{"id":"https://openalex.org/W4321360132","doi":"https://doi.org/10.3103/s0146411622070173","title":"NP-Completeness of the Minimum Spanning Tree Problem of a Multiple Graph of Multiplicity k \u2265 3","display_name":"NP-Completeness of the Minimum Spanning Tree Problem of a Multiple Graph of Multiplicity k \u2265 3","publication_year":2022,"publication_date":"2022-12-01","ids":{"openalex":"https://openalex.org/W4321360132","doi":"https://doi.org/10.3103/s0146411622070173"},"language":"en","primary_location":{"id":"doi:10.3103/s0146411622070173","is_oa":false,"landing_page_url":"https://doi.org/10.3103/s0146411622070173","pdf_url":null,"source":{"id":"https://openalex.org/S17203304","display_name":"Automatic Control and Computer Sciences","issn_l":"0146-4116","issn":["0146-4116","1558-108X"],"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/P4310320267","host_organization_name":"Pleiades Publishing","host_organization_lineage":["https://openalex.org/P4310320267","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Pleiades Publishing","Springer Nature"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Automatic Control and Computer Sciences","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":false,"oa_status":"closed","oa_url":null,"any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5019524013","display_name":"\u0410. \u0412. \u0421\u043c\u0438\u0440\u043d\u043e\u0432","orcid":"https://orcid.org/0000-0002-0980-2507"},"institutions":[{"id":"https://openalex.org/I198371094","display_name":"Yaroslavl State University","ror":"https://ror.org/044s2fj67","country_code":"RU","type":"education","lineage":["https://openalex.org/I198371094"]}],"countries":["RU"],"is_corresponding":true,"raw_author_name":"A. V. Smirnov","raw_affiliation_strings":["Demidov Yaroslavl State University, 150003, Yaroslavl, Russia"],"raw_orcid":"https://orcid.org/0000-0002-0980-2507","affiliations":[{"raw_affiliation_string":"Demidov Yaroslavl State University, 150003, Yaroslavl, Russia","institution_ids":["https://openalex.org/I198371094"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5019524013"],"corresponding_institution_ids":["https://openalex.org/I198371094"],"apc_list":null,"apc_paid":null,"fwci":0.3519,"has_fulltext":false,"cited_by_count":3,"citation_normalized_percentile":{"value":0.57925221,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":95},"biblio":{"volume":"56","issue":"7","first_page":"788","last_page":"799"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T14470","display_name":"Advanced Data Processing Techniques","score":0.9735000133514404,"subfield":{"id":"https://openalex.org/subfields/2207","display_name":"Control and Systems Engineering"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T14470","display_name":"Advanced Data Processing Techniques","score":0.9735000133514404,"subfield":{"id":"https://openalex.org/subfields/2207","display_name":"Control and Systems Engineering"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T12176","display_name":"Optimization and Packing Problems","score":0.9715999960899353,"subfield":{"id":"https://openalex.org/subfields/2209","display_name":"Industrial and Manufacturing Engineering"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10829","display_name":"Interconnection Networks and Systems","score":0.9018999934196472,"subfield":{"id":"https://openalex.org/subfields/1705","display_name":"Computer Networks and Communications"},"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/combinatorics","display_name":"Combinatorics","score":0.7474532127380371},{"id":"https://openalex.org/keywords/spanning-tree","display_name":"Spanning tree","score":0.7388025522232056},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.603276252746582},{"id":"https://openalex.org/keywords/tr\u00e9maux-tree","display_name":"Tr\u00e9maux