The structure of quasi-transitive graphs avoiding a minor with applications to the domino problem

TL;DR

The paper develops a canonical, symmetry-respecting decomposition theory for locally finite quasi-transitive graphs excluding a fixed minor, proving that such graphs admit a $\Gamma$-canonical tree-decomposition with adhesion at most $3$ whose torsos are either finite or planar minors. The approach combines a refined canonical tangle framework for order $4$, crossedge contraction techniques, and recent canonical-decomposition results to produce a decomposition robust to automorphisms, enabling broad applications. As consequences, it shows that these graphs attain their Hadwiger number, are vertex-accessible, and that minor-excluded groups are accessible and finitely presented; it also resolves the minor-excluded case of Ballier–Stein’s domino problem, establishing decidability precisely for virtually free groups. The methods synthesize Grohe’s finite-case program with Carmesin–Hamann–Miraftab’s canonical-tangle results, delivering both structural and algorithmic insights for highly symmetric graphs and their algebraic counterparts.

Abstract

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).

Figures (4)

• Figure 1: Left: a finite section of the $3$-connected infinite graph obtained by replacing in the infinite hexagonal planar grid each vertex by a triangle with three vertices of degree $3$. Right: a finite section of the quasi-$4$-connected torso $G\llbracket V_{z_0} \rrbracket$ of $(T,\mathcal{V})$. Note that it does not depend of the choice of $V_{z_0}$.
• Figure 2: A useful example.
• Figure 3: Interaction between minimal separations. The white zones represent empty sets while the grey represent potentially non-empty sets.
• Figure 4: Left: The graph $G$ when $S_2$ is incident to exactly $2$ crossedges. Here $t_2$ is part of no crossedge and $S_2$ and $S_3$ are crossing via the crossedge $r_2s_3$. Hence, $\mathrm{fc}(S_2) = \{s_1, t_2,s_3\}$. Right: The graph $G'$ obtained after contracting the crossedge $s_1s_2$. The dashed edges are edges that appear in $H$ and $H'$ respectively. The situation is identical when $S_2$ is incident to $3$ crossedges, but harder to illustrate in $2$ dimensions.

Theorems & Definitions (96)

• Theorem 1.1: see \ref{['thm: mainCTTD']}
• Theorem 1.2: see \ref{['thm: main2']}
• Theorem 1.3: see \ref{['thm: had']}
• Theorem 1.4: see \ref{['thm: access']}
• Corollary 1.5
• Theorem 1.6: see \ref{['cor: finpres']}
• Conjecture 1.7: Domino problem conjecture BS
• Theorem 1.8: see \ref{['thm: Domino']}
• Remark 2.1
• Theorem 2.2: TW