  A balanced partition is a clustering of a graph into a given number of
equal-sized parts. For instance, the Bisection problem asks to remove at most k
edges in order to partition the vertices into two equal-sized parts. We prove
that Bisection is FPT for the distance to constant cliquewidth if we are given
the deletion set. This implies FPT algorithms for some well-studied parameters
such as cluster vertex deletion number and feedback vertex set. However, we
show that Bisection does not admit polynomial-size kernels for these
parameters.
  For the Vertex Bisection problem, vertices need to be removed in order to
obtain two equal-sized parts. We show that this problem is FPT for the number
of removed vertices k if the solution cuts the graph into a constant number c
of connected components. The latter condition is unavoidable, since we also
prove that Vertex Bisection is W[1]-hard w.r.t. (k,c).
  Our algorithms for finding bisections can easily be adapted to finding
partitions into d equal-sized parts, which entails additional running time
factors of n^{O(d)}. We show that a substantial speed-up is unlikely since the
corresponding task is W[1]-hard w.r.t. d, even on forests of maximum degree
two. We can, however, show that it is FPT for the vertex cover number.
