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The polytope of non-crossing graphs on a planar point set. (English) Zbl 1063.68077
[An abridged version of this paper was published in Proc. 2004 Int. Symposium on Symbolic and Algebraic Computation, 250–257 (2004; Zbl 1134.05308).]
Summary: For any set $$\mathcal A$$ of $$n$$ points in $$\mathbb R^{2}$$, we define a $$(3n-3)$$-dimensional simple polyhedron whose face poset is isomorphic to the poset of “non-crossing marked graphs” with vertex set $$\mathcal A$$, 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 $$\mathcal A$$ in the interior of $$\text{conv}(\mathcal A)$$. The vertices of this polytope are all the pseudo-triangulations of $$\mathcal A$$, 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.

##### MSC:
 68R10 Graph theory (including graph drawing) in computer science 05C10 Planar graphs; geometric and topological aspects of graph theory 52B99 Polytopes and polyhedra
rigid graphs
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