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We introduce the idea of Assur graphs, a concept originally developed and exclusively employed in the literature of the kinematics community. The paper translates the terminology, questions, methods and conjectures from the kinematics terminology for one degree of freedom linkages to the terminology of Assur graphs as graphs with special properties in… (More)

Laman's characterization of minimally rigid 2-dimensional generic frameworks gives a matroid structure on the edge set of the underlying graph, as was first pointed out and exploited by L. Lovász and Y. Yemini. Global rigidity has only recently been characterized by a combination of two results due to T. Jordán and the first named author, and R. Connelly,… (More)

We consider the three forms of self-duality that can be exhibited by a planar graph G, map self-duality, graph self-duality and matroid self-duality. We show how these concepts are related with each other and with the connectivity of G. We use the geometry of self-dual polyhedra together with the structure of the cycle matroid to construct all self-dual… (More)

Pointed pseudo-triangulations are planar minimally rigid graphs embedded in the plane with <i>pointed</i> vertices (incident to an angle larger than <i>p</i>). 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… (More)

We expand on Tutte's theory of 3-blocks for 2-connected graphs, generalizing it to apply to infinite, locally finite graphs, and giving necessary and sufficient conditions for a labeled tree to be the 3-block tree of a 2-connected graph.

We show how to recursively construct all self–dual maps on the sphere together with their self–dualities, and classify them according to their edge–permutations. Although several well known classes of self–dual graphs, e.g., the wheels, have been known since the last century, [7], the general characteristics of self–dual graphs have only recently begun to… (More)

We prove that a planar graph is generically rigid in the plane if and only if it can be embedded as a pseudo-triangulation. This generalizes the main result of [4] which treats the minimally generically rigid case. The proof uses the concept of combinatorial pseudo-triangulation, CPT, in the plane and has two main steps: showing that a certain " generalized… (More)

In our previous paper, we presented the combinatorial theory for minimal isostatic pinned frameworks-Assur graphs-which arise in the analysis of mechanical linkages. In this paper we further explore the geometric properties of Assur graphs, with a focus on singular realizations which have static self-stresses. We provide a new geometric characterization of… (More)

Given a self–dual map on the sphere, the collection of its self– dual permutations generates a transformation group in which the map auto-morphism group appears as a subgroup of index two. A careful examination of this pairing yields direct constructions of self–dual maps and provides a classification of self–dual maps.