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

الأصل · EN

For any finite set of n points in 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, where a marked graph is defined as a geometric graph together with a subset of its vertices. The poset of non-crossing graphs on appears as the complement of the star of a face in that polyhedron. The polyhedron has a unique maximal bounded face, of dimension 2nᵢ +n -3 where nᵢ is the number of points of in the interior of (). The vertices of this polytope are all the pseudo-triangulations of, 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.