Ubiquity of graphs with nowhere‐linear end structure

Abstract A graph G is said to be ≼‐ubiquitous, where ≼ is the minor relation between graphs, if whenever Γ is a graph with nG≼Γ for all n∈N, then one also has ℵ0G≼Γ, where αG is the disjoint union of α many copies of G. A well‐known conjecture of Andreae is that every locally finite connected graph is ≼‐ubiquitous. In this paper we give a sufficient condition on the structure of the ends of a graph G which implies that G is ≼‐ubiquitous. In particular this implies that the full‐grid is ≼‐ubiquitous.


| INTRODUCTION
Given a graph G and some relation ◃ between graphs we say that G is ◃-ubiquitous if whenever Γ is a graph such that ◃ nG Γ for all ∈ n , then ◃ G ℵ Γ 0 , where αG denotes the disjoint union of α many copies of G. For example, a classic result of Halin [9] says that the ray is ⊆-ubiquitous, where ⊆ is the subgraph relation.
Examples of graphs which are not ubiquitous with respect to the subgraph or topological minor relation are known (see [2] for some particularly simple examples). In [1] Andreae initiated the study of ubiquity of graphs with respect to the minor relation ≼. He constructed a graph which is not ≼-ubiquitous, however the construction relied on the existence of a counterexample to the well-quasi-ordering of infinite graphs under the minor relation, for which only examples of size at least the continuum are known [13]. In particular, the question of whether there exists a countable graph which is not ≼-ubiquitous remains open. Most importantly, however, Andreae [1] conjectured that at least all locally finite graphs, those with all degrees finite, should be ≼-ubiquitous.
The Ubiquity Conjecture. Every locally finite connected graph is ≼-ubiquitous.
In [2] Andreae proved that his conjecture holds for a large class of locally finite graphs. The exact definition of this class is technical, but in particular his result implies the following. Theorem 1.1 (Andreae [2,Corollary 2]). Let G be a connected, locally finite graph of finite tree-width such that every block of G is finite. Then G is ≼-ubiquitous.
Note that every end in such a graph must have degree 1 one. Andreae's proof employs deep results about well-quasi-orderings of labelled (infinite) trees [12]. Interestingly, the way these tools are used does not require the extra condition in Theorem 1.1 that every block of G is finite and so it is natural to ask if his proof can be adapted to remove this condition. And indeed, it is the purpose of the present and subsequent paper [3], to show that this is possible, that is, that all connected, locally finite graphs of finite tree-width are ≼-ubiquitous.
The present paper lays the groundwork for this extension of Andreae's result. The fundamental obstacle one encounters when trying to extend Andreae's methods is the following: In the proof we often have two families of disjoint rays ∈ R i I = ( : ) i  and ∈ S j J = ( : ) j  in Γ, which we may assume all converge to a common end of Γ, and we wish to find a linkage between  and , that is, an injective function → σ I J : and a set  of disjoint finite paths P i from ∈ x R i i to ∈ y S σ i σ i ( ) ( ) such that the walks ∈ R x Py S i I = ( : ) formed by following each R i along to x i , then following the path P i to y σ i ( ) , then following the tail of S σ i ( ) , form a family of disjoint rays (see Figure 1). Broadly, we can think of this as "rerouting" the rays  to some subset of the rays in . Since all the rays in  and  converge to the same graph G. In this way, Theorem 1.2 can be thought of as a local structure theorem for the ends of a graph which does not contain a K ℵ 0 -minor.
In this way, Theorem 1.2 allows us to make some structural assumptions on the "host" graph Γ when considering the question of ≼-ubiquity. However, more importantly, it also allows us to make some structural assumptions about G. Roughly, if the ends of G do not have a particularly simple structure then the fact that ≼ nG Γ for each ∈ n  will imply that Γ must have a pebbly end.
Analysing this situation gives rise to the following definition: We say that an end ϵ of a graph G is linear if for every finite set  of at least three disjoint rays in G which converge to ϵ we can order the elements of  as R R R = { , , …, } n 1 2  such that for each ⩽ ⩽ k i n 1 < < ℓ , the rays R k and R ℓ belong to different ends of G V R − ( ) i . For example, the half-grid has a unique end and it is linear. On the other end of the spectrum, let us say that a graph G has nowhere-linear end structure if no end of G is linear.
Our main theorem in this paper is the following.
Finally, we will also show the following result about graphs that are not locally finite. For ∈ k , we let the k-fold dominated ray be the graph DR k formed by taking a ray together with k additional vertices, each of which we make adjacent to every vertex in the ray. For ⩽ k 2, DR k is a minor of the half-grid, and so ubiquitous by Theorem 1.5. In our last theorem, we show that DR k is ubiquitous for all ∈ k .
Theorem 1.7. The k-fold dominated ray DR k is ≼-ubiquitous for every ∈ k .
As mentioned before, it is the purpose of the present and subsequent paper [3], to show that all connected, locally finite graphs of finite tree-width are ≼-ubiquitous. As an example of such a graph for which the methods of this paper do not yet suffice, consider a graph which is the union of a sequence ∈ H n ( : ) n  of graphs, each isomorphic to n m if m n > + 1, see Figure 2. This graph has a unique end, which is of degree 2 and hence is linear. Moreover, it is nonplanar and hence not a subgraph of □  .
Furthermore, this graph is 2-connected and hence not already covered by Theorem 1.1, but it has tree-width 4 and hence will be covered by our result from [3]. The paper is structured as follows: In Section 2 we introduce some basic terminology for talking about minors. In Section 3 we introduce the concept of a ray graph and linkages between families of rays, which will help us to describe the structure of an end. In Sections 4 and 5 we introduce a pebble-pushing game which encodes possible linkages between families of rays and use this to give a sufficient condition for an end to contain a countable clique minor. In Sections 6 and 7 we prove Theorem 1.2, classifying the thick ends which are nonpebbly. In Section 8 we reintroduce some concepts from [4] and show that we may assume that the G-minors in Γ are concentrated towards some end ϵ of Γ. In Section 9 we use the results of the previous section to prove Theorem 1.5 and finally in Section 10 we prove Theorem 1.3 and its corollaries.

| PRELIMINARIES
In our graph-theoretic notation we generally follow the textbook of Diestel [6]. Given two graphs G and H the cartesian product □ G H is a graph with vertex set V G V H ( ) × ( ) with an edge between a b ( , ) and c d ( , ) if and only if a c = and ∈ bd E H ( ) or ∈ ac E G ( ) and b d = .
