Coarse geometry of quasi-transitive graphs beyond planarity

We study geometric and topological properties of infinite graphs that are quasi-isometric to a planar graph of bounded degree. We prove that every locally finite quasi-transitive graph excluding a minor is quasi-isometric to a planar graph of bounded degree. We use the result to give a simple proof of the result that finitely generated minor-excluded groups have Assouad-Nagata dimension at most 2 (this is known to hold in greater generality, but all known proofs use significantly deeper tools). We also prove that every locally finite quasi-transitive graph that is quasi-isometric to a planar graph is $k$-planar for some $k$ (i.e. it has a planar drawing with at most $k$ crossings per edge), and discuss a possible approach to prove the converse statement.


Introduction
Our work is motivated by a conjecture and a problem raised recently by Georgakopoulos and Papasoglu [GP23], lying at the intersection of metric graph theory and graph minor theory.Before we state them, we first need to introduce some terminology.
We say that a graph H is a k-fat minor of a graph G if there exists a family of connected subsets (M v ) v∈V (H) of V (G) such that (1) for each u ̸ = v ∈ V (H), d G (M u , M v ) ⩾ k; (2) for each e = uv ∈ E(H) there is a path P e whose two endpoints lie in M u and M v and internal vertices are not in v∈V (H) M v , and (3) for every e ̸ = e ′ ∈ E(H), d G (P e , P e ′ ) ⩾ k and for every e = uv ∈ E(H) and w / ∈ {u, v}, d G (P e , M w ) ⩾ k.A graph H is an asymptotic minor of G if for every k ⩾ 0, H is a k-fat minor of G.
Let (X, d X ) and (Y, d Y ) be two metric spaces.We say that X is quasi-isometric to Y if there is a map f : X → Y and constants ε ⩾ 0, λ ⩾ 1, and C ⩾ 0 such that (i) for any y ∈ Y there is x ∈ X such that d Y (y, f (x)) ⩽ C, and (ii) for every x 1 , x 2 ∈ X, It is not difficult to check that the definition is symmetric, and we often simply say that X and Y are quasi-isometric.If condition (i) is omitted in the definition above, we say that f is a quasi-isometric embedding of X in Y .We can view each graph G as a metric space, by considering the natural shortestpath metric associated to G. A graph is locally finite if every vertex has finite degree.Georgakopoulos and Papasoglu conjectured the following [GP23].
Conjecture 1.1 (Conjecture 9.3 in [GP23]).If G is locally finite, vertex-transitive and excludes some finite graph H as an asymptotic minor, then G is quasi-isometric to a planar graph.
It is natural to first prove this conjecture when G excludes some minor H (instead of an asymptotic minor, as defined above).This suggests the following: Question 1.2.Is it true that if G is locally finite, vertex-transitive and excludes some finite graph H as a minor, then G is quasi-isometric to a planar graph?
Our first result is a positive answer to this question, in a slightly stronger form.An infinite graph is quasi-transitive if its vertex set has finitely many orbits under the action of its automorphism group.Note that any vertex-transitive graph is quasitransitive, and that for quasi-transitive graphs, being locally finite is equivalent to having bounded degree.
The countable clique K ∞ is the graph with vertex set N in which every two vertices are adjacent (every graph which excludes a finite or countable graph H as a minor also excludes K ∞ as a minor).We prove the following.
Theorem 1.3.Every locally finite quasi-transitive K ∞ -minor free graph is quasiisometric to a planar graph of bounded degree.
The main technical tool that we use is a recent structural theorem on locally finite quasi-transitive graphs excluding the countable clique as a minor [EGLD23], which shows that such graphs have a canonical tree-decomposition in which all torsos are planar or finite (see the next section for the definitions).Most importantly, this result does not use the Robertson-Seymour graph minor structure theorem.
We note that the result which allows us to construct the quasi-isometry using the canonical tree-decomposition was also proved recently (and independently) by MacManus [Mac23] in a slightly different form (the "if" direction in his Corollary C).Our proof is very similar to his.
We now discuss several applications of Theorem 1.3.Application 1.Beyond planarity.A graph is k-planar if it has a drawing in the plane in which each edge is involved in at most k crossings (note that with this terminology, being planar is the same as being 0-planar).The local crossing number of a graph G, denoted by lcr(G), is the infimum integer k such that G is k-planar.
Problem 1.4 (Problem 9.4 in [GP23]).For any quasi-transitive graph G of bounded degree, G is quasi-isometric to a planar graph if and only if G has finite local crossing number.
