A min-max theorem for LIFO-search

We introduce a variant of the classic node search game called LIFO-search where searchers are assigned different numbers. The additional rule is that a searcher can be removed only if no searchers of lower rank are in the graph at that moment. We introduce the notion of shelters in graphs and we prove a min-max theorem implying their equivalence with the tree-depth parameter. As shelters provide escape strategies for the fugitive, this implies that the LIFO-search game is monotone and that the LIFO-search parameter is equivalent with the one of tree-depth.