{"id":"https://openalex.org/W4321854282","doi":"https://doi.org/10.48550/arxiv.2302.11862","title":"Bounds on Functionality and Symmetric Difference -- Two Intriguing Graph Parameters","display_name":"Bounds on Functionality and Symmetric Difference -- Two Intriguing Graph Parameters","publication_year":2023,"publication_date":"2023-02-23","ids":{"openalex":"https://openalex.org/W4321854282","doi":"https://doi.org/10.48550/arxiv.2302.11862"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2302.11862","is_oa":true,"landing_page_url":"http://arxiv.org/abs/2302.11862","pdf_url":"https://arxiv.org/pdf/2302.11862","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"},"type":"preprint","indexed_in":["arxiv","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2302.11862","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5084130815","display_name":"Pavel Dvo\u0159\u00e1k","orcid":"https://orcid.org/0000-0002-6838-1538"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Dvo\u0159\u00e1k, Pavel","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5080455350","display_name":"Luk\u00e1\u0161 Folwarczn\u00fd","orcid":"https://orcid.org/0000-0002-3020-6443"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Folwarczn\u00fd, Luk\u00e1\u0161","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5006649123","display_name":"Michal Opler","orcid":"https://orcid.org/0000-0002-4389-5807"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Opler, Michal","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5109848440","display_name":"Pavel Pudl\u00e1k","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Pudl\u00e1k, Pavel","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5081562141","display_name":"Robert \u0160\u00e1mal","orcid":"https://orcid.org/0000-0002-0172-6511"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"\u0160\u00e1mal, Robert","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5033663078","display_name":"Tung Anh Vu","orcid":"https://orcid.org/0000-0002-8902-5196"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Vu, Tung Anh","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":6,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":null,"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10374","display_name":"Advanced Graph Theory Research","score":0.9988999962806702,"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":0.9988999962806702,"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/T12292","display_name":"Graph Theory and Algorithms","score":0.9858999848365784,"subfield":{"id":"https://openalex.org/subfields/1707","display_name":"Computer Vision and Pattern Recognition"},"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/T10064","display_name":"Complex Network Analysis Techniques","score":0.9846000075340271,"subfield":{"id":"https://openalex.org/subfields/3109","display_name":"Statistical and Nonlinear Physics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.8191453218460083},{"id":"https://openalex.org/keywords/omega","display_name":"Omega","score":0.5625715255737305},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.4627215266227722},{"id":"https://openalex.org/keywords/graph","display_name":"Graph","score":0.46199774742126465},{"id":"https://openalex.org/keywords/bounded-function","display_name":"Bounded function","score":0.45943376421928406},{"id":"https://openalex.org/keywords/vertex","display_name":"Vertex (graph theory)","score":0.44623711705207825},{"id":"https://openalex.org/keywords/upper-and-lower-bounds","display_name":"Upper and lower bounds","score":0.4278546869754791},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.33037203550338745},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.3284922242164612}],"concepts":[{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.8191453218460083},{"id":"https://openalex.org/C2779557605","wikidata":"https://www.wikidata.org/wiki/Q9890","display_name":"Omega","level":2,"score":0.5625715255737305},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.4627215266227722},{"id":"https://openalex.org/C132525143","wikidata":"https://www.wikidata.org/wiki/Q141488","display_name":"Graph","level":2,"score":0.46199774742126465},{"id":"https://openalex.org/C34388435","wikidata":"https://www.wikidata.org/wiki/Q2267362","display_name":"Bounded function","level":2,"score":0.45943376421928406},{"id":"https://openalex.org/C80899671","wikidata":"https://www.wikidata.org/wiki/Q1304193","display_name":"Vertex (graph theory)","level":3,"score":0.44623711705207825},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.4278546869754791},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.33037203550338745},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.3284922242164612},{"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/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2302.11862","is_oa":true,"landing_page_url":"http://arxiv.org/abs/2302.11862","pdf_url":"https://arxiv.org/pdf/2302.11862","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":"doi:10.48550/arxiv.2302.11862","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2302.11862","pdf_url":null,"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":null,"is_accepted":false,"is_published":null,"raw_source_name":null,"raw_type":"article"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2302.11862","is_oa":true,"landing_page_url":"http://arxiv.org/abs/2302.11862","pdf_url":"https://arxiv.org/pdf/2302.11862","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"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G2601637665","display_name":"Dynamics