Let Pn be a set of n = 2m points that are the vertices of a convex polygon, and let Mmbe the graph having as vertices all the perfect matchings in the point set Pn whose edgesare straight line segments and do not cross, and edges joining two perfect matchings M1and M2 if M2 = M1 ¡ (a; b) ¡ (c; d) + (a; d) + (b; c) for some points a; b; c; d of Pn. Weprove the following results about Mm: its diameter is m ¡ 1; it is bipartite for every m;the connectivity is equal to m ¡ 1; it has no Hamilton path for m odd, m & 3; and finallyit has a Hamilton cycle for every m even, m&=4.