{"id":"https://openalex.org/W7155447180","doi":"https://doi.org/10.48550/arxiv.2604.20042","title":"On Threshold Compatibility Graphs","display_name":"On Threshold Compatibility Graphs","publication_year":2026,"publication_date":"2026-04-21","ids":{"openalex":"https://openalex.org/W7155447180","doi":"https://doi.org/10.48550/arxiv.2604.20042"},"language":null,"primary_location":{"id":"doi:10.48550/arxiv.2604.20042","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2604.20042","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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"article"},"type":"preprint","indexed_in":["datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://doi.org/10.48550/arxiv.2604.20042","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5032006903","display_name":"Sheikh Azizul Hakim","orcid":"https://orcid.org/0000-0002-3405-2402"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Hakim, Sheikh Azizul","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5063693993","display_name":"Md. Shamsuzzoha Bayzid","orcid":"https://orcid.org/0000-0002-5640-0615"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Bayzid, Md. Shamsuzzoha","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":2,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":false,"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":0.46369999647140503,"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":0.46369999647140503,"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/T12541","display_name":"Graph Labeling and Dimension Problems","score":0.20550000667572021,"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/T13664","display_name":"Genome Rearrangement Algorithms","score":0.0860000029206276,"subfield":{"id":"https://openalex.org/subfields/1311","display_name":"Genetics"},"field":{"id":"https://openalex.org/fields/13","display_name":"Biochemistry, Genetics and Molecular Biology"},"domain":{"id":"https://openalex.org/domains/1","display_name":"Life Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/expressive-power","display_name":"Expressive power","score":0.6614000201225281},{"id":"https://openalex.org/keywords/pairwise-comparison","display_name":"Pairwise comparison","score":0.5766000151634216},{"id":"https://openalex.org/keywords/constructive","display_name":"Constructive","score":0.5515999794006348},{"id":"https://openalex.org/keywords/vertex","display_name":"Vertex (graph theory)","score":0.41990000009536743},{"id":"https://openalex.org/keywords/graph","display_name":"Graph","score":0.37700000405311584},{"id":"https://openalex.org/keywords/comparability-graph","display_name":"Comparability graph","score":0.36800000071525574},{"id":"https://openalex.org/keywords/adjacency-list","display_name":"Adjacency list","score":0.3474999964237213},{"id":"https://openalex.org/keywords/chordal-graph","display_name":"Chordal graph","score":0.2881999909877777}],"concepts":[{"id":"https://openalex.org/C195818886","wikidata":"https://www.wikidata.org/wiki/Q5421724","display_name":"Expressive power","level":2,"score":0.6614000201225281},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6488999724388123},{"id":"https://openalex.org/C184898388","wikidata":"https://www.wikidata.org/wiki/Q1435712","display_name":"Pairwise comparison","level":2,"score":0.5766000151634216},{"id":"https://openalex.org/C2778701210","wikidata":"https://www.wikidata.org/wiki/Q28130034","display_name":"Constructive","level":3,"score":0.5515999794006348},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.5236999988555908},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.5209000110626221},{"id":"https://openalex.org/C80899671","wikidata":"https://www.wikidata.org/wiki/Q1304193","display_name":"Vertex (graph theory)","level":3,"score":0.41990000009536743},{"id":"https://openalex.org/C132525143","wikidata":"https://www.wikidata.org/wiki/Q141488","display_name":"Graph","level":2,"score":0.37700000405311584},{"id":"https://openalex.org/C198414033","wikidata":"https://www.wikidata.org/wiki/Q5155607","display_name":"Comparability graph","level":5,"score":0.36800000071525574},{"id":"https://openalex.org/C110484373","wikidata":"https://www.wikidata.org/wiki/Q264398","display_name":"Adjacency list","level":2,"score":0.3474999964237213},{"id":"https://openalex.org/C160446614","wikidata":"https://www.wikidata.org/wiki/Q1322892","display_name":"Chordal