513 research outputs found

    Flip Distance Between Triangulations of a Planar Point Set is APX-Hard

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    In this work we consider triangulations of point sets in the Euclidean plane, i.e., maximal straight-line crossing-free graphs on a finite set of points. Given a triangulation of a point set, an edge flip is the operation of removing one edge and adding another one, such that the resulting graph is again a triangulation. Flips are a major way of locally transforming triangular meshes. We show that, given a point set SS in the Euclidean plane and two triangulations T1T_1 and T2T_2 of SS, it is an APX-hard problem to minimize the number of edge flips to transform T1T_1 to T2T_2.Comment: A previous version only showed NP-completeness of the corresponding decision problem. The current version is the one of the accepted manuscrip

    The braided Ptolemy-Thompson group T∗T^* is asynchronously combable

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    The braided Ptolemy-Thompson group T∗T^* is an extension of the Thompson group TT by the full braid group B∞B_{\infty} on infinitely many strands. This group is a simplified version of the acyclic extension considered by Greenberg and Sergiescu, and can be viewed as a mapping class group of a certain infinite planar surface. In a previous paper we showed that T∗T^* is finitely presented. Our main result here is that T∗T^* (and TT) is asynchronously combable. The method of proof is inspired by Lee Mosher's proof of automaticity of mapping class groups.Comment: 45

    Flip-graph moduli spaces of filling surfaces

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    This paper is about the geometry of flip-graphs associated to triangulations of surfaces. More precisely, we consider a topological surface with a privileged boundary curve and study the spaces of its triangulations with n vertices on the boundary curve. The surfaces we consider topologically fill this boundary curve so we call them filling surfaces. The associated flip-graphs are infinite whenever the mapping class group of the surface (the group of self-homeomorphisms up to isotopy) is infinite, and we can obtain moduli spaces of flip-graphs by considering the flip-graphs up to the action of the mapping class group. This always results in finite graphs and we are interested in their geometry. Our main focus is on the diameter growth of these graphs as n increases. We obtain general estimates that hold for all topological types of filling surface. We find more precise estimates for certain families of filling surfaces and obtain asymptotic growth results for several of them. In particular, we find the exact diameter of modular flip-graphs when the filling surface is a cylinder with a single vertex on the non-privileged boundary curve.Comment: 52 pages, 29 figure

    The flip-graph of the 4-dimensional cube is connected

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    Flip-graph connectedness is established here for the vertex set of the 4-dimensional cube. It is found as a consequence that this vertex set has 92 487 256 triangulations, partitioned into 247 451 symmetry classes.Comment: 20 pages, 3 figures, revised proofs and notation

    The diameter of type D associahedra and the non-leaving-face property

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    Generalized associahedra were introduced by S. Fomin and A. Zelevinsky in connection to finite type cluster algebras. Following recent work of L. Pournin in types AA and BB, this paper focuses on geodesic properties of generalized associahedra. We prove that the graph diameter of the nn-dimensional associahedron of type DD is precisely 2n−22n-2 for all nn greater than 11. Furthermore, we show that all type BCDBCD associahedra have the non-leaving-face property, that is, any geodesic connecting two vertices in the graph of the polytope stays in the minimal face containing both. This property was already proven by D. Sleator, R. Tarjan and W. Thurston for associahedra of type AA. In contrast, we present relevant examples related to the associahedron that do not always satisfy this property.Comment: 18 pages, 14 figures. Version 3: improved presentation, simplification of Section 4.1. Final versio
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