Asymptotic Dimension of Minor-Closed Families and Assouad-Nagata Dimension of Surfaces

The asymptotic dimension is an invariant of metric spaces introduced by Gromov in the context of geometric group theory. In this paper, we study the asymptotic dimension of metric spaces generated by graphs and their shortest path metric and show their applications to some continuous spaces. The asymptotic dimension of such graph metrics can be seen as a large scale generalisation of weak diameter network decomposition which has been extensively studied in computer science. We prove that every proper minor-closed family of graphs has asymptotic dimension at most 2, which gives optimal answers to a question of Fujiwara and Papasoglu and (in a strong form) to a problem raised by Ostrovskii and Rosenthal on minor excluded groups. For some special minor-closed families, such as the class of graphs embeddable in a surface of bounded Euler genus, we prove a stronger result and apply this to show that complete Riemannian surfaces have Assouad-Nagata dimension at most 2. Furthermore, our techniques allow us to prove optimal results for the asymptotic dimension of graphs of bounded layered treewidth and graphs of polynomial growth, which are graph classes that are defined by purely combinatorial notions and properly contain graph classes with some natural topological and geometric flavours.

1. Introduction 1.1.Asymptotic dimension.Asymptotic dimension of metric spaces was introduced by Gromov [23] in the context of geometric group theory.Every metric space can be realized by a graph whose edges are weighted.In this paper, we address connections between structural graph theory and the theory of asymptotic dimension.In particular, we solve open problems in coarse geometry and geometric group theory via tools from structural graph theory.
Let (X, d) be a pseudometric 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.The asymptotic dimension of (X, d), denoted by asdim (X, d), 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 reader is referred to [7] for a survey on asymptotic dimension and its group theoretic applications, and to the lecture notes of Roe [42] on coarse geometry for more detailed proofs of some results of [23].
This paper is a combination of parts of two manuscripts [10,30] in the arXiv repository, where [30] has appeared as an extended abstract in a conference, and a number of results in the unpublished manuscript [10] are strengthened to Assouad-Nagata dimension in this paper.M. Bonamy   As the asymptotic dimension of a bounded space is 0, we are more interested in the asymptotic dimension of infinite pseudometric spaces or (infinite) families of (infinite or finite) pseudometric spaces.We define the asymptotic dimension of a family X of pseudometric spaces as the least n for which there exists a function D X : R + → R + which is an n-dimensional control function for all X ∈ X .
1.2.Graphs as (pseudo)metric spaces.A weighted graph (G, φ) consists of a graph G and a function φ : E(G) → R + .We call φ(e) the weight of e for each e ∈ E(G).The length in (G, φ) of a path P in G is the sum of the weights of the edges of P .Given two vertices x, y ∈ V (G), we define d (G,φ) (x, y) to be the infimum of the length in (G, φ) of a path between x and y; we define d (G,φ) (x, y) = ∞ if there exists no path between x and y.Notice that (V (G), d (G,φ) ) is a pseudometric space.
In this paper, we do not distinguish a weighted graph (G, φ) from the pseudometric space (V (G), d (G,φ) ).Therefore, the asymptotic dimension of a weighted graph is the asymptotic dimension of this pseudometric space.
Given an (unweighted) graph G, it can be viewed as a weighted graph in which each edge has weight 1.In particular, the asymptotic dimension of a graph is defined.
The main goal of our present work is to determine the asymptotic dimension of various (classes of weighted or unweighted) graphs.As observed above, a finite graph has asymptotic dimension 0, so this is only interesting for infinite graphs, or for infinite classes of (finite or infinite) graphs.On the other hand, every metric space can be realised by a weighted complete graph, so it will be more interesting to study graphs with restricted structure.
Indeed, the asymptotic dimension of (unweighted) graphs with structure restriction has attracted wide attention.For example, Gromov considered the asymptotic dimension of Cayley graphs of groups (see Section 1.7 for more details).In addition, Gromov [23] observed that d-dimensional Euclidean spaces have asymptotic dimension d, and it can easily be deduced from this that for any d 1, the class of d-dimensional grids (with or without diagonals) has asymptotic dimension d.On the other hand, it is implicit in the work of Gromov [24] (see also [42,Proposition 11.29]) that any infinite family of bounded degree expanders (in particular cubic expanders) has unbounded asymptotic dimension.A different proof, using vertex expansion instead of spectral expansion, is given in [25].Another example of a class of graphs with bounded degree and infinite asymptotic dimension is the class of lamplighter graphs of binary trees [5] (these graphs have maximum degree 4).This implies that bounding the degree is not enough to bound the asymptotic dimension.
The asymptotic dimension of graphs is studied in several research areas.It is a large-scale generalisation of weak diameter network decomposition which has been studied in distributed computing (see Section 1.9 for more details); a more refined notion of asymptotic dimension is called Assouad-Nagata dimension and its algorithmic form is related to weak sparse partition schemes, which are studied in theoretical computer science (see Sections 1.5 and 1.8 for more details).
A simple compactness argument (see Theorem A.2) shows that the asymptotic dimension of an infinite weighted graph is at most the asymptotic dimension of the class of its finite induced weighted subgraphs 1 .Hence in this paper, we only consider the asymptotic dimension of classes of finite weighted or unweighted graphs.
From now on, graphs, and more generally weighted graphs, are finite, unless stated otherwise.Note that the pseudometric space generated by a finite weighted or unweighted graph is a metric space.
1.3.Minor-closed classes of graphs.A (finite) graph H is a minor of a finite or infinite graph G if it can be obtained from a subgraph of G by contracting edges.We say that a class G of graphs is minor-closed if any minor of a graph from G is also in G.A minor-closed class G is proper if it does not contain all graphs.In particular, for every proper minor-closed family G, there exists a graph H such that H ∈ G, so G is a subclass of the class of H-minor free graphs.
Minor-closed classes are a far-reaching generalisation of classes of graphs with some geometric or topological properties.For example, any minor of a graph embeddable in a surface2 Σ is also embeddable in Σ, and thus classes of graphs embeddable in a fixed surface form a natural example of a proper minor-closed class.Other well-known examples of minor-closed classes include the class of linkless embeddable graphs and the class of knotless embeddable graphs.Moreover, a deep result in graph theory (the Graph Minor Theorem [41]) states that for every proper minor-closed class G, there exist finitely many graphs H 1 , H 2 , ..., H n such that G is the intersection of the classes of H i -minor free graphs.In order to prove this result, Robertson and Seymour obtained a structural description of any proper minor-closed class, which will be instrumental in the proof of Theorem 1.1 below.
Ostrovskii and Rosenthal [36] proved that for every integer t, the class of K t -minor free graphs has asymptotic dimension at most 4 t .Recently, Fujiwara and Papasoglu [21] proved that the class of planar graphs, which is one of the most extensively studied minor-closed classes, has asymptotic dimension at most 3 and asked whether there exists a uniform upper bound on the asymptotic dimension of proper minor-closed families, as follows.
Question 1 (Question 5.2 in [21]).Is there a constant k such that for any graph H, the class of H-minor free graphs has asymptotic dimension at most k? Can we take k = 2? Question 1 is a possible common strengthening of their result and the aforementioned result in [36].One of the main results of this paper is a complete solution of Question 1.
Theorem 1.1.For any graph H, the class of H-minor free graphs has asymptotic dimension at most 2. In particular, every proper minor-closed class has asymptotic dimension at most 2.
The bound in Theorem 1.1 is optimal if H is a non-planar graph, since the class of 2dimensional grids has asymptotic dimension 2 [23] and is a subclass of planar graphs.When H is planar, we prove that the bound for asymptotic dimension can be further reduced to 1.
Theorem 1.2.For every planar graph H, the asymptotic dimension of the class of H-minor free graphs is at most 1.In particular, every proper minor-closed class that does not contain all planar graphs has asymptotic dimension at most 1.
The bound in Theorem 1.2 is optimal.Bell and Dranishnikov [7] and Fujiwara and Papasoglu [21], respectively, showed that the class of trees and the class of cacti, respectively, have asymptotic dimension 1.Note that trees are K 3 -minor free and cacti are K 2,3 -minor free.
By a simple compactness argument (Theorem A.2), we obtain the following immediate corollary of Theorems 1.1 and 1.2.
Corollary 1.3.Let F be a class of finite or infinite graphs.
(1) For any (finite) graph H, if no member of F contains H as a minor, then asdim(F) 2.
(2) For any (finite) planar graph H, if no member of F contains H as a minor, then asdim(F) 1.
(3) If F is minor-closed and does not contain all (finite) graphs, then asdim(F) 2. (4) If F is minor-closed and does not contain all (finite) planar graphs, then asdim(F) 1.
As we will elaborate in Section 1.5, we are able to strengthen the statement in Theorem 1.1 for some graphs H so that it holds for weighted graphs and for a more refined notion of dimension which leads to results about Assouad-Nagata dimension for Riemannian surfaces.1.4.Treewidth and layered treewidth.A tree-decomposition of a graph G is a pair (T, X ) such that T is a tree and X is a collection (X t : t ∈ V (T )) of subsets of V (G), called the bags, such that • t∈V (T ) X t = V (G), • for every e ∈ E(G), there exists t ∈ V (T ) such that X t contains the ends of e, and • for every v ∈ V (G), the set {t ∈ V (T ) : v ∈ X t } induces a connected subgraph of T .For a tree-decomposition (T, X ), the adhesion of (T, X ) is max tt ∈E(T ) |X t ∩ X t |, and the width of (T, X ) is max t∈V (T ) |X t |−1.The treewidth of G is the minimum width of a tree-decomposition of G.
By the Grid Minor Theorem [39], excluding any planar graph as a minor is equivalent to having bounded treewidth.Hence we have Theorem 1.4 which is an equivalent form of Theorem 1.2.
Theorem 1.4.For every positive integer w, the asymptotic dimension of the class of graphs of treewidth at most w is at most 1.
The bound in Theorem 1.4 is optimal since it is easy to see that every path has treewidth 1 and the class of paths has asymptotic dimension at least 1.Note that the special case of Theorem 1.4 with the additional assumption of bounded maximum degree, which was proved in [10], can also be deduced from the work of Benjamini, Schramm and Timár [8].
A layering of a graph G is an ordered partition (V i ) i∈N or (V i ) i∈Z of V (G) into (possibly empty) subsets V i such that for every edge e of G, there exists i e such that V ie ∪ V ie+1 contains both ends of e.We call each set V i a layer.The layered treewidth of a graph G is the minimum w such that there exist a tree-decomposition of G and a layering of G such that the size of the intersection of any bag and any layer is at most w.
Layered treewidth is a common generalisation of treewidth and Euler genus of graphs.A number of classes of graphs with some geometric properties have bounded layered treewidth.For example, Dujmović, Morin and Wood [18] showed that for every nonnegative integer g, graphs that can be embedded in a surface of Euler genus at most g have layered treewidth at most 2g + 3.Moreover, the 2-dimensional grids with all diagonals have layered treewidth two while having unbounded treewidth and unbounded Euler genus as they can contain arbitrarily large complete minors.
Combining Theorem 1.4 with machinery developed by Brodskiy, Dydak, Levin and Mitra [12], we obtain the following result for graphs of bounded layered treewidth.
Theorem 1.5.For every positive integer w, the asymptotic dimension of the class of graphs of layered treewidth at most w is at most 2.
In fact, classes of graphs of bounded layered treewidth are of interest beyond minor-closed families.Let g, k be nonnegative integers.A graph is (g, k)-planar if it can be drawn in a surface of Euler genus at most g with at most k crossings on each edge.So (g, 0)-planar graphs are exactly the graphs of Euler genus at most g.It is well-known that (0, 1)-planar graphs (also known as 1-planar graphs in the literature) can contain an arbitrary graph as a minor.So the class of (g, k)-planar graphs is not a minor-closed family.On the other hand, Dujmović, Eppstein and Wood [16] proved that (g, k)-planar graphs have layered treewidth at most (4g + 6)(k + 1).
Hence the following is an immediate corollary of Theorem 1.5.
Corollary 1.6.For any nonnegative integers g and k, the class of (g, k)-planar graphs has asymptotic dimension at most 2.
Recall that Corollary 1.6 (and hence Theorem 1.5) is optimal since the class of 2-dimensional grids has asymptotic dimension 2. Other extensively studied graph classes that are known to have bounded layered treewidth include map graphs [13,16] and string graphs with bounded maximum degree [17].We refer readers to [16,17] for discussion of those graphs.
One weakness of layered treewidth is that adding apices can increase layered treewidth a lot.Note that for any vertex v in a graph G and for any layering of G, the neighbours of v must be contained in the union of three consecutive layers.So if a graph has bounded layered treewidth, then the subgraph induced by the neighbours of any fixed vertex must have bounded treewidth.However, consider the graphs that can be obtained from 2-dimensional grids by adding a new vertex adjacent to all other vertices.Since 2-dimensional grids can have arbitrarily large treewidth, such graphs cannot have bounded layered treewidth.
In contrast to the fragility of layered treewidth about adding apices, we show that adding a bounded number of apices does not increase the asymptotic dimension.Let F be a class of graphs.For every nonnegative integer n, define F +n to be the class of graphs such that for every G ∈ F +n , there exists Theorem 1.7.For every class of graphs F and nonnegative integer n, the asymptotic dimension of F +n equals the asymptotic dimension of F. This leads to the following strengthening of Theorem 1.5.
Corollary 1.8.Let k be a nonnegative integer.Let w be a positive integer.Let F be a class of graphs such that for every G ∈ F, there exists Z ⊆ V (G) with |Z| k such that G − Z has layered treewidth at most w.Then the asymptotic dimension of F is at most 2.
1.5.Assouad-Nagata dimension.Gromov [23] noticed that the notion of asymptotic dimension of a metric space can be refined by restricting the growth rate of the control function in its definition.Although this function can be chosen to be linear in many cases, its complexity can be significantly worse in general, as there exist (Cayley) graphs of asymptotic dimension n for which any n-dimensional control function D(r) grows as fast as Ω(exp exp • • • exp r k ), for any given height of the tower of exponentials [35].
A particularly interesting refinement of the asymptotic dimension is the Assouad-Nagata dimension introduced by Assouad [2] (see [29] for more results on this notion).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. A metric space (X, d) has Assouad-Nagata dimension at most n if X has an n-dimensional control function which is a dilation.The definition extends to families of metric spaces in a natural way.Observe that the Assouad-Nagata dimension is at least the asymptotic dimension.
We prove that when H = K 3,p (for any fixed p > 0), Theorem 1.1 can be extended to weighted graphs as well as in the setting of Assouad-Nagata dimension.Theorem 1.9.For any integer p > 0, the class of weighted (finite or infinite) graphs excluding the complete bipartite graph K 3,p as a minor has Assouad-Nagata dimension at most 2.

