  We study the parameterized complexity of the graph isomorphism problem when
parameterized by width parameters related to tree decompositions. We apply the
following technique to obtain fixed-parameter tractability for such parameters.
We first compute an isomorphism invariant set of potential bags for a
decomposition and then apply a restricted version of the Weisfeiler-Lehman
algorithm to solve isomorphism. With this we show fixed-parameter tractability
for several parameters and provide a unified explanation for various
isomorphism results concerned with parameters related to tree decompositions.
As a possibly first step towards intractability results for parameterized graph
isomorphism we develop an fpt Turing-reduction from strong tree width to the a
priori unrelated parameter maximum degree.
