Braced Triangulations and Rigidity

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