Globally Linked Pairs of Vertices in Generic Frameworks

A d-dimensional framework is a pair (G, p), where \(G=(V,E)\) is a graph and p is a map from V to \({\mathbb {R}}^d\). The length of an edge \(xy\in E\) in (G, p) is the distance between p(x) and p(y). A vertex pair \(\{u,v\}\) of G is said to be globally linked in (G, p) if the distance between p(u) and p(v) is equal to the distance between q(u) and q(v) for every d-dimensional framework (G, q) in which the corresponding edge lengths are the same as in (G, p). We call (G, p) globally rigid in \({\mathbb {R}}^d\) when each vertex pair of G is globally linked in (G, p). A pair \(\{u,v\}\) of vertices of G is said to be weakly globally linked in G in \({\mathbb {R}}^d\) if there exists a generic framework (G, p) in which \(\{u,v\}\) is globally linked. In this paper we first give a sufficient condition for the weak global linkedness of a vertex pair of a \((d+1)\)-connected graph G in \({\mathbb {R}}^d\) and then show that for \(d=2\) it is also necessary. We use this result to obtain a complete characterization of weakly globally linked pairs in graphs in \({\mathbb {R}}^2\), which gives rise to an algorithm for testing weak global linkedness in the plane in \(O(|V|^2)\) time. Our methods lead to a new short proof for the characterization of globally rigid graphs in \({\mathbb {R}}^2\), and further results on weakly globally linked pairs and globally rigid graphs in the plane and in higher dimensions.