Computing Branch Decomposition of Large Planar Graphs

A graph of small branchwidth admits efficient dynamic programming algorithms for many NP-hard problems on the graph. A key step in these algorithms is to find a branch decomposition of small width for the graph. Given a planar graph G of n vertices, an optimal branch decomposition of G can be computed in polynomial time, e.g., by the edge-contraction method in O(n) time. All known algorithms for the planar branch decomposition use Seymour and Thomas procedure which, given an integer β, decides whether G has the branchwidth at least β or not in O(n) time. Recent studies report efficient implementations of Seymour and Thomas procedure that compute the branchwidth of planar graphs of size up to one hundred thousand edges in a practical time and memory space. Using the efficient implementations as a subroutine, it is reported that the edge-contraction method computes an optimal branch decomposition for planar graphs of size up to several thousands edges in a practical time but it is still time consuming for graphs with larger size. In this paper, we propose divide-and-conquer based algorithms of using Seymour and Thomas procedure to compute optimal branch decompositions of planar graphs. Our algorithms have time complexity O(n). Computational studies show that our algorithms are much faster than the edge-contraction algorithms and can compute an optimal branch decomposition of some planar graphs of size up to 50,000 edges in a practical time.