Domination when the stars are out

We algorithmize the recent structural characterization for claw-free graphs by Chudnovsky and Seymour. Building on this result, we show that Dominating Set on claw-free graphs is (i) fixed-parameter tractable and (ii) even possesses a polynomial kernel. To complement these results, we establish that Dominating Set is not fixed-parameter tractable on the slightly larger class of graphs that exclude K1;4 as an induced subgraph. Our results provide a dichotomy for Dominating Set in K1;L-free graphs and show that the problem is fixed-parameter tractable if and only if L