Braced triangulations and rigidity

Cruickshank, James and Kastis, Eleftherios and Kitson, Derek and Schulze, Bernd (2023) Braced triangulations and rigidity. Discrete and Computational Geometry. ISSN 0179-5376 (In Press)

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Abstract

We consider the problem of finding an inductive construction, based on vertex splitting, of triangulated spheres with a fixed number of additional edges braces). We show that for any positive integer \( b \) there is such an inductive construction of triangulations with \( b \) braces, having finitely many base graphs. In particular we establish a bound for the maximum size of a base graph with $b$ braces that is linear in $b$. In the case that $b=1$ or $2$ we determine the list of base graphs explicitly. Using these results we show that doubly braced triangulations are generically) minimally rigid in two distinct geometric contexts arising from a hypercylinder in $\mathbb{R}^4$ and a class of mixed norms on $\mathbb{R}^3$.