{"?xml":{"@version":"1.0"},"edm:RDF":{"@xmlns:dc":"http://purl.org/dc/elements/1.1/","@xmlns:edm":"http://www.europeana.eu/schemas/edm/","@xmlns:wgs84_pos":"http://www.w3.org/2003/01/geo/wgs84_pos","@xmlns:foaf":"http://xmlns.com/foaf/0.1/","@xmlns:rdaGr2":"http://rdvocab.info/ElementsGr2","@xmlns:oai":"http://www.openarchives.org/OAI/2.0/","@xmlns:owl":"http://www.w3.org/2002/07/owl#","@xmlns:rdf":"http://www.w3.org/1999/02/22-rdf-syntax-ns#","@xmlns:ore":"http://www.openarchives.org/ore/terms/","@xmlns:skos":"http://www.w3.org/2004/02/skos/core#","@xmlns:dcterms":"http://purl.org/dc/terms/","edm:WebResource":[{"@rdf:about":"http://www.dlib.si/stream/URN:NBN:SI:doc-19AVH494/b977963e-1894-4637-8628-7f10371d10d2/PDF","dcterms:extent":"311 KB"},{"@rdf:about":"http://www.dlib.si/stream/URN:NBN:SI:doc-19AVH494/820a9f65-f2f0-4560-bc3c-89bd64245864/TEXT","dcterms:extent":"22 KB"}],"edm:TimeSpan":{"@rdf:about":"2008-2025","edm:begin":{"@xml:lang":"en","#text":"2008"},"edm:end":{"@xml:lang":"en","#text":"2025"}},"edm:ProvidedCHO":{"@rdf:about":"URN:NBN:SI:doc-19AVH494","dcterms:isPartOf":[{"@rdf:resource":"https://www.dlib.si/details/URN:NBN:SI:spr-UP1WMFAR"},{"@xml:lang":"sl","#text":"Ars mathematica contemporanea"}],"dcterms:issued":"2021","dc:creator":["Cabrera Martinez, Abel","Rodríguez-Velázquez, Juan Alberto"],"dc:format":[{"@xml:lang":"sl","#text":"številka:2"},{"@xml:lang":"sl","#text":"letnik:20"},{"@xml:lang":"sl","#text":"str. 233-241"}],"dc:identifier":["DOI:10.26493/1855-3974.2284.aeb","ISSN:1855-3966","COBISSID_HOST:92559619","URN:URN:NBN:SI:doc-19AVH494"],"dc:language":"en","dc:publisher":{"@xml:lang":"sl","#text":"Univerza na Primorskem, Fakulteta za matematiko, naravoslovje in informacijske tehnologije"},"dc:subject":[{"@xml:lang":"sl","#text":"leksikografski produkt grafov"},{"@xml:lang":"en","#text":"lexicographic product graph"},{"@xml:lang":"en","#text":"total domination"},{"@xml:lang":"en","#text":"total Roman domination"},{"@xml:lang":"sl","#text":"totalna dominacija"},{"@xml:lang":"sl","#text":"totalna rimljanska dominacija"}],"dcterms:temporal":{"@rdf:resource":"2008-2025"},"dc:title":{"@xml:lang":"sl","#text":"Closed formulas for the total Roman domination number of lexicographic product graphs|"},"dc:description":[{"@xml:lang":"sl","#text":"Let ?$G$? be a graph with no isolated vertex and ?$f\\colon V(G) \\to \\{0, 1, 2\\}$? a function. Let ?$V_i = \\{x \\in V(G) \\colon f(x) = i\\}$? for every ?$i \\in \\{0, 1, 2\\}$?. We say that ?$f$? is a total Roman dominating function on ?$G$? if every vertex in ?$V_0$? is adjacent to at least one vertex in ?$V_2$? and the subgraph induced by ?$V_1 \\cup V_2$? has no isolated vertex. The weight of ?$f$? is ?$\\omega (f)=\\sum_{v \\in V(G)}f(v)$?. The minimum weight among all total Roman dominating functions on ?$G$? is the total Roman domination number of ?$G$?, denoted by $\\gamma_{tR}(G)$. It is known that the general problem of computing $\\gamma_{tR}(G)$ is NP-hard. In this paper, we show that if ?