Definition 2.1. A one-way infinite path is called a ray and a two-way infinite path is called a double ray. For a path or ray P and vertices ∈ v w V P , ( ), let vPw denote the subpath of P with endvertices v and w. If P is a ray, let Pv denote the finite subpath of P between the initial vertex of P and v, and let vP denote the subray (or tail) of P with initial vertex v.
Given two paths or rays P and Q which are disjoint but for one of their endvertices, we write PQ for the concatenation of P and Q, that is, the path, ray or double ray ∪ P Q.  [6,Chap. 8]). An end of an infinite graph Γ is an equivalence class of rays, where two rays R and S are equivalent if and only if there are infinitely many vertex-disjoint paths between R and S in Γ. We denote by Ω(Γ) the set of ends of Γ.
We say that a ray ⊆ R Γ converges (or tends) to an end ϵ of Γ if R is contained in ϵ. In this case we call R an ϵ-ray.
Given an end ∈ ϵ Ω(Γ) and a finite set ⊆ X V(Γ) there is a unique component of X Γ − which contains a tail of every ray in ϵ, which we denote by C X ( , ϵ).
Note that this supremum is in fact a maximum, that is, for each end ϵ of Γ there is a set  of vertex-disjoint ϵ-rays with   = deg(ϵ)  , as proved by Halin [9,Satz 1]. If an end has finite degree, we call it thin. Otherwise, we call it thick.
A vertex ∈ v V(Γ) dominates an end ∈ ϵ Ω(Γ) if there is a ray ∈ R ϵ such that there are infinitely many v-R-paths in Γ that are vertex-disjoint apart from v.
We will use the following two basic facts about infinite graphs.
Note that every H which is an IG contains a subgraph H′ such that ) is a tidy IG, although this choice may not be unique. In this paper we will always assume without loss of generality that each IG is tidy.
Suppose R is a ray in some graph G. If H is a tidy IG in a graph Γ then in the restriction H R ( ) all rays which do not have a tail contained in some branch set will share a tail. Later in the paper we will want to make this correspondence between rays in G and Γ more explicit with use of the following definition: Definition 2.10 (Pullback). Let G be a graph, ⊆ R G a ray, and let H be a tidy IG. The )) is an IR.
Note that, since H is tidy, ↓ H R ( ) is well defined. As we shall see, ↓ H R ( ) will be a ray.
Lemma 2.11. Let G be a graph and let H be a tidy IG. If ⊆ R G is a ray, then the pullback ↓ H R ( ) is also a ray. such that RG( )  is the path n 1 2 … . In particular, for each i, we have that Suppose for a contradiction that there exists ⩽ ⩽ k i n 1 < < ℓ such that R k and R ℓ belong to the same end of G V R − ( ) i , and so there is an infinite family of vertex-disjoint Each of these paths must contain a subpath which goes from a ray R r for some ⩽ r i 1 < to a ray R s for some ⩽ i s n < , and which meets no other ray in . Since there are infinitely many paths, by the pigeon hole principle there is some ⩽ ⩽ r i s n 1 < < such that there are infinitely many vertex-disjoint paths from R r to R s in ⧹ G V R ( ) i which meet no other ray in , and so ∈ rs E(RG( ))  , a contradiction. □ We will also use the following lemma, whose proof is an easy exercise.  Definition 3.7 (Tail of a ray after a set). Given a ray R in a graph G and a finite set Definition 3.8 (Linkage of families of rays). Let such that is a collection of disjoint rays.
We say that  is obtained by transitioning from  to  along the linkage. We say the linkage  induces the mapping σ. Given a vertex set ⊆ X V G ( ) we say that the linkage is i ii for all ∈ i I and no other vertex in X is used by the members of  . We say that a function → σ I J : is a transition function from  to  if for any finite vertex set ⊆ X V G ( ) there is a linkage from  to  after X that induces σ.
We will need the following lemma from [4], which asserts the existence of linkages.

| A PEBBLE-PUSHING GAME
Suppose we have a family of disjoint rays ∈ R i I = ( : ) i  in a graph G and a subset ⊆ J I . Often we will be interested in which functions we can obtain as transition functions between ∈ R i J ( : ) We can think of this as trying to "reroute" the rays To this end, it will be useful to understand the following pebble-pushing game on a graph.
with ≠ i j. The pebble-pushing game (on G) is a game played by a single player. Given a game state Y y y y = ( , , …, ) , we imagine k labelled pebbles placed on the vertices y y y ( , , …, ) which does not contain a pebble, or formally, a Y -move is a game state Z z z z = ( , …, ) be a game state. The X -pebble-pushing game (on G) is a pebblepushing game where we start with k labelled pebbles placed on the vertices x x x ( , …, ) We say a game state Y is achievable in the X -pebble-pushing game if there is a sequence , that is, if it is a sequence of moves that pushes the pebbles from X to Y .
A graph G is k-pebble-win if Y is an achievable game state in the X -pebble-pushing game on G for every two game states X and Y .
The following lemma shows that achievable game states on the ray graph RG( )  yield transition functions from a subset of  to itself. Therefore, it will be useful to understand which game states are achievable, and in particular the structure of graphs on which there are unachievable game states.
j . Hence, it is sufficient to show the statement holds when σ is obtained from k (1, 2, …, ) by a single move, that is, there is some So, let ⊆ X V G ( ) be a finite set. We will show that there is a linkage from j after X that induces σ. By assumption, there is an edge tσ t which avoids X and all other S j .
Then the family P P P = ( , , …, ) We note that this pebble-pushing game is sometimes known in the literature as "permutation pebble motion" [11] or "token reconfiguration" [5]. Previous results have mostly focused on computational questions about the game, rather than the structural questions we are interested in. In [11] the authors give an algorithm that decides whether or not a graph is kpebble-win. From this result it should be possible to deduce the main result in this section, Lemma 4.9. However, since a direct derivation was shorter and self-contained, we will not use their results. We present the following simple lemmas without proof. Lemma 4.3. Let G be a finite graph and X a game state.
• If Y is an achievable game state in the X -pebble-pushing game on G, then X is an achievable game state in the Y -pebble-pushing game on G.
