Clique-like Dominating Sets

Continuing work by B acso and Tuza and Cozzens and Kelleher, we investigate dominating sets which induce subgraphs with small clique covering number, or small independence number. We show that if a graph G is connected and contains no induced subgraph isomorphic to P6 or Ht (the graph obtained by subdividing each edge of K1;t; t 3), then G has a dominating set which induces a connected graph with clique covering number at most t 1. We then show that if G has no isolated vertices and contains no induced subgraph isomorphic to mK2, then G contains a dominating set with independence number at most 2m 2. For both of these results, the bounds are best possible. Finally, we prove a theorem which implies similar bounds for a variety of classes of graphs, and we show that if G is connected and contains no induced subgraph isomorphic to Pk or Ht, then the domination number of G is bounded above by a constant that depends only on k, t, and the clique number of G.