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11
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 29 (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.
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 13 (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.
Minimum weight pseudotriangulations
 Proc. 24th Int. Conf. Foundations Software Tech. Theoretical Comput. Sci. (FSTTCS’04), volume 3328 of Lecture Notes in Computer Science
, 2004
"... Abstract. We consider the problem of computing a minimum weight pseudotriangulation of a set S of n points in the plane. We first present an O(n log n)time algorithm that produces a pseudotriangulation of weight O(log n·wt(M(S))) which is shown to be asymptotically worstcase optimal, i.e., there ..."
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Cited by 7 (0 self)
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Abstract. We consider the problem of computing a minimum weight pseudotriangulation of a set S of n points in the plane. We first present an O(n log n)time algorithm that produces a pseudotriangulation of weight O(log n·wt(M(S))) which is shown to be asymptotically worstcase optimal, i.e., there exists a point set S for which every pseudotriangulation has weight Ω(log n · wt(M(S))), where wt(M(S)) is the weight of a minimum spanning tree of S. We also present a constant factor approximation algorithm running in cubic time. In the process we give an algorithm that produces a minimum weight pseudotriangulation of a simple polygon. 1
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
Minimum weight triangulation by cutting out triangles
 In Proceeding of the 16th Annual International Symposium on Algorithms and Computation
, 2005
"... Abstract. We describe a fixed parameter algorithm for computing the minimum weight triangulation (MWT) of a simple polygon with (n − k) vertices on the perimeter and k hole vertices in the interior, that is, for a total of n vertices. Our algorithm is based on cutting out empty triangles (that is, t ..."
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Cited by 5 (1 self)
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Abstract. We describe a fixed parameter algorithm for computing the minimum weight triangulation (MWT) of a simple polygon with (n − k) vertices on the perimeter and k hole vertices in the interior, that is, for a total of n vertices. Our algorithm is based on cutting out empty triangles (that is, triangles not containing any holes) from the polygon and processing the parts or the rest of the polygon recursively. We show that with our algorithm a minimum weight triangulation can be found in time at most O(n 3 k! k), and thus in O(n 3) if k is constant. We also note that k! can actually be replaced by b k for some constant b. We implemented our algorithm in Java and report experiments backing our analysis. 1
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 3 (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 Fixed Parameter Algorithm for Minimum Weight Triangulation: Analysis and Experiments
 Proc. 22nd European Workshop Computational Geometry EWCG
, 2005
"... Abstract. We discuss and compare four fixed parameter algorithms for finding the minimum weight triangulation of a simple polygon with (n − k) vertices on the perimeter and k vertices in the interior (hole vertices), that is, for a total of n vertices. All four algorithms rely on the same abstract d ..."
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Cited by 3 (1 self)
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Abstract. We discuss and compare four fixed parameter algorithms for finding the minimum weight triangulation of a simple polygon with (n − k) vertices on the perimeter and k vertices in the interior (hole vertices), that is, for a total of n vertices. All four algorithms rely on the same abstract divideandconquer scheme, which is made efficient by a variant of dynamic programming. They are essentially based on two simple observations about triangulations, which give rise to triangle splits and paths splits. While each of the first two algorithms uses only one of these split types, the last two algorithms combine them in order to achieve certain improvements and thus to reduce the time complexity. By discussing this sequence of four algorithms we try to bring out the core ideas as clearly as possible and thus strive to achieve a deeper understanding as well as a simpler specification of these approaches. In addition, we implemented all four algorithms in Java and report results of experiments we carried out with this implementation. 1
Decomposing a Simple Polygon into PseudoTriangles and Convex Polygons ∗
, 2007
"... In this paper we consider the problem of decomposing a simple polygon into subpolygons that exclusively use vertices of the given polygon. We allow two types of subpolygons: pseudotriangles and convex polygons. We call the resulting decomposition PTconvex. We are interested in minimum decomposition ..."
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Cited by 1 (0 self)
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In this paper we consider the problem of decomposing a simple polygon into subpolygons that exclusively use vertices of the given polygon. We allow two types of subpolygons: pseudotriangles and convex polygons. We call the resulting decomposition PTconvex. We are interested in minimum decompositions, i.e., in decomposing the input polygon into the least number of subpolygons. Allowing subpolygons of one of two types has the potential to reduce the complexity of the resulting decomposition considerably. The problem of decomposing a simple polygon into the least number of convex polygons has been considered. We extend a dynamicprogramming algorithm of Keil and Snoeyink for that problem to the case that both convex polygons and pseudotriangles are allowed. Our algorithm determines such a decomposition in O(n 3) time and space, where n is the number of the vertices of the polygon. 1
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 1 (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
Modeling Virus SelfAssembly Pathways Using Computational Algebra and Geometry
 APPLICATIONS OF COMPUTER ALGEBRA (ACA2004)
, 2004
"... We develop a tractable model for elucidating the assembly pathways by which an icosahedral viral shell forms from 60 identical constituent protein monomers. This poorly understood process a remarkable example of macromolecular selfassembly occuring in nature and possesses many features that are d ..."
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We develop a tractable model for elucidating the assembly pathways by which an icosahedral viral shell forms from 60 identical constituent protein monomers. This poorly understood process a remarkable example of macromolecular selfassembly occuring in nature and possesses many features that are desirable while engineering selfassembly at the nanoscale. The model uses static geometric constraints to represent the driving (weak) forces that cause a viral shell to assemble and hold it together. The goal is to answer focused questions about the structural properties of a successful assembly pathway. Pathways and their