A (directed) temporal graph is a (directed) graph whose edges are available only at specific times during its lifetime $\tau$. Walks are sequences of adjacent edges whose appearing times are either strictly increasing or non-strictly increasingly (i.e., non-decreasing) depending on the scenario. Paths are temporal walks where each vertex is not traversed twice. A temporal vertex is a pair $(u,i)$ where $u$ is a vertex and $i\in[\tau]$ a timestep. In this paper we focus on the questions: (i) are there at least $k$ paths from a single source $s$ to a single target $t$, no two of which internally intersect on a temporal vertex? (ii) are there at most $h$ temporal vertices whose removal disconnects $s$ from $t$? Let $k^*$ be the maximum value $k$ for which the answer to (i) is YES, and let $h^*$ be the minimum value $h$ for which the answer to (ii) is YES. In static graphs, $k^*$ and $h^*$ are equal by Menger's Theorem and this is a crucial property to solve efficiently both (i) and (ii). In temporal graphs such equality has been investigated only focusing on disjoint walks rather than disjoint paths. We prove that, when dealing with non-strictly increasing temporal paths, $k^*$ is equal to $h^*$ if and only if $k^*$ is 1. We show that this implies a dichotomy for (i), which turns out to be polynomial-time solvable when $k\le 2$, and NP-complete for $k\ge 3$. In contrast, we also prove that Menger's version does not hold in the strictly increasing model and give hardness results also for this case. Finally, we give hardness results and an XP algorithm for (ii).

翻译：一个( 方向) 时间图是一个( 方向) 图表, 其边际只在其一生的特定时间 $\ tau美元 。 漫步是相邻边边缘的序列, 其显示的时间或严格增加( 即非下降) 取决于情景 。 路径是时间行走, 每个顶点不穿行两次 。 时间顶点是一对美元( u, i) 美元, 其中美元是一个顶点, 美元是一个顶点, 美元是双点 。 在本文中, 我们关注的问题是:( 一) 美元到一个单一来源, 美元到一个单一目标( 即不减少) 美元, 没有两次在时间顶点上交叉吗? (二) 时间顶点是最多为h美元, 美元断点是美元; 如果我们对此的答案是 Ye, 美元不是最低值, 美元是非点 美元 美元, 直线路面显示一个最低值 美元 。 ( 直值是正值是正数, 直值是正值是正值 。