{"id":"https://openalex.org/W7225120465","doi":"https://doi.org/10.5281/zenodo.23889801","title":"The inverse problem for the number of connected induced subgraphs","display_name":"The inverse problem for the number of connected induced subgraphs","publication_year":2026,"publication_date":"2026-11-02","ids":{"openalex":"https://openalex.org/W7225120465","doi":"https://doi.org/10.5281/zenodo.23889801"},"language":"en","primary_location":{"id":"doi:10.5281/zenodo.23889801","doi":"https://doi.org/10.5281/zenodo.23889801","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.23889801","pdf_url":null,"source":{"id":"https://openalex.org/S4306400562","display_name":"Zenodo (CERN European Organization for Nuclear Research)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I67311998","host_organization_name":"European Organization for Nuclear Research","host_organization_lineage":["https://openalex.org/I67311998"],"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"},"volume":null,"type":"preprint","is_traditional":true,"indexed_in":["datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://doi.org/10.5281/zenodo.23889801","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5159079366","display_name":"Lucas Alves de Almeida Lucas Alves de Almeida","orcid":"https://orcid.org/0009-0009-5606-7291","observed_orcids":["https://orcid.org/0009-0009-5606-7291"]},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Lucas Alves de Almeida","raw_affiliation_strings":["Independent researcher"],"raw_orcid":"https://orcid.org/0009-0009-5606-7291","affiliations":[{"raw_affiliation_string":"Independent researcher","institution_ids":[]}]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":["https://openalex.org/A5159079366"],"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":0.7103999853134155,"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":0.7103999853134155,"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/T10948","display_name":"Advanced Combinatorial Mathematics","score":0.1256999969482422,"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":0.1234000027179718,"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/inverse-problems","display_name":"inverse problems","score":0.7429999709129333},{"id":"https://openalex.org/keywords/connected-subgraphs","display_name":"connected subgraphs","score":0.5830000042915344},{"id":"https://openalex.org/keywords/graph-enumeration","display_name":"graph enumeration","score":0.6349999904632568},{"id":"https://openalex.org/keywords/finite-graphs","display_name":"finite graphs","score":0.4869999885559082},{"id":"https://openalex.org/keywords/cones","display_name":"cones","score":0.6069999933242798},{"id":"https://openalex.org/keywords/lean-4-formalization","display_name":"Lean 4 formalization","score":0.7519999742507935},{"id":"https://openalex.org/keywords/continuum-theory","display_name":"continuum theory","score":0.6949999928474426},{"id":"https://openalex.org/keywords/subtrees","display_name":"subtrees","score":0.49799999594688416},{"id":"https://openalex.org/keywords/connected-sets","display_name":"connected sets","score":0.5230000019073486},{"id":"https://openalex.org/keywords/path-graphs","display_name":"path graphs","score":0.40799999237060547},{"id":"https://openalex.org/keywords/claude-opus","display_name":"Claude Opus","score":0.8360000252723694}],"concepts":[{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.7735000252723694},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7530999779701233},{"id":"https://openalex.org/C45340560","wikidata":"https://www.wikidata.org/wiki/Q215382","display_name":"Disjoint sets","level":2,"score":0.6682000160217285},{"id":"https://openalex.org/C76444178","wikidata":"https://www.wikidata.org/wiki/Q72897900","display_name":"Connectivity","level":3,"score":0.6003000140190125},{"id":"https://openalex.org/C193435613","wikidata":"https://www.wikidata.org/wiki/Q2997928","display_name":"Connected component","level":2,"score":0.593500018119812},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.5651999711990356},{"id":"https://openalex.org/C80899671","wikidata":"https://www.wikidata.org/wiki/Q1304193","display_name":"Vertex (graph theory)","level":3,"score":0.5060999989509583},{"id":"https://openalex.org/C38754835","wikidata":"https://www.wikidata.org/wiki/Q2003238","display_name":"Strongly connected