Volume: 03, February 2014, Pages: 468-473 International Journal of Computing Algorithm Integrated Intelligent Research (IIR) 468 2-Rainbow Domination of Hexagonal Mesh Networks D.Antony Xavier, Elizabeth Thomas, S.Kulandai Therese Department of Mathematics, Loyola College, Chennai – 600 034, India. Email: elizathomas.25@gmail.com Abstract A 2-rainbow domination function of a graph G is a function f that assigns to each vertex a set of colors chosen from the set {1, 2} i.e. : ( ܩ) →({1,2}) , such that for any ݒ∈( ܩ), ( ݒ)= ∅; implies ⋃ ( ݑ) = {1,2}. ௨∈ே( ௩) The 2-rainbow domination number ߛଶ ( ܩ) of a graph G is the minimum ݓ( )= ∑ | ( )| , ௩( ) over all such functions f. The Hexagonal networks are popular mesh-derived parallel architectures. In this paper we present an upper bound for the 2-rainbow domination number of hexagonal networks. Keywords: Domination, Hexagonal network, Rainbow domination, 2- Rainbow domination. 1. Introduction Let G=(V(G); E(G)) be a simple graph of order n. For any vertex x∈ V, the open neighbourhood of x is the set ( ݔ)= { ݕ∈ | ݕݔ∈ ܧ} and the closed neighbourhood is the set [ ݔ]= ( ݔ) ∪ { ݔ}. If ⊆, then ( )= ⋃ ( ݒ) ௩∈ௌ denotes open neighbourhood of S and [ ]= ( ) ∪ denotes it’s closed neighbourhood. A set of vertices S in G is a dominating set, if [ ]= ( ܩ) . The domination number, γ (G), of G is the minimum cardinality of a dominating set of G. Domination and its related problems has vast application and has become a faster growing area in the field of graph theory. As the study of dominations in graph increased the research led to different types of dominations in graphs which are widely studied in [13,14]. The Rainbow domination is one such variant of classical domination which was introduced in [2]. A function is called a k-rainbow dominating function (kRDF) of G, if f assigns to each vertex a set of colors chosen from the set {1,…, } i.e., : ( ܩ) →({1,…, }) , such that for any vertex ݒ∈( ܩ), ( ݒ)= ∅⟹ ⋃ ( ݑ) = {1,…, }. ௨∈ே( ௩) The weight of f in G is ݓ( )= ∑ | ( )|. ௩( ) The minimum weight of a kRDF of G is called the k-rainbow dominating number of G and it is denoted by ߛ ( ܩ) . A function f is called a ߛ - function of G if ݓ( )= ߛ ( ܩ) . A 2-rainbow domination function of a graph G is a particular case of kRDF i.e. when k=2. The motivation for the introduction of this invariant was inspired by the following famous open problem [15]:Vizing’s Conjecture. In [1] a linear-time algorithm for determining a minimum weight 2-rainbow dominating function of an arbitrary tree was presented. In [3], Brešar and Šumenjak showed that the problem of determining whether a graph has a 2-rainbow dominating function of a given weight is NP-complete even when restricted to chordal graphs (or bipartite graphs). They also gave the exact values for paths, cycles and suns and upper and lower bounds for the generalized Petersen graphs. This concept has fascinated several authors and has be extensively studied in [1- 4,6,9,11,16,17]. In this paper we study this variant of the domination for hexagonal networks. Hexagonal networks are multiprocessor interconnection network based on regular triangular tessellations and this is widely studied in [9]. Hexagonal networks have been studied in a variety of contexts. They have been applied in chemistry to model benzenoid hydrocarbons[12], in image processing, in computer graphics[8], and in cellular networks[5]. An addressing scheme for hexagonal networks, and its corresponding routing and broadcasting algorithms were proposed by Chen et al.[10] 2. Preliminary results on 2-Rainbow domination Some known results and bounds for 2-rainbow domination in graphs are given below. Theorem 2.1[17]: Let G be a graph. Then ߛ( ܩ) ≤ ߛଶ ( ܩ) ≤ ߛோ ( ܩ) ≤ 2ߛ( ܩ).