{"id":"https://openalex.org/W7133297008","doi":"https://doi.org/10.1007/s00373-026-03030-y","title":"Isolation Game on Graphs","display_name":"Isolation Game on Graphs","publication_year":2026,"publication_date":"2026-03-03","ids":{"openalex":"https://openalex.org/W7133297008","doi":"https://doi.org/10.1007/s00373-026-03030-y"},"language":"en","primary_location":{"id":"doi:10.1007/s00373-026-03030-y","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s00373-026-03030-y","pdf_url":"https://link.springer.com/content/pdf/10.1007/s00373-026-03030-y.pdf","source":{"id":"https://openalex.org/S186126824","display_name":"Graphs and Combinatorics","issn_l":"0911-0119","issn":["0911-0119","1435-5914"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Graphs and Combinatorics","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"hybrid","oa_url":"https://link.springer.com/content/pdf/10.1007/s00373-026-03030-y.pdf","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5127915867","display_name":"Bo\u0161tjan Bre\u0161ar","orcid":null},"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"]},{"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":"Bo\u0161tjan Bre\u0161ar","raw_affiliation_strings":["Faculty of Natural Sciences and Mathematics, University of Maribor, Europe, Slovenia","Institute of Mathematics, Physics and Mechanics, Europe, Slovenia"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Faculty of Natural Sciences and Mathematics, University of Maribor, Europe, Slovenia","institution_ids":["https://openalex.org/I37696226"]},{"raw_affiliation_string":"Institute of Mathematics, Physics and Mechanics, Europe, Slovenia","institution_ids":["https://openalex.org/I4210106342"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5044816279","display_name":"Tanja Dravec","orcid":"https://orcid.org/0000-0003-0540-0938"},"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"]},{"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":true,"raw_author_name":"Tanja Dravec","raw_affiliation_strings":["Faculty of Natural Sciences and Mathematics, University of Maribor, Europe, Slovenia","Institute of Mathematics, Physics and Mechanics, Europe, Slovenia"],"raw_orcid":"https://orcid.org/0000-0003-0540-0938","affiliations":[{"raw_affiliation_string":"Faculty of Natural Sciences and Mathematics, University of Maribor, Europe, Slovenia","institution_ids":["https://openalex.org/I37696226"]},{"raw_affiliation_string":"Institute of Mathematics, Physics and Mechanics, Europe, Slovenia","institution_ids":["https://openalex.org/I4210106342"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5103053298","display_name":"Daniel Johnston","orcid":"https://orcid.org/0000-0002-8321-501X"},"institutions":[{"id":"https://openalex.org/I4401726811","display_name":"Hartford Financial Services (United States)","ror":"https://ror.org/00mwq1g96","country_code":null,"type":"company","lineage":["https://openalex.org/I4401726811"]},{"id":"https://openalex.org/I56722557","display_name":"Trinity College","ror":"https://ror.org/03j3dbz94","country_code":"US","type":"education","lineage":["https://openalex.org/I56722557"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Daniel P. Johnston","raw_affiliation_strings":["Department of Mathematics, Trinity College, Hartford, CT, USA"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Trinity College, Hartford, CT, USA","institution_ids":["https://openalex.org/I56722557","https://openalex.org/I4401726811"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5080518463","display_name":"Kirsti Kuenzel","orcid":null},"institutions":[{"id":"https://openalex.org/I4401726811","display_name":"Hartford Financial Services (United States)","ror":"https://ror.org/00mwq1g96","country_code":null,"type":"company","lineage":["https://openalex.org/I4401726811"]},{"id":"https://openalex.org/I56722557","display_name":"Trinity College","ror":"https://ror.org/03j3dbz94","country_code":"US","type":"education","lineage":["https://openalex.org/I56722557"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Kirsti Kuenzel","raw_affiliation_strings":["Department of Mathematics, Trinity College, Hartford, CT, USA"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Trinity College, Hartford, CT, USA","institution_ids":["https://openalex.org/I56722557","https://openalex.org/I4401726811"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5017681038","display_name":"Doug F. Rall","orcid":null},"institutions":[{"id":"https://openalex.org/I86115722","display_name":"Furman University","ror":"https://ror.org/04ytb9n23","country_code":"US","type":"education","lineage":["https://openalex.org/I86115722"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Douglas