Domination when the stars are out

Authors:
Danny Hermelin;Matthias Mnich;Erik Jan Van Leeuwen;Gerhard J. Woeginger
Affiliations:
Max-Planck-Institut für Informatik, Saarbrücken, Germany;International Computer Science Institute, Berkeley;Department of Informatics, University of Bergen, Norway;Department of Mathematics and Computer Science, TU Eindhoven, The Netherlands
Venue:
ICALP'11 Proceedings of the 38th international colloquim conference on Automata, languages and programming - Volume Part I
Year:
2011

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Abstract

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 K1,4 as an induced subgraph. Our results provide a dichotomy for Dominating Set in K1,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.