RAINBOW DOMINATION IN GRAPHS

Abstract

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 $\gamma_{{\rm r}k}(G)$, 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.