{"id":"https://openalex.org/W4327953199","doi":"https://doi.org/10.3390/sym15030743","title":"[k]-Roman Domination in Digraphs","display_name":"[k]-Roman Domination in Digraphs","publication_year":2023,"publication_date":"2023-03-17","ids":{"openalex":"https://openalex.org/W4327953199","doi":"https://doi.org/10.3390/sym15030743"},"language":"en","primary_location":{"id":"doi:10.3390/sym15030743","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym15030743","pdf_url":"https://www.mdpi.com/2073-8994/15/3/743/pdf?version=1679055611","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/2073-8994/15/3/743/pdf?version=1679055611","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5015234843","display_name":"Xinhong Zhang","orcid":"https://orcid.org/0000-0002-8833-2952"},"institutions":[{"id":"https://openalex.org/I46305995","display_name":"Taiyuan University of Science and Technology","ror":"https://ror.org/01wcbdc92","country_code":"CN","type":"education","lineage":["https://openalex.org/I46305995"]}],"countries":["CN"],"is_corresponding":true,"raw_author_name":"Xinhong Zhang","raw_affiliation_strings":["Department of Applied Mathematics, Taiyuan University of Science and Technology, Taiyuan 030024, China"],"raw_orcid":"https://orcid.org/0000-0002-8833-2952","affiliations":[{"raw_affiliation_string":"Department of Applied Mathematics, Taiyuan University of Science and Technology, Taiyuan 030024, China","institution_ids":["https://openalex.org/I46305995"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5026871802","display_name":"Xin Song","orcid":"https://orcid.org/0000-0001-9882-8356"},"institutions":[{"id":"https://openalex.org/I46305995","display_name":"Taiyuan University of Science and Technology","ror":"https://ror.org/01wcbdc92","country_code":"CN","type":"education","lineage":["https://openalex.org/I46305995"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Xin Song","raw_affiliation_strings":["Department of Applied Mathematics, Taiyuan University of Science and Technology, Taiyuan 030024, China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Applied Mathematics, Taiyuan University of Science and Technology, Taiyuan 030024, China","institution_ids":["https://openalex.org/I46305995"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5060356084","display_name":"Ruijuan Li","orcid":"https://orcid.org/0000-0002-2278-4622"},"institutions":[{"id":"https://openalex.org/I181877577","display_name":"Shanxi University","ror":"https://ror.org/03y3e3s17","country_code":"CN","type":"education","lineage":["https://openalex.org/I181877577"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Ruijuan Li","raw_affiliation_strings":["School of Mathematical Sciences, Shanxi University, Taiyuan 030006, China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Mathematical Sciences, Shanxi University, Taiyuan 030006, China","institution_ids":["https://openalex.org/I181877577"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5015234843"],"corresponding_institution_ids":["https://openalex.org/I46305995"],"apc_list":{"value":2000,"currency":"CHF","value_usd":2165},"apc_paid":{"value":2000,"currency":"CHF","value_usd":2165},"fwci":0.1896,"has_fulltext":false,"cited_by_count":1,"citation_normalized_percentile":{"value":0.50001667,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":91,"max":95},"biblio":{"volume":"15","issue":"3","first_page":"743","last_page":"743"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10374","display_name":"Advanced Graph Theory Research","score":0.9998000264167786,"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":0.9998000264167786,"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/T10720","display_name":"Complexity and Algorithms in Graphs","score":0.9810000061988831,"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/T10829","display_name":"Interconnection Networks and Systems","score":0.980400025844574,"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":0.8383679389953613},{"id":"https://openalex.org/keywords/domination-analysis","display_name":"Domination analysis","score":0.789162278175354},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7031877040863037},{"id":"https://openalex.org/keywords/digraph","display_name":"Digraph","score":0.6507846117019653},{"id":"https://openalex.org/keywords/vertex","display_name":"Vertex (graph theory)","score":0.556657075881958},{"id":"https://openalex.org/keywords/upper-and-lower-bounds","display_name":"Upper and lower bounds","score":0.4881797432899475},{"id":"https://openalex.org/keywords/path","display_name":"Path (computing)","score":0.4535974860191345},{"id":"https://openalex.org/keywords/function","display_name":"Function (biology)","score":0.43762046098709106},{"id":"https://openalex.org/keywords/graph","display_name":"Graph","score":0.114723801612854},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.06515336036682129}],"concepts":[{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.8383679389953613},{"id":"https://openalex.org/C83833204","wikidata":"https://www.wikidata.org/wiki/Q5290322","display_name":"Domination