{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:59:52Z","timestamp":1760147992478,"version":"build-2065373602"},"reference-count":16,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2023,3,17]],"date-time":"2023-03-17T00:00:00Z","timestamp":1679011200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Fundamental Research Program of Shanxi Province","award":["20210302123202","201901D211197"],"award-info":[{"award-number":["20210302123202","201901D211197"]}]},{"name":"Youth Foundation of Shanxi Province","award":["20210302123202","201901D211197"],"award-info":[{"award-number":["20210302123202","201901D211197"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Let D=(V(D),A(D)) be a finite, simple digraph and k a positive integer. A function f:V(D)\u2192{0,1,2,\u2026,k+1} is called a [k]-Roman dominating function (for short, [k]-RDF) if f(AN\u2212[v])\u2265|AN\u2212(v)|+k for any vertex v\u2208V(D), where AN\u2212(v)={u\u2208N\u2212(v):f(u)\u22651} and AN\u2212[v]=AN\u2212(v)\u222a{v}. The weight of a [k]-RDF f is \u03c9(f)=\u2211v\u2208V(D)f(v). The minimum weight of any [k]-RDF on D is the [k]-Roman domination number, denoted by \u03b3[kR](D). For k=2 and k=3, we call them the double Roman domination number and the triple Roman domination number, respectively. In this paper, we presented some general bounds and the Nordhaus\u2013Gaddum bound on the [k]-Roman domination number and we also determined the bounds on the [k]-Roman domination number related to other domination parameters, such as domination number and signed domination number. Additionally, we give the exact values of \u03b3[kR](Pn) and \u03b3[kR](Cn) for the directed path Pn and directed cycle Cn.<\/jats:p>","DOI":"10.3390\/sym15030743","type":"journal-article","created":{"date-parts":[[2023,3,20]],"date-time":"2023-03-20T05:46:42Z","timestamp":1679291202000},"page":"743","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["[k]-Roman Domination in Digraphs"],"prefix":"10.3390","volume":"15","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-8833-2952","authenticated-orcid":false,"given":"Xinhong","family":"Zhang","sequence":"first","affiliation":[{"name":"Department of Applied Mathematics, Taiyuan University of Science and Technology, Taiyuan 030024, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xin","family":"Song","sequence":"additional","affiliation":[{"name":"Department of Applied Mathematics, Taiyuan University of Science and Technology, Taiyuan 030024, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ruijuan","family":"Li","sequence":"additional","affiliation":[{"name":"School of Mathematical Sciences, Shanxi University, Taiyuan 030006, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,3,17]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Haynes, T.W., Hedetniemi, S.T., and Slater, P.J. (1998). Fundamentals of Domination in Graphs, Marcel Dekker, Inc.","DOI":"10.1002\/(SICI)1097-0037(199810)32:3<199::AID-NET4>3.0.CO;2-F"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Haynes, T.W., Hedetniemi, S.T., and Slater, P.J. (1998). Domination in Graphs: Advanced Topics, Marcel Dekker, Inc.","DOI":"10.1002\/(SICI)1097-0037(199810)32:3<199::AID-NET4>3.0.CO;2-F"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1053","DOI":"10.1016\/j.dam.2011.12.027","article-title":"Directed domination in oriented graphs","volume":"160","author":"Caro","year":"2012","journal-title":"Discret. Appl. Math."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"241","DOI":"10.1007\/s10878-012-9500-0","article-title":"Signed Roman domination in graphs","volume":"27","author":"Ahangar","year":"2014","journal-title":"J. Comb. Optim."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"2567","DOI":"10.1016\/j.disc.2008.04.001","article-title":"A lower bounds on the signed domination numbers of directed graphs","volume":"309","author":"Karami","year":"2009","journal-title":"Discret. Math."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1575","DOI":"10.1137\/070699688","article-title":"Extremal problems for Roman domination","volume":"23","author":"Chambers","year":"2009","journal-title":"SIAM J. Discret. Math."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"3447","DOI":"10.1016\/j.disc.2008.09.043","article-title":"On the Roman domination number of a graph","volume":"309","author":"Favaron","year":"2009","journal-title":"Discret. Math."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"155","DOI":"10.2298\/AADM140210003B","article-title":"The differential and the roman domination number of a graph","volume":"8","author":"Bermudo","year":"2014","journal-title":"Appl. Anal. Discret. Math."},{"key":"ref_9","first-page":"239","article-title":"Roman domination in graphs","volume":"266","author":"Cockayne","year":"2003","journal-title":"Discret. Math."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"539","DOI":"10.1016\/j.dam.2020.06.023","article-title":"On algorithmic complexity of double Roman domination","volume":"285","author":"Poureidi","year":"2020","journal-title":"Discret. Appl. Math."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"125444","DOI":"10.1016\/j.amc.2020.125444","article-title":"Triple Roman domination in graphs","volume":"391","author":"Ahangar","year":"2021","journal-title":"Appl. Math. Comput."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"228","DOI":"10.1016\/j.dam.2022.02.015","article-title":"Global triple Roman dominating function","volume":"314","author":"Pour","year":"2022","journal-title":"Discret. Appl. Math."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1907","DOI":"10.1007\/s40840-017-0582-9","article-title":"Double Roman Domination in Digraphs","volume":"42","author":"Hao","year":"2019","journal-title":"Bull. Malays. Math. Sci. Soc."},{"key":"ref_14","unstructured":"Zhang, X., Guo, Y., and Li, R. (Pure Appl. Func. Anal., 2023). The Roman domination of Kautz digraphs and generalized Kautz digraphs, Pure Appl. Func. Anal., in press."},{"key":"ref_15","unstructured":"Zhang, X., and Guo, Y. (Open Math., 2023). The Roman domination numbers of the directed de Bruijn and generalized directed de Bruijn graphs, Open Math., in press."},{"key":"ref_16","unstructured":"Harary, F., Norman, R.Z., and Cartwright, D. (1965). Structural Models, Wiley."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/3\/743\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T18:57:50Z","timestamp":1760122670000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/3\/743"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,3,17]]},"references-count":16,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2023,3]]}},"alternative-id":["sym15030743"],"URL":"https:\/\/doi.org\/10.3390\/sym15030743","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2023,3,17]]}}}