The graph exploration problem asks a searcher to explore an unknown graph.
This problem can be interpreted as the online version of the Traveling Salesman
Problem. The treasure hunt problem is the corresponding online version of the
shortest s-t-path problem. It asks the searcher to find a specific vertex in an
unknown graph at which a treasure is hidden.
Recently, the analysis of 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 ask for a tour in the graph or a s-t
path