513 research outputs found
Flip Distance Between Triangulations of a Planar Point Set is APX-Hard
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 in the Euclidean plane and two
triangulations and of , it is an APX-hard problem to minimize
the number of edge flips to transform to .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 is asynchronously combable
The braided Ptolemy-Thompson group is an extension of the Thompson
group by the full braid group 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 is finitely presented.
Our main result here is that (and ) 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
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
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
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 and , this paper focuses on geodesic properties of generalized
associahedra. We prove that the graph diameter of the -dimensional
associahedron of type is precisely for all greater than .
Furthermore, we show that all type 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 .
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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