EEnds as tangles
Jakob Kneip30th September 2019
Every end of an infinite graph G defines a tangle of infinite order in G .These tangles indicate a highly cohesive substructure in the graph if and onlyif they are closed in some natural topology.We characterize, for every finite k , the ends ω whose induced tangles oforder k are closed. They are precisely the tangles τ for which there is a set of k vertices that decides τ by majority vote. Such a set exists if and only if thevertex degree plus the number of dominating vertices of ω is at least k . Tangles were introduced by Robertson and Seymour in [66] and have since become oneof the central objects of study in the field of graph minor theory. So far most of thetheory of tangles deals with tangles in finite graphs. In [11], it was shown how the setΘ of tangles of infinite order of an arbitrary infinite graph can be used to compactifyit, much in the same way as the set Ω of ends of a connected locally finite graph canbe used to compactify it. Indeed, if a graph G is connected and locally finite, thesecompactifications | G | Θ and | G | Ω of G coincide. This is because every end ω of an infinitegraph G induces a tangle τ = τ ω of order ℵ in G , and for locally finite connected G themap ω τ ω is a bijection between the set Ω of ends of G and the set Θ of its ℵ -tangles.(Graphs that are not locally finite have ℵ -tangles that are not induced by an end.)In [11], a natural topology on the set −→ S = −→ S ℵ ( G ) of separations of finite order of G wasdefined. A tangle τ induced by an end of G is a closed set in this topology if and only if τ is defined by an ℵ -block in G , that is, if there is an ℵ -block K in G with K ⊆ B forall separations ( A, B ) in τ .Our research expands on this latter result. Every end ω of a graph induces not only atangle of infinite order in G , but for each k ∈ N the end ω induces a k -tangle in G . Theset −→ S k of all separations ( A, B ) of G with | A ∩ B | < k is a closed set in −→ S , and thus if thetangle τ induced by ω in G is a closed set in −→ S , the k -tangle τ ∩ −→ S k induced by ω will beclosed as well. However, it is possible that a tangle τ in G of infinite order fails to beclosed in −→ S , while its restrictions τ ∩ −→ S k to −→ S k are closed for some, or even all, k ∈ N . Inthis paper we characterize the ends of G by the behaviour of their tangles, as follows:1 a r X i v : . [ m a t h . C O ] S e p e show that, for an end ω and its induced tangle τ , the restriction τ ∩ −→ S k to −→ S k is aclosed set in −→ S if and only if deg( ω ) + dom( ω ) (cid:62) k, where deg( ω ) and dom( ω ) denote the vertex degree and number of vertices dominating ω ,respectively.We further show that τ is closed in −→ S if and only if ω is dominated by infinitely manyvertices.A question raised in [33] asks whether for a k -tangle τ in a finite graph G one can alwaysfind a set X of vertices which decides τ by majority vote, in the sense that ( A, B ) ∈ τ ifand only if | A ∩ X | < | B ∩ X | , for all ( A, B ) ∈ −→ S k . This problem is still open in general,although some process has been made recently ([55]). We establish an analogue in theinfinite setting: we show that for an end ω of G and its induced k -tangle τ ∩ −→ S k in G , theexistence of a finite set X which decides τ ∩ −→ S k in the above sense is equivalent to τ ∩ −→ S k being a closed set in −→ S .This paper is organized as follows: Section 22 contains the basic definitions and somenotation. Following that, in Section 33, we recall the core concepts and results from [11]that are relevant to our studies, including the topology defined on −→ S . Finally, in Section 44,we prove our main results Theorem 4.44.4 and Theorem 4.54.5. The first of these characterisesthe ends of a graph by the behaviour of their tangles, and the second shows that τ ∩ −→ S k being a closed set in −→ S for a k -tangle τ induced by some end ω of G is equivalent to bothdeg( ω ) + dom( ω ) (cid:62) k and to τ ∩ −→ S k being decided by some finite set of vertices. Throughout this paper G = ( V, E ) will be a fixed infinite graph. For any graph-theoreticalnotation not explained here we refer the reader to [22].A separation of G is a set { A, B } with A ∪ B = V such that G contains no edgebetween A (cid:114) B and B (cid:114) A . We call such a set { A, B } an unoriented separation withthe two orientations ( A, B ) and (
B, A ). Informally we think of the oriented separation(
A, B ) as pointing towards B and pointing away from A . The separator of a separation { A, B } is the set A ∩ B .The order of a separation ( A, B ) or { A, B } is the size | A ∩ B | of its separator. For acardinal κ we write S κ = S κ ( G ) for the set of all unoriented separations of G of order < κ .If S is a set of unoriented separations we write −→ S for the corresponding set of orientedseparations, that is, the set of all separations ( A, B ) with { A, B } ∈ S . Consequently wewrite −→ S κ for the set of all separations ( A, B ) of G with | A ∩ B | < κ .If S is a set of unoriented separations of G , an orientation of S is a set O ⊆ −→ S suchthat O contains precisely one of ( A, B ) or (
