  We give a fixed-parameter tractable algorithm that, given a parameter $k$ and
two graphs $G_1,G_2$, either concludes that one of these graphs has treewidth
at least $k$, or determines whether $G_1$ and $G_2$ are isomorphic. The running
time of the algorithm on an $n$-vertex graph is $2^{O(k^5\log k)}\cdot n^5$,
and this is the first fixed-parameter algorithm for Graph Isomorphism
parameterized by treewidth.
  Our algorithm in fact solves the more general canonization problem. We namely
design a procedure working in $2^{O(k^5\log k)}\cdot n^5$ time that, for a
given graph $G$ on $n$ vertices, either concludes that the treewidth of $G$ is
at least $k$, or: * finds in an isomorphic-invariant way a graph
$\mathfrak{c}(G)$ that is isomorphic to $G$; * finds an isomorphism-invariant
construction term --- an algebraic expression that encodes $G$ together with a
tree decomposition of $G$ of width $O(k^4)$.
  Hence, the isomorphism test reduces to verifying whether the computed
isomorphic copies or the construction terms for $G_1$ and $G_2$ are equal.
