  We prove that Graph Isomorphism and Canonization in graphs excluding a fixed
graph $H$ as a minor can be solved by an algorithm working in time $f(H)\cdot
n^{O(1)}$, where $f$ is some function. In other words, we show that these
problems are fixed-parameter tractable when parameterized by the size of the
excluded minor, with the caveat that the bound on the running time is not
necessarily computable. The underlying approach is based on decomposing the
graph in a canonical way into unbreakable (intuitively, well-connected) parts,
which essentially provides a reduction to the case where the given
$H$-minor-free graph is unbreakable itself. This is complemented by an analysis
of unbreakable $H$-minor-free graphs, performed in a second subordinate
manuscript, which reveals that every such graph can be canonically decomposed
into a part that admits few automorphisms and a part that has bounded
treewidth.
