867 research outputs found
Flip Graphs of Degree-Bounded (Pseudo-)Triangulations
We study flip graphs of triangulations whose maximum vertex degree is bounded
by a constant . In particular, we consider triangulations of sets of
points in convex position in the plane and prove that their flip graph is
connected if and only if ; the diameter of the flip graph is .
We also show that, for general point sets, flip graphs of pointed
pseudo-triangulations can be disconnected for , and flip graphs of
triangulations can be disconnected for any . Additionally, we consider a
relaxed version of the original problem. We allow the violation of the degree
bound by a small constant. Any two triangulations with maximum degree at
most of a convex point set are connected in the flip graph by a path of
length , where every intermediate triangulation has maximum degree
at most .Comment: 13 pages, 12 figures, acknowledgments update
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
Delaunay Edge Flips in Dense Surface Triangulations
Delaunay flip is an elegant, simple tool to convert a triangulation of a
point set to its Delaunay triangulation. The technique has been researched
extensively for full dimensional triangulations of point sets. However, an
important case of triangulations which are not full dimensional is surface
triangulations in three dimensions. In this paper we address the question of
converting a surface triangulation to a subcomplex of the Delaunay
triangulation with edge flips. We show that the surface triangulations which
closely approximate a smooth surface with uniform density can be transformed to
a Delaunay triangulation with a simple edge flip algorithm. The condition on
uniformity becomes less stringent with increasing density of the triangulation.
If the condition is dropped completely, the flip algorithm still terminates
although the output surface triangulation becomes "almost Delaunay" instead of
exactly Delaunay.Comment: This paper is prelude to "Maintaining Deforming Surface Meshes" by
Cheng-Dey in SODA 200
Triangle-Free Triangulations, Hyperplane Arrangements and Shifted Tableaux
Flips of diagonals in colored triangle-free triangulations of a convex
polygon are interpreted as moves between two adjacent chambers in a certain
graphic hyperplane arrangement. Properties of geodesics in the associated flip
graph are deduced. In particular, it is shown that: (1) every diagonal is
flipped exactly once in a geodesic between distinguished pairs of antipodes;
(2) the number of geodesics between these antipodes is equal to twice the
number of Young tableaux of a truncated shifted staircase shape.Comment: figure added, plus several minor change
Happy endings for flip graphs
We show that the triangulations of a finite point set form a flip graph that
can be embedded isometrically into a hypercube, if and only if the point set
has no empty convex pentagon. Point sets of this type include convex subsets of
lattices, points on two lines, and several other infinite families. As a
consequence, flip distance in such point sets can be computed efficiently.Comment: 26 pages, 15 figures. Revised and expanded for journal publicatio
Once punctured disks, non-convex polygons, and pointihedra
We explore several families of flip-graphs, all related to polygons or
punctured polygons. In particular, we consider the topological flip-graphs of
once-punctured polygons which, in turn, contain all possible geometric
flip-graphs of polygons with a marked point as embedded sub-graphs. Our main
focus is on the geometric properties of these graphs and how they relate to one
another. In particular, we show that the embeddings between them are strongly
convex (or, said otherwise, totally geodesic). We also find bounds on the
diameters of these graphs, sometimes using the strongly convex embeddings.
Finally, we show how these graphs relate to different polytopes, namely type D
associahedra and a family of secondary polytopes which we call pointihedra.Comment: 24 pages, 6 figure
Random lattice triangulations: Structure and algorithms
The paper concerns lattice triangulations, that is, triangulations of the
integer points in a polygon in whose vertices are also integer
points. Lattice triangulations have been studied extensively both as geometric
objects in their own right and by virtue of applications in algebraic geometry.
Our focus is on random triangulations in which a triangulation has
weight , where is a positive real parameter, and
is the total length of the edges in . Empirically, this
model exhibits a "phase transition" at (corresponding to the
uniform distribution): for distant edges behave essentially
independently, while for very large regions of aligned edges
appear. We substantiate this picture as follows. For sufficiently
small, we show that correlations between edges decay exponentially with
distance (suitably defined), and also that the Glauber dynamics (a local Markov
chain based on flipping edges) is rapidly mixing (in time polynomial in the
number of edges in the triangulation). This dynamics has been proposed by
several authors as an algorithm for generating random triangulations. By
contrast, for we show that the mixing time is exponential. These
are apparently the first rigorous quantitative results on the structure and
dynamics of random lattice triangulations.Comment: Published at http://dx.doi.org/10.1214/14-AAP1033 in the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
The geometry of flip graphs and mapping class groups
The space of topological decompositions into triangulations of a surface has
a natural graph structure where two triangulations share an edge if they are
related by a so-called flip. This space is a sort of combinatorial
Teichm\"uller space and is quasi-isometric to the underlying mapping class
group. We study this space in two main directions. We first show that strata
corresponding to triangulations containing a same multiarc are strongly convex
within the whole space and use this result to deduce properties about the
mapping class group. We then focus on the quotient of this space by the mapping
class group to obtain a type of combinatorial moduli space. In particular, we
are able to identity how the diameters of the resulting spaces grow in terms of
the complexity of the underlying surfaces.Comment: 46 pages, 23 figure
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