When p
3, 2-dimensional grids are K 3,p -minor free, so the bound in Theorem 1.9 is optimal.In addition, for any fixed integer g 0, the class of graphs embeddable in a surface of Euler genus g excludes K 3,2g+3 as a minor.So Theorem 1.9 immediately implies the following result.
Corollary 1.10.For any integer g 0, the class of weighted (finite or infinite) graphs embeddable in a surface of Euler genus g has Assouad-Nagata dimension 2.
Corollary 1.10 can be used for proving the following result about complete 2-dimensional connected Riemannian manifolds without boundary.
Theorem 1.11.The Assouad-Nagata dimension of any complete Riemannian surface of finite Euler genus is at most 2.
Note that Corollary 1.10 and Theorem 1.11 extend results of Jørgensen and Lang [27] on the plane.
Similar to the notion of dilation, a control function D X for a metric space X is said to be linear if there is a constant c > 0 such that D X (r) cr + c for any r > 0. We say that a metric space (X, d) has asymptotic dimension at most n of linear type if X has a linear n-dimensional control function.The definition extends to families of metric spaces in a natural way.This notion is sometimes called asymptotic dimension with Higson property [15].Obviously, the asymptotic dimension of linear type is between the asymptotic dimension and the Assouad-Nagata dimension.
Nowak [35] proved that the asymptotic dimension of linear type is not bounded by any function of the asymptotic dimension, by constructing (Cayley) graphs of asymptotic dimension 2 and infinite asymptotic dimension of linear type.We give another such example by showing that some classes of graphs of bounded layered treewidth do not have bounded asymptotic dimension of linear type, even though Theorem 1.5 shows that their asymptotic dimension is at most 2.
Theorem 1.12.There is no integer d such that the class of graphs of layered treewidth at most 1 has asymptotic dimension of linear type at most d.
So Theorem 1.12 states that Theorem 1.5 cannot be extended to asymptotic dimension of linear type or Assouad-Nagata dimension, even if we replace 2 by an arbitrary constant.This is in contrast with the case of K 3,p -minor free graphs and graphs with bounded Euler genus, which have bounded layered treewidth [18], yet have Assouad-Nagata dimension at most 2 by Theorem 1.9 and Corollary 1.10.
1.6.Growth rate.For any function f , a graph G has growth at most f if for any integer r, any vertex v ∈ V (G) has at most f (r) vertices at distance at most r.Similarly, we say that a class of graphs has growth at most f if all graphs in this class have growth at most f .It is known that vertex-transitive graphs of polynomial growth have bounded asymptotic dimension, while some classes of graphs of exponential growth have unbounded asymptotic dimension [25].
We prove the following result that not only removes the vertex-transitive requirement but also shows that polynomials are the fastest-growing growth rate that ensures finite asymptotic dimension.
(1) For any polynomial f , there exists d ∈ N, such that the class of graphs of growth at most f has bounded asymptotic dimension at most d.
(2) For any superpolynomial function3 f with f (r) 3r + 1 for every r ∈ N, the class of graphs of growth at most f has infinite asymptotic dimension.(3) For any function f : N → N for which there exists r 0 ∈ N with f (r 0 ) 3r 0 , the class of graphs of growth at most f has asymptotic dimension at most 1.
We remark that even though the growth rate is a pure combinatorial property, graphs with polynomial growth are closely related to bounded dimensional grids.See Section 9 for more details.1.7.Application 1: Asymptotic dimension of groups.Given a (finite or infinite) group Γ and a generating set S (assumed to be symmetric, in the sense that s ∈ S if and only if s −1 ∈ S), the Cayley graph Cay(Γ, S) is the (possible infinite) graph with vertex-set Γ, with an edge between two vertices u, v ∈ Γ if and only if u = vs for some s ∈ S. As observed by Gromov [23], when Γ is finitely generated, the asymptotic dimension of Cay(Γ, S) is independent of the choice of the finite generating set S, and thus the asymptotic dimension is a group invariant for finitely generated groups.The asymptotic dimension of a finitely generated group Γ is defined to be the asymptotic dimension of Cay(Γ, S) for some symmetric finite generating set S.
We say that Cay(Γ, S) is minor excluded if it is H-minor free for some (finite) graph H.
Question 2 (Problem 4.1 in [36]).Let Γ be a finitely generated group and S a finite generating set such that Cay(Γ, S) is minor excluded.Does it follow that Γ has asymptotic dimension at most 2?
An immediate corollary of Statement 1 of Corollary 1.3 gives a positive answer to Question 2, even when the group is not finitely generated.
Corollary 1.14.Let Γ be a group and S a symmetric (not necessarily finite) generating set such that Cay(Γ, S) is minor excluded.Then Cay(Γ, S) has asymptotic dimension at most 2. Furthermore, if S is finite, then Γ has asymptotic dimension at most 2.
Note that Corollary 1.14 is actually stronger.The finitely generated condition for Γ is only used for ensuring that the asymptotic dimension of Γ is independent of the choice of S. We remark that the choice of the generating set S of Γ affects whether Cay(Γ, S) is H-minor free or not.
Similarly, Corollary 1.6 leads to the following immediate corollary.
Corollary 1.15.For every group Γ with a symmetric (not necessarily finite) generating set S, if there exist nonnegative integers g and k such that the Cayley graph for (Γ, S) is (g, k)-planar, then the asymptotic dimension of Cay(Γ, S) is at most 2. In particular, if Γ is finitely generated, then Γ has asymptotic dimension at most 2.
1.8.Application 2: Sparse partitions.A ball of radius r (or r-ball ) centered in a point x in a metric space X, denoted by B r (x), is the set of points of X at distance at most r from x.
For a real r 0 and an integer n 0, a family U of subsets of elements of X has r-multiplicity at most n if each r-ball in X intersects at most n sets of U. It is not difficult to see that if D X (r) is an n-dimensional control function for a metric space X, then for any r > 0, X has a D X (2r)-bounded cover of r-multiplicity at most n + 1. Gromov [23] proved that a converse of this result also holds, in the sense that the asymptotic dimension of X is exactly the least integer n such that for any real number r > 0, there is a real number D X (r) such that X has a D X (r)-bounded cover of r-multiplicity at most n + 1.Moreover, the function D X has the same type as the n-dimensional control function D X of X: D X is linear if and only if D X is linear, and D X is a dilation if and only if D X is a dilation.As a consequence, the notion of Assouad-Nagata dimension is closely related to the wellstudied notions of sparse covers and sparse partitions in theoretical computer science.A weighted graph G admits a (σ, τ )-weak sparse partition scheme if for any r 0, the vertex-set of G has a partition into (σ • r)-bounded sets of r-multiplicity at most τ , and such a partition can be computed in polynomial time.As before, we say that a family of graphs admits a (σ, τ )-weak sparse partition scheme if all graphs in the family admit a (σ, τ )-weak sparse partition scheme.This definition was introduced in [26], and is equivalent to the notion of weak sparse cover scheme of Awerbuch and Peleg [4] (see [20]).Note that if a family of graphs admits a (σ, τ )-weak sparse partition scheme then its Assouad-Nagata dimension is at most τ − 1. Conversely, if a family of graphs has Assouad-Nagata dimension at most d and the covers can be computed efficiently, then the family admits a (σ, d + 1)-weak sparse partition scheme, for some constant σ.
All our proofs are constructive and yield polynomial-time algorithms to compute the corresponding covers.In particular, whenever we obtain that a fixed class has Assouad-Nagata dimension at most n, then we in fact obtain a (O(1), n + 1)-weak partition scheme for the class.So we have the following corollary from Theorem 1.9 and Corollary 1.10.
Corollary 1.16.For every integer g 0, there exists a real number N such that the following hold.
(1) There exists an (N, 3)-weak partition scheme for the class of weighted K 3,g -minor graphs.(2) There exists an (N, 3)-weak partition scheme for the class of weighted graphs embeddable in a surface of Euler genus g.
We remark that while the Assouad-Nagata dimension and sparse partition are almost equivalent, the emphasis is on different parameters.In the case of the Assouad-Nagata dimension, the goal is to minimise the dimension (or equivalently τ in the sparse partition scheme), while in the (σ, τ )-weak sparse partition scheme, the goal is usually to minimise a function of σ and τ which depends on the application.As an example, it was proved in [26] that if an n-vertex graph admits a (σ, τ )-weak sparse partition scheme, then the graph has a universal Steiner tree with stretch O(σ 2 τ log τ n), so in this case the goal is to minimise σ 2 • τ log τ .
1.9.Application 3: Weak diameter colouring and clustered colouring.The asymptotic dimension of a graph is closely related to weak diameter colourings of its powers.For a graph G, the weak diameter in G of a subset S of V (G) is the maximum distance in G between two vertices of S; the weak diameter in G of a subgraph H of G is the weak diameter in G of V (H) (thus we are taking distances in G rather than H).Given a colouring c of a graph G, a c-monochromatic component (or simply a monochromatic component if c is clear from the context) is a connected component of the subgraph of G induced by some colour class of c.
A graph G is k-colourable with weak diameter in G at most d if each vertex of G can be assigned a colour from {1, . . ., k} so that all monochromatic components have weak diameter in G at most d.
Weak diameter colouring is also studied under the name of weak diameter network decomposition in distributed computing (see [3]), although in this context k and d usually depend on |V (G)| (they are typically of order log |V (G)|), while here we will only consider the case where k and d are constants.Observe that the case d = 0 corresponds to the usual notion of (proper) colouring.Note also that weak diameter colouring should not be confused with the stronger notion that requires that each monochromatic component has bounded diameter, where the distance is computed in the monochromatic component instead of in G (see for instance [31,Theorem 4.1]).
For any integer 1, the -th power of a graph G, denoted by G , is the graph obtained from V (G) by adding an edge xy for each pair of distinct vertices of G with distance at most .Note that for any graph G, G Note that the case 2 of (1) is not a direct consequence of the case = 1 of (1), (2), or (3).This is because the 2nd power of a tree can contain arbitrarily large complete subgraphs, so the classes {G : G ∈ F} for 2 mentioned in Corollary 1.18 are not minor-closed families and have unbounded layered treewidth, even after the deletion of a bounded number of vertices.
Weak diameter colouring is related to clustered colouring which is a variation of the traditional notion of a proper colouring and has received wide attention.A graph G is kcolourable with clustering c if each vertex of G can be assigned a colour from {1, . . ., k} so that all monochromatic components have at most c vertices.The case c = 1 corresponds to the usual notion of (proper) colouring, and there is a large body of work on the case where c is a fixed constant.In this context, we say that a class G of graphs has clustered chromatic number at most k if there is a constant c such that every graph of G is k-colourable with clustering c (see [45] for a recent survey).
Corollary 1.19.Let ∆ 2 be an integer.If G is a class of graphs with asymptotic dimension at most k such that every graph in G has maximum degree at most ∆, then for every integer > 0, the clustered chromatic number of the class {G : G ∈ G} is at most k + 1.
Proof.By Proposition 1.17, for every integer > 0, there is a constant d such that for any graph G ∈ G, G is (k + 1)-colourable with weak diameter in G at most d.Consider such a colouring of G .Since G has maximum degree at most ∆, G has maximum degree at most ∆ +1 , so each monochromatic component of G has size at most ∆ ( +1)(d+1) .This implies that {G : G ∈ G} has clustered chromatic number at most k + 1.
A direct consequence of the case = 1 in Corollaries 1.18 and 1.19 is that under the bounded maximum degree condition, H-minor free graphs, bounded treewidth graphs, and graphs obtained from bounded layered treewidth graphs by adding a bounded number of apices are (respectively) 3-colourable, 2-colourable, and 3-colourable with bounded clustering (these results were originally proved in [31], [1], and [32], respectively).In addition, as the -th power of a graph of maximum degree ∆ has maximum degree at most ∆ +1 , the above result can be extended to any integer .1.10.Outline of the paper.The first goal of this paper is to prove Theorems 1.1 and 1.4,where the latter is equivalent to Theorem 1.2 by the Grid Minor Theorem and Theorem A.2.
The key tool (Theorem 3.1) that we develop in this paper to prove Theorems 1.1 and 1.4 might be of independent interest.It allows us to show that generating a new class of graphs from hereditary classes by using tree-decompositions of bounded adhesion does not increase the asymptotic dimension.
Using Proposition 1.17, we will prove Theorem 3.1 in its equivalent form stated in terms of weak diameter colouring.In order to do so, we will prove a stronger version that shows that we can extend any reasonable precolouring to a desired colouring.In Section 2, we build the machinery for extending such a precolouring and use this machinery to prove Theorem 1.7.In Section 3, we will prove Theorem 3.1 and use it to prove Theorem 1.4.
We require another essential tool (Theorem 4.3) to prove Theorem 1.1.Theorem 4.3 allows us to bound the asymptotic dimension if there exists a layering of graphs such that the asymptotic dimension of any subgraph induced by a union of finitely many consecutive layers is under control.We develop this machinery in Section 4. In Section 5, we will use this tool to derive our result on layered treewidth (Theorem 1.5) from the result on treewidth (Theorem 1.4) and prove Theorem 1.1 by combining Theorems 1.5, 1.7 and 3.1.
Sections 6, 7 and 8 address Assouad-Nagata dimension.We will prove Theorem 1.12 in Section 6, showing that no analogous result of Theorem 1.5 for Assouad-Nagata dimension can hold.In Section 7, we will prove the result for K 3,p -minor free graphs (Theorem 1.9) by using the tool developed in Section 4 and a notion called "fat minor".In Section 8, we show that the result for complete Riemannian surfaces (Theorem 1.11) follows from Theorem 1.9 by a simple argument about quasi-isometry.
Finally, in Section 9 we investigate the asymptotic dimension of classes of graphs with a low-dimensional representation and prove Theorem 1.13.
We conclude this paper in Section 10 with some open problems.
1.11.Remark on the notation.In this paper, R + denotes the set of all positive real numbers and N denotes the set of all positive integers.All graphs in this paper are simple.That is, there exist no parallel edges or loops.We recall that for a graph G and a subset S of V (G), the subgraph of G induced by S is the graph, denoted by G[S], whose vertex-set is S and whose edge-set consists of the edges of G with both ends in S; for a weighted graph (G, φ) and a subset S of V (G), the (weighted) subgraph of (G, φ) induced by S is the weighted graph (G[S], φ| E(G[S]) ).Note that the (weighted) subgraph induced by S is a (weighted) graph, so it defines a pseudometric space, but this pseudometric space is not necessarily equal to the induced metric which is the space S together with the distance function computed in G or (G, φ).

