The 2-Rainbow Domination of Sierpiński Graphs and Extended Sierpiński Graphs

Let G(V, E) be a connected and undirected graph with n-vertex-set V and m-edge-set E. For each v ∈ V, let N(v) = {u|v ∈ V and(u, v) ∈ E}. For a positive integer k, a k-rainbow dominating function of a graph G is a function f from V(G) to a k-bit Boolean string f(v) = fk(v)fk − 1(v) … f1(v), i.e., fi(v) ∈ {0, 1}, 1 ≤ i ≤ k, such that for any vertex v with f(v) = 0(k) we have ⋈u ∈ N(v)f(u) = 1(k), for all v ∈ V, where ⋈u ∈ Sf(u) denotes the result of taking bitwise OR operation on f(u), for all u ∈ S. The weight of f is defined as w(f)=∑v∈V∑i=1kfi(v)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$w(f) = {\sum }_{v\in V}{\sum }^{k}_{i=1} f_{i}(v)$\end{document}. The k-rainbow domination number γkr(G) is the minimum weight of a k-rainbow dominating function over all k-rainbow dominating functions of G. The 1-rainbow domination is the same as the ordinary domination. The k-rainbow domination problem is to determine the k-rainbow domination number of a graph G. In this paper, we determine γ2r(S(n, m)), γ2r(S+(n, m)), and γ2r(S++(n, m)), where S(n, m), S+(n, m), and S++(n, m) are Sierpiński graphs and extended Sierpiński graphs.