• If Y is an achievable game state in the X -pebble-pushing game on G and Z is an achievable game state in the Y -pebble-pushing game on G, then Z is an achievable game state in the X -pebble-pushing game on G.

Definition 4.4. Let G be a finite graph and let
( ) . We define the pebblepermutation group of G X ( , ) to be the set of permutations σ of k [ ] such that X σ is an achievable game state in the X -pebble-pushing game on G.
Note that by Lemma 4.3, the pebble-permutation group of G X ( , ) is a subgroup of the symmetric group S k . Lemma 4.5. Let G be a graph and let X be a game state. If Y is an achievable game state in the X -pebble-pushing game and σ is in the pebble-permutation group of Y , then σ is in the pebble-permutation group of X . Lemma 4.6. Let G be a finite connected graph and let X be a game state.
Conversely, since G is connected, for any game states X and Y there is some τ such that Y τ is an achievable game state in the X -pebble-pushing game, since we can move the pebbles to any set of k vertices, up to some permutation of the labels. We know by assumption that X τ −1 is an achievable game state in the X -pebble-pushing game. Therefore, by Lemma 4.3, Y is an achievable game state in the X -pebble-pushing game. □ Lemma 4.7. Let G be a finite connected graph and let X x x x = ( , , …, ) Then there is a bare path P p p p = … n . Furthermore, either every edge in P is a bridge in G, or G is a cycle.
such that both colour classes are nontrivial and for all the pebble-permutation group. Let us consider this as a 3-colouring For every achievable game state Z z z z = ( , , …, ) in the X -pebble-pushing game, we define a 3-colouring c Z given by c z c x ( ) = ( ) . We note that, for any achievable game state Z there is no ∈ z c r ( ) were, then by Lemma 4.3 X ij ( ) is an achievable game state in the X -pebble-pushing game, Since G is connected, for every achievable game state Z there is a path in G whose internal vertices have colour 0 (with respect to c Z ) and whose endvertices have distinct nonzero colours (with respect to c Z ). Let us consider an achievable game state Z for which G contains such a path P p p p = … m . Clearly v′ cannot be adjacent to p 1 or p m , since then we can push the pebble on p 1 or p m onto v′ and achieve a game state Z″ for which G contains a longer path than P with the required colouring. However, if v′ is adjacent to p ℓ with ⩽ ⩽ m 2 ℓ −1, then we can push the pebble on p 1 onto p ℓ and then onto v′, then push the pebble from p m onto p 1 and finally push the pebble on v′ onto p ℓ and then onto p m .
such that p i and p j are not adjacent in P. Then it is easy to find a sequence of moves which exchanges the pebbles on p 1 and p m , contradicting our assumptions on c Z .
Suppose then that p i is adjacent to a vertex v not in P. Then, However then, we can push the pebble on p m onto p i−1 , push the pebble on v onto p i and then onto p m and finally push the pebble on p i−1 onto p i and then onto v. As before, this contradicts our assumptions on c Z .
−1 is a bare path in G, and since every vertex in V V P − ( ′) is coloured using r or using b, there are at most k such vertices. □ Finally, suppose that there is some edge in P′ which is not a bridge of G, and so no edge of P′ is a bridge of G. Before we show that G is a cycle, we make the following claim: 1 and a vertex ∉ v C such that: • There exist distinct positive integers i j s , , and t such that c c r Proof of Claim 4.10. Suppose for a contradiction there exists such an achievable game state W . Since C is a cycle, we may assume without loss of generality (by possibly making moves along the cycle) that c , then we can push the pebble at v to c 2 and then to c 3 , push the pebble at c 1 to c 2 and then to v, and then push the pebble at c 3 to c 1 . This contradicts our assumptions on c W . The case where , then we can push the pebble at c 1 to c 2 and then to v, then push the pebble at c 4 to c 1 , then push the pebble at v to c 2 and then to c 4 . Again this contradicts our assumptions on c W . □ Since no edge of P′ is a bridge, it follows that G contains a cycle C containing P′. If G is not a cycle, then there is a vertex ∈ ⧹ v V C which is adjacent to C. However, by pushing the pebble on p 1 onto p 2 and the pebble on p m onto p m−1 , which is possible since ≥   V k + 2, we achieve a game state Z′ such that C and v satisfy the assumptions of the above claim, a contradiction.

| PEBBLY AND NONPEBBLY ENDS
Definition 5.1 (Pebbly). Let Γ be a graph and ω an end of Γ. We say ω is pebbly if for every ∈ k  there is an ≥ n k and a family If for some k there is no such family , we say ω is nonpebbly and in particular not k-pebble-win.
Clearly an end of degree k is not k-pebble-win, since no graph on at most k vertices is k-pebblewin, and so every pebbly end is thick. However, as we shall see, pebbly ends are particularly rich in structure in that they force a countable clique minor. The strategy to prove the following lemma is as follows. First, we fix a sequence of finite graphs G G , , … 1 2 such that G 1 is a single vertex graph, G n+1 extends G n by either exactly one vertex or exactly one edge, and  G n n is the countable clique. We assume inductively that we have constructed H n , which is an IG n such that each branch set contains the initial vertex of a ray from a family  of disjoint ω-rays for which RG( )  is   V G ( ) n -pebble-win, and H n is otherwise disjoint from any ray in . To extend H n to an IG n+1 , we distinguish two cases. If G n+1 extends G n by a vertex, we pick a linkage from  to a family of disjoint ω-rays whose ray graph is   V G ( ) n+1 -pebble-win, starting a new branch set at a ray that is not linked to from any of the rays extending a previous branch set. If G n+1 extends G n by an edge uv, we pick a linkage from  to itself inducing some transition function that links the ray corresponding to v to a neighbour of u in the ray graph. Then, we can extend each branch set of H n to obtain an IG n+1 by following the linkage.
Proof. By assumption, there exists a sequence , ,…   of families of disjoint ω-rays such that for each ∈ k , the ray graph RG( ) Let us enumerate the vertices and edges of K ℵ 0 with a bijection ∪ → σ : (2)    such then there will be tails T T T , , …, n 1 2 of n distinct rays in n  such that for every ∈ i n [ ], the tail T i meets H k in a vertex of the branch set of i, and is otherwise disjoint from H k . We will assume without loss of generality that T i is a tail of R i n .
