Braced triangulations and rigidity

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 R 4 and a class of mixed norms on R 3 . 2020 Mathematics Subject Classification. 52C25, 05C10, 46B20. E.K. and D.K.

Infinitesimal rigidity and prestress stability for frameworks in normed spaces

“…In the following, we denote the complete graph on six vertices minus any edge by K 6 − e, and the graph obtained by gluing two copies of K 5 at three vertices by K 5 ∪ K 3 K 5 (see Figure 6). The following result is proved in [6]. Theorem 4.2.…”

“…The rigidity of graphs in non-Euclidean normed spaces of dimension d ≥ 3 has previously been investigated in the settings of p -spaces [9,16], polyhedral spaces [14], matrix spaces [15], and for a class of mixed norms [6]. In each of these settings, necessary combinatorial conditions for rigidity have been derived but complete characterisations have remained elusive.…”