{"id":"https://openalex.org/W4410948883","doi":"https://doi.org/10.1007/s00373-025-02937-2","title":"Double Coalitions in Regular Graphs","display_name":"Double Coalitions in Regular Graphs","publication_year":2025,"publication_date":"2025-06-02","ids":{"openalex":"https://openalex.org/W4410948883","doi":"https://doi.org/10.1007/s00373-025-02937-2"},"language":"en","primary_location":{"id":"doi:10.1007/s00373-025-02937-2","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s00373-025-02937-2","pdf_url":"https://link.springer.com/content/pdf/10.1007/s00373-025-02937-2.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/s00373-025-02937-2.pdf","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5089473322","display_name":"Michael A. Henning","orcid":"https://orcid.org/0000-0001-8185-067X"},"institutions":[{"id":"https://openalex.org/I24027795","display_name":"University of Johannesburg","ror":"https://ror.org/04z6c2n17","country_code":"ZA","type":"education","lineage":["https://openalex.org/I24027795"]}],"countries":["ZA"],"is_corresponding":true,"raw_author_name":"Michael A. Henning","raw_affiliation_strings":["Department of Mathematics and Applied Mathematics, University of Johannesburg, Auckland Park, 2006, South Africa"],"raw_orcid":"https://orcid.org/0000-0001-8185-067X","affiliations":[{"raw_affiliation_string":"Department of Mathematics and Applied Mathematics, University of Johannesburg, Auckland Park, 2006, South Africa","institution_ids":["https://openalex.org/I24027795"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5059518359","display_name":"Doost Ali Mojdeh","orcid":"https://orcid.org/0000-0001-9373-3390"},"institutions":[{"id":"https://openalex.org/I4871159","display_name":"University of Mazandaran","ror":"https://ror.org/05fp9g671","country_code":"IR","type":"education","lineage":["https://openalex.org/I4871159"]}],"countries":["IR"],"is_corresponding":false,"raw_author_name":"Doost Ali Mojdeh","raw_affiliation_strings":["Department of Mathematics, Faculty of Mathematical Sciences, University of Mazandaran, Babolsar, Iran"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Faculty of Mathematical Sciences, University of Mazandaran, Babolsar, Iran","institution_ids":["https://openalex.org/I4871159"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5089473322"],"corresponding_institution_ids":["https://openalex.org/I24027795"],"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":0.12994264,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"41","issue":"4","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":0.9998999834060669,"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":0.9998999834060669,"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/T11476","display_name":"Graph theory and applications","score":0.9988999962806702,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"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/T12541","display_name":"Graph Labeling and Dimension Problems","score":0.9951000213623047,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.8280171155929565},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.7083098888397217},{"id":"https://openalex.org/keywords/chordal-graph","display_name":"Chordal graph","score":0.41788381338119507},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.3902994990348816},{"id":"https://openalex.org/keywords/graph","display_name":"Graph","score":0.29921257495880127}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8280171155929565},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.7083098888397217},{"id":"https://openalex.org/C160446614","wikidata":"https://www.wikidata.org/wiki/Q1322892","display_name":"Chordal graph","level":3,"score":0.41788381338119507},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.3902994990348816},{"id":"https://openalex.org/C132525143","wikidata":"https://www.wikidata.org/wiki/Q141488","display_name":"Graph","level":2,"score":0.29921257495880127}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1007/s00373-025-02937-2","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s00373-025-02937-2","pdf_url":"https://link.springer.com/content/pdf/10.1007/s00373-025-02937-2.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"}],"best_oa_location":{"id":"doi:10.1007/s00373-025-02937-2","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s00373-025-02937-2","pdf_url":"https://link.springer.com/content/pdf/10.1007/s00373-025-02937-2.