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Whiteley’s non-trivial vertex splitting conjecture for global rigidity

Prove that non-trivial vertex splitting preserves generic global rigidity in R^d; that is, show that if a graph G is obtained from a graph H by a vertex split of a vertex z into u and v with |N_G(u) ∩ N_G(v)| ≥ d−1 and deg_G(u), deg_G(v) ≥ d+1, and if H is globally rigid in R^d (equivalently, G/uv is globally rigid in R^d), then G is globally rigid in R^d.

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Background

Whiteley proved that vertex splitting preserves generic rigidity (Lemma 2.4 here). For global rigidity, several partial results are known: Connelly showed preservation under special coincident-vertex realizations; Jordan–Tanigawa proved it for sequences starting from graphs of maximum degree at most d+2.

A full, general proof would provide a powerful inductive construction for globally rigid graphs in higher dimensions.

References

Whiteley conjectures that an analogous result holds for global rigidity in $d$ as long as the vertex splitting is {\em non-trivial} i.e.~the degrees of $u$ and $v$ in $G$ are at least $d+1$.

— Rigidity of Graphs and Frameworks: A Matroid Theoretic Approach (2508.11636 - Cruickshank et al., 29 Jul 2025) in Inductive constructions — Vertex splitting