On perfect Italian domination in graphs

Abstract

For a vertex [Formula: see text] in a simple, finite and undirected graph [Formula: see text], the neighborhood [Formula: see text] of [Formula: see text] is the set consisting of all vertices of [Formula: see text] which are adjacent to [Formula: see text]. A perfect Italian dominating function on [Formula: see text] is a function [Formula: see text] such that for each [Formula: see text] with [Formula: see text], [Formula: see text]. The weight of a perfect Italian dominating function [Formula: see text] is the value [Formula: see text]. The perfect Italian domination number of [Formula: see text] is the minimum weight of a perfect Italian dominating function on [Formula: see text]. In this paper, we study the perfect Italian domination in graphs under some binary operations.