{"id":"https://openalex.org/W7126101409","doi":"https://doi.org/10.1016/j.tcs.2026.115789","title":"ALL ordinals are cop-robber ordinals","display_name":"ALL ordinals are cop-robber ordinals","publication_year":2026,"publication_date":"2026-01-29","ids":{"openalex":"https://openalex.org/W7126101409","doi":"https://doi.org/10.1016/j.tcs.2026.115789"},"language":"en","primary_location":{"id":"doi:10.1016/j.tcs.2026.115789","is_oa":true,"landing_page_url":"https://doi.org/10.1016/j.tcs.2026.115789","pdf_url":null,"source":{"id":"https://openalex.org/S90727058","display_name":"Theoretical Computer Science","issn_l":"0304-3975","issn":["0304-3975","1879-2294"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320990","host_organization_name":"Elsevier BV","host_organization_lineage":["https://openalex.org/P4310320990"],"host_organization_lineage_names":["Elsevier BV"],"type":"journal"},"license":"cc-by-nc-nd","license_id":"https://openalex.org/licenses/cc-by-nc-nd","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Theoretical Computer Science","raw_type":"journal-article"},"type":"article","indexed_in":["arxiv","crossref"],"open_access":{"is_oa":true,"oa_status":"hybrid","oa_url":"https://doi.org/10.1016/j.tcs.2026.115789","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5119648806","display_name":"JORGE CRUZ CHAPITAL","orcid":null},"institutions":[{"id":"https://openalex.org/I185261750","display_name":"University of Toronto","ror":"https://ror.org/03dbr7087","country_code":"CA","type":"education","lineage":["https://openalex.org/I185261750"]}],"countries":["CA"],"is_corresponding":false,"raw_author_name":"Jorge Cruz Chapital","raw_affiliation_strings":["Department of Mathematics, University of Toronto, Toronto, M5S 2E4, Canada"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, University of Toronto, Toronto, M5S 2E4, Canada","institution_ids":["https://openalex.org/I185261750"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5119648807","display_name":"TOM\u00c1\u0160 FL\u00cdDR","orcid":null},"institutions":[{"id":"https://openalex.org/I241749","display_name":"University of Cambridge","ror":"https://ror.org/013meh722","country_code":"GB","type":"education","lineage":["https://openalex.org/I241749"]}],"countries":["GB"],"is_corresponding":false,"raw_author_name":"Tom\u00e1\u0161 Fl\u00eddr","raw_affiliation_strings":["Peterhouse, University of Cambridge, CB2 1RD, UK"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Peterhouse, University of Cambridge, CB2 1RD, UK","institution_ids":["https://openalex.org/I241749"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5119648805","display_name":"Maria-Romina Ivan","orcid":null},"institutions":[{"id":"https://openalex.org/I97018004","display_name":"Stanford University","ror":"https://ror.org/00f54p054","country_code":"US","type":"education","lineage":["https://openalex.org/I97018004"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Maria-Romina Ivan","raw_affiliation_strings":["Department of Mathematics, Stanford University, 450 Jane Stanford Way, CA 94304, USA","Department of Pure Mathematics and Mathematical Statistics, Centre for Mathematical Sciences, Wilberforce Road, Cambridge, CB3 0WB, UK"],"raw_orcid":"https://orcid.org/0000-0003-0817-3777","affiliations":[{"raw_affiliation_string":"Department of Mathematics, Stanford University, 450 Jane Stanford Way, CA 94304, USA","institution_ids":["https://openalex.org/I97018004"]},{"raw_affiliation_string":"Department of Pure Mathematics and Mathematical Statistics, Centre for Mathematical Sciences, Wilberforce Road, Cambridge, CB3 0WB, UK","institution_ids":[]}]}],"institutions":[],"countries_distinct_count":3,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5119648805"],"corresponding_institution_ids":["https://openalex.org/I97018004"],"apc_list":{"value":2690,"currency":"USD","value_usd":2690},"apc_paid":{"value":2690,"currency":"USD","value_usd":2690},"fwci":0.0,"has_fulltext":false,"cited_by_count":0,"citation_normalized_percentile":{"value":0.22281614,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"1068","issue":null,"first_page":"115789","last_page":"115789"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10374","display_name":"Advanced Graph Theory Research","score":0.29510000348091125,"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":0.29510000348091125,"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/T11574","display_name":"Artificial Intelligence in Games","score":0.16419999301433563,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"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/T10991","display_name":"Game Theory and Voting Systems","score":0.14470000565052032,"subfield":{"id":"https://openalex.org/subfields/2002","display_name":"Economics and Econometrics"},"field":{"id":"https://openalex.org/fields/20","display_name":"Economics, Econometrics and Finance"},"domain":{"id":"https://openalex.org/domains/2","display_name":"Social Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/graph","display_name":"Graph","score":0.498199999332428},{"id":"https://openalex.org/keywords/integer","display_name":"Integer (computer science)","score":0.36320000886917114},{"id":"https://openalex.org/keywords/finite-graph","display_name":"Finite graph","score":0.3416000008583069},{"id":"https://openalex.org/keywords/finite-set","display_name":"Finite