math.CO — Combinatorics

Vertex splitting, global rigidity and nondegenerate equilibrium stresses

Contributed by James Cruickshank

We prove Whiteley's conjecture that every nontrivial vertex split of a generically globally rigid graph preserves generic global rigidity in $\mathbb{R}^d$, for all $d\geq1$. 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.