We prove that for any integer k, every bounded degree graph which is quasiisometric to a k-planar graph is k ′ -planar for some integer k ′ .In the particular case k = 0, we immediately obtain the "only if" direction of Problem 1.4 (we recently learned from Agelos Georgakopoulos that he also proved the case k = 0 independently).In Section 6, we raise a number of conjectures whose validity would imply a positive answer to the "if" direction of Problem 1.4.In the case k = 0, we also obtain our second application of Theorem 1.3: Theorem 1.5.Every locally finite quasi-transitive graph G which is K ∞ -minor-free has finite local crossing number.
The assumption that G is locally finite is necessary, as shown by the graph obtained from the square grid by adding a universal vertex (this graph is K 6 -minor free, but is not k-planar for any k < ∞).The assumption that G is quasi-transitive is also crucial: consider for each integer ℓ a graph G ℓ obtained from the square grid by adding an edge between two vertices at distance ℓ in the grid (if G ℓ is k-planar then k = Ω(ℓ)), and take the disjoint union of all graphs G ℓ , ℓ ∈ N.
Note that in the other direction, there exist 1-planar graphs that are vertextransitive and locally finite, but which contain all graphs as minors (the square grid with all diagonals is such an example).
Application 2. Assouad-Nagata dimension.Our second application of Theorem 1.3 requires the notions of asymptotic dimension and Assouad-Nagata dimension of metric spaces, which we introduce now.Let (X, d) be a metric space, and let U be a family of subsets of X.We say that U is D-bounded if each set U ∈ U has diameter at most D. We say that U is r-disjoint if for any a, b belonging to different elements of U we have d(a, b) > r.
We say that D X : R + → R + is an n-dimensional control function for (X, d) if for any r > 0, (X, d) has a cover U = n+1 i=1 U i , such that each U i is r-disjoint and each element of U is D X (r)-bounded.A control function D X for a metric space X is said to be a dilation if there is a constant c > 0 such that D X (r) ⩽ cr, for any r > 0.
The asymptotic dimension of (X, d), introduced by Gromov in [Gro93], is the least integer n such that (X, d) has an n-dimensional control function.If no such integer n exists, then the asymptotic dimension is infinite.The Assouad-Nagata dimension of (X, d), introduced by Assouad in [Ass82], is the least n such that (X, d) has an ndimensional control function which is a dilation.Clearly the asymptotic dimension is at most the Assouad-Nagata dimension.
It was proved in [BBE + 20] that every bounded degree graph excluding a minor has asymptotic dimension at most 2, and that any planar graph has asymptotic dimension at most 2.This was improved in [BBE + 23], where it was shown that any graph excluding a minor has asymptotic dimension at most 2, and any planar graph has Assouad-Nagata dimension at most 2.This was finally extended by Liu in [Liu23], who proved that any graph avoiding a minor has Assouad-Nagata dimension at most 2 (a different proof was then given by Distel in [Dis23]).All the results on graphs excluding a minor mentioned above (even for bounded degree graphs) crucially rely on the Graph minor structure theorem by Robertson and Seymour [RS03], a deep result proved in a series of 16 papers.
Using the invariance of Assouad-Nagata dimension under bilipschitz embedding [LS05], we will give a short proof of the fact that minor-excluded quasi-transitive graphs have Assouad-Nagata dimension at most 2. A finitely generated group is said to be minor-excluded if it has a Cayley graph which excludes a minor.Our result directly implies that finitely generated minor-excluded groups have Assouad-Nagata dimension at most 2, and thus asymptotic dimension at most 2 (which was originally conjectured by Ostrovskii and Rosenthal in [OR15]).We will only use Theorem 1.3 and a few simple tools from [BBE + 20, BBE + 23] based on the work of Brodskiy, Dydak, Levin and Mitra [BDLM08].In particular we will give a short proof of the fact that planar graphs of bounded degree have Assouad-Nagata dimension at most 2. Crucially, our proof for quasi-transitive graphs excluding a minor does not rely on the Graph minor structure theorem of Robertson and Seymour.

Preliminaries
All graphs in this paper are assumed to be infinite, unless stated otherwise.A tree-decomposition of a graph G is a pair (T, V) such that T is a tree and V is a collection (V t : t ∈ V (T )) of subsets of V (G), called the bags, such that • for every uv ∈ E(G), there exists t ∈ V (T ) such that u, v ∈ V t , and For a tree-decomposition (T, V), the width of (T, V) is sup t∈V (T ) |V t |−1 ∈ N∪{∞}.The treewidth of G is the minimum width of a tree-decomposition of G.