B, A ) for every { A, B } ∈ S . A tangle of S in G is an orientation τ of S such that there are no ( A , B ), ( A , B ), ( A , B ) ∈ τ forwhich G [ A ] ∪ G [ A ] ∪ G [ A ] = G .Properties of sets of separations of finite graphs, including their tangles, often generalizeto sets of separations of infinite graphs but not always. Those sets of separations to2hich these properties tend to generalize can be identified, however: they are the sets ofseparations that are closed in a certain natural topology [44]. Let us define this topologynext. It is analogous to the topology of a profinite abstract separation system definedin [44]. From here on we denote by S = S ℵ ( G ) the set of all (unoriented) separations of G offinite order. Thus −→ S = −→ S ℵ ( G ) is the set of all separations ( A, B ) of G with A ∩ B finite.We define our topology on −→ S by giving it the following basic open sets. Pick a finiteset Z ⊆ V and an oriented separation ( A Z , B Z ) of G [ Z ]. Then declare as open the set O ( A Z , B Z ) of all ( A, B ) ∈ −→ S such that A ∩ Z = A Z and B ∩ Z = B Z . We shall say thatthese ( A, B ) induce ( A Z , B Z ) on Z , writing ( A Z , B Z ) =: ( A, B ) (cid:22) Z , and that ( A, B ) and( A , B ) agree on Z if ( A, B ) (cid:22) Z = ( A , B ) (cid:22) Z .It is easy to see that the sets O ( A Z , B Z ) do indeed form the basis of a topology on −→ S .Indeed, ( A, B ) ∈ −→ S induces ( A , B ) on Z and ( A , B ) on Z if and only if it induceson Z = Z ∪ Z some separation ( A Z , B Z ) which in turn induces ( A i , B i ) on Z i for both i .Hence O ( A , B ) ∩ O ( A , B ) is the union of all these O ( A Z , B Z ).As we shall see, the intuitive property of tangles in finite graphs that they describe, ifindirectly, some highly cohesive region of that graph – however ‘fuzzy’ this may be interms of concrete vertices and edges – will extend precisely to those tangles of S that areclosed in −→ S . S We think of an oriented separation (
A, B ) ∈ −→ S as pointing towards B , or being orientedtowards B . In the same spirit, given an end ω of G , we say that ( A, B ) points towards ω ,and that ω lives in B , if some (equivalently: every) ray of ω has a tail in B . Furthermore,if ( A, B ) points to an end ω , then ( B, A ) points away from ω .Clearly, for every end of G and every { A, B } ∈ S , precisely one orientation of { A, B } points towards that end. In this way, every end ω of G defines an orientation of S byorienting each separation in S towards ω : τ = τ ω := { ( A, B ) ∈ −→ S | every ray of ω has a tail in B } It is reasy to see ([11]) that this is a tangle in G . We call it the end tangle induced on S by the end ω .Note that every end tangle contains all separations of the form ( A, V ) for finite A ⊆ V ,and thus no separation of the form ( V, B ). Furthermore, any two ends induce differentend tangles. Our aim in this section is to recall from [11] some properties of the end-tanglesof S that we shall later extend to its subsets S k . For the convenience of the reader, andalso in order to correct an inessential but confusing error in [11], we repeat some of thematerial from [11] here to make our presentation self-contained.Let us first see an example of an end tangle that is not closed in −→ S . Even though −→ S itself is not usually profinite in the sense of [44], the topology we define on −→ S is thesubspace topology of −→ S as a subspace of the (profinite) system of all oriented separations of G ,equipped with the inverse limit topology from [44]. xample 3.1. If G is a single ray v v . . . with end ω , say, then τ = τ ω is not closedin −→ S . Indeed, τ contains ( ∅ , V ), and hence does not contain ( V, ∅ ). But for every fi-nite Z ⊆ V the restriction ( Z, ∅ ) of ( V, ∅ ) to Z is also induced by the separation( { v , . . . , v n } , { v n , v n +1 , . . . } ) ∈ τ for every n large enough that Z ⊆ { v , . . . , v n − } . So( V, ∅ ) ∈ −→ S (cid:114) τ has no open neighbourhood in −→ S (cid:114) τ. (cid:3) Here is an example of an end tangle that is closed in −→ S . Unlike our previous example,it describes a highly cohesive part of G . Example 3.2. If K ⊆ V spans an infinite complete graph in G , then τ = { ( A, B ) ∈ −→ S | K ⊆ B } (1) is a closed set in −→ S . We omit the easy proof. But note that τ is indeed an end tangle: it is induced by theunique end of G which contains all the rays in K .Perhaps surprisingly, it is not hard to characterize the end tangles that are closed.They are all essentially like Example 3.23.2: we just have to generalize the infinite completesubgraph used appropriately. Of the two obvious generalizations, infinite completeminors [88] or subdivisions of infinite complete graphs [77], the latter turns out to be theright one.Let κ be any cardinal. A set of at least κ vertices of G is ( < κ )- inseparable if no twoofthem can be separated in G by fewer than κ vertices. A maximal ( < κ )-inseparable setof vertices is a κ - block . For example, the branch vertices of a T K κ are ( < κ )-inseparable.Conversely: Lemma 3.3.
