{"id":"https://openalex.org/W4404413650","doi":"https://doi.org/10.37236/11826","title":"Bootstrap Percolation in Strong Products of Graphs","display_name":"Bootstrap Percolation in Strong Products of Graphs","publication_year":2024,"publication_date":"2024-11-15","ids":{"openalex":"https://openalex.org/W4404413650","doi":"https://doi.org/10.37236/11826"},"language":"en","primary_location":{"id":"doi:10.37236/11826","is_oa":true,"landing_page_url":"http://dx.doi.org/10.37236/11826","pdf_url":null,"source":{"id":"https://openalex.org/S38448739","display_name":"The Electronic Journal of Combinatorics","issn_l":"1077-8926","issn":["1077-8926","1097-1440"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310317970","host_organization_name":"Electronic Journal of Combinatorics","host_organization_lineage":["https://openalex.org/P4310317970"],"host_organization_lineage_names":["Electronic Journal of Combinatorics"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"The Electronic Journal of Combinatorics","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"diamond","oa_url":"http://dx.doi.org/10.37236/11826","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5089443562","display_name":"Bo\u0161tjan Bre\u0161ar","orcid":"https://orcid.org/0000-0001-8471-4796"},"institutions":[{"id":"https://openalex.org/I37696226","display_name":"University of Maribor","ror":"https://ror.org/01d5jce07","country_code":"SI","type":"education","lineage":["https://openalex.org/I37696226"]}],"countries":["SI"],"is_corresponding":false,"raw_author_name":"Bo\u0161tjan Bre\u0161ar","raw_affiliation_strings":["University of Maribor,\r\nFaculty of Natural Sciences and Mathematics","University of Maribor,\nFaculty of Natural Sciences and Mathematics"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"University of Maribor,\r\nFaculty of Natural Sciences and Mathematics","institution_ids":["https://openalex.org/I37696226"]},{"raw_affiliation_string":"University of Maribor,\nFaculty of Natural Sciences and Mathematics","institution_ids":["https://openalex.org/I37696226"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5010057751","display_name":"Jaka Hed\u017eet","orcid":"https://orcid.org/0009-0009-5723-8603"},"institutions":[{"id":"https://openalex.org/I4210106342","display_name":"Institute of Mathematics, Physics, and Mechanics","ror":"https://ror.org/01eb3qa50","country_code":"SI","type":"facility","lineage":["https://openalex.org/I4210106342"]}],"countries":["SI"],"is_corresponding":false,"raw_author_name":"Jaka Hed\u017eet","raw_affiliation_strings":["Institute of Mathematics, Physics and Mechanics"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Institute of Mathematics, Physics and Mechanics","institution_ids":["https://openalex.org/I4210106342"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":0.2294,"has_fulltext":false,"cited_by_count":1,"citation_normalized_percentile":{"value":0.52797779,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":94,"max":97},"biblio":{"volume":"31","issue":"4","first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10064","display_name":"Complex Network Analysis Techniques","score":0.9884999990463257,"subfield":{"id":"https://openalex.org/subfields/3109","display_name":"Statistical and Nonlinear Physics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T10064","display_name":"Complex Network Analysis Techniques","score":0.9884999990463257,"subfield":{"id":"https://openalex.org/subfields/3109","display_name":"Statistical and Nonlinear Physics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11152","display_name":"Stochastic processes and statistical mechanics","score":0.9835000038146973,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"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/T11106","display_name":"Data Management and Algorithms","score":0.9815999865531921,"subfield":{"id":"https://openalex.org/subfields/1711","display_name":"Signal Processing"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/percolation","display_name":"Percolation (cognitive psychology)","score":0.6873217821121216},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.42892730236053467},{"id":"https://openalex.org/keywords/statistical-physics","display_name":"Statistical physics","score":0.3992652893066406},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.36346638202667236},{"id":"https://openalex.org/keywords/econometrics","display_name":"Econometrics","score":0.3452844023704529},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.22963657975196838},{"id":"https://openalex.org/keywords/psychology","display_name":"Psychology","score":0.17693182826042175}],"concepts":[{"id":"https://openalex.org/C2780457167","wikidata":"https://www.wikidata.org/wiki/Q17156484","display_name":"Percolation (cognitive psychology)","level":2,"score":0.6873217821121216},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.42892730236053467},{"id":"https://openalex.org/C121864883","wikidata":"https://www.wikidata.org/wiki/Q677916","display_name":"Statistical