Connected rigidity matroids and unique realizations of graphs
Journal of Combinatorial Theory Series B
European Journal of Combinatorics
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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 (2k-3)-connected then its underlying graph is k-vertex-connected. We also 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.