On Minimally Highly Vertex-Redundantly Rigid Graphs

Abstract

A graph $G=(V,E)$ is called $k$-rigid in $\mathbb {R}^{d}$ if $|V|\ge k+1$ and after deleting any set of at most $k-1$ vertices the resulting graph is rigid in $\mathbb {R}^{d}$. A $k$-rigid graph $G$ is called minimally $k$-rigid if the omission of an arbitrary edge results in a graph that is not $k$-rigid. B. Servatius showed that a 2-rigid graph in $\mathbb {R}^2$ has at least $2|V|-1$ edges and this bound is sharp. We extend this lower bound for arbitrary values of $k$ and $d$ and show its sharpness for the cases where $k=2$ and $d$ is arbitrary and where $k=d=3$. We also provide a sharp upper bound for the number of edges of minimally $k$-rigid graphs in $\mathbb {R}^d$ for all $k$.