Globally Linked Pairs of Vertices in Generic Frameworks

Abstract

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