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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 arethe same as in $(G,p)$. We call $(G,p)$ globally rigid in $\mathbb{R}^d$ wheneach vertex pair of $G$ is globally linked in $(G,p)$. A pair $\{u,v\}$ ofvertices 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 globallylinked.  In this paper we first give a sufficient condition for the weak globallinkedness 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 obtaina complete characterization of weakly globally linked pairs in graphs in$\mathbb{R}^2$, which gives rise to an algorithm for testing weak globallinkedness in the plane in $O(|V|^2)$ time. Our methods lead to a new shortproof for the characterization of globally rigid graphs in $\mathbb{R}^2$, andfurther results on weakly globally linked pairs and globally rigid graphs inthe plane and in higher dimensions.

Globally linked pairs of vertices in generic frameworks