  For an undirected graph G, we consider the following problems: given a fixed
graph H, can we partition the vertices of G into two non-empty sets A and B
such that neither the induced graph G[A] nor G[B] contain H (i) as a subgraph?
(ii) as an induced subgraph? These problems are NP-complete and are expressible
in monadic second order logic (MSOL). The MSOL formulation, together with
Courcelle's theorem implies linear time solvability on graphs with bounded
tree-width. This approach yields algorithms with running time f(|phi|, t) * n,
where |phi| is the length of the MSOL formula, t is the tree-width of the graph
and n is the number of vertices of the graph. The dependency of f(|phi|, t) on
|phi| can be as bad as a tower of exponentials.
  In this paper, we present explicit combinatorial algorithms for these
problems for graphs G whose tree-width is bounded. We obtain 2^{O(t^r)} * n
time algorithms when H is any fixed graph of order r. In the special case when
H = K_r, a complete graph on r vertices, we get an 2^{O(t+r \log t)} * n time
algorithm.
  The techniques can be extended to provide FPT algorithms to determine the
smallest number q such that V can be partitioned into q parts such that none of
the parts have H as a subgraph (induced subgraph).