When κ is infinite, every ( < κ ) -inseparable set of vertices in G containsthe branch vertices of some T K κ ⊆ G .Proof. Let K ⊆ V be ( < κ )-inseparable. Viewing κ as an ordinal we can find, inductivelyfor all α < κ , distinct vertices v α ∈ K and internally disjoint v α - v β paths in G for all β < α that also have no inner vertices among those v β or on any of the paths chosenearlier; this is because | K | (cid:62) κ , and no two vertices of K can be separated in G by the < κ vertices used up to that time.The original statement of Lemma 3.33.3 in [11, Lemma 5.4] asserted that for infinite κ a set K ⊆ V is a κ -block in G if and only if it is the set of branch vertices of some T K κ ⊆ G .It turns out that both directions of that assertion were false: the set of branch vertices ofa T K κ ⊆ G is certainly ( < κ )-inseparable, but might not be maximal with this propertyand hence not a κ -block. Conversely, if K is a κ -block, there might not be a T K κ ⊆ G whose set of branch vertices is precisely K : if | K | > κ this is certainly not possible, buteven if | K | = κ one might not be able to find a T K κ in G whose branch vertices are allof K . If for instance the graph G is a clique on κ vertices that is missing exactly oneedge, then K = V ( G ) is a κ -block in G but not the set of branch vertices of a T K κ ⊆ G .For our main result of this section we need one more observation:4 emma 3.4. If τ is an end tangle of G and ( A, B ) , ( C, D ) ∈ τ then ( A ∪ C , B ∩ D ) ∈ τ .Proof. Observe first that ( A ∪ C , B ∩ D ) is a separation of G with finite order and thuslies in −→ S . Moreover, if a ray of G has a tail in B and a tail in D , then that ray also has atail in B ∩ D , from which it follows that ( A ∪ C , B ∩ D ) ∈ τ , as claimed.We can now prove our main result of this section. Let us say that a set K ⊆ V defines an end tangle τ if τ satisfies (11). Theorem 3.5 ([11]) . Let G be any graph. An end tangle of G is closed in −→ S if and onlyif it is defined by an A -block.Proof. Suppose first that τ is an end tangle that is defined by an A -block K . To showthat τ is closed, we have to find for every ( A, B ) ∈ −→ S (cid:114) τ a finite set Z ⊆ V such that no( A , B ) ∈ −→ S that agrees with ( A, B ) on Z lies in τ . As ( A, B ) / ∈ τ , we have K ⊆ A ; pick z ∈ K (cid:114) B . Then every ( A , B ) ∈ −→ S that agrees with ( A, B ) on Z := { z } also also liesin −→ S (cid:114) τ , since z ∈ A (cid:114) B and this implies K B .Conversely, consider any end tangle τ and let K := \ { B | ( A, B ) ∈ τ } . No two vertices in K can be separated by in G by a finite-order separation: one orientation( A, B ) of this separation would be in τ , which would contradict the definition of K since A (cid:114) B also meets K . If K is infinite, it will clearly be maximal with this property, andhence be an A -block. This A -block K will define τ : by definition of K we have K ⊆ B for ever ( A, B ) ∈ τ , while also every ( A, B ) ∈ −→ S with K ⊆ B must be in τ : otherwise( B, A ) ∈ τ and hence K ⊆ A by definition of K , but K A ∩ B because this is finite.Hence τ will be defined by an A -block, as desired for the forward implication. It thus suffices to show that if K is finite then τ is not closed in −→ S , which we shall donext.Assume that K is finite. We have to find some ( A, B ) ∈ −→ S (cid:114) τ that is a limit pointof τ , i.e., which agrees on every finite Z ⊆ V with some ( A , B ) ∈ τ . We choose( A, B ) := (
V, K ), which lies in −→ S (cid:114) τ since ( K, V ) ∈ τ .To complete our proof as outlined, let any finite set Z ⊆ V be given. For every z ∈ Z (cid:114) K choose ( A z , B z ) ∈ τ with z ∈ A z (cid:114) B z : this exists, because z / ∈ K . As( K, V ) ∈ τ , by Lemma 3.43.4 we have ( A , B ) ∈ τ for A := K ∪ [ z ∈ Z (cid:114) K A z and B := V ∩ \ z ∈ Z (cid:114) K B z . As desired, ( A , B ) (cid:22) Z = ( A, B ) (cid:22) Z (which is ( Z, Z ∩ K ), since ( A, B ) = (
V, K )):every z ∈ Z (cid:114) K lies in some A z and outside that B z , so z ∈ A (cid:114) B , while every z ∈ Z ∩ K lies in K ⊆ A and also, by definition of K , in every B z (and hence in B ),since ( A z , B z ) ∈ τ . Whether or not τ is closed in −→ S is immaterial; we just did not use this assumption.
