generalized k-rainbow and generalized 2-rainbow domination in graphs

Assume we have a set of k colors and to each vertex of a graph g we assign an arbitry of these colors. if we require that each vertex to set is assigned has in its closed neighborhood all k colors, then this is called the generalized k-rainbow dominating function of a graph g. the corresponding γgkr, which is the minimum sum of numbers of assigned colores over all vertices of g, is called the gk-rainbow domination number of g. in this paper we present a linear algorithms for determining a minimum generalized 2-rainbow dominating set of a tree and on gp(n,2).