Rainbow Domination in Graphs

Assume we have a set of k colors and to each vertex of a graph G we assign an arbitrary subset of these colors. If we require that each vertex to which an empty set is assigned has in its neighborhood all k colors, then this is called the k-rainbow dominating function of a graph G. The corresponding invariant γ_ rk(G), which is the minimum sum of numbers of assigned colors over all vertices of G, is called the k-rainbow domination number of G. In this paper we connect this new concept to usual domination in (products of) graphs, and present its application to paired-domination of Cartesian products of graphs. Finally, we present a linear algorithm for determining a minimum $2$-rainbow dominating set of a tree.