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|
|Subtitle of host publication||38th International Colloquium, ICALP 2011, Zurich, Switzerland, July 4-8, 2011, Proceedings, Part I|
|Publication status||Published - 2011|
|Series||Lecture Notes in Computer Science|
Hermelin, D., Mnich, M., van Leeuwen, E. J., & Woeginger, G. J. (2011). Domination when the stars are out. In Automata, Languages and Programming: 38th International Colloquium, ICALP 2011, Zurich, Switzerland, July 4-8, 2011, Proceedings, Part I (pp. 462-473). Springer. Lecture Notes in Computer Science, Vol.. 6755 https://doi.org/10.1007/978-3-642-22006-7_39