5e conclude the section with the remark that an end tangle τ of an infinite graph G is defined by an A -block if and only if it is defined by some set X ⊆ V : for if τ is definedby some X ⊆ V , then X must be infinite, as otherwise { X, V } ∈ S would witness that X does not define τ . But if X is infinite then so is K := T { B | ( A, B ) ∈ τ } ⊇ X , andwe can follow the proof of Theorem 3.53.5 to show that K is an A -block defining τ . S k For k ∈ N we write S k for the subset of S containing all separations of G of order < k . Inthis section we shall extend Theorem 3.53.5 from the previous section to these S k , where wewill also be able to prove a stronger and wider statement which will allow us to classifyends by the behaviour of their end tangles – specifically, by the set of those k ∈ N forwhich their end tangle is closed in −→ S k .For every k ∈ N the set −→ S k is a closed set in −→ S . Given an end ω of G with end tangle τ = τ ω , the tangle induced by ω in S k is τ ∩ −→ S k . We say that ω induces a closed tangle in S k if τ ∩ −→ S k is a closed set in −→ S k , where the latter is equipped with the subspace topologyof −→ S .We seek to classify and characterize ends by the set of k for which their induced tanglesare closed in S k , similarly to Theorem 3.53.5. For l < k the set −→ S l is closed in −→ S k , and henceany end inducing a closed tangle in S k also induces a closed tangle in S l . Therefore, theends of G all fall into one of the following three categories:(i) Ends inducing a closed tangle in S ;(ii) ends whose induced tangle in S is not closed in −→ S , but whose induced tangle in S k is closed in −→ S k for every k ∈ N ;(iii) ends whose induced tangle in S k is not closed in −→ S k for some k ∈ N .The ends belonging to the first category are characterized in Theorem 3.53.5: they arethose ends whose tangle in −→ S ℵ is defined by an A -block. In the remainder of this sectionwe shall characterize the ends which fall into the latter two categories. Furthermore,for an end ω belonging to the third category, there exists a least k ∈ N for which thetangle induced by ω in S k is not closed in −→ S k . We shall determine this k and show that itdepends just on the vertex degree and number of dominating vertices of ω .Let us see examples of ends belonging to the third and second category, respectively: Example 4.1.
Let G be as in Example 3.13.1, that is, a single ray v v . . . with end ω .The same argument as in Example 3.13.1 shows that τ ∩ −→ S k is not closed in −→ S k for k (cid:62) ω does induce a closed tangle in S : the set τ ∩ −→ S = { ( ∅ , V ) } is closed in −→ S . Example 4.2.