graph","level":3,"score":0.2881999909877777},{"id":"https://openalex.org/C203776342","wikidata":"https://www.wikidata.org/wiki/Q1378376","display_name":"Line graph","level":3,"score":0.28349998593330383},{"id":"https://openalex.org/C180356752","wikidata":"https://www.wikidata.org/wiki/Q727035","display_name":"Adjacency matrix","level":3,"score":0.27570000290870667},{"id":"https://openalex.org/C54540088","wikidata":"https://www.wikidata.org/wiki/Q3498041","display_name":"Intersection graph","level":4,"score":0.27300000190734863},{"id":"https://openalex.org/C31170391","wikidata":"https://www.wikidata.org/wiki/Q188619","display_name":"Hierarchy","level":2,"score":0.266400009393692},{"id":"https://openalex.org/C146380142","wikidata":"https://www.wikidata.org/wiki/Q1137726","display_name":"Directed graph","level":2,"score":0.25760000944137573},{"id":"https://openalex.org/C170903109","wikidata":"https://www.wikidata.org/wiki/Q7844666","display_name":"Trivially perfect graph","level":5,"score":0.25589999556541443},{"id":"https://openalex.org/C64339825","wikidata":"https://www.wikidata.org/wiki/Q722659","display_name":"Graph property","level":5,"score":0.2558000087738037},{"id":"https://openalex.org/C67810366","wikidata":"https://www.wikidata.org/wiki/Q835942","display_name":"Interval graph","level":5,"score":0.2515999972820282}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.48550/arxiv.2604.20042","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2604.20042","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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":null,"raw_source_name":null,"raw_type":"article"}],"best_oa_location":{"id":"doi:10.48550/arxiv.2604.20042","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2604.20042","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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"article"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":0,"referenced_works":[],"related_works":[],"abstract_inverted_index":{"Pairwise":[0],"Compatibility":[1],"Graphs":[2],"(PCGs)":[3],"form":[4],"a":[5,49,101,133,144],"tree-metric":[6],"graph":[7,20,96],"class":[8,123],"that":[9,52,94,116],"originated":[10],"in":[11,19,28],"phylogeny":[12],"and":[13,41,85,160,176],"has":[14],"since":[15],"attracted":[16],"sustained":[17],"interest":[18],"theory.":[21],"Several":[22],"natural":[23],"generalizations":[24],"have":[25],"been":[26],"proposed":[27],"order":[29],"to":[30,149],"overcome":[31],"the":[32,71,76,89,111,122,166,170,173,177],"expressive":[33,77,146],"limitations":[34],"of":[35,79,124,172,179],"classical":[36],"PCGs,":[37],"including":[38,143,169],"$k$-interval-PCGs,":[39],"$k$-OR-PCGs,":[40],"$k$-AND-PCGs.":[42],"In":[43],"this":[44,80],"paper,":[45],"we":[46,92,114,135],"introduce":[47],"$(k,t)$-threshold-PCGs,":[48],"threshold-based":[50],"framework":[51],"unifies":[53],"these":[54],"generalized":[55],"notions:":[56],"adjacency":[57],"is":[58,100,126],"determined":[59],"by":[60],"whether":[61],"at":[62],"least":[63],"$t$":[64],"among":[65,129],"$k$":[66],"underlying":[67],"PCG":[68],"predicates":[69],"accept":[70],"vertex":[72],"pair.":[73],"We":[74,151],"investigate":[75],"power":[78],"model":[81],"from":[82,139],"both":[83],"constructive":[84],"asymptotic":[86],"viewpoints.":[87],"On":[88,110],"positive":[90],"side,":[91,113],"show":[93],"every":[95,104,118],"on":[97],"$n$":[98],"vertices":[99],"$(n,t)$-threshold-PCG":[102],"for":[103,117,165],"$1":[105],"\\le":[106,108],"t":[107],"n$.":[109],"negative":[112],"prove":[115],"fixed":[119],"pair":[120],"$(k,t)$,":[121],"$(k,t)$-threshold-PCGs":[125],"asymptotically":[127],"rare":[128],"all":[130],"graphs.":[131],"As":[132],"consequence,":[134],"obtain":[136],"sharp":[137],"separations":[138],"previously":[140],"studied":[141],"models,":[142],"strict":[145],"gap":[147],"relative":[148],"$k$-interval-PCGs.":[150],"also":[152],"study":[153],"explicit":[154],"obstruction":[155],"families":[156],"through":[157],"incidence":[158],"graphs":[159],"derive":[161],"additional":[162],"structural":[163],"consequences":[164],"conjunction":[167],"case,":[168],"strictness":[171],"$k$-AND-PCG":[174],"hierarchy":[175],"failure":[178],"closure":[180],"under":[181],"complement.":[182]},"counts_by_year":[],"updated_date":"2026-06-11T09:08:48.828518","created_date":"2026-04-24T00:00:00"}
