  An infinite graph is quasi-transitive if its vertex set has finitely many
orbits under the action of its automorphism group. In this paper we obtain a
structure theorem for locally finite quasi-transitive graphs avoiding a minor,
which is reminiscent of the Robertson-Seymour Graph Minor Structure Theorem. We
prove that every locally finite quasi-transitive graph $G$ avoiding a minor has
a tree-decomposition whose torsos are finite or planar; moreover the
tree-decomposition is canonical, i.e. invariant under the action of the
automorphism group of $G$. As applications of this result, we prove the
following.
  * Every locally finite quasi-transitive graph attains its Hadwiger number,
that is, if such a graph contains arbitrarily large clique minors, then it
contains an infinite clique minor. This extends a result of Thomassen (1992)
who proved it in the 4-connected case and suggested that this assumption could
be omitted.
  * Locally finite quasi-transitive graphs avoiding a minor are accessible (in
the sense of Thomassen and Woess), which extends known results on planar graphs
to any proper minor-closed family.
  * Minor-excluded finitely generated groups are accessible (in the
group-theoretic sense) and finitely presented, which extends classical results
on planar groups.
  * The domino problem is decidable in a minor-excluded finitely generated
group if and only if the group is virtually free, which proves the
minor-excluded case of a conjecture of Ballier and Stein (2018).
