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 Author, Editor(s)
 Author(s): Cygan, Marek Philip, Geevarghese Pilipczuk, Marcin Pilipczuk, Micha\l{} Wojtaszczyk, Jakub Onufry dblp dblp dblp dblp dblp Not MPG Author(s): Cygan, Marek Pilipczuk, Marcin Pilipczuk, Micha\l{} Wojtaszczyk, Jakub Onufry
 BibTeX cite key*: CyganPhilipPilipczukPilipczukWojtaszczyk2011

 Title
 Title*: Dominating set is fixed parameter tractable in claw-free graphs Attachment(s): domset-clawfree.pdf (427.52 KB)

 Journal

 Publisher
 Publisher's Name: Elsevier Publisher's URL: http://www.elsevier.com Publisher's Address: ISSN: 0304-3975

 Vol, No, pp, Date
 Volume*: 412 Number: 50 Publishing Date: November 2011 Pages*: 6982-7000 Number of VG Pages: Page Start: Page End: Sequence Number: DOI: 10.1016/j.tcs.2011.09.010

 Note: (LaTeX) Abstract: We show that the Dominating Set problem parameterized by solution size is fixed-parameter tractable (FPT) in graphs that do not contain the claw ($$K_{1,3}$$, the complete bipartite graph on four vertices where the two parts have one and three vertices, respectively) as an \emph{induced} subgraph. We present an algorithm that uses $2^{O(k^2)} n^{O(1)}$ time and polynomial space to decide whether a claw-free graph on $$n$$ vertices has a dominating set of size at most $$k$$. Note that this parameterization of Dominating Set is W[2]-hard on the set of all graphs, and thus is unlikely to have an FPT algorithm for graphs in general. The most general class of graphs for which an FPT algorithm was previously known for this parameterization of Dominating Set is the class of $$K_{i,j}$$-free graphs, which exclude, for some fixed $$i,j\in\mathbb{N}$$, the complete bipartite graph $$K_{i,j}$$ as a \emph{subgraph}. For $$i,j\ge 2$$, the class of claw-free graphs and any class of $$K_{i,j}$$-free graphs are not comparable with respect to set inclusion. We thus \emph{extend} the range of graphs over which this parameterization of Dominating Set is known to be fixed-parameter tractable. We also show that, in some sense, it is the presence of the claw that makes this parameterization of the Dominating Set problem hard. More precisely, we show that for any $$t\ge 4$$, the Dominating Set problem parameterized by the solution size is W[2]-hard in graphs that exclude the $$t$$-claw $$K_{1,t}$$ as an induced subgraph. Our arguments also imply that the related Connected Dominating Set and Dominating Clique problems are W[2]-hard in these graph classes. Finally, we show that for any $$t\in\mathbb{N}$$, the Clique problem parameterized by solution size, which is W[1]-hard on general graphs, is FPT in $$t$$-claw-free graphs. Our results add to the small and growing collection of FPT results for graph classes defined by excluded \emph{subgraphs}, rather than by excluded \emph{minors}. URL for the Abstract: http://www.sciencedirect.com/science/article/pii/S0304397511007766 Categories, Keywords: Fixed parameter tractability, Dominating Set, Claw-free graphs, Clique, Connected dominating set HyperLinks / References / URLs: Copyright Message: Copyright Elsevier 2011. Published in the journal Theoretical Computer Science, Volume 512, Issue 50. The original publication is available at http://www.sciencedirect.com/science/article/pii/S0304397511007766 Personal Comments: Download Access Level: Public

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 MPG Unit: Max-Planck-Institut für Informatik MPG Subunit: Algorithms and Complexity Group Appearance: MPII WWW Server, MPII FTP Server, MPG publications list, university publications list, working group publication list, Fachbeirat, VG Wort

BibTeX Entry:

@ARTICLE{CyganPhilipPilipczukPilipczukWojtaszczyk2011,
AUTHOR = {Cygan, Marek and Philip, Geevarghese and Pilipczuk, Marcin and Pilipczuk, Micha\l{} and Wojtaszczyk, Jakub Onufry},
TITLE = {Dominating set is fixed parameter tractable in claw-free graphs},
JOURNAL = {Theoretical Computer Science},
PUBLISHER = {Elsevier},
YEAR = {2011},
NUMBER = {50},
VOLUME = {412},
PAGES = {6982--7000},
MONTH = {November},
ISBN = {0304-3975},
DOI = {10.1016/j.tcs.2011.09.010},
}