Let M E 2 be an open, connected and bounded polygonal region with polygonal holes of dimension d {0, 1, 2}. For a given set of boundary points x 1 , ..., x n of M we derive the minimal number of convex pieces into which M can be divided such that for each x i , i = 1, ..., n, the boundary of the final convex partition contains segments of suitably prescribed directions having x i as a common starting point. The proofs are based on graph theoretic arguments and elementary topology.