  The dichotomy conjecture for the parameterized embedding problem states that
the problem of deciding whether a given graph $G$ from some class $K$ of
"pattern graphs" can be embedded into a given graph $H$ (that is, is isomorphic
to a subgraph of $H$) is fixed-parameter tractable if $K$ is a class of graphs
of bounded tree width and $W[1]$-complete otherwise.
  Towards this conjecture, we prove that the embedding problem is
$W[1]$-complete if $K$ is the class of all grids or the class of all walls.