tree","score":0.5905179381370544},{"id":"https://openalex.org/keywords/shortest-path-tree","display_name":"Shortest-path tree","score":0.5672566294670105},{"id":"https://openalex.org/keywords/minimum-spanning-tree","display_name":"Minimum spanning tree","score":0.5384663343429565},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.45841217041015625},{"id":"https://openalex.org/keywords/k-ary-tree","display_name":"K-ary tree","score":0.45811572670936584},{"id":"https://openalex.org/keywords/connected-dominating-set","display_name":"Connected dominating set","score":0.4543340802192688},{"id":"https://openalex.org/keywords/vertex","display_name":"Vertex (graph theory)","score":0.45002609491348267},{"id":"https://openalex.org/keywords/multiple-edges","display_name":"Multiple edges","score":0.41894158720970154},{"id":"https://openalex.org/keywords/graph","display_name":"Graph","score":0.34142789244651794},{"id":"https://openalex.org/keywords/line-graph","display_name":"Line graph","score":0.2721903920173645},{"id":"https://openalex.org/keywords/binary-tree","display_name":"Binary tree","score":0.18708482384681702},{"id":"https://openalex.org/keywords/tree-structure","display_name":"Tree structure","score":0.18658438324928284},{"id":"https://openalex.org/keywords/pathwidth","display_name":"Pathwidth","score":0.17265218496322632}],"concepts":[{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.7474532127380371},{"id":"https://openalex.org/C64331007","wikidata":"https://www.wikidata.org/wiki/Q831672","display_name":"Spanning tree","level":2,"score":0.7388025522232056},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.603276252746582},{"id":"https://openalex.org/C124302570","wikidata":"https://www.wikidata.org/wiki/Q7849146","display_name":"Tr\u00e9maux tree","level":5,"score":0.5905179381370544},{"id":"https://openalex.org/C23394520","wikidata":"https://www.wikidata.org/wiki/Q4919350","display_name":"Shortest-path tree","level":3,"score":0.5672566294670105},{"id":"https://openalex.org/C13743678","wikidata":"https://www.wikidata.org/wiki/Q240464","display_name":"Minimum spanning tree","level":2,"score":0.5384663343429565},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.45841217041015625},{"id":"https://openalex.org/C100560664","wikidata":"https://www.wikidata.org/wiki/Q3608019","display_name":"K-ary tree","level":4,"score":0.45811572670936584},{"id":"https://openalex.org/C37810922","wikidata":"https://www.wikidata.org/wiki/Q5161409","display_name":"Connected dominating set","level":3,"score":0.4543340802192688},{"id":"https://openalex.org/C80899671","wikidata":"https://www.wikidata.org/wiki/Q1304193","display_name":"Vertex (graph theory)","level":3,"score":0.45002609491348267},{"id":"https://openalex.org/C103257674","wikidata":"https://www.wikidata.org/wiki/Q6934937","display_name":"Multiple edges","level":3,"score":0.41894158720970154},{"id":"https://openalex.org/C132525143","wikidata":"https://www.wikidata.org/wiki/Q141488","display_name":"Graph","level":2,"score":0.34142789244651794},{"id":"https://openalex.org/C203776342","wikidata":"https://www.wikidata.org/wiki/Q1378376","display_name":"Line graph","level":3,"score":0.2721903920173645},{"id":"https://openalex.org/C197855036","wikidata":"https://www.wikidata.org/wiki/Q380172","display_name":"Binary tree","level":2,"score":0.18708482384681702},{"id":"https://openalex.org/C163797641","wikidata":"https://www.wikidata.org/wiki/Q2067937","display_name":"Tree structure","level":3,"score":0.18658438324928284},{"id":"https://openalex.org/C43517604","wikidata":"https://www.wikidata.org/wiki/Q7144893","display_name":"Pathwidth","level":4,"score":0.17265218496322632}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.3103/s0146411622070173","is_oa":false,"landing_page_url":"https://doi.org/10.3103/s0146411622070173","pdf_url":null,"source":{"id":"https://openalex.org/S17203304","display_name":"Automatic