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

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

2003-02-11

arxiv.org/abs/math/0302126v3

Discrete Comput. Geom. 33:2 (2005), 275-305

Mathematics

Combinatorics

28 pages, 16 figures. Main change from v1 and v2: Introduction has been reshaped

Scientific paper

10.1007/s00454-004-1143-1

For any finite 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.

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