SIAM Journal on Discrete Mathematics
Connectivity and network flows
Handbook of combinatorics (vol. 1)
A proof of Connelly's conjecture on 3-connected circuits of the rigidity matroid
Journal of Combinatorial Theory Series B
The Number of Embeddings of Minimally Rigid Graphs
Discrete & Computational Geometry
Connected rigidity matroids and unique realizations of graphs
Journal of Combinatorial Theory Series B
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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.