component","level":2,"score":0.47999998927116394},{"id":"https://openalex.org/C132525143","wikidata":"https://www.wikidata.org/wiki/Q141488","display_name":"Graph","level":2,"score":0.4767000079154968},{"id":"https://openalex.org/C108710211","wikidata":"https://www.wikidata.org/wiki/Q11538","display_name":"Mathematical proof","level":2,"score":0.47029998898506165},{"id":"https://openalex.org/C207467116","wikidata":"https://www.wikidata.org/wiki/Q4385666","display_name":"Inverse","level":2,"score":0.4300000071525574},{"id":"https://openalex.org/C97137487","wikidata":"https://www.wikidata.org/wiki/Q729138","display_name":"Integer (computer science)","level":2,"score":0.4171999990940094},{"id":"https://openalex.org/C311688","wikidata":"https://www.wikidata.org/wiki/Q2393193","display_name":"Time complexity","level":2,"score":0.35100001096725464},{"id":"https://openalex.org/C162392398","wikidata":"https://www.wikidata.org/wiki/Q272404","display_name":"Finite set","level":2,"score":0.34459999203681946},{"id":"https://openalex.org/C2777735758","wikidata":"https://www.wikidata.org/wiki/Q817765","display_name":"Path (computing)","level":2,"score":0.3424000144004822},{"id":"https://openalex.org/C2994516854","wikidata":"https://www.wikidata.org/wiki/Q141488","display_name":"Finite graph","level":3,"score":0.3395000100135803},{"id":"https://openalex.org/C167204820","wikidata":"https://www.wikidata.org/wiki/Q912058","display_name":"Simply connected space","level":2,"score":0.31619998812675476},{"id":"https://openalex.org/C203776342","wikidata":"https://www.wikidata.org/wiki/Q1378376","display_name":"Line graph","level":3,"score":0.28630000352859497},{"id":"https://openalex.org/C168291704","wikidata":"https://www.wikidata.org/wiki/Q902252","display_name":"Complement graph","level":5,"score":0.272599995136261},{"id":"https://openalex.org/C90119067","wikidata":"https://www.wikidata.org/wiki/Q43260","display_name":"Polynomial","level":2,"score":0.2653999924659729},{"id":"https://openalex.org/C2993105083","wikidata":"https://www.wikidata.org/wiki/Q141488","display_name":"Simple graph","level":3,"score":0.2648000121116638},{"id":"https://openalex.org/C94020503","wikidata":"https://www.wikidata.org/wiki/Q12916","display_name":"Real number","level":2,"score":0.26030001044273376}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.5281/zenodo.23889801","doi":"https://doi.org/10.5281/zenodo.23889801","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.23889801","pdf_url":null,"source":{"id":"https://openalex.org/S4306400562","display_name":"Zenodo (CERN European Organization for Nuclear Research)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I67311998","host_organization_name":"European Organization for Nuclear Research","host_organization_lineage":["https://openalex.org/I67311998"],"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"},{"id":"doi:10.5281/zenodo.23889802","doi":"https://doi.org/10.5281/zenodo.23889802","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.23889802","pdf_url":null,"source":{"id":"https://openalex.org/S4306400562","display_name":"Zenodo (CERN European Organization for Nuclear Research)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I67311998","host_organization_name":"European Organization for Nuclear Research","host_organization_lineage":["https://openalex.org/I67311998"],"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"},{"id":"doi:10.5281/zenodo.23899415","doi":"https://doi.org/10.5281/zenodo.23899415","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.23899415","pdf_url":null,"source":{"id":"https://openalex.org/S4306400562","display_name":"Zenodo (CERN European Organization for Nuclear Research)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I67311998","host_organization_name":"European Organization for Nuclear Research","host_organization_lineage":["https://openalex.org/I67311998"],"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":"doi:10.5281/zenodo.23889801","doi":"https://doi.org/10.5281/zenodo.23889801","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.23889801","pdf_url":null,"source":{"id":"https://openalex.org/S4306400562","display_name":"Zenodo (CERN European Organization for Nuclear Research)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I67311998","host_organization_name":"European Organization for Nuclear