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":[],"funders":[{"id":"https://openalex.org/F4320323959","display_name":"University of Johannesburg","ror":"https://ror.org/04z6c2n17"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4410948883.pdf","grobid_xml":"https://content.openalex.org/works/W4410948883.grobid-xml"},"referenced_works_count":23,"referenced_works":["https://openalex.org/W560340026","https://openalex.org/W1969343019","https://openalex.org/W2295287498","https://openalex.org/W2811376386","https://openalex.org/W3015959374","https://openalex.org/W3096372459","https://openalex.org/W3164994398","https://openalex.org/W3175576378","https://openalex.org/W3203804620","https://openalex.org/W4211174266","https://openalex.org/W4226425430","https://openalex.org/W4361000216","https://openalex.org/W4361001074","https://openalex.org/W4367459369","https://openalex.org/W4382138734","https://openalex.org/W4386289389","https://openalex.org/W4386289568","https://openalex.org/W4390954301","https://openalex.org/W4391655806","https://openalex.org/W4400427885","https://openalex.org/W4401214182","https://openalex.org/W4403027484","https://openalex.org/W4407298173"],"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/W2117848356","https://openalex.org/W2043410679","https://openalex.org/W1970195683","https://openalex.org/W2014290730"],"abstract_inverted_index":{"Abstract":[0],"A":[1,104,164],"set":[2,13,43,62,129,148,222,237],"S":[3,21,26,89,95,102],"of":[4,14,63,110,114,119,123,149,286,292,386],"vertices":[5,30,115],"in":[6,20,25,68,88,94,101,107,168,297],"a":[7,11,23,42,59,126,145,161,171,232,293],"graph":[8],"G":[9,15,38,64,108,120,150,169,287,298,390],"is":[10,39,58,125,144,170],"dominating":[12,61,128,147],"if":[16,33,65],"every":[17,66,92],"vertex":[18,67,93,172],"not":[19],"has":[22,83,96],"neighbor":[24,100],",":[27,90,121,220,249,285,353],"where":[28,250],"two":[29,86,111],"are":[31,35,157],"neighbors":[32,87],"they":[34],"adjacent.":[36],"If":[37],"isolate-free,":[40],"then":[41],"$$S":[44],"\\subseteq":[45],"V(G)$$":[46],"<mml:math":[47,72,136,178,210,224,239,256,276,314,327,345,362,399],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">":[48,73,137,179,211,225,240,257,277,315,328,346,363,400],"<mml:mrow>":[49,74,138,180,212,258,278,318,347,364,366,374,401,403],"<mml:mi>S</mml:mi>":[50,80],"<mml:mo>\u2286</mml:mo>":[51],"<mml:mi>V</mml:mi>":[52,75,185,190,197,227,242],"<mml:mo>(</mml:mo>":[53,76,280,367,404],"<mml:mi>G</mml:mi>":[54,77,281,368,405],"<mml:mo>)</mml:mo>":[55,78,282,369,406],"</mml:mrow>":[56,81,142,201,218,268,283,320,351,370,376,381,407],"</mml:math>":[57,82,143,202,219,230,245,269,284,323,333,352,382],"double":[60,105,127,146,162,165,233,272,294],"$$V(G)":[69],"\\setminus":[70,254],"S$$":[71],"<mml:mo>\\</mml:mo>":[79,264],"at":[84,97],"least":[85,98],"and":[91,117,155,324,391],"one":[99],".":[103,151,270,299],"coalition":[106,166,234,273,295],"consists":[109],"disjoint":[112],"sets":[113,153],"X":[116,154],"Y":[118,156],"neither":[122],"which":[124],"but":[130],"whose":[131],"union":[132],"$$X":[133],"\\cup":[134],"Y$$":[135],"<mml:mi>X</mml:mi>":[139],"<mml:mo>\u222a</mml:mo>":[140],"<mml:mi>Y</mml:mi>":[141],"Such":[152],"said":[158],"to":[159,304],"form":[160],"coalition.":[163],"partition":[167,173,296],"$$\\Psi":[174],"=":[175],"\\{V_1,V_2,\\ldots":[176],",V_k\\}$$":[177],"<mml:mi>\u03a8</mml:mi>":[181],"<mml:mo>=</mml:mo>":[182],"<mml:mo>{</mml:mo>":[183,265],"<mml:msub>":[184,189,196,226,241,329],"<mml:mn>1</mml:mn>":[186],"</mml:msub>":[187,192,199,229,244,332],"<mml:mo>,</mml:mo>":[188,193,195],"<mml:mn>2</mml:mn>":[191],"<mml:mo>\u2026</mml:mo>":[194],"<mml:mi>k</mml:mi>":[198,216,262],"<mml:mo>}</mml:mo>":[200,267],"such":[203,354],"that":[204,355],"for":[205,246,340,383],"all":[206],"$$i":[207],"\\in":[208,252],"[k]$$":[209],"<mml:mi>i</mml:mi>":[213,228,266],"<mml:mo>\u2208</mml:mo>":[214,260],"<mml:mo>[</mml:mo>":[215,261],"<mml:mo>]</mml:mo>":[217,263],"the":[221,289,302,308,384],"$$V_i$$":[223],"forms":[231],"with":[235],"another":[236],"$$V_j$$":[238],"<mml:mi>j</mml:mi>":[243,259],"some":[247],"j":[248],"$$j":[251],"[k]":[253],"\\{i\\}$$":[255],"The":[271],"number,":[274],"$$\\textrm{DC}(G)$$":[275],"<mml:mtext>DC</mml:mtext>":[279,365,402],"equals":[288],"maximum":[290],"order":[291],"We":[300],"discuss":[301],"problem":[303],"determine":[305],"or":[306],"estimate":[307],"best":[309],"possible":[310],"constants":[311],"$$\\theta":[312,325],"_{r}^{\\textrm{reg}}$$":[313],"<mml:msubsup>":[316,372],"<mml:mi>\u03b8</mml:mi>":[317,330,373],"<mml:mi>r</mml:mi>":[319,331,348,375,380],"<mml:mtext>reg</mml:mtext>":[321,377],"</mml:msubsup>":[322,378],"_{r}$$":[326],"(which":[334],"depend":[335],"only":[336],"on":[337],"r":[338,387],")":[339],"each":[341],"$$r":[342],"\\ge":[343],"3$$":[344],"<mml:mo>\u2265</mml:mo>":[349],"<mml:mn>3</mml:mn>":[350],"$$\\textrm{DC}(G)":[356,392],"\\le":[357,393],"\\theta":[358,394],"_{r}^{\\textrm{reg}}":[359],"\\times":[360,396],"r$$":[361],"<mml:mo>\u2264</mml:mo>":[371],"<mml:mo>\u00d7</mml:mo>":[379],"class":[385],"-regular":[388],"graphs":[389],"_{r}":[395],"\\Delta":[397],"(G)$$":[398]},"counts_by_year":[],"updated_date":"2026-03-27T05:58:40.876381","created_date":"2025-10-10T00:00:00"}
