The polytope of non-crossing graphs on a planar point set

For any set A of n points in R-2, we define a (3n - 3)- dimensional simple polyhedron whose face poset is isomorphic to the poset of "non-crossing marked graphs" with vertex set A, where a marked graph is defined as a geometric graph together with a subset of its vertices. The poset of non-crossing graphs on A appears as the complement of the star of a face in that polyhedron. The polyhedron has a unique maximal bounded face, of dimension 2n(i) + n - 3 where n(i) is the number of points of A in the interior of conv(A). The vertices of this polytope are all the pseudo-triangulations of A, and the edges are flips of two types: the traditional diagonal flips ( in pseudo-triangulations) and the removal or insertion of a single edge. As a by-product of our construction we prove that all pseudo-triangulations are infinitesimally rigid graphs.