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Journals and Conferences
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)
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 consider the three forms of self-duality that can be exhibited by a planar graph G, map self-duality, graph self-duality and matroid selfduality. 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)
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 2-dimensional generic rigidity matroid R(G) of a graph G. The notions of vertex and edge birigidity are introduced. We prove that vertex birigidity of G implies the connectivity of R(G) and that the connectivity of R(G) implies the edge birigidity of G. These implications are not equivalences. A class of minimal vertex birigid graphs is… (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)
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)
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. Mathematics Subject Classification: 05C40, 05C38, and 05C05.
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, , the general characteristics of self–dual graphs have only recently begun to… (More)
Given a self–dual map on the sphere, the collection of its self– dual permutations generates a transformation group in which the map automorphism 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.