Connected rigidity matroids and unique realizations of graphs
Journal of Combinatorial Theory Series B
Algorithms for the d-dimensional rigidity matroid of sparse graphs
JCDCG'04 Proceedings of the 2004 Japanese conference on Discrete and Computational Geometry
Efficient algorithms for the d-dimensional rigidity matroid of sparse graphs
Computational Geometry: Theory and Applications
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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) ≤ ½[(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.