Let G be the infinite grid, ω the unique end of G and τ the tangle inducedby ω in S = S ( G ). As G is locally finite it does not contain an A -block. Therefore τ cannot be defined by an A -block and is thus not closed in −→ S by Theorem 3.53.5. However,for every k ∈ N , it is easy to see that τ ∩ −→ S k is closed in −→ S k : indeed, for fixed k ∈ N , the size6f A is bounded in terms of k for every ( A, B ) ∈ τ ∩ −→ S k . Thus, for every ( A , B ) ∈ −→ S k (cid:114) τ ,any Z ⊆ V ( G ) with | Z ∩ A | sufficiently large witnesses that ( A , B ) does not lie in theclosure of τ : if Z is large enough that | Z ∩ A | > | A | for every ( A, B ) ∈ τ ∩ −→ S k , then no( A, B ) ∈ τ ∩ −→ S k will agree with ( A , B ) on Z .Example 4.24.2 shows that the end tangle τ of the infinite grid is not defined by a set X ⊆ V in the sense of (11). Moreover, even for any fixed k ∈ N (cid:62) , we cannot find a set X ⊆ V with X ⊆ B for all ( A, B ) ∈ τ ∩ −→ S k : for every x ∈ X the separation ( { x } ∪ N ( x ) , V (cid:114) { x } )lies in τ ∩ −→ S k . Thus, even the tangles that τ induces in −→ S k are not defined by any subsetof V . However, they come reasonably close to it: for X = V and any ( A, B ) ∈ τ , themajority of X lies in B , that is, we have | A ∩ X | < | B ∩ X | . In fact, for any fixed k ∈ N ,any finite set X ⊆ V that is at least twice as large as max {| A | | ( A, B ) ∈ τ ∩ −→ S k } has theproperty that | A ∩ X | < | B ∩ X | for every ( A, B ) ∈ τ ∩ −→ S k . Therefore, even though no τ ∩ −→ S k is defined by a X ⊆ V , we can for each k ∈ N find a (finite) X ⊆ V which ‘defines’ τ ∩ −→ S k by simple majority.To make the above observation formal, for an end tangle τ of G , let us call a set X ⊆ V a decider set for τ (resp., for τ ∩ −→ S k ) if we have | A ∩ X | < | B ∩ X | for every ( A, B ) ∈ τ (resp., ( A, B ) ∈ τ ∩ −→ S k ). Thus, if we have a decider set for an end tangle τ , given aseparation ( A, B ) ∈ −→ S , the decider set tells us which of ( A, B ) and (
B, A ) lies in τ by asimple majority vote. By Theorem 3.53.5, every end tangle that is closed in −→ S has a deciderset, as the A -block defining it is such a decider set. In analogy with this, we shall show inthe remainder of this section that for an end tangle τ its induced tangle τ ∩ −→ S k is closedin −→ S k if and only if τ ∩ −→ S k has a finite decider set. Such a finite decider set can be thoughtof as a local encoding of the tangle, or a local witness to the tangle being closed in −→ S k .In contrast, it is easy to see that every end tangle has an infinite decider set: Proposition 4.3.
For any end ω of G the end tangle τ induced by ω has an infinitedecider set.Proof. For any ray R ∈ ω its vertex set V ( R ) is a decider set for τ .As no end tangle of S can have a finite decider set the existence of decider sets forend tangles is thus only interesting for finite decider sets of the tangle’s restrictions tosome −→ S k .We shall complement this local witness of a given end tangle being closed with a moreglobal type of witness. For this we need to introduce some notation.The (vertex) degree deg( ω ) of an end ω of G is the largest size of a family of pairwisedisjoint ω -rays . A vertex v ∈ V dominates and end ω if it sends infinitely many disjointpaths to some (equivalently: to all) ray in ω . We write dom( ω ) for the number of verticesof G which dominate ω . An end ω is undominated if dom( ω ) = 0; it is finitely dominated if finitely many (including zero) vertices of G dominate ω ; and, finally, ω is infinitelydominated if dom( ω ) = ∞ . We will show that the category that an end ω belongs todepends just on these parameters deg( ω ) and dom( ω ). Concretely, we will show thefollowing: Here our notation deviates from that in [22], where d ( ω ) is used for the degree of ω . heorem 4.4. Let τ the end tangle induced by an end ω of G . Then the followingstatements hold: (i) τ is closed in −→ S if and only if dom( ω ) = ∞ . (ii) τ is not closed in −→ S but τ ∩ −→ S k is closed in −→ S k for every k ∈ N if and only if deg( ω ) = ∞ and dom( ω ) < ∞ . (iii) τ ∩ −→ S k is not closed in −→ S k for some k ∈ N if and only if deg( ω ) + dom( ω ) < ∞ . Theorem 4.44.4 will be a consequence of Theorem 3.53.5 and the following theorem, whichcharacterizes for which k ∈ N a given end tangle is closed in −→ S k and makes the connectionto finite decider sets: Theorem 4.5.
Let τ be the end tangle induced by an end ω of G and let k ∈ N . Thenthe following are equivalent: (i) τ ∩ −→ S k is closed in −→ S k ; (ii) deg( ω ) + dom( ω ) (cid:62) k ; (iii) τ ∩ −→ S k has a finite decider set; (iv) τ ∩ −→ S k has a decider set of size exactly k . Let us first prove Theorem 4.44.4 from Theorem 3.53.5 and Theorem 4.54.5:
Proof of Theorem 4.44.4.
