New results on quadruple Roman domination in graphs

Abstract

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–Gaddum bound on the quadruple Roman domination number of graphs.