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(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: |
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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 |
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