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. © 2011 Springer-Verlag.
CITATION STYLE
Hermelin, D., Mnich, M., Van Leeuwen, E. J., & Woeginger, G. J. (2011). Domination when the stars are out. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) (Vol. 6755 LNCS, pp. 462–473). https://doi.org/10.1007/978-3-642-22006-7_39
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