{"id":"https://openalex.org/W3109876496","doi":"https://doi.org/10.3390/sym12111929","title":"Total Domination in Rooted Product Graphs","display_name":"Total Domination in Rooted Product Graphs","publication_year":2020,"publication_date":"2020-11-23","ids":{"openalex":"https://openalex.org/W3109876496","doi":"https://doi.org/10.3390/sym12111929","mag":"3109876496"},"language":"en","primary_location":{"id":"doi:10.3390/sym12111929","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym12111929","pdf_url":"https://www.mdpi.com/2073-8994/12/11/1929/pdf?version=1606206632","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","doaj"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://www.mdpi.com/2073-8994/12/11/1929/pdf?version=1606206632","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5000212299","display_name":"Abel Cabrera Mart\u00ednez","orcid":"https://orcid.org/0000-0003-2806-4842"},"institutions":[{"id":"https://openalex.org/I55952717","display_name":"Universitat Rovira i Virgili","ror":"https://ror.org/00g5sqv46","country_code":"ES","type":"education","lineage":["https://openalex.org/I55952717"]}],"countries":["ES"],"is_corresponding":false,"raw_author_name":"Abel Cabrera Mart\u00ednez","raw_affiliation_strings":["Departament d\u2019Enginyeria Inform\u00e0tica i Matem\u00e0tiques, Universitat Rovira i Virgili, Av. Pa\u00efsos Catalans 26, 43007 Tarragona, Spain"],"raw_orcid":"https://orcid.org/0000-0003-2806-4842","affiliations":[{"raw_affiliation_string":"Departament d\u2019Enginyeria Inform\u00e0tica i Matem\u00e0tiques, Universitat Rovira i Virgili, Av. Pa\u00efsos Catalans 26, 43007 Tarragona, Spain","institution_ids":["https://openalex.org/I55952717"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5066088171","display_name":"Juan A. Rodr\u00edguez\u2010Vel\u00e1zquez","orcid":"https://orcid.org/0000-0002-9082-7647"},"institutions":[{"id":"https://openalex.org/I55952717","display_name":"Universitat Rovira i Virgili","ror":"https://ror.org/00g5sqv46","country_code":"ES","type":"education","lineage":["https://openalex.org/I55952717"]}],"countries":["ES"],"is_corresponding":true,"raw_author_name":"Juan A. Rodr\u00edguez-Vel\u00e1zquez","raw_affiliation_strings":["Departament d\u2019Enginyeria Inform\u00e0tica i Matem\u00e0tiques, Universitat Rovira i Virgili, Av. Pa\u00efsos Catalans 26, 43007 Tarragona, Spain"],"raw_orcid":"https://orcid.org/0000-0002-9082-7647","affiliations":[{"raw_affiliation_string":"Departament d\u2019Enginyeria Inform\u00e0tica i Matem\u00e0tiques, Universitat Rovira i Virgili, Av. Pa\u00efsos Catalans 26, 43007 Tarragona, Spain","institution_ids":["https://openalex.org/I55952717"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5066088171"],"corresponding_institution_ids":["https://openalex.org/I55952717"],"apc_list":{"value":2000,"currency":"CHF","value_usd":2165},"apc_paid":{"value":2000,"currency":"CHF","value_usd":2165},"fwci":0.583,"has_fulltext":true,"cited_by_count":9,"citation_normalized_percentile":{"value":0.74830839,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":97},"biblio":{"volume":"12","issue":"11","first_page":"1929","last_page":"1929"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10374","display_name":"Advanced Graph Theory Research","score":0.9991000294685364,"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":0.9991000294685364,"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":0.9979000091552734,"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":0.9447000026702881,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/domination-analysis","display_name":"Domination analysis","score":0.7229993939399719},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.5844161510467529},{"id":"https://openalex.org/keywords/product","display_name":"Product (mathematics)","score":0.5391474366188049},{"id":"https://openalex.org/keywords/graph-product","display_name":"Graph product","score":0.53572016954422},{"id":"https://openalex.org/keywords/graph","display_name":"Graph","score":0.5079308152198792},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.4975149929523468},{"id":"https://openalex.org/keywords/graph-theory","display_name":"Graph theory","score":0.48886650800704956},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.46522805094718933},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.37374770641326904},{"id":"https://openalex.org/keywords/pathwidth","display_name":"Pathwidth","score":0.18243318796157837},{"id":"https://openalex.org/keywords/line-graph","display_name":"Line graph","score":0.16307884454727173},{"id":"https://openalex.org/keywords/vertex","display_name":"Vertex (graph theory)","score":0.06935963034629822}],"concepts":[{"id":"https://openalex.org/C83833204","wikidata":"https://www.wikidata.org/wiki/Q5290322","display_name":"Domination analysis","level":4,"score":0.7229993939399719},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.5844161510467529},{"id":"https://openalex.org/C90673727","wikidata":"https://www.wikidata.org/wiki/Q901718","display_name":"Product (mathematics)","level":2,"score":0.5391474366188049},{"id":"https://openalex.org/C68103157","wikidata":"https://www.wikidata.org/wiki/Q569347","display_name":"Graph