Herman Servatius

Planar minimally rigid graphs and pseudo-triangulations

Pointed pseudo-triangulations are planar minimally rigid graphs embedded in the plane with pointed vertices (incident to an angle larger than p). 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---to the best of our knowledge---the first algorithmically effective result on graph embeddings with oriented matroid constraints other than convexity of faces.

Non-Crossing Frameworks with Non-Crossing Reciprocals

Abstract 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 stress on G; and a geometric condition on the stress vectors at some of the vertices. As in other recent papers where the interplay of non-crossingness and rigidity of straight-line plane graphs is studied, pseudo-triangulations show up as objects of special interest. For example, it is known that all planar Laman circuits can be embedded as a pseudo-triangulation with one non-pointed vertex. We show that for such pseudo-triangulation embeddings of planar Laman circuits which are sufficiently generic, the reciprocal is non-crossing and again a pseudo-triangulation embedding of a planar Laman circuit. For a singular (non-generic) pseudo-triangulation embedding of a planar Laman circuit, the reciprocal is still non-crossing and a pseudo-triangulation, but its underlying graph may not be a Laman circuit. Moreover, all the pseudo-triangulations which admit a non-crossing reciprocal arise as the reciprocals of such, possibly singular, stresses on pseudo-triangulation Laman circuits. All self-stresses on a planar graph correspond to liftings to piecewise linear surfaces in 3-space. We prove characteristic geometric properties of the lifts of such non-crossing reciprocal pairs.

Rigidity, global rigidity, and graph decomposition

The recent combinatorial characterization of generic global rigidity in the plane by Jackson and Jordan (2005) [10] recalls the vital relationship between connectivity and rigidity that was first pointed out by Lovasz and Yemini (1982) [13]. The Lovasz-Yemini result states that every 6-connected graph is generically rigid in the plane, while the Jackson-Jordan result states that a graph is generically globally rigid in the plane if and only if it is 3-connected and edge-2-rigid. We examine the interplay between the connectivity properties of the connectivity matroid and the rigidity matroid of a graph and derive a number of structure theorems in this setting, some well known, some new. As a by-product we show that the class of generic rigidity matroids is not closed under 2-sum decomposition. Finally we define the configuration index of the graph and show how the structure theorems can be used to compute it.

The generalized Reye configuration

Given an embedding in the plane of the vertices of a cube, we consider conditions on the embedded vertices such that they are the projection of the vertices of a 3 dimensional polyhedron combinatorially equivalent to the cube. We are particularly interested in conditions expressible as incidence conditions for the lines induced by the given vertex set in the plane. We show that the vertices and their incidences extend to a Reye configuration in the plane if and only if the vertices are the projection of a parallelepiped. More generally, the positions of the embedded vertices result from projecting a polyhedron combinatorially equivalent to the cube if and only if they belong to a plane configuration we call the generalized Reye configuration.

On the 2-sum in rigidity matroids

We show that the graph 2-sum of two frameworks is the underlying framework for the 2-sum of the infinitesimal and generic rigidity matroids of the frameworks. However, we show that, unlike the cycle matroid of a graph, these rigidity matroids are not closed under 2-sum decomposition.

A polynomial time algorithm for determining zero Euler---Petrie genus of an Eulerian graph

A dual-Eulerian graph is a plane graph which has an ordering defined on its edge set which forms simultaneously an Euler circuit in the graph and an Euler circuit in the dual graph. Dual-Eulerian graphs were defined and studied in the context of silicon optimization of cmos layouts. They are necessarily of low connectivity, hence may have many planar embeddings. We give a polynomial time algorithm to answer the question whether or not a planar multigraph admits an embedding which is dual-Eulerian and construct such an embedding, if it exists.