The Italian domination numbers of some generalized Sierpiński networks.

An Italian dominating function of a graph G = (V , E) with vertex set V is defined as a function f : V → { 0 , 1 , 2 } , which satisfies the condition that for every v ∈ V with f (v) = 0 , ∑ u ∈ N (v) f (u) ≥ 2. The weight of an Italian dominating function on G is the sum f (V) = ∑ u ∈ V f (v) and the Italian dominating number, and γ I (G) indicates the minimum weight of an Italian dominating function f. 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 S (G , t) is obtained, where G denotes any special class of Path P n , Cycle C n , Wheel W n , Star K 1 , n , and complete bipartite graph K m , n . [ABSTRACT FROM AUTHOR]