We will show (i) using Theorem 3.53.5 and (ii) with Theorem 4.54.5.The third statement is then an immediate consequence of the first two and the fact thatan end tangle which is closed in −→ S is also closed in −→ S k .To see that (i) holds, let us first suppose that τ is closed in −→ S . Then, by Theorem 3.53.5, τ is defined by an A -block K ⊆ V . It is easy to see that every vertex in K dominates ω ,hence dom( ω ) = ∞ as K is infinite.For the converse, suppose that ω is infinitely dominated, and let us show that τ is closedin −→ S . For every separation ( A, B ) ∈ τ every vertex dominating ω must lie in B . Therefore K := T { B | ( A, B ) ∈ τ } is infinite and thus, as seen in the proof of Theorem 3.53.5, an A -block defining τ , which is hence closed in −→ S .Let us now show (ii). By (i), τ is not closed in −→ S if and only if dom( ω ) < ∞ .On the other hand, by Theorem 4.54.5, τ ∩ −→ S k is closed in −→ S k for all k if and only ifdeg( ω ) + dom( ω ) (cid:62) k for all k ∈ N , that is, if at least one summand is infinite. Thus (ii)holds.Claim (iii) now follows either from claim (i) and (ii), or directly from Theorem 4.54.5.We will conclude this section by proving Theorem 4.54.5. For this we shall need twolemmas. The first lemma can be seen as an analogue of Menger’s Theorem between avertex set and an end. Given a set X ⊆ V and an end ω , we say that F ⊆ V separates X from ω if every ω -ray which meets X also meets F .8 emma 4.6. Let ω be an undominated end of G and X ⊆ V a finite set. The largestsize of a family of disjoint ω -rays which start in X is equal to the smallest size of a set T ⊆ V separating X from ω .Proof. Let T be a set separating X from ω of minimal size. Clearly, a family of disjoint ω -rays which all start in X cannot be larger than T , as every ray in that family mustmeet T . So let us show that we can find a family of | T | disjoint ω -rays starting in X .Observe that, as ω is undominated, for every vertex v ∈ V we can find a finite set T v ⊆ V (cid:114) { v } which separates v from ω . Thus, for every finite set Y ⊆ V we can find afinite set in V (cid:114) Y separating Y from ω : for instance, the set S { T v (cid:114) Y | v ∈ Y } .Pick a sequence of finite sets T n ⊆ V inductively by setting T := T and picking as T n a set of minimal size with the property that T n − ∩ T n = ∅ and that T n separates T n − from ω ; these sets exist by the above observation. Let C n be the component of G − T n that contains ω . Clearly T n ⊆ C n − .We claim that C := T n ∈ N C n = ∅ . To see this, consider any v ∈ C and a shortest v – X path in G . This path must pass through T n for every n ∈ N , which is impossible sincethe separators T n are pairwise disjoint. Therefore C must be empty.By the minimality of each T n , the sets T n are of non-decreasing size, and furthermoreMenger’s Theorem yields a family of | T n | many disjoint paths between T n and T n +1 foreach n ∈ N , as well as | T | many disjoint paths between X and T . By concatenatingthese paths we obtain a family of | T | = | T | many rays starting in X . To finish the proofwe just need to show that these rays belong to ω . To see this, let ω be another end of G ,and T a finite set separating ω and ω . As C = ∅ we have C n ∩ T = ∅ for sufficientlylarge n , which shows that the rays constructed do not belong to ω and hence concludesthe proof.An immediate consequence of Lemma 4.64.6 is that, for an undominated end ω , everyfinite set X ⊆ V can be separated from ω by at most deg( ω ) many vertices. In fact, wecan state a slightly more general corollary: Corollary 4.7.
Let ω be a finitely dominated end of G and X ⊆ V a finite set. Then X can be separated from ω by some T ⊆ V with | T | (cid:54) deg( ω ) + dom( ω ) .Proof. Let D be the set of vertices dominating ω and consider the graph G := G − D and the set X := X (cid:114) D . By Lemma 4.64.6 there is a set T ⊆ V ( G ) of size at most deg( ω )separating X from ω in G . Set T := T ∪ D . Then T separates X from ω in G and hassize | T | = | T | + | D | (cid:54) deg( ω ) + dom( ω ).The second lemma we shall need for our proof of Theorem 4.54.5 roughly states that, foran end of high degree, we can find a large family of disjoint ω -rays whose set of startingvertices is highly connected in G , even after removing the tails of these rays: Lemma 4.8.