The sets V t ∩ V t ′ for every tt ′ ∈ E(T ) are called the adhesion sets of (T, V) and the adhesion of (T, V) is the supremum of the sizes of its adhesion sets (possibly infinite).For t ∈ V (T ), the torso G V t is the graph obtained from G[V t ] (the subgraph of G induced by the bag V t ) by adding all edges uv with u, v ∈ V t for which there exist t ′ such that u and v lie in the adhesion set We say that a tree-decomposition (T, V) is canonical, if for every automorphism γ of G, γ sends bags of (T, V) to bags, and adhesion sets to adhesion sets.In other words, the automorphism group of G induces a group action on T .
A separation in a graph G = (V, E) is a triple (Y, S, Z) such that Y, S, Z are pairwise disjoint, V = Y ∪ S ∪ Z and there is no edge between vertices of Y and Z.The separation (Y, S, Z) is said to be tight if there are some components Consider a tree-decomposition (T, V) of a graph G, with V = (V t ) t∈V (T ) .Let A be an orientation of the edges of E(T ), i.e. a choice of either (t 1 , t 2 ) or (t 2 , t 1 ) for every edge t 1 t 2 of T .For a pair (t 1 , t 2 ) ∈ A, and for each i ∈ {1, 2}, let T i denote the component of T − {t 1 t 2 } containing t i .Then the edge-separation of G associated to We will need the main result of [EGLD23], which gives the structure of locally finite quasi-transitive graph excluding a minor (note that this result does not use the Graph minor structure theorem by Robertson and Seymour [RS03]).

Theorem 2.1 ([EGLD23]
).Let G be a locally finite quasi-transitive graph excluding K ∞ as a minor.Then there is an integer k such that G admits a canonical treedecomposition (T, V) of adhesion at most k, whose torsos have size at most k or are planar.Moreover, the edge-separations of (T, V) are tight.
The property that the edge-separations (T, V) are tight in the statement of Theorem 2.1 will be particularly useful in combination with the following result of Thomassen and Woess [TW93] (which was explicitly proved for transitive graphs, but the same proof also holds for quasi-transitive graphs).

Lemma 2.2 (Corollary 4.3 in [TW93]
).Let G be a locally finite graph.Then for every v ∈ V (G) and k ⩾ 1, there is only a finite number of tight separations (Y, S, Z) of order k in G such that v ∈ S.Moreover, if G is quasi-transitive then for any k ⩾ 1, there is only a finite number of orbits of tight separations of order at most k in G under the action of the automorphism group of G.

Proof of Theorem 1.3
We assume that G is connected, since otherwise we can consider each connected component separately.By Theorem 2.1, G has a canonical tree-decomposition (T, V) whose torsos G V t , t ∈ V (T ), are either planar or finite and whose adhesion sets have bounded size.For each u ∈ V (G), we let T u be the subtree of T with vertex set {t ∈ V (T ), u ∈ V t }.Note that as the edge-separations of (T, V) are tight, Lemma 2.2 implies that T u is finite for each u.
We let G ′ be the graph constructed as follow: for each t ∈ V (T ), we let V ′ t be a copy of V t and G ′ t (with vertex set Now for every edge st ∈ E(T ), we choose an arbitrary vertex u st ∈ V t ∩ V s (such a vertex exists, since G is assumed to be connected).We let: We also let T ′ be the 1-subdivision of T (the graph obtained from T by replacing each edge e = st by a two-edge path s, t e , t).Finally, for each e = st ∈ E(T ), we set st } and We observe that by definition, (T ′ , V ′ ) is a tree-decomposition of G ′ whose adhesion sets all have size 1.In particular, for every We also note that by the definition of G ′ and Lemma 2.2, G ′ has bounded degree.
Claim 3.1.For every graph G, if G has a tree-decomposition (T, V) such that every torso is planar and adhesion sets have size at most 1, then G is planar.
