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

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) = f k (v)f k − 1(v) … f 1(v), i.e., f i (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 ∈ S f(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 = 1 k f i ( v ) $w(f) = {\sum }_{v\in V}{\sum }^{k}_{i=1} f_{i}(v)$ . The k-rainbow domination number γ k r (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 γ 2r (S(n, m)), γ 2r (S +(n, m)), and γ 2r (S ++(n, m)), where S(n, m), S +(n, m), and S ++(n, m) are Sierpiński graphs and extended Sierpiński graphs. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Theory of Computing Systems Springer Journals

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

, Volume 61 (3) – Mar 13, 2017
14 pages

/lp/springer-journals/the-2-rainbow-domination-of-sierpi-ski-graphs-and-extended-sierpi-ski-ZyaIGuWiDO
Publisher
Springer Journals
Subject
Computer Science; Theory of Computation
ISSN
1432-4350
eISSN
1433-0490
D.O.I.
10.1007/s00224-017-9756-y
Publisher site
See Article on Publisher Site

### Abstract

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) = f k (v)f k − 1(v) … f 1(v), i.e., f i (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 ∈ S f(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 = 1 k f i ( v ) $w(f) = {\sum }_{v\in V}{\sum }^{k}_{i=1} f_{i}(v)$ . The k-rainbow domination number γ k r (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 γ 2r (S(n, m)), γ 2r (S +(n, m)), and γ 2r (S ++(n, m)), where S(n, m), S +(n, m), and S ++(n, m) are Sierpiński graphs and extended Sierpiński graphs.

### Journal

Theory of Computing SystemsSpringer Journals

Published: Mar 13, 2017

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