3,872 research outputs found
Four-connected triangulations of planar point sets
In this paper, we consider the problem of determining in polynomial time
whether a given planar point set of points admits 4-connected
triangulation. We propose a necessary and sufficient condition for recognizing
, and present an algorithm of constructing a 4-connected
triangulation of . Thus, our algorithm solves a longstanding open problem in
computational geometry and geometric graph theory. We also provide a simple
method for constructing a noncomplex triangulation of which requires
steps. This method provides a new insight to the structure of
4-connected triangulation of point sets.Comment: 35 pages, 35 figures, 5 reference
The polytope of non-crossing graphs on a planar point set
For any finite set \A of points in , we define a
-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
where 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.Comment: 28 pages, 16 figures. Main change from v1 and v2: Introduction has
been reshape
Transforming triangulations on non planar-surfaces
We consider whether any two triangulations of a polygon or a point set on a
non-planar surface with a given metric can be transformed into each other by a
sequence of edge flips. The answer is negative in general with some remarkable
exceptions, such as polygons on the cylinder, and on the flat torus, and
certain configurations of points on the cylinder.Comment: 19 pages, 17 figures. This version has been accepted in the SIAM
Journal on Discrete Mathematics. Keywords: Graph of triangulations,
triangulations on surfaces, triangulations of polygons, edge fli
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
Minimum-weight triangulation is NP-hard
A triangulation of a planar point set S is a maximal plane straight-line
graph with vertex set S. In the minimum-weight triangulation (MWT) problem, we
are looking for a triangulation of a given point set that minimizes the sum of
the edge lengths. We prove that the decision version of this problem is
NP-hard. We use a reduction from PLANAR-1-IN-3-SAT. The correct working of the
gadgets is established with computer assistance, using dynamic programming on
polygonal faces, as well as the beta-skeleton heuristic to certify that certain
edges belong to the minimum-weight triangulation.Comment: 45 pages (including a technical appendix of 13 pages), 28 figures.
This revision contains a few improvements in the expositio
Expansive Motions and the Polytope of Pointed Pseudo-Triangulations
We introduce the polytope of pointed pseudo-triangulations of a point set in
the plane, defined as the polytope of infinitesimal expansive motions of the
points subject to certain constraints on the increase of their distances. Its
1-skeleton is the graph whose vertices are the pointed pseudo-triangulations of
the point set and whose edges are flips of interior pseudo-triangulation edges.
For points in convex position we obtain a new realization of the
associahedron, i.e., a geometric representation of the set of triangulations of
an n-gon, or of the set of binary trees on n vertices, or of many other
combinatorial objects that are counted by the Catalan numbers. By considering
the 1-dimensional version of the polytope of constrained expansive motions we
obtain a second distinct realization of the associahedron as a perturbation of
the positive cell in a Coxeter arrangement.
Our methods produce as a by-product a new proof that every simple polygon or
polygonal arc in the plane has expansive motions, a key step in the proofs of
the Carpenter's Rule Theorem by Connelly, Demaine and Rote (2000) and by
Streinu (2000).Comment: 40 pages, 7 figures. Changes from v1: added some comments (specially
to the "Further remarks" in Section 5) + changed to final book format. This
version is to appear in "Discrete and Computational Geometry -- The
Goodman-Pollack Festschrift" (B. Aronov, S. Basu, J. Pach, M. Sharir, eds),
series "Algorithms and Combinatorics", Springer Verlag, Berli
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