Perfect double Italian and double Italian domination on Sierpiński graphs

For a given graph [Formula: see text], a Roman [Formula: see text]-dominating function or double Italian dominating function (abbreviated DIDF) is a function [Formula: see text] which satisfies the following condition: [Formula: see text], [Formula: see text]. The weight of a Roman [Formula: see text]-dominating function or double Italian dominating function is the sum [Formula: see text] and the minimum weight of a Roman [Formula: see text]-dominating function [Formula: see text] is the Roman [Formula: see text]-domination number or double Italian domination number, denoted by [Formula: see text] or [Formula: see text]. A perfect double Italian dominating function (abbreviated PDIDF) is a double Italian dominating function such that if [Formula: see text] then [Formula: see text], where [Formula: see text] is the set of vertices adjacent to [Formula: see text] and [Formula: see text]. The minimum weight of a perfect double Italian dominating function on [Formula: see text] is the perfect double Italian domination number of [Formula: see text], denoted by [Formula: see text]. In this paper, we find that [Formula: see text] for [Formula: see text], where [Formula: see text] is generalized Sierpiński graph. A comparison between [Formula: see text] and [Formula: see text] is assessed. Further [Formula: see text] and [Formula: see text] is found when [Formula: see text] is a graph with a single universal vertex and when [Formula: see text] is a graph having at least two universal vertices. We have examined a class of graphs for which [Formula: see text], provided [Formula: see text] has a single universal vertex.