{"id":"https://openalex.org/W7128786530","doi":"https://doi.org/10.48550/arxiv.2602.11716","title":"Cycles of Well-Linked Sets II: an Elementary Bound for the Directed Grid Theorem","display_name":"Cycles of Well-Linked Sets II: an Elementary Bound for the Directed Grid Theorem","publication_year":2026,"publication_date":"2026-02-12","ids":{"openalex":"https://openalex.org/W7128786530","doi":"https://doi.org/10.48550/arxiv.2602.11716"},"language":null,"primary_location":{"id":"pmh:doi:10.48550/arxiv.2602.11716","is_oa":true,"landing_page_url":null,"pdf_url":null,"source":{"id":"https://openalex.org/S4406922384","display_name":"Open MIND","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","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":null,"any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5012019509","display_name":"Meike Hatzel","orcid":"https://orcid.org/0000-0003-3249-1169"},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Hatzel, Meike","raw_affiliation_strings":[],"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5125918791","display_name":"Stephan Kreutzer","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Kreutzer, Stephan","raw_affiliation_strings":[],"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5079427577","display_name":"Marcelo Garlet Milani","orcid":"https://orcid.org/0000-0001-8398-4751"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Milani, Marcelo Garlet","raw_affiliation_strings":[],"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5048273965","display_name":"Irene Muzi","orcid":"https://orcid.org/0000-0003-2410-6523"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Muzi, Irene","raw_affiliation_strings":[],"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":4,"corresponding_author_ids":["https://openalex.org/A5012019509"],"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":0.7064999938011169,"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":0.7064999938011169,"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":0.05739999935030937,"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":0.03929999843239784,"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/treewidth","display_name":"Treewidth","score":0.5925999879837036},{"id":"https://openalex.org/keywords/grid","display_name":"Grid","score":0.5376999974250793},{"id":"https://openalex.org/keywords/digraph","display_name":"Digraph","score":0.5332000255584717},{"id":"https://openalex.org/keywords/directed-graph","display_name":"Directed graph","score":0.5098000168800354},{"id":"https://openalex.org/keywords/upper-and-lower-bounds","display_name":"Upper and lower bounds","score":0.5098000168800354},{"id":"https://openalex.org/keywords/function","display_name":"Function (biology)","score":0.4805999994277954},{"id":"https://openalex.org/keywords/conjecture","display_name":"Conjecture","score":0.47920000553131104}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7398999929428101},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.6205999851226807},{"id":"https://openalex.org/C132569581","wikidata":"https://www.wikidata.org/wiki/Q5067368","display_name":"Treewidth","level":5,"score":0.5925999879837036},{"id":"https://openalex.org/C187691185","wikidata":"https://www.wikidata.org/wiki/Q2020720","display_name":"Grid","level":2,"score":0.5376999974250793},{"id":"https://openalex.org/C2779145032","wikidata":"https://www.wikidata.org/wiki/Q191494","display_name":"Digraph","level":2,"score":0.5332000255584717},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.5181999802589417},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.5098000168800354},{"id":"https://openalex.org/C146380142","wikidata":"https://www.wikidata.org/wiki/Q1137726","display_name":"Directed