Let ω be an end of G and k (cid:54) deg( ω ) + dom( ω ) . Then there are a set X ⊆ V of k vertices and a set R of disjoint ω -rays with the following properties: everyvertex in X is either the start-vertex of a ray in R , or dominates ω and does not lie onany R ∈ R , and furthermore for any two sets A, B ⊆ X there are min( | A | , | B | ) manydisjoint A – B -paths in G whose internal vertices meet no ray in R and no vertex of X . roof. Pick a set D of vertices dominating ω and a set R of disjoint ω -rays not meeting D such that | D | + |R| = k ; we shall find suitable tails of the rays in R such that theirstarting vertices together with D are the desired set X .Using the fact that the vertices in D dominate ω and that the rays in R belong to ω ,we can pick for each pair x , x of elements of D ∪ R an x – x -path in G in such a waythat these paths are pairwise disjoint with the exception of possibly having a commonend-vertex in D . Let P be the set of these paths. Now for each ray in R pick a tail ofthat ray which avoids all the paths in P . Let R be the set of these tails and X the unionof their starting vertices and D . We claim that X and R are as desired.To see this, let us show that for any sets A, B ⊆ X we can find min( | A | , | B | ) manydisjoint A – B -paths in G whose internal vertices avoid D as well as V ( R ) for every R ∈ R . Clearly it suffices to show this for disjoint sets A, B of equal size. So let
A, B ⊆ X be two disjoint sets with n := | A | = | B | and let R A,B be the set of all raysin R that contain a vertex from A or B . For each pair ( a, b ) ∈ A × B there is a uniquepath P ∈ P such that each of its end-vertices either is a or b (if a ∈ D or b ∈ D ) or lieson a ray in R A,B which contains a or b ; let P a,b be the a – b -path obtained from P byextending it, for each of its end-vertices that is not either a or b , along the correspondingray in R up to a or b . Let P A,B be the set of all these paths P a,b . Note that the internalvertices of each path P a,b ∈ P A,B meet none of the rays in R or vertices in X .We claim that A and B cannot be separated by fewer than n = | A | vertices in G := [ P a,b ∈P A,B P a,b ;the claim will then follow from Menger’s Theorem. So suppose that some set T ⊆ V ( G )of size less than n is given. Every vertex from T can lie on at most n + 1 paths from P A,B : for any v ∈ T , if v does not lie in D or on some R ∈ R A,B , then v can meet atmost one path from P A,B as the paths in P A,B are pairwise internally disjoint; if v isa vertex in D then v meets at most the n paths in P A,B which have v as one of theirend-vertices; and finally, if v lies on a ray R ∈ R A,B , then v can meet all of the n pathsin P A,B which have an end-vertex on R , and additionally v can be the internal vertex ofup to one other path in P A,B that does not end on R .Thus, as | T | (cid:54) n −
1, the set T can meet at most | T | ( n + 1) (cid:54) ( n − n + 1) = n − P A,B . Since P A,B contains an a – b -path for every pair ( a, b ) in A × B ,there are | A × B | = n paths in P A,B , which proves that T cannot separate A and B in G .We can thus apply Menger’s Theorem to obtain n disjoint A – B -paths in G , which arethe desired disjoint paths in G whose internal vertices avoid the rays in R and verticesin X : the only vertices that are contained both in V ( G ) as well as in either X or a rayfrom R are vertices from A or B , which cannot be internal vertices of the n = | A | = | B | disjoint A – B -paths. 10e would like to thank Christian Elbracht for pointing out the counting argumentused in the final part of the proof of Lemma 4.84.8.We are now ready to prove Theorem 4.54.5: Theorem 4.54.5.
Let τ be the end tangle induced by an end ω of G and let k ∈ N . Thenthe following are equivalent: (i) τ ∩ −→ S k is closed in −→ S k ; (ii) deg( ω ) + dom( ω ) (cid:62) k ; (iii) τ ∩ −→ S k has a finite decider set; (iv) τ ∩ −→ S k has a decider set of size exactly k .Proof. We will show (i) ⇒ (ii) ⇒ (iv) ⇒ (iii) ⇒ (i).To see that (i) ⇒ (ii), let us first suppose that deg( ω ) + dom( ω ) < k and show that τ ∩ −→ S k is not closed in −→ S k . Let D be the set of dominating vertices of ω . As | D | =dom( ω ) < k the separation ( V, D ) lies in −→ S k . By definition of τ we have ( D, V ) ∈ τ and( V, D ) / ∈ τ . Thus it suffices to show that ( V, D ) lies in the closure of τ ∩ −→ S k in −→ S k . Thiswill be the case if for every finite set X ⊆ V there