The complex of non-crossing diagonals of a polygon

Given a convex n-gon P in R-2 with vertices in general position, it is well known that the simplicial complex theta(P) with vertex set given by diagonals in P and facets given by triangulations of P is the boundary complex of a polytope of dimension n - 3. We prove that for any non-convex polygonal region P with n vertices and h + 1 boundary components, theta(P) is a ball of dimension n + 3h - 4. We also provide a new proof that theta(P) is a sphere when P is convex with vertices in general position. (C) 2010 Elsevier Inc. All rights reserved.