analysis","level":4,"score":0.789162278175354},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7031877040863037},{"id":"https://openalex.org/C2779145032","wikidata":"https://www.wikidata.org/wiki/Q191494","display_name":"Digraph","level":2,"score":0.6507846117019653},{"id":"https://openalex.org/C80899671","wikidata":"https://www.wikidata.org/wiki/Q1304193","display_name":"Vertex (graph theory)","level":3,"score":0.556657075881958},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.4881797432899475},{"id":"https://openalex.org/C2777735758","wikidata":"https://www.wikidata.org/wiki/Q817765","display_name":"Path (computing)","level":2,"score":0.4535974860191345},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.43762046098709106},{"id":"https://openalex.org/C132525143","wikidata":"https://www.wikidata.org/wiki/Q141488","display_name":"Graph","level":2,"score":0.114723801612854},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.06515336036682129},{"id":"https://openalex.org/C78458016","wikidata":"https://www.wikidata.org/wiki/Q840400","display_name":"Evolutionary biology","level":1,"score":0.0},{"id":"https://openalex.org/C199360897","wikidata":"https://www.wikidata.org/wiki/Q9143","display_name":"Programming language","level":1,"score":0.0},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.0},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.3390/sym15030743","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym15030743","pdf_url":"https://www.mdpi.com/2073-8994/15/3/743/pdf?version=1679055611","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:fb5ba73846ae48e8aba7a67bd41d98e5","is_oa":false,"landing_page_url":"https://doaj.org/article/fb5ba73846ae48e8aba7a67bd41d98e5","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":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Symmetry, Vol 15, Iss 3, p 743 (2023)","raw_type":"article"},{"id":"pmh:oai:mdpi.com:/2073-8994/15/3/743/","is_oa":true,"landing_page_url":"https://dx.doi.org/10.3390/sym15030743","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; Volume 15; Issue 3; Pages: 743","raw_type":"Text"}],"best_oa_location":{"id":"doi:10.3390/sym15030743","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym15030743","pdf_url":"https://www.mdpi.com/2073-8994/15/3/743/pdf?version=1679055611","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":[{"score":0.44999998807907104,"display_name":"Reduced inequalities","id":"https://metadata.un.org/sdg/10"}],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4327953199.pdf"},"referenced_works_count":14,"referenced_works":["https://openalex.org/W1984228559","https://openalex.org/W2003370404","https://openalex.org/W2019578481","https://openalex.org/W2025942997","https://openalex.org/W2028111792","https://openalex.org/W2041964508","https://openalex.org/W2059179208","https://openalex.org/W2140596247","https://openalex.org/W2769897350","https://openalex.org/W3041916182","https://openalex.org/W3092092571","https://openalex.org/W4220923754","https://openalex.org/W6600020652","https://openalex.org/W6646452768"],"related_works":["https://openalex.org/W2964086477","https://openalex.org/W2034343496","https://openalex.org/W4300480352","https://openalex.org/W2963404550","https://openalex.org/W2794522194","https://openalex.org/W2082494488","https://openalex.org/W2054349495","https://openalex.org/W2998116189","https://openalex.org/W2081358954","https://openalex.org/W4210490117"],"abstract_inverted_index":{"Let":[0],"D=(V(D),A(D))":[1],"be":[2],"a":[3,9,17,37],"finite,":[4],"simple":[5],"digraph":[6],"and":[7,32,60,70,85,94,114,126,133],"k":[8],"positive":[10],"integer.":[11],"A":[12],"function":[13,20],"f:V(D)\u2192{0,1,2,\u2026,k+1}":[14],"is":[15,40,50],"called":[16],"[k]-Roman":[18,52,91,102],"dominating":[19],"(for":[21],"short,":[22],"[k]-RDF)":[23],"if":[24],"f(AN\u2212[v])\u2265|AN\u2212(v)|+k":[25],"for":[26,128],"any":[27,46],"vertex":[28],"v\u2208V(D),":[29],"where":[30],"AN\u2212(v)={u\u2208N\u2212(v):f(u)\u22651}":[31],"AN\u2212[v]=AN\u2212(v)\u222a{v}.":[33],"The":[34,42],"weight":[35,44],"of":[36,45,124],"[k]-RDF":[38,47],"f":[39],"\u03c9(f)=\u2211v\u2208V(D)f(v).":[41],"minimum":[43],"on":[48,89,100],"D":[49],"the":[51,65,71,86,90,98,101,121,129],"domination":[53,68,74,92,103,108,112,116],"number,":[54,75],"denoted":[55],"by":[56],"\u03b3[kR](D).":[57],"For":[58],"k=2":[59],"k=3,":[61],"we":[62,80,95,119],"call":[63],"them":[64],"double":[66],"Roman":[67,73],"number":[69,93,104,113],"triple":[72],"respectively.":[76],"In":[77],"this":[78],"paper,":[79],"presented":[81],"some":[82],"general":[83],"bounds":[84,99],"Nordhaus\u2013Gaddum":[87],"bound":[88],"also":[96],"determined":[97],"related":[105],"to":[106],"other":[107],"parameters,":[109],"such":[110],"as":[111],"signed":[115],"number.":[117],"Additionally,":[118],"give":[120],"exact":[122],"values":[123],"\u03b3[kR](Pn)":[125],"\u03b3[kR](Cn)":[127],"directed":[130,134],"path":[131],"Pn":[132],"cycle":[135],"Cn.":[136]},"counts_by_year":[{"year":2025,"cited_by_count":1}],"updated_date":"2026-05-22T06:13:13.366637","created_date":"2025-10-10T00:00:00"}