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

#### Abstract

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* ∈ *V*(*G*). We denote the directed graph obtained from *G* by directing all edges in both directions by *G*ʹ. *D* is a directed subgraph of *G*ʹ 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.

#### Keywords

ISSN: 1855-3974

Issues from Vol 6, No 1 onward are partially supported by the Slovenian Research Agency from the Call for co-financing of scientific periodical publications