Before revision:   In this paper we apply the concept of branch-decomposition to the vertex enumeration problem of polyhedra P = \{x : Sx = b, x \geq 0\}.
After revision:  It is a famous and long standing open question in polyhedral theory and computational geometry whether the vertices of a polytope (bounded polyhedron), described by a set of linear constraints, can be enumerated in total polynomial time. In this paper we apply the concept of branch-decomposition to the vertex enumeration problem of polyhedra P = \{x : Sx = b, x \geq 0\}.