Definitions
An unrooted binary tree is a connected undirected graph with no cycles in which each non-leaf node has exactly three neighbors. A branch-decomposition may be represented by an unrooted binary tree T, together with a bijection between the leaves of T and the edges of the given graph G = (V,E). If e is any edge of the tree T, then removing e from T partitions it into two subtrees T1 and T2. This partition of T into subtrees induces a partition of the edges associated with the leaves of T into two subgraphs G1 and G2 of G. This partition of G into two subgraphs is called an e-separation.
The width of an e-separation is the number of vertices of G that are incident both to an edge of E1 and to an edge of E2; that is, it is the number of vertices that are shared by the two subgraphs G1 and G2. The width of the branch-decomposition is the maximum width of any of its e-separations. The branchwidth of G is the minimum width of a branch-decomposition of G.
Read more about this topic: Branch-decomposition
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