Algorithms for the d-Dimensional Rigidity Matroid of Sparse Graphs

Abstract

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.