Proof of the Claim: If G contains K 5 or K 3,3 as a minor, then some torso of (T, V) must also contain K 5 or K 3,3 as a minor, which is a contradiction.The result then follows from Wagner's theorem [Wag37] stating that any graph excluding K 5 and K 3,3 is planar. ♢ We now construct a quasi-isometry f from G to G ′ .For each u ∈ V (G), we choose some t u ∈ V (T u ) and set f (u) := u (tu) .We also let which all exist by Lemma 2.2, as the edge-separations of (T, V) are tight.We note that for each We first show the following: Proof of the Claim: By Lemma 2.2, E(T ) has finitely many orbits under the action of the automorphism group of G. Hence, up to automorphism there are only finitely many pairs {u, v} such that u, v lie in a common adhesion set of (T, V).In particular, as G is connected this means that the set of values {d G (u, v), ∃st ∈ E(T ), u, v ∈ V s ∩ V t } admits a maximum C. The claim follows from this observation.♢ We now show that there is a constant α > 0 such that for every u, v ∈ V (G) and every f (u)f (v)-path P ′ in G ′ , there exists a uv-path P of size at most α Proof of the Claim: Assume first that s = t.♢ We now show that there exists a constant β > 0 such that for every u, v ∈ V (G) and every uv-path P in G, there exists a f (u)f (v)-path Proof of the Claim: First note that if V t is finite, then for each u, v ∈ V t we have: Now let u ∈ V (G) and s, t ∈ V (T u ).We let (s = t 0 , t 1 , . . ., t = t ℓ ) be the shortest st-path in T .Note that it is also a path in T u , hence ℓ ⩽ B. Recall that in the construction of G ′ , we have chosen for each edge st ∈ E(T ) a vertex u st ∈ V s ∩ V t and we have added an edge in G ′ between u for the sake of readability.Note that for each i ∈ [ℓ], x i might be equal to u and that both u and x i lie in the adhesion set To conclude the proof of the claim, let uv ∈ E(G).As (T, V) is a tree-decomposition, there exists some t ∈ V (T ) such that u, v ∈ V t .Then: thus by inequalities (1) and (2) we obtain To prove that f is a quasi-isometry, it remains to prove that each y ∈ V (G ′ ) is at bounded distance in G ′ from f (V (G)).For this, let y ∈ V (G ′ ) and t ∈ V (T ), u ∈ V (G) be such that y = u (t) .Then by inequality (2), d G ′ (y, f (u)) ⩽ (2A 2 +1)B so f is indeed a quasi-isometry.This concludes the proof of Theorem 1.3.□

Beyond planarity
In this section we prove Theorem 1.5.We first show that for graphs of bounded degree, having finite local crossing number is preserved under quasi-isometry.
Theorem 4.1.Let G be a graph of bounded degree which is quasi-isometric to a graph H of finite local crossing number.Then G also has finite local crossing number.
Note that in general, the property of being locally finite, or even of having countably many vertices is not preserved under quasi-isometry.The next lemma will be useful to make sure that we can restrict ourselves to locally finite graphs in the remainder of the proof.
Lemma 4.2.Let G be a graph of bounded degree which is quasi-isometric to a graph H. Then G is quasi-isometric to a subgraph H ′ of H of bounded degree.
Proof.We let f : and such that the A-neighborhood of f (V (G)) covers H.We also let ∆ ∈ N denote the maximum degree of G.Note that for each xy ∈ E(G), there exists a f (x)f (y)- We let H ′ be the subgraph of H given by the union of all such paths P xy .We first observe that H ′ is quasi-isometric to G, and that f gives the corresponding quasi-isometric embedding.Note that for every z ∈ V (H ′ ), by construction there must be some edge xy ∈ E(G) such that z ∈ P xy .In particular, d H ′ (z, f (x)) ⩽ 2A.Note that by construction we clearly have d H ′ (f (x), f (y)) ⩽ 2Ad G (x, y) for each x, y ∈ V (G), and as Note that as G degree at most ∆, we have |X| ⩽ ∆ 4A .In particular it implies that H ′ has degree at most ∆ 4A .□ Given a graph G and an integer k ⩾ 1, the k-th power of G, denoted by G k , is the graph with the same vertex set as G in which two vertices are adjacent if and only if they are at distance at most k in G.The k-blow-up of G, denoted by G ⊠ K k , is the graph obtained from G by replacing each vertex u by a copy C u of the complete graph K k , and by adding all edges between pairs C u , C v if and only if u and v are adjacent in G (so that each edge of G is replaced by a complete bipartite graph K k,k in G ⊠ K k ).Quasi-isometries of bounded degree graphs are related to graph powers and blow-ups by the following lemma.
Lemma 4.3.Let H be a graph, and let G be a graph of degree at most ∆ ∈ N which is quasi-isometric to H. Then there is an integer k such that G is a subgraph of Proof.We let A ⩾ 1 and f : V (G) → V (H) be such that for each x, x ′ ∈ V (G): and such that the A-neighborhood of f (V (G)) covers H.Note that for each for every y ∈ V (H).We now show that G is a subgraph of H ′ := H 2A ⊠ K B , which implies the lemma for k := max(2A, B).