product","level":5,"score":0.53572016954422},{"id":"https://openalex.org/C132525143","wikidata":"https://www.wikidata.org/wiki/Q141488","display_name":"Graph","level":2,"score":0.5079308152198792},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.4975149929523468},{"id":"https://openalex.org/C88230418","wikidata":"https://www.wikidata.org/wiki/Q131476","display_name":"Graph theory","level":2,"score":0.48886650800704956},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.46522805094718933},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.37374770641326904},{"id":"https://openalex.org/C43517604","wikidata":"https://www.wikidata.org/wiki/Q7144893","display_name":"Pathwidth","level":4,"score":0.18243318796157837},{"id":"https://openalex.org/C203776342","wikidata":"https://www.wikidata.org/wiki/Q1378376","display_name":"Line graph","level":3,"score":0.16307884454727173},{"id":"https://openalex.org/C80899671","wikidata":"https://www.wikidata.org/wiki/Q1304193","display_name":"Vertex (graph theory)","level":3,"score":0.06935963034629822},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.3390/sym12111929","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym12111929","pdf_url":"https://www.mdpi.com/2073-8994/12/11/1929/pdf?version=1606206632","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"},{"id":"pmh:oai:doaj.org/article:094dd6fe8f8a49ccbe0d28a068833f6d","is_oa":true,"landing_page_url":"https://doaj.org/article/094dd6fe8f8a49ccbe0d28a068833f6d","pdf_url":null,"source":{"id":"https://openalex.org/S4306401280","display_name":"DOAJ (DOAJ: Directory of Open Access Journals)","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-sa","license_id":"https://openalex.org/licenses/cc-by-sa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Symmetry, Vol 12, Iss 11, p 1929 (2020)","raw_type":"article"},{"id":"pmh:oai:mdpi.com:/2073-8994/12/11/1929/","is_oa":true,"landing_page_url":"https://dx.doi.org/10.3390/sym12111929","pdf_url":null,"source":{"id":"https://openalex.org/S4306400947","display_name":"MDPI (MDPI AG)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I4210097602","host_organization_name":"Multidisciplinary Digital Publishing Institute (Switzerland)","host_organization_lineage":["https://openalex.org/I4210097602"],"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":"Symmetry","raw_type":"Text"}],"best_oa_location":{"id":"doi:10.3390/sym12111929","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym12111929","pdf_url":"https://www.mdpi.com/2073-8994/12/11/1929/pdf?version=1606206632","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":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W3109876496.pdf","grobid_xml":"https://content.openalex.org/works/W3109876496.grobid-xml"},"referenced_works_count":24,"referenced_works":["https://openalex.org/W560340026","https://openalex.org/W573195588","https://openalex.org/W1594561525","https://openalex.org/W1994990018","https://openalex.org/W2003370404","https://openalex.org/W2045811688","https://openalex.org/W2125447390","https://openalex.org/W2549219973","https://openalex.org/W2592437091","https://openalex.org/W2765612660","https://openalex.org/W2916398069","https://openalex.org/W2951992432","https://openalex.org/W2980330514","https://openalex.org/W3004945501","https://openalex.org/W3017356100","https://openalex.org/W3033894122","https://openalex.org/W3094086239","https://openalex.org/W3101183505","https://openalex.org/W3104773112","https://openalex.org/W3107412252","https://openalex.org/W3120145815","https://openalex.org/W6616433025","https://openalex.org/W6634712007","https://openalex.org/W6729293389"],"related_works":["https://openalex.org/W2794522194","https://openalex.org/W2349109195","https://openalex.org/W2069797780","https://openalex.org/W2078360221","https://openalex.org/W1976768565","https://openalex.org/W2080000226","https://openalex.org/W2989120336","https://openalex.org/W3164121007","https://openalex.org/W2509241435","https://openalex.org/W4280538489"],"abstract_inverted_index":{"During":[0],"the":[1,11,34,50,61,68,75,85,104,116],"last":[2],"few":[3],"decades,":[4],"domination":[5,29,87,106],"theory":[6],"has":[7],"been":[8],"one":[9],"of":[10,15,36,47,52,57,63,89,108],"most":[12],"active":[13],"areas":[14],"research":[16],"within":[17],"graph":[18],"theory.":[19,76],"Currently,":[20],"there":[21,39,98],"are":[22,40,99],"more":[23],"than":[24],"4400":[25],"published":[26,43],"papers":[27,59],"on":[28],"and":[30,45,113],"related":[31],"parameters.":[32],"In":[33,93],"case":[35,51,62],"total":[37,86,105],"domination,":[38],"over":[41],"580":[42],"papers,":[44],"50":[46],"them":[48],"concern":[49],"product":[53,65,91,111],"graphs.":[54,66,92],"However,":[55],"none":[56],"these":[58,119],"discusses":[60],"rooted":[64,90,110],"Precisely,":[67],"present":[69],"paper":[70],"covers":[71],"this":[72],"gap":[73],"in":[74],"Our":[77],"goal":[78],"is":[79],"to":[80],"provide":[81],"closed":[82],"formulas":[83],"for":[84,103],"number":[88,107],"particular,":[94],"we":[95,114],"show":[96],"that":[97],"four":[100],"possible":[101],"expressions":[102],"a":[109],"graph,":[112],"characterize":[115],"graphs":[117],"reaching":[118],"expressions.":[120]},"counts_by_year":[{"year":2025,"cited_by_count":2},{"year":2024,"cited_by_count":3},{"year":2023,"cited_by_count":2},{"year":2022,"cited_by_count":1},{"year":2021,"cited_by_count":1}],"updated_date":"2026-05-21T06:26:12.895304","created_date":"2025-10-10T00:00:00"}