Centered sets
Given a set of vertices S ⊆ V (G), and an integer r 0, we denote by N r G (S) the set of vertices in V (G) which are at distance at most r from at least one vertex in S. Given an integer k 0, if a set of vertices Z is contained in N r G (S) for some set S of size at most k, we say that Z is a (k, r)-centered set in G.
In this section we prove that given some graph G and a (k, r)-centered set Z, the union of an arbitrary colouring of Z and a colouring of G − Z with bounded weak diameter in G − Z gives a colouring of G of bounded weak diameter in G.We will make extensive use of this result in the proof of Theorem 3.1, from which we derive all our results about the asymptotic dimension of minor-closed families.

Note that
• f (0, x, y) = y, and We shall prove it by induction on k.Since Z is (k, r)-centered, there exists a subset S of V (G) with size at most k such that Z ⊆ N r G (S).When k = 0, S = Z = ∅, and hence we are done since f (0, r, N ) N .So we may assume that k 1 and this lemma holds when k is smaller. Let

S ). We can apply the induction hypothesis on
, then all its vertices are at distance more than in G from any vertex of Z 1 with the same colour, so M is a c -monochromatic component and has weak diameter in , where each of them contains a vertex within distance in G at most from Z 1 , hence at distance in G at most 1 from Z 1 , and has weak diameter in ) from any vertex of M , and so the weak diameter in G of M is at most twice that value, which is f (k, r, N ), as desired.
Let > 0, m > 0, N > 0 be integers.We say a class F of graphs is (m, , N )-nice if for every G ∈ F, G is m-colourable with weak diameter in G at most N .
Recall that for every integer n 0, F +n is the class of graphs such that for every G ∈ F +n , there exists Z ⊆ V (G) with |Z| n such that G − Z ∈ F. Notice that such Z is an (n, 0)centered set in G. Through a direct application of Lemma 2.1 with (k, r) = (n, 0), we obtain the following observation.
Observation 2.2.For all integers n 0, N 1, there exists N * 1 such that for all integers Now we are ready to prove Theorem 1.7.The following is a restatement.
Theorem 2.3.Let F be a class of graphs and n 0 an integer.Then asdim(F +n ) = asdim(F).
A vertex-cover of a graph G is a subset S of V (G) such that G − S has no edge.As the class of edgeless graphs has asymptotic dimension 0, the following is a direct consequence of Theorem 2.3.
Observation 2.4.For any integer k, the class of all graphs with a vertex-cover of size at most k has asymptotic dimension 0.
By applying Lemma 2.1 with V (G) = Z and using the fact that a colouring of the empty graph has weak diameter 0, we obtain the following.
Observation 2.5.For any integers k 0, r 0, there exists an integer N * 1 such that for every graph G, and for all integers m 1,

