The Polytope of Non-Crossing Graphs on a Planar Point Set

Authors:
David Orden;Francisco Santos
Affiliations:
Departamento de Matemáticas, Universidad de Alcaá de Henares, 28871 Alcalá de Henares, Spain;Departamento de Matemáticas, Estadística y Computación, Universidad de Cantabria, 39005 Santander, Spain
Venue:
Discrete & Computational Geometry
Year:
2005

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Abstract

For any set A of n points in ℝ2, 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 2ni + n - 3 where ni 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.