A $\textit{resolving set}$ $R$ in a graph $G$ is a set of vertices such that every vertex of $G$ is uniquely identified by its distances to the vertices of $R$. Introduced in the 1970s, this concept has been since then extensively studied from both combinatorial and algorithmic point of view. We propose a generalization of the concept of resolving sets to temporal graphs, i.e., graphs with edge sets that change over discrete time-steps. In this setting, the $\textit{temporal distance}$ from $u$ to $v$ is the earliest possible time-step at which a journey with strictly increasing time-steps on edges leaving $u$ reaches $v$, i.e., the first time-step at which $v$ could receive a message broadcast from $u$. A $\textit{temporal resolving set}$ of a temporal graph $\mathcal{G}$ is a subset $R$ of its vertices such that every vertex of $\mathcal{G}$ is uniquely identified by its temporal distances from vertices of $R$. We study the problem of finding a minimum-size temporal resolving set, and show that it is NP-complete even on very restricted graph classes and with strong constraints on the time-steps: temporal complete graphs where every edge appears in either time-step 1 or 2, temporal trees where every edge appears in at most two consecutive time-steps, and even temporal subdivided stars where every edge appears in at most two (not necessarily consecutive) time-steps. On the other hand, we give polynomial-time algorithms for temporal paths and temporal stars where every edge appears in exactly one time-step, and give a combinatorial analysis and algorithms for several temporal graph classes where the edges appear in periodic time-steps.

翻译：图$G$中的$\textit{分辨集}$ $R$是一个顶点子集，使得$G$中每个顶点都能通过其到$R$中顶点的距离被唯一标识。这一概念于20世纪70年代提出，此后从组合与算法两个角度得到了广泛研究。我们提出将分辨集概念推广到时序图（即边集随时间步离散变化的图）上。在此设定下，从$u$到$v$的$\textit{时序距离}$是指沿严格递增时间步的旅程从$u$出发最早到达$v$的时间步，即$v$能接收到$u$广播消息的第一个时间步。时序图$\mathcal{G}$的$\textit{时序分辨集}$是其顶点子集$R$，使得$\mathcal{G}$中每个顶点都能通过其到$R$中顶点的时序距离被唯一标识。我们研究寻找最小规模时序分辨集的问题，并证明即使在受限图类及强时间步约束条件下该问题仍是NP完全的：包括所有边仅出现在时间步1或2的时序完全图、所有边至多出现在两个连续时间步的时序树，以及所有边至多出现在两个（不必然连续的）时间步的时序细分星图。另一方面，对每条边仅出现在一个时间步的时序路径和时序星图，我们给出了多项式时间算法，并对边以周期时间步出现的若干时序图类进行了组合分析并给出了算法。