{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,11]],"date-time":"2026-05-11T12:41:08Z","timestamp":1778503268615,"version":"3.51.4"},"reference-count":23,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2021,7,22]],"date-time":"2021-07-22T00:00:00Z","timestamp":1626912000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>This paper is devoted to the study of the quadruple Roman domination in trees, and it is a contribution to the Special Issue \u201cTheoretical computer science and discrete mathematics\u201d of Symmetry. For any positive integer k, a [k]-Roman dominating function ([k]-RDF) of a simple graph G is a function from the vertex set V of G to the set {0,1,2,\u2026,k+1} if for any vertex u\u2208V with f(u)&lt;k, \u2211x\u2208N(u)\u222a{u}f(x)\u2265|{x\u2208N(u):f(x)\u22651}|+k, where N(u) is the open neighborhood of u. The weight of a [k]-RDF is the value \u03a3v\u2208Vf(v). The minimum weight of a [k]-RDF is called the [k]-Roman domination number \u03b3[kR](G) of G. In this paper, we establish sharp upper and lower bounds on \u03b3[4R](T) for nontrivial trees T and characterize extremal trees.<\/jats:p>","DOI":"10.3390\/sym13081318","type":"journal-article","created":{"date-parts":[[2021,7,22]],"date-time":"2021-07-22T22:37:14Z","timestamp":1626993434000},"page":"1318","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":8,"title":["Quadruple Roman Domination in Trees"],"prefix":"10.3390","volume":"13","author":[{"given":"Zheng","family":"Kou","sequence":"first","affiliation":[{"name":"Institute of Computing Science and Technology, Guangzhou University, Guangzhou 510006, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Saeed","family":"Kosari","sequence":"additional","affiliation":[{"name":"Institute of Computing Science and Technology, Guangzhou University, Guangzhou 510006, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1267-696X","authenticated-orcid":false,"given":"Guoliang","family":"Hao","sequence":"additional","affiliation":[{"name":"College of Science, East China University of Technology, Nanchang 330013, China"},{"name":"College of Mathematics and Data Science, Minjiang University, Fuzhou 350108, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jafar","family":"Amjadi","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Azarbaijan Shahid Madani University, Tabriz 51368, Iran"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Nesa","family":"Khalili","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Azarbaijan Shahid Madani University, Tabriz 51368, Iran"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,7,22]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Haynes, T.W., Hedetniemi, S.T., and Slater, P.J. 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