  We give an isomorphism test for graphs of Euler genus $g$ running in time
$2^{O(g^4 \log g)}n^{O(1)}$. Our algorithm provides the first explicit upper
bound on the dependence on $g$ for an fpt isomorphism test parameterized by the
Euler genus of the input graphs. The only previous fpt algorithm runs in time
$f(g)n$ for some function $f$ (Kawarabayashi 2015). Actually, our algorithm
even works when the input graphs only exclude $K_{3,h}$ as a minor. For such
graphs, no fpt isomorphism test was known before.
  The algorithm builds on an elegant combination of simple group-theoretic,
combinatorial, and graph-theoretic approaches. In particular, we introduce
$(t,k)$-WL-bounded graphs which provide a powerful tool to combine
group-theoretic techniques with the standard Weisfeiler-Leman algorithm. This
concept may be of independent interest.
