Edge-maximal graphs of branchwidth : The -branches

Treewidth and branchwidth are two closely related connectivity parameters of graphs. Graphs of treewidth at most k have well-known alternative characterizations as subgraphs of chordal graphs and as partial k -trees. In this paper we give analogous alternative characterizations for graphs of branchwidth at most k . We first show that they are the subgraphs of chordal graphs where every maximal clique X has three subsets of size at most k each such that any two subsets have union X , with the property that every minimal separator contained in X is contained in one of the three subsets. Secondly, we give a characterization of the edge-maximal graphs of branchwidth k , that we call k -branches.