Braced triangulations and rigidity

TL;DR

This paper develops an inductive method for constructing triangulated spheres with a fixed number of braces, establishing bounds on base graphs and demonstrating minimal rigidity in certain geometric contexts.

Contribution

It introduces a new inductive construction for braced triangulations, providing bounds on base graph sizes and explicit classifications for small numbers of braces.

Findings

Existence of inductive constructions for any number of braces

Linear bounds on maximum size of base graphs with braces

Doubly braced triangulations are minimally rigid in specific geometric settings

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 $\mathbb{R}^3$.