We consider the worst-case query complexity of some variants of certain
\cl{PPAD}-complete search problems. Suppose we are given a graph G and a
vertex s∈V(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∈V(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. In order to do this, we prove a
separator theorem about grid graphs, which is interesting on its own right