Pointed pseudo-triangulations 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.

Abstract. We consider the problem of computing a minimum weight pseudo-triangulation of a set S of n points in the plane. We first present an O(n log n)-time algorithm that produces a pseudo-triangulation of weight O(log n·wt(M(S))) which is shown to be asymptotically worst-case optimal, i.e., there exists a point set S for which every pseudo-triangulation 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 pseudo-triangulation of a simple polygon. 1

A pseudo-triangle is a simple polygon with exactly three convex vertices, and a pseudo-triangulation is a face-to-face tiling of a planar region into pseudo-triangles. Pseudo-triangulations appear as data structures in computational geometry, as planar bar-and-joint 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, straight-line drawings from abstract versions called combinatorial pseudo-triangulations, algorithms and applications of pseudo-triangulations.

A pseudo-triangle is a simple polygon with exactly three convex vertices. A pseudo-triangulation of a finite point set S in the plane is a partition of the convex hull of S into interior disjoint pseudo-triangles whose vertices are points of S. A pointed pseudo-triangulation is one which has the least number of pseudo-triangles. We study the graph G whose vertices represent the pointed pseudo-triangulations and whose edges represent flips. We present an algorithm for

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

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 divide-and-conquer 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

Abstract We propose a combinatorial approach to planning non-colliding trajectories for a polyg-onal bar-and-joint framework with n vertices. It is based on a new class of simple motionsinduced by expansive one-degree-of-freedom mechanisms, which guarantee non-collisions by moving all points away from each other. Their combinatorial structure is captured by pointedpseudo-triangulations, 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 bar-and-joint planar polygonal linkage using only non-self-intersecting planarmotions. A step of the algorithm consists in moving a pseudo-triangulation-based mechanism along its unique trajectory in configuration space until two adjacent edges align. At thealignment event, a local alteration restores the pseudo-triangulation. 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 non-colliding 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 self-intersections 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 pseudo-triangulations, for which we prove a variety of combinatorial and rigidity theoretical properties. More prominently, a pointed pseudo-triangulation with a removed convex hull edge is a one-degree-of-freedom 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.

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 PT-convex. 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 dynamic-programming algorithm of Keil and Snoeyink for that problem to the case that both convex polygons and pseudo-triangles 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

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 non-crossing edge. Figure 1: A non-planar 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 bar-and-joint frameworks in dimension 2 are exactly the Laman graphs. A planar minimally rigid framework on a given generic two-dimensional point set p is a Laman

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 self-assembly occuring in nature and possesses many features that are desirable while engineering self-assembly 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