Braced Triangulations and Rigidity
James G. Cruickshank, Eleftherios Kastis, Derek Kitson, Bernd Schulze · Discrete & Computational Geometry · 2023
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 $$b=1$$ 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 $$\mathbb {R}^4$$ R 4 and a class of mixed norms on $$\mathbb {R}^3$$ R 3 .