As a common generalization of the domination number and the total domination number of a graph G, we study the 3686dfa757589c" title="Click to view the MathML source">k-component domination number γk(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 3686dfa757589c" title="Click to view the MathML source">k.
We show that for every positive integer 3686dfa757589c" title="Click to view the MathML source">k, and every graph G of order n at least k+1 and without isolated vertices, we have . Furthermore, we characterize all extremal graphs. We propose two conjectures concerning graphs of minimum degree 2, and prove a related result.