Abstract:
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It is known that a planar graph on n vertices has branch-width/tree-width bounded by alphasqrt{n}. In
many algorithmic applications it is useful to have a
small bound on the constant alpha. We give a proof of
the best, so far, upper bound for the constant alpha.
In particular, for the case of tree-width, alpha<3.182
and for the case of branch-width, alpha<2.122. Our proof
is based on the planar separation theorem of Alon,
Seymour & Thomas and some min-max theorem of the graph
minors series. Based on these bounds we introduce a new
method for solving different fixed parameter problems on
planar graphs. We prove that our method provides the best
so far exponential speed-up for fundamental problems on
planar graphs like Vertex Cover, Dominating Set, Independent Set
and many others. |