Domination When the Stars Are Out

We algorithmize the 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 unlikely to be fixed-parameter tractable on the slightly larger class of graphs that exclude K1,4 as an induced subgraph (K1,4-free graphs). 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. To complement that result, we show that Connected Dominating Set is unlikely to have a polynomial kernel on claw-free graphs and is unlikely to be fixed-parameter tractable on K1,4-free graphs. Combined, our results provide a dichotomy for Dominating Set and Connected Dominating Set on K1,ℓ-free graphs and show that the problem is fixed-parameter tractable if and only if ℓ ≤ 3.