Since σ(1) = 1 we can take H 1 to be the graph on the initial vertex of R 1 1 . Suppose then and we have already constructed H n−1 together with appropriate tails +1 after X by Lemma 3.9, and, after relabelling, we may assume this linkage induces the identity on r [ ]. Let us suppose the linkage consists of paths In the case that ∈ w r [ ], w v < , say, the game state is achievable in the r (1, 2, …, )-pebble-pushing game and we get, by a similar argument, all P x y P x y , , , ′, ′, ′ i i i i i i and P. We build H n from H n−1 by adjoining the following paths: • for each ≠ i v we add the path Tx Py Tx P y ′ ′ ′ i i i i i i i i to H n−1 , adding the vertices to the branch set of i; • we add P to H n−1 , adding the vertices of ⧹ V P y (ˆ) {ˆ} to the branch set of u; • we add the path T x P y T x P y We note that, since ∈ y yT x ′ v w v the branch sets for u and v are now adjacent. Hence H n is an IG n extending H n−1 . Finally, the rays y T ′ i i for ∈ i r [ ] are appropriate tails of the used rays of r  . □ As every countable graph is a subgraph of K ℵ 0 , a graph with a pebbly end contains every countable graph as a minor. Thus, as G ℵ 0 is countable, if G is countable, we obtain the following corollary: Corollary 5.3. Let Γ be a graph with a pebbly end ω and let G be a countable graph. Then So, at least when considering the question of ≼-ubiquity for countable graphs, Corollary 5.3 allows one to restrict one's attention to host graphs Γ in which each end is nonpebbly. For this reason it will be useful to understand the structure of such ends.
An immediate observation we can make is the following corollary of Lemma 4.9. in So, if ω is not pebbly, then the ray graph of every family of ω-rays is either close in structure to a path, or close in structure to a cycle. In fact, this dichotomy is not just true for each ray graph individually, but rather uniformly for each ray graph in the end. That is, we will show that either every ray graph of a family of ω-rays will be close in structure to a path, or every ray graph will be close in structure to a cycle. Furthermore, the structure of this end will restrict the possible transition functions between families of ω-rays.
As motivating examples consider the half-grid □   and the full-grid □  . Both graphs have a unique end ∕ ω ω h f and it is easy to show that the ray graph of every family of ω h -rays is a path, and the ray graph of every family of ω f -rays is a cycle (and so in particular □   is not 2pebble-win and □   is not 3-pebble-win).
There is a natural way to order any family of disjoint ω h -rays: by imagining them drawn in the half-plane their tails will appear in some order from left to right. Then, it can be shown that any transition function between two large enough families of ω h -rays must preserve this ordering.
Similarly, there is a natural way to cyclically order any family of disjoint ω f -rays. As before, it can be shown that any transition function between two large enough families of ω f -rays must preserve this ordering.
The aim of the next few sections is to demonstrate that the above dichotomy holds for all nonpebbly ends: that either every ray graph is close in structure to a path or close in structure to a cycle, and furthermore that in each of these cases the possible transition functions between families of rays are restricted in a similar fashion as those of the half-grid or full-grid, in which case we will say the end is half-grid-like or grid-like respectively. These results, whilst not used in this paper, will be a vital part of the proof in [3].
We note that, in principle, this trichotomy that an end of a graph is either pebbly, grid-like or half-grid-like, and the information that this implies about its finite ray graphs and the transitions between them, could be derived from earlier work of Diestel and Thomas [7], who gave a structural characterisation of graphs without a K ℵ 0 -minor. However, to introduce their result and derive what we needed from it would have been at least as hard as our work in Section 6, if not more complicated, and so we have opted for a straightforward and self-contained presentation.

| Polypods
It will be useful for our analysis of the structure of nonpebbly ends to consider the possible families of disjoint rays in the end with a fixed set of start vertices, and the relative structure of these rays. Definition 6.1. Given an end ϵ of a graph Γ, a polypod (for ϵ in Γ) is a pair X Y ( , ) of disjoint finite sets of vertices of Γ such that there is a family ∈ R y Y ( : ) y of disjoint ϵ-rays, where R y begins at y and all the R y are disjoint from X . Such a family ∈ R y Y ( : ) y is called a family of tendrils for X Y ( , ). The order of the polypod is   Y . The connection graph K X Y , of a polypod X Y ( , ) is a graph with vertex set Y . It has an edge between vertices v and w if and only if there is a family ∈ R y Y ( : ) y of tendrils for X Y ( , ) such that there is an R v -R w -path in Γ disjoint from X and from every other R y .
Note that the ray graph of any family of tendrils for a polypod must be a subgraph of the connection graph of that polypod. Definition 6.2. We say that a polypod X Y ( , ) for ϵ in Γ is tight if its connection graph is minimal amongst the connection graphs of polypods for ϵ in Γ with respect to the spanning isomorphic subgraph relation, that is, for no other polypod X Y ( ′, ′) for ϵ in Γ of isomorphic to a proper subgraph of K X Y , . (Let us write ⊂ H G if H is isomorphic to a subgraph of G.) We say that a polypod attains its connection graph if there is some family of tendrils for that polypod whose ray graph is equal to the connection graph. BOWLER ET AL. | 579 Lemma 6.3. Let X Y ( , ) be a tight polypod, ∈ R y Y ( : ) y a family of tendrils and for every ∈ y Y let v y be a vertex on R y . Let X′ be a finite vertex set disjoint from all v R y y and including X as well as each of the initial segments R v y y .
is a polypod with the same connection graph as X Y ( , ). In particular, X Y ( ′, ′) is tight.
Proof. The family ∈ v R y Y ( : ) y y witnesses that X Y ( ′, ′) is a polypod. Moreover every family of tendrils for X Y ( ′, ′) can be extended by the paths R v y y to obtain a family of tendrils for X Y ( , ). Hence if there is an edge v v y z in K ′ ′ X Y then there must also be the edge is tight we must have equality. Therefore X Y ( ′, ′) is tight as well. □ Lemma 6.4. Any tight polypod X Y ( , ) attains its connection graph.
Proof. We must construct a family of tendrils for X Y ( , ) whose ray graph is K X Y , . We will recursively build larger and larger initial segments of the rays, together with disjoint paths between them.
Precisely this means that, after partitioning  into infinite sets A e , one for each edge e of K X Y , , we will construct, for each ∈ n , a family ∈ P y Y ( : ) y n of disjoint paths, and also paths Q n such that for some arbitrary fixed ray ∈ R ϵ: • Each P y n starts at y. We write y n for the last vertex of P y n .
