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15
Planar Minimally Rigid Graphs and PseudoTriangulations
, 2003
"... Pointed pseudotriangulations are planar minimally rigid graphs embedded in the plane with pointed vertices (incident to an angle larger than π). In this paper we prove that the opposite statement is also true, namely that planar minimally rigid graphs always admit pointed embeddings, even under cer ..."
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Cited by 30 (14 self)
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Pointed pseudotriangulations are planar minimally rigid graphs embedded in the plane with pointed vertices (incident to an angle larger than π). In this paper we prove that the opposite statement is also true, namely that planar minimally rigid graphs always admit pointed embeddings, even under certain natural topological and combinatorial constraints. The proofs yield efficient embedding algorithms. They also provide—to the best of our knowledge—the first algorithmically effective result on graph embeddings with oriented matroid constraints other than convexity of faces.
A Lower Bound on the Number of Triangulations of Planar Point Sets
"... We show that the number of straightedge triangulations exhibited by any set of n points in general position in the plane is bounded from below by 4 :33 ). ..."
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Cited by 16 (3 self)
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We show that the number of straightedge triangulations exhibited by any set of n points in general position in the plane is bounded from below by 4 :33 ).
PseudoTriangulations  a Survey
 CONTEMPORARY MATHEMATICS
"... A pseudotriangle is a simple polygon with exactly three convex vertices, and a pseudotriangulation is a facetoface tiling of a planar region into pseudotriangles. Pseudotriangulations appear as data structures in computational geometry, as planar barandjoint frameworks in rigidity theory an ..."
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Cited by 14 (4 self)
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A pseudotriangle is a simple polygon with exactly three convex vertices, and a pseudotriangulation is a facetoface tiling of a planar region into pseudotriangles. Pseudotriangulations appear as data structures in computational geometry, as planar barandjoint frameworks in rigidity theory and as projections of locally convex surfaces. This survey of current literature includes combinatorial properties and counting of special classes, rigidity theoretical results, representations as polytopes, straightline drawings from abstract versions called combinatorial pseudotriangulations, algorithms and applications of pseudotriangulations.
On the number of plane graphs
 PROC. 17TH ANN. ACMSIAM SYMP. ON DISCRETE ALGORITHMS
, 2006
"... We investigate the number of plane geometric, i.e., straightline, graphs, a set S of n points in the plane admits. We show that the number of plane graphs is minimized when S is in convex position, and that the same result holds for several relevant subfamilies. In addition we construct a new extre ..."
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Cited by 11 (3 self)
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We investigate the number of plane geometric, i.e., straightline, graphs, a set S of n points in the plane admits. We show that the number of plane graphs is minimized when S is in convex position, and that the same result holds for several relevant subfamilies. In addition we construct a new extremal configuration, the socalled double zigzag chain. Most noteworthy this example bears Θ ∗ ( √ 72 n) = Θ ∗ (8.4853 n) triangulations and Θ ∗ (41.1889 n) plane graphs (omitting polynomial factors in both cases), improving the previously known best maximizing examples.
On the number of pseudotriangulations of certain point sets
 J. Combin. Theory Ser. A
, 2007
"... We pose a monotonicity conjecture on the number of pseudotriangulations of any planar point set, and check it on two prominent families of point sets, namely the socalled double circle and double chain. The latter has asymptotically 12 n n Θ(1) pointed pseudotriangulations, which lies significant ..."
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Cited by 9 (2 self)
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We pose a monotonicity conjecture on the number of pseudotriangulations of any planar point set, and check it on two prominent families of point sets, namely the socalled double circle and double chain. The latter has asymptotically 12 n n Θ(1) pointed pseudotriangulations, which lies significantly above the maximum number of triangulations in a planar point set known so far. ⋆ Parts of this work were done while the authors visited the Departament de
Enumerating pseudotriangulations in the plane
 COMPUT. GEOM. THEORY APPL
, 2005
"... A pseudotriangle is a simple polygon with exactly three convex vertices. A pseudotriangulation of a finite point set S in the plane is a partition of the convex hull of S into interior disjoint pseudotriangles whose vertices are points of S. A pointed pseudotriangulation is one which has the le ..."
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Cited by 7 (0 self)
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A pseudotriangle is a simple polygon with exactly three convex vertices. A pseudotriangulation of a finite point set S in the plane is a partition of the convex hull of S into interior disjoint pseudotriangles whose vertices are points of S. A pointed pseudotriangulation is one which has the least number of pseudotriangles. We study the graph G whose vertices represent the pointed pseudotriangulations and whose edges represent flips. We present an algorithm for
A Simple Sweep Line Algorithm for Counting Triangulations and Pseudotriangulations
, 2012
"... Let P ⊂ R 2 be a set of n points. In [1] and [2] an algorithm for counting triangulations and pseudotriangulations of P, respectively, is shown. Both algorithms are based on the divideandconquer paradigm, and both work by finding substructures on triangulations and pseudotriangulations that all ..."