Control and Computer Sciences","issn_l":"0146-4116","issn":["0146-4116","1558-108X"],"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/P4310320267","host_organization_name":"Pleiades Publishing","host_organization_lineage":["https://openalex.org/P4310320267","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Pleiades Publishing","Springer Nature"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Automatic Control and Computer Sciences","raw_type":"journal-article"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":11,"referenced_works":["https://openalex.org/W1530753374","https://openalex.org/W1965680834","https://openalex.org/W2011039300","https://openalex.org/W2117115743","https://openalex.org/W2401610261","https://openalex.org/W2540385142","https://openalex.org/W2572716357","https://openalex.org/W2891758599","https://openalex.org/W2922839089","https://openalex.org/W3124905242","https://openalex.org/W4246219036"],"related_works":["https://openalex.org/W2779648845","https://openalex.org/W2201964723","https://openalex.org/W2307044859","https://openalex.org/W2014991506","https://openalex.org/W2082872574","https://openalex.org/W1754127855","https://openalex.org/W130518292","https://openalex.org/W4232108645","https://openalex.org/W1587959445","https://openalex.org/W4321360132"],"abstract_inverted_index":{"Undirected":[0],"multiple":[1,20,62,71,107,112,126,143,157,164,189,248,255],"graphs":[2,249,256],"of":[3,15,26,34,80,84,94,103,122,133,152,155,217,219,227,234,240,242],"any":[4,167],"natural":[5],"multiplicity":[6,261],"k":[7,35,43,81,262],">":[8],"1":[9],"are":[10,13,269],"studied.":[11],"There":[12],"edges":[14,50,72,83,123],"three":[16],"types:":[17],"ordinary":[18,118],"edges,":[19,21,37],"and":[22,73,176,198,223],"multi-edges.":[23],"Each":[24],"edge":[25,102],"the":[27,77,91,100,120,135,173,184,188,193,199,215,220,228,238,258],"last":[28],"two":[29,168],"types":[30],"is":[31,58,90,109,128,161,191,211,237],"a":[32,56,61,88,110,125,142,149,156,163,180],"union":[33],"linked":[36,49,82],"which":[38],"connect":[39],"2":[40],"or":[41],"(":[42],"+":[44],"1)":[45],"vertices":[46,170],"correspondingly.":[47],"The":[48,131,231,265],"should":[51],"be":[52,66,76,99,139,206],"used":[53],"simultaneously.":[54],"If":[55,87,187],"vertex":[57,89],"incident":[59,68],"to":[60,69,213],"edge,":[63],"it":[64,74,97,210],"can":[65,75,138,205],"also":[67],"other":[70],"common":[78,92,101],"end":[79,93],"some":[85,95],"multi-edge.":[86,105],"multi-edge,":[96],"cannot":[98],"another":[104],"A":[106],"tree":[108,127,137,174,196,203,222,226],"connected":[111],"graph":[113,190],"with":[114],"no":[115],"cycles.":[116],"Unlike":[117],"trees,":[119],"number":[121],"in":[124,172,183,257],"not":[129],"fixed.":[130],"problem":[132,197,204],"finding":[134],"spanning":[136,146,153,195,202,221,225],"set":[140],"for":[141,246,253],"graph.":[144,158,186],"Complete":[145],"trees":[147,154],"form":[148],"special":[150],"class":[151],"Their":[159],"peculiarity":[160],"that":[162],"path":[165,181],"joining":[166],"selected":[169],"exists":[171,182],"if":[175,178],"only":[177],"such":[179,243],"initial":[185],"weighted,":[192],"minimum":[194,200],"complete":[201,224],"set.":[207],"As":[208],"well,":[209],"possible":[212],"formulate":[214],"problems":[216,245,268],"recognition":[218,244],"limited":[229],"weight.":[230],"main":[232],"result":[233],"this":[235],"article":[236],"proof":[239],"NP-completeness":[241],"arbitrary":[247],"as":[250,252],"well":[251],"divisible":[254],"case":[259],"when":[260],"\u2265":[263],"3.":[264],"corresponding":[266],"optimization":[267],"NP-hard.":[270]},"counts_by_year":[{"year":2025,"cited_by_count":1},{"year":2024,"cited_by_count":1},{"year":2023,"cited_by_count":1}],"updated_date":"2026-06-13T06:13:01.061226","created_date":"2025-10-10T00:00:00"}