• Each P y n has length at least n and there are at least n disjoint paths from P y n to R.
• For ⩽ m n, the path P y n extends P y m .
, , then Q n is a path from P v n to P w n .
, , then Q n meets no P y m with ∈ ⧹ y Y v w { , } for any ∈ m . • The Q n are pairwise disjoint. • All the P y n and all the Q n are disjoint from X .
• For any ∈ n , there is a family ∈ R y Y ( : ) y n of tendrils for X Y ( , ) such that each P y n is an initial segment of the corresponding R y n , and the R y n meet the Q m with ⩽ m n in P ẙ y n n .
Once the construction is complete, we obtain a family of tendrils by letting each R y be the union of all the P y n . Indeed, R y clearly is an ϵ-ray since there are arbitrarily many disjoint paths from R y to R. Furthermore, for any edge e of K X Y , the family ∈ Q n A ( : ) n e will witness that e is in the ray graph of this family. So that ray graph will be all of K X Y , , as required.
So it remains to show how to carry out this recursive construction. Let vw be the edge of K X Y , with ∈ A 1 vw . By the definition of the connection graph, there is a family ∈ R y Y ( : ) +1 , and such that there are at least n + 1 disjoint paths between P y n+1 and Rwhich is possible since both R and R y n+1 are ϵ-rays. This completes the recursion step, and so the construction is complete. □ Lemma 6.5. Let X Y ( , ) be a polypod for ϵ in Γ with connection graph K X Y , , let ∈ S y Y ( : ) y be a family of tendrils for X Y ( , ), and let ∈ R i I ( : ) i be a set of disjoint ϵ-rays. Then for any transition function σ from  to  and every pair ∈ y y Y , ′ such that there is a path from σ y Proof. Since σ is a transition function there exists a linkage from  to  after X which induces σ. This linkage gives us a family of tendrils ∈ S y Y ( ′ : ) y for X Y ( , ) such that S′ y and R σ y ( ) share a tail for each ∈ y Y . By Lemmas 3.2 and 3.6, if ∈ y y Y , ′ are such that there is a path from σ y ( ) to σ y ( ′) otherwise avoiding σ Y ( ) in ∈ R i I RG( : ) i , then S′ y and S ′ ′ y are adjacent in ∈ S y Y RG( ′ : ) y , and so y and y′ are adjacent in K X Y , . □ Corollary and Definition 6.6. Any two polypods for ϵ in Γ of the same order which attain their connection graphs have isomorphic connection graphs. We will refer to the graph arising in this way for polypods of order n for ϵ in Γ as the nth shape graph of the end ϵ.

| Frames
Given a family of tendrils ∈ R y Y ( : ) y for a polypod X Y ( , ), there may be different families of tendrils ∈ R y Y ( ′ : ) y for X Y ( , ) such that each R y shares a tail with some R ′ π y ( ) . To understand the possible transition functions between different families of rays in ϵ it will be useful to understand the possible functions π that arise in this fashion.
To do so we will consider frames, finite graphs L which contain a path family between two sets of vertices α Y ( ) and β Y ( ). For appropriate choices of α Y ( ) and β Y ( ), these will be the BOWLER ET AL.
| 581 subgraphs arising from a linkage from the family of tendrils ∈ R y Y ( : ) y to itself after X , each of which gives rise to a family ∈ R y Y ( ′ : ) y as above.
Some frames will contain multiple such path families, linking α Y ( ) to β Y ( ) in different ways. For appropriately chosen frames the possible ways we can link α Y ( ) to β Y ( ) will be restricted by the structure of K X Y , , which will allow us relate this to the possible transition functions from ∈ R y Y ( : ) y to itself, and from there to the possible transition functions between different families of rays.
consists of a finite graph L together with two injections α and β from Y to V L ( ). The set A α Y = ( ) is called the source set The weave pattern π  of is the bijection from Y to itself sending y to the inverse image under β of the endvertex of Q y . In other words, π  is the function so that every Q y is an α y ( )-β π y ( ( ))  path. The weave graph K  of  has vertex set Y and an edge joining distinct vertices u and v of Y precisely when there is a path from Q u to Q v in L disjoint from all other Q y . For a graph K with vertex set Y , we say that the Y -frame is K -spartan if all its weave graphs are subgraphs of K and all its weave patterns are automorphisms of K .
Connection graphs of polypods and weave graphs of frames are closely connected.
( , ) be a polypod for ϵ in Γ attaining its connection graph K X Y , and let ∈ R y Y = ( : ) y  be a family of tendrils for X Y ( , ). Let L be any finite subgraph of Γ disjoint from X but meeting all the R y . For each ∈ y Y let α y ( ) be the first vertex of R y in L and β y ( ) the last vertex of R y in L. Then the Y -frame L α β ( , , ) is K X Y , -spartan.
Proof. Since there is some family of tendrils ∈ S y Y ( : ) y attaining K X Y , and there is, by Lemma 3.9, a linkage from ∈ R y Y ( : ) y to ∈ S y Y ( : ) y after X and V L ( ), we may assume without loss of generality that ∈ R y Y RG( : ) y is isomorphic to K X Y , . For a given weave ∈ Q y Y = ( : ) y  , applying the definition of the connection graph to the rays R R α y Q β π y R ′ = ( ) ( ( )) and which does not meet any other R y , and so joins R ′ , , and so π  is an automorphism of K X Y , . □ Corollary 6.9. Let X Y ( , ) be a polypod for ϵ in Γ attaining its connection graph K X Y , and let ∈ R y Y = ( : ) y  be a family of tendrils for X Y ( , ). Then for any transition function σ from  to itself there is a K X Y , -spartan Y -frame for which both σ and the identity are weave patterns.
Proof. Let ∈ P y Y ( : ) y be a linkage from  to itself after X inducing σ, and let L be a finite subgraph graph of Γ containing ∈  P y Y y as well as a finite segment of each R y , such that each P y is a path between two such segments. Then the Y -frame on L which exists by Lemma 6.8 has the desired properties. □ Lemma 6.10. Let X Y ( , ) be a polypod for ϵ in Γ attaining its connection graph K X Y , and let ∈ R y Y = ( : ) y  be a family of tendrils for X Y ( , ). Then there is a K X Y , -spartan Y -frame for which both K X Y , and ∈ R y Y RG( : ) y are weave graphs.