graph","level":2,"score":0.5098000168800354},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.4805999994277954},{"id":"https://openalex.org/C2780990831","wikidata":"https://www.wikidata.org/wiki/Q319141","display_name":"Conjecture","level":2,"score":0.47920000553131104},{"id":"https://openalex.org/C2777831296","wikidata":"https://www.wikidata.org/wiki/Q12518","display_name":"Tower","level":2,"score":0.460999995470047},{"id":"https://openalex.org/C74197172","wikidata":"https://www.wikidata.org/wiki/Q1195339","display_name":"Directed acyclic graph","level":2,"score":0.33180001378059387},{"id":"https://openalex.org/C204911207","wikidata":"https://www.wikidata.org/wiki/Q860609","display_name":"Generating function","level":2,"score":0.3199999928474426},{"id":"https://openalex.org/C2777212361","wikidata":"https://www.wikidata.org/wiki/Q5127848","display_name":"Class (philosophy)","level":2,"score":0.304500013589859},{"id":"https://openalex.org/C2780586882","wikidata":"https://www.wikidata.org/wiki/Q7520643","display_name":"Simple (philosophy)","level":2,"score":0.2946000099182129},{"id":"https://openalex.org/C70501317","wikidata":"https://www.wikidata.org/wiki/Q462095","display_name":"Tree decomposition","level":5,"score":0.29249998927116394},{"id":"https://openalex.org/C182306322","wikidata":"https://www.wikidata.org/wiki/Q1779371","display_name":"Order (exchange)","level":2,"score":0.2919999957084656},{"id":"https://openalex.org/C145580432","wikidata":"https://www.wikidata.org/wiki/Q6497118","display_name":"Lattice graph","level":5,"score":0.25060001015663147}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:doi:10.48550/arxiv.2602.11716","is_oa":true,"landing_page_url":null,"pdf_url":null,"source":{"id":"https://openalex.org/S4406922384","display_name":"Open MIND","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Article"},{"id":"doi:10.48550/arxiv.2602.11716","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2602.11716","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":"pmh:doi:10.48550/arxiv.2602.11716","is_oa":true,"landing_page_url":null,"pdf_url":null,"source":{"id":"https://openalex.org/S4406922384","display_name":"Open MIND","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","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":{"In":[0,99],"2015,":[1],"Kawarabayashi":[2],"and":[3,31,121],"Kreutzer":[4],"proved":[5],"the":[6,11,14,34,65,75,78,94,97,108,123,127],"Directed":[7,109],"Grid":[8,17,110],"Theorem":[9,18,111],"-":[10,22],"generalisation":[12],"of":[13,49,57,74,77,96,107,134,140,149,158],"well-known":[15],"Excluded":[16],"to":[19,130],"directed":[20,50,160],"graphs":[21],"confirming":[23],"a":[24,42,54,61,87,131,144,163,170],"conjecture":[25],"by":[26],"Reed,":[27],"Johnson,":[28],"Robertson,":[29],"Seymour":[30],"Thomas":[32],"from":[33],"mid-nineties.":[35],"The":[36],"theorem":[37],"states":[38],"that":[39,46,155],"there":[40],"is":[41,84,113,143],"function":[43,67,73,128],"$f$":[44,129],"such":[45],"every":[47],"digraph":[48,157],"treewidth":[51,161],"$f(k)$":[52],"contains":[53,162,169],"cylindrical":[55,172],"grid":[56,79],"order":[58],"$k$":[59],"as":[60],"butterfly":[62],"minor.":[63,80],"However,":[64],"given":[66],"grows":[68],"faster":[69],"than":[70,86],"any":[71,156],"non-elementary":[72],"size":[76,95],"More":[81],"precisely,":[82],"it":[83],"larger":[85],"power":[88,132],"tower":[89,133],"whose":[90],"height":[91,135],"depends":[92],"on":[93],"grid.":[98,173],"this":[100],"paper,":[101],"we":[102],"present":[103],"an":[104],"alternative":[105],"proof":[106,142],"which":[112,166],"conceptually":[114],"much":[115],"simpler,":[116],"more":[117],"modular":[118],"in":[119,167],"composition":[120],"improves":[122],"upper":[124],"bound":[125],"for":[126],"$22$.":[136],"A":[137],"key":[138],"concept":[139],"our":[141],"new":[145],"structure":[146],"called":[147],"cycles":[148],"well-linked":[150],"sets":[151],"(CWS).":[152],"We":[153],"show":[154],"large":[159,164,171],"CWS,":[165],"turn":[168]},"counts_by_year":[],"updated_date":"2026-04-04T16:13:02.066488","created_date":"2026-02-14T00:00:00"}