Complexity of global rigidity for unit‑edge‑length graphs

Determine the computational complexity of deciding global rigidity for graphs whose edge lengths are all equal to one (crossings allowed); characterize whether the global‑rigidity decision problem is ∀R‑complete, coNP‑hard, or in a simpler class.

Background

The paper settles complexity for realization, rigidity, and global rigidity across several graph/linkage models, but the global‑rigidity problem for unit‑edge‑length graphs is the lone case left unresolved. Their results for {1,2}‑distance graphs show ∀R‑completeness, yet extending this to unit‑distance remains open and appears tied to the existence of nontrivial globally rigid unit‑distance graphs.

A resolution would complete the landscape summarized in Table 1 by filling the final open cell and clarifying the hardness of global rigidity under unit‑length constraints.

References

Table 1 settles most problems in this area, but a few interesting open problems remain. The one combination in the table that remains unsolved is the complexity of deciding global rigidity in graphs with unit edge lengths, allowing crossings.

— Who Needs Crossings?: Noncrossing Linkages are Universal, and Deciding (Global) Rigidity is Hard  (2510.17737 - Abel et al., 20 Oct 2025) in Section 7, Open Problems