Algorithms for the d-dimensional rigidity matroid of sparse graphs

• Authors: Sergey Bereg
• Venue: JCDCG'04 Proceedings of the 2004 Japanese conference on Discrete and Computational Geometry
• Year: 2004

Let ${\mathcal R}_{d}(G)$ be the d-dimensional rigidity matroid for a graph G=(V,E). Combinatorial characterization of generically rigid graphs is known only for the plane d=2 [11]. Recently Jackson and Jordán [5] derived a min-max formula which determines the rank function in ${\mathcal R}_{d}(G)$ when G is sparse, i.e. has maximum degree at most d + 2 and minimum degree at most d + 1. We present three efficient algorithms for sparse graphs G that (i) detect if E is independent in the rigidity matroid for G, and (ii) construct G using vertex insertions preserving if G is isostatic, and (iii) compute the rank of ${\mathcal R}_{d}(G)$. The algorithms have linear running time assuming that the dimension d is fixed.