TY - JOUR

T1 - Braced triangulations and rigidity

AU - Cruickshank, James

AU - Kastis, Eleftherios

AU - Kitson, Derek

AU - Schulze, Bernd

PY - 2023/7/2

Y1 - 2023/7/2

N2 - 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 $\mathbb{R}^3$.

AB - 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 $\mathbb{R}^3$.

M3 - Journal article

JO - Discrete and Computational Geometry

JF - Discrete and Computational Geometry

SN - 0179-5376

ER -