Menger's Theorem for Temporal Paths (Not Walks)

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. Temporal paths are temporal walks where each vertex is not traversed twice, i.e. time instants of each vertex, called temporal vertices, are visited consecutively. While on static graphs Menger's Theorem relies on disjoint paths, in temporal graphs the literature has focused on disjoint temporal walks. In this paper we focus on Menger's Theorem for temporal paths that are disjoint in temporal vertices. Given two vertices $s$ and $t$, let $k$ be equal to the maximum number of temporal vertex disjoint $s,t$-paths. We prove that $k$ is equal to the minimum number of temporal vertices to be removed to break all the $s,t$-paths, i.e. Menger's Theorem holds, if and only if $k=1$. The latter property also allows us to show that the related max-paths problem in temporal graphs is polynomial when $k\le 2$. This is best possible as we prove that such problem is \NP-hard when $k\ge 3$ for the directed case. Finally, we also give hardness results and an XP algorithm for the related min-cut problem.