Herman Servatius

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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)
Let Γ = (V, E) be a graph with vertex set V and edge set E. The graph group based on Γ, FΓ, 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, v1, . . . , vn, and n edges (vi, vi+1), indices modulo n. In this article we will show that if Γ has(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 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(More)
We study non-crossing frameworks in the plane for which the classical reciprocal on the dual graph is also non-crossing. We give a complete description of the self-stresses on non-crossing frameworks G whose reciprocals are non-crossing, in terms of: the types of faces (only pseudo-triangles and pseudo-quadrangles are allowed); the sign patterns in the(More)
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)
Higman has questioned which discrete hyperbolic groups [p, q] have representations onto almost all symmetric and alternating groups. We call this property 3tf and show that, except perhaps for finitely many values of/? and q, [p,q] has property JC. It is well known that the modular group F = (x,y\ x = y = 1> has the property that every alternating and(More)