Research","host_organization_lineage":["https://openalex.org/I67311998"],"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"},"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":0,"referenced_works":[],"related_works":["https://openalex.org/W3048211114","https://openalex.org/W3044627388","https://openalex.org/W7214521400","https://openalex.org/W2969173579","https://openalex.org/W2894533445","https://openalex.org/W4415194136","https://openalex.org/W2891696629","https://openalex.org/W2131724314","https://openalex.org/W1771019176","https://openalex.org/W2952628886"],"abstract_inverted_index":{"For":[0],"a":[1,16,22,90,141,146,178,186,201,217,225,240,254,275,305],"finite":[2,30,276],"graph":[3,32,92,148,202],"G,":[4],"let":[5],"c(G)":[6,27,204],"be":[7],"the":[8,60,85,120,128,144,156,234,264,268,290,319,337,362,370,374,386,390,394],"number":[9,269],"of":[10,87,99,188,219,256,270,297,318,321],"nonempty":[11],"vertex":[12],"sets":[13,167],"that":[14,21,190,253],"induce":[15],"connected":[17,31,91,132,147,166,272],"subgraph.":[18],"We":[19,250,283],"show":[20],"positive":[23],"integer":[24,77],"k":[25,38,49,172,206],"equals":[26],"for":[28,71,116,267],"some":[29],"G":[33],"if":[34,37],"and":[35,64,104,115,127,185,230,259,278,324,332,336,344,346,355,372,392],"only":[36,111,328],"\u2209":[39],"{2,":[40],"4,":[41],"5,":[42],"8,":[43],"9,":[44],"16}.":[45],"Every":[46],"attainable":[47],"value":[48,86,137],"\u2260":[50],"25":[51,65],"is":[52,84,245],"attained":[53],"on":[54,89,122,130,149,261],"\u230alog2":[55],"k\u230b":[56],"+":[57],"1":[58,83],"vertices,":[59,126],"least":[61,161],"possible":[62],"number,":[63],"needs":[66],"six":[67,123],"vertices.":[68,96],"More":[69],"precisely,":[70],"every":[72],"m":[73,95],"\u2265":[74,151,173],"6":[75],"each":[76],"from":[78,181,195,216],"2m\u22121":[79],"to":[80,124,237,274,368,388],"2m":[81],"\u2212":[82,163],"c":[88],"with":[93,203,334,385],"exactly":[94],"The":[97,136,170,213,243,364],"proofs":[98],"these":[100,287],"results":[101,288,320],"are":[102,175],"short":[103],"use":[105],"no":[106,295,356],"computer;":[107],"computer":[108,241],"calculations":[109],"serve":[110],"as":[112],"independent":[113,357],"checks,":[114],"two":[117],"auxiliary":[118],"results:":[119],"values":[121,171,235],"eight":[125],"result":[129],"all":[131,271],"subgraphs":[133,273],"described":[134],"below.":[135],"16":[138],"falls":[139],"into":[140],"gap":[142],"above":[143],"path:":[145],"n":[150,162],"4":[152,307],"vertices":[153],"other":[154],"than":[155,168],"path":[157],"Pn":[158],"has":[159,359],"at":[160],"3":[164],"more":[165],"Pn.":[169],"17":[174],"realized":[176],"by":[177,227,239],"recursive":[179],"construction":[180],"cones,":[182],"disjoint":[183],"unions":[184],"family":[187],"graphs":[189],"we":[191,279,293],"call":[192],"near-cones,":[193],"starting":[194],"seven":[196],"small":[197],"graphs;":[198],"it":[199],"produces":[200],"=":[205],"in":[207,210,221,247,289],"time":[208],"polynomial":[209],"log":[211],"k.":[212],"question":[214],"arose":[215],"problem":[218,266,371],"Ordo\u00f1ez":[220,229],"continuum":[222],"theory,":[223],"through":[224],"reduction":[226],"Camargo,":[228],"Ram\u00edrez,":[231],"who":[232],"found":[233],"up":[236],"255":[238],"search.":[242],"proof":[244],"formalized":[246],"Lean":[248,306],"4.":[249],"also":[251],"observe":[252],"theorem":[255],"Czabarka,":[257],"Sz\u00e9kely":[258],"Wagner":[260],"subtrees":[262],"reduces":[263],"corresponding":[265],"computation,":[277],"report":[280],"its":[281],"outcome.":[282],"did":[284],"not":[285,351],"find":[286],"literature,":[291],"but":[292],"make":[294],"claim":[296],"novelty":[298],"or":[299],"priority.":[300],"Supplementary":[301],"material.":[302],"LaTeX":[303],"source;":[304],"/":[308],"Mathlib":[309,313],"formalization":[310],"(Lean":[311],"4.35.0,":[312],"commit":[314],"6015dbb,":[315],"library":[316],"ConnectedSets)":[317],"Sections":[322],"2\u20136":[323],"Lemma":[325],"7.2,":[326],"using":[327],"Lean's":[329],"standard":[330],"axioms":[331],"checked":[333],"leanchecker":[335],"external":[338],"type":[339],"checker":[340],"nanoda;":[341],"programs":[342],"(Python":[343],"C)":[345],"their":[347],"outputs.":[348],"Status.":[349],"Preprint;":[350],"yet":[352,360],"peer":[353],"reviewed,":[354],"expert":[358],"verified":[361],"proofs.":[363],"main":[365],"model":[366],"used":[367,384],"solve":[369],"prove":[373],"theorems":[375],"was":[376,383],"Claude":[377],"Opus":[378],"5.5":[379],"(Anthropic);":[380],"ChatGPT-Astra":[381],"(OpenAI)":[382],"author":[387],"read":[389],"manuscript":[391],"check":[393],"arguments.":[395],"See":[396],"AI_DISCLOSURE.md.":[397]},"counts_by_year":[],"updated_date":"2026-11-04T10:54:05.112534","created_date":"2026-11-03T00:00:00"}
