Graph exploration is one of the fundamental tasks performed by a mobile agent in a graph. An $n$-node graph has unlabeled nodes, and all ports at any node of degree $d$ are arbitrarily numbered $0,\dots, d-1$. A mobile agent, initially situated at some starting node $v$, has to visit all nodes of the graph and stop. In the absence of any initial knowledge of the graph the task of deterministic exploration is often impossible. On the other hand, for some families of graphs it is possible to design deterministic exploration algorithms working for any graph of the family. We call such families of graphs {\em explorable}. Examples of explorable families are all finite families of graphs, as well as the family of all trees. In this paper we study the problem of which families of graphs are explorable. We characterize all such families, and then ask the question whether there exists a universal deterministic algorithm that, given an explorable family of graphs, explores any graph of this family, without knowing which graph of the family is being explored. The answer to this question turns out to depend on how the explorable family is given to the hypothetical universal algorithm. If the algorithm can get the answer to any yes/no question about the family, then such a universal algorithm can be constructed. If, on the other hand, the algorithm can be only given an algorithmic description of the input explorable family, then such a universal deterministic algorithm does not exist.

翻译：图探索是移动智能体在图中所执行的基本任务之一。考虑一个包含n个节点的图，其中节点无标签，且任意度为d的节点上的所有端口被任意编号为0,…,d-1。初始位于某个起始节点v的移动智能体需要访问图中所有节点并停止。在缺乏图结构初始信息的情况下，确定性探索任务往往不可行。另一方面，对于某些图族，可以设计出适用于该族中任意图的确定性探索算法。我们将此类图族称为“可探索图族”。可探索图族的例子包括所有有限图族以及所有树构成的图族。本文研究哪些图族是可探索的这一问题。我们刻画了所有此类图族，进而提出疑问：是否存在一个通用确定性算法，在给定一个可探索图族后，能够探索该族中的任意图，而无需知晓当前被探索的是族中哪个具体图？该问题的答案取决于可探索图族以何种方式提供给假设存在的通用算法。若算法能够回答关于该图族的任意是非问题，则此类通用算法可以构建。反之，若算法仅能获得输入可探索图族的算法描述，则此类通用确定性算法不存在。