Optimal branch-decomposition of planar graphs in O(n3) Time

We give an <i>O</i>(<i>n</i><sup>3</sup>) time algorithm for constructing a minimum-width branch-decomposition of a given planar graph with <i>n</i> vertices. This is achieved through a refinement to the previously best known algorithm of Seymour and Thomas, which runs in <i>O</i>(<i>n</i><sup>4</sup>) time.