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2-Rainbow Domination and Its Practical Variation on Weighted Graphs

  • Chung-Kung Yen
Conference paper
Part of the Smart Innovation, Systems and Technologies book series (SIST, volume 20)

Abstract

Let G(V, E) be a simple graph with n-vertex-set V and m-edge-set E. Two positive weights, w 1 (v) and w 2 (v), are assigned to each vertex v. For each vV, let N(v) = {u | uV and (u, v) ∈ E} and N[v] = {v} ∪ N(v). A 2-rainbow function of G is a function f mapping each vertex v to f(v) = f 2(v) f 1(v), f 2(v), f 1(v) ∈ {0, 1}. The weight of f is defined as \(w(f) = \sum_{v \in V[f_1(v)w_1(v) + f_2(v)w_2(v)]}\). A 2-rainbow function f of G is called a 2-rainbow dominating function if Θ u ∈ N(v) f(u) = 11, for all vertices v with f(v) = 00, where Θ u ∈ N(v) f(u) is the result of performing bit-wise Boolean OR on f(u), for all uN(v). Our problem is to obtain a 2-rainbow dominating function f of G such that w(f) is minimized. This paper first proves that the problem is NP-hard on unweighted planar graphs. Then, an O(n)-time algorithm for the problem on trees is proposed using the dynamic programming strategy. Finally, a practical variation, called the weighted minimum tuple 2-rainbow domination problem, is proposed and the relationship between it and the weighted minimum domination problem is established.

Keywords

2-rainbow domination functions tuple 2-rainbow domination functions planar graphs trees NP-hard 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Department of Information ManagementShih Hsin UniversityTaipeiTaiwan

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