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Configurations of points in the plane constrained by directions only or by lengths alone lead to equivalent theories known as parallel drawings and infinitesimal rigidity of plane frameworks. We combine these two theories by introducing a new matroid on the edge set of the complete graph with doubled edges to describe the combinatorial properties of… (More)

Configurations of points in the plane constrained by only directions or by lengths alone lead to equivalent theories known as parallel drawings and infinitesimal rigidity of plane frameworks. We combine these two theories by introducing a new matroid on the edge set of the complete graph with doubled edges to describe the combinatorial properties of… (More)

Let Γ = (V, E) be a graph with vertex set V and edge set E. The graph group based on Γ, F Γ , is the group generated by V , with defining relations xy = yx, one for each pair (x, y) of adjacent vertices in Γ. For n ≥ 3, the n-gon is the graph with n vertices, v 1 ,. .. , v n , and n edges (v i , v i+1), indices modulo n. In this article we will show that if… (More)

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)

The question of which groups admit planar Cayley graphs goes back over 100 years, being settled for finite groups by Maschke in 1896. Since that time, various authors have studied infinite planar Cayley graphs which satisfy additional special conditions. We consider the question of which groups possess any planar Cayley graphs at all by categorizing such… (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)

Rigidity We are all familiar with frameworks of rods attached at joints. A rod and joint framework gives rise to a simple mathematical model consisting of line segments in Euclidean 3-space with common endpoints. A deformation is a continuous one-parameter family of such frameworks. If a framework has only trivial deformations , e.g. translations and… (More)

- Yevgen Amigud, Geoffrey Archer, Janelle Smith, Melissa Szymanski, Brigitte Servatius
- 2002

—A web-enabled laboratory provides remote access to state of the art equipment. The use of web-enabled laboratories in undergraduate education has dramatically increased over the past six years, but no criteria have been established to assess the quality of this new teaching tool. We identify ten vital components any successful web-enabled laboratory should… (More)