Highly connected rigidity matroids have unique underlying graphs (Q691582)

Whitney showed that 3-connected graphs are uniquely determined by the edge sets of their cycles, i.e., a 3-connected graph is uniquely determined by its cycle matroid. By interpreting the cycle matroid of a graph as a 1-dimensional rigidity matroid, the authors outline how to extend Whitney's result to higher dimensions. In particular they prove that 7-(vertex) connected graphs are uniquely determined by their 2-dimensional generic rigidity matroid, and if a 2-dimensional rigidity matroid is \(2k-3\)-(vertically) connected then its underlying graph is \(k\)-connected. The key ingredients in the proof are that 6-connected graphs are rigid in the plane and that 3-connected redundantly rigid graphs have 2-connected rigidity matroids in dimension 2. The 3 dimensional versions of these results are still conjectures, see \textit{L. Lovász} and \textit{Y. Yemini} [SIAM J. Algebraic Discrete Methods 3, 91--98 (1982; Zbl 0497.05025)], and \textit{B. Jackson} and \textit{T. Jordán} [J. Comb. Theory, Ser. B 94, No. 1, 1--29 (2005; Zbl 1076.05021)]. As a general result holding for all dimensions it is shown that for complete graphs on \(n\) vertices, where \(n \geq d+2\), the connectivity of the \(d\)-dimensional rigidity matroid is \(n-d\), hence in any dimension there exist graphs with arbitrarily highly connected rigidity matroid.