Branchwidth of chordal graphs

This paper revisits the ‘branchwidth territories’ of Kloks, Kratochvíl and Müller [T. Kloks, J. Kratochvíl, H. Müller, New branchwidth territories, in: 16th Ann. Symp. on Theoretical Aspect of Computer Science, STACS, in: Lecture Notes in Computer Science, vol. 1563, 1999, pp. 173–183] to provide a simpler proof, and a faster algorithm for computing the branchwidth of an interval graph. We also generalize the algorithm to the class of chordal graphs, albeit at the expense of exponential running time. Compliance with the ternary constraint of the branchwidth definition is facilitated by a simple new tool called kk-troikas: three sets of size at most kk each are a kk-troika of set SS, if any two have union SS. We give a straightforward O(m+n+q2)O(m+n+q2) algorithm, computing branchwidth for an interval graph on mm edges, nn vertices and qq maximal cliques. We also prove a conjecture of Mazoit [F. Mazoit, A general scheme for deciding the branchwidth, Technical Report RR2004-34, LIP — École Normale Supérieure de Lyon, 2004. http://www.ens-lyon.fr/LIP/Pub/Rapports/RR/RR2004/RR2004-34.pdf], by showing that branchwidth can be computed in polynomial time for a chordal graph given with a clique tree having a polynomial number of subtrees.