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 k 1,4 as an induced subgraph. Our results provide a dichotomy for dominating set in k 1,l-free graphs and show that the problem is fixed-parameter tractable if and only if l = 3. Finally, we show that our algorithmization can also be used to show that the related connected dominating set problem is fixed-parameter tractable on claw-free graphs