Generalized connected domination in graphs

As a generalization of connected domination in a graph G we consider domination by sets having at most k components. The order γ c k (G) of such a smallest set we relate to γ c (G), the order of a smallest connected dominating set. For a tree T we give bounds on γ c k (T) in terms of minimum valency and diameter. For trees the inequality γ c k (T)≤ n-k-1 is known to hold, we determine the class of trees, for which equality holds.