{"id":"https://openalex.org/W7220958060","doi":"https://doi.org/10.48550/arxiv.2610.09926","title":"Minimal examples of Cohen-Macaulay independence complexes that fail to be shellable or vertex decomposable","display_name":"Minimal examples of Cohen-Macaulay independence complexes that fail to be shellable or vertex decomposable","publication_year":2026,"publication_date":"2026-10-07","ids":{"openalex":"https://openalex.org/W7220958060","doi":"https://doi.org/10.48550/arxiv.2610.09926"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2610.09926","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.09926","pdf_url":"https://arxiv.org/pdf/2610.09926","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"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"},"volume":null,"type":"preprint","is_traditional":true,"indexed_in":["arxiv","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2610.09926","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5025446621","display_name":"Adam Van Tuyl","orcid":"https://orcid.org/0000-0002-6799-6653","observed_orcids":["https://orcid.org/0000-0002-6799-6653"]},"institutions":[{"id":"https://openalex.org/I98251732","display_name":"McMaster University","ror":"https://ror.org/02fa3aq29","country_code":"CA","type":"education","lineage":["https://openalex.org/I98251732"]}],"countries":["CA"],"is_corresponding":true,"raw_author_name":"Van Tuyl, Adam","raw_affiliation_strings":["Department of Mathematics and Statistics McMaster University , Hamilton , ON L8S 4L8 , Canada"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics and Statistics McMaster University , Hamilton , ON L8S 4L8 , Canada","institution_ids":["https://openalex.org/I98251732"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5025446621"],"corresponding_institution_ids":["https://openalex.org/I98251732"],"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/T12069","display_name":"Commutative Algebra and Its Applications","score":0.9955999851226807,"subfield":{"id":"https://openalex.org/subfields/2602","display_name":"Algebra and Number Theory"},"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/T12069","display_name":"Commutative Algebra and Its Applications","score":0.9955999851226807,"subfield":{"id":"https://openalex.org/subfields/2602","display_name":"Algebra and Number Theory"},"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/T10948","display_name":"Advanced Combinatorial Mathematics","score":0.002099999925121665,"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":9.999999747378752e-05,"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/independence-complexes","display_name":"independence complexes","score":0.9430000185966492},{"id":"https://openalex.org/keywords/cohen-macaulay-complexes","display_name":"Cohen-Macaulay complexes","score":0.9860000014305115},{"id":"https://openalex.org/keywords/shellability","display_name":"shellability","score":0.9700000286102295},{"id":"https://openalex.org/keywords/simplicial-complex","display_name":"simplicial complexes","score":0.8730000257492065},{"id":"https://openalex.org/keywords/connected-graphs","display_name":"connected graphs","score":0.6620000004768372},{"id":"https://openalex.org/keywords/graph-enumeration","display_name":"graph enumeration","score":0.6470000147819519},{"id":"https://openalex.org/keywords/minimal-counterexample","display_name":"minimal counterexample","score":0.5820000171661377}],"concepts":[{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.819100022315979},{"id":"https://openalex.org/C80899671","wikidata":"https://www.wikidata.org/wiki/Q1304193","display_name":"Vertex (graph theory)","level":3,"score":0.7882000207901001},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6409000158309937},{"id":"https://openalex.org/C35651441","wikidata":"https://www.wikidata.org/wiki/Q625303","display_name":"Independence (probability theory)","level":2,"score":0.39899998903274536},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.39800000190734863},{"id":"https://openalex.org/C132525143","wikidata":"https://www.wikidata.org/wiki/Q141488","display_name":"Graph","level":2,"score":0.39010000228881836},{"id":"https://openalex.org/C2984853995","wikidata":"https://www.wikidata.org/wiki/Q1060343","display_name":"Independence number","level":3,"score":0.37049999833106995},{"id":"https://openalex.org/C160446614","wikidata":"https://www.wikidata.org/wiki/Q1322892","display_name":"Chordal graph","level":3,"score":0.32739999890327454},{"id":"https://openalex.org/C88230418","wikidata":"https://www.wikidata.org/wiki/Q131476","display_name":"Graph theory","level":2,"score":0.31949999928474426},{"id":"https://openalex.org/C161677786","wikidata":"https://www.wikidata.org/wiki/Q2478475","display_name":"Neighbourhood (mathematics)","level":2,"score":0.30790001153945923},{"id":"https://openalex.org/C18359143","wikidata":"https://www.wikidata.org/wiki/Q7888149","display_name":"Maximal independent set","level":5,"score":0.27790001034736633}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2610.09926","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.09926","pdf_url":"https://arxiv.org/pdf/2610.09926","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"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.2610.09926","doi":"https://doi.org/10.48550/arxiv.2610.09926","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2610.09926","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"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":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2610.09926","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.09926","pdf_url":"https://arxiv.org/pdf/2610.09926","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"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":[],"study_designs":[],"awards":[],"funders":[{"id":"https://openalex.org/F4320331257","display_name":"Alliance de recherche num\u00e9rique du Canada","ror":"https://ror.org/010r6td27"},{"id":"https://openalex.org/F4320334593","display_name":"Natural Sciences and Engineering Research Council of Canada","ror":"https://ror.org/01h531d29"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7220958060.pdf","grobid_xml":"https://content.openalex.org/works/W7220958060.grobid-xml"},"referenced_works_count":22,"referenced_works":["https://openalex.org/W1545519674","https://openalex.org/W1849958228","https://openalex.org/W1969151603","https://openalex.org/W1990600049","https://openalex.org/W1992807466","https://openalex.org/W2020703424","https://openalex.org/W2024478699","https://openalex.org/W2052882152","https://openalex.org/W2065019132","https://openalex.org/W2089328508","https://openalex.org/W2095895362","https://openalex.org/W2096737814","https://openalex.org/W2118624390","https://openalex.org/W2126402198","https://openalex.org/W2243778077","https://openalex.org/W2261931716","https://openalex.org/W2890117438","https://openalex.org/W2963187742","https://openalex.org/W2963898523","https://openalex.org/W3002957086","https://openalex.org/W4241015214","https://openalex.org/W7196923222"],"related_works":["https://openalex.org/W2095895362","https://openalex.org/W2261931716","https://openalex.org/W2269982198","https://openalex.org/W1728868014","https://openalex.org/W2166676844","https://openalex.org/W7207048043","https://openalex.org/W1891351328","https://openalex.org/W4381611283","https://openalex.org/W3176795680","https://openalex.org/W2029836625"],"abstract_inverted_index":{"We":[0],"report":[1],"on":[2,11],"an":[3],"exhaustive":[4],"census":[5],"of":[6,19],"all":[7],"1,006,700,565":[8],"connected":[9],"graphs":[10,20,41,60],"$11$":[12],"vertices":[13],"to":[14],"find":[15],"the":[16,52],"minimal":[17],"examples":[18],"whose":[21],"independence":[22],"complexes":[23],"are":[24,38],"Cohen-Macaulay":[25,47],"but":[26,31,48,66],"not":[27,32,49,55,67],"shellable,":[28,50],"or":[29],"shellable":[30,65],"vertex":[33,68],"decomposable.":[34,69],"In":[35],"particular,":[36],"there":[37],"exactly":[39,58],"two":[40],"$G$":[42],"where":[43,61],"${\\rm":[44,62],"Ind}(G)$":[45,63],"is":[46,54,64],"when":[51],"characteristic":[53],"two,":[56],"and":[57],"six":[59]},"counts_by_year":[],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-10-09T00:00:00"}
