Graphs, branchwidth, and tangles! Oh my!

Branch decomposition-based algorithms have been used in practical settings to solve some NP-hard problems like the travelling salesman problem (TSP) and general minor containment. The notions of branch decompositions and branchwidth were introduced by Robertson and Seymour to assist in proving the Graph Minors Theorem. Given a connected graph G and a branch decomposition of G of width k where k is at least 3, a practical branch decomposition-based algorithm to test whether a graph has branchwidth at most k − 1 is given. The algorithm either constructs a branch decomposition of G of width at most k − 1 or constructs a tangle basis of order k, which offers a lower bound on the branchwidth ofG. The algorithm is utilized repeatedly in a practical setting to find an optimal branch decomposition of a connected graph, whose branchwidth is at least 2, given an input branch decomposition of the graph from a heuristic. This is the first algorithm for the optimal branch decomposition problem for general graphs that has been shown to be practical. Computational results are provided to illustrate the effectiveness of finding optimal branch decompositions. A tangle basis is related to a tangle, a notion also introduced by Robertson and Seymour; however, a tangle basis is more constructive in nature. Furthermore, it is shown that a tangle basis of order k is coextensive to a tangle of order k. © 2004 Wiley Periodicals, Inc. NETWORKS, Vol. 45(2), 55–6