### Abstract

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.

Original language | English |
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Pages (from-to) | 240-247 |

Number of pages | 8 |

Journal | European Journal of Combinatorics |

Volume | 34 |

Issue number | 2 |

DOIs | |

Publication status | Published - Feb 1 2013 |

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics

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## Cite this

*European Journal of Combinatorics*,

*34*(2), 240-247. https://doi.org/10.1016/j.ejc.2012.09.001