A \emph{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 points of view. We propose a generalization of the concept of resolving sets to temporal graphs, \emph{i.e.}, graphs with edge sets that change over discrete time-steps. In this setting, the \emph{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 $v$, \emph{i.e.}, the first time-step at which $v$ could receive a message broadcast from $u$. A \emph{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\emph{分辨集} \(R\)是一个顶点子集，使得\(G\)中每个顶点由其到\(R\)中各顶点的距离唯一确定。这一概念自20世纪70年代提出以来，已从组合与算法角度被广泛研究。我们提出将分辨集概念推广至时序图，即边集随时间步离散变化的图。在此设定下，\emph{从\(u\)到\(v\)的时序距离}是指从\(u\)出发、沿严格递增时间步的边构成的旅程能够到达\(v\)的最早时间步，即\(v\)可能收到从\(u\)广播消息的首个时间步。时序图\(\mathcal{G}\)的\emph{时序分辨集}是其顶点子集\(R\)，使得\(\mathcal{G}\)中每个顶点由其到\(R\)中各顶点的时序距离唯一标识。我们研究了寻找最小时序分辨集的问题，并证明该问题即使在高度受限的图类及严格时间步约束下也是NP完全的：包括每条边仅出现在时间步1或2的时序完全图、每条边至多出现在两个连续时间步的时序树，甚至每条边至多出现在两个（不一定连续）时间步的时序细分星图。另一方面，我们针对每条边仅出现在单一时间步的时序路径和时序星图给出了多项式时间算法，并对若干边以周期性时间步出现的时序图类进行了组合分析与算法设计。