Search for the end of a path in the d-dimensional grid and in other graphs

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 $s \in V(G)$. 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 $v\in V(G)$, 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.