Simple、strict、proper、happy：时间图中的可达性研究
Simple, strict, proper, happy: A study of reachability in temporal graphs
AI总结：
本文系统研究了时间路径的strict、proper和simple特征对时间图可达性表达能力的影响，揭示了表达能力层次，并倡导将happy时间图作为证明否定结果的首选设置。
摘要：
动态网络是一个复杂的主题。它们不仅继承了静态网络的复杂性（作为特例）；而且对定义上的细微差别非常敏感，这常常导致文献中结果的混淆和不可比性。在本文中，我们退一步，更详细地考察了三个这样的方面，并系统地探讨了它们的影响；即，时间路径是否需要是 strict（即路径上的时间必须是递增的，而不仅仅是递减的），时间标记是否是 proper（两条相邻边不能同时存在），以及时间标记是否是 simple（一条边只能有一个存在时间）。特别是，我们研究了这些特征的不同组合如何影响图在可达性方面的表达能力。我们的结果揭示了由此产生的设置的表达能力层次，阐明了在考虑任一组合时所造成的通用性损失。有些设置比预期的更通用；特别是，proper 时间图证明与允许非 strict 路径的一般时间图具有同等的表达能力。此外，我们展示了最简单的设置，即 happy 时间图（即同时是 proper 和 simple 的），在某种（受限但有用的）意义上，仍然具有足够的表达能力来模拟一般时间图的可达性。此外，这种设置被提倡作为证明否定结果的首选目标。我们通过将两个已知结果加强到 happy 图来证明这一点（即稀疏 spanner 的不存在性，以及计算时间分量的硬度）。总体而言，我们希望本文能被视为在不同时间图设置之间进行选择的指南，同时意识到这些选择影响通用性的方式。
英文摘要：
Dynamic networks are a complex subject. Not only do they inherit the complexity of static networks (as a particular case); they are also sensitive to definitional subtleties that are a frequent source of confusion and incomparability of results in the literature. In this paper, we take a step back and examine three such aspects in more details, exploring their impact in a systematic way; namely, whether the temporal paths are required to be \emph{strict} (i.e., the times along a path must increasing, not just be non-decreasing), whether the time labeling is \emph{proper} (two adjacent edges cannot be present at the same time) and whether the time labeling is \emph{simple} (an edge can have only one presence time). In particular, we investigate how different combinations of these features impact the expressivity of the graph in terms of reachability. Our results imply a hierarchy of expressivity for the resulting settings, shedding light on the loss of generality that one is making when considering either combination. Some settings are more general than expected; in particular, proper temporal graphs turn out to be as expressive as general temporal graphs where non-strict paths are allowed. Also, we show that the simplest setting, that of \emph{happy} temporal graphs (i.e., both proper and simple) remains expressive enough to emulate the reachability of general temporal graphs in a certain (restricted but useful) sense. Furthermore, this setting is advocated as a target of choice for proving negative results. We illustrates this by strengthening two known results to happy graphs (namely, the inexistence of sparse spanners, and the hardness of computing temporal components). Overall, we hope that this article can be seen as a guide for choosing between different settings of temporal graphs, while being aware of the way these choices affect generality.
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