{"id":"https://openalex.org/W4408293941","doi":"https://doi.org/10.3390/sym17030405","title":"On the Existence of Perfect (1, 2)-Dominating Sets in Graphs","display_name":"On the Existence of Perfect (1, 2)-Dominating Sets in Graphs","publication_year":2025,"publication_date":"2025-03-08","ids":{"openalex":"https://openalex.org/W4408293941","doi":"https://doi.org/10.3390/sym17030405"},"language":"en","primary_location":{"id":"doi:10.3390/sym17030405","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym17030405","pdf_url":"https://www.mdpi.com/2073-8994/17/3/405/pdf?version=1741424489","source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Symmetry","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://www.mdpi.com/2073-8994/17/3/405/pdf?version=1741424489","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5079723727","display_name":"Urszula Bednarz","orcid":"https://orcid.org/0000-0002-4856-9613"},"institutions":[{"id":"https://openalex.org/I111664937","display_name":"Rzesz\u00f3w University of Technology","ror":"https://ror.org/056xse072","country_code":"PL","type":"education","lineage":["https://openalex.org/I111664937"]}],"countries":["PL"],"is_corresponding":true,"raw_author_name":"Urszula Bednarz","raw_affiliation_strings":["The Faculty of Mathematics and Applied Physics, Rzesz\u00f3w University of Technology, al. Powsta\u0144c\u00f3w Warszawy 12, 35-959 Rzesz\u00f3w, Poland"],"affiliations":[{"raw_affiliation_string":"The Faculty of Mathematics and Applied Physics, Rzesz\u00f3w University of Technology, al. Powsta\u0144c\u00f3w Warszawy 12, 35-959 Rzesz\u00f3w, Poland","institution_ids":["https://openalex.org/I111664937"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5079723727"],"corresponding_institution_ids":["https://openalex.org/I111664937"],"apc_list":{"value":2000,"currency":"CHF","value_usd":2165},"apc_paid":{"value":2000,"currency":"CHF","value_usd":2165},"fwci":1.6414,"has_fulltext":false,"cited_by_count":1,"citation_normalized_percentile":{"value":0.82250859,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":97,"max":99},"biblio":{"volume":"17","issue":"3","first_page":"405","last_page":"405"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10374","display_name":"Advanced Graph Theory Research","score":0.9994000196456909,"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":0.9994000196456909,"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/T12541","display_name":"Graph Labeling and Dimension Problems","score":0.9864000082015991,"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/T12288","display_name":"Optimization and Search Problems","score":0.9858999848365784,"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/dominating-set","display_name":"Dominating set","score":0.5623800158500671},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.5431979298591614},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.5144262909889221},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.4036247134208679},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.33182579278945923},{"id":"https://openalex.org/keywords/graph","display_name":"Graph","score":0.20396876335144043},{"id":"https://openalex.org/keywords/vertex","display_name":"Vertex (graph theory)","score":0.12197187542915344}],"concepts":[{"id":"https://openalex.org/C146661039","wikidata":"https://www.wikidata.org/wiki/Q2915204","display_name":"Dominating set","level":4,"score":0.5623800158500671},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.5431979298591614},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5144262909889221},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.4036247134208679},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.33182579278945923},{"id":"https://openalex.org/C132525143","wikidata":"https://www.wikidata.org/wiki/Q141488","display_name":"Graph","level":2,"score":0.20396876335144043},{"id":"https://openalex.org/C80899671","wikidata":"https://www.wikidata.org/wiki/Q1304193","display_name":"Vertex (graph theory)","level":3,"score":0.12197187542915344}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.3390/sym17030405","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym17030405","pdf_url":"https://www.mdpi.com/2073-8994/17/3/405/pdf?version=1741424489","source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Symmetry","raw_type":"journal-article"}],"best_oa_location":{"id":"doi:10.3390/sym17030405","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym17030405","pdf_url":"https://www.mdpi.com/2073-8994/17/3/405/pdf?version=1741424489","source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Symmetry","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4408293941.pdf"},"referenced_works_count":17,"referenced_works":["https://openalex.org/W1969343019","https://openalex.org/W2061040604","https://openalex.org/W2745566800","https://openalex.org/W3009096111","https://openalex.org/W3083943252","https://openalex.org/W4200309633","https://openalex.org/W4205426181","https://openalex.org/W4210590032","https://openalex.org/W4221065207","https://openalex.org/W4247549312","https://openalex.org/W4248814197","https://openalex.org/W4382138734","https://openalex.org/W4385011276","https://openalex.org/W4389056422","https://openalex.org/W4392376841","https://openalex.org/W4405646534","https://openalex.org/W6774656756"],"related_works":["https://openalex.org/W2899084033","https://openalex.org/W2748952813","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/W4207084262"],"abstract_inverted_index":{"In":[0],"this":[1],"paper,":[2],"we":[3,62,125],"investigate":[4],"the":[5,96,99,111,118,127,137,160,165,169],"existence":[6,112,161],"of":[7,17,31,53,82,113,121,162,171],"perfect":[8,24],"(1,2)-dominating":[9,25,33],"sets":[10,39],"((1,2)-PDS)":[11],"in":[12,87,115,164],"graphs":[13,65,83],"with":[14,36,66],"three":[15,138],"vertices":[16,139],"maximum":[18,119],"degree":[19,54,69,120],"equal":[20,73],"to":[21,74,117],"n\u22122.":[22],"A":[23],"set":[26],"is":[27],"a":[28,32,42,46,59,67,77,90,93,122],"special":[29],"case":[30],"set.":[34],"Graphs":[35],"such":[37],"dominating":[38],"may":[40,104],"exhibit":[41],"symmetric":[43,78,100],"structure.":[44],"If":[45],"graph":[47,91],"has":[48,58,92],"at":[49],"least":[50],"one":[51],"vertex":[52],"one,":[55],"then":[56],"it":[57],"(1,2)-PDS.":[60,94],"Hence,":[61],"consider":[63],"only":[64],"minimum":[68],"greater":[70],"than":[71],"or":[72,79,101,142,150],"2.":[75],"Therefore,":[76],"asymmetric":[80,102],"structure":[81,103],"can":[84],"be":[85,105],"useful":[86],"determining":[88],"whether":[89,145],"On":[95],"other":[97],"hand,":[98],"even":[106],"more":[107],"helpful":[108],"when":[109],"studying":[110],"(1,2)-PDS":[114,132,163],"relation":[116],"graph.":[123],"Moreover,":[124],"analyze":[126],"structural":[128],"conditions":[129,158],"under":[130],"which":[131],"exist,":[133],"considering":[134],"cases":[135],"where":[136],"are":[140,148],"adjacent":[141],"nonadjacent,":[143],"and":[144,156,173],"their":[146],"neighborhoods":[147],"identical":[149],"distinct.":[151],"Our":[152],"study":[153],"provides":[154],"necessary":[155],"sufficient":[157],"for":[159],"given":[166],"cases,":[167],"extending":[168],"understanding":[170],"(1,2)-domination":[172],"its":[174],"applications.":[175]},"counts_by_year":[{"year":2026,"cited_by_count":1}],"updated_date":"2025-12-21T23:12:01.093139","created_date":"2025-10-10T00:00:00"}
