## Dániel Gerbner, Balázs Keszegh, Dömötör
Pálvölgyi, Günter Rote, and Gábor Wiener:

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

*Ars
Mathematica Contemporanea* **12** (no. 2) (2017), 301–314.
### 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` 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.

Last update: February 28, 2017.