As in the definition of a blow-up, for each v ∈ V (H) we denote by C v the associated clique of size B in H ′ .For every v ∈ V (H) we fix an arbitrary injection g v : f −1 (v) → C v , and define an injective mapping g : V (G) → V (H ′ ) by letting g(x) := g f (x) (x) for each x ∈ V (G).In other words every two vertices of G having the same image v by f are sent by g in the same clique C v in H ′ .By construction g is injective, so we just need to check that it defines a graph homomorphism to conclude that G is a subgraph of H ′ .Let xy ∈ E(G).Then d H (f (x), f (y)) ⩽ 2A so in particular every vertex in V f (x) is at distance at most 2A to every vertex in V f (y) in H ′ .In particular, this means that g(x)g(y) ∈ E(H ′ ), as desired.□ The next observation will allow us to slightly simplify the statement of Lemma 4.3.
Observation 4.4.For every graph H of bounded maximum degree, and every integer k, there exists a graph G of bounded maximum degree such that lcr(G) = lcr(H) and Proof.Let G be the graph obtained from H by attaching to each vertex k pendant vertices of degree 1. , where F 2 is the planar graph obtained from F 1 by placing a new vertex at each crossing (and note that if F 1 has bounded degree, then F 2 also has bounded degree).It follows that there is a planar graph H of bounded degree and an integer k, such that G is a subgraph of H k .□ We now prove that bounded powers of planar graphs of bounded degree are ℓplanar for some ℓ.This was proved for finite graphs in [DMW23, Lemma 12].

Lemma 4.6 ([DMW23]
).Let H be a finite planar graph of maximum degree at most ∆ and let G be a subgraph of H k , for some integer k.Then G is ℓ-planar, for ℓ := 2k(k + 1)∆ k .
However, we need a version of Lemma 4.6 for infinite locally finite graphs.The first option is to simply follow the proof of [DMW23], which starts with a planar drawing of G, and adds for any path P of length at most k, an edge between the endpoints of P , drawn in a close neighborhood around P .However in the locally finite case this approach requires that the original planar drawing has the property that every edge has a small neighborhood which does not intersect any other vertices or edges of the graph.Such a drawing always exists but it requires a little bit of work.So instead, we chose to extend Lemma 4.6 to infinite locally finite graphs using a simple compactness argument.
Lemma 4.7.Let G be a locally finite graph.If there is an integer ℓ such that all finite induced subgraphs of G are ℓ-planar, then G is also ℓ-planar.
Proof.We first observe that any ℓ-planar embedding of a graph H can be described combinatorially, by considering the planar graph H + obtained from H by replacing all crossings by new vertices.The corresponding planar embedding of H + can be completely described (up to homeomorphism) by its rotation system (the clockwise cyclic ordering of the neighbors around each vertex), and there are only finitely many such rotation systems if H (and thus H + ) is finite.
We are now ready to prove the lemma.We can assume that G is connected, since otherwise we can consider each connected component independently.Since G is locally finite and connected, it is countable and we can write V (G) = {v 1 , v 2 , . ..}.We define a rooted tree T as follows.The root of T is the unique ℓ-planar embedding of G[{v 1 }], up to homeomorphism.For every k ⩾ 1, and any ℓ-planar embedding of G k := G[{v 1 , . . ., v k }] we add a node in the tree and connect it to the node corresponding to the resulting ℓ-planar embedding of G k−1 = G k −v k (the embedding obtained by deleting v k in the embedding of G k ).The resulting tree T is infinite (since every graph G k is ℓ-planar by assumption), and locally finite (since every graph G k has only finitely many different ℓ-planar embeddings, up to homeomorphism).By König's infinity lemma [Kön27] (or by repeated applications of the pigeonhole principle), T has an infinite path starting at the root.This infinite path corresponds to a sequence of ℓ-planar embeddings of G k , k ⩾ 0, with the property that for every k ⩾ 0, the ℓ-planar embedding of G k can be obtained from the ℓ-planar embedding of G k+1 by deleting v k+1 (and all edges incident to v k+1 ).By taking the union of all the ℓ-planar embeddings of G k , k ⩾ 0, we thus obtain an ℓ-planar embedding of G, as desired. □ We obtain the following as a direct consequence.
Corollary 4.8.Let H be a locally finite planar graph of maximum degree at most ∆ and let G be a subgraph of H k , for some integer k.Then G is ℓ-planar, for ℓ := 2k(k + 1)∆ k .
