In this extended abstract we state some definitions and results from our earlier paper [C.Paul and J.A.Telle. New tools and simpler algorithms for branchwidth. In European Symposium on Algorithm - ESA 2005. To appear in Lecture Notes in Computer Science, 2005. Full draft avalaible as LIRMM RR 05-017] and use these to characterize the class of edge-maximal graphs of branchwidth k. Similarly to the maximal graphs of treewidth k (the k-trees) they turn out to be a subclass of chordal graphs where every minimal separator has size k.