{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,3,30]],"date-time":"2022-03-30T22:19:18Z","timestamp":1648678758710},"reference-count":13,"publisher":"World Scientific Pub Co Pte Lt","issue":"03","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Discrete Math. Algorithm. Appl."],"published-print":{"date-parts":[[2014,9]]},"abstract":"<jats:p> Let G = (V, E) be a graph and let k be a positive integer. A Roman k-dominating function ( R k-DF) on G is a function f : V(G) \u2192 {0, 1, 2} such that every vertex u for which f(u) = 0 is adjacent to at least k vertices v<jats:sub>1<\/jats:sub>, v<jats:sub>2<\/jats:sub>, \u2026, v<jats:sub>k<\/jats:sub> with f(v<jats:sub>i<\/jats:sub>) = 2 for i = 1, 2, \u2026, k. The weight of an R k-DF is the value f(V(G)) = \u2211<jats:sub>u\u2208V(G)<\/jats:sub> f(u) and the minimum weight of an R k-DF on G is called the Roman k-domination number \u03b3<jats:sub>kR<\/jats:sub>(G) of G. In this paper, we present relations between \u03b3<jats:sub>kR<\/jats:sub>(G) and \u03b3<jats:sub>R<\/jats:sub>(G). Moreover, we give characterizations of some classes of graphs attaining equality in these relations. Finally, we establish a relation between \u03b3<jats:sub>kR<\/jats:sub>(G) and \u03b3<jats:sub>R<\/jats:sub>(G) for {K<jats:sub>1,3<\/jats:sub>, K<jats:sub>1,3<\/jats:sub>+e}-free graphs and we characterize all such graphs G with \u03b3<jats:sub>kR<\/jats:sub>(G) = \u03b3<jats:sub>R<\/jats:sub>(G)+t, where [Formula: see text]. <\/jats:p>","DOI":"10.1142\/s1793830914500451","type":"journal-article","created":{"date-parts":[[2014,4,21]],"date-time":"2014-04-21T21:43:01Z","timestamp":1398116581000},"page":"1450045","source":"Crossref","is-referenced-by-count":2,"title":["Relations between the Roman k-domination and Roman domination numbers in graphs"],"prefix":"10.1142","volume":"06","author":[{"given":"Ahmed","family":"Bouchou","sequence":"first","affiliation":[{"name":"Department of Electrical Engineering and Computer Science, University of M\u00e9d\u00e9a, Algeria"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Mostafa","family":"Blidia","sequence":"additional","affiliation":[{"name":"LAMDA-RO Laboratory, Department of Mathematics, University of Blida, B.P. 270, Blida, Algeria"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Mustapha","family":"Chellali","sequence":"additional","affiliation":[{"name":"LAMDA-RO Laboratory, Department of Mathematics, University of Blida, B.P. 270, Blida, Algeria"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"219","published-online":{"date-parts":[[2014,6,16]]},"reference":[{"key":"rf1","doi-asserted-by":"publisher","DOI":"10.1137\/070699688"},{"key":"rf2","doi-asserted-by":"publisher","DOI":"10.1016\/j.disc.2003.06.004"},{"key":"rf3","unstructured":"J. F.\u00a0Fink and M. S.\u00a0Jacobson, Graph Theory with Applications to Algorithms and Computer (John Wiley and Sons, New York, 1985)\u00a0pp. 283\u2013300."},{"key":"rf4","unstructured":"J. F.\u00a0Fink and M. S.\u00a0Jacobson, Graph Theory with Applications to Algorithms and Computer (John Wiley and Sons, New York, 1985)\u00a0pp. 301\u2013311."},{"key":"rf5","doi-asserted-by":"publisher","DOI":"10.1007\/BF01848079"},{"key":"rf6","first-page":"247","volume":"44","author":"Hansberg A.","journal-title":"Australas. J. Combin."},{"key":"rf7","doi-asserted-by":"publisher","DOI":"10.4134\/JKMS.2009.46.6.1309"},{"key":"rf8","doi-asserted-by":"publisher","DOI":"10.1090\/coll\/038"},{"key":"rf9","doi-asserted-by":"publisher","DOI":"10.1002\/jgt.3190060104"},{"key":"rf10","doi-asserted-by":"publisher","DOI":"10.1016\/S0012-365X(98)00103-4"},{"key":"rf11","doi-asserted-by":"publisher","DOI":"10.2307\/2589113"},{"key":"rf12","doi-asserted-by":"publisher","DOI":"10.1038\/scientificamerican1299-136"},{"key":"rf13","first-page":"1","volume":"216","author":"Xu B.","journal-title":"Discrete Math."}],"container-title":["Discrete Mathematics, Algorithms and Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.worldscientific.com\/doi\/pdf\/10.1142\/S1793830914500451","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,8,6]],"date-time":"2019-08-06T13:16:16Z","timestamp":1565097376000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.worldscientific.com\/doi\/abs\/10.1142\/S1793830914500451"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2014,6,16]]},"references-count":13,"journal-issue":{"issue":"03","published-online":{"date-parts":[[2014,6,16]]},"published-print":{"date-parts":[[2014,9]]}},"alternative-id":["10.1142\/S1793830914500451"],"URL":"https:\/\/doi.org\/10.1142\/s1793830914500451","relation":{},"ISSN":["1793-8309","1793-8317"],"issn-type":[{"value":"1793-8309","type":"print"},{"value":"1793-8317","type":"electronic"}],"subject":[],"published":{"date-parts":[[2014,6,16]]}}}