Proof. By adding finitely many vertices to X if necessary, we may obtain a superset X′ of X such that for any two of the R y , if there is any path between them disjoint from all the other rays and X′, then there are infinitely many disjoint such paths. Let ∈ S y Y ( : ) y be any family of tendrils for X Y ( , ) with connection graph K X Y , . For each edge e uv = of RG( )  let P e be a path from R u to R v disjoint from all the other R y and from X′. Similarly for each edge f uv = of K X Y , let Q f be a path from S u to S v disjoint from all the other S y and from X′. Let ∈ P y Y ( ′ : ) y be a linkage from the S y to the R y after Let the initial vertex of P′ y be γ y ( ) and the end vertex be β y ( ). Let π be the permutation of Y by setting π y ( ) to be the element of Y with β y ( ) on R π y ( ) . Let L be the graph given by the union of all paths of the form S γ y ( ) y and R β y ( ) π y ( ) together with P′ y , P e and Q e . Letting α be the identity function on Y , it follows from Lemma 6.8 The paths Q f witness that the weave graph for the paths S γ y P ( ) ′ y y includes K X Y , and so, by K X Y , -spartanness, must be equal to K X Y , . The paths P e witness that the weave graph for the paths R β y ( ) y includes the ray graph RG( )  . However conversely, since V L ( ) is disjoint from X′, if two of the R y are joined in L by a path disjoint from the other rays in  then they are joined by infinitely many, and hence adjacent in RG( )  . It follows that the weave graph is equal to RG( )  . □ Hence to understand ray graphs and the transition functions between them it is useful to understand the possible weave graphs and weave patterns of spartan frames. Their structure can be captured in terms of automorphisms and cycles.
1 is a cycle of K , and we call such cycles turnable. If t = 2 then we call the edge z z 1 2 of K flippable. We say that an automorphism of K is locally generated if it is a product of local automorphisms. Remark 6.12. A cycle C in K is turnable if and only if all its vertices have the same neighbourhood in K E C − ( ), and whenever a chord of length ∈ ℓ , that is, a chord whose endvertices have distance ℓ on C, is present in K C [ ], then all chords of length ℓ are present. Similarly an edge e of K is flippable if and only if its two endvertices have the same neighbourhood in K e − . Thus, if K is connected and contains at least three vertices, no vertex of degree one or cutvertex of K can lie on a turnable cycle or a flippable edge. So vertices of degree one and cutvertices in such graphs are preserved by locally generated automorphisms. BOWLER ET AL. | 583 Lemma 6.13. Let L α β = ( , , )  be a K -spartan Y -frame. Then for any two of its weave patterns π and π′ the automorphism ⋅ π π′ −1 of K is locally generated. Furthermore, if K is a weave graph for  then each weave graph for  contains a turnable cycle or a flippable edge of K .
Proof. Let us suppose, for a contradiction, that the conclusion does not hold and let be weaves for  such that either ≠ π π   and ⋅ π π −1   is not locally generated, or K K =  and K  does not contain a turnable cycle or a flippable edge of K . Each edge of L is in one path of  or  since otherwise we could simply delete it.
Similarly no edge appears in both  and  since otherwise we could simply contract it. No vertex appears on just one of P y or Q y since otherwise we could contract one of the two incident edges. Vertices of L appearing in neither  nor  are isolated and so may be ignored. Thus we may suppose that each edge of L appears in precisely one of  or , and that each vertex of L appears in both.
Let Z be the set of those ∈ y Y for which ∉ α y β Y ( ) ( ). For any ∈ z Z, let γ z ( ) be the second vertex of P z , that is, the neighbour of α z ( ) on P z , and let Furthermore, Z is nonempty as  and  are distinct. Let z be any element of Z. Then since Z is finite there must be i j This preserves all of π  , π  and K  , and can only make K  bigger, contradicting the minimality of our counterexample. So we must have ≥ t 2. on Y . Let L′ be the graph obtained from L by deleting all vertices of the form α z ( ) i . Let α′ be the injection from Y to V L ( ′) sending z i to γ z ( ) i for ⩽ i t and sending any other ∈ y Y to α y ( ). Then L α β ( ′, ′, ) is a Y -frame. For any weave ∈ P y Y (ˆ: ) is a weave in L α β ( , , ) with the same weave pattern and whose weave graph includes that of ∈ P y Y (ˆ: ) i is an edge of K  , and hence, since  is K -spartan, also an edge of K , and so σ is a local automorphism of K . It follows that K  includes a turnable cycle or a flippable edge. Finally, by the minimality of   E L ( ) we know that ⋅ π π ′ ′ −1   is locally generated and hence so is . This is the desired contradiction. □ Finally, the following two lemmas are the main conclusions of this section: Lemma 6.14. Let X Y ( , ) be a polypod attaining its connection graph K X Y , such that K X Y , is a cycle of length at least 4. Then for any family of tendrils  for this polypod the ray graph is K X Y , . Furthermore, any transition function from  to itself preserves each of the cyclic orientations of K X Y , .
Proof. By Lemma 6.10 there is some K X Y , -spartan Y -frame for which both K X Y , and the ray graph RG( )  are weave graphs. Since K X Y , is a cycle of length at least 4 and hence has no flippable edges, the ray graph must include a cycle by Lemma 6.13 and so since it is a subgraph of K X Y , it must be the whole of K X Y , . Similarly Lemma 6.13 together with Corollary 6.9 shows that all transition functions must be locally generated and so must preserve the orientation. □ Lemma 6.15. Let X Y ( , ) be a polypod attaining its connection graph K X Y , such that K X Y , includes a bare path P whose edges are bridges. Let  be a family of tendrils for X Y ( , ) whose ray graph is K X Y , . Then for any transition function σ from  to itself, the restriction of σ to P is the identity.
Proof. By Lemmas 6.9 and 6.13 any transition function must be a locally generated automorphism of K X Y , , and so by Remark 6.12 it cannot move the vertices of the bare path, which are vertices of degree one or cutvertices. □

| GRID-LIKE AND HALF-GRID-LIKE ENDS
We are now in a position to analyse the different kinds of thick ends which can arise in a graph in terms of the possible ray graphs and the transition functions between them. The first kind of ends is the pebbly ends, in which, by Corollary 5.3, for any n we can find a family of n disjoint rays whose ray graph is K n and for which every function → σ n n : [ ] [ ] is a transition function. So, in the following let us fix a graph Γ with a thick nonpebbly end ϵ and a number ∈ N , where ≥ N 3, such that ϵ is not N -pebble-win. Under these circumstances we get nontrivial restrictions on the ray graphs and the transition functions between them. There are two essentially different cases, corresponding to the two cases in Corollary 5.4: the grid-like and the half-gridlike case.