and Structure of Networks","funder_award_id":"810115","funder_id":"https://openalex.org/F4320320300","funder_display_name":"European Commission"},{"id":"https://openalex.org/G2722083710","display_name":null,"funder_award_id":"810115","funder_id":"https://openalex.org/F4320321006","funder_display_name":"Grantov\u00e1 Agentura \u010cesk\u00e9 Republiky"},{"id":"https://openalex.org/G3255793682","display_name":null,"funder_award_id":"24-12046S","funder_id":"https://openalex.org/F4320321006","funder_display_name":"Grantov\u00e1 Agentura \u010cesk\u00e9 Republiky"},{"id":"https://openalex.org/G5664323314","display_name":null,"funder_award_id":"19-27871X","funder_id":"https://openalex.org/F4320321006","funder_display_name":"Grantov\u00e1 Agentura \u010cesk\u00e9 Republiky"}],"funders":[{"id":"https://openalex.org/F4320320300","display_name":"European Commission","ror":"https://ror.org/00k4n6c32"},{"id":"https://openalex.org/F4320321006","display_name":"Grantov\u00e1 Agentura \u010cesk\u00e9 Republiky","ror":"https://ror.org/01pv73b02"}],"has_content":{"pdf":true,"grobid_xml":false},"content_urls":{"pdf":"https://content.openalex.org/works/W4321854282.pdf"},"referenced_works_count":0,"referenced_works":[],"related_works":["https://openalex.org/W238609530","https://openalex.org/W2734192830","https://openalex.org/W3016395638","https://openalex.org/W1636163329","https://openalex.org/W4292947165","https://openalex.org/W3045121788","https://openalex.org/W3201655533","https://openalex.org/W3134442314","https://openalex.org/W2281248943","https://openalex.org/W3102538728"],"abstract_inverted_index":{"Functionality":[0],"($\\mathrm{fun}$)":[1],"is":[2,79,183,232,260],"a":[3,53,61,107,136,239],"graph":[4,8,42,92,138,173,199,245],"parameter":[5,139],"that":[6,73,90,167,227,254],"generalizes":[7],"degeneracy":[9],"defined":[10],"by":[11,88,113,163],"Alecu":[12,164],"et":[13,165,187],"al.":[14,166,188],"[JCTB,":[15],"2021].":[16],"They":[17,175],"research":[18],"the":[19,38,74,129,143,155,178,211,217],"relation":[20],"of":[21,40,64,116,131,145,152,154,180,222,229,242,256],"functionality":[22,39,76,124,179,218],"to":[23,52],"many":[24],"other":[25],"graphs":[26,115,182,231,259],"parameters":[27],"(tree-width,":[28],"clique-width,":[29],"VC-dimension,":[30],"etc.).":[31],"Extending":[32],"their":[33],"research,":[34],"we":[35,105,134,237,252],"completely":[36],"characterize":[37],"random":[41],"$G(n,p)$":[43],"for":[44,60,71,81,123,171],"all":[45,150],"possible":[46],"$p$.":[47,65],"We":[48,85,225],"provide":[49,238],"matching":[50,109],"(up":[51],"constant":[54],"factor)":[55],"lower":[56,121],"and":[57,104,194,219,236],"upper":[58],"bound":[59,111],"large":[62],"range":[63],"It":[66,160],"follows":[67],"from":[68],"our":[69],"bounds":[70,122],"$G(n,p)$,":[72],"maximum":[75],"(roughly":[77],"$\\sqrt{n}$)":[78],"achieved":[80],"$p":[82],"\\approx":[83],"1/\\sqrt{n}$.":[84],"complement":[86],"this":[87,192],"showing":[89],"every":[91,172],"$G$":[93,200,246],"on":[94],"$n$":[95],"vertices":[96,153],"have":[97],"$\\mathrm{fun}(G)":[98,168],"\\le":[99,169],"O(\\sqrt{":[100],"n":[101],"\\ln":[102],"n})$":[103],"give":[106],"nearly":[108],"$\u03a9(\\sqrt{n})$-lower":[110],"provided":[112],"incident":[114],"projective":[117],"planes.":[118],"Previously":[119],"known":[120],"were":[125],"only":[126],"logarithmic":[127],"in":[128],"number":[130],"vertices.":[132],"Further,":[133,251],"study":[135],"related":[137],"symmetric":[140,220],"difference":[141,221],"($\\mathrm{sd}$),":[142],"minimum":[144],"$|N(u)":[146],"~\u0394~":[147],"N(v)|$":[148],"over":[149],"pairs":[151],"``worst":[156],"possible''":[157],"induced":[158],"subgraph.":[159],"was":[161],"observed":[162],"\\mathrm{sd}(G)+1$":[170],"$G$.":[174],"asked":[176],"whether":[177],"interval":[181,198,223,230,244],"bounded.":[184],"Recently,":[185],"Dallard":[186],"[RiM,":[189],"2024]":[190],"answered":[191],"positively":[193],"they":[195,207,215],"constructed":[196],"an":[197,243],"with":[201,247],"$\\mathrm{sd}(G)":[202,248],"=":[203,249],"\u0398(\\sqrt[4]{n})$":[204],"(even":[205],"though":[206],"did":[208],"not":[209],"mention":[210],"explicit":[212],"bound),":[213],"i.e.,":[214],"separate":[216],"graphs.":[224],"show":[226,253],"$\\mathrm{sd}$":[228,255],"at":[233],"most":[234],"$O(\\sqrt[3]{n})$":[235],"different":[240],"example":[241],"\u0398(\\sqrt[4]{n})$.":[250],"circular":[257],"arc":[258],"$\u0398(\\sqrt{n})$.":[261]},"counts_by_year":[],"updated_date":"2026-06-14T07:44:22.658603","created_date":"2025-10-10T00:00:00"}
