Edge-maximal graphs of branchwidth kk: The kk-branches

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