Higher-dimensional characterization of rigid graphs

Determine a combinatorial characterization of rigid graphs for generic embeddings into Euclidean space R^d for all dimensions d ≥ 3, analogous to the Pollaczek–Geiringer–Laman theorem that characterizes rigidity in the plane (d = 2).

Background

Graph rigidity asks whether a generic embedding of a graph’s vertices into Rd is determined up to rigid motions by edge lengths. For d = 2, the Pollaczek–Geiringer–Laman theorem gives a complete combinatorial characterization of rigid graphs. The paper notes that such a characterization is not known for dimensions d ≥ 3, highlighting a longstanding open direction in rigidity theory.

This problem connects to several matroidal formulations of rigidity studied in the paper, where a higher-dimensional characterization would have implications for symmetric matrix completion and related algebraic matroids.

References

When $d=2$, the Pollaczek-Geiringer--Laman theorem \cites{PollaczekGeiringer,Laman} gives a simple characterization of rigid graphs. There is no known generalization to embeddings of graphs into $\mathbb{R}d$ for any $d \ge 3$.

— Rigidity matroids and linear algebraic matroids with applications to matrix completion and tensor codes  (2405.00778 - Brakensiek et al., 2024) in Introduction

Determining which graphs are rigid in a given dimension is a well-studied problem. When $d\leq2$ complete descriptions are known \cites{Lam70,PG27} whereas when $d\geq 3$ it is the central open problem in rigidity theory to give a combinatorial characterisation of rigid graphs.

— Rigidity of complements of bounded-degree graphs  (2609.05058 - Haslegrave et al., 4 Sep 2026) in Section 1, Introduction