We consider the worst-case query complexity of some variants of certain PPAD-complete search problems. Suppose we are given a graph G and a vertex s in G. We denote the directed graph obtained from G by directing all edges in both directions by G'. D is a directed subgraph of G' which is unknown to us, except that it consists of vertex-disjoint directed paths and cycles and one of the paths originates in s. Our goal is to find an endvertex of a path by using as few queries as possible. A query specifies a vertex v in G and the answer is the set of the edges of D incident to v, together with their directions.
We also show lower bounds for the special case when D consists of a single path. Our proofs use the theory of graph separators. Finally, we consider the case when the graph G is a grid graph. In this case, using the connection with separators, we give asymptotically tight bounds as a function of the size of the grid, if the dimension of the grid is considered as fixed. For this, we prove a separator theorem about grid graphs, which is interesting on its own right.