Gluing along a tree
In this section, we prove one of the main technical results of this paper, from which we derive our results about the asymptotic dimension of classes of graphs of bounded treewidth, and of minor-closed families of graphs.This result roughly states that if every graph in a class G has a tree-decomposition with bounded adhesion where each bag belongs to some family F, then the asymptotic dimension of G is essentially that of F -or more precisely that of a class of graphs that can be obtained from the graphs of F by adding vertices and edges in a specific way.
A class F of graphs is hereditary if for every G ∈ F, every induced subgraph of G belongs to F. Theorem 3.1.Let C be a hereditary class of graphs, and let θ 1 be an integer.Let G be a class of graphs such that for every G ∈ G, there exists a tree-decomposition (T, X ) of G of adhesion at most θ, where X = (X t : t ∈ V (T )), such that C contains all graphs which can be obtained from any G[X t ] by adding, for each neighbour t of t in T , a set of pairwise non-adjacent new vertices whose neighbourhoods are contained in X t ∩ X t .Then asdim(G) max{asdim(C), 1}.
3.1.Sketch of the proof of Theorem 3.1.The proof of Theorem 3.1 follows a rather technical induction, whose precise statement is that of Lemma 3.2.For convenience of the reader, and in order to ease the understanding of the purpose of the incoming set-up, we begin by sketching the main steps of the proof.
Let G ∈ G and (T, X ) be a tree-decomposition of G as stated in Theorem 3.1.As (T, X ) has bounded adhesion, we can treat T as a rooted tree so that the bag of the root, denoted by X t * , has size at most the adhesion, up to creating a redundant bag if necessary.We shall prove a stronger statement: for every Z ⊆ N 3 G (X t * ), every precolouring c| Z on Z with at most m colours extends to an m-colouring of G with bounded weak diameter in G , by induction on the adhesion of (T, X ), and subject to this, induction on |V (G)|.(See Lemma 3.2 for a precise statement.) By first extending c| Z to N 3 G (X t * ), we may assume Hence the subgraph T 0 of T induced by the nodes whose bags intersect Z is a subtree of T containing t * .Let G 0 be the subgraph of G induced by the bags of the nodes in T 0 .Let U E be the set of edges of T with exactly one end in T 0 .For every e ∈ U E , let G e be the subgraph of G induced by the bags of the nodes in the component of T − e disjoint from T 0 .Then for each e ∈ U E , there is a partition are not far from each other in G e if and only if they are contained in the same part of P e .This can be done as |V (G 0 ) ∩ V (G e )| is bounded by the adhesion of (T, X ).
Note that Z is an (|X t * |, 3 )-centered set, and G 0 − Z has a tree-decomposition of smaller adhesion by the definition of G 0 and T 0 .So (G 0 − Z) has a colouring by induction.Hence the precolouring c Z on Z can be extended to G 0 by Lemma 2.1.However, it is troublesome to further extend the colouring to G , as no information about G e for e ∈ U E can be seen from G 0 and edges of G with ends in V (G 0 ) cannot be completely told from G 0 .To overcome this difficulty, we add "gadgets" to G 0 to obtain a graph H such that extending c Z from Z to H gives sufficient information about how to further extend it to G .The gadgets we add to form H are a vertex v Y for each e ∈ U E and each part Y ∈ P e , and the edges between v Y and Y .
However, H − Z possibly does not have a tree-decomposition of smaller adhesion, so the induction hypothesis cannot be applied to H − Z. Instead, we setup a more technical induction hypothesis to overcome this difficulty.This is the motivation of (η, θ, F, F )-constructions mentioned in Section 3.2.So we can extend c Z to an m-colouring of H by using this technical setting.
Note that no vertex in Z is in e∈U E G e .Then for each e ∈ U E , we colour vertices in Call the set of vertices coloured in this step Z 1 .Then we colour every uncoloured vertex in G e that has distance in G e at most from Z 1 colour 1.Call the set of vertices coloured in this step Z 2 − Z 1 , and let Z 2 contain Z 1 .Then we colour every uncoloured vertex in G e that has distance in G e at most from Z 2 colour 2. This ensures that no matter how we further colour the other vertices, every monochromatic component intersecting G 0 must be contained in V (G 0 ) ∪ Z 1 ∪ Z 2 ; and we can show that such monochromatic components have small weak diameter due to our definition of P e and v Y .
At this point, for every e ∈ U E , the vertices coloured in G e are contained in Hence for each e ∈ U E , we can extend this precolouring to an m-colouring of G e with bounded weak diameter in G e by induction, since G e has fewer vertices than G. Every monochromatic component not intersecting G 0 must be contained in G e for some e ∈ U E and hence has small weak diameter.This completes the sketch of the proof of Lemma 3.2 (and Theorem 3.1, which follows as a simple consequence).
3.2.Key lemma for proving Theorem 3.1.Let , N, m 1 be integers.Recall that a class F of graphs is (m, , N )-nice if for every G ∈ F, G is m-colourable with weak diameter in G at most N .
Let G be a graph, and let (T, X ) be a tree-decomposition of G, where X = (X t : t ∈ V (T )).For every S ⊆ V (T ), we define X S = t∈S X t ; when T is a subgraph of T , we write X T instead of X V (T ) .
A rooted tree is a directed graph whose underlying graph is a tree where all vertices have indegree 1, except one which has in-degree 0 and that we call the root.A rooted tree-decomposition of a graph G is a tree-decomposition (T, X ) of G such that T is a rooted tree.
We denote F +1 by F + .Let F and F be classes of graphs.Let η, θ be nonnegative integers with η θ.A rooted tree-decomposition (T, X ) of a graph G is called an (η, θ, F, F )-construction of G if it has adhesion at most θ and satisfies the following additional properties: • for every edge tt ∈ E(T ), if |X t ∩ X t | > η, then one end of tt has no child, say t , and the set X t − X t contains at most 1 vertex, and -F contains all graphs which can be obtained from G[X t ] by adding, for each child t of t in T , vertices whose neighbourhoods are contained in X t ∩ X t .(Note that this final property only applies to nodes t that have children, since otherwise the precondition "for each child t of t" is void.)We say that a graph G is (η, θ, F, F )-constructible if there exists an (η, θ, F, F )-construction of G.
For every rooted tree T , define I(T ) to be the set of nodes of T that have at least one child.
Let G be a graph and m > 0 an integer.Let S ⊆ V (G).Let c : S → [m] be a function.Let c be an m-colouring of G such that c (v) = c(v) for every v ∈ S. Then we say that c can be extended to c .
Recall that a class F of graphs is hereditary if for every G ∈ F, every induced subgraph of G belongs to F. Note that if F is hereditary, then so is F + .Lemma 3.2.For any integers 1, N 1, m 2, θ 0, there exists a function f * : N ∪ {0} → N such that the following holds.Let F and F be (m, , N )-nice hereditary classes.Let η be a nonnegative integer with η θ.Let G be an (η, θ, F, F )-constructible graph with an (η, θ, F, F )-construction (T, X ).Denote X by (X t : t ∈ V (T )).Let t * be the root of T .For every Proof.A visual summary of some of the notation introduced throughout the proof is depicted in Figure 1.Let 1, N 1, m 2, θ 0 be integers.By Observation 2.2, there exists an integer N F + (that depends only on N ) such that F + is (m, , N F + )-nice.Note that F ⊆ F + , so we may assume that N F + N .
We define the following.
Proof.Assume that η = 0. Note that for every component and it follows from the fact that (T, X ) is an (η, θ, F, F )-construction that one end of tt has no child, say t , and Henceforth we assume that we have proven the lemma for all instances for which (η, ) is lexicographically smaller, and assume η 1.
Claim 2. We may assume that G is connected.
and hence the induction hypothesis applies to C. So we may assume t * ∈ V (T C ).Then Z ∩ V (C) = ∅.Let T C be the rooted tree obtained from T C by adding a node t * C adjacent to the root of T C , where t * C is the root of T C .Let the bag at t * C be the set consisting of a single vertex in the intersection of V (C) and the bag of the root of T C .Then since η 1, we obtain an (η, θ, F, F )-construction of C with underlying tree T C .(Note that η 1 implies that t * C satisfies the last condition of being an (η, θ, F, F )-construction, since stars are the only graphs that can be obtained by adding new vertices to the bag of t * C , and η 1 implies that F contains stars.)Since so the induction hypothesis applies to C. ♦ So henceforth we may assume that G is connected.
Claim 3. We may assume that so by the induction hypothesis, c (and hence c Z ) can be extended to an m-colouring of G with weak diameter in G at most f * (η).Hence we may assume that A snapshot of the setting of Lemma 3.2 and the notations used in the proof.Note that by Claim 3 we know that we can assume that Z = N 3 G (X t * ), but we only include the original setting of this lemma in the picture.
For each v ∈ X t * , let T v be the subgraph of T induced by {t ∈ V (T ) : 1, right, where the edges of U E are depicted with thin lines).Note that for every vertex t ∈ V (T 0 ) incident with an edge e ∈ U E , the component of T − e disjoint from t is disjoint from T 0 .
For each e ∈ E(T ), define X e to be the intersection of the bags of the ends of e.Since the tree-decomposition (T, X ) has adhesion at most θ, |X e | θ for every e ∈ E(T ).For each e ∈ E(T ), let T e be the component of T − e disjoint from t * .Since G is connected and X t * = ∅, we may assume that X e = ∅ for every e ∈ E(T ), for otherwise X Te = ∅ so that we can delete T e from T to decrease |V (T )| and apply induction. Let and G e = G[X Te ] for every edge e ∈ U E (see Figure 1 for a visual summary of some of the notation introduced in this paragraph).By Claim 3, Z ∩ X t = ∅ for every t ∈ V (T 0 ), and Z ∩ X t = ∅ for every t ∈ V (T 1 ).
A naive plan is to first extend c Z to G 0 by using the induction hypothesis, and then further extend this to G 1 using the induction hypothesis on the graphs G e for e ∈ U E .An issue of this naive plan is that some edges of G with both ends in V (G 0 ) cannot be told from G 0 , so colouring G 0 without knowing those edges in G might make some monochromatic component in G contain arbitrarily many monochromatic components in G 0 .To overcome this difficulty as well as other potential issues, we add "gadgets" to G 0 to obtain a graph H such that extending c Z to H gives sufficient information about how to extend it to G .
For each e ∈ U E , we define a partition P e of X e such that each part of P e is a connected component of the subgraph of (G e ) 7 induced by X e .In other words, two vertices x, y ∈ X e are in the same part of P e if and only if there exists a sequence a 1 , . . ., a θ of (not necessarily distinct) elements of X e such that a 1 = x, a θ = y, and for every i ∈ [θ − 1], there exists a path in G e from a i to a i+1 of length at most 7 .
Define H to be the graph obtained from G 0 by adding, for each e ∈ U E and Proof.We first show that it is enough to give an (η − 1, θ, F, F )-construction of H − Z.By the induction hypothesis, this would imply that there exists an m-colouring c H of (H − Z) The remainder of the proof of this claim is devoted to showing the existence of an (η − 1, θ, F, F )-construction of H − Z.We start by showing the existence of an (η, θ, F, F )construction for H.
Define T to be the rooted tree obtained from T 0 by adding, for each e ∈ U E and Y ∈ P e , a node t Y adjacent to the end of e in V (T 0 ).Note that for every e ∈ U E , since X e = ∅, the end of e in V (T 0 ) has a child in both T and T .For each t ∈ V (T 0 ), define X t = X t ; for each t ∈ V (T ) − V (T 0 ), t = t Y for some e ∈ U E and Y ∈ P e , and we define Clearly, (T , X ) is a tree-decomposition of H of adhesion at most max e∈U E {θ, |X e |} = θ.For each tt ∈ E(T ), say t is a child of t, if tt ∈ E(T 0 ), then X t = X t , X t = X t , t has a child in both T and T , and t has a child in T if and only if t has a child in T (since X e = ∅ for every e ∈ U E ); if tt ∈ E(T 0 ), then t ∈ V (T 0 ) and t ∈ V (T 0 ), and |X t − X t | = 1.Hence for every tt ∈ E(T ), if |X t ∩ X t | > η, then one end of tt , say t , has no child, and by adding, for each child t of t in T , new vertices whose neighbourhoods are contained in X t ∩ X t .Therefore, (T , X ) is an (η, θ, F, F )-construction of H.
For every t ∈ V (T ), let X t = X t − Z.Let X = (X t : t ∈ V (T )).So (T , X ) is a tree-decomposition of H − Z of adhesion at most θ.Since for every t ∈ V (T 0 ), X t ∩ Z = ∅ and Z = N 3 G (X t * ), we know that for every e ∈ E(T 0 ), If η − 1 = 0, then let T = T and X = X ; otherwise, let t 0 be a node of T with X t 0 = ∅ closest to t * , let v 0 be a vertex in X t 0 , let T be the rooted tree obtained from T by adding a new node t * 0 adjacent to t * , where t * 0 is the root of T , and let X = (X t : t ∈ V (T )), where 0 and t is in the path in T between t * and t 0 , and X t = X t otherwise.
Then (T , X ) is a tree-decomposition of H − Z. Since (T , X ) is an (η, θ, F, F )construction of H, and G is connected, and F, F and F + are hereditary, (T , X ) is an Ge (Y ), and for every there exists a path P in G 1 from v to X e internally disjoint from X e of length at most i .Since P is internally disjoint from X e , P is a path in G e .Let y be the vertex in V (P ) ∩ X e .Let Y be the member of P e containing y.So v ∈ N i Ge (Y ).Let Y be any member of P e − {Y }.If v ∈ N i Ge (Y ), then there exists a walk in G e from Y to Y of length at most 2i 6 , so Y = Y by the definition of P e , a contradiction.Hence Ge v (Y v ).Let c H be the function from Claim 4. This assigns a colour to all vertices from V (H) and in particular all vertices from V (G 0 ).We now use the structure of H to extend this colouring of G 0 to G 1 .Let c 3 : So we may assume that |X e | η.We may also assume that X e = ∅, since if X e = ∅, then V (G e ) = ∅ (since G is connected and X t * = ∅).Define Q e to be the rooted tree obtained from T e by adding a node r e adjacent to the end of e in V (T e ), where r e is the root of Q e .Let W re = X e ; for every t ∈ V (T e ), let and F and F are hereditary, (Q e , W) is an (η, θ, F, F )-construction of G e .
Note that I(Q e ) = {r e } ∪ (I(T ) ∩ V (T e )).Note that every vertex that belongs to the shortest directed path in T containing t * and an end of e belongs to I(T ) − V (T e ).So |I(Q e )| |I(T )|, and equality holds only when t * is an end of e.If t * is an end of e, then since X e = ∅, and Z ⊇ X t * (by Claim 3), X t ∩ Z ⊇ X t ∩ X t * = ∅, where t is the end of e other than t * , so t ∈ V (T 0 ), a contradiction.Hence |I(Q e )| < |I(T )|.
Recall that Z e ⊆ N 3 Ge (X e ) = N 3 Ge (W re ).Hence by the induction hypothesis, c e can be extended to an m-colouring of G e with weak diameter in G e at most f * (η).♦ For every e ∈ U E , let c e be the m-colouring from Claim 6. Define To prove this lemma, if suffices to show that c has weak diameter in G at most f * (η).Let M be a c-monochromatic component of G .The rest of the proof is devoted to showing that M has weak diameter in G at most f * (η).
We may assume that for every e Proof.Assume that V (M ) ⊆ V (G e ) − Z 1 for some e ∈ U E .Then every edge in M ⊆ G is an edge in G e .So M is a c e -monochromatic component in G e .Hence the weak diameter in G e of M is at most f * (η).That is, for any vertices x, y in M , there exists a path P x,y in G e between x and y of length at most f * (η).Since G e ⊆ G, G e ⊆ G .So for any vertices x, y in M , P x,y is a path in G between x and y of length at most f * (η).Hence the weak diameter in G of M is at most f * (η).♦ In particular, Claim 8 shows that for all e ∈ U E , if M contains a vertex from V (G e ) − X e , then we can assume that it also contains a vertex from Z 1 ∩ (V (G e ) − X e ), since the distance between V (G e ) − Z 1 and X e is greater than .Let , there exists e ∈ B with x ∈ X Te ∩ Z 2 − X e , and there exists a unique Y x ∈ P e such that x ∈ N 2 Ge (Y x ).Moreover, we have . By Claim 9, there exists e ∈ B such that x ∈ X Te ∩ Z 2 − X e .So by Claim 5, there exists a unique Y x ∈ P e such that x ∈ N 2 Ge (Y x ).Furthermore, We further choose such a vertex x such that the distance in M between x and x is as small as possible.(Note that it is possible that x = x .)Let P be a shortest path in M ⊆ G between x and x .By the choice of x , V (P ) − {x } ⊆ V (M ) ∩ X Te ∩ Z 2 − Z 1 .Denote P by x 1 x 2 . . .x |V (P )| .So for every i ∈ [|V (P )| − 1], there exists a path in G e between x i and x i+1 with length in G e at most .By Claim 5, there exists a unique Y x ∈ P e such that x ∈ N Ge (Y x ), and for each v ∈ V (P ) − {x }, there exists a unique , then there exists a walk in G e from Y x i to Y x i+1 with length at most 2 + +2 5 , so Y x i = Y x i+1 by the definition of P e , a contradiction.So , define Y x to be the set mentioned in Claim 10.Let M be the graph obtained from M by identifying, for each e ∈ B and Y ∈ P e , all vertices Observe that since M is connected, M is also connected.Note that there is a natural injection from V (M ) to V (H) obtained by the identification mentioned in the definition of M .So we may assume Note that for any e ∈ U E , distinct vertices x, y ∈ X e and path P in G e between x and y internally disjoint from X e of length at most having at least one internal vertex, the ends of P are contained in the same part (say Y ) of P e , so there exists a path P = xv Y y in H of length two between x and y; since P has at least one internal vertex, the length of P is at most the length of P .Hence, for every path P in G of length at most between two distinct vertices in V (G 0 ), we can replace each maximal subpath P of P of length at least two whose all internal vertices are not in V (G 0 ) by P to obtain a walk P in H of length at most the length of P having the same ends as P .
Hence if xy is an edge of M with x, y ∈ V (G 0 ), then x, y ∈ V (M ), and since M ⊆ G , there exists a path P xy in G of length at most between x and y, so P xy is a walk in H of length at most between x and y, so xy ∈ E(H ).Now assume that there exists an edge xy of M with x ∈ V (G 0 ) and y ∈ V (G 0 ).Since y ∈ V (G 0 ), there exists y 0 ∈ V (M ) with v Yy 0 = y such that xy 0 ∈ E(M ), and there exists e y ∈ U E such that y 0 ∈ X Te y − X ey .Since M ⊆ G , there exists a path P xy in G of length at most between x and y 0 .Let y be the vertex in V (P xy ) ∩ X ey such that the subpath of P xy between y 0 and y is contained in G ey .Then y ∈ Y y 0 by Claim 5.So xP xy y y is a walk in H of length at most , where P xy is the subpath of P xy between x and y .So xy ∈ E(H ).
Hence every edge of M incident with a vertex of V (G 0 ) is an edge of H . Finally, assume that there exist distinct e 1 , e 2 ∈ B, Y 1 ∈ P e 1 and Y 2 ∈ P e 2 such that v Y 1 v Y 2 ∈ E(M ).So for each i ∈ [2], there exists x i ∈ V (M ) ∩ Z 2 ∩ X Te i − X e i by Claim 10 such that Y x i = Y i , and there exists a path P x 1 x 2 in G of length at most between x 1 and x 2 .For each i ∈ [2], let y i be the vertex in V (P x 1 x 2 ) ∩ X e i such that the subpath of P x 1 x 2 between x i and y i is contained in G e i .Then v Y 1 y 1 P x 1 x 2 y 2 v Y 2 is a walk in H of length at most , where P x 1 x 2 is the subpath of P x 1 x 2 between y 1 and y This proves E(M ) ⊆ E(H ), and hence M is contained in a c H -monochromatic component in H . ♦ In particular, the weak diameter of M in H is at most f 1 (f * (η − 1)) by Claim 4. We shall use this fact to bound the weak diameter in G of M .We now give a relation between the weak diameter in G and in H . Claim 12.For every pair of vertices u, v ∈ V (G 0 ), if P is a path in H between u and v, then there exists a walk P in G between u and v of length at most 7θ |E(P )|.
Proof.Let Q be a path in H such that there exists e ∈ U E such that Q is from X e to X e , internally disjoint from X e , and contains at least one internal vertex corresponding to a vertex in X Te − X e .Then Q has length two, and there exists Y Q ∈ P e containing both ends of Q.So there exists a path Q in G e between the ends of Q of length at most 7θ by the definition of Let P be a path in H between u and v. Consider each subpath P of P in which there exists e P ∈ U E such that P is from X e P to X e P , internally disjoint from X e P , and contains at least one internal vertex corresponding to a vertex in X Te P − X e P , and replace it with P as obtained in the above paragraph.We obtain a walk in G between u and v of length at most 7θ |E(P )|.♦ For a vertex u ∈ V (M ) − V (G 0 ) and a path P in M on at least 2 vertices having v Yu as an end, we let h P (u) be the neighbour of v Yu in P .Note that since all the vertices v Y are pairwise non-adjacent in H, it follows that h P (u) ∈ Y u .
Claim 13.For a vertex u ∈ V (M ) − V (G 0 ) and a path P in M on at least 2 vertices having v Yu as an end, there exists a path in G from u to h P (u) of length in G at most (7θ + 2) .
Proof.By Claim 10, there exists e ∈ U E such that u ∈ (X Te − X e ) ∩ N 2 Ge (Y u ), so there exists a path P u in G e from u to Y u with length at most 2 .Let u be the end of P u in Y u .Since Y u contains both u and h P (u), there exists a path in G from u to h P (u) of length at most 7θ by the definition of P e .So there exists a walk in G from u to h P (u) of length in G at most (7θ + 2) .♦ Claim 14.For every pair of vertices x, y ∈ V (M ), there exists a path in G between x and y with length in G at most f * (η).
Proof.For every vertex u ∈ V (M ), if u ∈ V (G 0 ), then define h 0 (u) = u; otherwise, define h 0 (u) = v Yu .Let x, y ∈ V (M ).We may assume that x = y, for otherwise we are done.By Claims 4 and 11, there exists a path P 0 in H between h 0 (x) and h 0 (y) of length in H at most f 1 (f * (η − 1)).So there exists a path P in H of length in H at most • f 1 (f * (η − 1)) between h 0 (x) and h 0 (y).
For every u ∈ {x, y}, if u ∈ V (G 0 ), then let h(u) = u; if u ∈ V (G 0 ), then let h(u) = h P (u).For every u ∈ {x, y}, let Q u be a shortest path in G from u to h(u).Note that if u ∈ V (G 0 ), then Q u has length 0; if u ∈ V (G 0 ), then by Claim 13, Q u has length in G at most (7θ + 2) .
Since h(x) and h(y) are in V (G 0 ), and the subpath of P between h(x) and h(y) is a path in H of length in G at most • f 1 (f * (η − 1)), by Claim 12, there exists a walk W in G between h(x) and h(y) of length in G at most 7θ ♦ By Claim 14, the weak diameter in G of M is at most f * (η).Therefore, the weak diameter in G of M is at most f * (η).This proves the lemma.

