New upper bounds on the decomposability of planar graphs and fixed parameter algorithms

It is known that a planar graph on n vertices has branch-width/tree-width bounded by α √ n. In many algorithmic applications it is useful to have a small bound on the constant α. We give a proof of the best, so far, upper bound for the constant α. In particular, for the case of treewidth, α < 3.182 and for the case of branch-width, α < 2.122. Our proof is based on the planar separation theorem of Alon, Seymour & Thomas and some min-max theorems of Robertson & Seymour from 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 and many others.