Rigidity, global rigidity, and graph decomposition

The recent combinatorial characterization of generic global rigidity in the plane by Jackson and Jordán (2005) [10] recalls the vital relationship between connectivity and rigidity that was first pointed out by Lovász and Yemini (1982) [13]. The Lovász–Yemini result states that every 6-connected graph is generically rigid in the plane, while the Jackson–Jordán 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.