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Globally linked pairs in the plane

Characterize, for a given finite graph G, exactly which unordered vertex pairs {u,v} are globally linked in generic realizations in R^2 (i.e., have the same distance in every non-congruent framework with the same edge lengths).

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Background

Jordan and Villányi showed that global rigidity in the plane is equivalent to every pair being weakly globally linked, and they fully characterized weakly globally linked pairs. However, the exact characterization of globally linked pairs (a stronger notion) remains unresolved.

Several partial results and a conjectured characterization are known (see JJS1, JJS2 referenced in the paper).

References

The related problem of characterising when two vertices in a graph are globally linked in $2$ is one of the few remaining open problems for generic rigidity in $2$.

— Rigidity of Graphs and Frameworks: A Matroid Theoretic Approach (2508.11636 - Cruickshank et al., 29 Jul 2025) in Section 3.2 (Global rigidity in 2D)