### 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 |
---|---|

Pages (from-to) | 240-247 |

Number of pages | 8 |

Journal | European Journal of Combinatorics |

Volume | 34 |

Issue number | 2 |

DOIs | |

Publication status | Published - Feb 2013 |

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### ASJC Scopus subject areas

- Geometry and Topology
- Theoretical Computer Science
- Computational Theory and Mathematics

### Cite this

*European Journal of Combinatorics*,

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

**Highly connected rigidity matroids have unique underlying graphs.** / Jordán, T.; Kaszanitzky, Viktória E.

Research output: Contribution to journal › Article

*European Journal of Combinatorics*, vol. 34, no. 2, pp. 240-247. https://doi.org/10.1016/j.ejc.2012.09.001

}

TY - JOUR

T1 - Highly connected rigidity matroids have unique underlying graphs

AU - Jordán, T.

AU - Kaszanitzky, Viktória E.

PY - 2013/2

Y1 - 2013/2

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=84867135349&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84867135349&partnerID=8YFLogxK

U2 - 10.1016/j.ejc.2012.09.001

DO - 10.1016/j.ejc.2012.09.001

M3 - Article

VL - 34

SP - 240

EP - 247

JO - European Journal of Combinatorics

JF - European Journal of Combinatorics

SN - 0195-6698

IS - 2

ER -