3.3.
Proof of Theorems 3.1 and 1.4.We are now ready to prove Theorem 3.1, which we restate here for the convenience of the reader.Theorem 3.1.Let C be a hereditary class of graphs, and let θ 1 be an integer.Let G be a class of graphs such that for every G ∈ G, there exists a tree-decomposition (T, X ) of G of adhesion at most θ, where X = (X t : t ∈ V (T )), such that C contains all graphs which can be obtained from any G[X t ] by adding, for each neighbour t of t in T , a set of new vertices whose neighbourhoods are contained in X t ∩ X t .Then asdim(G) max{asdim(C), 1}.
Proof.By Proposition 1.17, there exists a function f : N → N such that C is (asdim(C) + 1, , f ( ))-nice for every ∈ N. Define g : N → N to be the function such that for every Let G ∈ G.So there exists a tree-decomposition (T, X ) of G of adhesion at most θ, where X = (X t : t ∈ V (T )), such that C contains all graphs which can be obtained from any G[X t ] by adding, for each neighbour t of t in T , a set of new vertices whose neighbourhoods are contained in X t ∩ X t (in particular for every t ∈ V (T ), G[X t ] ∈ C).Let t 0 be a node of T with X t 0 = ∅, and let v 0 be a vertex in X t 0 .Let T be the rooted tree obtained from T by adding a new node t 0 adjacent to t 0 , where t 0 is the root of T .Let X t 0 = {v 0 }; for every t ∈ V (T ), let X t = X t .Let X = (X t : t ∈ V (T )).Then (T , X ) is a (θ, θ, C, C)construction of G.For every ∈ N, applying Lemma 3.2 by taking ( , N, m, θ, F, F , η, Z) = ( , f ( ), max{asdim(C), 1} + 1, θ, C, C, θ, ∅), G is (max{asdim(C), 1} + 1)-colourable with weak diameter in G at most g( ).
We can now prove Theorem 1.4, which we also restate here for convenience.Theorem 3.3.For any integer w, the class of graphs of treewidth at most w has asymptotic dimension at most 1.
Proof.Let F be the class of graphs of treewidth at most w, and let C be the class of graphs that have a vertex-cover of size at most w + 1.Note that C is a hereditary class and asdim(C) = 0 by Observation 2.4.
Note that for every graph G of treewidth at most w, there exists a tree-decomposition (T, X ) of G of adhesion at most w, where X = (X t : t ∈ V (T )), such that C contains all graphs which can be obtained from any G[X t ] by adding, for each neighbour t of t in T , a set of new vertices whose neighbourhoods are contained in X t ∩ X t (since X t is a vertex-cover of size at most w + 1 of such graphs).Hence by Theorem 3.1, asdim(F) max{asdim(C), 1} = 1.

Control functions and layerings
In this section we introduce some notation and state a result of Brodskiy, Dydak, Levin and Mitra [12] which is an extension of a result of Bell and Dranishnikov [6], and we will use it to derive a result about graph layering.4.1.Real projections and layerings.Given a metric space (X, d), and a real c > 0, a function f : X → R is c-Lipschitz if for any x, y ∈ X, |f (x) − f (y)| c • d(x, y) (such functions can be defined between any two metric spaces, but here we will only consider R as the codomain).When f is 1-Lipschitz, we call it a real projection of (X, d).
In the context of graphs, an interesting example of real projections comes from layerings.Recall that a layering L = (L i ) i∈N or L = (L i ) i∈Z of a graph G is an ordered partition of V (G) into (possibly empty) sets (called the layers), such that for any edge uv of G, u and v lie in the same layer or in a union of two consecutive layers (i.e. in L i ∪ L i+1 for some i).Note that a layering can also be seen as a function L : V (G) → N or L : V (G) → Z such that for any edge uv, |L(u) − L(v)| 1.In particular, a layering can be seen as a real projection by the triangle inequality.4.2.r-components and (r, s)-components.Let (X, d) be a metric space.Recall that a subset S ⊆ X is r-bounded if for any x, x ∈ S, d(x, x ) r.Given a subset A ⊆ X, two points x, x ∈ A are r-connected5 in A if there are points x 1 = x, x 2 , . . ., x = x in A, for some ∈ N, such that for any 1 i − 1, d(x i , x i+1 ) r.A maximal set of r-connected points in A is called an r-component of A. Note that these r-components form a partition of A. Observe that in an unweighted graph G, the 1-components of a subset U ⊆ V (G) of vertices are exactly the vertex-sets of the connected components of G[U ], the subgraph of G induced by U .(See Figure 2(a), for an example of 2-components of a subset A of vertices.)Recall that D X : R + → R + is an n-dimensional control function for X if for any r > 0, X 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.Observe that D X : R + → R + is an n-dimensional control function for X if and only if for any r > 0, X is a union of n + 1 sets whose r-components are D X (r)-bounded.In this section it will be convenient to work with this definition of control functions.
Let (X, d) be a metric space and let f : X → R be a real projection of X.A subset A ⊆ X is said to be (r, s)-bounded with respect to f (and d) if for all x, x ∈ A we have d(x, x ) r and |f (x) − f (x )| s (when f is clear from the context we often omit "with respect to f ", and similarly for d).Two vertices x, x of A are (r, s)-connected in A if there are vertices x 1 = x, x 2 , . . ., x = x in A, for some ∈ N, such that for any 1 i − 1, {x i , x i+1 } is (r, s)-bounded (i.e.d(x i , x i+1 ) r and |f (x) − f (x )| s).A maximal set of (r, s)-connected vertices in A is called an (r, s)-component of A. Note that these (r, s)-components form a partition of A.
Note that by the definition of a real projection, any r-bounded set is also (r, r)-bounded, and similarly being r-connected is equivalent to being (r, r)-connected, and an r-component is the same as an (r, r)-component.Observe that in Figure 2(b), the vertices u and v are 4-connected in {u, v} but they are not (4, 3)-connected in {u, v} with respect to f , where f is defined to be the function f (x) = d(u, x) for any x.This shows that the notions of r-components and (r, s)-components differ when s < r.

4.3.
Control functions for real projections.We have seen above that D X : R + → R + is an n-dimensional control function for a metric space (X, d) if for any r > 0, X is a union of n + 1 sets whose r-components are D X (r)-bounded.It will be convenient to extend this definition to real projections, as follows.For a metric space (X, d) and a real projection f : X → R, we say that D f : R + × R + → R + is an n-dimensional control function for f if for any real numbers r, S > 0, any (∞, S)-bounded subset A ⊆ X is a union of n + 1 sets whose r-components are D f (r, S)-bounded.We say that the control function D f is linear if there are constants a, b, c > 0 such that D f (r, S) ar + bS + c for any r, S > 0. We say that the control function D f is a dilation if there are constants a, b > 0 such that D f (r, S) ar + bS for any r, S > 0.
The following is a special case of a combination of Proposition 4.7 in [12] and Theorem 4.9 in [12]. 6heorem 4.1 ([12]).Let (X, d) be a metric space and f : X → R be a real projection of (X, d).
If f admits an n-dimensional control function D f , then X admits an (n + 1)-dimensional control function D X such that D X (r) only depends on r, D f and n.Moreover, if D f is linear then D X is also linear; if D f is a dilation then D X is also a dilation.

4.4.
Intrinsic control of real projections.Let (G, φ) be a weighted graph, and let A be a subset of V (G).The weighted subgraph of (G, φ) induced by A is the weighted graph d (G,φ) ) with respect to L and d (G,φ) , A is a union of n + 1 sets whose r-components are D L (r, S)-bounded, where the definitions of r-components and D L (r, S)-bounded are with respect to the metric Since every (unweighted) graph can be viewed as a weighted graph whose weight on each edge is 1, the definition for intrinsic control function is also defined for graphs G and corresponding function L.
As before, we say that an intrinsic control function D L for a real projection is linear if there are constants a, b, c > 0 such that D L (r, S) ar + bS + c, for any real numbers r, S > 0. We also say that D L is a dilation if there are constants a, b > 0 such that D L (r, S) ar + bS, for any real numbers r, S > 0.
We now prove that intrinsic control functions can be transformed into (classical) control functions.
Lemma 4.2.Let (G, φ) be a weighted graph.Let L : V (G) → R be a real projection of (V (G), d (G,φ) ) such that L admits an n-dimensional intrinsic control function D. Then D (r, S) := D(r, S + 2r) for any r, S > 0 is an n-dimensional control function for L. In particular, if D is linear, then D is also linear, and if D is a dilation, then D is a dilation.
Proof.Note that we may assume that D is a non-decreasing function.Let S > 0 be a real number, and let X be an (∞, S)-bounded subset of vertices of G with respect to L and d (G,φ) .It follows that there is some a > 0 such that for any x ∈ X, L(x) ∈ [a, a + S].Fix some r > 0, and denote by X + the preimage of the interval [a − r, a + S + r] under L. Note that X + ⊇ X and X + is a maximal (∞, r )-bounded subset of V (G) with respect to L and d (G,φ) for some real number r S + 2r.Let Consider now an r-component C of U i , for some 1 i n + 1 (where the distance in the definition of r-components is with respect to the metric d (G,φ) ).For any u, v ∈ C, there are u 0 = u, u 1 , . . ., u t = v in U i , for some t ∈ N, such that for any 0 j t − 1, d (G,φ) (u j , u j+1 ) r.It follows that u j and u j+1 are two vertices in X connected by a path P j (of length at most r) in G, so V (P j ) ⊆ X + , and thus u j and u j+1 are also connected by ).Hence, u j and u j+1 lie in the same r-component of U + i (where the distance in the definition of r-component is with respect to d H + ).Since all r-components of U + i (with respect to d H + ) are D(r, S + 2r)-bounded with respect to d H + , they are also D(r, S + 2r)-bounded with respect to d G .This shows that D (r, S) := D(r, S + 2r) is an n-dimensional control function for L. 4.5.Layerability.In this section, we will consider graph layerings (or more generally real projections) such that any constant number of consecutive layers induce a graph from a "simpler" class of graphs.For instance, any d-dimensional grid has a layering in which each layer induces a (d − 1)-dimensional grid, and moreover any constant number of consecutive layers induce a graph that is a d-dimensional grid where the height in one dimension is bounded and hence is significantly simpler than a general d-dimensional grid.
Given a class C of weighted graphs and a sequence of classes L = (L i ) i∈N of weighted graphs with L 1 ⊆ L 2 ⊆ • • • , we say that C is L-layerable if there is a function f : R + → N such that any weighted graph (G, φ) from C has a real projection L : V (G) → R such that for any S > 0, any maximal (∞, S)-bounded set in (G, φ) with respect to L and d (G,φ) induces a weighted graph from L f (S) .If there is a constant c > 0 such that f (S) max{cS, 1} for any S > 0, we say that C is c-linearly L-layerable.Proof.For any i 1, let D i be an n-dimensional control function for the graphs in L i .Since L 1 ⊆ L 2 ⊆ • • • , we may assume that for every x, D 1 (x) D 2 (x) • • • .Since C is L-layerable, there exists a function f : R + → N such that for any weighted graph (G, φ) ∈ C, there exists a real projection L (G,φ) : V (G) → R such that for any S > 0, any maximal (∞, S)-bounded set in (G, φ) with respect to L (G,φ) and d (G,φ) induces a weighted graph in L f (S) .
Note that D(r, S)

