Temporal graphs are dynamic graphs where the edge set can change in each time step, while the vertex set stays the same. Exploration of temporal graphs whose snapshot in each time step is a connected graph, called connected temporal graphs, has been widely studied. In this paper, we extend the concept of graph automorphisms from static graphs to temporal graphs for the first time and show that symmetries enable faster exploration: We prove that a connected temporal graph with $n$ vertices and orbit number $r$ (i.e., $r$~is the number of automorphism orbits) can be explored in $O(r n^{1+\epsilon})$ time steps, for any fixed $\epsilon>0$. For $r=O(n^c)$ for constant $c<1$, this is a significant improvement over the known tight worst-case bound of $\Theta(n^2)$ time steps for arbitrary connected temporal graphs. We also give two lower bounds for temporal exploration, showing that $\Omega(n \log n)$ time steps are required for some inputs with $r=O(1)$ and that $\Omega(rn)$ time steps are required for some inputs for any $r$ with $1\le r\le n$. Moreover, we show that the techniques we develop for fast exploration can be used to derive the following result for rendezvous: Two agents with different programs and without communication ability are placed by an adversary at arbitrary vertices and given full information about the connected temporal graph, except that they do not have consistent vertex labels. Then the two agents can meet at a common vertex after $O(n^{1+\epsilon})$ time steps, for any constant $\epsilon>0$. For some connected temporal graphs with the orbit number being a constant, we also present a complementary lower bound of $\Omega(n\log n)$ time steps.

翻译：时间图是一类动态图，其边集在每个时间步可发生变化，而顶点集保持不变。针对每个时间步快照均为连通图的时间图（称为连通时间图）的探索问题已被广泛研究。本文首次将图自同构的概念从静态图扩展至时间图，并证明对称性能够实现更快的探索：我们证明，对于具有$n$个顶点和轨道数$r$（即自同构轨道数量）的连通时间图，可在$O(r n^{1+\epsilon})$个时间步内完成探索（其中$\epsilon>0$为任意固定常数）。当$r=O(n^c)$且常数$c<1$时，该结果显著优于任意连通时间图已知的紧最坏情况界$\Theta(n^2)$个时间步。我们同时给出两个时间图探索下界：对某些$r=O(1)$的输入，需要$\Omega(n \log n)$个时间步；而对任意满足$1\le r\le n$的$r$，某些输入需要$\Omega(r n)$个时间步。此外，我们证明为快速探索而发展的技术还可推导出如下相遇结果：由敌手将两个具有不同程序且无通信能力的智能体放置在任意顶点上，两者虽获取连通时间图的完整信息，但缺乏一致的顶点标签，则在任意常数$\epsilon>0$下，两智能体可在$O(n^{1+\epsilon})$个时间步内于同一顶点相遇。对于轨道数为常数的某些连通时间图，我们还给出互补的下界$\Omega(n\log n)$个时间步。