Proof.Let X be a finite subset of V (G) ⊆ V (H) and for any pair x, x ′ ∈ X with d H (x, x ′ ) ⩽ k, consider a path P x,x ′ of length at most k between x and x ′ in H. Let Y be the union of X and the vertex sets of all the paths P x,x ′ defined above.Then H[Y ] is a finite planar graph, and G[X], the finite subgraph of G induced by X, is a subgraph of H[Y ] k .By Lemma 4.6, G[X] is ℓ-planar with ℓ := 2k(k + 1)∆ k .Since this holds for any finite set X, it follows from Lemma 4.7 that G itself is ℓ-planar, as desired.□ Theorem 4.1 is now a direct consequence of Corollary 4.5 and Corollary 4.8.By combining Theorems 1.3 and 4.1, we then immediately deduce Theorem 1.5.

Assouad-Nagata dimension of minor-excluded groups
For two graphs G and H, we say that a function f : When such a function f exists we also say that the graph G has a bilipschitz embedding in H. Since graphs are uniformly discrete metric spaces (in the sense that any two distinct elements lie at distance at least 1 apart), for any quasi-isometric embedding of a bounded degree graph G in a graph H, there is a bilipschitz embedding of G in a graph obtained from H by adding, for each vertex v of H, a bounded number of vertices of degree 1 adjacent to v (note that in the journal version of the paper, it is incorrectly stated that with the same assumptions there is a bilipschitz embedding of G in H).Hence, Theorem 1.3 has the following immediate consequence.
Corollary 5.1.Every locally finite quasi-transitive K ∞ -minor free graph has a bilipschitz embedding in a planar graph of bounded degree.
We first prove that planar graphs of bounded degree have Assouad-Nagata dimension at most 2 (a stronger version of this result, without the bounded degree assumption, was proved in [BBE + 23]).
We need the following result, first proved in [BST12] in a slightly different form.Another proof can be found in [BBE + 20, BBE + 23] based on a result of Ding and Oporowski [DO95] which states that every graph of treewidth at most t and maximum degree at most ∆ is a subgraph of the strong product of a tree with a complete graph on 24t∆ vertices.The proofs of all these results are fairly short.
A layering of a graph G is a partition of the vertex set of G into sets L 0 , L 1 , . .., called layers, so that any pair of adjacent vertices in G either lies in the same layer or in consecutive layers (i.e.layers L i , L i+1 for some i ⩾ 1).A simple example of layering is given by a BFS-layering of G, obtained by choosing one root vertex v C in each connected component C of G, and then defining L i (for all i ⩾ 0) as the set of vertices of G at distance exactly i from one of the vertices v C .
It was proved by Bodlaender [Bod88] that planar graphs of bounded diameter have bounded treewidth.This directly implies the following.
Lemma 5.3.For any BFS-layering of a planar graph G, and any integer k, the subgraph of G induced by k consecutive layers of L has bounded treewidth.
Proof.Let L 0 , L 1 , . . .be a BFS-layering of G. Consider the planar subgraph H of G induced by k consecutive layers L i , L i+1 , . . ., L i+k−1 .If i = 0, then H is a disjoint union of graphs of radius at most k, and thus has bounded treewidth by Bodlaender's result [Bod88].Assume now that i ⩾ 1, and let H + be the supergraph of H obtained by adding a vertex r that dominates all the vertices of L i .Note that H + can be obtained from the subgraph of G induced by the layers L 0 , L 1 , . . ., L i+k−1 , by contracting all layers L 0 , L 1 , . . ., L i−1 (which induce a connected subgraph of G, by definition of a BFS-layering) into a single vertex.Thus H + is a minor of G, which implies that H + is planar.Moreover, it follows from the definition of a BFS-layering that each vertex of H + lies at distance at most k from r, and thus H + has diameter at most 2k.By Bodlaender's result [Bod88], H + has bounded treewidth, and thus H (as a subgraph of H + ) also has bounded treewidth.□ The next result appears as Theorem 4.3 in [BBE + 23], and is a simple application of the main result in [BDLM08] (which has a nice and short combinatorial proof).We immediately deduce the following.
Corollary 5.5.Every planar graph G of bounded degree has Assouad-Nagata dimension at most 2.
Proof.Consider a BFS-layering L of G.By Lemma 5.3, for any k ⩾ 1, the disjoint union of all subgraphs of G induced by k consecutive layers of L has bounded treewidth.As G has bounded degree, this disjoint union of subgraphs of G also has bounded degree, and thus by Theorem 5.2 it has Assouad-Nagata dimension at most 1.Hence, it follows from Theorem 5.4 that G itself has Assouad-Nagata dimension at most 2. □ We are now ready to prove the main result of this section.Recall that a stronger version (without the quasi-transitivity assumption) was proved in [Liu23,Dis23], but the version below has a reasonably simple proof that does not rely on the Robertson-Seymour graph minor structure theorem.
