Brigitte 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.

Constraining Plane Configurations in Computer-Aided Design: Combinatorics of Directions and Lengths

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 direction-length designs.

Combinatorial characterization of the Assur graphs from engineering

Abstract We introduce the idea of Assur graphs, a concept originally developed and exclusively employed in the literature of the kinematics community. This 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 rigidity theory. Exploiting the recent works in combinatorial rigidity theory we provide mathematical characterizations of these graphs derived from ‘minimal’ linkages. With these characterizations, we confirm a series of conjectures posed by Offer Shai, and offer techniques and algorithms to be exploited further in future work.

Configurations from a Graphical Viewpoint

Preface.- Introduction.- Graphs.- Groups, Actions, and Symmetry.- Maps.- Combinatorial Configurations.- Geometric Configurations.- Index.- Bibliography.

Geometric properties of Assur graphs

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 Assur graphs, based on special singular realizations. These singular positions are then related to dead-end positions in which an associated mechanism with an inserted driver will stop or jam.

Self-dual graphs

Abstract 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 graphs.

Birigidity in the plane

The two-dimensional generic rigidity matroid

Non-Crossing Frameworks with Non-Crossing Reciprocals

R(G)

The 24 symmetry pairings of self-dual maps on the sphere

of a graph G is considered. The notions of vertex and edge birigidity are introduced. It is proved that vertex birigidity of G implies the connectivity of

Self-dual maps on the sphere

R(G)