Layered treewidth and minor-closed families
We now apply the tool developed in the previous section to prove that graphs of bounded layered treewidth have asymptotic dimension at most 2 (Theorem 1.5).
Lemma 5.1.For any integer w > 0, the class of graphs of layered treewidth at most w is (L i ) i∈N -layerable, where for every i ∈ N, L i is the class of graphs of treewidth at most i.
Proof.Let w > 0. Let f : R + → N be the function such that f (x) = wx for every real x.Let G be a graph with layered treewidth at most w.So it has a tree-decomposition (T, X ) and a layering V = (V 1 , V 2 , . . . ) such that each bag of (T, X ) intersects each layer of V in at most w vertices.Note that for any > 0 and any set U which is a union of at most consecutive layers, the intersection of U and any bag of (T, X ) has size at most w , so the treewidth of For any > 0, any (∞, )-bounded set with respect to L and d G is a subset of a union of at most consecutive layers, so it induces a graph in L f ( ) .Therefore, F is (L i ) i∈N -layerable.Now we are ready to prove Theorem 1.5, which we restate here.Theorem 5.2.For any integer w > 0, the class of graphs of layered treewidth at most w has asymptotic dimension at most 2.
Proof.Let F be the class of graphs of layered treewidth at most w.For every integer i > 0, let L i be the class of graphs of treewidth at most i.By Theorem 3.3, asdim(L i ) 1 for each i ∈ N. By Lemma 5.1, F is (L i ) i∈N -layerable.So by Theorem 4.3, asdim(F) 2.
To prove Theorem 1.1, we need the following lemma.Lemma 5.3.Let p > 0 be an integer.For every integer x > 0, let F x be the class of graphs of layered treewidth at most x.Let W be the class of graphs such that for every G ∈ W, G can be obtained from a graph G ∈ F +p p by adding new vertices and edges incident with these vertices, where the neighbourhood of each new vertex is contained in a clique in G .Then W ⊆ F +p p+1 .
Proof.Let G ∈ W. So there exists H ∈ F +p p such that G can be obtained from H by adding new vertices whose neighbourhoods are contained in cliques of H. Since H ∈ F +p p , there exists Since H − Z ∈ F p , there exist a layering (V 1 , V 2 , . . . ) of H − Z and a tree-decomposition (T, X = (X t : t ∈ V (T )) of H − Z such that the intersection of any V i and any X t has size at most p.
For each integer i > 0, define Let T be the tree obtained from T by adding, for every Let i > 0 be an integer, and let Therefore, the layered treewidth of We can now prove Theorem 1.1.The following is a restatement.
Theorem 5.4.For any graph H, the class of H-minor free graphs has asymptotic dimension at most 2.
Proof.Let F be the class of H-minor free graphs.For every integer x > 0, let F x be the class of graphs of layered treewidth at most x.By [40, Theorem 1.3] and [18,Theorem 20], there exists an integer p > 0 such that for every graph G ∈ F, there exists a tree-decomposition (T, X ) of G of adhesion at most p such that for every t ∈ V (T ), the torso 7 at t belongs to F +p p .Let W be the class of graphs such that for every G ∈ W, G can be obtained from a graph G ∈ F +p p by adding new vertices and edges incident with these vertices, where the neighbourhood of each new vertex is contained in a clique in G .By Lemma 5.3, W ⊆ F +p p+1 .Note that F +p p+1 is closed under taking subgraphs.Hence for every G ∈ F, there exists a tree-decomposition (T, X ) of G of adhesion at most p, where X = (X t : t ∈ V (T )), such that for every t ∈ V (T ), F +p p+1 contains all graphs which can be obtained from any G[X t ] by adding, for each neighbour t of t in T , a set of new vertices whose neighbourhoods are contained in X t ∩ X t .
7 Given a tree-decomposition (T, X ) of G, where X = (Xt : t ∈ V (T )), the torso at t is the graph obtained from G[Xt] by adding edges such that Xt ∩ X t is a clique for each neighbour t of t in T .

Assouad-Nagata dimension of bounded layered treewidth graphs
A natural question is whether the asymptotic dimension can be replaced by the Assouad-Nagata dimension in Theorems 1.1, 1.4, or 1.5.We now show that the answer to this stronger question is negative in the case of layered treewidth.That is, Theorem 1.5 cannot be extended by replacing asymptotic dimension by Assouad-Nagata dimension.
Given a graph G and an integer k 0, the k-subdivision of G, denoted by G (k) , is the graph obtained from G by replacing each edge of G by a path on k + 1 edges.We start with the following simple observation.Lemma 6.1.Let k 0 and n 0 be integers.Let c > 0 be a real number.Let D : R + → R + with D(r) cr for every r > 0. Let G be a graph.If D is an n-dimensional control function of G (k) , then the function f (x) := cx for every x > 0 is an n-dimensional control function of G.
Proof.Fix some real r > 0. Let U 1 , . . ., U n+1 be subsets of Note that Lemma 6.1 does not assume any independence between k and G, and in particular we will later apply it with k = |E(G)|.
Recall that a graph is 1-planar if it has a drawing in the plane so that each edge contains at most one edge-crossing (see [37] for more details on k-planar graphs).Note that Corollary 1.6 directly implies that the class of 1-planar graphs has asymptotic dimension 2. We now prove that the analogous result does not hold for the Assouad-Nagata dimension.Lemma 6.2.There is no integer d such that the class of 1-planar graphs has Assouad-Nagata dimension at most d.
Proof.Take any family F of graphs of unbounded Assouad-Nagata dimension (for instance, grids of increasing size and dimension).For each graph G ∈ F, let Note that G * is 1-planar (this can be seen by placing the vertices of G in general position in the plane, joining adjacent vertices in G by straight-line segments, and then subdividing each edge at least once between any two consecutive crossings involving this edge).Define F * := {G * | G ∈ F}, and note that all the graphs of F * are 1-planar.
Suppose to the contrary that there exists an integer d such that the class of 1-planar graphs has Assouad-Nagata dimension at most d.Then F * has Assouad-Nagata dimension at most d.So there exists a d-dimensional control function D of all graphs in F * such that D is a dilation.Hence there exists a constant c > 0 such that D(r) cr for every r > 0, and for every Note that [16,Theorem 3.1] implies that every 1-planar graph has layered treewidth at most 12.So Lemma 6.2 implies a weaker version of Theorem 1.12 that replaces the number 1 by 12. Inspired by a comment of an anonymous referee of an earlier version of this paper, the number 12 can be dropped to 1 by using a standard argument about quasi-isometry and [11, Lemma 3].
Two metric spaces (X, d X ) and (Y, d Y ) are quasi-isometric if there is a map f : X → Y and constants 0, λ 1, and C 0 such that for any y ∈ Y there is x ∈ X such that d Y (y, f (x)) C, and for every It is not difficult to check that the definition is symmetric.Moreover, if for every r > 0, X has a cover by n sets whose r-components are D X (r)-bounded and there exists a map f : X → Y as above, then for every r > 0, Y has a cover by n-sets whose r-components are D Y (r)-bounded, where D Y only depends on D X and the constants λ, , and C in the definition of f .Moreover, D X is linear if and only if D Y is linear.This implies that asymptotic dimension (of linear type) is invariant under quasi-isometry.Moreover, if each member of a family X of metric spaces is quasi-isometric to some metric space in a family Y of metric spaces, with uniformly bounded constants λ, , and C in the definition of the quasi-isometry map, then asdim X asdim Y, and the same holds for the asymptotic dimension of linear type.
We can now derive the following, which is a restatement of Theorem 1.12.Corollary 6.3.There is no integer d such that the class of graphs of layered treewidth at most 1 has asymptotic dimension of linear type at most d.
Proof.Let Y be the class of graphs of layered treewidth at most 1.Suppose to the contrary that there exists an integer d such that Y has asymptotic dimension of linear type at most d.
Let X be the class of graphs of layered treewidth at most 12.By [11, Lemma 3(a)] (with k = 12), for every graph G with layered treewidth at most 12, G can be made into a graph G with layered treewidth at most 1 by replacing each edge by a path on at most 24 vertices.So the mapping ι : V (G) → V (G ) with ι(v) = v for every v ∈ V (G) satisfies that for every y ∈ V (G ), there exists x ∈ V (G) such that d G (y, ι(x)) 11, and for every x 1 , x 2 ∈ V (G), . That is, every member of X is quasi-isometric to a member of Y with uniformly bounded constants.Since Y has asymptotic dimension of linear type at most d, so does X .
Hence X has a d-dimensional control function D defined by D(x) := cx + c for some real number c > 0. Since every edge has weight 1, the function f (x) := 2cx is a d-dimensional control function for X .Hence the class of graphs of layered treewidth at most 12 has Assouad-Nagata dimension at most d.By [16, Theorem 3.1], every 1-planar graph has layered treewidth at most 12, and so the Assouad-Nagata dimension of the class of 1-planar graphs is at most d.This contradicts Lemma 6.2.
7. K 3,p -minor free graphs 7.1.Terminology.Let (G, φ) be a weighted graph.For two subsets A, B of V (G), we define We say that a subset S of vertices of G is connected if G[S] is connected.7.2.Fat minors.An equivalent way to define minors is the following: a (weighted or unweighted) graph G contains a graph H as a minor if V (G) contains |V (H)| vertex-disjoint subsets {T v | v ∈ V (H)}, each inducing a connected subgraph in G, and such that for every edge uv in H, T u and T v are connected by an edge in G.We will also need the following interesting metric variant of minors: for some integer q 1, a weighted graph (G, φ) contains a graph H as a q-fat minor if • each subset T v induces a connected subgraph of G; • any two sets T u and T v are at distance at least q apart in (G, φ); • for every edge uv in H, T u and T v are connected by a path P uv (of length in (G, φ) at least q) in G, such that for any pair of distinct edges uv and xy of H (possibly sharing a vertex), the paths P uv and P xy are at distance at least q in (G, φ), and for any edge uv in H and any vertex w distinct from u and v, P uv is at distance at least q from T w in (G, φ).Note that if (G, φ) contains H as a q-fat minor, then G contains H as a minor.An example of q-fat K 3 -minor is depicted in Figure 3, where the last property (stating that sets T w are far from paths P uv ) is not mentioned explicitly, for the sake of readability.
For a real interval I, we say a weighted graph is an I-weighted graph if the weight of each edge is in I.Note that for (0, 1]-weighted graphs, the assumption that two sets A and B are at distance at least q implies that any path connecting A and B contains at least q vertices.≥ q ≥ q ≥ q ≥ q Figure 3.A q-fat K 3 -minor. In this section, we will prove a technical lemma on graphs with no q-fat K 2,p -minors, generalising a result of Fujiwara and Papasoglu (Theorem 3.1 in [21]) about similar objects called "thetas" in their Lemma 7.1.Let p 2 and q 1 be integers and r > 0, κ > 0 be real numbers.Let (G, φ) be a (0, κ]-weighted graph.If (G, φ) does not contain the complete bipartite graph K 2,p as a q-fat minor, then (G, φ) has a cover by two sets whose r-components are (5r + 9q + 9κ)p-bounded.
Proof.Let (G, φ) be a (0, κ]-weighted graph with no q-fat K 2,p -minor.We may assume that G is connected, for otherwise we can consider each weighted connected component of (G, φ) separately.We fix a root vertex v ∈ V (G), and recall that an annulus is a set of the form A(a, b) = {u ∈ V (G) | a d (G,φ) (u, v) < b} for some 0 < a < b.For any integer k 1, let A k := A(kr, (k + 1)r).For any integer k, we say that the annuli A k and A k+1 are consecutive.
We define C 0 := {A k | k = k 0 + i, i 0 is an even integer} and define C 1 := A 0 ∪ {A k | k = k 0 + i, i 0 is an odd integer}.Note that these two sets clearly cover V (G), so we only need to show that each r-component in C 0 or C 1 is (5r + 9q + 9κ)p-bounded.By the definition of the annuli A k , observe that each r-component of C 0 or C 1 is contained in A 0 or in some annulus A k with k k 0 .Since A 0 is (5r + 9q + 9κ)p-bounded, it thus suffices to show that for any k k 0 , each r-component of A k is also (5r + 9q + 9κ)p-bounded.
For each 1 i p, let P i be a shortest path in (G, φ) from x ι(i) to the root v.For each 1 i − 1, let R i be a shortest path in (G, φ) from x i to x i+1 .Note that each R i has length in (G, φ) at most r.We define the following (see also Figure 4 for an illustration).
• Let A be the set of vertices in 1 i p V (P i ) with distance in (G, φ) at most (k − 1)r − (3q + 3κ) from v (note that since k k 0 , we have kr r + 3q + 3κ, so v ∈ A and hence A = ∅).• Let D be the set of vertices in 1 i p V (P i ) with distance in (G, φ) at least (k − 1)r from v.
Note that A and B induce connected subgraphs of G. Since the vertices x 1 , x 2 , . . ., x are in A k , they are at distance at least kr from v. Since each R i has length at most r, B is at distance in (G, φ) at least (k − 1)r from v and thus at distance in (G, φ) at least (k − 1)r − ((k − 1)r − (3q + 3κ)) = 3q + 3κ from A. Since each P i is a shortest path between v and x ι(i) , for every x ∈ V (P i ), we have For each 1 i p, • let a i be the vertex in P i such that the distance in (G, φ) between a i and A is at most q + κ, and subject to this, the subpath of P i between a i and V (P i ) ∩ V (B) is as short as possible, • let b i be the vertex in the subpath of P i between a i and V (P i ) ∩ B such that the distance in (G, φ) between b i and B is at most q + κ, and subject to this, the subpath of P i between b i and V (P i ) ∩ V (B) is as long as possible, • let Q i be the subpath of P i between a i and b i (see again Figure 4 for an illustration), The construction of a q-fat K 2,p -minor with p = 3 in the proof of Lemma 7.1.The figure is for illustrating the notation only and should not reflect any level of generality: for instance r can be either much larger or much smaller than q + κ, and the vertices in A or B closest to a i or b i are not necessarily in P i .
• let P A i be a path in (G, φ) from A to a i whose length in (G, φ) equals the distance in (G, φ) between A and a i , and • let P B i be a path in (G, φ) from B to b i whose length in (G, φ) equals the distance in (G, φ) between B and b i .Note that each P A i and P B i have length at most q + κ.Since A and B are at distance at least 3q + 3κ apart in (G, φ), the length of Q i is at least (3q + 3κ) − 2(q + κ) q + κ, so Q i contains at least 2 edges and least one internal vertex.
We shall prove that the vertex sets {A, B, V (Q i ) : 1 i p} together with the connecting paths {P A i , P B i : 1 i p} form a q-fat K 2,p -minor.Since (G, φ) is (0, κ]-weighted, by the minimality of the length of the subpath of P i between a i and V (B) ∩ V (P i ), the distance in (G, φ) between A and V (Q i ) is at least (q + κ) − κ = q; similarly, the distance in (G, φ) between B and V (Q i ) is at least q.In particular, {A, B, V (Q i ) : 1 i p} is a collection of pairwise disjoint sets inducing connected subgraphs of G.
By the definition of A, we know that for each 1 i p and r−(3q+3κ)) = 2r + 3q + 3κ.If there exist 1 i < j p such that the distance in (G, φ) between some vertex x in Q i and some vertex y in Q j is less than 3q + 3κ, then d (G,φ) (x ι(i) , x ι(j) ) d (G,φ) (x ι(i) , x) + d (G,φ) (x, y) + d (G,φ) (y, x ι(j) ) < (2r + 3q + 3κ) + (3q + 3κ) + (2r + 3q + 3κ) = 4r + 9q + 9κ, a contradiction.So any two distinct paths Q i , Q j are at distance at least 3q + 3κ in (G, φ).This implies that {A, B, V (Q i ) : 1 i p} is a collection of sets with pairwise distance in (G, φ) at least q.Furthermore, this together with the fact that each P A i and P B i has length at most q + κ imply that for distinct i, j, the distance between For any X ∈ {A, B} and Y ∈ {V (P A i ), V (P B i ) : 1 i p} with X ∩ Y = ∅, the distance between X and Y is at least 2q + 2κ, for otherwise, since the length of each of P A i and P B i is at most q + κ, the distance between X and Y is less than 3q + 3κ, a contradiction.This together with the fact that each P A i and P B i has length at most q + κ imply that for any (not necessarily distinct) i, j, the distance between P A i and P B j is at least (2q + 2κ) − (q + κ) q.This shows that {A, B, V (Q i ) : 1 i p} together with {P A i , P B i : 1 i p} form a q-fat K 2,p -minor.This contradiction completes the proof.
Note that a graph containing a q-fat K 2,2 -minor also contains a chordless cycle of length at least 4q.It follows that graphs without chordless cycles of length at least 4q have no q-fat K 2,2 -minor, so by Lemma 7.1 (or directly by Theorem 3.1 in [21]), for any 4 the class of graphs without chordless cycles of length at least has asymptotic dimension 1 (note that the asymptotic dimension is at least 1 because this class contains all trees).Lemma 7.2.Let H be a graph.Let H * be the graph obtained from H by adding a new vertex adjacent to all other vertices.Let (G, φ) be a weighted graph such that G has no H * minor.For any vertex v ∈ V (G), and any real numbers 0 < s < t and q > 2(t − s), the weighted subgraph of (G, φ) induced by the vertices {u ∈ V (G) | s d (G,φ) (u, v) t} does not contain H as a q-fat minor.
Proof.Let v be a fixed vertex of G.
Let (G , φ ) be the weighted subgraph of (G, φ) induced by B. Assume for the sake of contradiction that (G , φ ) contains H as a q-fat minor.Let {T u | u ∈ V (H)} and {P uw | uw ∈ E(H)} be as in the definition of a q-fat minor.For each u ∈ V (H), let P u be a path in G from T u to v such that the length in (G, φ) of P u equals d (G,φ) (T u , {v}), and let P + u be the maximal subpath of P u containing the vertex in V (P u ) ∩ T u and contained in G[B].Note that the length in (G , φ ) of P + u is at most t − s.For any u ∈ V (H), set T + u := T u ∪ V (P + u ).Each of the sets T + u is connected and disjoint from A, and two sets T + u and T + w are at distance in (G , φ ) at least q − 2(t − s) > 0 (in particular the sets are pairwise disjoint).For each u ∈ V (H), since the length in (G, φ) of P u is d (G,φ) (T u , {v}) t, V (P u ) ⊆ A ∪ B, so there exists an edge of P u between A and T + u .For any edge uw in H, let P + uw be a minimum subpath of P uw between T + u and T + w .Since each path P uw is at distance in (G , φ ) at least q − (t − s) > t − s from all the sets T x with x ∈ {u, w}, the subpaths P + uw are disjoint from all sets T + x with x ∈ {u, w}.Moreover, since the paths in {P uw : uw ∈ E(H)} are pairwise vertexdisjoint and disjoint from A, the paths in {P + uw : uw ∈ E(H)} are also pairwise vertex-disjoint and disjoint from A. It follows that G contains H * as a minor, where A is the connected subset corresponding to the vertex in V (H * ) − V (H), which is a contradiction.7.3.K 3,p -minor free graphs.Recall that Fujiwara and Papasoglu [21] proved that unweighted planar graphs have Assouad-Nagata dimension at most 3.We now prove that the dimension can be reduced to 2, and the class of graphs can be extended to all graphs avoiding K 3,p as a minor, for any fixed integer p 1. Moreover the result below holds for weighted graphs (this will be used to derive our result on Riemannian surfaces in Section 8).
The following observation is obvious.
Observation 7.3.Let (G, φ) be a weighted graph and let (G , φ ) be obtained from (G, φ) by subdividing an edge e of G once (i.e.replacing e by a path with two edges) and assigning weights summing to φ(e) to the two newly created edges, while all the other edges of G retain their subsets U 1 , U 2 , . . ., U n+1 of X with n+1 i=1 U i = X such that for each 1 Let S be an r-component of U i (in the metric d X ) for some 1 i n + 1.Note that S is contained in a k r r-component of U i (in the metric k r d X ).So the weak diameter in (X, k r d X ) of S is at most f (k r r) = f (1).
Let u, v be two points in S. Since the weak diameter in (X, k r d X ) of S is at most f (1), k r d X (u, v) f (1).Hence d X (u, v) f (1)  kr = f (1)r = g(r).Therefore, the weak diameter in (X, d X ) of S is at most g(r).That is, U 1 , U 2 , . . ., U n+1 are subsets of X with n+1 i=1 U i = X such that for each 1 i n + 1, every r-component (in the metric d X ) of U i is g(r)-bounded (in the metric d X ).Hence g is an n-dimensional control function of F.
We can now prove Theorem 1.9.The following is a restatement.Corollary 7.6.For any integer p 1, the class of K 3,p -minor free weighted graphs has Assouad-Nagata dimension at most 2.
Proof.It immediately follows from Lemmas 7.4 and 7.5.

