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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 2 / 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.

Author comments: updated to match version at https://arxiv.org/abs/2611.10919

Provenance statement

The proofs of all the new results in this paper were discovered by ChatGPT 6.1 Sol, which also produced the first draft of the paper. The author directed ChatGPT to investigate the problem and contributed to the revision of the manuscript to its current form. This included requesting clarifications and rewrites of the proofs, and making decisions about the organisation and presentation of the paper. The author has checked the arguments presented here and takes full responsibility for 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 depositSupersededNov 06, 2026
  2. v2 · 16 pagesSubmitted by James Cruickshankupdated to match version at https://arxiv.org/abs/2611.10919CurrentNov 12, 2026