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.
|Title of host publication||Automata, Languages and Programming (38th International Colloquium, ICALP 2011, Zürich, Switzerland, July 4-8, 2011. Proceedings, Part I)|
|Editors||L. Aceto, M. Henzinger, J. Sgall|
|Place of Publication||Berlin|
|Publication status||Published - 2011|
|Name||Lecture Notes in Computer Science|