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,`-free graphs and show that the problem is fixed-parameter tractable if and only if ` ≤ 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. ∗Max-Planck-Institut für Informatik, Saarbrücken, Germany, [email protected]. †International Computer Science Institute, Berkeley, USA, [email protected]. ‡Department of Informatics, University of Bergen, Norway, [email protected]. §Department of Mathematics and Computer Science, TU Eindhoven, The Netherlands, [email protected]. 1 ar X iv :1 01 2. 00 12 v1 [ cs .D S] 3 0 N ov 2 01 0