Theorem 5.6.Every locally finite quasi-transitive K ∞ -minor free graph has Assouad-Nagata dimension at most 2. Proof.By Corollary 5.1, every locally finite quasi-transitive K ∞ -minor free graph G has a bilipschitz embedding in some planar graph H of bounded degree, which has Assouad-Nagata dimension at most 2. Since Assouad-Nagata dimension is invariant under bilipschitz embedding [LS05], G has Assouad-Nagata dimension at most 2. □

Open problems
We now discuss possible extensions or variants of Theorem 1.3.
Quasi-isometries to quasi-transitive planar graphs.It was proved by Mac-Manus [Mac23] that if a finitely generated group has a Cayley graph which is quasiisometric to a planar graph, then it is quasi-isometric to a planar Cayley graph.In the same spirit, it is natural to ask whether we can require the planar graph of bounded degree in Theorem 1.3 to be quasi-transitive, or even a Cayley graph.Our current proof does not preserve symmetries, as we do a number of non-canonical choices for the images of the vertices.As remarked by a referee, the stronger question above, whether we can require the planar graph in Theorem 1.3 to be a Cayley graph, is a special case of a Problem of Woess [Woe91, Problem 1], which asked whether every transitive graph is quasi-isometric to a Cayley graph.This turned out to have a negative answer [EFW12] in general, but the question restricted to (quasi-)transitive graphs excluding a minor might still have a positive answer.
A finite list of obstructions.MacManus [Mac23] recently gave a precise characterization of quasi-transitive graphs which are quasi-isometric to a planar graph, in terms of the existence of a canonical tree-decomposition sharing similarities with that of Theorem 2.1.It would be interesting to also find a characterization in terms of obstructions.For instance, it was conjectured in [GP23] that a graph is quasi-isometric to a planar graph if and only if it does not contain K 5 or K 3,3 as an asymptotic minor (see also Question 1.1 for a version of this problem in the particular case of transitive graphs).
Examples of quasi-transitive graphs that are not quasi-isometric to any planar graph include Cayley graphs of a group of Assouad-Nagata dimension at least 3, for instance any grid in dimension 3.This rules out any generalization of Theorem 1.3 using classes of polynomial growth or expansion.This example also shows that we cannot extend Theorem 1.3 to all families of bounded queue-number or stacknumber.
Here is perhaps a more interesting example.The strong product G ⊠ H of two graphs G and H has vertex set V (G) × V (H), and two distinct vertices (u, x) and (v, y) are adjacent if and only if (u = v or uv ∈ E(G)) and (x = y or xy ∈ E(H)).Consider the strong product T ⊠ P of the infinite binary tree T and the infinite 2-way path P .Using Theorems 5.2 and Lemma 5.4, this graph has Assouad-Nagata dimension at most 2. As it contains a quasi-isometric copy of a 2-dimensional grid, the Assouad-Nagata dimension of T ⊠ P is indeed equal to 2. On the other hand, we observe that for any integer k, T ⊠ P contains the complete bipartite graph K k,k as an asymptotic minor.To see this, remark that T contains an infinite k-claw (the graph obtained by gluing k infinite 1-way paths at their starting vertex) as an asymptotic minor (obtained by contracting a subtree of T with k leaves into a single vertex, and pruning the additional branches).The strong product of this infinite k-claw with P consists of k copies of an infinite 2-dimensional grid (restricted to the upper half-plane, say), glued on a common infinite path π.On this path we can select k vertices, arbitrarily far apart, and on each infinite grid we can select a single vertex, arbitrarily far from π, and connect it to the k vertices of π via disjoint paths.By taking a ball of sufficiently large radius around each of the 2k vertices, we obtain K k,k as an r-fat minor for arbitrarily large r, and thus K k,k as an asymptotic minor.This is illustrated for k = 3 in Figure 1.Since containing a graph H as an asymptotic minor is invariant under quasi-isometry, any graph G which is quasi-isometric to T ⊠ P also contains every K k,k as an asymptotic minor, and thus every finite graph as a minor (in particular, G cannot be planar).Recall that any graph excluding a minor has Assouad-Nagata dimension at most 2 [Liu23,Dis23].It is natural to wonder whether some sort of converse holds, that is whether any graph of Assoud-Nagata dimension at most 2 is quasi-isometric to a graph excluding a minor (this would be a natural extension of Theorem 1.3).The example above shows that this is false, even for vertex-transitive graphs.