$G$? is a graph with no isolated vertex and ?$H$? is a nontrivial graph, then the total Roman domination number of the lexicographic product graph ?$G \\circ H$? is given by ?$$\\gamma_{tR}(G)(G \\circ H) = \\begin{cases} 2 \\gamma_{tR}(G) & \\text{if} \\; \\gamma(H) \\ge 2, \\\\ \\xi(G) & \\text{if} \\; \\gamma(H)=1,\\end{cases}$$? where ?$\\gamma(H)$? is the domination number of ?$H$?, ?$\\gamma_{t}(G)$? is the total domination number of ?$G$? and ?$\\xi (G)$? is a domination parameter defined on ?$G$?"},{"@xml:lang":"sl","#text":"Naj bo ?$G$? graf brez izoliranih vozlišč in ?$f\\colon V(G) \\to \\{0, 1, 2\\}$? funkcija. Naj bo ?V$V_i = \\{x \\in V(G) \\colon f(x) = i\\}$? za vsak ?$i \\in \\{0, 1, 2\\}$?. Funkcija ?$f$? se imenuje totalna rimljanska dominacijska funkcija na ?$G$?, če je vsako vozlišče množice ?$V_0$? sosednje najmanj enemu vozlišču množice ?$V_2$? in če podgraf, induciran z ?$V_1 \\cup V_2$?, nima nobenega izoliranega vozlišča. Teža funkcije ?$f$? je ?$\\omega (f)=\\sum_{v \\in V(G)}f(v)$?. Minimalno težo, ki jo lahko dosežejo totalno rimljansko dominacijske funkcije na ?$G$?, imenujemo totalno rimljansko dominacijsko število grafa ?$G$? in označimo z ?$\\gamma_{tR}(G)$?. Znano je, da je splošni problem izračuna ?$\\gamma_{tR}(G)$? NP-težek. V članku pokažemo, da če je ?$G$? graf, ki nima nobenega izoliranega vozlišča, ?$H$? pa je netrivialen graf, potem je totalno rimljansko dominacijsko število leksikografskega produkta grafov ?$G \\circ H$? podano z izrazom ?$$\\gamma_{tR}(G)(G \\circ H) = \\begin{cases} 2 \\gamma_{tR}(G) & \\text{if} \\; \\gamma(H) \\ge 2, \\\\ \\xi(G) & \\text{if} \\; \\gamma(H)=1,\\end{cases}$$? kjer je ?$\\gamma(H)$? dominacijsko število grafa ?$H$?, ?$\\gamma_{t}(G)$? je totalno dominacijsko število grafa ?$G$?, ?$\\xi (G)$? pa je dominacijski parameter, ki je definiran na grafu ?$G$?"}],"edm:type":"TEXT","dc:type":[{"@xml:lang":"sl","#text":"znanstveno časopisje"},{"@xml:lang":"en","#text":"journals"},{"@rdf:resource":"http://www.wikidata.org/entity/Q361785"}]},"ore:Aggregation":{"@rdf:about":"http://www.dlib.si/?URN=URN:NBN:SI:doc-19AVH494","edm:aggregatedCHO":{"@rdf:resource":"URN:NBN:SI:doc-19AVH494"},"edm:isShownBy":{"@rdf:resource":"http://www.dlib.si/stream/URN:NBN:SI:doc-19AVH494/b977963e-1894-4637-8628-7f10371d10d2/PDF"},"edm:rights":{"@rdf:resource":"http://creativecommons.org/licenses/by/4.0/"},"edm:provider":"Slovenian National E-content Aggregator","edm:intermediateProvider":{"@xml:lang":"en","#text":"National and University Library of Slovenia"},"edm:dataProvider":{"@xml:lang":"sl","#text":"Univerza na Primorskem, Fakulteta za naravoslovje, matematiko in informacijske tehnologije"},"edm:object":{"@rdf:resource":"http://www.dlib.si/streamdb/URN:NBN:SI:doc-19AVH494/maxi/edm"},"edm:isShownAt":{"@rdf:resource":"http://www.dlib.si/details/URN:NBN:SI:doc-19AVH494"}}}}