| Grid-like ends
The first case focuses on ends which behave like that of the infinite grid. In this case, all large enough ray graphs are cycles and all transition functions between them preserve the cyclic order.
Formally, we say that the end ϵ is grid-like if the N ( + 2)nd shape graph for ϵ is a cycle. For the rest of this subsection we will assume that ϵ is grid-like. Let us fix some polypod X Y ( , ) of order N + 2 attaining its connection graph. Let ∈ S y Y ( : ) y be a family of tendrils for X Y ( , ) whose ray graph is the cycle C K = N XY +2 , . , which is a cycle by Lemma 6.14. Hence, ij is not a bridge of K J , and it is easy to see that this implies that ij is not a bridge of K . Hence, K is a cycle. □ Given a cycle C, a cyclic orientation of C is an orientation of the graph C which does not have any sink. Note that any cycle has precisely two cyclic orientations. Given a cyclic orientation and three distinct vertices x y z , , , we say that they appear consecutively in the order x y z ( , , ) if y lies on the unique directed path from x to z. Given two cycles C C , ′, each with a cyclic orientation, we say that an injection → f V C V C : ( ) ( ′) preserves the cyclic orientation if whenever three distinct vertices x y , and z appear on C in the order x y z ( , , ) then their images appear on C′ in the order f x f y f z ( ( ), ( ), ( )). We will now choose cyclic orientations of every large enough ray graph such that the transition functions preserve the cyclic orders corresponding to those orientations. To that end, we fix a cyclic orientation of K X Y , . We say that a cyclic orientation of the ray graph for a family ∈ R i I ( : ) i of at least N + 3 disjoint ϵ-rays is correct if there is a transition function σ from the S y to the R i which preserves the cyclic orientation of K X Y , .
Lemma 7.2. For any family ∈ R i I ( : ) i of at least N + 3 disjoint ϵ-rays there is precisely one correct cyclic orientation of its ray graph.
Proof. We first claim that there is at least one correct cyclic orientation. By Lemma 3.9, there is a transition function σ from the S y to some subset J of I , and we claim that there is some cyclic orientation of the ray graph K of ∈ R i I ( : ) i such that σ preserves the cyclic orientation of K X Y , . We first note that the ray graph K J of ∈ R i J ( : ) i is a cycle by Lemma 7.1, and it is obtained from K by subdividing edges, which does not affect the cyclic order. Hence it is sufficient to show that there is some cyclic orientation of K J such that σ preserves the cyclic orientation of K X Y , . Since each linkage inducing σ gives rise to a family of tendrils ∈ S y Y ( ′ : ) y where S′ y shares a tail with R σ y ( ) , it follows that if σ y ( ) and σ y ( )′ are adjacent in K J then y and y′ are adjacent in K X Y , . Since both K J and K X Y , are cycles, it follows that there is some cyclic orientation of K J such that σ preserves the cyclic orientation of K X Y , . Suppose for a contradiction that there are two, and let σ and σ′ be transition functions witnessing that both orientations of the ray graph are correct. By Lemma 4.2 we may assume without loss of generality that the images of σ and σ′ are the same. Call this common image I′. Since the ray graphs of ∈ R i I ( : ) i and ∈ R i I ( : ′) i are both cycles, the former is obtained from the latter by subdivision of edges. Since this does not affect the cyclic order, we may assume without loss of generality that I I ′ = . By Lemma 3.9 again, there is some transition function τ from the R i to the S y . By Lemma 6.14, both ⋅ τ σ and ⋅ τ σ′ must preserve the cyclic order, which is the desired contradiction. □ It therefore makes sense to refer to the correct orientation of a ray graph. Corollary 7.3. Any transition function between two families of at least N + 3 ϵ-rays preserves the correct orientations of their ray graphs.
Proof. Suppose that ∈ R i I = ( : ) i  and ∈ T j J = ( : ) j  are families of at least N + 3 rays and σ is a transition function from  to  .
Let us fix some transition function τ from ∈ S y Y ( : ) y to  and let  be a linkage from ∈ S y Y ( : ) and which induces σ. For every ∈ y Y let r τ y ( ) denote the endvertex of the path ∈ P y  on R τ y ( ) . For every ∈ i I let r′ i be the initial vertex of the path ∈ P′ ′ i  on R i .
to  which is after X and induces ⋅ σ τ. It follows that ⋅ σ τ is a transition function from ∈ S y Y ( : ) y to  . However, by the definition of correct orientation and Lemma 7.2, τ and ⋅ σ τ both preserve the cyclic orientation of K X Y , , and hence σ must preserve the correct orientation of the ray graphs of  and  . □

| Half-grid-like ends
In this subsection we suppose that ϵ is thick but neither pebbly nor grid-like. We shall call such ends half-grid-like, since as we shall shortly see in this case the ray graphs and the transition functions between them behave similarly to those for the unique end of the half-grid. Note that this implies Theorem 1.2.
We will need to carefully consider how the ray graphs are divided up by their cutvertices. In particular, for a graph K and vertices x and y of K we will denote by C K ( ) xy the union of all components of K x − which do not contain y, and we will denote by K xy the graph . We will refer to K xy as the part of K between x and y. As in the last subsection, let X Y ( , ) be a polypod of order N + 2 attaining its connection graph and let ∈ S y Y ( : ) y be a family of tendrils for X Y ( , ) with ray graph K X Y , , which by assumption is not a cycle. By Corollary 5.4 there is a bare path of length at least 1 in K X Y , all of whose edges are bridges. Let y y 1 2 be any edge of that path. Without loss of generality we have ≠ ∅ C K ( ) y y X Y , 1 2 . Throughout the remainder of this section we will always consider arbitrary families ∈ R i I = ( : ) i  of disjoint ϵ-rays with ≥   I N + 3. We will write K to denote the ray graph of .
Remark 7.4. For any transition function σ from the S y to the R i , we have the inclusions meet precisely in σ y ( ) 1 and σ y ( ) 2 .
