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Vertex splitting, global rigidity and nondegenerate equilibrium stresses

Primarily AI-generated textHuman understanding: all partsmath.CO — Combinatoricscs.DM — Discrete Mathematics

Contributed by James Cruickshank ↗

Version 1 / Nov 06, 2026 / CC BY 4.0

Abstract

We prove Whiteley's conjecture that every nontrivial vertex split of a generically globally rigid graph preserves generic global rigidity in Rd\mathbb{R}^d, for all d≥1d\geq1. We give two proofs of the stronger assertion that the split graph is infinitesimally rigid at every generic coincident realisation of the two split vertices. Both proofs use maximum-rank equilibrium stress matrices and then apply the coincident-rigidity criterion of Cruickshank, Jackson and Tanigawa. The second construction also shows that every generically globally rigid graph admits a nondegenerate equilibrium stress of maximum rank at every generic realisation, establishing a conjecture of Jord\'an and Tanigawa.

Provenance statement

All the proofs of the main results presented in this paper were created by ChatGPT 6.1 Sol. ChatGPT is also responsible for most of the exposition. The second author directed ChatGPT to investigate the problem, checked the correctness of the arguments and contributed to the revision of the manuscript. This included requesting clarifications and further checks of the arguments, and making decisions about the organisation and presentation of the paper.

References

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Show all 12 referencesShow fewer references
  1. Cruickshank, James; Jackson, Bill; Tanigawa, Shin-ichi. Global Rigidity of Triangulated Manifolds. vol. 458DOI
  2. Cruickshank, James; Jackson, Bill; Jordán, Tibor; Tanigawa, Shin-ichi. Rigidity of Graphs and Frameworks: A Matroid Theoretic Approach. Surveys in Combinatorics 2026, no. 505, pp. 189–230DOI

Version history

  1. v1 · 16 pagesSubmitted by James CruickshankInitial depositCurrentNov 06, 2026