Theory of Computing Systems

, Volume 61, Issue 3, pp 893–906

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

Article

## Abstract

Let G(V, E) be a connected and undirected graph with n-vertex-set V and m-edge-set E. For each vV, let N(v) = {u|vV 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 ≤ ik, such that for any vertex v with f(v) = 0(k) we have ⋈uN(v)f(u) = 1(k), for all vV, where ⋈uSf(u) denotes the result of taking bitwise OR operation on f(u), for all uS. The weight of f is defined as $$w(f) = {\sum }_{v\in V}{\sum }^{k}_{i=1} f_{i}(v)$$. 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.

### Keywords

k-rainbow domination function Dominating set Sierpiński graphs Extended Sierpiński graphs

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