Temporal graphs naturally model graphs whose underlying topology changes over time. Recently, the problems TEMPORAL VERTEX COVER (or TVC) and SLIDING-WINDOW TEMPORAL VERTEX COVER(or $\Delta$-TVC for time-windows of a fixed-length $\Delta$) have been established as natural extensions of the classic problem VERTEX COVER on static graphs with connections to areas such as surveillance in sensor networks. In this paper we initiate a systematic study of the complexity of TVC and $\Delta$-TVC on sparse graphs. Our main result shows that for every $\Delta\geq 2$, $\Delta$-TVC is NP-hard even when the underlying topology is described by a path or a cycle. This resolves an open problem from literature and shows a surprising contrast between $\Delta$-TVC and TVC for which we provide a polynomial-time algorithm in the same setting. To circumvent this hardness, we present a number of exact and approximation algorithms for temporal graphs whose underlying topologies are given by a path, that have bounded vertex degree in every time step, or that admit a small-sized temporal vertex cover.

翻译：时态图自然模拟了底层拓扑随时间变化的图。近期，问题TEMPORAL VERTEX COVER（简称TVC）和SLIDING-WINDOW TEMPORAL VERTEX COVER（简称$\Delta$-TVC，针对固定长度$\Delta$的时间窗口）已被确立为经典问题VERTEX COVER在静态图上的自然扩展，并与传感器网络中的监控等领域相关联。本文首次系统性地研究了TVC和$\Delta$-TVC在稀疏图上的复杂性。我们的主要结果表明，对于每个$\Delta\geq 2$，即使底层拓扑由一条路径或一个环描述，$\Delta$-TVC也是NP难的。这解决了文献中的一个开放问题，并揭示了$\Delta$-TVC与TVC之间的惊人对比——对于TVC，我们在相同设置下提供了一个多项式时间算法。为规避这一困难，我们提出了若干精确和近似算法，适用于底层拓扑为路径、每个时间步顶点度数有界或允许小规模时态顶点覆盖的时态图。