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.
ASJC Scopus subject areas
- Discrete Mathematics and Combinatorics