Extremal families of redundantly rigid graphs in three dimensions

A rigid graph G is said to be k-vertex (resp. k-edge) rigid in Rd if it remains rigid after the removal of less than k vertices (resp. edges). The definition of k-vertex (resp. k-edge) globally rigid graphs in Rd is similar. We study each of these four versions of redundant (global) rigidity and determine the smallest number of edges in a k-vertex (resp. k-edge) rigid (resp. globally rigid) graph on n vertices in R3 for all positive integers k, except for four special cases, where we provide a close-to-tight bound.(c) 2022 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).