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

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 2005 |

### Fingerprint

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Theoretical Computer Science

### Cite this

*Journal of Combinatorial Theory. Series B*,

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

**The d-dimensional rigidity matroid of sparse graphs.** / Jackson, Bill; Jordán, T.

Research output: Contribution to journal › Article

*Journal of Combinatorial Theory. Series B*, vol. 95, no. 1, pp. 118-133. https://doi.org/10.1016/j.jctb.2005.03.004

}

TY - JOUR

T1 - The d-dimensional rigidity matroid of sparse graphs

AU - Jackson, Bill

AU - Jordán, T.

PY - 2005/9

Y1 - 2005/9

N2 - Let Rd (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 Rd (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 Rd (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 Rd (G). We use the case d = 4 to show that, if G is 10-connected, then G can be made rigid in ℝ3 by pinning down approximately three quarters of its vertices.

AB - Let Rd (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 Rd (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 Rd (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 Rd (G). We use the case d = 4 to show that, if G is 10-connected, then G can be made rigid in ℝ3 by pinning down approximately three quarters of its vertices.

UR - http://www.scopus.com/inward/record.url?scp=23244468113&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=23244468113&partnerID=8YFLogxK

U2 - 10.1016/j.jctb.2005.03.004

DO - 10.1016/j.jctb.2005.03.004

M3 - Article

AN - SCOPUS:23244468113

VL - 95

SP - 118

EP - 133

JO - Journal of Combinatorial Theory. Series B

JF - Journal of Combinatorial Theory. Series B

SN - 0095-8956

IS - 1

ER -