A pair u,v of vertices is said to be globally linked in a d-dimensional framework (G,p) if

the distance between u and v is the same in all d-dimensional frameworks (G,q) equivalent to (G,p).

Global linkedness, which is a refined, local version of global rigidity, is not a generic property.

For a given dimension d, we say that u,v is globally linked (resp. weakly globally linked) in G if u,v is globally linked in every (resp. some)

generic framework (G,p).

The problem of finding a combinatorial characterization (and efficient algorithm) for d-dimensional (weak) global linkedness

of vertex pairs in graphs is one of the few remaining major problems in combinatorial rigidity which are unsolved even for d=2.

In this talk I will present some new results on globally linked pairs and globally rigid graphs in the plane and in higher dimensions.

(Based on joint papers with Daniel Garamvolgyi and Soma Villanyi)