Treasure hunt and rendezvous are fundamental tasks performed by mobile agents in graphs. In treasure hunt, an agent has to find an inert target (called treasure) situated at an unknown node of the graph. In rendezvous, two agents, initially located at distinct nodes of the graph, traverse its edges in synchronous rounds and have to meet at some node. We assume that the graph is connected (otherwise none of these tasks is feasible) and consider deterministic treasure hunt and rendezvous algorithms. The time of a treasure hunt algorithm is the worst-case number of edge traversals performed by the agent until the treasure is found. The time of a rendezvous algorithm is the worst-case number of rounds since the wakeup of the earlier agent until the meeting. To the best of our knowledge, all known treasure hunt and rendezvous algorithms rely on the assumption that degrees of all nodes are finite, even when the graph itself may be infinite. In the present paper we remove this assumption for the first time, and consider both above tasks in arbitrary connected graphs whose nodes can have either finite or countably infinite degrees. Our main result is the first universal treasure hunt algorithm working for arbitrary connected graphs. We prove that the time of this algorithm has optimal order of magnitude among all possible treasure hunt algorithms working for arbitrary connected graphs. As a consequence of this result we obtain the first universal rendezvous algorithm working for arbitrary connected graphs. The time of this algorithm is polynomial in a lower bound holding in many graphs, in particular in the tree all of whose degrees are infinite.

翻译：宝藏搜索与会合是移动智能体在图结构中执行的基本任务。在宝藏搜索任务中，智能体需找到位于图中未知节点处的静止目标（称为宝藏）。而在会合任务中，两个初始位于不同节点的智能体在同步轮次中遍历图的边，最终必须在某个节点相遇。我们假设图是连通的（否则这两种任务均不可行），并研究确定性宝藏搜索与会合算法。宝藏搜索算法的时间复杂度定义为智能体在找到宝藏前最坏情况下的边遍历次数；会合算法的时间复杂度则定义为从较早唤醒的智能体开始到相遇时刻的最坏情况轮次数量。据我们所知，现有所有宝藏搜索与会合算法均依赖"所有节点度数有限"的假设，即便图本身可能是无限的。本文首次突破这一限制，在节点度数可为有限或可数无限的任意连通图中研究上述两类任务。我们的主要成果是首个适用于任意连通图的通用宝藏搜索算法。我们证明，该算法的时间复杂度在所有适用于任意连通图的宝藏搜索算法中具有最优量级。基于该成果，我们进一步得到首个适用于任意连通图的通用会合算法。该算法的时间复杂度与许多图（特别是所有节点度数均为无限的树）中的理论下界呈多项式关系。