Lemma 7.5. For any transition function σ from the S y to the R i , the graph K σ y σ y ( ) ( ) 1 2 is a path from σ y ( ) 1 to σ y ( ) 2 . This path is a bare path in K and all of its edges are bridges.
Proof. Since K is connected, K σ y σ y ( ) ( ) 1 2 must include a path P from σ y ( ) 1 to σ y ( ) 2 . If it is not equal to that path then it follows from Lemma 4.2 that the function σ′, which we BOWLER ET AL.
| 587 define to be just like σ except for σ y σ y ′( ) = ( ) 1 2 and σ y σ y ′( ) = ( ) 2 1 , is a transition function from the S y to the R i . But then by Remark 7.4 we have a contradiction. The last sentence of the lemma follows from the definition of K σ y σ y Given a path P with endvertices s and t we say the orientation of P from s to t to mean the total order ⩽ on the vertices of P where ⩽ a b if and only if a lies on sPb, in this case we say that a lies before b. Note that every path with at least one edge has precisely two orientations. Now, we fix a transition function σ max from the S y to the R i so that the path ≔ P K σ y σ y ( ) ( ) max 1 max 2 is as long as possible. We call P the central path of K and the orientation of P from σ y ( ) max 1 to σ y ( ) max 2 the correct orientation. We first note that, for large enough families of rays almost all of the ray graph lies on the central path.
Lemma 7.6. At most N vertices of K are not on the central path.
Proof. By Remark 7.4 we have . If it were a proper subset, then we would be able to use Lemma 4.2 to produce a transition function in which this path is longer. So we must have σ . However, since y y 1 2 is a bridge, and so at most N vertices of K are not on the central path. □ We call P the central path of K and the orientation of P from σ y ( ) max 1 to σ y ( ) max 2 the correct orientation. We note the following simple corollary, which will be useful in later work. Proof. The "if" direction is clear by applying Lemma 4.2 to σ max . For the "only if" direction, we begin by setting We . Then for any N ( + 2)-tuple x x ( , …, ) N max 1 max +2 -pebble-pushing game on K we must have the following three properties, since they are preserved by any single move: . Now let σ be any transition function from the S y to the R i . Let x x ( , …, ) Thus the central path and the correct orientation depend only on our choice of y 1 and y 2 . Hence, we get the following corollary. Corollary 7.9. Each ray graph on at least N + 3 vertices contains a unique central path with a correct orientation and every transition function between two families of at least N + 3 ϵ-rays sends vertices of the central path to vertices of the central path and preserves the correct orientation.
Proof. Consider the family ∈ R i I = ( : ) i  with its ray graph K and another family ∈ T j J = ( : ) j  of at least N + 3 rays, with ray graph K  , and let τ be a transition function from  to  .
Let v v , 1 2 be two vertices in the central path P of K with v 1 before v 2 . By Lemma 7.8 there is transition function σ from ∈ S y Y ( : ) y to  with σ y v ( ) = Proof. That σ x ( ) is an inner vertex of the central path of RG( )  follows from Corollary 7.9. We note, by Lemma 6.5, given any family of rays  and a transition BOWLER ET AL.
| 589 function γ from  to , if y separates x from z in K X Y , then γ y ( ) separates γ x ( ) from γ z ( ) in RG( )  .
 be a transition function with τ y x ( ) = 1 which exists by Lemma 7.8. Since x is an inner vertex of the central path of RG( )  , there are exactly two components of x RG( ) −  , one containing v 1 and one containing v 2 . Furthermore, by Lemma 6.5, it follows that τ C K ( ( )) y y X Y , 1 2 and are contained in different components of x RG( ) −  . Hence, by Lemma 4.2 we may assume without loss of generality that However, by the remark above applied to the transition function ⋅ σ τ we conclude

| G-TRIBES AND CONCENTRATION OF G-TRIBES TOWARDS AN END
To show that a given graph G is ≼-ubiquitous, we shall assume that ≼ nG Γ holds for every ∈ n  an show that this implies ≼ G ℵ Γ 0 . To this end we use the following notation for such collections of nG in Γ, most of which we established in [4].
• A G-tribe in Γ (with respect to the minor relation) is a family  of finite collections F of disjoint subgraphs H of Γ such that each member H of  is an IG. • A G-tribe  in Γ is called thick, if for each ∈ n  there is a layer ∈ F  with ≥   F n; otherwise, it is called thin.
• A G-tribe ′  in Γ is a G-subtribe 2 of a G-tribe  in Γ, denoted by ≼ ′   , if there is an if there is such an injection Ψ satisfying ⊆ F F ′ Ψ( ′).
• A thick G-tribe  in Γ is concentrated at an end ϵ of Γ, if for every finite set X of vertices of Γ, the G-tribe is a thin subtribe of  . It is strongly concentrated at ϵ if additionally, for every finite vertex set X of Γ, every member H of  intersects C X ( , ϵ).
We note that every thick G-tribe  contains a thick subtribe ′  such that every ∈  H  is a tidy IG. We will use the following lemmas from [4].
When G is clear from the context we will often refer to a G-subtribe as simply a subtribe.
Let ∈ k  be the degree of ϵ. By [8,Corollary 5.5] there is a sequence of vertex sets ∈ S n ( : ) n  such that:

| UBIQUITY OF MINORS OF THE HALF-GRID
Here, and in the following, we denote by  the infinite, one-ended, cubic hexagonal half-grid (see Figure 3). The following theorem of Halin is one of the cornerstones of infinite graph theory. In [10], Halin used this result to show that every topological minor of  is ubiquitous with respect to the topological minor relation ⩽. In particular, trees of maximum degree 3 are ubiquitous with respect to ⩽.
However, the following argument, which is a slight adaptation of Halin's, shows that every connected minor of  is ubiquitous with respect to the minor relation. In particular, the dominated ray, the dominated double ray, and all countable trees are ubiquitous with respect to the minor relation.
The main difference to Halin's original proof is that, since he was only considering locally finite graphs, he was able to assume that the host graph Γ was also locally finite.
We will need the following result of Halin.
Theorem 1.5. Any connected minor of the half-grid □   is ≼-ubiquitous.
Proof. Suppose ≼ □ G   is a minor of the half-grid, and Γ is a graph such that ≼ nG Γ for each ∈ n . By Lemma 8.4 we may assume there is an end ϵ of Γ and a thick G-tribe  F I G U R E 3 The hexagonal half-grid .