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).

翻译：无限图称为准传递图，若其顶点集在自同构群作用下仅有有限个轨道。本文建立了避免子式的局部有限准传递图的结构定理，该定理与Robertson-Seymour图子式结构定理相呼应。我们证明每个避免子式的局部有限准传递图$G$均具有树分解，其躯干为有限图或平面图；此外该树分解是典范的，即在$G$的自同构群作用下保持不变。作为该结果的应用，我们证明了以下结论：*每个局部有限准传递图均能达到其Hadwiger数，即若该图包含任意大的团子式，则它包含无穷大团子式。这推广了Thomassen(1992)在四连通情形下的结论，其曾指出该假设可被省略。*避免子式的局部有限准传递图是可达的（在Thomassen与Woess意义下），这将平面图已有结论推广至任意真子式闭族。*排除子式的有限生成群是可达的（在群论意义下）且有限呈现，这推广了平面群经典结论。*排除子式的有限生成群上的多米诺问题可判定当且仅当该群本质自由，这证实了Ballier与Stein(2018)猜想中排除子式的情形。