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Cited by 4 (4 self)
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Let P ⊂ R 2 be a set of n points. In [1] and [2] an algorithm for counting triangulations and pseudotriangulations of P, respectively, is shown. Both algorithms are based on the divideandconquer paradigm, and both work by finding substructures on triangulations and pseudotriangulations that allow the problems to be split. These substructures are called triangulation paths for triangulations, or Tpaths for short, and zigzag paths for pseudotriangulations, or PTpaths for short. Those two algorithms have turned out to be very difficult to analyze, to the point that no good analysis of their running time has been presented so far. The interesting thing about those algorithms, besides their simplicity, is that they experimentally indicate that counting can be done significantly faster than enumeration. In this paper we show two new algorithms, one to compute the number of triangulations of P, and one to compute the number of pseudotriangulations of P. They are also based on Tpaths and PTpaths respectively, but use the sweep line paradigm and not divideandconquer. The important thing about our algorithms
Pseudotriangulations, rigidity and motion planning
 Discrete and Computational Geometry
, 2005
"... Abstract We propose a combinatorial approach to planning noncolliding trajectories for a polygonal barandjoint framework with n vertices. It is based on a new class of simple motionsinduced by expansive onedegreeoffreedom mechanisms, which guarantee noncollisions by moving all points away fr ..."
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Abstract We propose a combinatorial approach to planning noncolliding trajectories for a polygonal barandjoint framework with n vertices. It is based on a new class of simple motionsinduced by expansive onedegreeoffreedom mechanisms, which guarantee noncollisions by moving all points away from each other. Their combinatorial structure is captured by pointedpseudotriangulations, a class of embedded planar graphs for which we give several equivalent characterizations and exhibit rich rigidity theoretic properties.The main application is an efficient algorithm for the Carpenter's Rule Problem: convexify a simple barandjoint planar polygonal linkage using only nonselfintersecting planarmotions. A step of the algorithm consists in moving a pseudotriangulationbased mechanism along its unique trajectory in configuration space until two adjacent edges align. At thealignment event, a local alteration restores the pseudotriangulation. The motion continues for O(n3) steps until all the points are in convex position. 1 Introduction We present a combinatorial solution to the Carpenter's Rule Problem: how to plan noncolliding reconfigurations of a planar robot arm. The main result is an efficient algorithm for the problem of continuously moving a simple planar polygon to any other configuration with the same edgelengths and orientation, while remaining in the plane and never creating selfintersections along the way. This is done by first finding motions that convexify both configurations with expansive motions (which never bring two points closer together) and then taking one path in reverse. All of the constructions are elementary and are based on a novel class of planar embedded graphs called pointed pseudotriangulations, for which we prove a variety of combinatorial and rigidity theoretical properties. More prominently, a pointed pseudotriangulation with a removed convex hull edge is a onedegreeoffreedom expansive mechanism. If its edges are seen as rigid bars (maintaining their lengths) and are allowed to rotate freely around the vertices (joints), the mechanism follows (for a well defined, finite time interval) a continuous trajectory along which no distance between a pair of points ever decreases. The expansive motion induced by these mechanisms provide the building blocks of our algorithm.
A Simple Aggregative Algorithm for Counting Triangulations of Planar Point Sets and Related Problems
"... We give an algorithm that determines the number tr(S) of straight line triangulations of a set S of n points in the plane in worst case time O(n 2 2 n). This is the the first algorithm that is provably faster than enumeration, since tr(S) is known to be Ω(2.43 n) for any set S of n points. Our algor ..."
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Cited by 1 (1 self)
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We give an algorithm that determines the number tr(S) of straight line triangulations of a set S of n points in the plane in worst case time O(n 2 2 n). This is the the first algorithm that is provably faster than enumeration, since tr(S) is known to be Ω(2.43 n) for any set S of n points. Our algorithm requires exponential space. The algorithm generalizes to counting all triangulations of S that are constrained to contain a givenset ofedges. It can alsobe used to compute an optimal triangulationofS (unconstrained or constrained) for a reasonably wide class of optimality criteria (that includes e.g. minimum weight triangulations). Finally, the approach can also be used for the random generation of triangulations of S according to the perfect uniform distribution. The algorithm has been implement and is substantially faster than existing methods on a variety of inputs. 1