简单、严格、正常与有向：有向与无向时间图中的可达性比较
Simple, Strict, Proper, and Directed: Comparing Reachability in Directed and Undirected Temporal Graphs
AI总结：
本文首次全面分析有向时间图的层级结构，揭示其单链层级，并证明无向类可在可达性等价下转化为有向类，反之不行，为时间图计算问题提供统一分析框架。
摘要：
我们首次对有向时间图的时间设置进行了全面分析，完全解决了它们在支撑、可达性和诱导可达性等价方面的层级关系。这些概念由Casteigts、Corsini和Sarkar提出，捕捉了时间图类之间不同层次的等价性。他们的分析聚焦于无向图，并沿三个维度展开：严格与非严格（路径上的时间是否严格递增）、正常与任意（相邻边是否可同时出现）、以及简单与多标签（一条边是否可多次出现）。在本工作中，我们通过增加有向与无向这一基本区分来扩展他们的框架。我们的结果揭示了有向图的单链层级，其中严格且简单类最具表达力，而正常且简单类表达力最弱。相比之下，无向图形成双链层级，其中严格且多标签类最具表达力，而正常且简单类表达力最弱。这两条链由非严格且简单类和严格且简单类构成，我们证明这两类是不可比较的。除了考察有向图和无向图类的内部层级外，我们还对两者进行了比较。我们证明，在可达性等价下，每个无向类都可以转化为其对应的有向类，而没有任何有向类可以转化为无向类。我们的发现对时间图上计算问题的研究具有重要意义。在更具表达力的图类中的正面结果，只要问题在可达性等价下保持不变，就可以扩展到较弱的类。反之，对表达力较弱的类的困难性结果会传播到更强的类。我们希望这些发现能激发一种统一的方法，用于在不同设置下分析时间图。
英文摘要：
We present the first comprehensive analysis of temporal settings for directed temporal graphs, fully resolving their hierarchy with respect to support, reachability, and induced-reachability equivalence. These notions, introduced by Casteigts, Corsini, and Sarkar, capture different levels of equivalence between temporal graph classes. Their analysis focused on undirected graphs under three dimensions: strict vs. non-strict (whether times along paths strictly increase), proper vs. arbitrary (whether adjacent edges can appear simultaneously), and simple vs. multi-labeled (whether an edge can appear multiple times). In this work, we extend their framework by adding the fundamental distinction of directed vs. undirected. Our results reveal a single-strand hierarchy for directed graphs, with strict & simple being the most expressive class and proper & simple the least expressive. In contrast, undirected graphs form a two-strand hierarchy, with strict & multi-labeled being the most expressive and proper & simple the least expressive. The two strands are formed by the non-strict & simple and the strict & simple class, which we show to be incomparable. In addition to examining the internal hierarchies of directed and of undirected graph classes, we compare the two. We show that each undirected class can be transformed into its directed counterpart under reachability equivalence, while no directed class can be transformed into any undirected one. Our findings have significant implications for the study of computational problems on temporal graphs. Positive results in more expressive graph classes extend to weaker classes as long as the problem is independent under reachability equivalence. Conversely, hardness results for a less expressive class propagate to stronger classes. We hope these findings will inspire a unified approach for analyzing temporal graphs under the different settings.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
