  Let Q_n be the graph of n times n times n cube with all non-decreasing
diagonals (including the facial ones) in its constituent unit cubes. Suppose
that a subset S of V(Q_n) separates the left side of the cube from the right
side. We show that S induces a subgraph of tree-width at least n/sqrt{18}-1. We
use a generalization of this claim to prove that the vertex set of Q_n cannot
be partitioned to two parts, each of them inducing a subgraph of bounded
tree-width.
