2010/11/30 by Danny Hermelin, Matthias Mnich, Hermelin, Danny +5 · 1 citation
Computer Science · #Data Structures and Algorithms (cs.DS) #Discrete Mathematics (cs.DM) #FOS: Computer and information sciences #cs.DM #cs.DS
paper · pdf · doi:10.48550/arxiv.1012.0012
Revised some proofs compared to v2. Significantly expanded proofs and several additional results compared to v1
arxiv created 2019/01/14 · arxiv updated 2019/01/16
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 (K1,4-free graphs). 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. To complement that result, we show that Connected Dominating Set has no polynomial kernel on claw-free graphs and is not fixed-parameter tractable on K1,4-free graphs. Combined, our results provide a dichotomy for Dominating Set and Connected Dominating Set on K1,L-free graphs and show that the problem is fixed-parameter tractable if and only if L <= 3.