Characterize generic rigidity and global rigidity for d ≥ 3

Determine combinatorial characterizations of finite graphs that are (i) generically rigid and (ii) generically globally rigid in Euclidean R^d for every fixed dimension d ≥ 3.

Background

For d = 1 and d = 2, complete combinatorial characterizations are known: forests characterize rigidity on the line, and Laman’s theorem characterizes planar rigidity; Hendrickson’s conditions are sufficient for global rigidity in the plane. For d ≥ 3, deciding rigidity or global rigidity is NP-hard in general, but in the generic setting the property depends only on the graph. Nevertheless, no purely combinatorial characterization is known.

This problem underpins many structural and algorithmic questions in rigidity theory and affects applications across CAD, molecular conformation, and sensor networks.