Domination When the Stars Are Out
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 K_{1,4} as an induced subgraph (K_{1,4}-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 K_{1,4}-free graphs. Combined, our results provide a dichotomy for Dominating Set and Connected Dominating Set on K_{1,L}-free graphs and show that the problem is fixed-parameter tractable if and only if L <= 3.
Cite
@article{arxiv.1012.0012,
title = {Domination When the Stars Are Out},
author = {Danny Hermelin and Matthias Mnich and Erik Jan van Leeuwen and Gerhard Woeginger},
journal= {arXiv preprint arXiv:1012.0012},
year = {2019}
}
Comments
Revised some proofs compared to v2. Significantly expanded proofs and several additional results compared to v1