Iterated colorings of graphs

For a graph property P, in particular maximal independence, minimal domination and maximal irredundance, we introduce iterated P-colorings of graphs. The six graph parameters arising from either maximizing or minimizing the number of colors used for each property, are related by an inequality chain, and in this paper we initiate the study of these parameters. We relate them to other well-studied parameters like chromatic number, give alternative characterizations, find graph classes where they differ by an arbitrary amount, investigate their monotonicity properties, and look at algorithmic issues.