Surfaces
8.1.Graphs on surfaces.Recall that in this paper, a surface is a non-null connected 2-dimensional manifold without boundary.
A compact surface can be orientable or non-orientable.The compact orientable surface of genus h is obtained by adding h 0 handles to the sphere; while the compact non-orientable surface of genus k is formed by adding k 1 cross-caps to the sphere.By the Surface Classification Theorem, any compact surface is one of these two types (up to homeomorphism).The Euler genus of a compact surface Σ is defined as twice its genus if Σ is orientable, and as its non-orientable genus otherwise.
A compact surface with boundary of Euler genus g is obtained from a compact surface of Euler genus g by deleting a finite number of open disks bounded by pairwise disjoint contractible simple closed curves on the surface.
In this section, all the non-compact surfaces we consider are metrisable (or equivalently triangulable).The Euler genus of a non-compact surface S is the supremum Euler genus of all compact subsurfaces (with boundary) of S (note that the Euler genus of a non-compact surface might be infinite).There also exists a classification theorem for non-compact surfaces, but we will not need it in this paper (see Section 3.5 in [33] for more details about non-compact surfaces).
Corollary 8.1.For any integer g 0, the class of finite or infinite weighted graphs embeddable in a surface of Euler genus at most g has Assouad-Nagata dimension at most 2.
Proof.By Theorem A.2, we can restrict ourselves to finite weighted graphs, and we can thus assume that the surface under consideration is compact (with boundary).For any integer g 0, K 3,2g+3 cannot be embedded in a compact surface of Euler genus at most g (with or without boundary) by a simple consequence of Euler's Formula (see Proposition 4.4.4 in [33]).Since the class of graphs embeddable in a surface of Euler genus at most g is closed under taking minors, every graph embeddable in a surface of Euler genus at most g is K 3,2g+3 -minor free.Therefore, this corollary follows as an immediate consequence of Corollary 7.6.

8.2.
From graphs on surfaces to Riemannian surfaces.A Riemannian manifold M is a manifold M together with a metric defined by a scalar product on the tangent space of every point.Recall that a surface is a non-null connected 2-dimensional manifold without boundary.So a Riemannian surface S is a surface S together with a metric defined by a scalar product on the tangent space of every point.An important property that we will need is that for any point p ∈ S, there is a small open neighbourhood N containing p that is strongly convex, i.e. any two points in N are joined by a unique shortest path.A Riemannian surface S with metric d is complete if the metric space (S, d) is complete.By the Hopf-Rinow theorem, in a complete Riemannian surface S, subsets that are bounded and closed are compact, and for any two points p, q ∈ S, there is a length-minimising geodesic connecting p and q (i.e. a path of length d(p, q)).Note that compact Riemannian surfaces are also complete.For more background on Riemannian surfaces, interested readers are referred to the standard textbook [44].
The following result appears to be well known in the area.For instance it can be deduced from the work of Saucan [43].Here we include a simple proof (suggested to us by Gaël Meignez) in dimension 2 for completeness of the paper.Lemma 8.2.Let S be a complete Riemannian surface with distance function d.Then there is a finite or countable locally finite 8 infinite (0, 1]-weighted graph (G, φ) embedded in S such that any point of S is at distance at most 2 from a vertex of G in S and for any vertices x, y ∈ V (G), d(x, y) d (G,φ) (x, y) 5d(x, y) + 2.
Proof.We consider an inclusion-wise maximal set P of points of S that are pairwise at distance at least 1  5 apart in (S, d).A simple area computation shows that any bounded region R of S contains a finite number of points of P (by compactness, the region has bounded area and it follows from the Bertrand-Diguet-Puiseux Theorem [9,14,38] that there is a uniform lower bound on the areas of the balls of radius 1  10 centered in P ∩ N 1/10 (R), while these balls are pairwise disjoint), and thus it follows that P is countable.By maximality of P , the open balls of radius 25 centered in P cover S. Let G be the graph with vertex-set P , in which two points p, q ∈ P are adjacent if their closed balls of radius 1  2 intersect.For each such pair p, q, we join p to q by a shortest path (of length d(p, q) 1) on the surface S. Note that any two such shortest paths intersect in a finite number of points and segments (since otherwise we could find an arbitrarily small neighbourhood containing two points joined by two distinct shortest paths, contradicting the property that sufficiently small neighbourhoods are strongly convex), and each such shortest path can intersect only finitely many other such shortest paths.For each intersection point between two paths and each end of an intersecting segment between the paths, we add a new vertex to G .Let G be the resulting graph (where two vertices of G are adjacent if they are consecutive on some shortest path between vertices of G ).By definition, G is countable and locally finite, and properly embedded in S. Note that each edge e of G corresponds to a shortest path between the two endpoints of e in S (we denote the length of this shortest path by e ).So (G, φ) is a weighted graph, where φ maps each edge e of G to e .Note that by definition, all the weights are in the interval (0, 1].For any two vertices p, q in G, we clearly have d(p, q) d (G,φ) (p, q).Consider now a length-minimising geodesic γ between p and q in S, and take k 5d(p, q) + 2 points p 1 , . . ., p k (in this order) on γ, with p 1 = p, p k = q, and such that d(p i , p i+1 ) 1  5 for any 1 i k − 1. Recall that each point of S is at distance at most 2 5 from a point of P = V (G ) ⊆ V (G).For each i, we let r i be a point in P such that d(p i , r i ) Theorem 8.3.For any integer g 0, the class of complete Riemannian surfaces of Euler genus at most g has Assouad-Nagata dimension at most 2.
Proof.Since g is finite, it suffices to prove that the class F of complete Riemannian surfaces of Euler genus exactly g has Assouad-Nagata dimension at most 2. By Lemma 8.2, every complete Riemannian surface S of genus g is quasi-isometric to some weighted finite or infinite weighted graph embeddable in S, with constants in the quasi-isometry that are uniform (in fact the constants are even independent of g).Since the asymptotic dimension of the class of weighted finite or infinite graphs embeddable in S is at most 2 by Corollary 8.1, asdim(F) 2.
Since F is scaling-closed, by Lemma 7.5, the Assouad-Nagata dimension of F is at most 2.

