Let [Formula: see text] be a finite and 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 quadruple Roman dominating function (QRDF) is a function [Formula: see text] satisfying the condition that for any vertex [Formula: see text] with [Formula: see text]. The weight of a QRDF is [Formula: see text]. The quadruple Roman domination number [Formula: see text] of [Formula: see text] is the minimum weight of a QRDF on [Formula: see text]. In this paper, we investigate the properties of the quadruple Roman domination number of graphs, present bounds on [Formula: see text] and give exact values for some graph families. In addition, complexity results are also obtained.

This paper is devoted to the study of the quadruple Roman domination in trees, and it is a contribution to the Special Issue “Theoretical computer science and discrete mathematics” 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,…,k+1} if for any vertex u∈V with f(u)<k, ∑x∈N(u)∪{u}f(x)≥|{x∈N(u):f(x)≥1}|+k, where N(u) is the open neighborhood of u. The weight of a [k]-RDF is the value Σv∈Vf(v). The minimum weight of a [k]-RDF is called the [k]-Roman domination number γ[kR](G) of G. In this paper, we establish sharp upper and lower bounds on γ[4R](T) for nontrivial trees T and characterize extremal trees.

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.

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