A temporal graph is a graph whose edges are available only at specific times. In this scenario, the only valid walks are the ones traversing adjacent edges respecting their availability, i.e. sequence of adjacent edges whose appearing times are non-decreasing.

Given a graph G and vertices s and t of G, Menger’s Theorem states that the maximum number of (internally) vertex disjoint s,t-paths is equal to the minimum size of a subset X for which G-X contains no s,t-path. This is a classical result in Graph Theory, taught in most basic Graph Theory courses, and it holds also when G is directed and when edge disjoint paths and edge cuts are considered instead. A direct translation of Menger’s Theorem to the temporal context has been known not to hold since an example was shown in the seminal paper by Kempe, Kleinberg and Kumar (STOC’00). In this talk, an overview of possible temporal versions of Menger’s Theorem will be discussed, as well as the complexity of the related problems.