set","score":0.3391999900341034},{"id":"https://openalex.org/keywords/time-complexity","display_name":"Time complexity","score":0.3183000087738037}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6945000290870667},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.5776000022888184},{"id":"https://openalex.org/C132525143","wikidata":"https://www.wikidata.org/wiki/Q141488","display_name":"Graph","level":2,"score":0.498199999332428},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.4634000062942505},{"id":"https://openalex.org/C97137487","wikidata":"https://www.wikidata.org/wiki/Q729138","display_name":"Integer (computer science)","level":2,"score":0.36320000886917114},{"id":"https://openalex.org/C2994516854","wikidata":"https://www.wikidata.org/wiki/Q141488","display_name":"Finite graph","level":3,"score":0.3416000008583069},{"id":"https://openalex.org/C162392398","wikidata":"https://www.wikidata.org/wiki/Q272404","display_name":"Finite set","level":2,"score":0.3391999900341034},{"id":"https://openalex.org/C311688","wikidata":"https://www.wikidata.org/wiki/Q2393193","display_name":"Time complexity","level":2,"score":0.3183000087738037},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.28360000252723694},{"id":"https://openalex.org/C61797465","wikidata":"https://www.wikidata.org/wiki/Q1188986","display_name":"Term (time)","level":2,"score":0.2612000107765198},{"id":"https://openalex.org/C83833204","wikidata":"https://www.wikidata.org/wiki/Q5290322","display_name":"Domination analysis","level":4,"score":0.2533000111579895}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.1016/j.tcs.2026.115789","is_oa":true,"landing_page_url":"https://doi.org/10.1016/j.tcs.2026.115789","pdf_url":null,"source":{"id":"https://openalex.org/S90727058","display_name":"Theoretical Computer Science","issn_l":"0304-3975","issn":["0304-3975","1879-2294"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320990","host_organization_name":"Elsevier BV","host_organization_lineage":["https://openalex.org/P4310320990"],"host_organization_lineage_names":["Elsevier BV"],"type":"journal"},"license":"cc-by-nc-nd","license_id":"https://openalex.org/licenses/cc-by-nc-nd","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Theoretical Computer Science","raw_type":"journal-article"},{"id":"pmh:oai:arXiv.org:2509.04270","is_oa":true,"landing_page_url":"http://arxiv.org/abs/2509.04270","pdf_url":"https://arxiv.org/pdf/2509.04270","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"}],"best_oa_location":{"id":"doi:10.1016/j.tcs.2026.115789","is_oa":true,"landing_page_url":"https://doi.org/10.1016/j.tcs.2026.115789","pdf_url":null,"source":{"id":"https://openalex.org/S90727058","display_name":"Theoretical Computer Science","issn_l":"0304-3975","issn":["0304-3975","1879-2294"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320990","host_organization_name":"Elsevier BV","host_organization_lineage":["https://openalex.org/P4310320990"],"host_organization_lineage_names":["Elsevier BV"],"type":"journal"},"license":"cc-by-nc-nd","license_id":"https://openalex.org/licenses/cc-by-nc-nd","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Theoretical Computer Science","raw_type":"journal-article"},"sustainable_development_goals":[{"id":"https://metadata.un.org/sdg/16","display_name":"Peace, Justice and strong institutions","score":0.5589262247085571}],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":1,"referenced_works":["https://openalex.org/W4288740300"],"related_works":[],"abstract_inverted_index":{"The":[0,32],"game":[1,33,93],"of":[2,59,88,109,142,166,193],"cops":[3],"and":[4,20,153,196],"robbers,":[5],"played":[6],"on":[7,39],"a":[8,14,68,72,129,167,191],"fixed":[9],"graph":[10,75],"G":[11,66,76,110],",":[12],"is":[13,67,103,123,145],"two-player":[15],"game,":[16],"where":[17],"the":[18,21,36,40,49,53,60,81,89,92,105,124,140,155,162],"cop":[19,37,50,54],"robber":[22],"(the":[23],"players)":[24],"take":[25],"turns":[26],"in":[27,96],"moving":[28],"to":[29],"adjacent":[30],"vertices.":[31],"finishes":[34],"if":[35,148],"lands":[38],"robber\u2019s":[41],"vertex.":[42],"In":[43,137,176],"that":[44,48,65,119,185],"case":[45],"we":[46,63,77,117,179],"say":[47,64],"wins.":[51],"If":[52],"can":[55,78,160],"always":[56],"win,":[57],"regardless":[58,87],"starting":[61,90],"positions,":[62,91],"cop-win":[69,74,130,135,168],"graph.":[70,131],"For":[71],"finite":[73,115],"ask":[79],"for":[80,128],"minimum":[82],"number":[83,102],"n":[84,99],"such":[85],"that,":[86],"will":[94],"end":[95],"at":[97,114],"most":[98],"steps.":[100],"This":[101],"called":[104,173],"maximum":[106,125,163],"capture":[107,126,143,164],"time":[108,127,144,165],".":[111],"By":[112],"looking":[113],"paths,":[116],"see":[118],"any":[120],"non-negative":[121],"integer":[122],"What":[132],"about":[133],"infinite":[134],"graphs?":[136],"this":[138,177,182],"case,":[139],"notion":[141],"nicely":[146],"generalised":[147],"one":[149],"works":[150],"with":[151],"ordinals,":[152],"so":[154],"question":[156,192],"becomes":[157],"which":[158],"ordinals":[159,171,187],"be":[161],"graph?":[169],"These":[170],"are":[172,188],"CR":[174],"(Cop-Robber)-ordinals.":[175],"paper":[178],"fully":[180],"settle":[181],"by":[183],"showing":[184],"all":[186],"CR-ordinals,":[189],"answering":[190],"Bonato,":[194],"Gordinowicz":[195],"Hahn.":[197]},"counts_by_year":[],"updated_date":"2026-02-04T23:10:29.248076","created_date":"2026-01-30T00:00:00"}
