Pointed pseudo-triangulations are planar minimally rigid graphs embedded in the plane with pointed vertices (adjacent 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 the first algorithmically effective result on graph embeddings with oriented matroid constraints other than convexity of faces. These constraints are described by combinatorial pseudo-triangulations, first defined and studied in this paper. Also of interest are our 32 R. Haas et al. / Computational Geometry 31 (2005) two proof techniques, one based on Henneberg inductive constructions from combinatorial rigidity theory, the other on a generalization of Tutte's barycentric embeddings to directed graphs.
Abstract.Given a graph T , define the group Fr to be that generated by the vertices of T, with a defining relation xy -yx for each pair x, y of adjacent vertices of T. In this article, we examine the groups Fr-, where the graph T is an H-gon, (n > 4). We use a covering space argument to prove that in this case, the commutator subgroup Ff contains the fundamental group of the orientable surface of genus 1 + (n -4)2"-3 . We then use this result to classify all finite graphs T for which Fp is a free group.To each graph T = ( V, E), with vertex set V and edge set E, we associate a presentation PT whose generators are the elements of V, and whose relations are {xy = yx\x ,y adjacent vertices of T).
We show that there is a bar-and-joint framework G(p) which has a configuration p in the plane such that the component of p in the space of all planar configurations of G has a cusp at p. At the cusp point, the mechanism G(p) turns out to be third-order rigid in the sense that every third-order flex must have a trivial first-order component. The existence of a third-order rigid framework that is not rigid calls into question the whole notion of higher-order rigidity.
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