Highly connected rigidity matroids have unique underlying graphs

Let M be a d -dimensional generic rigidity matroid of some graph G . We consider the following problem, posed by Brigitte and Herman Servatius in 2006: is there a (smallest) integer k d such that the underlying graph G of M is uniquely determined, provided that M is k d -connected? Since the one-dimensional generic rigidity matroid of G is isomorphic to its cycle matroid, a celebrated result of Hassler Whitney implies that k 1 = 3 . We extend this result by proving that k 2 ≤ 11 . To show this we prove that (i) if G is 7-vertex-connected then it is uniquely determined by its two-dimensional rigidity matroid, and (ii) if a two-dimensional rigidity matroid is ( 2 k − 3 ) -connected then its underlying graph is k -vertex-connected. o prove the reverse implication: if G is a k -connected graph for some k ≥ 6 then its two-dimensional rigidity matroid is ( k − 2 ) -connected. Furthermore, we determine the connectivity of the d -dimensional rigidity matroid of the complete graph K n , for all pairs of positive integers d , n .