As explained above, it was conjectured in [GP23] that graphs that are quasiisometric to a planar graph can be characterized by a finite list of forbidden asymptotic minors.A natural question is whether this can be replaced by a finite list of forbidden quasi-isometric graphs, at least in the case of quasi-transitive graphs.We do not have a good candidate for such a finite list, but it should contain at least the two examples mentioned above: 3-dimensional grids and the product of the binary tree with a path.One difficulty is that no such list is even known (or conjectured to exist) for Assouad-Nagata dimension at most 2, or asymptotic dimension at most 2. k-planar graphs.We conjecture the following variant of Theorem 1.3 for k-planar graphs.
Conjecture 6.1.Every quasi-transitive graph of bounded local crossing number and degree is quasi-isometric to a planar graph.
Using Theorem 4.1, Conjecture 6.1 would imply a positive answer to Problem 1.4.We observe that it is enough to prove Conjecture 6.1 for 1-planar graphs.Conjecture 6.2.Every quasi-transitive 1-planar graph of bounded degree is quasiisometric to a planar graph.
To see that the case lcr ⩾ 2 reduces to the case lcr = 1, observe that for every k-planar graph G with k ⩾ 2, its (k −1)-subdivision G (k−1) (the graph obtained from G by replacing every edge by a path on k edges) is locally finite, quasi-transitive, quasi-isometric to G, and 1-planar.To see the last point, consider any embedding of G in R 2 in which every edge is involved in at most k crossings and assume without loss of generality that the crossing points between every two edges are all pairwise distinct.Then for every edge e ∈ E(G), one can add the k −1 corresponding vertices of G (k−1) subdividing e in the drawing by putting at least one vertex on each of the curves connecting two consecutive crossing points of e with other edges.
We note that Conjecture 6.2 (and thus Conjecture 6.1) would be a direct consequence of the following.Conjecture 6.3.Let G be a quasi-transitive 1-planar graph of bounded degree.Then there is an integer k and an embedding of G in the plane with at most 1 crossing per edge such that for every pair of crossing edges uv, xy in G, we have d G (u, x) ⩽ k.
In a previous version of this manuscript we were conjecturing something stronger, namely that for any embedding of G with at most 1 crossing per edge, there is an integer k such that all pairs of crossing edges lie at distance at most k in G.But this is false (as shown by the two-way infinite path, drawn in such a way that it self-intersects at more and more distant points).
In this paper we have mainly considered graphs with finite local crossing number.A natural generalization is the following: a graph is (< ω)-planar if it has a drawing in the plane in which each edge is involved in finitely many crossings.This raises the following question.Question 6.4.Let G be a quasi-transitive graph of bounded degree which is (< ω)planar.Is it true that G has finite local crossing number?
It was observed by Kolja Knauer (personal communication) that Question 6.4 has a negative answer, as indeed any infinite locally finite graph G is (< ω)-planar.To see this, consider an ordering v 1 , v 2 , . . . of V (G), map each vertex v i to the point with coordinates (i, i 2 ) in the plane, and each edge v i v j as a segment joining v i and v j .Note that by convexity of the function x → x 2 , every edge crossing an edge v i v j must have an endpoint v k with i < k < j.As for every pair i < j there are only finitely many such vertices v k and each of them has finite degree, the edge v i v j is crossed by only finitely many other edges.
As there exist quasi-transitive graphs of bounded degree that have unbounded local crossing number (the 3-dimensional grid for instance, see [DEW17]), the paragraph above implies that Question 6.4 has a negative answer.
and thus d G (u, v) ⩽ C by Claim 3.2.Assume now that s ̸ = t.Then by definition of G ′ , we must have st ∈ E(T ) and u = v, and thus d G (u, v) = 0.
Theorem 5.4 ([BBE + 23]).If a graph G has a layering L = (L 0 , L 1 , . ..) such that for any integer k, the disjoint union of all subgraphs of G induced by k consecutive layers of L has Assouad-Nagata dimension at most n, then G has Assouad-Nagata dimension at most n + 1.
Corollary 4.5.If a graph G with bounded degree is quasi-isometric to a graph of bounded local crossing number, then there exists a planar graph H of bounded maximum degree and an integer k, such that G is a subgraph of H k .
Note that the graph H k ⊠K k is a subgraph of G k+2 .To see this, one can bijectively map each clique C v of H for v ∈ V (H) to the k pendant vertices we attached to v in G, and observe that it gives an isomorphism between the graph induced by these vertices in G 3 , and H ⊠ K k .Since adding pendant vertices does not change the local crossing number, lcr(G) = lcr(H).Moreover G has bounded maximum degree if and only if H has bounded maximum degree.□Wecan now combine the results above to deduce the following corollary.