Dominating sets inducing large components.

As a common generalization of the domination number and the total domination number of a graph G , we study the k -component domination number γ k ( G ) of G defined as the minimum cardinality of a dominating set D of G for which each component of the subgraph G [ D ] of G induced by D has order at least k . We show that for every positive integer k , and every graph G of order n at least k + 1 and without isolated vertices, we have γ k ( G ) ≤ k n k + 1 . Furthermore, we characterize all extremal graphs. We propose two conjectures concerning graphs of minimum degree 2 , and prove a related result. [ABSTRACT FROM AUTHOR]