is a separation ( A, B ) ∈ τ ∩ −→ S k with( A, B ) (cid:22) X = ( V, D ) (cid:22) X .So let X be a finite subset of V . By Corollary 4.74.7 some set T of size at mostdeg( ω ) + dom( ω ) separates X from ω . Let C be the component of G containing ω . Wedefine the separation ( A, B ) by setting A := V (cid:114) C and B := T ∪ C . Then ( A, B ) is aseparation of G with | ( A, B ) | = | T | (cid:54) deg( ω ) + dom( ω ) < k, so ( A, B ) ∈ −→ S k . In fact ( A, B ) lies in τ ∩ −→ S k as ω lives in B . Furthermore we have X ⊆ A and D ⊆ B since no vertex dominating ω can be separated from ω by T . Therefore( A, B ) (cid:22) X = ( X, X ∩ D ) = ( V, D ) (cid:22) X , showing that ( V, D ) lies in the closure of τ ∩ −→ S k in −→ S k .Let us now show that (ii) ⇒ (iv). So let us assume that deg( ω ) + dom( ω ) (cid:62) k . Thenby Lemma 4.84.8 we find a set X ⊆ V of size k and a family R of ω -rays such that everyvertex of X either dominates ω or is the start-vertex of a ray in R , and such that forany A, B ⊆ X we can find min( | A | , | B | ) many disjoint A – B -paths in G whose internalvertices meet neither X nor any ray in R .We claim that X is the desired decider set for τ ∩ −→ S k . To see this, let ( A, B ) beany separation in τ ∩ −→ S k ; we need to show that | A ∩ X | < | B ∩ X | . Let us write X A (cid:114) B := ( A (cid:114) B ) ∩ X and X B (cid:114) A := ( B (cid:114) A ) ∩ X as well as X A ∩ B := ( A ∩ B ) ∩ X . Itthen suffices to prove | X A (cid:114) B | < | X B (cid:114) A | .So suppose to the contrary that | X A (cid:114) B | (cid:62) | X B (cid:114) A | . Note first that no vertex in X A (cid:114) B dominates ω as witnessed by the finite-order separation ( A, B ) ∈ τ ⊆ S k . Therefore, forevery vertex in X A (cid:114) B , we have a ray in R starting at that vertex. Each of those disjointrays must pass through the separator A ∩ B , and none of them hits X A ∩ B . Furthermore11y Lemma 4.84.8 there are | X B (cid:114) A | many disjoint X A (cid:114) B – X B (cid:114) A -paths whose internal verticesavoid R and X . These paths, too, must pass the separator A ∩ B without meeting X A ∩ B or any of the rays above. Thus we have | A ∩ B | (cid:62) | X A ∩ B | + | X A (cid:114) B | + | X B (cid:114) A | = | X | = k, a contradiction since ( A, B ) ∈ S k and hence | A ∩ B | < k . Therefore we must have | X A (cid:114) B | < | X B (cid:114) A | , which immediately implies | A ∩ X | < | B ∩ X | .Finally, let us show that (iii) ⇒ (i). So let X ⊆ V ( G ) be a finite decider set for τ ∩ −→ S k .We need to show that no ( A, B ) ∈ S k (cid:114) τ lies in the closure of τ ∩ −→ S k . For this let( A, B ) ∈ S k (cid:114) τ be given; then X witnesses that ( A, B ) does not lie in the closureof τ . To see this, let any ( C, D ) ∈ τ be given. As X is a decider set for τ we have | C ∩ X | < | D ∩ X | , and since ( A, B ) / ∈ τ we have | A ∩ X | (cid:62) | B ∩ X | . Therefore ( A, B )and (
C, D ) do not agree on X , which thus witnesses that ( A, B ) does not lie in theclosure of τ ∩ −→ S k in S k .Note that in our proof above that (iii) implies (i) we did not make use of the assumptionthat the tangle τ was induced by an end of G : indeed, every orientation of S k that has afinite decider set is closed in −→ S k .For an end tangle τ that is closed in −→ S we can say slightly more about its decider setsin −→ S k : for every k ∈ N the restriction τ ∩ −→ S k has a decider set of size exactly k which is a( < k )-inseparable set. Finding these ( < k )-inseparable decider sets is straightforward:such an end tangle τ is defined by an A -block K by Theorem 3.53.5, and every subset X ⊆ K of size k is a ( < k )-inseparable decider set for τ ∩ −→ S k . However, having a ( < k )-inseparabledecider set for τ ∩ −→ S k for all k ∈ N is not a characterizing property for the closed endtangles of G : Example 4.9.
For n ∈ N let K n be the complete graph on n vertices. Let G be thegraph obtained from a ray R = v v . . . by replacing each vertex v n with the completegraph K n . Then G has a unique end ω ; let τ be the end tangle induced by ω . Since ω isundominated τ is not closed in −→ S = −→ S ( G ) by Theorem 4.44.4. However, for every k ∈ N , thetangle τ ∩ −→ S k has a ( < k )-inseparable decider set of size k : the clique K k which replacedthe vertex v k of R is such a ( < k )-inseparable decider set. References [1] R. Diestel. Ends and tangles.
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