The Italian domination numbers of some generalized Sierpiński networks

Abstract

An Italian dominating function of a graph [Formula: see text] with vertex set [Formula: see text] is defined as a function [Formula: see text], which satisfies the condition that for every [Formula: see text] with [Formula: see text], [Formula: see text]. The weight of an Italian dominating function on [Formula: see text] is the sum [Formula: see text] and the Italian dominating number, and [Formula: see text] indicates the minimum weight of an Italian dominating function [Formula: see text]. In this paper, the structure of the generalized Sierpiński networks is investigated using the bounds of Italian domination number of graphs and the methods of mathematical induction and reduction to absurdity. Then, the Italian domination on the generalized Sierpiński networks [Formula: see text] is obtained, where [Formula: see text] denotes any special class of Path [Formula: see text], Cycle [Formula: see text], Wheel [Formula: see text], Star [Formula: see text], and complete bipartite graph [Formula: see text].