Variations of Roman domination in Kneser graphs

Let [Formula: see text] be a graph. The weight of a function [Formula: see text] defined on the vertex set of [Formula: see text] is [Formula: see text]. A Roman dominating function (RDF) of [Formula: see text] is a function [Formula: see text] such that every vertex [Formula: see text] for which [Formula: see text] has a neighbor [Formula: see text] with [Formula: see text]. Many variations of RDF are available in the literature and among that a double Roman dominating function (DRDF) is a function [Formula: see text] having the property that if [Formula: see text], then vertex [Formula: see text] must have at least two neighbors assigned 2 under [Formula: see text] or one neighbor with [Formula: see text], and if [Formula: see text], then vertex [Formula: see text] must have at least one neighbor with [Formula: see text] and an Italian dominating function (IDF) on a graph [Formula: see text] is a function [Formula: see text], satisfying the property that if [Formula: see text] for a vertex [Formula: see text], then [Formula: see text] with [Formula: see text], that is, either [Formula: see text], with [Formula: see text], or at least two vertices [Formula: see text] with [Formula: see text]. The minimum weights among all RDF, DRDF and IDF on a graph [Formula: see text] are called the Roman domination number, the double Roman domination number and the Italian domination number of the graph [Formula: see text], respectively. In this paper, we find the double Roman domination number and Italian domination number of Kneser graphs.