Domination When the Stars Are Out

Abstract

We algorithmize the structural characterization for claw-free graphs by Chudnovsky and Seymour. Building on this result, we show that D ominating S et on claw-free graphs is (i) fixed-parameter tractable and (ii) even possesses a polynomial kernel. To complement these results, we establish that D ominating S et is unlikely to be fixed-parameter tractable on the slightly larger class of graphs that exclude K 1,4 as an induced subgraph ( K 1,4 -free graphs). We show that our algorithmization can also be used to show that the related C onnected D ominating S et problem is fixed-parameter tractable on claw-free graphs. To complement that result, we show that C onnected D ominating S et is unlikely to have a polynomial kernel on claw-free graphs and is unlikely to be fixed-parameter tractable on K 1,4 -free graphs. Combined, our results provide a dichotomy for D ominating S et and C onnected D ominating S et on K 1,ℓ -free graphs and show that the problem is fixed-parameter tractable if and only if ℓ ≤ 3.