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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 39 (15 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.
Flips in Planar Graphs
, 2006
"... We review a selection of results concerning edge flips in triangulations and planar graphs concentrating mainly on various aspects of the following problem: Given two different planar graphs of the same size, how many edge flips are necessary and sufficient to transform one graph into another. We st ..."
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Cited by 29 (5 self)
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We review a selection of results concerning edge flips in triangulations and planar graphs concentrating mainly on various aspects of the following problem: Given two different planar graphs of the same size, how many edge flips are necessary and sufficient to transform one graph into another. We study the problem both from a combinatorial perspective (where only a combinatorial embedding of the graph is specified) and a geometric perspective (where the graph is embedded in the plane, vertices are points and edges are straightline segments). We highlight both the similarities and differences of the two settings, describe many extensions and generalizations, outline several applications and mention open problems.
The zigzag path of a pseudotriangulation
 IN PROC. 8TH INTERNATIONAL WORKSHOP ON ALGORITHMS AND DATA STRUCTURES (WADS
, 2003
"... We define the path of a pseudotriangulation, a data structure generalizing the path of a triangulation of a point set. This structure allows us to use divideandconquer type of approaches for suitable (i.e. decomposable) problems on pseudotriangulations. We illustrate this method by presenting a ..."
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Cited by 18 (6 self)
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We define the path of a pseudotriangulation, a data structure generalizing the path of a triangulation of a point set. This structure allows us to use divideandconquer type of approaches for suitable (i.e. decomposable) problems on pseudotriangulations. We illustrate this method by presenting a novel algorithm that counts the number of pseudotriangulations of a point set.
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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Cited by 7 (0 self)
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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 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
Enumerating planar minimally rigid graphs
 Proc. 12th Annual International Computing and Combinatorics Conference (COCOON 2006
, 2006
"... Motivated by the work of Kawamoto et al. [5], who first suggested the use of graph enumeration techniques as an engineering tool for finding an optimum mechanism design, we give an algorithm for enumerating all the planar Laman graphs embedded on a given generic set p of n points. Our algorithm is b ..."
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Cited by 2 (0 self)
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Motivated by the work of Kawamoto et al. [5], who first suggested the use of graph enumeration techniques as an engineering tool for finding an optimum mechanism design, we give an algorithm for enumerating all the planar Laman graphs embedded on a given generic set p of n points. Our algorithm is based on the Reverse search paradigm of Avis and Fukuda [1]. In particular, we obtain that the set of all planar Laman graphs on a given point set is connected by flips which remove an edge and then restore the Laman property with the addition of a noncrossing edge. Figure 1: A nonplanar Laman graph. A graph on n vertices is a Laman graph if it has exactly 2n − 3 edges and every subset of n ′ < n vertices spans at most 2n ′ − 3 edges. A classical result in Rigidity Theory [3], due to Laman, states that the underlying graphs of generic minimally rigid barandjoint frameworks in dimension 2 are exactly the Laman graphs. A planar minimally rigid framework on a given generic twodimensional point set p is a Laman
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"... If P is a set of points in a plane and S is a subset of nonhull points, then T P S denotes the set of all pseudotriangulations in which S vertices are nonpointed. It is shown that the flip graph on T P S pseudotriangulations is connected and no flip edge exists between pseudotriangulations of T P ..."
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If P is a set of points in a plane and S is a subset of nonhull points, then T P S denotes the set of all pseudotriangulations in which S vertices are nonpointed. It is shown that the flip graph on T P S pseudotriangulations is connected and no flip edge exists between pseudotriangulations of T P