We say that a pair of vertices u,v of a graph G is globally linked in a d-dimensional bar-joint framework (G,p) if the distance of u and v is the same in every d-dimensional framework (G,q) equivalent to (G,p). For a given dimension d, we say that u,v is (generically) globally linked in G if u,v is globally linked in every generic d-dimensional framework (G,p). The notion of globally linked pairs is a natural analogue of global rigidity (or "unique reconstructibility"), but not much is known about it in general. In particular, we do not have a polynomial algorithm for recognizing generically globally linked pairs, even in the two-dimensional case.

Interestingly, in almost all graph classes where we do have a combinatorial characterization of globally linked pairs, it is in terms of the number of internally disjoint paths between u and v. In this talk I will discuss this phenomenon, as well as a number of recent results about globally linked pairs.

(Based on joint work with Tibor Jordán.)