  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 $\mathbb{R}^3$.
