High Quality Content by WIKIPEDIA articles! In graph theory, a branch-decomposition of an undirected graph G is a hierarchical clustering of the edges of G, represented by an unrooted binary tree T with the edges of G as its leaves. Removing any edge from T partitions the edges of G into two subgraphs, and the width of the decomposition is the maximum number of shared vertices of any pair of subgraphs formed in this way. The branchwidth of G is the minimum width of any branch-decomposition of G; branchwidth is closely related to tree-width and many graph optimization problems may be solved efficiently for graphs of small branchwidth. Branch-decompositions and branchwidth may also be generalized from graphs to matroids.