords/split-graph","display_name":"Split graph","score":0.5708820819854736},{"id":"https://openalex.org/keywords/bipartite-graph","display_name":"Bipartite graph","score":0.5608072876930237},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.5325747132301331},{"id":"https://openalex.org/keywords/maximal-independent-set","display_name":"Maximal independent set","score":0.4926948845386505},{"id":"https://openalex.org/keywords/disjoint-sets","display_name":"Disjoint sets","score":0.49169689416885376},{"id":"https://openalex.org/keywords/cograph","display_name":"Cograph","score":0.4734841585159302},{"id":"https://openalex.org/keywords/enumeration","display_name":"Enumeration","score":0.46139097213745117},{"id":"https://openalex.org/keywords/chordal-graph","display_name":"Chordal graph","score":0.4174138903617859},{"id":"https://openalex.org/keywords/indifference-graph","display_name":"Indifference graph","score":0.41351717710494995},{"id":"https://openalex.org/keywords/lexicographical-order","display_name":"Lexicographical order","score":0.4106701910495758},{"id":"https://openalex.org/keywords/graph","display_name":"Graph","score":0.3834376931190491},{"id":"https://openalex.org/keywords/1-planar-graph","display_name":"1-planar graph","score":0.31047672033309937}],"concepts":[{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.8709596395492554},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8208013772964478},{"id":"https://openalex.org/C122818955","wikidata":"https://www.wikidata.org/wiki/Q1060343","display_name":"Independent set","level":3,"score":0.6020305156707764},{"id":"https://openalex.org/C8554925","wikidata":"https://www.wikidata.org/wiki/Q3893853","display_name":"Split graph","level":5,"score":0.5708820819854736},{"id":"https://openalex.org/C197657726","wikidata":"https://www.wikidata.org/wiki/Q174733","display_name":"Bipartite graph","level":3,"score":0.5608072876930237},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.5325747132301331},{"id":"https://openalex.org/C18359143","wikidata":"https://www.wikidata.org/wiki/Q7888149","display_name":"Maximal independent set","level":5,"score":0.4926948845386505},{"id":"https://openalex.org/C45340560","wikidata":"https://www.wikidata.org/wiki/Q215382","display_name":"Disjoint sets","level":2,"score":0.49169689416885376},{"id":"https://openalex.org/C59824394","wikidata":"https://www.wikidata.org/wiki/Q5141281","display_name":"Cograph","level":5,"score":0.4734841585159302},{"id":"https://openalex.org/C156340839","wikidata":"https://www.wikidata.org/wiki/Q2704791","display_name":"Enumeration","level":2,"score":0.46139097213745117},{"id":"https://openalex.org/C160446614","wikidata":"https://www.wikidata.org/wiki/Q1322892","display_name":"Chordal graph","level":3,"score":0.4174138903617859},{"id":"https://openalex.org/C74133993","wikidata":"https://www.wikidata.org/wiki/Q3115472","display_name":"Indifference graph","level":3,"score":0.41351717710494995},{"id":"https://openalex.org/C159254197","wikidata":"https://www.wikidata.org/wiki/Q1144915","display_name":"Lexicographical order","level":2,"score":0.4106701910495758},{"id":"https://openalex.org/C132525143","wikidata":"https://www.wikidata.org/wiki/Q141488","display_name":"Graph","level":2,"score":0.3834376931190491},{"id":"https://openalex.org/C102192266","wikidata":"https://www.wikidata.org/wiki/Q4545823","display_name":"1-planar graph","level":4,"score":0.31047672033309937}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.37236/656","is_oa":true,"landing_page_url":"https://doi.org/10.37236/656","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v18i1p169/pdf","source":{"id":"https://openalex.org/S38448739","display_name":"The Electronic Journal of Combinatorics","issn_l":"1077-8926","issn":["1077-8926","1097-1440"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310317970","host_organization_name":"Electronic Journal of Combinatorics","host_organization_lineage":["https://openalex.org/P4310317970"],"host_organization_lineage_names":["Electronic Journal of Combinatorics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"The Electronic Journal of Combinatorics","raw_type":"journal-article"},{"id":"pmh:oai:digitalcommons.unl.edu:mathfacpub-1143","is_oa":false,"landing_page_url":"https://digitalcommons.unl.edu/mathfacpub/143","pdf_url":null,"source":{"id":"https://openalex.org/S4306400577","display_name":"Lincoln (University of Nebraska)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I114395901","host_organization_name":"University of Nebraska\u2013Lincoln","host_organization_lineage":["https://openalex.org/I114395901"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Department of Mathematics: Faculty Publications","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.372.8608","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.372.8608","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.combinatorics.org/ojs/index.php/eljc/article/viewFile/v18i1p169/pdf/","raw_type":"text"}],"best_oa_location":{"id":"doi:10.37236/656","is_oa":true,"landing_page_url":"https://doi.org/10.37236/656","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v18i1p169/pdf","source":{"id":"https://openalex.org/S38448739","display_name":"The Electronic Journal of Combinatorics","issn_l":"1077-8926","issn":["1077-8926","1097-1440"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310317970","host_organization_name":"Electronic Journal of Combinatorics","host_organization_lineage":["https://openalex.org/P4310317970"],"host_organization_lineage_names":["Electronic Journal of Combinatorics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"The Electronic Journal of Combinatorics","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2100842600.pdf","grobid_xml":"https://content.openalex.org/works/W2100842600.grobid-xml"},"referenced_works_count":14,"referenced_works":["https://openalex.org/W1632119805","https://openalex.org/W1986646471","https://openalex.org/W2004655495","https://openalex.org/W2011504778","https://openalex.org/W2013568176","https://openalex.org/W2037520118","https://openalex.org/W2083618772","https://openalex.org/W2115164934","https://openalex.org/W2317161824","https://openalex.org/W2963597942","https://openalex.org/W3104688290","https://openalex.org/W3124913715","https://openalex.org/W3136243343","https://openalex.org/W6603860290"],"related_works":["https://openalex.org/W2898526288","https://openalex.org/W2083238533","https://openalex.org/W2607123191","https://openalex.org/W2061237693","https://openalex.org/W2042018534","https://openalex.org/W2905516113","https://openalex.org/W2978720059","https://openalex.org/W4286287454","https://openalex.org/W2045381948","https://openalex.org/W2113511711"],"abstract_inverted_index":{"The":[0],"study":[1,92],"of":[2,5,21,27,47,55,71,84,107,120,124,148],"the":[3,24,45,60,65,68,75,85,100,104,118,122,142,145],"number":[4,26,54,70,106,123,132,147],"independent":[6,28,72,109,149],"sets":[7,29,73,110,126,150],"in":[8,127],"a":[9,12,52],"graph":[10,66,102],"has":[11,103],"rich":[13],"history.":[14],"Recently,":[15],"Kahn":[16],"proved":[17],"that":[18,64,99],"disjoint":[19],"unions":[20],"$K_{r,r}$'s":[22],"have":[23],"maximum":[25,69,105],"amongst":[30],"$r$-regular":[31,39],"bipartite":[32],"graphs.":[33,40],"Zhao":[34],"extended":[35],"this":[36,89],"to":[37,49],"all":[38],"If":[41],"we":[42,91,97,116,140],"instead":[43],"restrict":[44],"class":[46],"graphs":[48,128,143,152],"those":[50],"on":[51],"fixed":[53],"vertices":[56,155],"and":[57,156],"edges,":[58],"then":[59],"Kruskal-Katona":[61],"theorem":[62],"implies":[63],"with":[67,129,144,153],"is":[74],"lex":[76,101],"graph,":[77],"where":[78],"edges":[79],"form":[80],"an":[81],"initial":[82],"segment":[83],"lexicographic":[86],"ordering.":[87],"In":[88],"paper,":[90],"three":[93],"related":[94],"questions.":[95],"Firstly,":[96],"prove":[98],"weighted":[108],"for":[111,137,151],"any":[112],"appropriate":[113],"weighting.":[114],"Secondly,":[115],"solve":[117],"problem":[119],"maximizing":[121],"independents":[125],"specified":[130],"independence":[131],"or":[133],"clique":[134],"number.":[135],"Finally,":[136],"$m\\leq":[138],"n$,":[139],"find":[141],"minimum":[146],"$n$":[154],"$m$":[157],"edges.":[158]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2025,"cited_by_count":1},{"year":2022,"cited_by_count":2},{"year":2021,"cited_by_count":4},{"year":2020,"cited_by_count":2},{"year":2019,"cited_by_count":1},{"year":2018,"cited_by_count":3},{"year":2017,"cited_by_count":3},{"year":2016,"cited_by_count":3},{"year":2015,"cited_by_count":2},{"year":2014,"cited_by_count":1},{"year":2013,"cited_by_count":7},{"year":2012,"cited_by_count":3}],"updated_date":"2026-06-11T09:08:48.828518","created_date":"2025-10-10T00:00:00"}