Algorithmic generation of graphs of branchwidth

Branchwidth is a connectivity parameter of graphs closely related to treewidth. Graphs of treewidth at most k can be generated algorithmically as the subgraphs of k-trees: starting with Kk+1 one repeatedly chooses a k-clique C and adds a new vertex adjacent to vertices in C. In this paper we give an analogous algorithm for generating the graphs of branchwidth at most k. To this end we first investigate the family of edge-maximal graphs of branchwidth k, that we call k-branches. The k-branches are, just as the k-trees, a subclass of the chordal graphs where all minimal separators have size k. However, a striking difference arises when considering subgraph-minimal members of the family. Whereas Kk+1 is the only minimal k-tree, we show that for any k ≥ 7 a minimal k-branch having q maximal cliques exists for any value of q 6∈ {3, 5}, except for k = 8, q = 2. We give a precise characterization of minimal k-branches for all values of k. Our investigation culminates in a non-deterministic generation algorithm, that adds one or two new maximal cliques in each step, yielding as output exactly the k-branches. Full draft is avalaible as LIRMM RR05-048 at http://www.lirmm.fr/~paul CNRS LIRMM, Montpellier, France, [email protected] CIS Dept., Univ of Oregon, USA Dept. of Informatics, Univ. of Bergen, Norway, [email protected] Research conducted while on sabbatical at LIRMM