F. Rall","raw_affiliation_strings":["Emeritus Professor of Mathematics, Furman University, Greenville, SC, USA"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Emeritus Professor of Mathematics, Furman University, Greenville, SC, USA","institution_ids":["https://openalex.org/I86115722"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":5,"corresponding_author_ids":["https://openalex.org/A5044816279"],"corresponding_institution_ids":["https://openalex.org/I37696226","https://openalex.org/I4210106342"],"apc_list":{"value":2290,"currency":"EUR","value_usd":2890},"apc_paid":{"value":2290,"currency":"EUR","value_usd":2890},"fwci":0.0,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":{"value":0.48458666,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"42","issue":"2","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":0.3806000053882599,"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":0.3806000053882599,"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/T11152","display_name":"Stochastic processes and statistical mechanics","score":0.058400001376867294,"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/T11329","display_name":"Limits and Structures in Graph Theory","score":0.04280000180006027,"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/graph","display_name":"Graph","score":0.5152999758720398},{"id":"https://openalex.org/keywords/continuation","display_name":"Continuation","score":0.45680001378059387},{"id":"https://openalex.org/keywords/invariant","display_name":"Invariant (physics)","score":0.40560001134872437},{"id":"https://openalex.org/keywords/set","display_name":"Set (abstract data type)","score":0.34310001134872437},{"id":"https://openalex.org/keywords/independent-set","display_name":"Independent set","score":0.3321000039577484},{"id":"https://openalex.org/keywords/existential-quantification","display_name":"Existential quantification","score":0.32089999318122864}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8066999912261963},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.7116000056266785},{"id":"https://openalex.org/C132525143","wikidata":"https://www.wikidata.org/wiki/Q141488","display_name":"Graph","level":2,"score":0.5152999758720398},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.4880000054836273},{"id":"https://openalex.org/C88626702","wikidata":"https://www.wikidata.org/wiki/Q1128903","display_name":"Continuation","level":2,"score":0.45680001378059387},{"id":"https://openalex.org/C190470478","wikidata":"https://www.wikidata.org/wiki/Q2370229","display_name":"Invariant (physics)","level":2,"score":0.40560001134872437},{"id":"https://openalex.org/C177264268","wikidata":"https://www.wikidata.org/wiki/Q1514741","display_name":"Set (abstract data type)","level":2,"score":0.34310001134872437},{"id":"https://openalex.org/C122818955","wikidata":"https://www.wikidata.org/wiki/Q1060343","display_name":"Independent set","level":3,"score":0.3321000039577484},{"id":"https://openalex.org/C136979486","wikidata":"https://www.wikidata.org/wiki/Q773483","display_name":"Existential quantification","level":2,"score":0.32089999318122864},{"id":"https://openalex.org/C83833204","wikidata":"https://www.wikidata.org/wiki/Q5290322","display_name":"Domination analysis","level":4,"score":0.3082999885082245},{"id":"https://openalex.org/C203776342","wikidata":"https://www.wikidata.org/wiki/Q1378376","display_name":"Line graph","level":3,"score":0.30399999022483826},{"id":"https://openalex.org/C146661039","wikidata":"https://www.wikidata.org/wiki/Q2915204","display_name":"Dominating set","level":4,"score":0.2996000051498413},{"id":"https://openalex.org/C8554925","wikidata":"https://www.wikidata.org/wiki/Q3893853","display_name":"Split graph","level":5,"score":0.2874000072479248},{"id":"https://openalex.org/C32972407","wikidata":"https://www.wikidata.org/wiki/Q1187620","display_name":"Perfect graph","level":5,"score":0.2867000102996826}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.1007/s00373-026-03030-y","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s00373-026-03030-y","pdf_url":"https://link.springer.com/content/pdf/10.1007/s00373-026-03030-y.pdf","source":{"id":"https://openalex.org/S186126824","display_name":"Graphs and Combinatorics","issn_l":"0911-0119","issn":["0911-0119","1435-5914"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Graphs and Combinatorics","raw_type":"journal-article"},{"id":"pmh:oai:d.cobiss.net/repository/si:270312963","is_oa":true,"landing_page_url":"https://link.springer.com/article/10.1007/s00373-026-03030-y","pdf_url":null,"source":{"id":"https://openalex.org/S7407055177","display_name":"dCOBISS.SI Digital Repository","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":"COBISS-ID: 