### Abstract

Let R_{d} (G) be the d-dimensional rigidity matroid for a graph G = (V, E). For X ⊆ V let i (X) be the number of edges in the subgraph of G induced by X. We derive a min-max formula which determines the rank function in R_{d} (G) when G has maximum degree at most d + 2 and minimum degree at most d + 1. We also show that if d is even and i (X) ≤ 1/2 [(d + 2) X - (2d + 2)] for all X ⊆ V with X ≥ 2 then E is independent in R_{d} (G). We conjecture that the latter result holds for all d ≥ 2 and prove this for the special case when d = 3. We use the independence result for even d to show that if the connectivity of G is sufficiently large in comparison to d then E has large rank in R_{d} (G). We use the case d = 4 to show that, if G is 10-connected, then G can be made rigid in ℝ^{3} by pinning down approximately three quarters of its vertices.

Original language | English |
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Pages (from-to) | 118-133 |

Number of pages | 16 |

Journal | Journal of Combinatorial Theory. Series B |

Volume | 95 |

Issue number | 1 |

DOIs | |

Publication status | Published - Sep 1 2005 |

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### ASJC Scopus subject areas

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics

### Cite this

*Journal of Combinatorial Theory. Series B*,

*95*(1), 118-133. https://doi.org/10.1016/j.jctb.2005.03.004