Domination When the Stars Are Out

Danny Hermelin, Matthias Mnich, E.J. van Leeuwen, Gerhard J. Woeginger

Research output: Contribution to journalArticleAcademicpeer-review

Abstract

We algorithmize the 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 unlikely to be 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 is unlikely to have a polynomial kernel on claw-free graphs and is unlikely to be 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.
Original languageEnglish
Article number25
Pages (from-to)25:1-25:90
JournalACM Transactions on Algorithms
Volume15
Issue number2
DOIs
Publication statusPublished - 2019

Keywords

  • claw-free graphs
  • dominating set
  • polynomial kernel
  • connected dominating seet
  • fixed parameter tratable

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