The Graph Exploration problem asks a searcher to explore an unknown environment. The environment is modeled as a graph, where the searcher needs to visit each vertex beginning at some vertex $s$. Furthermore, Treasure Hunt problems are a variation of Graph Exploration, in which the searcher needs to find a hidden treasure, which is located at some vertex $t$. In these online problems, any online algorithm performs poorly because it has too little knowledge about the instance to react adequately to the requests of the adversary. Thus, the impact of a priori knowledge is of interest. In graph problems, one form of a priori knowledge is a map of the graph. We survey the graph exploration and treasure hunt problem with an unlabeled map, which is an isomorphic copy of the graph, that is provided to the searcher. We formulate decision variants of both problems by interpreting the online problems as a game between the online algorithm (the searcher) and the adversary. The map, however, is not controllable by the adversary. The question is, whether the searcher is able to explore the graph fully or find the treasure for all possible decisions of the adversary. We prove the PSPACE-completeness of these games, whereby we analyze the variations which ask for the mere existence of a tour through the graph or path to the treasure and the variations that include costs. Additionally, we analyze the complexity of related problems that relax the path constraint, allowing multiple visits of vertices or edges, or have additional constraints, like requiring to visit specific edges.

翻译：图探索问题要求搜索者探索未知环境。该环境被建模为一张图，搜索者需从某个起始顶点 $s$ 出发访问每个顶点。宝藏搜寻问题是图探索的一种变体，其中搜索者需要找到位于某个顶点 $t$ 的隐藏宝藏。在这些在线问题中，任何在线算法的表现都很差，因为其对实例的了解甚少，无法充分应对对手的请求。因此，先验知识的影响值得关注。在图问题中，先验知识的一种形式是图的映射。我们研究了带未标记映射（即图的同构副本）的图探索与宝藏搜寻问题，该映射会提供给搜索者。我们通过将在线问题解释为在线算法（搜索者）与对手之间的博弈，来构建这两个问题的判定变体。然而，映射不受对手控制。问题在于：对于对手的所有可能决策，搜索者是否能完全探索该图或找到宝藏？我们证明了这些博弈的PSPACE完全性，其中分析了仅要求存在一条遍历图的巡游或通往宝藏的路径的变体，以及包含成本的变体。此外，我们还分析了放宽路径约束（允许多次访问顶点或边）或添加额外约束（例如要求访问特定边）的相关问题的复杂性。