{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T14:29:28Z","timestamp":1753885768584,"version":"3.41.2"},"reference-count":19,"publisher":"World Scientific Pub Co Pte Ltd","issue":"02","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Discrete Math. Algorithm. Appl."],"published-print":{"date-parts":[[2023,2]]},"abstract":"<jats:p> Let [Formula: see text] be an integer and [Formula: see text] be a simple graph with vertex set [Formula: see text]. Let [Formula: see text] be a function that assigns label from the set [Formula: see text] to the vertices of a graph [Formula: see text]. For a vertex [Formula: see text], the active neighborhood of [Formula: see text], denoted by [Formula: see text], is the set of vertices [Formula: see text] such that [Formula: see text]. A [Formula: see text]-RDF is a function [Formula: see text] satisfying the condition that for any vertex [Formula: see text] with [Formula: see text], [Formula: see text]. The weight of a [Formula: see text]-RDF is [Formula: see text]. The [Formula: see text]-Roman domination number [Formula: see text] of [Formula: see text] is the minimum weight of an [Formula: see text]-RDF on [Formula: see text]. The case [Formula: see text] is called quadruple Roman domination number. In this paper, we first establish an upper bound for quadruple Roman domination number of graphs with minimum degree two, and then we derive a Nordhaus\u2013Gaddum bound on the quadruple Roman domination number of graphs. <\/jats:p>","DOI":"10.1142\/s1793830922500781","type":"journal-article","created":{"date-parts":[[2022,4,13]],"date-time":"2022-04-13T06:10:18Z","timestamp":1649830218000},"source":"Crossref","is-referenced-by-count":3,"title":["New results on quadruple Roman domination in graphs"],"prefix":"10.1142","volume":"15","author":[{"given":"J.","family":"Amjadi","sequence":"first","affiliation":[{"name":"Department of Mathematics, Azarbaijan Shahid Madani University, Tabriz, I. R. Iran"}]},{"given":"N.","family":"Khalili","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Azarbaijan Shahid Madani University, Tabriz, I. R. 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