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

Authors:
David Orden;Francisco Santos
Affiliations:
University of Alcalá de Henares, Alcalá de Henares (Madrid) SPAIN;University of Cantabria, (Cantabria) SPAIN
Venue:
ISSAC '04 Proceedings of the 2004 international symposium on Symbolic and algebraic computation
Year:
2004

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Abstract

For any finite set A of n points in general position 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 pointed 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 3n-3-2n;b; where n;b; is the number of convex hull points of 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.