New Tools and Results for Branchwidth

We provide new tools, such as k-troikas and good subtree-representations, that allow us to give fast and simple algorithms computing branchwidth. We show that a graph G has branchwidth at most k if and only if it is a subgraph of a chordal graph in which every maximal clique has a k-troika respecting its minimal separators. Moreover, if G itself is chordal with clique tree T then such a chordal supergraph exists having clique tree a minor of T . We use these tools to give a straightforward O(m+n+ q2) algorithm computing branchwidth for an interval graph on m edges, n vertices and q maximal cliques. We also prove a conjecture of F. Mazoit [11] by showing that branchwidth is polynomial on a chordal graph given with a clique tree having a polynomial number of subtrees.