25536512","raw_type":"info:eu-repo/semantics/publishedVersion"}],"best_oa_location":{"id":"doi:10.1007/s00373-026-03030-y","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s00373-026-03030-y","pdf_url":"https://link.springer.com/content/pdf/10.1007/s00373-026-03030-y.pdf","source":{"id":"https://openalex.org/S186126824","display_name":"Graphs and Combinatorics","issn_l":"0911-0119","issn":["0911-0119","1435-5914"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Graphs and Combinatorics","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G3894112125","display_name":null,"funder_award_id":"P1-0297","funder_id":"https://openalex.org/F4320322554","funder_display_name":"Javna Agencija za Raziskovalno Dejavnost RS"}],"funders":[{"id":"https://openalex.org/F4320322554","display_name":"Javna Agencija za Raziskovalno Dejavnost RS","ror":"https://ror.org/059bp8k51"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7133297008.pdf","grobid_xml":"https://content.openalex.org/works/W7133297008.grobid-xml"},"referenced_works_count":15,"referenced_works":["https://openalex.org/W1725657036","https://openalex.org/W1899855728","https://openalex.org/W2044002450","https://openalex.org/W2073093064","https://openalex.org/W2136564671","https://openalex.org/W2900115293","https://openalex.org/W2963808569","https://openalex.org/W2999386192","https://openalex.org/W3010685994","https://openalex.org/W3154464724","https://openalex.org/W4313334182","https://openalex.org/W4367301867","https://openalex.org/W4391926715","https://openalex.org/W4393192673","https://openalex.org/W4399376019"],"related_works":[],"abstract_inverted_index":{"Abstract":[0],"Given":[1],"a":[2,6,75,82],"graph":[3,142,292],"G":[4,104,293],"and":[5,30,193,232],"family":[7],"of":[8,67,128,224],"graphs":[9,225],"$$\\mathcal":[10,18,68,95,116,147,185,200,226,233,308],"F$$":[11,19,69,96,117,148,186,201,227],"<mml:math":[12,20,37,53,70,97,118,149,159,187,202,228,235,257,310],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">":[13,21,38,54,71,98,119,150,160,188,203,229,236,258],"<mml:mi>F</mml:mi>":[14,22,72,99,120,151,170,189,204,230,238,270,285],"</mml:math>":[15,23,47,63,73,100,121,152,174,190,205,231,241,289],",":[16,242],"an":[17,94],"-isolating":[24,101,122],"set,":[25],"as":[26,74,132,136],"introduced":[27],"by":[28],"Caro":[29],"Hansberg,":[31],"is":[32,131,216,298],"any":[33,291],"set":[34,102,123],"$$S\\subset":[35],"V(G)$$":[36],"<mml:mrow>":[39,55,161,166,259,264,269,281],"<mml:mi>S</mml:mi>":[40,60],"<mml:mo>\u2282</mml:mo>":[41],"<mml:mi>V</mml:mi>":[42],"<mml:mo>(</mml:mo>":[43,167,265,282],"<mml:mi>G</mml:mi>":[44,56,168,266,283],"<mml:mo>)</mml:mo>":[45,171,274,286],"</mml:mrow>":[46,62,172,173,271,275,287,288],"such":[48],"that":[49,113,178,194,248],"$$G":[50],"-":[51],"N[S]$$":[52],"<mml:mo>-</mml:mo>":[57],"<mml:mi>N</mml:mi>":[58],"<mml:mo>[</mml:mo>":[59],"<mml:mo>]</mml:mo>":[61],"contains":[64],"no":[65],"member":[66],"subgraph.":[76],"In":[77],"this":[78],"paper,":[79],"we":[80,243],"introduce":[81],"game":[83,146,199,215],"in":[84,103,183],"which":[85,138,304],"two":[86,221],"players":[87],"with":[88],"opposite":[89],"goals":[90],"are":[91],"together":[92],"building":[93],".":[105,175,294],"Following":[106],"the":[107,114,126,129,141,145,179,184,195,198,214,301],"domination":[108],"games,":[109],"Dominator":[110,210],"(Staller)":[111],"wants":[112],"resulting":[115],"obtained":[124],"at":[125,217],"end":[127],"game,":[130,192,303],"small":[133],"(as":[134],"big)":[135],"possible,":[137],"leads":[139],"to":[140],"invariant":[143],"called":[144],"-isolation":[153,191,206],"number,":[154],"denoted":[155],"$$\\iota":[156,250],"_\\textrm{g}(G,\\mathcal":[157],"F)$$":[158],"<mml:msub>":[162,260,277],"<mml:mi>\u03b9</mml:mi>":[163,261,278],"<mml:mtext>g</mml:mtext>":[164,262,279],"</mml:msub>":[165,263,280],"<mml:mo>,</mml:mo>":[169,267,284],"We":[176],"prove":[177],"Continuation":[180],"Principle":[181],"holds":[182],"difference":[196],"between":[197,246],"numbers":[207],"when":[208,307],"either":[209],"or":[211],"Staller":[212],"starts":[213],"most":[218],"1.":[219],"Considering":[220],"arbitrary":[222],"families":[223],"F'$$":[234],"<mml:msup>":[237,268],"<mml:mo>\u2032</mml:mo>":[239,272],"</mml:msup>":[240,273],"find":[244],"relations":[245],"them":[247],"ensure":[249],"_\\textrm{g}(G,{\\mathcal":[251,255],"{F}}')":[252],"\\le":[253],"\\iota":[254],"{F}})$$":[256],"<mml:mo>\u2264</mml:mo>":[276],"for":[290],"A":[295],"special":[296],"focus":[297],"given":[299],"on":[300],"isolation":[302],"takes":[305],"place":[306],"{F}=\\{K_2\\}$$":[309],"xmlns:mml=\"http":[311]},"counts_by_year":[],"updated_date":"2026-03-14T06:41:57.775601","created_date":"2026-03-04T00:00:00"}