# The d-dimensional rigidity matroid of sparse graphs

Bill Jackson, T. Jordán

Research output: Contribution to journalArticle

10 Citations (Scopus)

### Abstract

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.

Original language English 118-133 16 Journal of Combinatorial Theory. Series B 95 1 https://doi.org/10.1016/j.jctb.2005.03.004 Published - Sep 2005

### Fingerprint

Sparse Graphs
Matroid
Rigidity
Independence Results
Minimum Degree
Min-max
Maximum Degree
Subgraph
Connectivity
Graph in graph theory

### ASJC Scopus subject areas

• Discrete Mathematics and Combinatorics
• Theoretical Computer Science

### Cite this

In: Journal of Combinatorial Theory. Series B, Vol. 95, No. 1, 09.2005, p. 118-133.

Research output: Contribution to journalArticle

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