  We give a new fpt algorithm testing isomorphism of $n$-vertex graphs of tree
width $k$ in time $2^{k\operatorname{polylog} (k)}\operatorname{poly} (n)$,
improving the fpt algorithm due to Lokshtanov, Pilipczuk, Pilipczuk, and
Saurabh (FOCS 2014), which runs in time $2^{\mathcal{O}(k^5\log
k)}\operatorname{poly} (n)$. Based on an improved version of the
isomorphism-invariant graph decomposition technique introduced by Lokshtanov et
al., we prove restrictions on the structure of the automorphism groups of
graphs of tree width $k$. Our algorithm then makes heavy use of the group
theoretic techniques introduced by Luks (JCSS 1982) in his isomorphism test for
bounded degree graphs and Babai (STOC 2016) in his quasipolynomial isomorphism
test. In fact, we even use Babai's algorithm as a black box in one place.
  We also give a second algorithm which, at the price of a slightly worse
running time $2^{\mathcal{O}(k^2 \log k)}\operatorname{poly} (n)$, avoids the
use of Babai's algorithm and, more importantly, has the additional benefit that
it can also used as a canonization algorithm.
