Planar branch decompositions II: The cycle method

This is the second of two papers dealing with the relationship of branchwidth and planar graphs. Branchwidth and branch decompositions, introduced by Robertson and Seymour, have been shown to be beneficial for both proving theoretical results on graphs and solving NP-hard problems modeled on graphs. The first practical implementation of an algorithm of Seymour and Thomas for computing optimal branch decompositions of planar hypergraphs is presented. This algorithm encompasses another algorithm of Seymour and Thomas for computing the branchwidth of any planar hypergraph, whose implementation is discussed in the first paper. The implementation also includes the addition of a heuristic to decrease the run times of the algorithm. This method, called the cycle method, is an improvement on the algorithm by using a ‘divide-and-conquer’ approach.