physics","level":1,"score":0.3992652893066406},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.36346638202667236},{"id":"https://openalex.org/C149782125","wikidata":"https://www.wikidata.org/wiki/Q160039","display_name":"Econometrics","level":1,"score":0.3452844023704529},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.22963657975196838},{"id":"https://openalex.org/C15744967","wikidata":"https://www.wikidata.org/wiki/Q9418","display_name":"Psychology","level":0,"score":0.17693182826042175},{"id":"https://openalex.org/C169760540","wikidata":"https://www.wikidata.org/wiki/Q207011","display_name":"Neuroscience","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.37236/11826","is_oa":true,"landing_page_url":"http://dx.doi.org/10.37236/11826","pdf_url":null,"source":{"id":"https://openalex.org/S38448739","display_name":"The Electronic Journal of Combinatorics","issn_l":"1077-8926","issn":["1077-8926","1097-1440"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310317970","host_organization_name":"Electronic Journal of Combinatorics","host_organization_lineage":["https://openalex.org/P4310317970"],"host_organization_lineage_names":["Electronic Journal of Combinatorics"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"The Electronic Journal of Combinatorics","raw_type":"journal-article"}],"best_oa_location":{"id":"doi:10.37236/11826","is_oa":true,"landing_page_url":"http://dx.doi.org/10.37236/11826","pdf_url":null,"source":{"id":"https://openalex.org/S38448739","display_name":"The Electronic Journal of Combinatorics","issn_l":"1077-8926","issn":["1077-8926","1097-1440"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310317970","host_organization_name":"Electronic Journal of Combinatorics","host_organization_lineage":["https://openalex.org/P4310317970"],"host_organization_lineage_names":["Electronic Journal of Combinatorics"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"The Electronic Journal of Combinatorics","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":0,"referenced_works":[],"related_works":["https://openalex.org/W4391375266","https://openalex.org/W1979597421","https://openalex.org/W2007980826","https://openalex.org/W2061531152","https://openalex.org/W3002753104","https://openalex.org/W2077600819","https://openalex.org/W2142036596","https://openalex.org/W3102564941","https://openalex.org/W2949171345","https://openalex.org/W1966598944"],"abstract_inverted_index":{"Given":[0],"a":[1,43,107,111,175],"graph":[2],"$G$":[3,10,37,50,64,81,200],"and":[4,103,171,201,230],"assuming":[5],"that":[6,52,96,159,188],"some":[7],"vertices":[8,48,156],"of":[9,36,42,45,63,75,78,90,113,116,133,181,184,222,236,239],"are":[11,192],"infected,":[12],"the":[13,39,56,83,121,211,223],"$r$-neighbor":[14,57],"bootstrap":[15,58],"percolation":[16,59,73,234],"rule":[17,60],"makes":[18],"an":[19],"uninfected":[20],"vertex":[21,62],"$v$":[22,25],"infected":[23,30,47],"if":[24,198],"has":[26],"at":[27,137,142,153,204],"least":[28,138,143,154,205],"$r$":[29],"neighbors.":[31],"The":[32],"$r$-percolation":[33],"number,":[34],"$m(G,r)$,":[35],"is":[38,82],"minimum":[40],"cardinality":[41],"set":[44],"initially":[46],"in":[49,135],"such":[51,187],"after":[53],"continuously":[54],"performing":[55],"each":[61],"eventually":[65],"becomes":[66],"infected.":[67],"In":[68,146],"this":[69,218],"paper,":[70],"we":[71,94,109,157,172,208,231],"consider":[72,233],"numbers":[74,191,235],"strong":[76,84,114,131,182,212,237],"products":[77,115,132,183,238],"graphs.":[79],"If":[80],"product":[85],"$G_1\\boxtimes":[86],"\\cdots":[87],"\\boxtimes":[88,161,163],"G_k$":[89],"$k$":[91,117,185],"connected":[92,118],"graphs,":[93,229],"prove":[95,158],"$m(G,r)=r$":[97],"as":[98,100],"soon":[99],"$r\\le":[101],"2^{k-1}$":[102],"$|V(G)|\\ge":[104],"r$.":[105],"As":[106],"dichotomy,":[108,176],"present":[110],"family":[112],"graphs":[119,134,186],"with":[120],"$(2^{k-1}+1)$-percolation":[122],"number":[123],"arbitrarily":[124,193],"large.":[125,194],"We":[126],"refine":[127],"these":[128],"results":[129,224],"for":[130,167,216],"which":[136,217],"two":[139],"factors":[140,150],"have":[141,152,203],"three":[144,155,206],"vertices.":[145],"addition,":[147],"when":[148],"all":[149,168],"$G_i$":[151],"$m(G_1":[160],"\\dots":[162],"G_k,r)\\leq":[164],"3^{k-1}":[165],"-k$":[166],"$r\\leq":[169],"2^k-1$,":[170],"again":[173],"get":[174],"since":[177],"there":[178],"exist":[179],"families":[180],"their":[189],"$2^{k}$-percolation":[190],"While":[195],"$m(G\\boxtimes":[196],"H,3)=3$":[197],"both":[199],"$H$":[202],"vertices,":[207],"also":[209],"characterize":[210],"prisms":[213],"$G\\boxtimes":[214],"K_2$":[215],"equality":[219],"holds.":[220],"Some":[221],"naturally":[225],"extend":[226],"to":[227],"infinite":[228,241],"briefly":[232],"two-way":[240],"paths.":[242]},"counts_by_year":[{"year":2026,"cited_by_count":1}],"updated_date":"2026-05-21T06:26:12.895304","created_date":"2025-10-10T00:00:00"}