In the study of temporal graphs, only paths respecting the flow of time are relevant. In this context, many concepts of walks disjointness were proposed over the years, and the validity of Menger's Theorem, as well as the complexity of related problems, has been investigated. In this paper, we introduce and investigate a type of disjointness that is only time dependent. Two walks are said to be snapshot disjoint if they are not active in a same snapshot (also called timestep). The related paths and cut problems are then defined and proved to be W[1]-hard and XP-time solvable when parameterized by the size of the solution. Additionally, in the light of the definition of Mengerian graphs given by Kempe, Kleinberg and Kumar in their seminal paper (STOC'2000), we define a Mengerian graph for time as a graph $G$ that cannot form an example where Menger's Theorem does not hold in the context of snapshot disjointness. We then give a characterization in terms of forbidden structures and provide a polynomial-time recognition algorithm. Finally, we also prove that, given a temporal graph $(G,\lambda)$ and a pair of vertices $s,z\in V(G)$, deciding whether at most $h$ multiedges can separate $s$ from $z$ is NP-complete.
翻译:在时间图研究中,只有与时间流相关的路径才具有相关性。在这方面,多年来提出了许多行走脱节的概念,对门杰理论的有效性以及相关问题的复杂性进行了调查。在本文中,我们介绍并调查了一种仅取决于时间的脱节类型。据说,如果两行不在同一快照(也称为时间步骤)中活动,则会快照脱节。随后,相关路径和切断问题被定义并被证明为W[1]硬和XP-时间可按解决方案大小参数来比较的脱节概念。此外,根据肯佩、克莱伯格和库马尔在其基本文件(STOC'2000年)中给出的门杰尔图表定义,我们把门杰尔图定义为一个时间的图表$G$,但不能成为在快照脱节的背景下不维持 Menger的神志。我们随后对禁制结构进行定性,并提供一个以美元为美元的综合数字识别。最后,我们还要证明一个以美元/美元为美元的平面图,是否以美元为美元。