Geometric graphs and graphs of polynomial growth
We now explore the asymptotic dimension of geometric graph classes and graph classes of polynomial growth.
For an integer d 1, and some real C 1, let D d (C) be the class of graphs G whose vertices can be mapped to points of R d such that • any two vertices of G are mapped to points at (Euclidean) distance at least 1 apart, and • any two adjacent vertices of G are mapped to points at distance at most C apart.The following is a simple consequence of results on coarse equivalence in R d (see Section 3.2 in [7]), but we include a self-contained proof for the sake of completeness.Recall that a graph G has growth at most f if for every r > 0, every r-ball in the metric space generated by G contains at most f (r) vertices; a graph class has growth at most f is all graphs in the class have growth at most f .A graph has growth rate at most d if it has growth at most f , where f (r) = r d (for r > 1).Hence classes of polynomial growth are exactly the classes of graphs whose growth rate is uniformly bounded.Krauthgamer and Lee [28] proved that any graph G with growth rate at most d is in the class D O(d log d) (2).We thus obtain the following result as a direct corollary of Theorem 9.1.This proves the first part of Theorem 1.13.Corollary 9.2.For any real d > 1, the class of graphs of growth rate at most d has asymptotic dimension O(d log d).In particular, classes of graphs of polynomial growth have bounded asymptotic dimension.
Again, Corollary 9.2 is a fairly natural extension of the fact that d-dimensional grids have bounded asymptotic dimension (d-dimensional grids form the basic example of graphs of polynomial growth).
Let us now argue why the assumption that the growth is polynomial in Corollary 9.2 cannot be weakened, showing the second part of Theorem 1.13.Recall that a function f is said to be superpolynomial if it can be written as f (r) = r g(r) with g(r) → ∞ when r → ∞.
Given p 0, a p-subdivided 3-regular tree is obtained from a tree in which all internal vertices have degree 3 by subdividing each edge p times.Given two integers k 1 and p 1 and a graph G, we say that a graph G is a (k, p)-stretch of G if it is obtained from G as follows.For each vertex v ∈ V (G) we define T v to be a p-subdivided 3-regular tree with d G (v) leaves, and subject to this, such that the radius of T v is as small as possible; and each leaf of T v is indexed by a different neighbour of v in G. Define G to be the graph obtained from the disjoint union of these trees T v (over all v ∈ V (G)) by adding, for each edge uv of G, a path with k internal vertices between the leaf of T v indexed u and the leaf of T u indexed v (see Figure 5 for an illustration).For any set S of vertices of G , the projection of S in G is the set of vertices v of G with V (T v ) ∩ S = ∅.We note that the intriguing assumption of Theorem 9.5, requiring that f (r) 3r + 1 for every r ∈ N, turns out to be necessary.If f (r 0 ) 3r 0 for some r 0 ∈ N, then graphs with growth at most f do not contain the (r 0 − 1)-subdivision of K 1,3 as a subgraph and hence as a minor.So the class of graphs with growth at most f has asymptotic dimension at most 1 by Theorem 1.2.This proves the third part of Theorem 1.13.

Conclusion and open problems
For some function f , we say that a class of graphs G has expansion at most f if any graph obtained by contracting pairwise disjoint connected subgraphs of radius at most r in a subgraph of a graph of G has average degree at most f (r) (see [34] for more details on this notion).Note that every proper minor-closed family has constant expansion.In Section 9 we have proved that classes of graphs of polynomial growth have bounded asymptotic dimension.Note that if a class has bounded (polynomial, and superpolynomial, respectively) growth, then it has bounded (polynomial, and superpolynomial) expansion.
Question 3. Is it true that every class of graphs of polynomial expansion has bounded asymptotic dimension?
Observe that polynomial expansion would again be best possible here, as we have constructed classes of graphs of (barely) superpolynomial growth (and therefore expansion) with unbounded asymptotic dimension.It should be noted that there are important connections between polynomial expansion and the existence of strongly sublinear separators [19].On the other hand, Hume [25] proved that classes of graphs of bounded growth and bounded asymptotic dimension have sublinear separators.
A class of graphs C is monotone if any subgraph of a graph from C lies in C. David Wood asked a weaker form of the converse of Question 3 in private communication after preliminary forms of this paper [10,30] appeared in arXiv.He asked whether every monotone class of graphs of bounded asymptotic dimension has bounded expansion.We remark that the monotonicity is required in Wood's question.The class consisting of all complete graphs has asymptotic dimension 0 (as all the graphs in this class have diameter at most 2) but does not have bounded expansion.This also shows that monotonicity cannot be replaced by the weaker property that the class is hereditary (i.e.closed under taking induced subgraphs).
It is known that monotone classes admitting strongly sublinear separators are exactly the monotone classes with polynomial expansions [19].Hence one can ask a stronger version of Wood's question as follows.
Question 4. Is it true that every monotone class of graphs of bounded asymptotic dimension has polynomial expansion?Note that positive answers of both Questions 3 and 4 would give a characterization of monotone graph classes with bounded asymptotic dimension in terms of polynomial expansion.However, during the review process of an earlier version of this paper, some of us (unpublished) were able to find a negative answer to Question 4: for every function f , there exists a monotone class of graphs with asymptotic dimension 1 and with expansion greater than f .Note that the control functions in our negative answer for Question 4 depend on f , so they do not dispute Wood's question for bounded expansion.Lemmas 7.1 shows that for any p, q, the class of graphs with no q-fat K 2,p -minors has asymptotic dimension at most 1.On the other hand, Theorem 1.1 shows that for any t, the class of K t -minor free graphs has asymptotic dimension at most 2. A natural question is whether this can be extended to q-fat minors, as follows.
Question 5. Is it true that there is a constant d such that for any integer q and graph H, the class of graphs with no q-fat H-minor has asymptotic dimension at most d?

Claim 6 .
and let c e : Z e → [m] such that c e (v) = c 3 (v) for every v ∈ Z e .By Claim 5, for every e ∈ U E , Z e ⊆ N 3 Ge (X e ).For every e ∈ U E , c e can be extended to an m-colouring c e of G e with weak diameter in G e at most f * (η).Proof.We aim to apply the induction hypothesis to G e .Let e ∈ U E .If |X e | > η, then since (T, X ) is an (η, θ, F, F )-construction of G, |V (G e )| |X e | + 1 θ + 1, so c e can be extended to an m-colouring of G e with weak diameter in G e at most |V (G e )| θ + 1 f * (η).
Now assume that there exist e ∈ B and distinct Y, Y ∈ P e such that v Y v Y ∈ E(M ).So there exists ab ∈ E(M ) such that a, b ∈ V (M ) ∩ X Te − X e , Y a = Y and Y b = Y .By Claim 10, there exist a path P a in G e from a to Y a with length at most 2 and a path P b in G e from b to Y b of length at most 2 .Since ab ∈ E(M ) ⊆ E(G ), there exists a path P ab in G of length at most from a to b.If V (P ab ) ⊆ X Te , then P a ∪ P ab ∪ P b is a walk in G e from Y a to Y b of length at most 2 + + 2 < 7 , contradicting that Y a = Y and Y b = Y are distinct parts of P e .So V (P ab ) ⊆ X Te .In particular, there exist distinct a , b ∈ V (P ab ) ∩ X e such that the subpath P a of P ab between a and a and the subpath P b of P ab between b and b are contained in G e .Since the length of P a and P b are at most , a ∈ Y a and b ∈ Y b by Claim 5.So v Y a P b v Y is path in H of length at most the length of P ab , where P is the subpath of P ab between a and b .Hence v Y v Y ∈ H .

Figure 2 .
Figure 2. (a) The two 2-components A 1 and A 2 of a subset A, and (b) an example of a set B = {u, v} which has one 4-component but two (4, 3)components.

Theorem 4 . 3 .
Let L = (L 1 , L 2 , . ..) be a sequence of classes of weighted graphs of asymptotic dimension at most n.Let C be an L-layerable class of weighted graphs.Then the asymptotic dimension of C is at most n + 1.In addition, assume moreover that the class C is c-linearly L-layerable for some c > 0. If there exist a, b, d 0 such that each class L i has an n-dimensional control function D i with D i (r) ar + b(i − 1) + d for every r > 0, then C has asymptotic dimension at most n + 1 of linear type.If moreover d = 0, then C has Assouad-Nagata dimension at most n + 1.
r) is an n-dimensional intrinsic control function for L (G,φ) for any (G, φ) ∈ C. By Lemma 4.2, D (r, S) := D(r, S + 2r) is an n-dimensional control function for L (G,φ) for any (G, φ) ∈ C. By Theorem 4.1, for every (G, φ) ∈ C, (G, φ) admits an (n + 1)-dimensional control function D (G,φ) such that D (G,φ) (r) only depends on r, D and n, and hence only depends on r, f, (D i ) i∈N and n.Let (G 0 , φ 0 ) ∈ C. Then D (G 0 ,φ 0 ) is an n-dimensional control function for all members of C.This shows that C has asymptotic dimension at most n + 1.If the class C is c-linearly L-layerable for some c > 0, then it holds that D (r, S) = D(r, S + 2r) D max{c(S+2r),1} (r).If moreover there exist a, b, d 0 such that D i (r) ar + b(i − 1) + d for every i ∈ N and r > 0, then D (r, S) ar + b • (max{c(S + 2r), 1} − 1) + d ar + b • c(S + 2r) + d = (a + 2bc)r + bcS + d, which means that D is linear.In this case it follows from Theorem 4.1 that D (G,φ) is linear, and thus C has asymptotic dimension at most n + 1 of linear type.If moreover d = 0, we have D (r, S) (a + 2bc)r + bcS.By Theorem 4.1, C has Assouad-Nagata dimension at most n + 1.

Theorem 9 . 1 .
For any integer d 1 and real C 1, the class D d (C) has asymptotic dimension at most d.Proof.Fix an integer and a graph G ∈ D d (C), together with a mapping π of the vertices of G in R d as in the definition of D d (C).Since R d has asymptotic dimension d, there is a function f (independent of G) and a covering of R d by d + 1 sets C 1 , • • • , C d+1 , such that for any 1 i d + 1, C i consists of a union of open sets of diameter at most f ( ), such that any two open sets in the union are at distance more than C apart.For any vertex v of G, color v with an arbitrary integer 1 i d + 1 such that π(v) ∈ C i .Let F be a monochromatic component in G , say of color i.Note that for any edge uv in F ⊆ G , π(u) and π(v) are at Euclidean distance at most C in R d , and thus π(u) and π(v) lie in the same open set of C i .It follows that the images of all the vertices of F lie in the same open set of C i , of diameter at most f ( ).Since any two elements of F must be mapped to points at distance at least 1 apart, a simple volume computation shows that F must contain at most g(d, , C) vertices, for some function g.Therefore F has diameter at most g(d, , C) in G , which shows that D d (C) has asymptotic dimension at most d.

For every d 2 ,Figure 5 . 1 . 9 . 3 .
Figure 5.A local view of the (k, p)-stretch of a graph, for k = 5 and p = 1.Lemma 9.3.For any integers d 2, k 1, p 1, the class G k,p d has asymptotic dimension at least d.Proof.Assume for a contradiction that there is a function D such that for every r ∈ N, every graph in G k,p d can be covered by d sets whose r-components are D (r)-bounded.Consider a graph G ∈ G d , and a (k, p)-stretch G ∈ G k,p d of G. Let f (k, p, r) = (10p log 2 d)(r + 1) + (k + 1)r.By assumption, there are d sets U 1 , . . ., U d that cover G so that for each 1 i d, each f (k, p, r)-component of U i is D (f (k, p, r))-bounded.For 1 i d, let U i be the projection of U i in G.Note that U 1 , . . ., U d forms a cover of G, and every r-component of some U i is a subset of the projection of a f (k, p, r)-component of U i in G. Therefore, for every i, every r-component of U i is D (f (k, p, r))-bounded.By setting D(r) = D (f (k, p, r)), this shows that D is a (d − 1)-dimensional control function for G d , a contradiction.
Let F be a class of graphs.Let m > 0 be an integer.Then asdim(F) m − 1 if and only if there exists a function f : N → N such that for every G ∈ F and ∈ N, G is m-colourable with weak diameter 4 in G at most f ( ).We say that a class G of graphs has weak diameter chromatic number at most k if there is a constant d such that every graph G in G is k-colourable with weak diameter in G at most d.