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：有向时间图是一种仅在生命周期$\tau$内的特定时刻边可用的有向图。游走是相邻边的序列，其出现时间根据场景严格递增或非严格递增（即非递减）。路径是指每个顶点不被遍历两次的时间游走。时间顶点是配对$(u,i)$，其中$u$为顶点，$i\in[\tau]$为时间步。本文聚焦于以下问题：(i) 是否存在至少$k$条从单一源点$s$到单一汇点$t$的路径，且任意两条路径不在同一时间顶点上内部相交？(ii) 是否存在至多$h$个时间顶点，使得移除它们后$s$与$t$不再连通？设$k^*$为使问题(i)答案为“是”的最大$k$值，$h^*$为使问题(ii)答案为“是”的最小$h$值。在静态图中，根据门格尔定理$k^*$与$h^*$相等，这是高效求解(i)和(ii)的关键性质。在时间图中，该相等性此前仅针对不相交游走而非不相交路径进行了研究。我们证明：在处理非严格递增时间路径时，$k^* = h^*$当且仅当$k^* = 1$。这表明问题(i)存在二分性：当$k \le 2$时可在多项式时间内求解，而当$k \ge 3$时则为NP完全问题。此外，我们还证明了在严格递增模型下门格尔定理不成立，并给出该情形的难度结论。最后，我们给出了问题(ii)的难度结论与一个XP算法。